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Guided -mode resonance reflection and transmission filters in the optical and microwave spectral ranges

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GUIDED-MODE RESONANCE REFLECTION AND TRANSMISSION
FILTERS IN THE OPTICAL AND MICROWAVE
SPECTRAL RANGES
The members o f the Committee approve the doctoral
dissertation o f Sorin Tibuleac
Robert Magnusson
Supervising Professor
Theresa A. Maldonado
Alan Davis
~~
CL
fY\ GJteLfYia-dLo
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/
Kambiz Alavi
Truman Black
Dean of the Graduate School
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Dedicated to my parents Dumitru and Elena Tibuleac
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GUIDED-MODE RESONANCE REFLECTION AND TRANSMISSION
FILTERS IN THE OPTICAL AND MICROWAVE
SPECTRAL RANGES
by
SORIN TIBULEAC
Presented to the Faculty of the Graduate School of
The University of Texas at Arlington in Partial Fulfillment
of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
THE UNIVERSITY OF TEXAS AT ARLINGTON
August 1999
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ACKNOWLEDGMENTS
I wish to express my deepest gratitude to my supervising professor, Dr. Robert
Magnusson, for his highly competent guidance and constant support during the past six years
of my education. I am also grateful to Dr. Theresa A. Maldonado for her support,
encouragement and pertinent advice. Furthermore, I would like to thank my graduate
committee members Dr. Kambiz Alavi, Dr. Truman Black, and Dr. Alan Davis for their help
and valuable comments.
I am indebted to Dr. Cinzia Zuffada, Jet Propulsion Laboratory, Caltech, for
introducing me to genetic algorithms, for inviting me to work at JPL, and collaborating on
developing new genetic algorithm optim ization programs.
I have been fortunate to work in an excellent group of graduate students from whom I
have learned a lot and with whom I have fruitfully collaborated on a number o f research
projects. Among them I am most grateful to Debra Wawro for our collaboration on fiberendface grating devices, Zhongshan Liu for working with me on experimental aspects of
guided-mode resonance reflection filters, Preston Young for his contributions and constant
support of the experimental activity in microwave and optical guided-mode resonance filters;
and Dongho Shin for the rigorous coupled-wave analysis code utilized in development of
new computer programs described in this dissertation. I also wish to acknowledge the support
of Timothy R. Holzheimer, Raytheon Systems, for spectral measurements on microwave
waveguide-grating filters.
My education in optics has begun at the Institute of Atomic Physics, Bucharest,
Romania, in the Holography Laboratory led by Dr. Valentin I. Vlad, to whom I wish to
iv
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express my sincere appreciation. I would also like to thank my former colleagues Dr. Adrian
Petris and Ion Apostol for their friendship and for all that I have learned while working with
them.
Finally, I would like to express my warmest gratitude to my wife, Camelia, for her
active role in editing this dissertation and, most importantly, for her encouragement,
understanding, and complete cooperation that has helped me succeed in finalizing my PhD.
July 15, 1999
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ABSTRACT
GUIDED-MODE RESONANCE REFLECTION AND TRANSMISSION
FILTERS IN THE OPTICAL AND MICROWAVE
SPECTRAL RANGES
Publication N o.____________
Sorin Tibuleac, Ph.D.
The University of Texas at Arlington, 1999
Supervising Professor: Robert Magnusson
In this dissertation, new reflection and transmission filters are developed and
characterized in the optical and microwave spectral regions. These guided-mode resonance
(GMR) filters are implemented by integrating diffraction gratings into classical thin-film
multilayers to produce high efficiency filter response and low sidebands extended over a
large spectral range. Diffraction from phase-shifted gratings and gratings with different
periods is analyzed using rigorous coupled-wave theory yielding a new approach to filter
linewidth broadening, line-shaping, and multi-line filters at normal incidence. New single­
grating transmission filters presented have narrow linewidth, high peak transmittance, and
low sideband reflectance. A comparison with classical thin-film filters shows that GMR
devices require significantly fewer layers to obtain narrow linewidth and high peak response.
All-dielectric microwave frequency-selective surfaces operating in reflection or transmission
are shown to be realizable with only a few layers using common microwave materials.
Single-layer and multilayer waveguide gratings operating as reflection and transmission
vi
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filters, respectively, were built and tested in the 4-20 GHz frequency range. The presence of
GM R notches and peaks is clearly established by the experimental results, and their spectral
location and lineshape found to be in excellent agreement with the theoretical predictions. A
new computer program using genetic algorithms and rigorous coupled-wave analysis was
developed for optimization of multilayer structures containing homogeneous and diffractive
layers. This program was utilized to find GMR filters possessing features not previously
known. Thus, numerous examples of transmission filters with peaks approaching 100%,
narrow linewidths (-0.03%), and low sidebands have been found in structures containing
only 1-3 layers. A new type o f GMR device integrating a waveguide grating with
subwavelength period on the endface of an optical fiber is developed for high-resolution
biomedical or chemical sensors and spectral filtering applications. Diffraction gratings with
submicron periods exhibiting high efficiencies have been recorded for the first time on coated
and uncoated endfaces of single-mode and multimode fibers. Guided-mode resonance
transmittance notches of -18% were experimentally obtained with structures consisting of
photoresist gratings on thin
films of SisN4 deposited on optical fiber endfaces.
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TABLE OF CONTENTS
ACKNOWLEDGMENTS ..............................................................................................
iv
ABSTRACT ...................................................................................................................
vi
LIST OF ILLUSTRATIONS ..........................................................................................
xii
Chapter
1. INTRODUCTION ................................................................................................
I
2. PRINCIPLES OF GUIDED-MODE
RESONANCE FILTERS ....................................................................................
14
2.1 Introduction ..........................................................................................
14
2.2 Waveguide properties .........................................................................
17
2.2.1 Single-layer waveguide grating ...........................................
17
2.2.2 Multilayer waveguide grating .............................................
21
2.3 Thin-film properties ............................................................................
24
2.3.1 Antireflection structures .......................................................
25
2.3.2 High-reflection structures .....................................................
27
2.4 Rigorous coupled-wave analysis of diffraction
by waveguide gratings ....................................................................
28
2.4.1 Single-period structures .......................................................
28
2.4.2 Double-period structures ......................................................
33
3. OPTICAL GUIDED-MODE
RESONANCE FILTERS
................................................................................
40
3.1 Introduction ..........................................................................................
40
3.2
Reflection filters ..................................................................................
40
3.2.1 Single-layer reflection filters ...............................................
40
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3.2.2 Double-layer reflection filters .............................................
42
3.2.3 Triple-layer reflection filters ...............................................
48
3.3 Transmission filters .............................................................................
51
3.3.1 Double-grating transmission filter ......................................
51
3.3.2 Single-grating transmission filters .....................................
53
3.4 Multi-line filters ...................................................................................
58
3.4.1 Techniques for obtaining multi-line filters .......................
58
3.4.2 Double-period waveguide gratings ....................................
59
3.5 Control of filter characteristics ............................................................
64
3.5.1 Linewidth control mechanisms ...........................................
64
3.5.2 Phase-shifted gratings for filter line control ......................
67
3.6 Comparison with homogeneous thin-film filters ..............................
73
3.6.1 Reflection filters ...................................................................
73
3.6.2 Transmission filters ..............................................................
75
3.7 Filter design using the “direct” approach ...........................................
80
3.8 Experimental results ............................................................................
83
3.8.1 Background ..........................................................................
83
3.8.2 High-efficiency guided-mode resonance
reflection filter ...............................................................
84
3.8.3 Guided-mode resonance laser mirror .................................
86
4. MICROWAVE GUIDED-MODE
RESONANCE FILTERS ...................................................................................
91
4.1 Introduction ..........................................................................................
91
4.2 Reflection filters ...................................................................................
92
4.3 Transmission filters .............................................................................
97
4.4 Experimental results ............................................................................ 100
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4.4.1 Notch filters ........................................................................... 101
4.4.2 Bandpass filters .................................................................... 104
4.4.3 Fabry-Perot filters ................................................................ 106
5. GUIDED-MODE RESONANCE FILTERS DESIGNED
WITH GENETIC ALGORITHMS ................................................................... 109
5.1 Introduction ........................................................................................... 109
5.2 Background ........................................................................................... 110
5.2.1 Numerical optimization methods ........................................ 110
5.2.2 Principles of genetic algorithm search
and optimization ............................................................. 113
5.2.3 Review of genetic algorithms in optics .............................. 117
5.3 Genetic algorithm program for multilayer
waveguide gratings ....................................................................... 120
5.3.1 Program description ............................................................. 120
5.3.2 Convergence tests ................................................................. 128
5.4 Reflection filters ................................................................................... 137
5.5 Transmission filters .............................................................................. 143
5.5.1 Double-layer/double-grating filters .................................... 143
5.5.2 Single-layer/single-grating filters ........................................ 146
5.5.3 Triple-layer/single-grating filters ........................................ 147
6. FIBER-ENDFACE GUIDED-MODE
RESONANCE DEVICES .................................................................................. 156
6.1 Introduction ........................................................................................... 156
6.2 Guided-mode resonance sensors ......................................................... 157
6.3 Experimental results ............................................................................. 166
7. CONCLUSIONS ........................................................................................................ 172
7.1 Contributions ........................................................................................ 172
x
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7.2 Future research ..................................................................................... 178
REFERENCES ................................................................................................................ 184
BIOGRAPHICAL INFORMATION ............................................................................. 194
xi
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LIST OF ILLUSTRATIONS
Figure
Page
1.1. TE polarization spectral response of a triple-layer guided-mode
2
resonance reflection filter .........................................................................
1.2. Infrared bandpass guided-mode resonance filter spectral response,
centered at Ac = 10.6 (im ..........................................................................
6
2.1. Single-layer grating with rectangular profile of high (nu) and low
(n0 refractive indices with period A and thickness d ...........................
16
2.2. Guided-mode resonance at A, = 550 nm in the reflectance of a
single-layer waveguide-grating ...............................................................
16
2.3. Antireflection coating designs using 2 layers (V-coating and
W-coating) and 3 layers ...........................................................................
27
2.4. Schematic of a multiple-layer thin-film structure with an arbitrary
number of binary gratings of the same grating period but
different modulations ...............................................................................
29
2.5. Schematic of a multiple-layer thin-film structure with an arbitrary
number of gratings possessing either period Ai, or period A 2 ..............
34
3.1.
Single-layer guided-mode resonance filter ...................................................
44
3.2. Guided-mode resonance in reflectance of the single-layer
waveguide grating of figure 3.1 ..............................................................
45
3.3.
3.4.
3.5.
3.6.
Reflectance of the single-layer guided-mode resonance filter over
the entire visible range for TE and TM-polarized incident
wave ...........................................................................................................
45
Reflectance versus angle of incidence at fixed wavelength (0.55
(im) for the device with the spectral response of figure 3.3 ..................
46
Reflectance of a double-layer waveguide-grating filter with V-type
antireflection coating ................................................................................
46
Reflectance of a double-layer waveguide-grating filter with
W-type antireflection coating ..................................................................
47
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3.7. Reflectance of a double-layer structure with a surface-relief
grating ......................................................................................................
47
3.8. Triple-layer guided-mode resonance filter response. The optical
thicknesses of the layers are A/4- A/2 -A/4 at 0.55 pm .........................
50
3.9. Triple-layer filter response with surface-relief grating. Thicknesses
are all quarter-wave at 0.55 pm ..............................................................
50
3.10. Transmittance of a 6-layer structure with the grating in the first
layer and five high-low homogeneous A/4-layers .................................
53
3.11. Spectral response of a 9-layer transmission filter with gratings in
the top and the bottom layers .................................................................
55
3.12. Double-grating transmission filter with 11 layers (caption) and its
spectral response ......................................................................................
55
3.13. A 9-layer structure with one grating in the center layer (top) and
its spectral response (bottom) .................................................................
56
3.14. Single-grating transmittance filter of figure 3.13 over a narrower
spectral range. The layer structure is illustrated in the inset ................
57
3.15. Transmittance of a 9-layer single-grating resonance filter that uses
only 2 materials ........................................................................................
57
3.16. Single-layer guided-mode resonance filter at 3° and 5° angle of
incidence ..................................................................................................
61
3.17. Reflectance of a double-layer waveguide grating at normal
incidence with two identical gratings shifted by 0.05A .......................
62
3.18. Double-line reflection filter response at normal incidence. The
structure has 2 gratings and 1 homogeneous center layer ....................
62
3.19. The same structure as in figure 3.17 is shown here for zero
thickness of the buffer layer ...................................................................
63
3.20. Sum of the diffraction efficiencies in the non-zero orders in the
case of the 2-layer/2-grating structure of figure 3.19 ............................
63
3.21. Linewidth dependence on the grating modulation in the case of a
single-layer structure ...............................................................................
66
3.22. Linewidth dependence on the average refractive index of the
grating in a single-layer reflection filter ................................................
67
3.23. Linewidth dependence on the grating fill factor of a single-layer
reflection filter ..........................................................................................
67
xiii
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3.24. Guided-mode resonance filter with phase-shifted gratings ........................
70
3.25. Comparison between the filter response of the Tt-phase-shifted
versus non-shifted waveguide grating .....................................................
71
3.26. The relative linewidth (AA.(tc)/AA.(0)) dependence on the refractive
index difference between the grating and the substrate ........................
71
3.27. Linewidth narrowing of a transmittance resonance filter with
7t-phase shifted grating versus a non-shifted grating ..............................
72
3.28. Double-layer waveguide grating with two identical gratings
phase-shifted by 7t/2 exhibiting im proved reflection
filter lineshape ...........................................................................................
72
3.29. Notch filter with 150 alternating quarter-wave layers of silica and
BaF 2 on silica substrate with air as cover ................................................
78
3.30. Double-layer guided-mode resonance filter in V-coating design ..............
78
3.31. Transmittance of a Fabry-Perot filter with 11 layers of alternating
ZnS and MgF 2 ............................................................................................
79
3.32. Transmittances of a 2-grating/l 1-layer waveguide grating and of a
l-grating/9-layer guided-mode resonance filter .....................................
79
3.33. Flow chart of guided-mode resonance filter design using the
“direct” method .........................................................................................
82
3.34. Lloyd’s mirror interference setup for holographic recording of
diffraction gratings in photoresist ...........................................................
88
3.35. Scanning electron micrograph of the double-layer guided-mode
resonance structure ....................................................................................
88
3.36. Experimental setup for measuring the spectral characteristics of
guided-mode resonance reflectance filters at normal
incidence ....................................................................................................
89
3.37. Theoretical and experimental spectral response of a double-layer
guided-mode resonance reflection filter ..................................................
89
3.38. Schematic representation of a folded-cavity dye laser operating
with a guided-mode resonance mirror as output coupler ......................
90
3.39. Spectral line of the dye laser operating with the guided-mode
resonance mirror measured with a spectrum analyzer ..........................
90
4.1.
Reflectance of a single-layer waveguide grating in the microwave
spectral range ............................................................................................
xiv
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93
4.2.
4.3.
4.4.
4.5.
4.6.
Spectral response of a triple-layer guided-mode resonance
reflection filter for TE and TM polarization of the
incident wave ............................................................................................
93
Single-layer guided-mode resonance reflection filters with (a)
narrow-line and (b) broad-band response ...............................................
96
Single-layer response of a guided-mode resonance filter in air
with plexiglas/air grating ..........................................................................
96
Dual-line reflection filter response. The structure has 3 layers
with 2 gratings in the first and third layer ..............................................
99
Guided-mode resonance transmission filter spectral response
with the structure illustrated in the inset ................................................
99
4.7.
Experimental setup in an anechoic chamber for transmittance
measurements of microwave waveguide gratings ................................. 102
4.8.
Calculated and measured transmittance notches due to guidedmode resonances occurring in a single-layer plexiglas/air
waveguide grating for a TE-poIarized incident wave ........................... 102
4.9.
Theoretical and experimental transmittance of the single-layer
plexiglas/air waveguide grating of figure 4.8 for a TMpolarized incident wave ............................................................................ 103
4.10. Theoretical and experimental spectral response of a G 10
fiberglass and air waveguide grating exhibiting
guided-mode resonance peaks ................................................................. 106
4.11. Theoretical transmittance o f a 5-layer Fabry-Perot with alternating
layers of G10 fiberglass and air ............................................................... 108
4.12. Experimental frequency response of a 5-layer fiberglass/air FabryPerot for TE-polarized incident wave .................................................... 108
5.1.
Flow chart of a genetic algorithm using rigorous coupled-wave
analysis for merit function evaluation .................................................... 115
5.2.
Crossover and mutation operations illustrated for chromosomes
composed of 6 genes ................................................................................ 116
5.3.
Example of a diffractive structure consisting of two gratings in
two separate layers, with physical parameters shown in (a)
and corresponding chromosome in (b) ................................................... 124
5.4.
Merit (residual) function dependence on the population replaced
per generation after 400 iterations ........................................................... 134
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5.5. Merit (residual) function dependence on the mutation probability
after 400 iterations ................................................................................... 135
5.6. Convergence history for binary (a), Gray (b), and real (c) encoding
of the waveguide-grating parameters ..................................................... 136
5.7. Reflectance of a guided-mode resonance filter generated by the
genetic algorithm and reference reflectance used as
input data .................................................................................................. 139
5.8. The difference between the two reflectance curves of figure 5.7 ............... 139
5.9. Reflectance of a guided-mode resonance filter generated by the
genetic algorithm (with nH = 2.5, nL = 2.4, d = 221.1 nm,
f = 0.369) and reference reflectance ....................................................... 140
5.10. The difference between the two reflectance curvesof figure 5.9 .............. 140
5.11. Reflectance of a guided-mode resonance filter generated by the
genetic algorithm (with nn = 2.0, nL = 1.6) and reference
reflectance ................................................................................................ 141
5.12. The difference between the two reflectance curves of figure 5.11 ............ 141
5.13. Reflectance of a guided-mode resonance filter generated by the
genetic algorithm (with nn = 2.5, nL = 2.4, d = 220.8 nm,
f = 0.583) and reference reflectance ....................................................... 142
5.14. The difference between the two reflectance curvesof figure 5.13 ............ 142
5.15. Transmittance of a bandpass guided-mode resonance filter (0.5
nm linewisth) in the optical spectral region generated by
genetic algorithm optimization of the 2-grating structure
of figure 5.3a ............................................................................................ 149
5.16. The filter response of figure 5.15 over a larger wavelength range ............ 149
5.17. Guided-mode resonance transmission filter response (0.2 nm
linewidth) of a 9-layer structure illustrated in the inset ....................... 150
5.18. Guided-mode resonance transmission filter (linewidth = 0.2 nm)
with a 2-layer/2-grating structure (nn,i = nn,2 = 2.35, nm =
nL.2 = 1-65) designed with the genetic algorithm .................................. 150
5.19. Transmittance of the 9-layer/l-grating structure o f figure 5.17
plotted in dB ............................................................................................. 151
5.20. Transmittance of the 2-layer/2-grating structure o f figure 5.18
plotted in dB ............................................................................................. 151
xvi
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5.21. Transmittance of the 9-layer/1-grating device o f figure 5.17 over a
200 nm spectral range ............................................................................. 152
5.22. Transmittance of the 2-layer/2-grating structure o f figure 5.18
over a 200 nm spectral range ................................................................... 152
5.23. Guided-mode resonance transmission filter (linewidth = 0.2 nm)
with a 2-layer/2-grating structure (nH,i = 2.5, n H ,2 = 2.4, nm
= 1.75, nm = 1-35) designed with the genetic algorithm ..................... 153
5.24. The filter response of figure 5.23 illustrated over a 200 nm
spectral range ........................................................................................... 153
5.25. Infrared bandpass guided-mode resonance filter spectral response
centered at Ac = 10.6 pm ......................................................................... 154
5.26. Filter response o f figure 5.25 expressed in dB ............................................ 154
5.27. Spectral responses of two infrared bandpass filters (b) at 1.55 pm
employing the same structure (a) and the same materials but
different grating periods, fill factors, and thicknesses .......................... 155
6.1. Spectral shift of a guided-mode resonance peak (b) exhibited by a
double-layer waveguide grating (a) as 20 nm (dotted line) and
40 nm (dashed line) of material (n = 1.4) is deposited on top ............. 161
6.2. Spectral shift a double-layer guided-mode resonance generated by
20 nm and 40 nm of material (n = 1.4) deposited on top ..................... 162
6.3. Single-layer device formed with ZnSe and Si02 performing the
same sensing task as the examples of figures 6.1 and 6.2 .................... 162
6.4. Resonance spectral shifts (b) in a single-layer surface-relief
waveguide grating (a) made with Si on silica substrate ........................ 163
6.5. Guided-mode resonance peaks generated with a single-layer
surface-relief grating made with ZnSe and using water
as cover medium ...................................................................................... 164
6.6. Single-layer guided-mode resonance device with a surface-relief
Si grating sensing changes in the refractive index of the cover
region ......................................................................................................... 164
6.7. Single-layer guided-mode resonance device indicating shifts in
resonant wavelength for different refractive indices of the
cover region ............................................................................................... 165
6.8.
Resonance wavelengths versus refractive index of the cover
region for the device of figure 6.7 ........................................................... 166
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6.9.
Calculated TE and TM-polarization spectral response (b) of a
guided-mode resonance filter (a) with photoresist grating
and Si3 N 4 waveguide on a silica substrate ............................................. 169
6.10. Scanning electron micrographs of photoresist diffraction gratings
on optical fiber endfaces .......................................................................... 170
6.11. Experimental set-up for spectral transmittance measurements of a
guided-mode resonance filter fabricated on the endface of an
optical fiber ............................................................................................... 171
6.12. Transmittance measurements of a double-layer guided-mode
resonance filter fabricated on a multimode optical fiber
endface ...................................................................................................... 171
xviii
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CHAPTER 1
INTRODUCTION
The guided-mode resonance effect occurring in waveguide gratings is receiving
increasing interest generated by the physics of the resonance in itself, as well as by the
multitude of potential applications [1-81]. Guided-mode resonances are a class of grating
resonances, also referred to as grating anomalies. In general, grating resonances represent
rapid variations in the electromagnetic field, reflected or transmitted, from a diffraction
grating with respect to a varying parameter such as the wavelength, angle of incidence, or any
of the physical parameters of the grating (refractive indices, thicknesses, grating period, or fill
factor). There are two principal types of dielectric grating anomalies: the Rayleigh type
[1,82], which is the classical Wood’s anomaly [83], and the resonance type [1,18]. The
Rayleigh type is caused by one of the spectral orders becoming an evanescent wave at the
grazing angle (or an evanescent wave becoming a propagating wave) causing a redistribution
of energy between the remaining propagating orders. The resonance anomaly is caused by
possible guided modes supportable by the waveguide grating. It is the latter type that is
studied throughout this dissertation and employed in the development of optical and
microwave guided-mode resonance devices for spectral filtering, reffactive-index and
thickness sensing, and laser resonator applications.
Recent studies of resonance anomalies in waveguide gratings have demonstrated
unique filtering capabilities of such structures [22], This new type of optical filter combines
principles of diffraction by periodic structures with waveguide properties and anti-reflection
thin-film characteristics to yield filters with 100% reflectance at a desired wavelength. The
filter characteristics such as linewidth, sideband reflectance and range can be tailored by
1
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2
appropriate choice of the multilayer waveguide-grating parameters (figure 1.1) [24,27]. It has
been theoretically demonstrated that waveguide-grating structures with only 3 layers can
provide narrow-linewidth reflection peaks with low sidebands extending over the entire
visible spectral range [29],
1
0 .9 -
0 .8 -^
0.7 -j
0 .6 - .
O 0.5 i
S
E 0.4-3
S
•
* 0 .3 -
0 . 2 -i
o . i -i
0
'
0.4
^
1
»
0.45
I
I
4
V
0.5
0.55
0.6
0.65
W A V EL E N G T H (|Xm)
I
I
.
.
.
.
I
!
0.7
Figure 1.1. Guided-mode resonance reflection filter response of the waveguide
grating illustrated in the inset for a TE-polarized, normally incident wave [48].
The grating periods are: a) A = 0.29 fim, b) 0.31 |im and c) 0.33 |im. The optical
thicknesses of the layers are A/4—A/2 —A/4 at 0.55 pm. Refractive indices are ni =
1.38, nn,2 = 2.1, nL,2 = 2.0, n 3 = 1.62, no = 1.0, and ns = 1.52.
During the past few years, guided-mode resonance reflection filters utilizing various
materials, layer configurations and fabrication techniques have been built. These exhibit
advantageous features such as: peak reflectances exceeding 98% [66], line widths between
0.016% [42] and 6.5% [63] (full width at half maximum) of the central wavelength,
sidebands below 2% [42, 62], filter ranges of 400 nm in the visible range [17], and
polarization sensitivity or independence [35]. However, these characteristics have been found
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3
separately in different devices and current effort is directed towards fabrication of devices
that exhibit the advantageous features predicted by the theoretical calculations. The properties
o f guided-mode resonance devices are fundamentally similar regardless of the spectral range,
with differences arising from grating feature size, fabrication techniques, materials employed,
testing methods, and target applications. Guided-mode resonance reflection filters have been
fabricated with central wavelengths in the visible [6,17,57], infrared [35,42,62,63,74], and
microwave [26,78,79] spectral ranges.
While most studies to date have concentrated on reflection filters, it has been shown
that guided-mode resonances can also yield transmission (bandpass) filters [34]. Utilizing a
structure with multiple layers satisfying a high reflection condition, grating resonances can be
used to design high-efficiency transmission bandpass filters again with narrow linewidth,
symmetrical response, and low sidebands [34,47].
The unique properties of guided-mode resonance filters make these devices attractive
for a multitude of applications, such as spectral and angular filtering with a wide variety of
lineshapes, polarization components at any angle of incidence, laser resonator mirrors and
tuning elements, mirrors and phase-locking elements for vertical-cavity surface-emitting laser
arrays, sensors for bio-medical and environmental engineering, anti-counterfeiting and
security devices, wavelength division multiplexing, and electro-optic modulators, switches,
filters and waveplates. [24,76].
This dissertation continues theoretical and experimental studies o f guided-mode
resonance filters in the optical and microwave spectral ranges. Theoretical developments
include new techniques to implement a large variety of filters with novel characteristics,
emphasizing the broad range of filter types that can be realized with waveguide-grating
structures
[48,68].
Specific
contributions
include
thin-film
antireflection
designs
implemented in waveguide gratings, to generate low and extended sideband reflectance of
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4
guided-mode resonance filters (figure 1.1) [48]; phase shifted gratings used for linewidth
control or lineshape improvement [32,79]; double-line filters at normal incidence with
structures containing gratings of different periods [32,79]; single-grating transmission filters
with high-reflectance thin-film homogeneous layers [43,47]; comparisons between guidedmode resonance filters and classical thin-film designs [48]; microwave guided-mode
resonance filters operating in reflection or transmission [39,49,69,79]. A variety of
waveguide-grating devices can be designed with a recently developed search and
optimization computer program based on genetic algorithms and rigorous coupled-wave
analysis [77]. Using this optimization program, thin-film diffractive structures requiring only
1-3 layers are found, exhibiting bandpass filtering spectral responses with narrow linewidths
(<0.1 %) and low sidebands (<0.1 %) in a specified wavelength range (figure 1.2) [65,71].
Multiple solutions to the same problem are found with the genetic algorithm search
and optimization routine [65,71]. Among the experimental contributions, are the first
experimental demonstration o f guided-mode transmission resonances with excellent
matching between the theoretical and measured data [78]; high-efficiency guided-mode
resonance filters [72,74]; laser output-coupling mirrors using the guided-mode resonance
effect [66,73]; recording of high-efficiency subwavelength gratings on optical fiber endfaces
[80,81]; and development of guided-mode resonance devices on optical fiber endfaces for
applications in bio-medical sensing and spectral filtering for optical communications [80].
The concepts underlying the properties of guided-mode resonance filters arise at the
confluence of more mature optical fields such as grating diffraction, waveguide propagation,
and thin-film interference. Fundamental aspects of diffraction, guided waves, and multilayer
interference that are directly related to the guided-mode resonance effect, and its applications
to spectral filtering are presented in chapter 2. Physical insight can be gained, and filter
characteristics can be derived from studying the waveguide properties o f zero-order
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5
waveguide-grating structures. The resonance anomalies occurring in the filters under study
are due to possible guided modes that would be supported by the waveguide grating in the
absence of the modulation. The periodic modulation o f the guide makes the structure leaky,
preventing sustained propagation of modes in the waveguide and coupling the waves out into
the substrate and cover. An equivalent waveguide approximation presented in section 2.2 can
be used to predict the range of wavelengths (or angles of incidence) within which the
resonances can occur, the approximate spectral (or angular) resonance locations, and the free
spectral (or angular) ranges o f the resonances [29]. In the single-layer case, the resonance
location is found by solving the eigenvalue equation of a slab waveguide, where the
propagation vector component along the boundary is given by the grating equation, and the
refractive index of the waveguiding layer is replaced by the average refractive index of the
grating [24]. Similarly, for multilayer waveguide gratings, the resonance locations can be
found by solving the corresponding homogeneous multilayer eigenvalue equation with
coefficients given by the characteristic matrix of the structure [29].
The reflectance spectrum in the region of a guided-mode resonance typically exhibits
a high peak (approaching 100%) preceded or followed by a notch (approaching 0). The
variation has, in general, an asymmetrical lineshape with sideband levels unsuitably high for
either reflection or transmission filters. An important development in the study o f guidedmode resonance phenomena has been the application of thin-film interference properties (and
results) to improve the lineshape symmetry and reduce the sideband levels [27,29]. This has
enabled high-efficiency reflection or transmission filters with low, extended sidebands. The
thin-film interference properties of waveguide-gratings are presented in section 2.3.
Antireflection coatings with 1, 2 or 3 layers presented in this section are employed in chapter
3 for design of guided-mode resonance filters operating in reflection mode (also referred to as
notch, minus, or bandstop filters in the thin-film literature) [48]. The high-reflectance
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6
properties of thin-film high/low refractive index stacks with quarter-wave thicknesses are
discussed in view of their application to guided-mode resonance bandpass filters [34,47].
Grating
—
.5
Homogeneous
9
9.5
10 10.5 11 11.5 12 12.5
WAVELENGTH (pm)
Figure 1.2. The spectral response of an infrared transmission (bandpass) guidedmode resonance filter spectral response, centered at Ac = 10.6 pm (solid line). The
physical parameters o f this filter were found with a genetic algorithm optimization
routine. This response is generated by a single-layer waveguide-grating with the
cross-section illustrated in the inset and possessing the following physical
parameters: A = 6.91 pm, f = 0.42, d = 3.7 pm, nH = 4.0, nL = 2.65, nc = 1.0, and
ns = 1.4. The dashed curve represents the transmittance of a homogeneous layer
with the same thickness and average refractive index (n = 3.285).
The rigorous coupled-wave theory derived from Maxwell’s equations is a general
method that can be used to calculate the electromagnetic fields within, and diffracted by, any
periodic structure. Section 2.4.1 reviews the rigorous coupled-wave theory applied to
multilayer binary (rectangular refractive-index profile) gratings following the derivations of
references 84-86. The rigorous coupled-wave analysis described in section 2.4.1 is extended
in section 2.4.2 for the case of multilayer structures containing gratings with two different
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7
periods, Ai or A 2 in different layers [40]. All theoretical computations of waveguide-grating
devices presented in this dissertation are performed using computer codes based on the
rigorous coupled-wave analysis. These codes have been developed in detail within the
Electro Optics Research Center at The University o f Texas at Arlington.
Chapter 3 discusses theory, properties, design, and experimental results o f guidedmode resonance filters in the optical spectral region. Numerous examples of reflection and
transmission filters are presented. Although a waveguide grating with a single layer exhibits
filtering properties as shown in section 3.2.1, the filter response can be significantly
improved with one or two additional homogeneous layers as discussed in sections 3.2.2. and
3.2.3, respectively. Filters with multiple layers have lower sidebands extending over larger
wavelength ranges, and their response is less sensitive to variations in waveguide-grating
parameters. Guided-mode resonance reflection filters that integrate gratings into multilayers
of well-known anti-reflection coatings are presented in chapter 2 [29,48]. The availability of
these design techniques (such as “V” and “W”-structures [48]) allow increased flexibility in
the choice of materials for fabrication of practical filter devices compared to previously
reported devices [29]. The response of the resulting structure combines the 100% peak
reflectance characteristic for the waveguide grating with the antireflection properties of the
equivalent homogeneous thin-film design responsible for low sidebands over a large
wavelength range.
The guided-mode resonance effect can be used to design transmission bandpass filters
by combining the resonance effect of a waveguide grating with the high-reflectance thin-film
stacks presented in chapter 2. Section 3.3.1 presents the double-grating design that provides
narrow-band transmission filters with almost
100
% efficiency and low sidebands over a wide
spectral range [34]. A transmission filter design consisting of a single grating in the center
layer bordered by two dielectric mirrors composed of high/low quarter-wave layers is
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8
presented in section 3.3.2 [47]. It is found that the transmittance curve of the single-grating
filter obtained with fewer layers has a narrower linewidth than a comparable double-grating
design while maintaining the same high peak transmittance and low sidebands.
Multi-line filters can be achieved with waveguide gratings by several means that are
discussed in section 3.4.1. One method to obtain double-line filters is to break the
symmetrical coupling that occurs at normal incidence between the
±1
evanescent diffracted
waves and the waveguide modes. This occurs, for instance, in any waveguide-grating device
operated at non-normal incidence. At normal incidence multi-line filters are achieved in
structures containing gratings with different periods [32,79]. Section 3.4.2 presents a filter
consisting of two gratings and an intermediate homogeneous layer. An advantage of this
method of attaining a double-peak response resides in the full control over the two center
wavelengths determined by the periods of the two gratings. It is shown that the reflectance
peaks approach
(typically
~2
100
% provided that the two gratings are decoupled by a sufficiently thick
wavelengths) homogeneous layer.
For implementation of specific reflection and transmission filters, the spectral
characteristics are tailored by adjusting the parameters of the device. The center wavelength
of the filter is determined by the grating period and the refractive indices of the grating. The
sidebands can be made arbitrarily low over a specified spectral range by adding layers with
refractive indices and thicknesses, obeying antireflection (for reflection filters) or high
reflection (for transmission filters) conditions. The filter linewidth, discussed in section 3.5.1,
is controlled by the modulation of the grating, the difference between the average refractive
index in the grating region and the refractive indices of the surrounding media, and the
grating fill factor [24,40]. The linewidth dependence on the grating parameters can be
explained in terms of the guided-mode confinement and leakage of the waveguide gratings.
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9
The influence of phase-shifted gratings on the filter response is illustrated in section
3.5.2 [32,79]. It is shown that while arbitrary phase shifts distort a symmetrical filter
response, broader (for reflection filters), or narrower (for transmission filters) linewidths can
be obtained for specific values of the phase shifts depending on the waveguide-grating
parameters. The spectral response of a guided-mode resonance filter typically has a
Lorentzian shape that is particularly undesirable in narrow linewidth filters. Phase shifts of
7t/2 are shown to alter the shape of the reflectance curve of single-layer filters, bringing it
closer to an ideal rectangular shape.
A comparative study of guided-mode resonance filters with classical designs based on
homogeneous thin-film layers is presented in section 3.6 [48]. It is shown in section 3.6.1
how typical reflection filter designs require considerably more layers to yield equivalent
narrow-band linewidths (in the nm range), that can be obtained with only 2 or 3 layers in a
waveguide-grating structure [48]. Guided-mode resonance transmission filters are compared
to classical Fabry-Perot designs in section 3.6.2.
A typical design procedure for guided-mode resonance filters is presented in section
3.7 [49,68]. The design of reflection and transmission filters is tantamount to finding the
physical parameters of the thin-film diffractive structure (such as grating(s) period(s) and fill
factor(s), number o f layers, refractive indices, thicknesses) to yield the desired central
wavelength, linewidth, sideband level and sideband spectral range. This entails repeated
iterations in which approximate waveguide eigenvalue equations, thin-film antireflection and
high-reflection properties, and rigorous coupled-wave analysis are used to generate the
specified filter line. This method of design is referred to as the “direct” method, as opposed to
an “inverse” method based on genetic algorithms and presented in chapter 5.
Experimental results on guided-mode resonance filters are presented in section 3.8.
Section 3.8.2 describes fabrication and spectral measurements of a guided-mode resonance
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10
filter exhibiting the highest peak reflectance reported to date (—98% at 860 nm) [74]. The
device consists of a photoresist grating on a Hf 0 2 waveguide fabricated on a fused silica
substrate. The grating is obtained by holographic recording in a Lloyd’s mirror interference
set-up. Testing of the spectral properties of this filter is performed with a dye laser tunable in
the range 800-920 nm, and excellent matching to the theoretical predictions is found. This
high-efficiency reflection filter is used as the output-coupling mirror in a commercial dye
laser [6 6 ]. The laser beam generated with the resonance-mirror cavity has a power of 100
mW with Argon laser pump power of 5 W. The linewidth of the laser beam (-0.3 nm) is
determined by the width of the resonance peak at the threshold output-mirror reflectance
required for laser oscillation to occur [6 6 ].
While guided-mode resonance filters for the optical region have received considerable
interest lately, there has been no comparable study of the filtering characteristics of these
structures in the microwave spectral region. Chapter 4 demonstrates how the design
principles for the optical guided-mode resonance filters can be applied to produce ideal or
almost ideal microwave reflection and transmission filters in the frequency range 4 —20 GHz
[39.49.79]. Furthermore, due to the larger sizes of the gratings and the availability of
materials with larger dielectric constants, there is an increased flexibility in the design of
microwave filters, as compared to their optical counterparts. Microwave filters can have large
linewidths and improved antireflection or high-reflection properties to reduce sidebands
while maintaining the center line frequency response close to 100%. Examples of single-layer
and multilayer narrow and broadband reflection and transmission filters are given in sections
4.2 and 4.3, respectively. Experimental measurements of guided-mode resonance filters
performed in an anechoic microwave chamber at 4 —20 GHz are presented in section 4.4
[78.79]. Teflon lenses are used to collimate the incident beam on the waveguide gratings and
to focus the diffracted beam towards the receiver. Measurements on single-layer waveguide
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11
gratings constructed with plexiglas bars in air exhibit transmittance notches at wavelengths
close to theoretically predicted values for both TE and TM incident wave polarizations [79].
Guided-mode resonances in transmittance are obtained in 5-layer structures containing a G10
fiberglass/air grating in the center layer, and a homogeneous G 1 0 layer at a distance in air on
each side of the grating [78]. The measured spectral response of the filter is accurately
matched by the theoretical calculations. Fabry-Perot filters with 5 layers of alternating G10
fiberglass and air have also been fabricated and tested [69]. As predicted by theory,
transmittance measurements of Fabry-Perot filters are shown to yield larger linewidths than
guided-mode resonance filters with an equal number of layers and using the same materials.
Chapter 5 presents a new method of guided-mode resonance filter design based on
genetic algorithm routines [77]. Genetic algorithms are search and optimization procedures
that follow the natural selection process, operating on a large population of candidate
solutions, and utilizing the genetic operators of selection, crossover and mutation to improve
the merit function from one generation to another [104]. Section 5.2 is an introduction to the
concepts and important features o f genetic algorithm optimization. A variety of optimization
problems employing genetic algorithms have been reported in the optics literature in optical
signal and image processing, diffractive optics, thin-film optics, image formation, and
tomography [112-126]. These applications of genetic algorithms in optics are reviewed in
section 5.3 [77]. An outline of the genetic algorithm program developed for optimization of
multilayer structures, containing homogeneous thin films and diffraction gratings is presented
in section 5.4. The program employs rigorous coupled-wave analysis for calculation o f the
reflected and transmitted diffraction efficiencies [84-86] and hence, for evaluation o f the
merit function for the generated structures. The software library PGAPack [110] performs
specific genetic algorithm operations (chromosome generation, ranking, selection, crossover,
mutation, etc.) The algorithm seeks to find the physical parameters of the diffractive structure
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12
that generates the spectral dependence of the zero-order reflected (or transmitted) diffraction
efficiency provided by the user in a reference data file. It is important to study the evolution
of the optimization process (i.e., the convergence) for various starting conditions, to
determine the influence o f the input parameters on the optimization procedure, and to find
some guidelines for selecting the appropriate set o f input parameters for a specific
application. In section 5.5, the convergence of the merit function is studied as a function of
key genetic-algorithm parameters, such as the population replaced at each iteration, mutation
probability, type of encoding, number of generations, population size [77].
This program is versatile and can be applied for design of a variety of diffractive
devices, such as fan-out gratings, anti-reflection coatings, high-reflectors, polarizing
elements, beam-splitters, and edge filters. In the present work, the program is used to design
new reflection and transmission filters based on guided-mode resonances, illustrated in
sections 5.6 and 5.7, respectively. Transmission bandpass filters found by the genetic
algorithm possess features not previously known [64,65,71]. Thus, calculated examples
demonstrate that guided-mode resonance bandpass filters with narrow linewidths (<0 . 1 %),
peak transmittance close to
100
%, and low sidebands ( - 0 . 1 %) can be obtained with thin-film
grating structures containing only 1-3 layers. It is demonstrated that the low sidebands of
these filters are not a result of the high-reflectance properties of equivalent homogeneous thin
films, as in the case o f the previously reported transmission filters (section 3.3). The highmodulation gratings employed in the design of transmission filters with reduced number of
gratings provide the high transmittance peak, as well as the low sideband response. Multiple
solutions to a given filter design problem are found using the genetic algorithm routine,
which may facilitate fabrication and implementation o f guided-mode resonance filters in
applications.
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13
A new type of guided-mode resonance device consisting of waveguide gratings
fabricated on optical fiber endfaces is proposed is chapter
6
[80]. Submicron diffraction
gratings with high efficiency have been recorded for the first time on optical fiber endfaces.
Waveguide gratings have been fabricated, consisting of a photoresist grating on a silicon
nitride thin film deposited on a multimode fiber endface. Spectral measurements of these
devices performed with a titanium sapphire laser indicate notches in the transmittance curve
due to guided-mode resonances. Although the preliminary experiments indicate notches with
only -18% efficiency, these represent the first demonstration of a guided-mode resonance
device integrated with an optical fiber. More generally, it is the first experimental study of
guided-mode resonance devices embedded in a waveguide (i.e. with a guided-mode incident
wave). These fiber-endface filters have potential applications in optical communications,
fiber-optic sensors, and fiber-laser technology.
The conclusions are presented in chapter 7, stressing the novel contributions to the
understanding and development of guided-mode resonance filters brought about by the
present dissertation. The present study of guided-mode resonances has enhanced the
knowledge of electromagnetic phenomena occurring in periodic, thin-film structures and
widened the path to the development of a new class of optical and microwave devices with a
large number of potential applications. Further theoretical and experimental research is
needed for the guided-mode resonance devices to be used in practical applications in the
optical and microwave spectral region. The results o f this dissertation research contribute to
future developments of real devices.
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CHAPTER 2
PRINCIPLES OF GUIDED-MODE RESONANCE FILTERS
2.1 Introduction
The guided-mode resonance effect occurs in thin-film structures containing
waveguides and diffraction gratings. It consists of rapid variations in the intensities o f the
reflected and transmitted waves, as the wavelength, angle of incidence, or any one o f the
structural parameters is varied around their resonance values [18]. The resonance conditions
are satisfied as the incident wave is coupled to leaky modes of the diffractive device. The
effect is present in wide variety of structural configurations containing a single layer or
multiple layers [29], one or more gratings of equal [34] or different grating periods [32], in
one-dimensional or two-dimensional [38] diffractive devices, with gratings formed by
rectangular, sinusoidal or other refractive index or thickness profiles [6 8 ].
The basic structure of a single-layer guided-mode resonance filter is shown in figure
2.1. The planar, rectangular grating is composed of dielectric materials with refractive indices
nu and nL, thickness d, and grating period A. The grating fill factor f is defined as the fraction
of the period containing the high-refractive index material. The modulation of the grating is
determined by the variation of the refractive index within a grating period, An = nn - nL- For
a grating to possess waveguide properties, its average refractive index nav, must be greater
than the refractive indices of the cover (medium containing the incident and reflected waves)
nc and substrate (medium containing the transmitted waves) ns- The average refractive index
is calculated as
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15
The main purpose of this study is to apply the guided-mode resonance effect to highefficiency filter devices. Therefore, to maximize the intensity variation of the propagating
fields at resonance, a sufficiently small grating period is chosen to obtain only zero
diffraction orders propagating in reflection (Ro) and transmission (To).
A typical TE-polarization (i.e., the electric vector normal to the plane of incidence of
figure 2 . 1 ) reflectance spectrum for the device geometry of figure
2 .1
is illustrated in figure
2.2 [48]. For the greater part of the spectrum, the zero-order grating has the reflectance and
transmittance of a thin film, with a refractive index equal to the average refractive index of
the grating. At specific values of the wavelength and incident angle, the diffractive element
enables the incident electromagnetic wave to couple to the waveguide modes supportable by
the structure. The periodic modulation o f the guide makes the structure “leaky,” preventing
sustained propagation of modes in the waveguide, but coupling the waves out into the
substrate and cover. As the wavelength is varied around the resonance, rapid variations in the
phases of the re-radiated waves occur. The re-radiated waves interfere with the directly
reflected and transmitted fields, to generate rapid variations in the intensity of the externally
observable electromagnetic fields with respect to the wavelength or angle of incidence of the
incident wave. A similar behavior is found in multilayer structures (figure 2.4) containing
gratings and homogeneous layers. Due to the waveguide nature o f the diffraction grating
resonance phenomenon, physical insight can be gained by studying the waveguide properties
of zero-order waveguide-grating structures, as discussed in section 2 .2 .
The reflectance spectrum in the region of a guided-mode resonance typically exhibits
a high peak (approaching 100%) preceded or followed by a notch (approaching 0%). The
variation has, in general, an asymmetrical lineshape with sideband levels unsuitably high for
either reflection or transmission filters.
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16
Incident wave
i rz
Figure 2.1. Single-layer grating with rectangular profile of high (nH) and low (nO
refractive indices with period A and thickness d. The cover and substrate
refractive indices are nc and ns, respectively. The plane o f incidence is the x,z
plane.
1
-
0.9
---- Grating
0 .8
----Homogeneous
B) 0.7-E
O
Sj °-6~O
w 05-.
0.4
^ 0.3
0 .2
I
. ~
0J
0 1
► —
1 1
1 r |
1 1
1 1
|
1 1 1 1
|
I I
1 1 I
r
1 I
__
1 1 I
I
I
I
1 1 1 1 1 1 1 1 I
I 'I
0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59
WAVELENGTH (pm )
Figure 2.2. Guided-mode resonance at X, = 550 nm in the reflectance of a single­
layer waveguide-grating with period A = 308 nm, thickness d = 150 nm and
refractive indices nn = 2.1, Ol = 2.0, ns = 1.52, and nc = 1.0. The equivalent
homogeneous layer has d = 150 nm and n = 2.05.
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17
By applying multilayer thin-film interference coatings, improvements in the lineshape
symmetry and sideband levels o f guided-mode resonance filters can be obtained [27,29]. This
enables high-efficiency reflection or transmission filters with low, extended sidebands,
potentially useful in a number o f applications [24].
2.2 Waveguide Properties
2.2.1 Single-layer waveguide grating
Exact calculation of the spectral response of waveguide gratings and design of devices
based on these structures is performed using rigorous electromagnetic theories, such as the
rigorous coupled-wave analysis presented in section 2.4. However, this numerical method
does not provide an analytic solution for the diffraction efficiency of the gratings and,
therefore, does not yield detailed insight into the physical processes occurring inside a
multilayer waveguide-grating structure. The waveguide properties of the thin-film gratings
under study are emphasized by expressing the total field in the grating region, Ey(x,z), in a
A
sum of inhomogeneous plane waves with amplitudes, S(.( z ) , propagating along the x
direction (along the boundary as shown in figure 2.1) [84]
oo
Ey (X,
z) =
S { (z) exp[- 7 <jlVtx]
where cr = k2 - i K is the wavevector in the grating region, k 2 = kQ
(2.2)
is the wavevector
magnitude in the grating region of the zero-diffracted order (i = 0 ), ko = 2nfk, X is the freespace wavelength of light, and £aVis the average relative permittivity in the grating region,
given by £m =£Hf + £L(l - / ) . The magnitude of the grating vector is K=2rc/A, where A is
the grating period. It is assumed that Ey(x,z) is oriented along the y direction (TE
polarization) in this case. The electric field given by (2.2) is substituted in the wave equation
V zE y(xJz) + k 2e(x)Ey(x,z) = 0
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(2.3)
where e(x) is the relative permittivity of the grating with periodic modulation. This can be
expanded in Fourier series as
£(x)= £ f Aexp [jhKx]
(2.4)
where £h are the coefficients of the Fourier expansion. The resulting set of coupled-wave
equations is written as [18]
where 8 c is the relative permittivity of the cover region, £h* is the complex conjugate of Eh,
0
is the incidence angle, and A e = n H
2 —n \ is the modulation of the relative permittivity in the
grating region.
A
Note that this second-order differential equation contains no first derivatives 5f. . In
the case o f unslanted gratings, this expansion leads to constant-coefficient differential
equations. However, this would not be valid in the general case of slanted diffraction gratings
where z-dependent coefficients make the system more difficult to solve numerically [84].
The coupled-wave equations in the form given by (2.5) can be related to the wave
equation inside the slab region of an unmodulated dielectric waveguide given by
(2.6)
dz
where k 2 =
is the plane-wave propagation constant in the waveguiding layer,
£2
is the
relative permittivity o f the waveguiding region, and P is the propagation constant of the
guided slab waveguide modes (the component of k^ along the x-axis in figure 2.1). A similar
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19
expression is obtained from the coupled-wave equations in the case of van ish in g ly small
grating modulation Ae—>0.
S,-
s in # '
(z) +
i
\2
S,(z) = 0
(2.7)
The propagation constant inside a waveguide grating is written as [18,22,24]
Pi =
si11
(2.8)
~ M / A)
By analogy with waveguide theory, a guided mode can exist if
(2.9)
k 2^em - Q e ^ s i n d ' - i A . / a ) 2J > 0
In addition, the corresponding forward- and backward-diffracted waves in the cover
and substrate regions are required to be nonpropagating waves.
This condition can be
expressed as
rj-O y/^T sin # ' - U / A
) 3
<0
(2. 10)
where £s is the substrate relative permittivity. From inequalities (2.9) and (2.10), the
condition for the guided waves to exist inside a grating can be represented as [18,24]
m a x { y i7 » V ^ 7 14 ^ c sin # ' —i A / a | <
(2 . 11)
In a waveguide grating this condition defines the range of wavelengths X or incident angles 0'
for which modes can exist. The existence of modes enables the coupling of the diffraction
orders to these modes, leading to resonance behavior in waveguide gratings.
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20
Therefore, the first step in designing a waveguide-grating filter would be to use
condition (2 . 1 1 ) to find the wavelength range of the grating resonance, once the grating
materials, the grating period, and the angle of incidence have been selected. Within this
range, one can use the properties of the equivalent unmodulated waveguide to find an
approximate location of the resonance.
For a classical, homogeneous slab waveguide, the eigenvalue equation can be written
in the form [87]
tan(«f) =
K (y+ S)
k
(2 . 12)
2 -y S
where d is the thickness of the slab waveguide,
(2.13)
are the transverse (i.e., z component) wavenumbers in the cover, grating, and substrate,
respectively, and (3* is the wavevector component along the boundary (i.e., along x)
corresponding to mode i (figure 2.1). In the homogeneous waveguide case, the eigenvalue
equation is solved numerically for Pi leading to a full description of the mode propagation
inside the waveguide, hi the waveguide-grating case the real part of the propagation constant
Pi is given by equation (2 .8 ) and the relative permittivity of the homogeneous waveguide
layer
is replaced by the average relative permittivity 8 aVof the high and low regions of the
82
grating.
The waveguide eigenvalue equation can thus be used to determine the approximate
wavelength locations of the grating resonances A^s, corresponding to the modes that can be
supported by the waveguide; the first resonance peak corresponds to the fundamental (zeroth)
mode and the next resonances correspond to consecutively higher-order modes. Once the
resonance wavelengths are known, an important parameter for filter design can be
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21
determined: the resonance-free spectral range (i.e., the difference between consecutive
resonance wavelengths,
= AV+I ~ K > where v = (0 , 1 ,2 ,...) are the integers labeling the
waveguide modes). Since we have assumed Ae—>0, the solution will be a better
approximation for a smaller modulation. A similar procedure can be used to determine
resonance locations for the TM polarization [20].
2.2.2 M ultilayer waveguide grating
For a multilayer waveguide grating, the resonance wavelength locations can be
determined from the corresponding homogeneous multilayer eigenvalue equation [88,89].
PcA + PCPSB + C + PsD —0
(2.14)
A, B, C, and D are the elements of the N-layer characteristic matrix calculated as a product of
the individual layer characteristic matrices. Pc, Ps are given below by equations (2 .2 1 ) and
(2 .22 ).
(2.15)
The elements of the characteristic matrix of the n-th layer are given as [8 8 ]
m Ujl =cos (kdnPn)
m \2,n
= [ - j s m ( k d nPn)]/Pn
(2.16)
(2.17)
" h i = - j P n sin(kdnPn)
(2.18)
m22<n=cos(kdnPn)
(2.19)
where dn is the thickness of the n-th layer.
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22
The wavevector components perpendicular to the layer boundaries in the waveguide
grating (Pn), cover (Pc), and substrate (Ps) can be defined as in the case of homogeneous
layers [29]
Pn=[£av, n - ( f i i ^ ) 2]U2
(2 -2 0 )
P c ^ S c - d P i / k ) 2]1' 2
(2.21)
Ps = [£ s -(A -/* 0 2]I/2
(2-22)
where the component of the propagation vector along the boundary, Pi, is given by the same
expression as in the single-layer waveguide grating equation (2 .8 ), and Eav.n is the average
relative permittivity of the n-th layer.
As in the single-layer waveguide case, the eigenvalue equation (2.14) can be solved
numerically for X to find the approximate location of the guided-mode resonances and the
free spectral range of a multilayer waveguide. Knowing the approximate location of the
resonance, one can use the coupled-wave theory describedearlier to accurately determine the
wavelength
o f the waveguide-grating resonance, as well as all other features of the filter
devices.
The spectral and angular widths of guided-mode resonances are other important filter
features, and their dependence on the physical parameters of the structure has been studied
with rigorous coupled-wave analysis computations. Numerical simulations have determined
that the linewidth of guided-mode resonance filters broadens with increased modulation of
the grating [24]. This can be explained by the increased leakage of the waveguide mode. The
mode confinement (determined by the difference in refractive index between the waveguide
and the adjacent layers) and the grating fill factor also determine the linewidth of the guidedmode resonance filters. The linewidth is maximum for a grating with fill factor f = 0.5 and
decreases for smaller or larger values of f [40]. This can also be explained by the waveguide
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23
properties of the structure. Calculations o f radiation losses from a waveguide with an overlaid
grating have demonstrated that, in the case of TE polarization, the leakage versus f exhibits a
cosine dependence with a peak at, or close to, f = 0.5 [90]. Higher order modes have been
found to have increased radiation losses [90]. Conversely, TM polarization modes exhibit
maximum leakage at values that differ from 0.5, with increasing shift from this value in the
case o f higher order modes [90]. Moreover, TM modes can exhibit two peaks in the leakage
versus fill factor dependence [90].
A direct relation between the radiation loss and the width of the guided-mode
resonances has also been established under simplifying assumptions using waveguide
coupled-mode theory [46]. A single-layer waveguide grating with sinusoidal variation of the
refractive index, allowing only one mode to couple to the output radiation field, yields the
following spectral (AX) and angular (A0) linewidths of guided-mode resonance filters [46].
(2.23)
(2.24)
where X,- and 0r are the resonance wavelength and angle o f incidence, respectively. The
waveguide coupling loss y can be calculated with waveguide coupled-mode theory. The
reflectance around a guided-mode resonance was approximated with a Lorentzian lineshape.
The spectral and angular linewidths calculated with coupled-mode theory are close to the
values calculated with rigorous-coupled wave analysis for a range of structural parameter
values that yield a Lorentzian lineshape of the resonance. Linewidth control of guided-mode
resonance filters is further discussed in this dissertation, and calculated examples in the
optical and microwave spectral ranges are illustrated in chapters 3 and 4, respectively.
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24
Waveguide theory has been employed to explain guided-mode resonance phenomena
in diffractive structures consisting of a thin film with two corrugated boundaries
[7,8,10,13,16]. Specific phase shift values between the upper and lower corrugations
(dependent on the structural parameters) were calculated to yield unidirectional output
coupling of the wave in the substrate. These phase-shift values were shown to extinguish the
guided-mode resonance peak, otherwise present in the spectrum o f the reflected wave [16].
It can be concluded that waveguide theory not only provides a useful tool in
determining the grating resonance characteristics, but more importantly, offers insight into
the wave interactions occurring inside a waveguide grating, explaining the origin and some
properties of the resonance phenomena associated with these structures.
2.3 Thin-Film Properties
The rapid reflected and transmitted intensity variations due to guided-mode
resonances are associated with sideband reflectance and transmittance values too high for
many practical applications. It was found that the sideband response of a guided-mode
resonance can be calculated with thin-film interference theory, by replacing the grating (or
gratings) of a single (or multilayer) waveguide grating structure with equivalent
homogeneous layers [27,29,48]. An equivalent homogeneous layer has the same thickness as
the grating, and a refractive index equal to the average refractive index of the grating,
calculated according to (2.1). Such an example is illustrated in figure 2.2. The equivalent
homogeneous layer (dashed line) features the same reflectance as the waveguide grating
(solid line) over the entire spectral range except for the resonance region. The homogeneous
thin-film approximation is valid for a wide range of grating refractive indices but fails at high
modulations of the gratings for both reflection and transmission filters. This discrepancy will
be shown in transmission filter examples of chapter 5. However, within the limits of its
validity, this approximation enables the design of reflection and transmission filters with low
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25
and extended sidebands, by applying known results from thin-film interference theory
[27,29,34,48],
2.3.1 Antireflection structures
Guided-mode resonance reflection filters are designed by embedding the resonant
grating in a structure possessing low reflectance (i.e., in an antireflection design). In the case
o f a single-layer waveguide grating, in the limit of small grating modulation, the reflectance
at normal incidence is given by [27]
D_ n l ( n c - n s ) z cos 2 k 2d + (ns n c - n 2m ) 2 sin 2 k2d
K
~
2----------------------2--------- 2---------
2---- 2-------- 2---------
nav(nc + ns') cos k 2d + {ns nc Jt-nm) sin k2d
( 2 .2 5
where nav is the average refractive index of the grating, k 2 = kn2av is the wavevector
magnitude in the layer, ns and nc are the substrate and cover refractive indices, respectively,
and d is the thickness.
An ideal antireflection structure would thus have equal substrate and cover refractive
indices (nc = ns) and a thickness equal to a multiple of half o f the central wavelength (d =
mA/2nav) [91]. This result is effectively used in the microwave spectral range where the
waveguide grating does not require a supporting substrate and nc = ns = 1.0 (air). However,
at optical wavelengths, the cover and substrate refractive indices are typically different. In
this situation, the reflectance can be minimized, while maintaining the waveguide nature of
the grating (nav > nc, ns), by selecting a half-wavelength thick waveguide grating [91].
Multiple layers can be used to further decrease the sideband reflectance and/or to
widen the low reflectance spectral (or angular) range. The antireflection characteristics of the
structure are maintained if one or more layers are replaced by low-modulation gratings.
Double-layer structures can yield zero reflectance at a desired wavelength, if the layers are
quarter-wavelength thick and the average refractive indices of the layers satisfy the condition
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26
*Jnc / ns [92,93]. In most practical cases, this refractive index condition can not be
met exactly due to the limited values of refractive indices available in thin-film materials, as
well as to fabrication constraints. A more general antireflection condition that allows more
freedom in the choice of refractive indices can also be obtained. The thicknesses of the two
layers are determined by the expressions [92]
t a n Z j - (ns - n c ) ( n l - n cns )nt
‘ Cni n m ~ n cnl){nc ns - n f )
(2.26)
(2.27)
(«1 n m ~ n C n 2 X W2 “ n C n S )
where 8 , = 27tnidjA,, i = 1,2. The refractive indices have to satisfy conditions imposed by the
existence of real values for 8 i and
82
. The antireflection solutions (2.26) and (2.27) provide a
spectral variation of the reflectance with a zero at a single wavelength and increasing
reflectance on either side of the minimum. Such a design, commonly named “V-coating”
[93,94] is illustrated in figure 2.3 (solid line).
An antireflection coating with a wider wavelength range can be obtained if two
minima are generated in the spectral response of the two-layer structure. This design, referred
to in the literature as “W-coating,” is illustrated in figure 2.3 (dashed line) [93,94]. Further
reduction in reflectance can be achieved with additional layers. Figure 2.3 (dotted line)
illustrates the reflectance of a three layer antireflection design determined by thicknesses of
the layers equal to quarter-, half-, and quarter-wavelength, respectively [92-94]. In general,
multilayer antireflection coatings are designed by numerical optimization o f the thicknesses
and refractive indices to achieve the desired low reflection characteristics. These
antireflection designs can provide the low-sidebands required in guided-mode resonance
filter design as demonstrated by the calculated examples of chapter 3.
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27
0.04V-coating
0.035t
W-coating
0.032 0.025
O 0.02
u
ft 0.015
0.005
0
0.45
0.5
0.55
0.6
WAVELENGTH (pm )
0.65
Figure 2.3. Antireflection coating designs using 2 layers (V coating and W
coating) and 3 layers. The refractive indices of the layers are ni = 1.38, n2 = 2.05,
and for the 3-layer coating, n 3 = 1.62. The thicknesses are di = 0.073 pm, d2 =
0.115 pm (V coating), dt = 0.1 pm, d2 = 0.134 pm (W coating), and di = 0.1 pm,
d 2 = 0.134 pm, and d3 = 0.085 pm (3-layer coating).
2.3.2 High reflectance structures
Transmission filters can be generated in waveguide gratings by embedding resonant
gratings in the homogeneous thin-film structures with a high-reflectance property. The most
common high-reflectance design consists of alternating thin films of high and low refractive
index materials with quarter-wave thicknesses [92-95]. The quarter-wave mirror reflectance
at the central wavelength of a thin-film structure with an even number of layers and a high
refractive index first layer is given approximately by [96]
(2.28)
where N is the number of quarter-wave layers.
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28
For an odd number of layers with a high refractive index material in the first layer, the
reflectance is [96]
n r n o f n- L \ 2N
R ~ 1—45
n a \ n HJ
(2.29)
These expressions demonstrate that, in both cases, the reflectance increases with the
number of layers N and with the ratio between the low and the high refractive indices (ni/nn)Another important parameter in filter design is the width of the wavelength region with low
transmittance. From thin-film interference theory, the width of the high-reflectance zone at
normal incidence is given by [96]
f
Xr -/L, =A0 n
1
1
/r-a rcc o s(-£ )
tu+ arccos(-£)
- n t + n t —6nHn r
£=—
^
CnH+nL)
(2.30)
(2.31)
where A* and Ai are the right and left spectral limits of the high-reflectance region,
respectively, and Ao is the central wavelength. Note that this formula does not depend on the
order of high and low refractive index alteration and, more importantly, the relative width o f
the boundary wavelengths depends only on the nH/ n L ratio, and not on the number of layers
considered.
2.4 Rigorous Coupled-Wave Analysis of
Diffraction by Waveguide Gratings
2.4.1 Single-period structures
Accurate description of resonance properties in waveguide-grating devices requires
rigorous diffraction analysis routines that obtain exact solutions of Maxwell’s equations,
taking into account second order field derivatives, propagating and evanescent waves, and
boundary conditions at interfaces. The numerical method used in this work, rigorous coupled-
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29
wave analysis, is accurate, efficient and stable, and has been applied to a wide variety o f
gratings
(h o lo g ra p h ic ,
binary, single or multilayer, surface relief, dielectric or metallic,
isotropic or anisotropic) for arbitrary polarization and angle o f the incident beam. The
diffraction efficiencies of these structures can be calculated with computer codes developed
in the Electro Optics Research Center at The University of Texas at Arlington. In the present
dissertation this formalism is employed to calculate diffraction efficiencies of a multilayer
structure consisting o f binary gratings with rectangular permittivity profile and homogeneous
layers as illustrated in figure 2.4. This section presents the rigorous coupled-wave analysis for
TE-polarized incident waves.
Figure 2.4. Schematic of a multiple-layer thin-film structure with an arbitrary
number of binary gratings o f the same grating period but different modulations.
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30
In the rigorous coupled-wave approach the tangential electric (Ey,) and magnetic
(Hx,/) fields inside the Z-th grating layer are expanded in terms o f spatial harmonics
oo
E y.l
(*» z )
H x ,l
=
X
S y .u
U ) exPi - j k x j X ]
f
xi/2
z)= -j
( 2 .3 2 )
£0
where kx,i is the wavevector component along x given by the Floquet condition
kxJ = kQnr sin 6 —i K , ko = 2 k/X is the wavenumber in free space, and
0
is the angle of
incidence. K is the grating vector magnitude, K= 2k /A, where A is the grating period, ni is the
refractive index of the input medium, £o and |io are the free-space permittivity and
permeability, and Sy,/,i (z) and Ux,/,i (z) are the normalized amplitudes of the i-th spaceharmonic fields.
These field expansions must satisfy Maxwell’s equations assuming harmonic time
dependence and uniform field along the y direction.
dEyj (x, z) _
dz
jO)fi0H xl(x,z)
_x
2
dz
(O/Lq
(2.33)
dhr
where the periodic variation o f the permittivity in layer Z, e/(x) is expressed in terms of its
Fourier components as in (2.4). Substituting the field expressions (2.32) and the relative
permittivity expansion (2.4) in equations (2.33) and eliminating the amplitudes Ux,/,i one
obtains an infinite set of second-order coupled-wave equations given by
d d*(’J£
z)
where z' = koz, for
-} = k0
a=-~
7
< i < <» and 1 < I < L.
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<2-34>
31
In order to obtain a numerical solution, the infinite number of coupled-wave
equations are truncated to a finite number of harmonic orders N and the eigenvectors and
eigenvalues of the resulting NxN system matrix are calculated. The solution to the system of
second-order coupled-wave equations can be written as
S y .u
N
(*) = I
fctm exp(-fc0<?, m(z - Dt_{)) + Clm exp(k0qlm (z - D,))]
f7T=I
(2.35)
D,_l < z < D [
A = 5 X
p = i
where d/ is the thickness of the /-th layer, q/,m is the positive root of the m-th eigenvalue,
W/im is the corresponding eigenvector, while Cfm and C,~m are unknown amplitude
constants to be determined from the boundary conditions.
The tangential electric and
magnetic fields Ey,/ and Hx,/ in the grating region can then be obtained from relations (2.32)
and Maxwell’s equations (2.33).
Aim exp(-*0^/ m(z - A - i)) +
i= ( A /- I ) /2
E,,= S
lx ...
(= —( A M ) / 2
m=l
/= (A /—1 ) /2
s
t= —(AT—1 ) /2
►exp( ~ j k xix)
Aim eXP(^0^/,m(z —A ))
N
m=l
(2.36)
Aim exp(-fc04/.m (Z - A -. )) -
exp ( ~ jk xix)
A7meXP ( M / . m ( Z - A ) )
In the homogeneous layers the tangential electric and magnetic fields are written as a
sum of forward and backward propagating plane waves with unknown constant field
amplitudes P/,i and Q/,i.
(a^ / 2 f/> . e x p ^ y . , (z - A _,» +1
= ,J
„
. ..
,
„ 4 a J exP( ^ n , ( Z- A ) )
= ( - ! _ ) '" f n c - k r
|2[,i exp(fc0^ (z - Z>,))
(2.37)
J
P 7 " )
where y, f. = j ( n f —n, sin 2 0) 1 / 2
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32
In the input (medium I) and output region (medium IE), the electric field can be
expressed in terms of the incident and diffracted propagating waves in the form
(jv-n/2
E , j = exP(-J**.0z) + £ Ri exPUkzjjZ) exP( r f c XJx)
(A f-l)/2
E y .m =
^
2
Ti e x P ( - j b z j t u
i= - ( A f - 1 ) /2
e x P (- J K
j
'
x )
where R, and Ti are the normalized electric field amplitudes or the reflected and transmitted
diffraction orders respectively, and D is the thickness of the structure. The wavevector
components along x and z are determined from phase matching requirements and imposing
the Floquet condition kxJ = k0[n, sin8 - i(A / A )], kz pJ = (k* —kxj)112 where p = I or HI.
Imposing the boundary conditions at the interfaces between the L adjacent layers and
between the structure and the input and output media, a system of 2N(L+1) equations is
obtained, with 2NL unknown field amplitudes in the layers, N unknown reflected wave
amplitudes R; and N unknown transmitted wave amplitudes T;.
This system can be solved directly to find all unknowns simultaneously, but this
would be highly inefficient, particularly for large L. To increase the efficiency of the
numerical computation, the number of equations can be reduced by eliminating the 2NL
unknown amplitudes C^m and C[m in the L layers resulting in a system of 2N equations for
unknowns Ri and Ti. To prevent numerical instabilities associated with inversion of illconditioned matrices and increase the accuracy of the computation, the enhanced
transmittance approach presented in reference
86
is utilized to calculate the reflected and
transmitted wave amplitudes.
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33
Once the reflected and transmitted diffraction orders have been found, the diffraction
efficiencies are determined from the formulas
l.zj
DEr j = Re
i*r
(2.39)
IfI g j
DE tj = Re
2.4.2 D ouble-period structures
In this section, the rigorous coupled-wave analysis described in section 2.4.1 is
extended for the case of multilayer structures containing gratings with two different periods
Ai or A 2 in different layers, as illustrated in figure 2.5 [40]. The coupled-wave theory
derivation follows the same steps as in the previous section. A major difference that arises in
the present case is that each grating period generates its own diffraction spectrum.
Consequently, to account for the double-diffraction spectrum, the electric and magnetic fields
in each layer o f the diffractive structure are expanded in a double summation over the plane
waves amplitudes, Sy,/tn,i2 and Ux,/,ii,i2 -
E y j(*>z ) = I I S ylJli2 (z )e x p [- y ^ . ,2 x]
n=—'2=r
V'2
_£o_
1^0
00
(2.40)
00
q=-«oi2=-
where kx,ii,i2 is the wavevector component along x given by the Roquet condition for double­
period gratings kxii i2 = k0n, sin 6 —ixK x —i2K 2; ni is the refractive index of the cover
medium; ko = 2itfk, is the free-space wave number; K\=2tUK\ and K2=27t/A2 are the grating
vectors of the two gratings, and eo and |io are the free-space permittivity and permeability.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.5. Schematic o f a multiple-layer thin-film structure with an arbitrary
number of binary gratings possessing either period Ai, or period A2 .
The relative permittivities o f the grating layers are expanded in Fourier series
according to the periodicity of the layers (Ai or A2 ).
e l ( x ) = ^ e lhcxp[jhKlx\
*=-~
(2.41)
if layer I has a grating with period Ai, and
00
£ 1 0 ) = X e u>QXp\jhK2x]
A= —00
(2.42)
if layer I has a grating with period A 2 , where £/,h are the coefficients of the Fourier expansion
for layer I. Substituting the electric and magnetic field expressions
(2.40) and the
permittivities expansions (2.41) in Maxwell’s equations (2.33), the following infinite sets of
second-order coupled-wave equations result. For layers with grating period Ai
d- a(
^ Zv) z-
K0
(
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2-43>
35
and for layers with grating period A2
(z44)
where z' = koz and -«> < ij, i2 <
A solution is obtained by truncating the infinite-dimensional system of coupled-wave
equations to a finite number o f harmonic orders N2. The eigenvectors and eigenvalues of the
resulting N 2xN 2 system matrix are calculated and the solution to the system (2.43) and (2.44)
can be written as
n
2
r
i
S ,MJl W = H WlA.i2.mtCtrn eXP
(* ~ D l-l )) +
eXP(M/.m (* “ A ))J
m =l
(2.45)
1
dp
Dl_l < z < D l
p
=1
where d/ is the thickness o f the /-th layer, q/,m is the positive root of the m-th eigenvalue,
W/ii i2 ,m is the corresponding eigenvector, while C;*m and CJm are unknown amplimde
constants to be determined from the boundary conditions. The tangential electric and
magnetic fields Ey,/ and Hx,/ in the grating region can then be obtained from relations (2.45)
and Maxwell’s equations (2.33). Thus, the electric and magnetic fields in the input medium,
output medium, and Z-th grating or homogeneous layer (where the gratings can have periods
Ai or A2) are written as follows:
In the input region, z < 0
Ey.i = I X K
q *2
/2
exp(A ./.i,-2*)+<?/,,-2.o exp(-yZ: z / 0 z)]exp(-yfcx. ,-2 x)
KnJi eXP(A .M .,2d
n
where 8 ;1,12,0 =
1
‘2
1
exp (~ jk xA,i2x)
_ + ^ ,' 2 .oexP(-y^z./.Q2 ).
for ii = i2 = 0 , and 0 otherwise.
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^
36
In a grating Z-th layer, D m < z <D/
a
AT
Q.m exp( kqQl.m (Z D(_, )) +
y w ‘.q.‘
, - .2 ’n
‘CXV(~ J^x.q ,i2
m= I
C/7m eXP(^0^,,m(Z - Dl ))
n
a
‘2
N
Q!m exp(-fc0^ ; m(Z - D,_, )) Z ~ ko(ll.mWl,ii.i2'n
m=l
Q7meXP (^ 0 ^ .m (Z -A ))
(2.47)
txp(-jk.-x)
For a homogeneous Z-th layer, D m < z <D/
E
= y y
\ p i;n.i2 ^ ( r K Y i . q . n
^
tt
exP(^or/.n,2 (.Z-D,))
,/2
_/
1
(z" A-i))+
V’v’ , ,
(2.48)
J ^ , 2 exP ( - ^ o ^ ,- 2 ^ - A - 1) ) - ]
hi the output region, z > D, where D is the total thickness of the structure
E y.m
=I I 7i
,,/2
n
exp(-yZ:r ///. /2 (z-D ))exp(-yfcit. ^
x )
'2
(2.49)
H x.m = ( t ^ E X ^ A . / / ; , , . / ,
q
where the summation
) ^ ,2
exV ( - j k z.nr.il.i2 (Z - D))exp(-yZ:_c . i.2 x)
(2
indices ii and i2 take values in the range [-(N -l)/2, (N -l)/2];
k zf 0 = Z:0«/ cos 6and k zffl0 = k0nm cos 6 are the z-componentsof the propagation
vector in
the input and output regions, respectively. Ru,i2 and T,i .i2 are the normalized electric field
amplitudes or the reflected and transmitted diffraction orders, respectively. The wavevector
components along x and z are determined from phase matching requirements and imposing
the Floquet condition
kx,h,i2 = konr sin 6 - ixK x - i2K 2
k z.P.h.n = ( ^ - A:x,/I,-2) 1/2» kp = konp,
where p = I or HI (input or output medium).
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(2.50)
(2.51)
37
In the region of the layers, the z component of the wavevector is
(2.52)
where n/ is the refractive index of layer I.
The boundary conditions require that the tangential components of the electric and
magnetic fields (Ey and Hx for TE polarization) be continuous at the interfaces. The following
equations result. At z = 0, the electric and magnetic field boundary conditions for a grating
first layer are
(2.53)
and for a homogeneous first layer are
<5f,./2 .0 + R n . i 2 = P \.i\ ,/2 + 2 l . q , ' 2 e X P ( - fc0 ? I .q ,- 2 ^ l )
At the interface between layers (1-1) and /, (z = D/_0 the boundary conditions, depending on
whether the layers are grating or homogeneous, are written as follows:
For a grating/grating interface
m=l
(2.55)
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38
For a homogeneous (/-l) layer and a grating I layer
,/2 exP(
n2
k o7 i - i
,/2d i - 1) +
Q i- Uh,,-2 J
r
r-'/.m + Q.m eXP(—
—
i
)J
m=l
(2.56)
? 7 - I . / t ,/2 [^1-1,11,12
eXP( —1k o Y l - l . i i ^ l - l ) —Q l-lJ i ,«2 ]
at2
LQ.m —Q,m eXP(—
-j
)J
/n=l
For a grating (/-l) layer and a homogeneous I layer
[Q-Ubi exp(
k 0 <2 l - l . m d l - l ) + C l - l , m \
~
m =l
[p/,n + 2
, 2
; , ', , 2
exP(-*o r i A j 2 d i ) ]
at2
^ L ^ / - l , f | . / 2 ,ro # y -l,m t - 'r - I . m e X P (
m=l
[
r,.n .i2 k
.,^ ,2
~ Q l.h .i2
(2.57)
'k QCl l - \ , n A l - \ ) — Q - I . m ]
eXP(—
,i2 d l )]
For a homogeneous/homogeneous interface
I/*'-i.fi ,i2 exP(
7 i- ij t,-2^/-i)
+
,/2 J —
&W2 + 2 ,,1.,2 &xP(~k oTi.n,i2 )]
Yl-UlJZ
(2.58)
l,fj,<2 eXP(~k oYl-\.i\.i2 <^ l-\ ) —Q l-\.n,i2 ] =
7 i .n .i 2 [P i.n.i2 ~ Q i.ii ,'2 e x P ( - £ o r / . , , . , 2 ^ ) ]
A t the interface between the final layer and the output region (z = D l), in the case o f a grating
final (L) layer, the boundary conditions for the electric and magnetic fields are written as
yv2
k
^
(2‘59)
*2
J
K0
exp(-*0? t. , A ) - C ; J
m=l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
and, in the case of a homogeneous final (L) layer, as
=K
,,./2
e x p ( - £ jLJi4d L) + QUiJi J
(2.60)
TkJk =
k , , , 2 exp(-*oYu ^ d L) - Q U i.z ]
The relations (2.53)-(2.60) form a system of 2N2 (L+1) equations with 2N2L unknown
field amplitudes in the layers, N2 unknown reflected wave amplitudes R u # and N 2 unknown
transmitted wave amplitudes Tu#. As in the single-grating period case, the efficiency of
calculations can be increased by eliminating the 2N2L unknown amplitudes in the layers C jm
and CJm resulting in a system of 2N2 equations for unknowns R u# and Tii,i2 . The enhanced
transmittance approach is then utilized to calculate the reflected and transmitted wave
amplitudes [8 6 ]. Once the reflected and transmitted diffraction orders have been found, the
diffraction efficiencies are determined by the following relations:
Jr
DErjx ,i2 ~ R e
D E TJiJ2 = R e
z , r , h .*2
k z.i.o
R n -‘2
(2.61)
n.i
2
l zM I.0
The accuracy of calculations depends on the number of waves retained in the coupledwave expansion. Most examples presented in this dissertation involve zero-order, single­
period gratings, with TE-polarized waves at normal incidence, and lossless structures with
thin layers. For such structures, 15 field harmonics ensure a high accuracy of diffraction
efficiency calculations [85,86]. A larger number of field harmonics were employed for TM
polarization and in computations of structures with high modulation, and/or thick layers, as
for instance in some of the examples of chapter 5.
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CHAPTER 3
OPTICAL GUIDED-MODE RESONANCE FILTERS
3.1
Introduction
A guided-mode resonance filter structure with arbitrary parameters exhibits rapid
variations of the reflected and transmitted fields with respect to the wavelength or angle of
incidence. The appropriate grating parameters have to be found to transform the response of a
grating into an almost ideal filter response. This is achieved by combining the diffraction
effects o f a zero-order grating with the mode properties o f waveguides and the
reflection/transmission characteristics provided by interference in multilayer thin films as
described in chapter 2 .
3.2
Reflection Filters
The design of ideal reflection filters based on the guided-mode resonance effect
involves specifying the filter parameters (thicknesses and the refractive indices of each layer,
grating period, substrate and cover refractive index) aiming to achieve
100
% reflectance at
the desired wavelength with abrupt, symmetrical line shapes of specified linewidth and low
reflectance around the passband wavelength region. In the thin-film literature [91-96] these
filters are also called “rejection” or “minus” filters, or “notch” filters. Several design types
consisting of single- and multilayer devices are presented in the following paragraphs.
3.2.1 Single-layer reflection Alters
A single-layer waveguide-grating is depicted in figure 3.1. The average refractive
index of the grating layer has to be higher than the cover and substrate refractive indices, for
waveguiding and grating resonances to occur. To obtain highly efficient filters, a high spatial
40
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41
frequency grating (A<A/ns, A/nc) is chosen to prevent higher diffraction orders from
propagating.
An example of a guided-mode resonance reflection filter response centered at 0.55
|im with the structure of figure 3.1 is illustrated in figure 3.2 for TE incident polarization
[48]. The grating materials have refractive indices o f 2.1 and 2.0 corresponding
approximately to zirconium dioxide (ZrC>2 ) and hafnium dioxide (Hf 0 2 ) [92], respectively,
and silica as the substrate (n = 1.52). The waveguide thickness is important in determining
filter characteristics such as sideband reflectance, lineshape, and free spectral range. In
general, the lineshape of the guided-mode resonance is asymmetrical, as illustrated by the
dashed line of figure 3.2. A symmetrical line-shape with lower sideband reflectance is
achieved by choosing the grating thickness to be near a multiple of half-wavelength [27] (i.e.,
the resonance wavelength) in the layer (d = 0.134 (im). The resonance wavelength is set by
the value of the grating period.
Figure 3.3 shows the calculated reflectance of this device over the entire visible range.
It can be seen that a single-layer waveguide-grating can act as a filter with 100% reflectance
at the desired wavelength and a symmetrical lineshape. The filter response can be attributed
to both waveguide-grating resonances and thin-film interference effects. At the resonance
wavelength, the strong coupling between the external propagating waves and the adjacent
evanescent waves, which produces a rapid variation in the reflectance, is a dominant effect
over the thin-film interference effects. Away from resonance, the sideband reflectance is
given by the equivalent homogeneous thin-film properties. The TE and TM polarized waves
couple to the leaky waveguide mode at different wavelengths, thus making the guided-mode
resonance filter a polarization sensitive device. In general, guided-mode resonance filters
possess different resonance locations, linewidths, and free spectral ranges for TE and TM
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
polarized incident waves. This is illustrated in figure 3.3 for the single-layer guided-mode
resonance filter.
Guided-mode resonances occur whenever any one or more physical parameters (i.e.,
wavelength, angle of incidence, thickness, refractive index, grating period, or fill factor) are
varied around their waveguide-coupling values. The main focus in this dissertation is on the
spectral variation of the reflected and transmitted waves. The angular dependence of the zeroorder diffracted waves determines an angular aperture of the spectral filters based on guidedmode resonances. Figure 3.4 illustrates the reflectance versus angle of incidence for the
single-layer guided-mode resonance filter with the spectral response of figure 3.3, at the
resonance wavelength (A, = 0.55 pm). The influence o f the limited angular aperture of
fabricated guided-mode resonance devices on the peak reflectance for optical and microwave
filters is discussed in sections 3.8.2 and 4.4.1, respectively.
For the filter with the response plotted in figure 3.3, the reflectance outside the filter
passband is around 5% and increases further away from resonance. With silica glass as a
substrate and with grating of a higher average refractive index to form the resonating
waveguide, the side-band reflectance of a single-layer thin film can not be made lower than
approximately 4% in practical design. However, this limitation can be overcome in multilayer
designs, as shown in the next section.
3.2.2 Double-layer reflection filters
The single-layer reflection filter characteristics can be greatly improved using
additional homogeneous thin films in antireflection design. Existing thin-film coating
techniques can be combined with waveguide-grating resonances, to reduce the reflectance in
the sidebands to arbitrarily low values and over extended wavelength ranges. As in
homogeneous thin-film design, limitations are imposed by the refractive indices of the optical
coating materials (in the visible range commonly lying between 1.3 —2.6) with small losses
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43
and the various physical, chemical, and thermal properties of the materials that "will affect the
adherence and stability of the thin-film stack.
A double-layer filter is obtained from a single-layer filter by adding a homogeneous
layer either between the grating and substrate, or on top of the grating. Using the single-layer
filter presented in the previous section as a starting point, two possible antireflectionimproved designs employing 2 layers are shown in Figures 3.5 and 3.6. M agnesium fluoride
(MgF2 ) is used as a second layer deposited on top of the ZrC>2/ Hf 0 2 grating, due to its low
refractive index (n = 1.38).
Figure 3.5 shows the reflectance of a double-layer filter with
homogeneous layers having non-quarter-wave thicknesses equal to
di
grating and
= 73
nm
(0.73xquarter-wave), d 2 = 115 nm (1.72xquarter-wave), respectively [48]. In the thin-film
literature [91-95] this is called a “V-coating,” since it provides a zero reflectance at a single
wavelength. The waveguide-grating filter response has a very low reflectance in the
sidebands close to the resonance wavelength o f 550 nm (less than 0.3 % over the range 520 —
580 nm). Towards the ends of the visible range an increase in the sideband reflectance is
observed.
A “W-coating” [93-95] that uses the same materials as the “V-coating” in figure 3.5
with the reflectance shown in figure 3.6 provides low reflectance over a w ide wavelength
range, at the expense of a higher sideband reflectance around the central wavelength [48].
The thicknesses of the layers in this case are di = 100 nm (A/4-thickness) and d 2 = 134 nm
(X/2 -thickness), respectively.
It has been shown [27] that the resonance effect exists both in structures where the
grating is the high-refractive index material providing the waveguiding (as in the filter of
figures 3.5 and 3.6), and in structures with a homogeneous waveguiding layer a nd an adjacent
grating o f lower average refractive index. The latter type of waveguide grating is illustrated
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44
here by a surface-relief grating, a simpler, and thus a more practical device using only one
material both as the high-index material and as the second layer material. As an illustration,
figure 3.7 shows the calculated response of a double-layer filter with yttrium oxide (Y 2 O 3) (n
= 1.8) [92] and air, forming a grating on a homogeneous layer of Y 2 O 3 [48].
The filter example of figure 3.7 has thicknesses and average refractive indices
satisfying antireflection requirements at the specified wavelength (i.e., optical thicknesses are
quarter-wave and refractive indices are related by the condition [92] n 2 2 /ni ,a v 2 = ns/nc where
ni,av is the average refractive index of the top layer, and n 2 is the refractive index of the
second layer). The large modulation of the grating can not be accurately modeled by the
average refractive-index approximation. As a result, although the average refractive indices
and layer thicknesses are very close to satisfying the antireflection condition (n 2 /ni,av =
1.528, ns/nc = 1.520), the reflectance curve is somewhat asymmetrical, with higher sidebands
than in the previous cases. The larger linewidth of this filter is due to the larger modulation of
the grating.
Incident wave
Figure 3.1. Single-layer guided-mode resonance filter consisting o f a grating with
rectangular refractive-index profile o f high (nn) and low (nO refractive indices
with period A, fill factor f, and thickness d.
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45
d = 110 nm
0.9
d = 134 nm
0.8
0.7
0.6
5 0.5
S
0.4
0.3
0.2
0.1
0.52
0.53
0.54
0.55
0.56
0.57
0.58
WAVELENGTH (Jim)
Figure 3.2. Guided-mode resonance in reflectance of the single-layer waveguide
grating of figure 3.1. The sideband response is asymmetrical (for a thickness d =
1 1 0 nm) and becomes symmetrical for d = 0.134 (im (half-wave at A, = 0.5 pm).
W ith the notations of figure 3.1, the parameters are A = 0.314 fim, f = 0.5, nH =
2.1 (Z r02), nL = 2.0 (H f02), nc = 1.0, and ns = 1.52.
a
0.9
TM
0.8
TE
0.7
o
5 0.6
O 0.5
U
:
Ei! 0 .4
^ 0 .3 - i
0 . 2 -i
0 . 1 -i
0 .4
0 .4 5
0.5
0.55
0.6
0 .65
0.7
WAVELENGTH (|im )
Figure 3.3. Reflectance of the single-layer guided-mode resonance filter over the
entire visible range for TE and TM polarization of the incident wave. The
parameters of the structure are the same as in figure 3.2 with d = 0.134 pm.
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46
0.9
0.8
0.7
0.6
U 0.5
0.3
0.2
0.1
■2
-1.5 -1 -0.5 0 0.5
1
ANGLE OF INCIDENCE (DEG.)
1.5
2
Figure 3.4. Reflectance versus angle of incidence at fixed wavelength (k = 0.55
pm) for the device with the spectral response of figure 3.3 for TE polarization.
0.5
0.55
0.6
0.65
WAVELENGTH (|im )
Figure 3.5. Reflectance o f a double-layer waveguide-grating filter with V-type
antireflection coating. The layer thicknesses are di = 0.073 pm (0.73xquarterwave thickness) and dz = 0.115 pm (1.72xquarter-wave thickness). The refractive
indices are ni = 1.38, nH^ = 2.1, nL,2 = 2.0, nc = 1.0, ns = 1.52.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.4
0.45
0.5
0.55
0.6
0.65
WAVELENGTH (l« n )
0.7
Figure 3.6. Reflectance of a double-layer waveguide-grating filter with W-type
antireflection coating. Layer thicknesses are di = 0.1 (im (^/4-thickness) and
=
0.134 (im (A/2-thickness). Refractive indices are ni = 1.38, nn,2 = 2.1, n ^ = 2.0,
nc = 1.0, ns= 1.52.
-
A 4
r*-*i
t R
uw
ow
Mi l l / ll l i I i 1l I|
0.4
0.45
... .
0.6
0.65
0.5
0.55
WAVELENGTH (p.m)
0.7
Figure 3.7. Reflectance of a double-layer structure with a surface-relief grating.
The parameters are A = 0.36 pm, di = 0.0936 (im (quarter-wave), d 2 = 0.0755 (im
(quarter-wave), nn.i = 1.8, nm = 1.0, n2 = 1.8, ns = 1.52, nc = 1.0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
3.2.3 Triple-layer reflection filters
Adding a third layer in an antireflection design can further improve the filter
characteristics by reducing the sideband reflection, increasing the low-reflectance range,
allowing for change in linewidth, and providing robustness and flexibility in design. A first
triple-layer example uses the V and W-types o f double-layer filters as a starting design. The
first two layers are the same as in the V and W filters and a third layer of
A I2 O 3
(n = 1.62)
[92] is added between the grating (second layer) and the substrate. The optical thicknesses of
the layers are X/4—A/2-A/4. This filter with the response shown in figure 3.8 has a reflectance
o f less than 0.25% over almost the entire visible range (450 — 720 nm), indicating a
substantial improvement from the double-layer designs [48].
An additional advantage of the 3-layer design is that the resonance location can be
shifted linearly by a change in the grating period while maintaining all other physical
parameters of the structure constant, without affecting the range or reflectance of the
sidebands. This can be seen in figure 3.8, where grating periods o f 290 nm, 310 nm, and 330
nm correspond to resonance peaks at 522 nm, 553 nm, and 584 nm, respectively.
Another type o f antireflection design for triple-layer waveguide-grating reflection
filters uses quarter-wave layer thicknesses for all three layers, and materials with refractive
indices obeying the condition [92] ni 2 n 32/n 22 = ncns- Similar to the other multilayer designs,
the waveguide-grating resonances appear at the same wavelength, regardless o f which layer
contains the grating. However the linewidth, lineshape, and sideband reflectance may vary
[29]. From a practical viewpoint a surface-relief grating (grating in first layer) may be easier
to fabricate than an embedded grating. Such an example is illustrated in figure 3.9, where the
first layer contains the surface relief grating, while the second and third layers are
homogeneous [48]. The average refractive index of the first layer (nn.i = 1.6, n y = 1.0) and
the refractive indices of the second (n2 = 2.0) and third layers (n 3 = 1.85) satisfy the above
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
antireflection equation (ni2 n32/n 22 = 1.523). The resulting reflectance curve has low sidebands
(< 0.25 % over 180 nm range) and an asymmetry due to the high grating modulation.
However, it is less asymmetrical than the double-layer surface relief grating in figure 3.7.
This shows that validity of the average refractive-index approximation employed in the
antireflection design of the triple-layer devices holds better than in the double-layer filters.
This observation leads to the conclusion that the thin-film antireflection effects in a
multilayer waveguide-grating stack are increasingly robust with respect to the magnitude of
the modulation index, as the number of layers increases. The larger linewidth o f this filter
compared to the previous triple-layer design is due to the larger modulation of the grating.
Guided-mode resonance filters with lower sideband reflectance and extensive
wavelength ranges are obtainable by the addition of thin-film layers in antireflection design.
However, the filter range can be limited by additional resonating modes appearing with
increased waveguide thickness. The increased number of design parameters in multilayer
waveguide-grating structures allows for a greater flexibility in the choice of refractive
indices, to achieve the desired filter response, and for an increased robustness to variations in
the filter parameters [29].
The linewidth of a single-layer guided-mode resonance filter has been shown to
increase with the modulation of the grating [24]. The numerical aperture of the filter also
increases with the grating modulation. However, at high modulations the linewidth may
become asymmetrical, and the antireflection thin-film equivalence breaks down. With
multilayer waveguide gratings, the thin-film antireflection design holds for higher grating
modulations [29]. Therefore, with multilayer structures, it is possible to design filters with
larger linewidths (due to larger grating modulation) and larger numerical apertures, while
maintaining a symmetrical lineshape and low sidebands extended over a large spectral region.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.4
0.45
0.5
0.55
0.6
0.65
WAVELENGTH (pm)
0.7
Figure 3.8. Triple-layer guided-mode resonance filter response. The grating
periods are: (a) A = 0.29 pm, (b) 0.31 pm and (c) 0.33 pm. The optical
thicknesses of the layers are A/4- A/2 -A/4 at 0.55 pm. Refractive indices are ni =
1.38, nn.2 = 2.1, nr ? = 2.0, n3 = 1.62, nc = 1.0, ns = 1.52.
l0.9 -j
0.8
w o.7-i
u
° - 6 “
c j 0.5 -
w
Ed 0.4
s
„.3.;
0 . 2 -i
0.1-1
0 f T 1 i i I I I I I r I I I I*| I T ^ i I I I I I | II I r0.4
0.45
0.5
0.55
0.6
0.65
0.7
WAVELENGTH (p m )
Figure 3.9. Triple-layer filter response with surface-relief grating. Thicknesses are
all quarter-wave at 0.55 pm. Other
parameters are A = 0.33pm, nH,i = 1.6, nm =
1.0, n 2 = 2.0, n 3 = 1.85, nc = 1.0, ns = 1.52.
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51
3.3
Transm ission Filters
3.3.1 Double-grating transmission filter
The starting point in the design o f a guided-mode resonance transmission filter is, as
in the case o f reflection filters, a single-layer zero-order grating with average refractive index
higher than the cover and substrate refractive indices. The response in reflectance is an
asymmetrical curve with rapid reflectance and transmittance variations from
0
to
100
% and
non-zero sidebands. To obtain good reflection filter characteristics, homogeneous thin films
are added with all thicknesses and refractive indices chosen to satisfy anti-reflection
requirements, hi contrast, in the design o f transmission filters, the transmission peak that
corresponds to the resonance minimum in the reflection curve is used [34], and thin-film
layers are added in a high-reflectance design, to reduce the transmittance away from the
resonance wavelength.
As shown previously, at wavelengths away from resonance the waveguide grating
behaves effectively like a thin film with refractive index equal to the average refractive index
of the high and low refractive-index materials of the grating. Therefore, thin-film design
techniques can be applied to a stack o f homogeneous or grating layers, to reduce the
transmittance sidebands while maintaining the high-transmittance at a desired wavelength
due to the guided-mode resonance. The most common high-reflectance design consists of
alternating thin films of high and low-refractive index materials with quarter-wave
thicknesses.
Materials that allow a large difference between the refractive indices of successive
layers are used (nhigh = 2.35, niow = 1.38) in order to obtain a high reflectance at non-resonant
wavelengths and a large relative width of the high-reflectance region. The sideband
transmittance can be reduced to arbitrarily low levels by increasing the number of high/low
refractive-index quarter-wave layers. The grating replaces a homogeneous layer to generate
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52
high transmittance resonance peaks. The refractive indices of the grating materials must yield
an average refractive index that preserves the high or low-refractive index nature of the
original homogeneous layer, in order to maintain the high-reflectance properties of the thinfilm stack.
Figure 3.10 illustrates the transmittance o f a
6
-layer structure with the first layer
containing the grating with refractive indices nn = 2.5 and nL = 2.2 and five high/low quarterwave layers. With additional pairs, the sidebands can be reduced to arbitrarily-low levels.
However, the increase in the number of layers is accompanied by an increase in the number
o f modes that can be supported by the increasingly thick waveguide. This has a negative
effect on the filter response, since it limits the filter spectral range by allowing new
resonances to appear in the high-reflectance wavelength region [34]. It is found in this case
that the resonance transmission peaks are attenuated, as more layers are added to the thin-film
stack and the reflectance is increased.
The resonance efficiency can be enhanced, by substituting one or more of the
homogeneous layers with gratings. Such an example is shown in figure 3.11, where the
resonance transmittance of a 9-layer high/low refractive-index structure has been increased,
by replacing the high-index layers next to the cover and substrate with gratings. The TE 2
mode gives a transmission resonance close to
100
% and can be used as a transmission filter
within a range limited by neighboring resonance peaks.
By adding two more layers a grating-enclosed transmission filter is obtained, with
high transmittance at the resonant wavelength and low sidebands (less than 1.5%) over a
range of 60 nm (as illustrated in figure 3.12) [48]. Selecting the resonance corresponding to
the TE 2 mode as central wavelength, one obtains a highly efficient (almost
line transmission filter over a range of approximately 60 nm (figure 3.14).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
%) narrow-
53
0 " T — '— i— i” i — |— r — i— i
0.45
0.5
i 1 1
i - i — i-"
|
i
i
0.55
0.6
WAVELENGTH (M-m)
i— i—
0.65
Figure 3.10. Transmittance of a 6 -layer structure with the grating in the first layer
and five high-low homogeneous X/4-layers. The grating parameters are A = 0.33
Jim, d = 0.0584 |i.m, nn = 2.5, and nL = 2.2. The thicknesses and refractive indices
of the homogeneous layers are d = 0.0585 pm, n = 2.35 (odd layers), d = 0.1 |im,
n = 1.38 (even layers).
3.3.2 Single-grating transm ission filters
A transmission filter with similar characteristics as the grating-enclosed design but
with only one grating instead of two would be desirable to simplify device fabrication. A
single grating placed in the first layer was seen (figure 3.10) to produce only low-efficiency
resonances. Similar results are obtained if the grating is placed in other layers. However,
almost
100
% transmittance peaks can be obtained if the grating is located in the central layer
of the high-reflectance stack (figure 3.13) [47]. In this situation, the grating is surrounded by
high-reflectance layers enhancing the field in the grating layer, and thus, the leakage of the
waveguide grating. The situation is comparable to the Fabry-Perot filter with dielectric
mirrors on both sides of the etalon layer. Compared to the grating-enclosed design with the
same number of layers (figure 3.11), the single-grating design has a narrower bandwidth, due
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
to the reduced leakage obtained with only one grating placed in the center of the center layer
of the structure.
The transmission filters presented above make use of four different materials, two for
the grating, and two for the high/low homogeneous layers. A. considerable simplification is
brought about by the use of the same two materials (for example ZnS and MgF) for both
grating and high/low quarter-wave layers. Such an example is the single-grating 9-layer
structure with the calculated response illustrated in figure 3.15 for two different grating fill
factors (f = 0.5 and f = 0.75) [48]. The grating fill factor is defined here as the ratio between
the width of the high-refractive index region of the grating and the grating period. The grating
has refractive indices corresponding to ZnS and MgF 2 and is bounded on both sides by four
layers of ZnS (n = 2.35) and MgF 2 (n = 1.38) [92]. The high-reflectance property of the thinfilm device is not greatly affected by the relatively lower average refractive index o f the
grating layer. The effect of a fill factor different than 0.5 is a decrease in linewidth and a shift
in the resonance location of the transmittance peak.
Experimental demonstration of the single-grating transmission filter (discussed in
chapter 4) has been accomplished in the microwave spectral range [78]. As shown in chapter
4, larger refractive-index values available at microwave frequencies enable transmission
filters with low sideband response built with fewer (5) layers than in the visible spectral
range. Further decrease in number of layers is obtained with non-quarter-wave devices
obtained by genetic algorithm optimization. Such examples are presented in chapter 5 and
compared to the quarter-wave designs illustrated in this section.
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55
0.9 -i
te
0.8
2
g 0.7-i
«j
0 .6
TE
-i
H
Z 0.4 H
0.30.2 0 . 1-
0.45
0.5
0.55
0.6
WAVELENGTH (H-Hl)
0.65
Figure 3.11. Spectral response of a 9-layer transmission filter with gratings in the
top and the bottom layers. The gratings parameters are A = 0.33 (im, nH,i = n H.9 =
2.5, nm = m , 9 = 2.2, di = dg = 0.0584 pm . Homogeneous layers have n = 2.35
(odd layers) and n = 1.38 (even layers) and quarter-wave thicknesses d = 0.0585
pm (odd layers) d = 0 . 1 pm (even layers).
l0.9 H
j
|
i
0.52
i
i
i
{ i
0.53
i
t
i
|
i
i
i
i
|
i
i
i
i
j
i
i
i i |
r *! 1 t
0.54 0.55 0.56 0.57
WAVELENGTH (pin)
f
0.58
Figure 3.12. Double-grating transmission filter (caption) and its spectral response.
The thicknesses and refractive indices are the same as in the structure of figure
3.11 with two additional homogeneous high-low A/4-layers. The grating period is
A = 0.345 pm.
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56
0.9
0.6
t/3
0.4
0.3
0.2
0.1
0.45
0.5
0.55
WAVELENGTH
0.6
0.65
(}Xra)
Figure 3.13. A 9-layer structure with one grating in the center layer (top) and its
spectral response (bottom). All layers are quarter-wave thick at 0.55 pm. The
grating has the parameters A = 0.33 |im, nn.5 = 2.5, n^s = 2.2, d 5 = 0.0584 |im.
The homogeneous layers have the parameters n = 2.35 and d = 0.0585 pm (odd
layers), and n = 1.38 and d = 0.0996 pm (even layers); nc = 1.0, ns = 1.52.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.53
0.54
0.55
0.56
WAVELENGTH (Jim)
0.57
Figure 3.14. Single-grating transmittance filter of figure 3.13 over a narrower
spectral range. The layer structure is illustrated in the inset.
8
0.9
f = 0.5
0.8
f = 0.75
0-7
<! 0.6
E ->
go.5
Z 0.4
0.1
0.53
0.54
0.55
0.56
WAVELENGTH (pm)
0.57
Figure 3.15. Transmittance of a 9-layer single-grating resonance filter that uses 2
materials only. The parameters are: A = 0.34 pm, nn = n^d = 2.35, nL = neVen = 1.38,
ns = 1.52, nc = 1.0. The grating fill factor is f = 0.5 (solid line) and f = 0.75 (dashed
line). All optical layer thicknesses are quarter-wave at 0.55 pm.
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58
3.4
M ulti-Line Filters
3.4.1 Techniques for obtaining m ulti-line filters
The reflection and transmission filter examples presented in the two preceding
sections possess spectral characteristics exhibiting a single high reflectance (or transmittance)
peak, with low side reflectance (or transmittance) extended over a certain wavelength range.
Waveguide-grating devices can also generate a different type o f filter response with two or
more high-efficiency peaks and low response between and outside the spectral region
containing the peaks. This can be accomplished either by breaking the coupling symmetry o f
the external waves to the waveguide mode, or by selecting waveguide gratings that can
support more than one guided mode. In the latter option, adjusting the waveguide-grating
structural parameters to provide peaks at the desired wavelengths may be a more involved
procedure. If more than two peaks are desired, the multi-mode waveguide may be the option
of choice, and the design procedure can be facilitated by search and optimization routines,
such as the genetic algorithm presented in chapter 5. An example o f multi-line filters using
multi-order waveguides is presented in figure 3.13, where a transmission filter exhibits 4
peaks in the visible range.
A straightforward manner of breaking the coupling symmetry, thus obtaining a
double-peak filter, is to rotate a single-peak filter in the incidence plane, away from normal
incidence. At non-normal incidence, the coupling of the +1 and -1 diffraction orders to the
waveguide mode occurs at different wavelengths, thus giving rise to two peaks. The distance
between the two peaks can be adjusted by changing the incident angle. Such an example is
illustrated in figure 3.16 indicating different spectral locations o f the double-peak filter for
different angles of incidence. As in the single-line filters, the center wavelengths can be
shifted by appropriate selection of the grating period. The linewidths of the two peaks are
smaller than the linewidth of the filter at normal incidence. At small angles of incidence the
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59
two peaks have different linewidths. Further away from normal incidence, the linewidths
become approximately equal [74]. In figure 3.16, all four peaks have linewidths (full width at
half maximum (FWHM)), AA. -0 .2 nm.
Double peaks are also obtained if the coupling symmetry is broken through phaseshifted gratings or slanted gratings [32,54]. A typical response of a phase-shifted waveguide
grating obtained by translating half of the grating along the grating vector direction is
illustrated in figure 3.17. The two peaks are closely spaced and their linewidths differ
significantly. Control over the locations and widths o f the peaks is limited. Phase-shifting
also provides a mechanism for linewidth and lineshape control for single-line filters
[32,43,79], as shown in section 3.5.
3.4.2 Double-period waveguide gratings
The diffraction by multiple gratings of different periods and orientations in the same
layer has been studied previously for some cases of interest in holographic applications (large
angular separations between gratings or small Bragg angles). Tu et al. [97,98] analyzed
superposed gratings with no restrictive assumptions for the grating parameters, using a
multiple-scattering formulation that provides accurate quantitative results, as well as physical
insight into the diffraction process. More recently, Lemarchand et al. [67] demonstrated
theoretically that the angular aperture of guide-mode resonance filters can be significantly
increased with doubly-periodic structures without a simultaneous increase in spectral
linewidth. This is achieved with two gratings in two separate layers, where the top grating
period is two times larger than that of the lower grating.
In this work there is interest in the fundamental physical properties, as well as in new
design capabilities that emerge by use of structures with gratings of different periods in
successive layers. For instance, a device with two gratings o f slightly different periods in the
three-layer geometry shown in the inset of figure 3.18 can act as a two-line filter with the
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60
center frequencies determined approximately by the resonances of the individual single-layer
waveguide gratings [32,40]. Figure 3.18 shows the filter response of a three-layer device with
a difference between the two grating periods of
20
nm, exhibiting approximately
100
%
reflection peaks 31.6 nm apart. The reflection spectra of the double-line filters were
computed following the coupled-wave approach developed in section 2.4.2 [40]. The two
peaks can be centered independently on the desired center wavelengths by appropriate choice
of the two grating periods.
Generalizing these observations to more than two gratings of different periods, one
can anticipate that multi-line filters with high efficiency and good control over the bandwidth
and sidebands can be realized by designing individual single-line filters and stacking these
with buffer layers in between [40]. For small grating period difference, resulting in partly
overlapping peaks, this approach can also provide yet another linewidth broadening
mechanism. However, this may also lead to a distorted lineshape of the filter [40].
The buffer layer plays an important role in weakening the coupling between the
evanescent waves of the two gratings. At normal incidence, selecting the grating periods to be
less than the wavelength in the cover and substrate (A < A/ns, A < A/nc) will ensure that only
the zero diffraction order propagates. However, an evanescent wave from the first grating
diffracting off the second grating can produce propagating higher-order diffracted waves. A
diffracted wave, denoted by indices (ii, ii), is propagating or evanescent depending on
whether kAPji,i2 , given by equation (2.51), is real or imaginary, respectively. Propagating
waves, for normal incidence and equal cover and substrate refractive indices satisfy the
following relation:
JVi+A^l
X
A,A 2
where Ai and A2 are the grating periods of the two gratings.
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(3.!)
61
Figure 3.19 illustrates the zero-order reflectance and transmittance of the structure of figure
3.18 with no buffer layer between the two gratings. At non-resonance wavelengths the
incident wave is mostly reflected or transmitted in the zero diffraction order. At resonance
wavelengths an increased amount of the incident wave is diffracted into higher orders, as
shown in figure 3.20. A total of 121 waves are retained in the calculations o f figures 3.19 and
3.20 with 11 satisfying the propagating-wave condition (3.1) in the illustrated wavelength
range. Thus, non-zero diffraction orders can not be avoided in such a superimposed double­
grating structure. However, the buffer layer separating the gratings can reduce the power
diffracted in the higher orders (i.e., decouple the gratings) to obtain two almost independent
zero-order gratings. The resulting three-layer structure possesses two grating resonances with
peak reflectance close to 100% [40,79].
0.9
3 deg.
0.8
5 deg.
0.7
0.6
U 0.5
w
fa 0.4
0.3
0.1
0.4
0.45
0.5
0.55
0.6
0.65
WAVELENGTH (pm )
0.7
Figure 3.16. Single-layer guided-mode resonance filter at 3° (solid line) and 5°
(dashed line) angle of incidence. The parameters of the device are A = 0.314 pm, f
= 0.5, d = 0.134 pm, nH = 2.1, nL = 2.0, nc = 1.0, and ns = 1.52. The response of
this waveguide grating at normal incidence is shown in figure 3.3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.544 0.546 0.548 0.55 0.552 0.554 0.556
WAVELENGTH (Jim)
Figure 3.17. Reflectance of a double-layer waveguide grating at normal incidence
with two identical gratings shifted by 0.05A. The structure is obtained from the
single-layer waveguide grating with the response of figure 3.3 by laterally shifting
half of the grating along the grating vector.
0.49
0.51
0.53
0.55
0.57
WAVELENGTH (pm )
0.59
Figure 3.18. Double-line reflection filter response at normal incidence. The 2grating/3-layer structure has the following parameters Ai = 0.30 pm, A 2 = 0.32
pm , di = d3 = 0.137 pm, d2 = 0.723 pm (2A,), nH,i = n H,3 = 2.1, nm = n U 3 = 1.9, n2
= nc = n s= 1-52.
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63
£ 0.7
0.6
0.5
< 0 .4 '
W
:
? 0-3 :
S 0 . 2 ■;
W
:
5? 0 . 1 -
0.49
0.51
0.53
0.55
0.57
WAVELENGTH (jJUl)
0.59
Figure 3.19. The same structure as in figure 3.18 is shown here for zero thickness
o f the buffer layer (cfe = 0). This corresponds to a 2-layer/2-grating waveguide
grating.
0.15
0.1
0.05
0-=H
0.49
0.51
0.53
0.55
0.57
WAVELENGTH (jim )
0.59
Figure 3.20. Sum of the diffraction efficiencies in the non-zero orders in the case
o f the 2-layer/2-grating structure of figure 3.19.
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64
3.5
Control o f Filter Characteristics
It has been shown in sections 3.2 and 3.3 how basic features of a filter (such as
sideband reflectance or transmittance and filter wavelength range) can be controlled by
appropriate choice of the thicknesses and refractive indices of the grating, and the
homogeneous layers of a multilayer structure.
The grating period is the main design parameter used to determine the center
wavelength of the guided-mode resonance filter. The sensitivity o f the peak spectral location
to the angle of incidence can also be used effectively to tune the central wavelength o f the
filter to the desired value. Angle tuning may not be possible in applications that require a
specific angle of incidence on the filter. Whenever it is acceptable, angle tuning can be used
to compensate for the resonance wavelength shift due to errors in manufacturing. Other
important features in the design of reflection and transmission bandpass filters are the
linewidth and the lineshape.
3.5.1 Linewidth control mechanisms
It has been demonstrated that several linewidth control mechanisms are available in
the design of guided-mode resonance filters [24]. Figure 3.21 illustrates the linewidth
dependence of a single-layer reflection filter on the modulation amplitude.
The average refractive index o f the grating is constant while the refractive index
modulation (nn - n j varies from 0.018 to 0.63 corresponding to exponentially increasing
linewidths between 0.017 nm and 19 nm [40]. High grating modulations imply stronger
coupling between the propagating and evanescent orders and “leakier” waveguide-grating
producing broader resonance peaks. For lower modulation, the guided modes induced in the
grating couple out less efficiently and the filter has a narrower linewidth around the central
resonance wavelength.
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65
Another important factor determining the linewidth is the mode confinement of the
waveguide-grating stack [24]. In the case of a single-layer waveguide grating, this reduces to
the difference between the average refractive index of the grating and the refractive indices o f
the cover and substrate.
For a single-layer waveguide-grating supporting a single mode, the linewidth has been
shown to increase with the confinement of the mode determined by the refractive-index
difference for a constant modulation index Ae/Sav and relatively small refractive-index
differences [24]. The linewidth vanishes for average grating index approaching the substrate
index, since the resonance effect is due to the waveguiding properties of the grating layer.
Calculated values of the linewidth as a function of the average refractive index of the
grating, nav, are plotted in figure 3.22, for a single-layer symmetric (nc = ns) reflection filter
[40]. The grating period, thickness, and modulation (nn - nO of the grating are fixed. The
linewidth dependence on the grating refractive index is shown in figure 3.22 to be non­
monotonic. At low values of nav, the linewidth increases with nav due to greater overlap of the
waveguide-mode field with the grating, which results in stronger output-coupling loss. An
optimum overlap is reached at some value of nav. Further increase in nav produces stronger
mode confinement and smaller output-coupling losses.
The mode confinement in a waveguide is also determined by the waveguide
thickness, with increasing thickness providing stronger mode confinement. Non-monotonic
variations of the linewidth with waveguide-grating thickness have been reported, and
interpreted on the basis of the overlap integral between the guided-mode field and the
radiation mode field, which determines the coupling loss in the coupled-mode analysis [46].
Only one mode can propagate in the equivalent waveguide for the range o f grating
refractive indices given in figure 3.22. In waveguides supporting more than one mode,
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66
different waveguide confinement of the individual modes leads to different linewidths o f the
corresponding reflection peaks.
The grating fill factor may also be used to adjust the linewidth to the desired value
[40,68]. Figure 3.23 shows the linewidth dependence o f a reflection single-layer filter on the
grating fill factor [40]. As expected, the linewidth decreases to zero as the grating vanishes
(fill factor approaches zero or one). The maxim um linewidth corresponds to the value o f the
fill factor which provides the highest leakage of the waveguide grating, as discussed in
section 2.2.2. An increase in linewidth can also be realized by designing multi-resonance
structures (obtained by mechanisms discussed in section 3.4) to merge two or more resonance
peaks into a broadened filter line [32,40]. An additional linewidth control design method, to
be discussed in the next section, is provided by adjacent phase-shifted gratings
lOOi
e£
l(h
X
H
Q
£
I
-> 0.1o.oi0.01
0.1
1
MODULATION
Figure 3.21. Linewidth dependence on the grating modulation (the difference
between the high and low refractive indices of the grating) in the case of a single­
layer structure. The average refractive index of the grating is constant and equal to
1.81. The cover and substrate refractive indices are nc = ns = 1.52; the thickness is
d = 156 nm.
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67
0. 02 0
M
1.5
M
I I I I . |
1.6
L 1 I L I 'I I . [ | 1 I 1 [ | 1 1 1' 1 - | I 1 1 . | I 1 I I
1.7
1.8
1.9
2
2.1
2.2
2.3
AVERAGE REFRACTIVE INDEX
Figure 3.22. Linewidth dependence on the average refractive index of the grating
in a single-layer reflection filter. The modulation is nn - nL = 0.1; other
parameters are d = 150 nm, A = 350 nm, no = ns = 1.52.
0 .5
0.45
0.4
-£ ?0 3 5
| °J
Q 0.25
S
z
J
0.2
0.15
0.1
0.0 5
0
0
0.1
03
03
0 .4
05
0 .6
0 .7
0 .8
0 .9
1
FILL FACTOR
Figure 3.23. Linewidth dependence on the grating fill factor of a single-layer
reflection filter. The refractive indices are nn = 1.9, nL = 1.8, nc = 1.0, ns = 1.52.
The thickness is d = 153 nm and the period A = 350 nm.
3.5.2 Phase-shifted gratings for filter line control
Corrugated gratings fabricated on both boundaries of a waveguide and phase-shifted
with respect to each other have been experimentally demonstrated to exhibit high (-78%)
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68
input coupling efficiency [99]. Conversely, slab waveguides with phase-shifted corrugated
surfaces have also been shown theoretically and experimentally to act as highly-efficient
unidirectional grating output couplers of the guided mode by suppressing the wave exiting
into the substrate, inherent in any grating output coupler [10,16,19].
The basic principle is that the waveguide mode that reaches the dual-corrugation area
couples out on each corrugation independently in both directions, where the radiated waves
interfere. The phase-shifted gratings induce a phase shift between the interfering waves that
can lead to a constructive interference in one direction and a destructive interference in the
other. A lateral shift between the gratings, Az, induces a phase change of -KAz (where K is
the grating vector magnitude) in the first-order diffracted wave, but no phase change in the
zero-order diffracted wave [100]. Analytical expressions for the radiative losses into the
cover and substrate have been derived in the approximation of sinusoidal weakly-corrugated
gratings, and optimum values of the phase shift and relative modulations of the two gratings
have been determined for 100% efficiency output coupling into the cover [99,100]. The
waveguide properties of the phase-shifted corrugated gratings have been used to study the
guided-mode resonances occurring under free-space diffraction [16]. Numerical calculations
of the reflectance spectrum for various values of the phase-shift were performed and
interpreted in terms of guided-mode interference and waveguide radiative losses [16],
hi the present work, phase-shifted waveguide gratings with cross sections illustrated
in figure 3.24 are studied in view of their applications for filter design. The two adjacent
phase-shifted gratings have been chosen to be identical and with quarter-wave thickness, to
maintain antireflection condition required for symmetrical, low sideband reflectance. An
arbitrary phase-shift between the gratings produces two resonance peaks in the reflection
spectrum (figure 3.17) [40]. The separation between the two peaks and their linewidths
changes with the phase shift. For phase shifts less than tc/2, the maximum reflectance for both
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69
peaks reaches 100%. It was found that the two peaks merge at phase-shift values that depend
on the thicknesses o f the two gratings. In the case o f identical gratings, with thicknesses equal
to quarter-wavelength at the resonance wavelength, the two peaks o f a guided-mode
resonance reflection filter merge at phase shifts of tt/2 and k [40]. This is illustrated in figures
3.25 and 3.28. For optical reflection filter design with quarter-wave thick layers for low
sideband reflectance, the most interesting case is when the phase shift between the gratings is
equal to K. This phase-shift value induces a broadening of the reflection spectrum of the
unshifted grating filter. Waveguide theory for a double-grating output coupler demonstrates
that for two identical gratings with quarter-wave thicknesses, m axim um radiative losses
occur when the phase shift is equal to 0 and k [10,19,100]. The increased leakage o f these
structures at Tt-phase shift accounts for the linewidth broadening of the reflection filter
spectral response. Figure 3.25 superimposes the spectral response of the filters for 0 and 7t
phase shifts, showing a significant increase in linewidth [32,40].
The linewidth broadening has been found to increase linearly with the refractive index
difference between the grating and the substrate [32,40]. Figure 3.26 shows calculated points
for the relative linewidth of the k phase-shifted grating for different average refractive index
of the grating, while keeping the grating modulation constant. A three fold increase in
linewidth at the wavelength of 550 nm is obtained using a waveguide-grating filter with Kshifted gratings, having an average refractive index of 2.4 on a silica substrate (with
refractive index, n = 1.52) [40]. The relative increase in bandwidth has been found to be
nearly independent o f the grating modulation in the range
0 .0 1
< (nn - ul) <
1
for a fixed
average refractive index of the grating [40]. The phase-shift corresponding to a single
reflectance peak with broadened linewidth depends on the thickness o f the waveguide
grating. A waveguide grating with a thickness that is not a multiple of a quarter-wavelength,
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70
and therefore, exhibiting peak broadening at a phase shift that differs from
k
is presented at
microwave frequencies in chapter 4.
The same phase-shifts that induce an increase in the linewidth o f reflection filters can
cause a decrease in the linewidth o f guided-mode resonance transmission filters [43]. The
phase-shift responsible for a constructive interference in reflection produces a destructive
interference in transmission. This leads to reduced output coupling in the substrate, which
explains the narrower linewidth. Figure 3.27 shows the linewidth narrowing effect of a k phase shifted grating in the center layer o f a single-grating transmission filter [43]. The
linewidth of the guided-mode resonance transmission filter is reduced from
0 .1 1
nm to 0.018
nm, as half of the grating is translated by half a period along the grating vector.
A 71/2 phase shift between the gratings can produce an improved lineshape of a
reflection filter response [32,40]. This effect is more clearly visible in the case of a symmetric
filter with higher modulation, as illustrated by figure 3.28. The 7t/ 2 -shifted grating has lower
sidebands and a steeper variation around the resonant wavelength bringing the typically
observed triangular (approximately Lorentizan) line-shape o f the resonant filter closer to the
rectangular line-shape of an ideal filter.
Figure 3.24. Guided-mode resonance filter with phase-shifted gratings.
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71
0.9
0.%-.
0.7
~
0.6
U 0.5
sE 0 .4 9
,:
* 0 .3 0.2
O.lH
0.544 0.546 0.548 0.55 0.552 0.554 0.556
WAVELENGTH (M-m)
Figure 3.25. Comparison between the filter response of the 7t-phase-shifted versus
non-shifted waveguide grating. The non-shifted structure has the parameters of
figure 3.3. The phase-shifted device is realized by lateral displacement of half of
the grating as shown in figure 3.24.
2.5
0.5
0
0.2 0.4 0.6 0.8
1
1.2
1.4
Figure 3.26. The relative linewidth (AA,(7t)/AA.(0)) dependence on the refractive
index difference between the grating and the substrate. The modulation is constant
and equal to 0.1. The other parameters are: A = 350 nm, di = d2 = 72 nm, nc =
1.0, ns = 1.52.
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72
0.9
0.8
0.1
0.498
0.499
0.5
0.501
WAVELENGTH (^im)
0.502
Figure 3.27. Linewidth narrowing of a guided-mode resonance transmittance filter with
7t-phase shifted grating (dotted line) versus a non-shifted grating (solid line). The layer
structure is illustrated in the inset, hi the non-shifted waveguide grating all layers are
quarter-wave thick at 0.5 pm. The grating period is A = 0.3 pm and the refractive
indices are nH = 2.5, nL = 2.2, n ^ j = 2.35, Ueven = 1-38, ns = 1.52, and nc = 1.0.
1
0.9 r
0.8 -j
0 .7 0.6
cj 0.5 u3 0.4r
5
:
06 0 .3 -
0.2 -j
0.1 r
00 46
■' i 1 ' 1 1 i 1 '
0.5
f 1 r<
i ' 1 i ' 1 11 '■
0.54 0.58
0.62
WAVELENGTH (p.m)
0.66
Figure 3.28. Double-layer waveguide grating with two identical gratings phaseshifted by n il exhibiting improved reflection filter lineshape. The parameters are
A = 350 nm, di = d2 = 78 nm, nn = 2.0, hl = 1.6, nc = ns = 1.52.
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73
3.6 Comparison with Homogeneous
Thin-Film Filters
It is o f practical interest to compare the recently developed guided-mode resonance
filters to classical filters, to determine their most important advantages, and hence the
applications in which resonance filters might enhance the state of the art. Filters with
linewidths less than
10
% of the central wavelength are referred to as narrow-linewidth filters
[91-93,96] and it is in this category that guided-mode resonance filters can be generally
included. The linewidth can be broader in high modulation gratings as in the case of surfacerelief devices, but often at the expense of a distorted lineshape and higher sidebands.
The following parameters are used to describe transmission (reflection) narrow
linewidth filter properties [96]: Tmax (Rmax) - maximum transmittance (reflectance) in a high
transmittance (reflectance) zone, Xmax - wavelength where T ^ (Rmax) is achieved, Xt and Xr the wavelengths corresponding to the transmittance (reflectance) level of
10
% at the extreme
left and right ends of the low transmittance (reflectance) region, Xr - Xi — the filter range,
Xi/Xmax, Xt/Xmax and (Xr - Xi) Xmax - the relative values of the filter wavelength range, AX0.5 full width of the high transmittance (reflectance) band at the 0.5Tmax (0.5Rmax) level, AX0.1 full width of the high transmittance (reflectance) band at the O.lTmax (O.IRmax) level,
AXo.5/Xmax and AXo.i/Xmax - the relative linewidths, r\ = AX0 .1/AX0 .5 -
a coefficient
characterizing the abruptness of the transition from passband to stopband.
3.6.1 Reflection filters
Reflection filters, also known as notch or minus filters, are among the most difficult
filters to design using only homogeneous thin films [91]. To achieve specified filter
parameters, some designs use periodic layered media, while others use varying thicknesses of
the layers determined through optimization techniques; some designs use only two materials,
while others use several to simulate variable refractive index media [93,94,101].
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74
A classical notch filter design that uses only two materials consists o f a stack o f
alternating high and low refractive index materials. The difference between the refractive
indices o f the two materials determines the linewidth of the filter, while the number of highlow periods determines the peak reflectance. Small refractive-index difference yields a
narrow linewidth filter. A large number of periods yields a high-efficiency filter. A narrowline notch filter with high-low quarter-wave layers using materials with refractive indices nodd
= 1.52 (silica) and neven = 1.47 (BaF2 ) [94] results in a filter response with a peak reflectance
o f approximately 96% for 150 layers and 77% for 100 layers [40,48]. The reflection filter
with 150 layers has a linewidth o f 16 nm. A narrower linewidth than that o f a quarter-wave
coating is possible using a three-quarter-wave stack. Such a design has the response shown in
figure 3.29, using the same materials as in the quarter-wave thickness design [48]. The
reflection filter of figure 3.29 has a linewidth at 0.5Rmax, AAo.5 = 5.3 nm (AXo^A™* = 0.96%),
a linewidth at 0.1 Rmax, AAo.i = 18.7 nm (AXo.i/Xmax = 3.40%), and a line-shape factor r\ =
AXo.i/AAo.5 = 3.53 [48]. The sideband reflectance of this filter exhibits ripples with an average
value of 4%.
Guided-mode resonance filters are particularly suitable for filter applications where a
narrow linewidth is required. Using two materials with the same refractive index difference
as in figure
3 .2 9 ,
linewidths of less than
0 .2
nm can be achieved with only one layer
[4 8 ] .
The
single-layer resonance filter has 100% peak reflectance, similar sideband reflectance, but no
ripples. It has been shown in section
3 .2
that using two- and three-layer waveguide-grating
structures, reflection filters can be designed with very low sidebands over the whole visible
range. The three-layer design of figure
(A X o.s/X m ax = 0 .2 4 % ),
shape factor T| =
3 .3 0
3 .9
a linewidth at 0 . 1 R m ax,
A X 0.1/A X 0.5 = 3 . 1 1 [4 8 ] .
has a linewidth at
A A o.t = 8 . 4
nm
0 .5 Rmax» A X 0.5 =
(A X o.i/X m ax = 0 .7 5 % ) ,
The v-coating design of figure
3 .5 ,
2 .7
nm
and a line-
shown in figure
for a wavelength region around the central peak, uses only two layers to achieve a
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75
sideband reflectance below -0.3% within a 60 nm range. The filter parameters are: linewidths
AA0 .5 = 0.44 nm (AAoVAmax = 0.04%), AAo.i = 1 .4 nm (AAo.i/^max = 0.12%), and line-shape
factor rj = 3.11 [48]. This combination of very narrow linewidth, 100% peak reflectance, and
very low sidebands is particularly difficult to achieve with a small number of homogeneous
layers only. The line-shape factor is approximately the same for all resonance filters with
rectangular grating profile, Tj -3.1. This value is characteristic o f normalized Lorentzian
functions which have a ratio between the widths at 10% and 50% equal to 3. An ideal filter
should possess
T|
—1- One solution to improve this filter parameter is to use phase-shifted
guided-mode resonant gratings [32,40]. With a spatial phase-shift of tc/2 between two
gratings, a reduction of i) to 1.75 was achieved in the example of figure 3.28.
Compared to homogeneous-layer reflection filters, guided-mode resonance filters
achieve similar or better filtering characteristics in narrow-linewidth applications. The small
number of layers (typically 2 or 3) required to obtain narrow-linewidth, high-efficiency filters
is far smaller than for the classical filters, which is an important advantage in practical
fabrication.
3.6.2 Transmission filters
Transmission filters can be very roughly divided into broad-band filters and narrow­
band filters. Since the guided-mode resonance filters are inherently narrow-band filters this
section will focus on comparing them to existing narrow-line filters that use only
homogeneous thin films.
The standard thin-film narrow-band filter is a Fabry-Perot etalon, consisting of two
quarter-wave mirror stacks with a half-wave layer placed between them [91-96]. The
linewidth of the Fabry-Perot filters is narrower for increased refractive-index difference
between the quarter-wave layers and for a larger number of layers. For comparison with
guided-mode resonance transmission filters, the response of an 11-layer Fabry-Perot filter
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76
using ZnS (n = 2.35) and Mgfo (n = 1.38) as high- and low-refractive index materials is
shown in figure 3.31 [48]. The thin-film formula for this design is (HL) 3 (LH) 3 where H and L
denote quarter-wave thicknesses of the high- and low-refractive index material.
In comparison, an 11-layer resonance waveguide-grating filter (double-grating design)
with the response shown in figure 3.32 uses materials with the same refractive indices in the
homogeneous layers (layers 2 - 10) as the Fabry-Perot filter. Comparing the two types of
transmission filters it is found that the resonance filter has a lower sideband transmission and
a slightly higher peak transmission. The lineshape factor is approximately i\ - 3 for both
filters. The filter range however is larger in the Fabry-Perot filter. The smaller range o f the
guided-mode resonance filters is due to other resonating waveguide modes. The linewidth of
the resonance filter (AX0 .5 =
0 .8
nm) is more than
10
times narrower for the same number of
layers. To achieve a comparable linewidth, a Fabry-Perot filter needs 21 layers [48].
The single-grating resonance filter can provide narrower linewidths with a smaller
number of layers than the double-grating design using the same materials for the
homogeneous layers, grating and substrate [47]. Figure 3.32 also shows the 9-layer singlegrating transmission filter with a linewidth (AX0.5 = 0.12 nm) approximately 7 times smaller
than the 11-layer structure [43,48]. The Fabry-Perot filter has much broader response (AX0 .5 =
18.8 nm) and higher sidebands than guided-mode resonance filters with equal number of
layers (9) that use the same materials in the homogeneous layers. To achieve the same
linewidth, 27 layers are required in a Fabry-Perot filter. The drawback of the waveguidegrating filter is again a smaller filter range. Apparently the Fabry-Perot design allows for an
arbitrary linewidth, provided that enough high-low layers are added on both sides of the half­
wave central layer. In practice however, filters with very small linewidth have to satisfy
severe layer thickness uniformity requirements [92].
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77
Another method to reduce the linewidth is to use a higher-order for the Fabry-Perot
etalon (i.e., thicker spacer layers). In practice, spacer layers higher than 4th order exhibit
roughness that broadens the pass band and reduces the peak transmittance so much that any
advantage o f the higher order is lost [92]. The guided-mode resonance filter can achieve
small linewidths with fewer layers relaxing the uniformity requirements imposed on
homogeneous thin-film designs.
Genetic algorithm optimization can be utilized to obtain similar narrow bandpass
characteristics with only 1-3 layers in structures with non-quarter-wave layer thicknesses. The
linewidth can be made narrower by forming the grating with materials having a smaller
refractive index difference in the grating layer. Another advantage of the guided-mode
resonance filter is the high (theoretically close to 100%) peak transmittance. However, a
drawback of guided-mode resonance filters is the increased complexity of fabrication due to
the introduction of the sub wavelength grating.
Both the guided-mode resonance filter and thin-film Fabry-Perot filter have line-shape
factor (rj ~ 3) that denotes a triangular pass band, hi this respect the Fabry-Perot filter can be
improved by coupling simple filters in series to form multiple-cavity filters with a more
rectangular line shape. In the case of the guided-mode resonance filters, it may be possible to
alter the lineshape by use of phase-shifted gratings as shown earlier for the case of reflection
filters [40].
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78
0.9
0.8
W 0.7-i
CJ
<1 ° - 6 i
U 0.5
CU
Ed 0.4
§ 0.3-1
0.2
0.1
0.4
0.45
0.5
0.55
0.6
0.65
WAVELENGTH (p m )
0.7
Figure 3.29. Notch filter with 150 alternating three-quarter-wave layers of silica
(nodd = 1.52) and BaF 2 (neven = 1-47) on silica substrate with air as cover.
0.9
0.8
0.7
0.52 0.53 0.54 0.55 0.56 0.57 0.58
WAVELENGTH (pm )
Figure 3.30. Double-layer guided-mode resonance filter in v-coating design (di =
0.073 p m , d 2 = 0.1156 pm). The materials are MgF (ni = 1.38) in the first layer,
ZrC>2 (nn ,2 = 2.1) and HfC>2 (nL,2 = 2.0) in the second layer containing the grating,
and silica (ns = 1.52) and air as substrate and cover materials, respectively.
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79
0.9 -j
UJ
0.8
O 0 .7 Z
£
0 . 6 -j
2 0.5 4
co
z 0 .4 -i
$h- 0.3 H
0.2
0.1 -I
0.45
0.5
0.55
0.6
WAVELENGTH (Hm)
0.65
Figure 3.31. Transmittance of a Fabry-Perot filter with 11 layers of alternating
ZnS (nodd = 2.35) and MgF2 (neven = 1-38); nc = 1.0, ns = 1.52. The center layer is
half-wave thick; the other layer thicknesses are all quarter-wave at 0.55 (im.
0.9
11 layers
2 gratings
0.8
9 layers
1 grating
11 i
0 0.7
z
£
0.6
1 0.5
co
z- 0.4
£ 0.3
H
0.2
0.535
0.54
0.545
0.55
WAVELENGTH (pm)
0.555
Figure 3.32. Spectral response of an 11-layer waveguide grating with gratings in
layers 1 and 11 (left) and a 9-layer guided-mode resonance filter with one grating
in layer 5 (right). The grating periods are A = 336 nm (left) and A = 330 nm
(right). The refractive indices and thicknesses are the same in both cases. The
grating layers have the parameters na = 2.5, nL = 2.2, d = 58.4 nm. For oddnumbered layers: d = 58.5 nm, n = 2.35. For even-numbered layers d = 99.6 nm, n
= 1.38. The substrate is silica (ns = 1.52) and the cover is air.
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80
3.7 Filter Design Using the “Direct”
Approach
A typical procedure for design o f reflection or transmission filters based on guidedmode resonances is illustrated by the flow chart of figure 3.33 [49,68]. The design is an
iterative approach that utilizes grating diffraction, waveguide, and thin-film properties of
waveguide-grating structures to obtain the desired filter line. In this work, it is referred to as
the “direct” method, to distinguish it from the “inverse” method of design presented in
chapter 5, that employs a genetic algorithm search and optimization method.
The design starts by distinguishing between the parameters that are fixed and those
that can be changed during the search procedure. Fixed parameters may be given by the
operating conditions such as incident wave polarization, angle o f incidence, cover and
substrate materials or by the device fabrication constraints. An initial waveguide-grating
structure is selected, largely based on the designers’ experience. This involves choosing the
number o f layers, refractive indices, thicknesses, grating period, and fill factor o f the device.
This initial structure undergoes modifications during the search procedure for the desired
filter response.
Typically, simpler structures (i.e., few layers, single, weakly-modulated grating) are
desirable as starting designs, since they allow known thin-film and waveguide techniques to
be used in adjusting the filter line. A fill factor of 0.5 is often convenient in grating
fabrication. Furthermore, if the initial structure exhibits a larger linewidth than necessary this
can be decreased through changes in fill factor.
Approximate eigenvalue equations discussed in chapter 2 are used to evaluate the
center wavelength and range o f the filter. The grating period is changed until the peak is
located at the desired wavelength. The next step is to use rigorous coupled-wave analysis, to
determine the exact response of the waveguide grating.
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81
The sideband response of the filter is reduced to acceptable levels by adding
homogeneous layers with refractive indices and thicknesses obeying antireflection (for
reflection filters) or high reflection (for transmission filters) conditions. This leads to a shift
in resonance location, which has to be subsequently repositioned back to the correct
wavelength by appropriate modification of the grating period. The next step is to adjust the
linewidth to the specified value. This can be accomplished, as discussed in section 3.5, by
changing the materials forming the grating (i.e., modulation) or the layers adjacent to the
grating (i.e., mode confinement), through modifications in fill factor, or by phase-shifted
gratings. These structural changes alter the sidebands as well as the resonance locations.
Thus, the procedure described above has to be repeated until a filter with the specified
linewidth and sidebands has been obtained. The final step is to adjust the center wavelength
o f the filter by changing the grating period.
The procedure described in this section has been utilized to obtain the guided-mode
resonance reflection and transmission filters presented in chapters 3 (at optical wavelengths)
and 4 (at microwave wavelengths). This design method is, in effect, a search and
optimization process, which in some cases, can take many iterations to perform. The designer
needs to be involved at every step, taking decisions based on acquired knowledge and
experience. A natural step towards a more efficient and effective design procedure is to use
computer optimization algorithms that are able to perform a global search and find the
optimum solution to a given problem. Such an approach is described in chapter 5.
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82
Select (starting)
- structure
- materials: ns, nH, nL
- Center,
:enter> d, 0, f=0.5
Solve Eigenvalue
equation to find A
RCWA to find exact
R (X ),
Sidebands?
T(X)
Change
layer d
(AR,HR)
Add homogeneous
layers (AR, HR)
Change f, materials (nH-nL),
phase shift, or structure
Change A
to fine-tune :enter
Desired filter response
R(k), T(k)
Figure 3.33.
Flow chart of guided-mode resonance filter design using the
“direct” method.
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83
3.8
Experimental Results
3.8.1 Background
Experimental realization of guided-mode resonance filters has been attempted with a
variety of waveguide-grating structures and fabrication techniques. Mashev and Popov [6]
fabricated a resonance structure consisting of a glass surface-relief grating coated with a thinfilm layer obtained by Ag+ ion exchange in molten AgNCb. The resulting corrugated- •
waveguide (i.e., the waveguide layer had nonplanar boundaries) device exhibited a -35%
reflection efficiency in the visible spectral region. Employing a similar design, Avrutsky et al.
[13] obtained a maximum reflectance o f -70% at the HeNe laser wavelength with a SiC>2 NbaOs corrugated grating on a CaF 2 substrate. Gale et al. [17] illustrated a guided-mode
resonance response of a waveguide grating (refractive index, n = 2) embedded in a material
with n = 1.5, obtaining a TE peak efficiency of -95% in the visible spectral region,
accompanied by high, asymmetrical sideband reflectance.
Peng and Morris [38] utilized a photoresist grating on a SisN4 waveguide, to reach a
peak efficiency of -92% for a TE polarized incident wave in the near-infrared range. Sharon
et al. [42] demonstrated light modulation at 1.55 pm in a multilayer InP/InGaAsP waveguide
grating by modulating the refractive index of the waveguide; their device exhibited a guidedmode resonance response with 85% efficiency and -1.5% sideband reflectance. A guidedmode resonance filter with a Si surface relief grating on a sapphire substrate was used by
Brundrett et al. [63] to achieve a broad reflectance peak of -80% at 1.68 pm. Magnusson et
al. [62] obtained a -94% resonance efficiency with a TM-polarized laser beam at Brewster
angle o f incidence with a surface-relief photoresist grating on a HfC>2 waveguide layer. The
following section describes fabrication and testing of a surface-relief structure that yields the
highest peak reflectance reported in the literature, 98% for a TE polarized incident wave [74].
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84
3.8.2 High-efficiency guided-m ode resonance
reflection filter
A waveguide-grating device has been fabricated by depositing a layer of HfC>2 with a
thickness d 2 = 270 nm on a fused-silica substrate by e-beam evaporation and subsequent
recording of a holographic grating in photoresist on top of the HfC>2 layer. The grating with a
period A = 487 nm and a thickness of d = 160 nm was recorded with an Ar+ laser (X = 364
nm) in a Lloyd mirror interference setup [88] illustrated in figure 3.34. An example of a
fabricated double-layer structure is illustrated by the scanning electron micrograph of figure
3.35. The spectral characteristics of the device are measured with a dye laser operating in the
800-900 nm wavelength range pumped with an Ar+ laser (X = 514 nm). The beam reflected
from the guided-mode resonance filter is measured automatically, while the laser wavelength
is scanned across the dye range in increments of 0.1 nm with an intracavity birefringent filter
under computer control, as shown schematically in figure 3.36 [74]. The specified linewidth
o f the dye laser system is 0.05 nm. The reflectance of the guided-mode resonance filter is
obtained by normalizing the power reflected from the device with the power of the incident
beam. Figure 3.37 illustrates the experimentally measured spectral response of the two-layer
guided-mode resonance filter for a TE-polarized probing beam incident at normal incidence
on the device [74]. The guided-mode resonance filter exhibits a peak reflectance exceeding
98% at the wavelength X = 860 nm with a linewidth (FWHM) of AX ~ 2.2 nm and low
sidebands (< 5%) over the wavelength range provided by the dye. The guided-mode
resonance corresponding to a TM-polarized wave at normal incidence is not seen
experimentally, which is consistent with theoretical calculations that show it to occur outside
the measurement spectral range. The theoretical reflectance curve presented in figure 3.37 is
obtained using the rigorous coupled-wave analysis [84-86] and assuming a grating with a
rectangular profile for simplicity. The thicknesses and refractive indices used in the
theoretical modeling are determined by spectral reflectometry measurements. A fill factor,
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85
defined as the fraction of the period occupied by the high index material, f = 0.3 (consistent
with the value measured from figure 3.35) is assumed in the calculations to closely match the
experimental curves with the theoretical calculations as seen in figure 3.37. The main source
of measurement error is slight fluctuation of the dye-laser output power; repeated
measurements have established the total experimental error to be within ±1% .
A splitting of the resonance at non-normal incidence (between 0° - 4.5°) has also
been measured and the results are in agreement with the theoretical predictions [74]. The
reflectance maxima shift away from the 8 = 0° value by -35 nm for the short wave peak (-1
diffracted order) and -3 0 nm for the long wave peak (+1 diffracted order), as the angle of
incidence on the filter is varied from 0° to 4.5°. It was found that the linewidths of the “+1”
and “-1” resonances are different from one another and both are smaller at angles 9 ^ 0 ° than
at normal incidence. The theoretical linewidth of the “-1” reflection peak increases from 0.8
nm at 0 = 1° to 0.9 nm at 0 = 4.5°, while the linewidth of the “+1” reflection peak decreases
from 1.2 nm to 1.0 nm in the same range of incident angles. The corresponding linewidths
determined experimentally are 0.8 nm at 0 = 1° and 1.1 nm at 0 = 4.5° for the “-1” reflection
maxima, and 1.6 nm at 0 = 1° and 1.3 nm at 0 = 4.5° for the “+1” reflection peaks [74].
The angular aperture of the guided-mode resonances varies with the angle of
incidence following the same trend as the spectral linewidth. The reduced angular aperture of
the filter at oblique incidence coupled with the non-zero divergence of the dye laser beam is
one cause for the slight decrease in efficiency of the filter from the theoretical 100% peak
reflectance. The angular components of a diverging probe beam have slightly different
resonance wavelengths, which results in a smearing of the reflectance curve and a reduction
of the peak response. Thus, the lowest experimental guided-mode resonance reflectance R =
82.2%, (found at 0 = 1° for the “-1” resonance) corresponds to the smallest angular aperture
calculated to be A0 = 0.1° (FWHM) [74]. hi contrast, the high-efficiency resonance peak at
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86
normal incidence has an angular aperture almost 5 times larger. The different linewidths and
angular apertures of the resonance peaks corresponding to the
+1
and
-1
diffraction orders is
a consequence of the different coupling efficiencies of the incident wave to the corresponding
leaky modes with opposite directions of propagation. Other factors contributing to the lower
experimental peak reflectance compared to the theoretical predictions are the scattering and
absorption losses of the waveguide grating.
3.8.3 Guided-mode resonance laser mirror
Guided-mode resonance filters operating in reflection can be utilized as laser mirrors
provided the reflectance is sufficiently high for the gain of the laser medium to compensate
for the losses in the cavity. Theoretically, with the high peak reflectances attainable in
guided-mode resonance filters, laser action can be sustained in a variety of active media,
including lasers possessing low gain as, for instance, gas lasers. This novel laser mirror has
unique wavelength and polarization sensitivity. The polarization selectivity of the guidedmode resonance filter used as a laser mirror provides a linearly polarized output beam
eliminating the need for intracavity elements oriented at Brewster angle. Furthermore, the
direction of polarization of the laser beam can be conveniently changed by rotating the
guided-mode resonance filter about the optical axis of the resonator. The narrow linewidths
of the guided-mode resonance laser mirrors may be useful in applications requiring a
reduction in the linewidth of a broadband laser source. The angular sensitivity of the
resonance wavelength enables a wavelength tuning method that requires no additional
filtering elements within the cavity. Avrutsky et al. [13] demonstrated a spatial broadening of
the resonance peak in the direction of propagation of the waveguide mode, for Gaussian
incident beams with diameters, a, comparable to the inverse of the radiative losses otr, in the
waveguides (aOr ~1). A mirror with such a beam broadening effect produces an increase in
spatial coherence of the laser beam [13]. The spatial beam broadening also increases the
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87
mode volume in a laser. Finally, it is worth mentioning that guided-mode resonance mirrors
with etched surface-relief gratings may prove a viable alternative to multilayer thin-film
stacks as high-power laser mirrors.
In the first experiments of this kind, Avrutsky et al. [13] used a guided-mode
resonance filter with -50% peak reflectance as a mirror in a laser cavity containing a
Rhodamine
6
G dye cell. Lasing occurred simultaneously at the guided-mode resonance
wavelength, as well as in a wide spectral band around the peak o f the Rhodamine
6
G
fluorescence curve. The spurious, wide-spectrum lasing was caused by the high gain of the
dye that allowed laser oscillation even with low mirror reflectances provided by Fresnel
reflections on the surface of the waveguide grating. A sim ilar device with -50% peak
reflectance was used as an external cavity mirror in an InGaAs semiconductor laser [16]. The
mirror induced a linewidth narrowing of the laser beam by reducing the multimode spectrum
to a single longitudinal mode.
In the present research, the guided-mode resonance filter described in section 3.8.2,
with 98% reflectance at 860 nm, was used as an output-coupling mirror in a commercial dye
laser [66,75]. The flat output mirror of the Coherent 599 Dye laser with broadband output
(800 - 920 nm with the dye LDS 821) was replaced with the guided-mode resonance filter
and lasing was achieved at the wavelength of 860 nm. Figure 3.38 shows a schematic
representation of the dye laser operating with a guided-mode resonance mirror. The laser
power was -100 mW when pumped with an Ar+ laser emitting a power of 5 W at the
wavelength of 514 nm. The linewidth o f the laser beam without the bireffingent filter in the
cavity was measured with an Anritsu spectrum analyzer to be -0.3 nm (figure 3.39). The
linewidth of the laser is set by the guided-mode resonance filter linewidth at the threshold
reflectance for laser oscillation to occur, hi the case of the dye LDS921, the threshold
reflectance o f the output coupler for laser oscillation (as specified by the manufacturer) varies
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88
between 93% at the peak fluorescence wavelength of 840 nm and 99% at the end
wavelengths of the operating spectral range (800 and 920 nm). It is estimated that the
threshold reflectance at 860 nm is ~95%. Thus, the measured linewidth of the laser is
consistent with the linewidth of the guided-mode resonance filter measured at 95%
reflectance value from figure 3.37.
Substrate mount
Ar+ laser, X —365 nm
Spatial filter
Mirror
Figure 3.34. Lloyd’s mirror interference setup for holographic recording of
diffraction gratings in photoresist.
G rati n e
Waveguide
Substrate
Figure 3.35. Scanning electron micrograph of the double-layer guided-mode
resonance structure consisting o f a photoresist grating and a HfC>2 waveguide
layer on a fused-silica substrate [74].
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89
Collimating
Lens
Beamsplitter
Reflection filter
or Mirror
Detector.
P C
Power meter
oo
Figure 3.36. Experimental setup for measuring the spectral characteristics of
guided-mode resonance filters operating in reflection at normal incidence. A
mirror with known reflectance vs. wavelength replaces the filter for incident beam
intensity measurements.
0 .9 -
Theory
0.8
Experiment
0 .7 -
0.6 U 0 .5 0 .4 0 .3 -
0 .2 -
0.H
840
845
850
855
860
865
870
875
880
WAVELENGTH (nm)
Figure 3.37. Theoretical and experimental spectral response of a double-layer
guided-mode resonance reflection filter with the structure illustrated in figure 3.35
[74]. The parameters o f the device used in the theoretical modeling are A = 487
nm, Uc = 1.0, ns = 1.48, nm = 1.63, niL = 1.0, n 2 = 1.98, di = 160 nm, d2 = 270 nm.
The grating is assumed to have a rectangular profile with a fill factor f = 0.3.
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90
High reflector
GMR mirror
Laser Output
< ---------
High reflector
t
Pump Beam
Dye Jet
Figure 3.38. Schematic representation o f a folded-cavity dye laser operating with
a guided-mode resonance mirror as output coupler.
SPECTRUM
1 0 0 j w / d iv
lm-
FAST
9 8 -1 0 -0 3 0 0 :1 6
AMR
1
I
500
*:
9 7 4 .8
rat io
0 -5
4 8 7 -4
0:
0 .8 5 4 5
W1 0 .8 5 9 1 4
RES 0 .2 run
W2—
W1
0 .8 5 9 5
0.28TMR
0 .8 6 4 5
U2 0 -8 5 9 4 2
lnn/div
Figure 3.39. Spectral line of the dye laser operating with the guided-mode
resonance mirror measured with a spectrum analyzer.
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CHAPTER 4
MICROWAVE GUIDED-MODE RESONANCE FILTERS
4.1
Introduction
Employing the same theoretical principles as for optical filter design has enabled
guided-mode resonance filters to be also applied in the microwave region [26,78,79].
Reflection filters with 100% reflectance at a desired wavelength and arbitrary-low sidebands
over extended frequency ranges can be designed as in the optical region. At microwave
frequencies, the grating period is on the order of
1
cm with grating fabrication and assembly
possible with ordinary machine tools. Furthermore, it is possible to use air as cover as well as
substrate, hereby forming a single-layer symmetric guided-mode resonance filter with low,
symmetrical sidebands. Microwave dielectric materials have a larger range of refractive
indices, which can be used to design guided-mode filters with larger linewidths than in the
optical region. The available materials with high dielectric constants also allow for improved
high-reflectance design in multilayer structures leading to high-efficiency transmission filters
with fewer layers than in the optical wavelength region [40].
The desired application at microwave frequencies is an integrated filter-antenna
system. The guided-mode resonance filter could be included in the antenna (or antennas)
aperture to act as a frequency-selective surface [69]. This type o f filter is sim ilar to traditional
frequency selective surfaces, but is all dielectric having no metallic components incorporated
within the filter structure. This would be particularly advantageous at millimeter-wave
communications where metallic frequency-selective surfaces exhibit high losses [56]. The
guided-mode resonance filter may thus allow the deletion of the filter that would normally be
placed in front of the first stage of amplification, and which is typically located at the output
91
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92
o f the antenna [69]. This novel amplification stage could be used for interference reduction,
either in reception or transmission of communication signals. The guided-mode resonance
filter could potentially cover a reflector antenna or a phased array aperture for interference
reduction. Interfering signals at the same frequency emanating from directions other than the
main beam of the antenna and filter combination would be rejected by the filter and not reach
the antenna. This could work in transmitting mode as well, limiting interference to other
antenna systems. In particular, such an interference reduction system could be used, both for
reception and transmission, for a ground station pointed towards a satellite [69]. The use of
the guided-mode resonance filters as a diagnostic or calibration tool is also possible. One use,
for instance, could be to verify the quiet zone in an anechoic chamber [69].
4.2
Reflection Filters
A typical single-layer reflection filter with air as cover and substrate, Ec = 6 s = 1.0,
has the calculated response shown in figure. 4.1 [79]. The grating materials have dielectric
constants £h= 2.59 and £ l= 2.05 (corresponding approximately to the parameters of plexiglas
and teflon, respectively), and a grating period A = 1.65 cm. As shown in figure 4.1(a),
arbitrary thicknesses of the grating layer yield reflectance curves with asymmetrical
lineshape. Using a thickness of half wavelength, determined by the resonance frequency, v
-1 4 GHz, a symmetrical diffraction efficiency with small sideband reflection (figure 4.1(b))
in the spectral range 1 2 - 1 6 GHz results. Adding homogeneous layers, as discussed in
chapter 3 for optical guided-mode resonance filters, can reduce the sideband reflectance.
Thus, adding two more layers with dielectric constants
£2
= 6.13 (E-glass) and
£3
= 2.59
(plexiglas), approximately satisfying the three-layer antireflection condition £i,eff2 £32/£22 = 1 >
and with quarter-wave thicknesses of the three layers, a high efficiency filter is realized with
a broad low-reflectance response (figure 4.2). For TE polarization R < 1 % (-20 dB) in the
range 1 2 —16 GHz and R < 0.1 % (-30 dB) in the range 13.0 - 14.9 GHz. A change in grating
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93
period shifts the resonance wavelength approximately linearly within the low-reflectance
spectral region with insignificant changes in the filter lineshape or sideband reflectance.
1
0.9
CJ
(a)
(b)
0.8 -j
‘I t kR
0.7 H
0.6
E-
05 r
O
ui
-J
[x. 0.4
\
0 .3 -i
0 -2 *i
0.1 -i
J
n i |' i i i
12
12.5
i i [ i i i i | i i i i 1 1 i i'f [ i i i i | i i
135 14 14.5
FREQUENCY (GHz)
13
15
15.5
16
Figure 4.1. Reflectance of a single-layer waveguide grating with grating period, A
= 1.65 cm and dielectric constants, Eh = 2.59 (plexiglas), El = 2.05 (teflon), and £c
= £s = 1.0. The thickness is (a) d = 0.5 cm and (b) d = 0.7 cm (half-wavelength).
TE
0.8
TM
R
w
-
A
l l
U
m
L
i
|
t
0.1
........ j
10
11
12
13
14
15
16
17
18
FREQUENCY (GHz)
Figure 4.2. Spectral response of a triple-layer guided-mode resonance reflection
filter for TE and TM polarization of the incident wave. The parameters are: A =
1.3 cm, £h,i = 2.59 (plexiglas), Em = 2.05 (teflon), e2 = 6.13 (E-glass), £ 3 = 2.59,
Be = £s = 1.0, d[ = 0.35 cm, d 2 = 0.22 cm, and d 3 = 0.33 cm.
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94
A grating period, A = 1.3 cm, generates the TE and TM polarization spectral response
illustrated in figure 4.2 [79]. The TM and TE resonances typically have similar sideband
reflectance but different frequency location, linewidth and lineshape. This structure is thus
more robust parametrically than the single-layer waveguide grating, since a slight change in
one grating parameter has a reduced effect on the filter response. Almost identical filter
characteristics can be obtained with the grating being contained in the second or third layer,
provided that the antireflection condition is satisfied [29]. Multiple-layer filters allow for a
greater flexibility in the choice of dielectric constants to achieve desired filter parameters as
well as an increased control over the pass-band and the angular aperture of the filter.
An important goal in filter design is spectral linewidth control. The linewidth of
guided-mode resonance reflection filters has been shown to increase with the modulation of
the grating and the dielectric-constant difference between the grating and substrate [24]. The
increase with modulation is due to increased leakage of the waveguide grating about the
resonance wavelength. A guided-mode resonance response with symmetrical lineshape can
be approximated by a Lorentzian function with the center frequency determined by the phasematching condition of the external field to the waveguide mode and with the linewidth
proportional to the waveguide coupling loss. Therefore, an increased leakage of the grating
generates a larger linewidth of the guided-mode resonance filter. A higher dielectric constant
of the grating region leads to increased confinement of the modes in the associated
unmodulated waveguide, thus resulting in a broadening of the linewidth [24]. The influence
of the modulation on the bandwidth is illustrated in figure 4.3 by two examples of simple,
single-layer filters of fixed thickness (d = 0.67 cm) and grating period (A = 1.61 cm) at
central frequencies close to 14 GHz [79]. A narrow-band filter response (figure 4.3(a)) is
obtained with the grating made of plexiglas (Eh = 2.59) and polystyrene (El = 2.54) resulting
in a modulation of Ae = Eh - £l = 0.05, a central frequency v ~13.74 GHz, and a linewidth
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95
(full width at half maximum) Av -1 .5 MHz. A larger modulation achieved with alternating
bars of glass laminate (Eh = 3.0) and teflon (el = 2.05) yields a modulation of Ae = 0.95, a
center wavelength v -1 4 GHz, and a filter linewidth Av -162.4 MHz. The sideband
reflectance of these filters can be reduced by adding layers with antireflection design, as
demonstrated by the previous three-layer example.
An important advantage in the microwave spectral region is provided by the
possibility of placing a grating in air and using air as the low dielectric constant medium (El =
1.0) as well as the surrounding medium (Ec = £s = 1)- Figure 4.4 (with 4> = 0) shows a single­
layer guided-mode resonance filter made of plexiglas bars (e = 2.59) in air [80]. This periodic
set of rectangular bars has a large modulation as well as a large dielectric-constant difference
between the grating region and the substrate. The grating with a period of A = 3.0 cm
generates a resonance at the frequency v = 8.65 GHz with a bandwidth Av = 8 MHz. Larger
or smaller bandwidths can be obtained with larger or smaller dielectric constant materials,
respectively.
The linewidths of guided-mode resonance filters are limited by the available low-loss
microwave materials with the highest dielectric constants. Further increase in linewidth can
be accomplished with spatially phase-shifted gratings obtained by translating one half of the
grating with respect to the other, as discussed in section 3.5.2. Figure 4.4 (<I> = nJ6) illustrates
the spectral response of such a two-layer structure with tc/6 shifted gratings [79]. The
resonance is broadened to Av = 24.4 MHz and slightly shifted in frequency. Other values of
the phase shift lead to a split-peak resonance caused by the different coupling conditions of
the +1 and -1 diffracted orders to the waveguide modes of the device. At a certain value of
the phase shift that depends on the waveguide-grating parameters (in this case 7t/6) the two
peaks merge into a broadened reflectance peak.
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96
10.9
0.8
:
(a)
j
(b)
0.7
cw
_>
—
0.6 -3
A
cj 0.5
w
■
0.4
^ t R
i in
s05 0.3,
I
0.2
_________ ^
0.1 "
0-
t
---------------- >
12
12.5
13 13.5 14 14.5 15
FREQUENCY (GHz)
15.5
16
Figure 4.3. Single-layer guided-mode resonance reflection filters with a) narrow-line
and b) broad-band response. The dielectric constants are 8h = 2.59 (plexiglas) and El
= 2.54 (polystyrene) for the narrow-line filter and Eh = 3.0 (glass laminate) and El =
2.05 (teflon) for the broad-band filter. Both structures have, A = 1.61 cm, d = 0.67
cm, and £c = £s = 1.0.
l
0.9
0.8
cdJ °-7
C
<! 06
0.5
s 0.4
* 0.3
0.2
0.1
0
8
8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9
FREQUENCY (GHz)
9
Figure 4.4. Single-layer response of a guided-mode resonance filter in air (£c = £s =
1.0) with plexiglas as the high-dielectric-constant medium (Eh = 2.59) and air as a
low-dielectric medium (El = 1.0). The period is A = 3.0 cm and thickness, d = 2.58
cm. The dotted curve shows the reflectance of the two-layer structure obtained by
shifting half of the grating by 7t/6 with respect to the other half, as shown in the inset.
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97
As discussed in section 3.4.2, new possibilities emerge by use of structures with
gratings of different periods in successive layers. For instance, a device with two gratings o f
slightly different periods in the three-layer geometry shown in the inset of figure 4.5 can act
as a double-line filter at normal incidence with the center frequencies determined by the
resonances of the individual single-layer waveguide gratings. The buffer layer placed
between the two gratings should be thick enough to weaken the coupling between the
evanescent waves of the two gratings. Figure 4.5 shows the response of a three-layer device
using two plexiglas/air gratings of different periods separated by an air gap buffer [79]. The
difference between the two grating periods is AA = 0.05 cm, exhibiting nearly 100%
reflection peaks with a frequency difference between them o f 126.6 MHz (corresponding to a
wavelength difference of 0.05 cm). The frequency difference between the two peaks can be
adjusted by changing the periods of the gratings. For small grating period difference,
resulting in partly overlapping resonance peaks, this approach can also provide yet another
linewidth broadening technique. However, this may also lead to a distorted filter lineshape.
4.3
Transmission Filters
Transmission filters can be obtained by superimposing the resonance of a waveguide
grating on the high-reflectance response of a homogeneous multilayer structure [34,47]. This
is achieved by replacing one or more high-dielectric-constant layers of the high/low
homogeneous quarter-wave stack with gratings of approximately equal average dielectric
constant to maintain the low off-resonance transmittance of the device. The resonances that
were used to generate reflection filters are now forced by the high-reflectance design to
produce transmission filters. A higher difference between dielectric constants of the
successive layers determines a larger width of the low-transmittance spectral region affecting
the free spectral range of the filter. An increased ratio between high and low average
dielectric constants of the quarter-wave stack also implies fewer layers to achieve the same
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98
sideband transmittance. Since materials with relatively high dielectric constants are available
at microwave frequencies and air can be used effectively as the low-dielectric constant
medium, high-efficiency transmission filters with few layers and low sideband transmittance
can be designed. Single-grating transmission filters can be obtained with a dielectric
multilayer mirror on each side of the grating [47,79].
Figure 4.6 shows an example of a 5-layer transmission filter at v -1 4 GHz with the
structure illustrated in the inset using E-glass (e = 6.13) and silica (8 = 3.7) as grating
materials. All layers are quarter-wave thick at the frequency v = 14 GHz to obtain a high
reflectance at frequencies away from resonance. The filter bandwidth is Av = 45 MHz.
As in the case o f reflection filters, the bandwidth of the transmission filters can be
made narrower or broader by forming the grating with materials having a smaller or higher
difference between the dielectric constants within a period of a waveguide grating. The filter
range of guided-mode resonance transmission filters is limited by peaks due to other
resonating modes and by the range of the high-reflectance spectral region of the multilayer
structure. For example, with the response of figure 4.6, the waveguide supports three TE
modes with the peak at 14 GHz caused by coupling of the evanescent diffracted waves to the
TE 2 waveguide mode. The filter range is limited by the TE[ resonance at 11.94 GHz and by a
Rayleigh anomaly (produced by the onset of higher-order diffracted waves) at 18.98 GHz.
Arbitrarily small sideband transmittance can be achieved with an increase in the number of
quarter-wave homogeneous layers. However, as the structure thickness increases additional
modes can propagate in the equivalent homogeneous multilayer waveguide, thereby
decreasing the filter range. New avenues for design of guided-mode resonance transmission
filters are opened by structures containing highly modulated, thick gratings (d -A,), with
asymmetric fill factors (M).5). Such devices can yield high-efficiency transmission filters
with yet fewer layers [49, 64].
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8 .4
8 .5
8 .6
8 .7
8 .8
8 .9
9
9.1
9 .2
FREQUENCY (GHz)
Figure 4.5. Dual-line reflection filter response. The structure has 3 layers with 2
gratings in the first and third layer. The parameters are: Ai = 2.95 cm, A 2 = 3.0 cm,
eH.i = £h,3 = 2.59, £m = £ 2 = £l ,3 = £c = £s = 10, di = d3 = 2.58 cm, and d 2 = 5.2 cm.
Guided-mode resonance peaks are at frequencies Vt = 8.65 GHz and V2 = 8.78 GHz.
0.9
—
-
'I t'
[U
H
H
-
0 .4
0.2
i x
1
-rm -f
1 2.5
13
1 3 .5
14
1 >1 1| , 1 1 1 | 1 1 1 i
1 4 .5
15
15 .5
16
FREQUENCY (GHz)
Figure 4.6. Guided-mode resonance transmission filter spectral response with the
structure illustrated in the inset. The peak frequency is v -1 4 GHz. The filter
parameters are: A = 1.58 cm, £ H ,3 = 6.13 (E-glass), £l ,3 = 3.7 (silica), £i = £ 5 = 6.13, £ 2
= £ 4 = £ c = £s = 1-0, di = d5 = 0.22 cm, d 2 = cU = 0.53 cm, and d 3 = 0.24 cm.
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100
However, due to the high modulation of the grating, the average dielectric-constant
approximation employed in the design of high-reflectance structures does not hold as well.
Therefore, unlike the previous transmission filter example, the physical parameters of the
layers are more efficiently determined by an elaborate search procedure such as genetic
algorithm optimization [64].
4.4
Experim ental Results
The filtering properties of waveguide gratings based on guided-mode resonances are
experimentally demonstrated with single-layer and multilayer devices designed to operate as
reflection or transmission filters. Spectral measurements of the waveguide-grating
transmittance are made in an anechoic chamber with the setup illustrated in figure 4.7 [78]. A
plano-convex teflon lens, with a diameter D = 35.6 cm and focal length fi = 121.9 cm, is
positioned with the flat side towards the transmitter horn antenna, at a distance from the
transmitter equal to the lens focal length. Thus, an approximately planar wavefront is incident
on the waveguide-grating. A second microwave lens, identical to the first, focuses the
transmitted zero-order diffracted wave on the receiver horn antenna located in the focal plane
o f the lens. The transmitted power is measured in the spectral range 4-20 GHz in increments
o f 2.5 MHz, thus acquiring 6400 data points. The transmittance of the guided-mode
resonance filter is obtained by normalizing the microwave power measured with the setup of
figure 4.7 with the power received without the filter. The raw measured data contains rapidly
varying oscillations due to interference effects caused by multiple reflections between
components in the path of the beam. Therefore, to obtain a smoother experimental response
and to compare the data with the theoretical predicted spectral characteristics, each filter
transmittance value is replaced with the average transmittance of 20 neighboring frequency
points.
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101
4.4.1
Notch filters
A single-layer guided-mode resonance device has been fabricated with rectangular
plexiglas bars in air forming a grating with period, A = 3.0 cm, fill factor, f = 0.5, and
thickness, d = 0.87 cm. The spectral response of this device, measured with the setup of
figure 4.7, exhibits a reflection peak at the waveguide-grating resonance frequency
corresponding to a sharp notch in the high-transmittance region. Figure 4.8 compares the
calculated response using rigorous coupled-wave analysis w ith the experimentally measured
frequency dependence of the transmittance for a TE-polarized incident wave [79]. The
dielectric constant and loss tangent used in calculations are £ = 2.59, and tan8 = 0.0067,
respectively [102]. Assuming an incident angle of the plane waves on the structure, 0 = 1°,
the two notches in the transmittance curve observed in the experimental data at frequencies
8.70 GHz and 9.20 GHz are closely matched by the theoretical calculations that yield
resonance frequencies 8.74 GHz and 9.24 GHz, respectively. The splitting of the guidedmode resonance, visible in figure 4.8, is due to asymmetry arising in coupling of the +1 and —
1 evanescent diffracted orders to the leaky modes of the waveguide as the angle of incidence
shifts away from zero. Thus, at non-normal incidence, the resonances corresponding to the +1
and -1 diffracted orders occur at different frequencies. T he coupling efficiencies of the
incident wave to the corresponding leaky modes with opposite directions of propagation are
different causing unequal linewidths and transmittance values in the resonance notches.
The theoretical and experimental spectral response o f the same plexiglas structure,
under identical experimental conditions, is shown in figure 4.9 for a TM-polarized incident
wave [79]. A TM-polarization guided-mode resonance was measured at the frequency 9.30
GHz, in excellent agreement with the theoretically calculated value of 9.29 GHz. Identical
theoretical waveguide-grating parameters match the experimental results for both TE and TM
polarizations.
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102
Spherical
wavefront
Teflon
Lens
Transmits
121.9 cm
Guided-mode
Spherical
resonance filter
wavefront
Zero-order
Planar
■
diffracted Teflon
wavefront I n
wavefront Lens
Receiver
152.4 cm
152.4 cm
121.9 cm
Figure 4.7. Experimental setup in an anechoic chamber for transmittance
measurements of waveguide gratings in the spectral range 4 - 2 0 GHz. The
dimensions of the filters are 91.4 cm (along the grating vector) by 61.0 cm.
03
CU
C
J
E-
Theory
Experiment
8
10
12
14
FREQUENCY (GHz)
16
20
Figure 4.8. Calculated and measured transmittance notches due to guided-mode
resonances occurring in a single-layer plexiglas/air waveguide grating for a TEpolarized incident wave. The grating parameters are A = 3.0 cm, f = 0.5, d = 0.87
cm, £h = 2.59, 6l = £c = £s = 10. The calculated plot was obtained with a loss
tangent, tanS = 0.0067, and an incident angle, 0 = 1°.
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103
Theory
-8 —
4
i
i i | " i i i |i i i |
6
i
i i |
ii i |
8 10
12
14 16
FREQUENCY (GHz)
i i i |
18
ii i |
i i i
20
Figure 4.9. Theoretical and experimental transmittance of the single-layer
plexiglas/air waveguide grating of figure 4.8 for a TM-polarized incident wave.
In both polarization cases, the measured transmittance values at resonance are higher
than the theoretical predictions. This can be explained in part by the imperfect collimation of
the microwave beam with a divergence that exceeds the angular aperture of the device. The
angular components of a diverging beam generate guided-mode resonances at slightly
different frequencies, thus contributing to the decrease of the filter efficiency. The
discrepancy between the theoretical and experimental resonance transmittance is more
pronounced in the case o f TM polarization which is due, in part, to the narrower linewidth
and angular aperture in comparison with the TE guided-mode resonances. Thus, the
calculated angular widths (at —3dB) of the transmittance notches are AGte = 7.3° for the TEpolarization case and A 0 tm = 0.25° for the TM-polarization. The guided-mode resonance
frequency varies with changes in geometrical parameters or dielectric constants o f the
waveguide gratings. Therefore, variations of the dielectric constants and geometrical
parameters (thicknesses, grating period, fill factor) across the device may also contribute to
decrease in filter efficiency. Yet another factor that can affect the filter efficiency and lead to
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104
deviations from the theoretical data is the interaction of the non-planar incident wave with a
limited number of grating periods (-12 periods) of this waveguide grating while the theory
assumes an infinite structure and infinite number of periods.
4.4.2
Bandpass filters
The concept of transmission guided-mode resonance filters at microwave frequencies
discussed in section 4.3 is demonstrated with 5-layer waveguide gratings employing G10
fiberglass as the high dielectric constant material and air as the low dielectric constant
medium, for both the homogeneous layers as well as for the grating layer [78,79]. The
devices, with the cross-section illustrated in the inset of figure 4.10, consist of a grating
formed with rectangular bars of G10 fiberglass equally spaced in air and homogeneous G10
sheets placed on each side of the grating as shown. The thicknesses of the G10 sheets and the
air gaps between the grating and the G10 sheets are chosen to be equal to a quarterwavelength at the central frequency of the filter, thus providing the low transmittance
sidebands required for a transmission filter design.
The waveguide-grating with the response illustrated in figure 4.10 has G10 sheet
thickness, d = 0.317 cm with ± 0.03 cm variation according to manufacturer’s specification.
This thickness is equal to a quarter-wavelength at the frequency v = 11.14 GHz (for a
refractive index of G10 fiberglass n = 2.12 [103]). The high-reflectance design is
accomplished with air gap thicknesses between the grating and the G10 homogeneous sheets
twice as thick as the G10 layers. The grating period (A = 2.0 ± 0.05 cm) is selected to
generate a guided-mode resonance within the high-reflectance spectral range of the structure.
Figure 4.10 demonstrates an excellent agreement between the theoretically calculated
transmittance for a TE-polarized incident wave and the spectral response measured with the
experimental set up shown in figure 4.7 [78]. A TM-polarization guided-mode resonance
peak, predicted theoretically at v = 13.87 GHz to exhibit significantly smaller linewidth and
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105
angular aperture was not observed experimentally. The grating period used in the calculation
o f figure 4.10 (A = 2.03 cm) is the average value determined through measurement on the
waveguide grating after assembly. A thickness of the fiberglass layer d = 0.31 cm and a loss
tangent value tanS = 0.01 were used to yield the close fit between the theory and
measurements. Guided-mode resonance transmission peaks are determined experimentally
(after smoothing the curve by averaging over 20 neighboring points) at frequencies v = 10.50
GHz and v = 14.51 GHz; the theoretically predicted values are v = 10.50 GHz, and v = 14.34
GHz, respectively. Transmission notches accompanying each guided-mode resonance peak
are measured at frequencies and with transmittance values close to the theoretical predictions.
The split-notch visible in the experimental data is caused by a small nonzero angle of
incidence on the device, which has not been accounted for in the theoretical calculations
given for normal incidence. The lower experimental peak transmittances compared to the
calculated values can be attributed, in part, (as in the reflection filter case) to the phase
curvature o f the incident beam, to the sensitivity of the resonance frequencies to waveguide
grating parameter variations across the device, and to the finite-size effect (the microwave
beam covers -1 8 periods here) of the device [79].
The presence of guided-mode transmission resonances is emphasized by comparison
of the G10 waveguide grating spectral response with the transmittance of an identical
structure with the grating replaced by a homogeneous layer with dielectric constant equal to
the average dielectric constant of the grating (figure 4.10). The spectral response of the
structure containing only homogeneous layers (dashed curve) possesses a high-reflectance
spectral region similar to the transmittance curve of the waveguide grating, except for the
peaks and notches due to the guided-mode resonances [79].
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106
Theory
Experiment
4
6
8
10
12
14
FREQUENCY (GHz)
16
18
20
Figure 4.10. Theoretical and experimental spectral response of a G10 fiberglass and
air waveguide grating exhibiting guided-mode resonance peaks. The parameters of the
structure are: A = 2.03 cm, f = 0.5, di = d 3 = ds = 0.31 cm, d 2 = dr = 0.62 cm, 8H,3 = £i
= £5 = 4.49, and £ l ,3 = e2 = £ 4 = £c = £s = 1-0. A value for the loss tangent, tan8 =
0.01, was used in calculations. The dashed curve represents the transmittance o f a
structure identical to the one presented in the inset, but with the grating replaced by a
homogeneous layer with dielectric constant, 8 = 2.75.
4.4.3
Fabry-Perot filters
Dielectric Fabry-Perot transmission filters consist o f a central layer bounded on both
sides by two mirrors obtained with alternating high and low dielectric constant layers [88,92].
Their design is similar to the guided-mode resonance filter represented in figure 4.10 with the
quarter-wave thick grating replaced by a half-wave thick homogeneous layer. Five-layer
Fabry-Perot filters, with G10 fiberglass as the high-dielectric constant material and air as the
low-dielectric constant material, were built and tested using the set-up of figure 4.7 [69].
The calculated transmittance plot for the G10 fiberglass Fabry-Perot filter is
illustrated in figure 4.11 [69]. Calculations with a loss tangent tanS = 0.002 yielded a
maximum transmittance of 96.16% at the center frequency o f the filter v = 11.32 GHz and a
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107
linewidth Av = 0.97 GHz. A higher loss tangent (tan5 = 0.01) results in a transmittance peak
of 82.78%, a central frequency v = 11.31 GHz and a linewidth Av = 1.05 GHz.
The spectral response of Fabry-Perot filters for a normally-incident wave, measured in
the frequency range 4-20 GHz with microwave lenses for beam collimation is illustrated in
figure 4.12 for the G10 fiberglass device [69]. The G10 fiberglass Fabry-Perot filter
possesses a high experimentally measured peak transmittance (-71.5% for TE polarization
and -73% for TM polarization) at the frequency v = 11.48 GHz with a linewidth Av = 1.07
GHz. As expected [92], the central frequencies and the linewidths are the same for TE and
TM polarizations.
The experimental transmittance curve compares very well with the calculated spectral
response and the central frequency and linewidth of the filter are in excellent agreement with
the theoretical predicted values. The experimental results confirm the narrower linewidths
that can be obtained with guided-mode resonance filters compared to Fabry-Perot filters built
with the same number of layers and the same materials. For low-loss materials, guided-mode
resonance filters should exhibit similar sideband response and comparable peak
transmittance. However, guided-mode resonance filters are more sensitive to losses than
Fabry-Perot structures.
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108
-
U
2-
-4 -
0 .3 -
0.2
-
10-
0 . 1-12
4
6
8
10
12
14
16
18
20
4
6
8
FREQUENCY (GHz)
10
12
14
16
18
20
FREQUENCY (GHz)
Figure 4.11. Theoretical transmittance of a 5-layer Fabry-Perot at normal
incidence with dielectric constant of the odd layers £ = 4.5 (G10 fiberglass),
dielectric constant of the even layers £ = 1.0 (air), and thicknesses of the layers di
= ds = 0.317 cm, and dz = d3 = cU = 0.635 cm. The loss tangent is tan5 = 0.002
(dotted line) and tanS = 0.01 (solid line).
0 .9 -
2-
0 .5 -i
0 .4 H
0 .3 -
0.2 -
-
10 -
0 . 1-12
4
6
8
10
12
14
16
FREQUENCY (GHz)
18
20
4
6
8
10
12
14
16
18
20
FREQUENCY (GHz)
Figure 4.12. Experimental frequency response of a 5-layer fiberglass/air FabryPerot for TE-polarized incident wave. The theoretical plot is shown in figure 4.11.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTERS
GUIDED-MODE RESONANCE FILTERS DESIGNED
WITH GENETIC ALGORITHMS
5.1
Introduction
Genetic algorithms are numerical optimization programs based on the principles and
mechanisms of natural evolution and genetics [104-108]. The ability of the genetic algorithm
to perform a parallel search in a parameter space and find the global minimum of a merit
function in an effective and efficient manner has led in recent years to its widespread
application and increased popularity. Genetic algorithms are simple, stochastic methods of
optimization that do not require prior knowledge about the search space and therefore can be
applied to a variety of optimization problems in science, engineering and business [104-108].
This dissertation focuses on the application of genetic algorithms in diffractive optics and
particularly in the study and design of reflection and transmission filters based on the guidedmode resonance effect in waveguide gratings [49,64,65,71]. Diffractive optics is a rapidly
growing area of research with numerous applications in imaging, optical image processing,
optical computing, et cetera. However, in most practical devices, the spatial distribution of
thicknesses and refractive indices o f a diffractive element that produce the desired
transformation of an optical beam can not be easily determined with analytical expressions
and require the use of numerical optimization methods. Guided-mode resonance filter design
is such an example where the desired filter characteristics (center wavelength, linewidth,
sideband level, filter range, and lineshape) are determined by iteratively solving coupled
differential equations with different waveguide-grating parameters. Current knowledge about
the guided-mode resonance effect provides guidelines in the search for a desired filter
109
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110
response as described in chapter 2. However, practical restrictions imposed on the selection
of waveguide-grating parameters (such as, for instance, the limited number of refractive
indices available in a particular spectral range) and the complex dependence of the filter
characteristics on several physical parameters makes the search procedure time consuming.
Furthermore, once a satisfactory design has been found there is no certainty that there exists
no other simpler structure (with perhaps fewer layers of utilizing fewer materials) that can
perform equally well. For some types of filters that have yet to be fully understood such as
transmission filters with extended wavelength ranges, wide-linewidth reflection filters, multiline filters, and filters with improved lineshape, the design process is largely based on trial
and error. Therefore, a global optimization technique would aid not only in the design of new
filters, but also in the study of the guided-mode resonance effect, as well as more generally in
the study of diffraction by sub-wavelength periodic structures. A computer program that uses
genetic algorithms as a search method and rigorous coupled-wave analysis as a computational
method for the merit function optimization of diffractive optics devices has been developed
and is presented in this chapter. This program is used to design reflection and transmission
bandpass filters with features not previously known to be possible. Such an example is the
transmission filter obtained in structures with only one or two layers, possessing low
sideband transmittance generated by diffraction instead of thin-film interference effects.
5.2
5.2.1
Background
Num erical optimization methods
The wide variety of existing optimization techniques can broadly be classified as
calculus-based, enumerative, and random [104]. Calculus-based methods typically search for
the extremum of a merit function by calculating its gradient and finding the location in the
search space where the gradient is zero. These methods have been applied extensively in thinfilm design and are capable of localizing the local optima efficiently and with very good
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accuracy [111]. However, a major disadvantage of these methods is that they are susceptible
to converge to a local extremum of the merit function instead of the global extremum. Thus,
if the starting point of the gradient-based optimization procedure is in the vicinity of a low
peak, representing a local extremum, the algorithm may reach this peak and miss the higher,
neighboring peak representing the main event [104]. Another drawback of these methods
stems from the assumption that derivatives of the merit function must exist and have a finite
value in the parameter space, which may not be true in some applications.
Enumerative schemes are simply iterative calculations of all values of the merit
function in the parameter space [104]. While this method is effective in finding a global
extremum it lacks efficiency and becomes prohibitive for large parameter spaces as are often
encountered in practical problems.
Random search methods, such as genetic algorithms and simulated annealing
algorithms, attempt to combine the efficiency of the calculus-based techniques and the
effectiveness of the enumerative schemes using random numbers to perform a global search
in the parameter space. It is important to note that random methods do not imply random
searches through the parameter space but a guided exploration based on certain rules and
exploiting historical information about the merit function values.
Simulated annealing [106, 109] is a mathematical procedure that mimics the physical
process by which a collection o f atoms is heated and then cooled at a low enough rate to
allow the atoms to settle in the lowest energy state. The energy of the system represents the
cost function to be minimized by the algorithm. In the search for the minimum energy state,
small random changes are made to the parameters and the resulting energy change AE is
evaluated. If AE < 0, the change is accepted and the new parameter values are used as a
starting point for the next search iteration. If AE > 0, the new state is accepted with a
probability P(AE) = expC-AE/ksT), where T, the temperature of the system is a control
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112
parameter, and ks is the Boltzmann constant [109]. B y retaining a number of configurations
with increased energy and thus allowing uphill moves, entrapment in local m inim a is
avoided. The success in finding the lowest energy state depends on the choice o f the
annealing schedule consisting of the successive set o f decreasing temperatures and on the
time that the system is permitted to reach equilibrium at a given temperature.
Genetic algorithms are optimization procedures that follow the natural selection
process operating on a large population of candidate solutions and utilizing the genetic
operators of selection, crossover and mutation to improve the merit function from one
generation to another [104]. Similarly to the simulated annealing procedure, genetic
algorithms rely on probabilistic rather than deterministic rules to guide the search in the
complex parameter space. As opposed to the optimization methods mentioned above, the
genetic algorithm does not require an initial guess as a starting point. In gradient-based
techniques the initial guess determines whether the algorithm reaches a local or a global
extremum of the merit function. Genetic algorithms operate on a population o f points
performing extremum searches in several hyperplanes simultaneously. Therefore, the
procedure is less likely to get stuck in local minima o f the merit function than optimization
methods based on a point-to-point search. The genetic algorithm is robust since it requires no
auxiliary knowledge about the function (like derivatives in gradient-based methods) but just
the function evaluation at different points in space. T he simplicity of the algorithm has made
it applicable to a wider range of problems than gradient-based techniques. However calculusbased methods may utilize more knowledge about the specific problem to be optimized and
therefore may perform better than genetic algorithms but in a narrower problem domain. A
genetic algorithm combined with a local optimization routine would more than likely perform
better than the individual routines by themselves [77]. The main drawback of the genetic
algorithms are the relatively long computing time required to reach the optimum solution,
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113
particularly as the number of parameters increases. The inherent parallelism o f the search
procedure enables the implementation of the genetic algorithm on parallel multi-processor
computers, which reduces the computational tim e significantly [64].
5.2.2 Principles o f genetic algorithm search
and optim ization
An important characteristic feature of genetic algorithms is that they operate on a
coding (e.g., a binary coding) of the parameters rather than on the parameters themselves
[77,104-108]. Thus, the first task of a genetic algorithm is to generate a set of random
numbers in a particular encoding that corresponds to the variables of the problem. Each
variable is called a “gene” or “allele” and represents a particular feature or character as, for
instance, the thickness or refractive index of a layer in a diffractive optics problem.
Combining several genes one obtains a string called a “chromosome” which represents a
candidate solution. For example, in a homogeneous-layer thin-film optimization procedure a
chromosome would be composed of the thicknesses and the refractive indices o f all the layers
in a structure. Many such candidate solutions are generated simultaneously by a genetic
algorithm. Together they form a “population” and successive populations generated by the
genetic algorithm are referred to as “generations.”
There are many implementations of genetic algorithms but they all have in common
the basic operators of selection, crossover and mutation [77,104-108]. All these operators are
applied on the population of a generation to create the next generation. A typical flow chart
for a genetic algorithm optimization procedure is presented in figure 5.1. An initial
population is generated randomly with each gene spanning its allowed range o f values. The
domain can be discrete for some genes and continuous for others, the only restriction
consisting in the locations of the genes in the chromosome, which must remain the same for
all chromosomes. A merit (sometimes referred to as cost, fitness, or residual) function is
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114
calculated for each chromosome. This merit function is problem specific and the success of
the optimization procedure depends largely on the choice of the merit function. The
chromosomes are ranked in terms of their performance evaluated by the merit function.
Successive generations are then created by retaining a part of the chromosomes from one
generation to the next and by forming new chromosomes through recombination of the best
chromosomes in the old population. The greater the fitness value of a chromosome, the more
likely it is to participate in the recombination process. Some algorithm s retain a fixed number
o f chromosomes [110] while others are more problem-specific and retain all chromosomes
with fitness better than a user-defined value [111].
The recombination process consists of applying either or both crossover and mutation
operators. In the crossover operation, two chromosomes exchange portions of their encoded
representation. The three types of crossover mechanisms encountered in genetic algorithms
are illustrated in figure 5.2 [112]. A single-point crossover is realized by choosing a point in
the chromosome chains at random and exchanging the data to the right o f this point between
the parent chromosomes. In the two-point crossover the data between two randomly selected
points is swapped while in the multiple-point crossover data is exchanged at random between
the two parent chromosomes. Higher ranked strings will be more likely to participate in the
crossover and thus form new chromosomes.
The crossover operation is a random but structured information exchange between
chromosomes and represents the essential tool in local searches, (i.e., in exploring points
within the hyperplanes already represented in the population) [105]. However, crossover
alone would produce convergence in local extrema and, to explore other points in space and
avoid “premature convergence,” the mutation operator is introduced in the genetic algorithm.
A mutation is the random change of a gene from one value to another. Mutation is
carried out with a user-defined probability and according to the statistical rules implemented
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115
in the genetic algorithm program. Mutation has a very important role in the search process
ensuring variability in the population and hence, avoiding the entrapment of the algorithm in
local extrema of the merit function.
Generate initial
population
Calculate R or T
vs. A, or 0 using
RCW Analysis
Calculate merit
function
Rank according to
merit function
Generate new
subpopulation (by
mutation and
crossover)
Retain portion of
current population
(Selection)
Stopping
criteria
are met ?
Optimum filter
parameters
Figure 5.1. Flow chart of a genetic algorithm using rigorous coupled-wave
analysis for merit function evaluation [77]. The program uses the library
PGAPACK [110] to perform specific genetic algorithm operations such as
mutation, crossover, selection, ranking, and generation of new chromosomes.
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Single-point
crossover
—
»
A
nL nH
d
XL
xH
Two-Doint
crossover
A
nL nH
d
xL
xH
HHHBHI nH 1 d
A
xL W m
nL H H B H H
xH
Uniform
m
1 A 1nL
nH
d
Xl
nL
nH W
A f iS u S S f l
XH
d
ts a m
xh
----
IS 3 E
Xl
mBbhB
Mutation
A
nL nH
d
xL
xh
a
M U
nH
xL
Figure 5.2. Crossover and mutation operations illustrated for chromosomes
composed of 6 genes encoded as real numbers. In the 3 types of crossover
operations shown here genes of the parent chromosomes (white and grey) are
exchanged to yield new chromosomes. In the mutation operation, one or more
genes are randomly changed from one value to another.
The operators of selection, crossover and mutation are independent of the application
and only the merit function contains domain-specific knowledge. Another operator utilized in
some genetic algorithms to diversify the search is the restarting operator. After a number of
generations the best string is retained while all the others are discarded and the whole
population is reseeded as mutant variations of the best string. For the same purpose of
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117
avoiding premature convergence, other genetic operators introduce a random disturbance in
every chromosome of a population after a number of generations or when all chromosomes
have reached the same set of genes.
The genetic algorithm ends, after a user-determined fixed number of iterations, or
when the merit function has reached an extremum that is close enough to the desired value,
or when all chromosomes in a population have merit functions within a small enough range.
5.2.3
Review of genetic algorithm s in optics
Optics applications often require optimization of functions involving a large number
of variables, performing searches in extensive continuous and/or discrete parameter spaces,
tasks that may render gradient-based search routines ineffective but are managed successfully
by genetic algorithms. A variety of optimization problems employing genetic algorithms have
been reported in the literature in optical signal and image processing, diffractive optics, thinfilm optics, image formation and tomography [112-126].
Genetic algorithms have been utilized for optim ization of the quantized phase or
amplitude in each element of a two-dimensional array to achieve the desired far-field
intensity distribution. These display devices may find practical applications in optical
information processing, optical pattern recognition, optical interconnections and spatial
filtering. Yoshikawa et al. [113] have generated a 64x64 element array with 16 phase levels
that reconstructs a desired simple image upon illumination with a uniform, monochromatic
light source. The design of a spatial filter with a binary amplitude distribution used in a jointtransform optical correlator to discriminate between two given images has been reported by
Mahlab et al. [114]. Takaki et al. [115] improved the pattern discrimination ability in an
optical filtering system employing a liquid-crystal active lens by utilizing a genetic algorithm
to find the optimum 8-level phase distribution in a 64x64 pixel array. An optimization o f the
spatial amplitude filter in a generalized optical Fourier transform processor for a simple
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118
pattern recognition task was presented in reference 116. The problem of image deconvolution
utilizing a micro-genetic algorithm was addressed by Johnson and Abushagur [117].
Numerical examples illustrated the reconstruction o f two binary images from their
convolution without any prior knowledge about the two distributions except their regions of
support.
The design of diffractive optical elements may require the encoding of large non­
periodic phase arrays that would involve a large chromosome chain in the genetic algorithm
thus becoming computationally expensive. To overcome this problem Brown and Kathman
[118] reduced the number of variables by encoding the phase function of the diffractive
element by its Fourier coefficients and used a genetic algorithm with floating point variables
and variable mutation variance to design single and multiple, cascaded diffractive elements
for laser beam shaping.
An optical implementation of a genetic algorithm was demonstrated by Friedman et
al. [119] in a hybrid electro-optical system that exploits the inherent parallelism of optical
architectures. Allowing the vector-matrix multiplications and summations of binary elements
to be performed optically while the rest of the algorithm is executed by a digital computer can
thus increase the computational speed.
Promising results for application of genetic algorithms to tomography to achieve
increased accuracy over conventional reconstruction methods for a reduced number of
projections have been reported recently by Kihm and Lyons [120,121]. To improve the
convergence and reduce the computation time for the reconstruction of the unknown field
from its measured projections, a hybrid genetic algorithm was devised where each new
generation is partially created by the conventional genetic algorithm operators and partially
by a concurrent Simplex operator [121].
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119
Genetic algorithms have been proposed by Betensky [122] as useful tools in
optimization of Gaussian optics systems for improved lens design. To minimize the merit
function, the algorithm selects the best set of structural changes to be applied to a starting
design from a predefined list of permissible operators. Chen and Yamamoto [123] have
designed a camera lens with seven elements by encoding the physical parameters of each lens
in a chromosome. The authors found the definition o f the merit function to be more important
in convergence towards an optimum solution rather than the evolution strategy itself and used
a two level merit function that includes the desired lens specifications (focal lengths, fnumber and field of view). In a non-imaging application of genetic algorithms, Ashdown
[124] has discussed the optimization of the near-field and far-field photometric distribution
for the design of direct and indirect illuminating systems.
The design of diffractive optical elements using a genetic [125] and microgenetic
algorithm [112] was reported by Johnson et al. Fan-out gratings for optical interconnection
applications have been obtained with the desired intensity in each diffracted order by
encoding the phase of each cell of an NxN matrix of cells and using the fast Fourier
transform to calculate the diffraction pattern. A rigorous coupled-mode analysis was
employed to compute diffraction efficiencies for binary periodic structures
with
sub wavelength feature sizes and multiple layers. These corrugated dielectric gratings were
optimized with the microgenetic algorithm to achieve reflection and transmission
characteristics of various diffractive optical elements such as polarizing beamsplitters,
antireflection coatings, resonant reflectors, and fan-out gratings. The resonance reflector
generated with the microgenetic algorithm was found by optimizing a dual-surface corrugated
grating to yield 100% reflectance at one given wavelength. This subwavelength structure can
form the basis of narrow-band reflection and transmission filters exploiting the guided-mode
resonance effect. However, to obtain filter characteristics with specified linewidth, high
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120
efficiency, symmetrical lineshape and low sidebands, the optimization process must be
performed for a set of wavelength points spanning the entire spectral range of interest and
allowing the algorithm to search in a wider parameter space to find the period and fill factor
of the grating and the optimum thicknesses and refractive indices of a multilayer structure.
Michielssen et al. [126] have designed practical homogeneous-layer lowpass and
highpass optical filters using a real-code genetic algorithm. The dielectric constants and the
number o f homogeneous layers were fixed allowing the genetic algorithm to optimize the
thicknesses of the layers to obtain the specified response. This genetic algorithm
implementation can be applied to design other types of optical components.
However, the use of only homogeneous layers restricts the range of devices that can
be designed with the genetic algorithm. Furthermore, structures containing subwavelength
gratings as well as homogeneous layers can produce optical components such as bandpass
filters, antireflection surfaces, wave plates, and polarization-selective mirrors with
significantly fewer layers than the corresponding devices that employ homogeneous layers
only.
5.3 Genetic Algorithm Program for
M ultilayer W aveguide Gratings
5.3.1
Program description
A genetic algorithm program has been developed for optimization of diffractive optics
structures with multiple homogeneous and grating layers and incident TE polarized plane
waves [65,71,77]. The program employs rigorous coupled-wave analysis for calculation of
the reflected and transmitted diffraction efficiencies [84-86] and hence, for evaluation of the
merit function for the generated structures. The software library PGAPack [110] performs
specific genetic algorithm operations (chromosome generation, ranking, selection, crossover,
mutation, etc.)
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121
The algorithm seeks to find the physical parameters of the diffractive structure that
generates the spectral dependence of the zero-order reflected (or transmitted) diffraction
efficiency provided by the user in a reference data file. Alternatively, the optimization can be
performed for the angular dependence of the diffraction efficiency and at a fixed wavelength
of the incident light. The physical parameters to be found in the optimization process are the
grating period, the refractive indices and the thicknesses of the layers, the fill factors, and
relative spatial phase shifts of the gratings. The refractive indices o f the cover and substrate,
the angle of incidence (or the wavelength for angular dependence optimization), the
maximum number of layers, and the m inim um and m axim um values for the thicknesses, fill
factors, and grating period are required as input parameters to the program. The refractive
indices o f the candidate solutions are selected from a list o f discrete values supplied by the
user in a separate input file. All other physical parameters (grating period, fill factors,
thicknesses) are allowed to vary continuously within the ranges established by the user.
Therefore, the program seeks the minimum of the merit function in a mixed discrete
and continuous parameter space. This is a practical approach since in fabrication of
diffractive optical structures only a limited number of materials can be used in a given
spectral range while the thicknesses, fill factors, and the grating period can be varied
continuously within a range, and within the accuracy limitations o f the equipment. In some
applications one or more of the physical parameters may be fixed due to either fabrication
constraints (e.g., fill factor of the grating equal to 0.5) or user knowledge about the physics of
the problem (e.g., known grating period for center wavelength o f resonance filters). This
feature is included in the program and will expedite the search procedure by reducing the
dimension of the parameter space. A priori information can also simplify the search
procedure by reducing the range over which a parameter can vary dining optimization and
thereby reducing the total number of points in the parameter space.
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122
The physical parameters of the diffractive structure can be encoded as binary, binary
Gray, or real numbers [110]. For binary and binary Gray representations of real and integer
numbers the user must specify the number of bits for encoding the thicknesses, the grating
period, the fill factors and the refractive indices. The number o f bits allocated for each
variable determines the accuracy of the representation of real numbers. Increasing the
accuracy allows a better solution to be found but at the same time increases the total number
of points in the parameter space decreasing the convergence. The binary and the binary Gray
encodings allow the genetic algorithm to access and operate on individual bits of a gene,
instead of the gene as a whole as in the real encoding [110]. For instance, a single-point
crossover operation may take place with the crossover point in the middle of a gene in the
binary encodings but only between genes in the real encoding. The Gray binary encoding
differs from the binary encoding in that consecutive integer numbers differ by only one bit.
This difference induces different paths in the genetic algorithm optimization procedure. For
instance mutation of one bit in a gene produces an incremental change in the value of the
corresponding physical parameter if it is represented in Gray code but may lead to a large
variation in the case of binary encoding.
The program starts by randomly generating a population of chromosomes in the
specified encoding and range of values for each variable. As an example, figure 53(b) shows
the chromosome of a double-layer grating with its genes corresponding to the physical
parameters of the structure illustrated in figure 53(a) [65,71]. The chromosome has (5Nl + 1)
genes where N l is the number of layers of the diffractive structure. Each layer is assumed to
be a grating with the same period A, but with different refractive indices nn and nL,
thicknesses d, and coordinates (relative to the grating period) of the high-refractive index
region o f each grating XL and XH. To select the refractive indices in each layer, the algorithm
generates integer random numbers, which represent pointers to refractive index values in the
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123
corresponding input file. Homogeneous layers are generated either when the same refractive
index is selected for both regions of the binary grating, or when the fill factor defined as (X h
—X l) is smaller or greater than the values specified by the user in the input file fmm and fmaxFor (X h - X l) < fmin the layer is considered as homogeneous with the refractive index nL,
while for (X h - X l) > fmax the layer is homogeneous with refractive index Hh- Different
values o f X l in different layers generate phase-shifted layers. The number of layers N l is
fixed and provided by the user. However the program can analyze structures with fewer
layers whenever it selects a layer thickness that is smaller than the minimum layer thickness
(from the input file). In this case the thickness is set to zero and the number of layers
decreases by one.
The number sequence forming a population of strings is unique for each run of the
program. A feature is included that allows the same number sequence to be generated each
time for debugging or reproducibility purposes [110]. In binary representation, each bit of a
string has equal probabilities of being set to 0 or 1. In the real encoding the genes are set to a
value selected uniformly within the user-specified range.
The population (i.e., the total number of chromosomes generated in the beginning,
which is to remain constant after each iteration) is established by the user depending on the
dimension of the search space and the length of the chromosome. An increased number of
genes and/or a large range of variation for the genes may require a large population for
effective optimization. Operating with larger populations, the genetic algorithm is more likely
to find the global minimum of the merit function since it searches more regions of the space
simultaneously. However, this is achieved at the expense o f an increase in computational
time, which imposes a practical limitation on the population size.
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124
Incident
wave
■
Reflected
▲ wave
Xl.1 xh. i
di
dz
(a)
Chromosome
A
nm
nH,i
di
xui
x H,i
nu2
nH.2
d2
XL.2
Xh .2
genes (alleles)
(b)
Figure 5.3. Example of a diffractive structure consisting o f two gratings in two
separate layers, with physical parameters shown in (a) and corresponding
chromosome represented in (b). The chromosome is a candidate solution in the
optimization process. A set o f chromosomes forms a population. The total
population of chromosomes at a given iteration is called a generation. In this case,
the parameters to be optimized are the grating period A, the thicknesses di, d 2 ,
refractive indices, nm, nn.i, n^ 2 , n ^ , and relative positions of the high-refractive
index materials within a grating period, X y , Xh . i , Xl .2, and x h .2-
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125
The initially generated population is evaluated by calculating a merit function for each
chromosome as the deviation between the synthesized value of reflected (or transmitted)
zero-order diffraction efficiency and the desired one. The genetic algorithm searches for the
global minimum of the following merit function
where DEoA,i are zero order reflected (or transmitted) diffraction efficiency values calculated
with rigorous-coupled wave analysis (chapter 2) for the structure generated by the genetic
algorithm, DEre^ are the reference data points, M is the total number of target values, Wi are
the weight factors, and n is the power index of the merit function. The target points represent
either a wavelength or an angular dependence of a diffraction order efficiency. Any
diffraction order may be selected for optimization but for the applications of interest to this
work concerning only zero-order gratings, the zero-order efficiencies are utilized. The power
index of the merit function can take any integer values but in thin-film optics optimization
routines the most common value is n = 2. Different values of n can affect the optimization
results due to changes induced in the relative contributions of individual target deviation
points IDEca,; —DEref.il to the merit function. For larger values of n higher deviations will be
emphasized and the merit function becomes more sensitive to nonequal deviations forcing the
genetic algorithm to find a more uniform approximation to the reference data [96].
Once the merit function has been calculated, the chromosomes are ranked from the
best-fit to the least-fit, with the best-fit possessing the lowest merit function. A number of
chromosomes are retained while others are replaced by newly generated chromosomes. The
selection mechanism typically used is the tournament selection consisting in retaining the
best chromosomes o f a population. Other selection mechanisms such as probabilistic
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126
tournament (with an associated probability of selecting a chromosome), proportional and
stochastic universal selection can also be chosen for use in the optimization procedure [110].
The number o f chromosomes replaced is an input parameter to the program and has
an important influence on the optimization progress. A high percentage of chromosomes
replaced provides more new points for fitness testing which is beneficial in the search
procedure, but it will also increase the computation time. It is also possible that a large
replacement will cause the elimination of certain chromosomes that, after subsequent
crossover and mutation, would have generated the optimum solution. Therefore, several
convergence tests need to be performed to establish the optimum population replacement for
a specific problem [77].
The new population that replaces the discarded chromosomes is formed by crossover
and mutation of the chromosomes that are retained from the old generation [110]. The
chromosomes that survive become parents and generate enough chromosomes to maintain the
total population constant from one generation to another. The algorithm allows the user to
decide whether a string can undergo both crossover and mutation or just one of the two
operations.
In the case where either crossover or mutation is carried out the probability of going
towards one or the other operation is decided by a random logical variable which has an
associated flip probability (provided in the input file) of returning a logical value “true.” A
probability of 0.5 corresponds to flipping an unbiased coin. In the case when both mutation
and crossover are performed, the random logical value of the flip probability determines
whether crossover is executed first followed by a mutation operation or vice versa [110].
Crossover takes place by pairing the chromosomes selected to survive from the old
generation into the new one from top to bottom o f the list (with best-ranked strings at the
top). The crossover operation is performed with a probability defined by the user in the input
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Ill
file. The algorithm has the options o f single-point, two-point or uniform crossover (figure
5.2). For the latter type of crossover, the probability of swapping two parent bits (or genes in
case of real encoding) called uniform crossover probability, must be specified in the input
file.
Mutation takes place with a probability defined by the user in the input file. For
binary encoding, mutation is performed by replacing one or more of the bits of a chromosome
with its complement. For real encoding, the mutation occurs for one or more genes o f a
chromosome and can be one of several types: “range,” “constant,” “uniform,” or “Gaussian”
[110]. If the mutation is of the “range” type, the gene will be replaced with a number selected
with equal probability from the allowed range of variation for the gene. In the other three
mutation types the gene g is replaced by g ± p x g where the value o f p is determined
differently for each mutation operator. For constant mutation, p is a constant provided by the
user. Uniform mutation occurs when p is selected uniformly from an interval [0 —M„] where
Mu is an input parameter. In the Gaussian type of mutation p is generated by a Gaussian
distribution with mean 0 and standard deviation a given in the input file.
After generating a new pair of chromosomes through crossover and/or mutation, the
algorithm performs a verification to determine whether they are different from their parent
chromosomes. If the new chromosome is identical to the parent chromosome, the mutation
operator is applied to the new chromosome until at least one mutation has occurred.
After evaluation of the newly generated chromosomes and ranking the new
generation, the process of selection, crossover and mutation is repeated until the stopping
criteria is met. This can be determined by a fixed number of iterations, no change in the best
string after a number of iterations, or when certain fraction of the population has the same
merit function [110]. The genetic algorithm also has the restarting option by which the best
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128
string is kept and all others are generated as mutants of the best string. The number of
iterations between restarting operations is defined by the user.
The program prints the best N0„t chromosomes and their corresponding merit
functions in the output file, where Nout is an input parameter. By printing a number of the top
chromosomes, the user can asses the distribution of solutions and hence the degree of
convergence of the algorithm. A large dispersion in the gene values o f the final chromosomes
indicates that the algorithm has not yet converged and changes need to be made in the input
parameters of a future run. Typical changes would be to try more iterations, larger
populations or impose more constraints according to a priori knowledge about the physics of
the problem [111]. However changes in other genetic algorithm input parameters can also
improve the convergence and the effectiveness of the optimization.
5.3.2
Convergence tests
The genetic algorithm developed in this work has a general applicability. The genetic
operators and optimization procedure can be utilized in any optimization task involving
multilayer structures containing gratings and homogeneous layers, with minor modifications
pertaining to the encoding and decoding of the chromosomes. The merit function evaluation
subroutines can be applied to optimization of any structure that can be modeled with the
rigorous coupled-wave analysis.
However, the optimum set of the program-input parameters may be problem specific
due to the dimension of the solution space and the particular variation o f the merit function in
the parameter space. To determine the influence of the input parameters on the optimization
procedure and final result and to find some guidelines for selecting the appropriate set of
input parameters for a specific application, it is important to study the evolution of the
optimization process (i.e., the convergence) for various starting conditions [65,77].
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129
In this section, the convergence of the merit function is studied as a function of key
genetic-algorithm parameters such as the population replaced at each iteration, mutation
probability, type o f encoding, number of generations, population size, for the same problem.
In all tests discussed here, the program is required to design a single-layer guided-mode
resonance reflection filter with the response specified in the input file by the spectral
dependence of the zero-order reflection diffraction efficiency. This reference data is
generated with the rigorous coupled-wave theory for a single-layer grating with the following
physical parameters: grating period A = 314 nm, thickness d = 134 nm, fill factor f = 0.5,
refractive indices of the grating: nn = 2.1 and m, = 2.0, refractive indices of the cover and
substrate: nc = 1.0 and ns = 1.52, and normally incident, TE polarized plane wave (figure 3.2
and 3.3). The optimization is performed in terms of the layer thickness, fill factor and
refractive indices of the grating over a wavelength range 0.546 - 0.554 nm. The grating
period is fixed at the value A = 314 nm, the cover and substrate refractive indices have
constant values of nc = 1.0 and ns = 1.52, respectively, and the incident angle is set at 0 = 0°.
The allowed range for fill factor optimization is between 0.1 - 0.9 and the thickness range is
50 - 350 nm). Throughout the tests the algorithm uses the same set of 13 refractive indices
with values from 1.3 - 2.5 in increments of 0.1. The materials are assumed to be lossless
although the program can handle lossy grating structures as well.
Comparing the merit function values for tests with the three different types of
crossover, it was found that multiple crossover yields the best results in comparison with the
single-point and two-point crossover operators. The crossover probability was maintained at
0.8 and the uniform crossover probability was 0.5 for all tests performed. Other genetic
algorithm parameters kept constant for all tests, were the flip probability equal to 0.5, the
tournament selection type, and maximum iteration as the stopping criterion.
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130
The tests performed with populations of 500 and 1000 chromosomes indicate that
although fast convergence and low merit function values are also possible w ith a smaller
number of chromosomes for certain values of the genetic algorithm parameters, the larger
population is generally expected to yield lower merit functions for all other parameters being
constant. However, in some cases the increased computation time for larger populations may
not be rewarded by a substantial decrease in merit function and an optimum population has to
be determined for a typical chromosome length and search space. It has been observed that
larger populations also provide the algorithm with less sensitivity to the other input
parameters and therefore fewer trials are required to determine the optimum set of genetic
algorithm parameters.
A more detailed investigation of the convergence sensitivity to genetic algorithm
parameters was performed for the population replacement and the mutation probability,
contrasting real versus binary and Gray encoding performance [77]. Hence, the population
size was fixed at 500, the crossover type was uniform with probability fixed at 0.8, and the
number of generations and mutation probability were varied. For the real encoding the
mutation type was chosen to be Gaussian with standard deviation <7= 0.1. In all cases the
newly created strings were specified to undergo both crossover and mutation. W hen using
binary or Gray encoding 10 bits were chosen to represent the thickness, 10 bits to represent
the fill factor and 4 bits to represent the pointer to the set of materials. In this case uniform
crossover with probability 0.8 was chosen. The number of generations was taken to be 400,
to ensure that convergence had been reached.
The results are illustrated in figures 5.4-5.6, after performing at least four runs for the
same set-up, with different random number sequences in the genetic algorithm functions to
seed the population, in order to construct a statistically significant sample of the outcomes.
Figure 5.4 illustrates the behavior of the merit function after 400 generations as a function of
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131
number of chromosomes replaced per generation, showing the spread over the runs. It is
noted that the real encoding produces smaller m erit functions, therefore indicating that for
this problem it is better suited than the others. Overall, the sensitivity to replacement values
over the range 50 - 250 chromosomes is not very strong when viewing the total distribution
o f results. Li the case of real encoding, figure 5.4 (c) indicates, on average that the lowest
merit function is obtained for a replacement value of 150. In the case o f binary and Gray
encodings, slightly smaller merit functions are achieved for the replacement value o f 250.
However, this replacement also produces a large spread o f the residuals indicating that a poor
solution can obtained as well as a good one.
Figure 5.5 shows the behavior of the residual as a function of increasing values of
mutation probability. For the binary and Gray encoding a replacement value of 50 was used,
whereas for the real encoding the value was 200. Otherwise, the same genetic algorithm
parameters as discussed above were retained. The binary and Gray encoding results are not
very sensitive to the value of mutation probability. However, on average, the binary encoding
favors lower mutation probabilities than the real encoding which benefits from an increase in
the value of this parameter to 1. In the real encoding the mutation operator is applied to the
gene as a whole which, for the structures studied here, are represented by 4 numbers. In the
binary encoding mutation takes place at the level o f individual bits (e.g., for a representation
of the filter parameters (d, f, nn, ni,) with (10, 10, 4, 4) bits, the mutation is applied
individually to all of the 28 bits of the chromosome with the specified mutation probability).
Therefore, for the mutation to be active in the search procedure with the real
encoding, the mutation probability must be higher than in the binary case where a low
probability is compensated by the larger number o f elements (bits) to which it is applied. The
differences observed between the binary and the real encodings are also due to the different
manner in which mutation is carried out. In the real encoding, the Gaussian mutation does not
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132
change the gene value by an arbitrarily large amount as in the binary case, but applies a
random change in a lim ited range (determined by a) around the value of the gene.
Thus, in the real encoding, mutation performs a more localized search alongside the
crossover operator before shifting to a different region of space and is able to find a global
minimum with greater accuracy. The results o f figure 5.5 illustrate that mutation probabilities
of 1 are detrimental in the binary encoding due to the rapid changes occurring in the best
chromosomes which prevents the fine tuning performed by crossover. On average, very low
mutation probabilities (0.001) produce equally poor results by precluding the algorithm from
exploring new regions in the parameter space.
Figure 5.6 reports the convergence history for three different values of number of
chromosomes replaced at each generation, (i.e., 50, 150 and 250). When increasing this
number, note that the value of the merit function at convergence is reached after a decreasing
number of generations. While the computational burden increases with the increase of the
chromosomes replaced per generation, this is offset by the ability to reach convergence in
fewer generations.
Therefore, the total amount of calculations necessary to reach convergence is similar
in all cases. Note that the real encoding generally provides lower merit functions than the
binary and Gray encodings. The high sensitivity of the guided-mode resonances to structural
parameters is equivalent to a mgged search space for the genetic algorithm with multiple and
narrow local extrema. Therefore, the improved fine-tuning performed by the mutation
operator in the real encoding leads to superior convergence results.
The number of reference reflectance vs. wavelength points has been found to
significantly influence the final result of the optimization process. It is well known from
homogeneous thin-film optimization techniques that more “targets” typically lead to
improved results due to the additional information supplied by the user [94].
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133
The number of data points is an even more critical parameter in the design o f grating
devices that exhibit sharp variations in the reflectance and transmittance spectral dependence.
In this case the global minima of the merit function can not be reached if insufficient target
data is provided, hi reference 112, for instance, the micro-genetic algorithm uses only one
reference point with zero-order reflectance equal to 1 at the wavelength X = 1.0. The resulting
structure exhibits an alm ost 100% peak at A. = 1.0 but other high-efficiency peaks are also
present in the proximity of the desired peak thus limiting the filter range. The side peaks can
be eliminated in the optimization process by providing the merit function with more reference
reflectance points, hi the present work, between 40-80 reference data points are used. The
major drawback of the increased number of target data is the increased computation time.
The distribution of reference reflectance points in the spectral range of interest is also
important in the search for an optimum design. In the case of the rapidly varying reflectance
characteristics studied here, it is advantageous to use unequally spaced data points with more
reflectance values in the resonance spectral region and less in the sidebands. Utilizing this
type of distribution, the algorithm is able in all tests (performed with the reference data o f
figure 5.7) to find a resonance and typically within ± 0 .1 nm of the reference reflectance
peak.
A different method to emphasize some reference points over others is to introduce
different weight factors. Increased values of the weight factors in some reference points will
raise the accuracy in these spectral regions at the expense of a larger target deviation in
wavelength regions considered of lesser importance.
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134
0 .2 5
0 .2 5
RESIDUAL FUNCTION
Binary encoding
Gray encoding
0.2
0 .1 5 -
0 .1
0 .1 5 -
-
0 .0 5 -
0 .0 5 -
0
r 1' I' ■
"
50
10 0
1 50
200
250
300
50
REPLACED POPULATION
10 0
150
200
250
300
REPLACED POPULATION
(b)
Real encoding
0 .1 5 -
p
0.1
0 .0 5 -
111' ' I ' 9 ' 11111 1 t
50
100
1 50
200
250
300
REPLACED POPULATION
(C)
Figure 5.4. Merit (residual) function dependence on the population replaced per
generation after 400 iterations. Each dot corresponds to a different random
number seeding o f the initial population. Results for binary (a), Gray (b) and real
(c) encoding of the waveguide-grating parameters (thickness, fill factor, high- and
low-refractive indices of the grating) are illustrated. The mutation probability is
fixed at 0.005 for binary and Gray encoding and 0.5 for the real encoding.
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135
0.25
0.25
RESIDUAL FUNCTION
Binary encoding
Gray encoding
0. 2 -
0. 2-0
H
0
0.15
0. 1 -
0.15
o
3 o-H
o
a
o
O g
8
o
C/3
0.05-
0.001
0.05-
0.01
0.001
MUTATION PROBABILITY
0.01
0.1
MUTATION PROBABILITY
(a)
(b)
0.25
0.2 -
§
O
5*
3
a
Real encoding
_ o
8
° §
o
0.15o.i
C/3
S 0.05 H
o
o
o
o
o
g
o
<
g
T
0.001
0.01
g {
<
0.1
MUTATION PROBABILITY
(C)
Figure 5.5. M erit (residual) function dependence on the mutation probability after
400 iterations. Each dot corresponds to a different random number seeding o f the
initial population. Results for binary (a), Gray (b) and real (c) encoding o f the
waveguide-grating parameters (thickness, fill factor, high- and low-refractive
indices of the grating) are illustrated. The population replaced per generation is
fixed at 50 for binary and Gray encoding, and 200 for the real encoding.
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1
136
0.3
03
0-25'
025
RESIDUAL
FUNCTION
z
\
1
0 . 1-
0.05
r
..
o
g
Binary
encoding
replace 250
Binary
encoding
replace 150
02-
3i 0^
9 o.i
Binary
encoding
replace 50
L
0.05
■11' ■111
>' I " ' 11 " " I " ' 11" " I " " I " " T '
SO 100 150 200 250 300 350 40C
ITERATION
50
100 150 200 250 300 350 40C
ITERATION
100 150 200 250 300 350 40C
ITERATION
(a)
RESIDUAL
FUNCTION
03-
03-
Gray encoding
replace 150
Gray encoding
replace 50
0.15-
03
0.15-
0.15-
S 0.1-
0. 1-
0.050
Gray encoding
replace 250
025-
0.05-
0.05-
50
100 150 200 250 300 350 4 a
ITERATION
0
ITERATION
50
100 150 200 250 300 350 4 a
ITERATION
(b)
encoding
replace 50
encoding
replace 250
encoding
replace 150
0.15-
RESIDUAL
FUNCTION
0.25-
0.05'T " '" T
100 150 200 250 300 350 4 a
ITERATION
<11■I■|'l l■I
100 ISO 2 a 250 300 350 4 a
ITERATION
ia
150 2a 250
ITERATION
3a
350
(c)
Figure 5.6. Convergence history for binary (a), Gray (b), and real (c) encoding of
the waveguide-grating parameters. In each plot the different curves correspond to
a different random number seeding of the initial population. The replaced
population values per generation are 50 (teft column), 150 (center column), and
250 (right column).
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4a
137
5.4
Reflection Filters
Examples of some of the guided-mode resonance reflection filters found in the tests
described in the previous section are presented below. The input reference data consists of a
set of 50 unequally distributed reflectance-versus-wavelength points with equal weight
factors, selected from the reflection filter response of figure 3.3. Figure 5.7 illustrates the
filter response of a grating generated by the genetic algorithm optimization routine [65]. The
optimization is performed in terms of the refractive indices, the thickness and the fill factor of
the grating with the filter input parameters and data files specified in section 5.3.2. The
genetic algorithm parameters use to generate this result are: binary encoding where the
number of bits for (d, f, nn, nO are (10, 10, 4, 4), population = 500, replaced population =
100, generations = 100, mutation probability = 0.01, merit function index n = 2. The
algorithm is able to find the refractive indices that generated the reference data, the thickness
with a 0.15% error, and the fill factor with a 9.5% error. The merit function after 100
iterations is MF = 0.027. The difference between the reflectance of the genetic algorithm­
generated grating and the reference reflectance, henceforth called the “deviation from target,”
is illustrated in figure 5.8. The deviations are seen to arise mostly from the region of the
resonance, which is explained by the high sensitivity of the peak location and linewidth to
slight changes in layer thickness and fill factor.
Alternative structures that fit the reference data with good approximation can also be
generated with the genetic algorithm program by utilizing different genetic algorithm
parameters or the same input parameters but a different seeding o f the initial population [65].
Real encoding has been used to produce the results shown in figures 5.9 - 5.14.
Figures 5.9 and 5.11 illustrate the response of two gratings generated with the same
random-number seeding of the initial population, and the same input parameters (population
= 500, replaced population = 200, generations = 200, mutation probability = 0.5) except for
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138
the different merit function index. The filter in figure 5.9 was generated using an index n = 2
resulting in a merit function value MF = 0.026 while the structure with the response in figure
5.11 has an index n = 1 and merit function value MF = 0.029.
The deviations from target corresponding to figures 5.9 and 5.11 are presented in
figures 5.10 and 5.12, respectively. The effect of the merit function index on the search path
taken by the genetic algorithm in the two cases can be observed by comparing the deviations
from target. A power index n = 1 stresses all deviations between the reference and the
generated reflectance points equally when calculating the merit function expression (5.1). For
higher power indices, larger deviations such as those occurring in the resonance region o f
figure 5.11 are emphasized in the merit function calculation and the algorithm seeks to
minimize the maximum deviations. Consequently, the maxim um deviation is lower in figure
5.10 (max[DEGA4 - DEref.i] = 0.049) than in figure 5.12 (max[DEGA,i —DEref.J = 0.088).
The filter with the response in figure 5.13 and the deviation from target in figure 5.14
also exemplify this effect. The filter illustrated in figure 5.13 was generated with the same
input parameters as the filter of figure 5.9 except for a mutation probability equal to 1.
Compared to the filter of figure 5.9, the merit function value is lower (MF = 0.021) and the
sidebands are closer to the reference reflectance data.
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139
0.9
0.8
Binary encoding
MF index n = 2
generated
by GA
reference
0.5
0.4
0.3
0.2
o.i0.546
0.548
0.55
0.552
WAVELENGTH (flm )
0.554
Figure 5.7. Reflectance of a guided-mode resonance filter generated by the
genetic algorithm and reference reflectance used as input data to the program. The
parameters of the grating used to generate the reference reflectance are: nn = 2.1,
nL = 2.0, d = 134 nm, f = 0.5. The parameters of the grating generated by the
genetic algorithm are: njj = 2.1, ^ = 2.0, d = 133.8 nm, f = 0.547.
0.15
0 . 1-
0.05-
u
-0.05- 0 . 1-
-0.15-
0.2
0.546
0.55
0.552
0.548
WAVELENGTH (lim )
0.554
Figure 5.8. The difference between the two reflectance curves of figure 5.7.
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140
l0.9
0.8 r
W°-7^
u
Real encoding
MF index n = 2
r
J
*
generated
by GA
* reference
«
I 0-6'
O 0.5 d
3
:
i°-4=
0 .3 -j
0. 2 ~
0.H
I
0.546
0.548
0.55
0.552
WAVELENGTH (fim)
0.554
Figure 5.9. Reflectance of a guided-mode resonance filter generated by the
genetic algorithm and reference reflectance used as input data to the program. The
parameters of the grating used to generate the reference reflectance are: nn = 2.1,
nL = 2.0, d = 134 nm, f = 0.5. The parameters of the grating generated by the
genetic algorithm are: nn = 2.5, nL = 2.4, d = 221.1 nm, f = 0.369.
0.05
0.04 -i
0.03
0 .0 2 - .
5
g
o .o i-i
°:
g -0.01-
| -0.02-i
-0.03 -■
-0.04
-0.05
0.546
0.552
0.548
0.55
WAVELENGTH (pjn)
0.554
Figure 5.10. The difference between the two reflectance curves of figure 5.9.
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141
l
Real encoding
MF index n = 1
0.9
0.8
•
^
■ generated
by GA
reference
« 0-7
Z 0.6
So.5
£E 0.4
i ■
0.3
0.2
0.1
0
• ■
« «
1— '— I— I— '— '— 1
0.546
0.548
1
I— r - ■ 1
ii
1—
0.55
0.552
WAVELENGTH (pm )
0.554
Figure 5.11. Reflectance of a guided-mode resonance filter generated by the
genetic algorithm and reference reflectance used as input data to the program. The
parameters of the grating used to generate the reference reflectance are: nH = 2.1,
nL = 2.0, d = 134 nm, f = 0.5. The parameters of the grating generated by the
genetic algorithm are: nn = 2.0, nL= 1.6, d = 278.5 nm, f = 0.811.
o.i
0.082
0.06-
1
0.04-J
E0
0 .0 2 -
o-
0.02
0.546
0.55
0.552
0.548
WAVELENGTH (pm )
0.554
Figure 5.12. The difference between the two reflectance curves of figure 5.11.
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1
generated
by GA
0.9-
Real encoding
0 . 8 - MF index n = 2
0.7- Mutation prob. = 1
reference
0. 6 -
t-H
uw 0.5jU.
0.4-
■:
M
0.30.2 -
0. 1-
0
0. 546
• «
1 '~r ' T
0.548
0.55
0.552
WAVELENGTH (}im)
■T
0.554
Figure 5.13. Reflectance of a guided-mode resonance filter generated by the
genetic algorithm and reference reflectance used as input data to the program. The
parameters of the grating used to generate the reference reflectance are: nn = 2.1,
nL = 2.0, d = 134 nm, f = 0.5. The parameters of the grating generated by the
genetic algorithm are: nn = 2.5, nL = 2.4, d = 220.8 nm, f = 0.583.
0.05
0 .041
M
0.03:
EO
0 .0 2 -
O
0.01o-
-
0.01
0.546
0.552
0.548
0.55
WAVELENGTH (lim)
0.554
Figure 5.14. The difference between the two reflectance curves of figure 5.13.
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143
5 i
5.5.1
Transmission Filters
Double-layer/double-grating filters
It has been shown in chapter 3 that guided-mode resonance transm ission filters
require a larger number of layers than reflection filters in order to obtain low sideband
transmittance. In practice, this can increase interface scattering and volume absorption losses.
Losses in multilayer waveguide gratings cause a reduction in peak efficiency of the filter [68]
which can offset the advantage brought by the guided-mode resonance devices of obtaining a
high-efficiency, narrowband response. Furthermore, multilayer waveguide-grating stacks
support multiple resonant modes, which may limit the filter’s free spectral range. Thus, it is
of interest to find waveguide gratings with fewer layers that nevertheless exhibit low
transmittance in the spectral region outside the resonance. This section demonstrates that
highly modulated waveguide gratings with reduced number of layers can provide both a high
peak transmittance through the guided-mode resonance effect, as well as low sideband
transmittance in a limited spectral range outside the filter linewidth.
The capability of the genetic algorithm program to design narrow bandpass filters
with given characteristics is illustrated by an example in the visible region (figure 5.15) that
requires the optimization of all physical parameters of a 2-layer diffractive device (figure
5.3(a)) to yield a Lorentzian line centered at Ac = 0.55 pm, with a linewidth (FWHM) AX =
0.5 nm (relative linewidth, AA/Ac = 0.09%). The solution is sought in the physical parameter
space defined by the following ranges for grating period 0.28 pm < A < 0.35 pm, thickness
0.05 pm < d < 0.35 pm, and fill factor 0.1 < f = Xr -
xl <
0.9, and using a set o f refractive
index values ranging from 1.3 — 2.5 in increments of 0.1. hi all optimization examples
presented in this chapter, the incident angle, cover and substrate refractive indices are fixed.
Figure 5.15 illustrates the reference Lorentzian curve and the response o f the structure found
by the genetic algorithm program with the physical parameter values given in the figure
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144
caption. The choice of a Lorentzian lineshape for the filter examples presented in this
dissertation is justified by the tendency of the guided-mode resonance filters with
symmetrical response around the peak wavelength to attain this type of wavelength
dependence. The reference response is defined at 51 wavelength values covering the
optimization wavelength range 0.545 - 0.555 pm. A close fit between the desired and the
genetic algorithm-generated transmittance is obtained by assigning higher weight factors to
the wavelength points at and around the resonance. The most significant genetic algorithm
parameters of the optimization routine have the following values: the population of
chromosomes is equal to 1000, with 400 being replaced at each iteration, the crossover
probability is 0.8, and the mutation probability is 0.001. A Gray code binary representation of
the chromosomes was utilized. The filter response over a larger wavelength range is
presented in figure 5.16. The spectral range of low sideband transmittance is limited by the
available low-loss materials with highest refractive indices in the visible wavelength region.
A direct comparison between the transmission filters designed with the “direct”
method described in chapter 3 and the genetic-algorithm-generated filters is presented in the
following. Figure 5.17 illustrates a transmission filter at 0.55 pm with 0.23 nm linewidth
obtained by the “direct” method of design described in section 3.3.2. The device is obtained
by replacing the center layer of a thin-film, all-homogeneous layer structure with a diffraction
grating. The homogeneous layers have quarter-wave thicknesses and a high refractive index
difference (nodd = 2.35 and n^eii = 1-38) to attain low sideband transmittance. The grating
parameters are chosen to maintain the high-reflectance properties of the structure. Therefore,
the grating possesses a thickness equal to a quarter-wave at 0.55 pm and refractive indices
that maintain the high-refractive index nature of the center layer (with nn = 2.5 and nL = 2.1).
The same filter performance is sought with the genetic algorithm program. The
genetic algorithm reference transmittance is a Lorentzian line with 0.2 nm linewidth defined
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145
at 53 points in the wavelength range 0.545 - 0.555 Jim. The refractive index input file
consisted of 25 values between 1.3 - 2.5 in increments o f 0.05. The grating period, fill
factors, and thicknesses were optimized in the ranges 0.28 pm < A < 0.35 pm, 0.1 < f = xH —
xL < 0.9, and 0.01 pm < d < 0.4 pm, respectively. The genetic algorithm found a structure
with two phase-shifted gratings (in different layers), with different fill factors employing the
same materials in both layers (nn = 2.35 and nL = 1.65). The transmittance of the device
produced by the genetic algorithm is illustrated in figure 5.18. This result was obtained with a
real encoding, a population o f 3000 chromosomes, with half replaced at each iteration, a
Gaussian mutation probability of 1 with <7=1, and a crossover probability of 0.8, after 180
iterations. The device found by the genetic algorithm achieves, with only two layers, a
slightly higher peak (98.7%), a more symmetrical lineshape, and lower transmittance (< 0.5%
in the range of figure 5.18) than the structure designed with the “direct” method. The
difference between sideband responses o f the two filters within the 10-nm optimization range
is illustrated in figures 5.19 and 5.20 where the transmittances are represented in dB. The
transmittance of the two structures over a larger wavelength range (0.45 - 0.65 pm) is plotted
in figures 5.21 and 5.22. An important difference between the two types of devices is
emphasized by the dashed curves of figures 5.21 and 5.22 illustrating the response of the
equivalent homogeneous structures. These are obtained by replacing each grating with a
homogeneous layer possessing the same thickness and a refractive index equal to the average
refractive index of the grating. The homogeneous 9-layer device exhibits a response almost
overlapping the transmittance of the guided-mode resonant device except for the resonance
regions. In contrast, the sideband transmittance of the genetic-algorithm structure of figure
5.21 is much lower than that of the corresponding homogeneous-structure. In both cases, the
filter range is limited by the adjacent resonant peaks. The peak at 0.532 pm is due to a
Rayleigh resonance. However, the reduced number of layers employed by the genetic-
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146
algorithm device allows only 2 resonant modes as opposed to 5 modes in the 9-layer
structure.
The genetic algorithm routine is able to find several filter structures with 2 grating
layers exhibiting similar response over the 10-nm optimization range. Figures 5.23 and 5.24
illustrate one such transmission filter example with -0 .2 nm linewidth where the two
resonance peaks are spaced further apart than in the solutions of figures 5.21 and 5.22. Thus,
a larger free spectral range is achieved with sideband transmittance below 10% over a range
of -100 nm, and below 1% within a -80 nm spectral range.
5.5.2
Single-layer/single-grating filters
A more practical device with a single grating layer as shown in the inset of figure
5.25, is employed for the design of an infrared bandpass filter, centered at Ac = 10.6 pm with
filter linewidth AX, = 12.7 nm, and relative linewidth, AXfkc = 0.12%. The simpler geometry
compared to the visible-range filters presented in section 5.5.1 is achieved with higher
refractive indices of the grating (nt* = 4.0 and nL = 2.65) on a substrate with a lower refractive
index (ns = 1.4). The optimization is performed at 25 wavelength points in a 40 nm
wavelength range around the central wavelength using a Lorentzian function as a reference
response.
The physical parameters optimized are the two refractive indices, the period, fill
factor, and thickness of the grating. The refractive indices were selected from a set of 8
values, n = 1.0, 1.3, 1.35, 1.7, 2.15, 2.35, 2.65, and 4.0, corresponding to materials
transparent in the operating wavelength range (air,
Y F 3, C aF 2 , Y 2O 3,
ZnS, ZnSe, CdTe, and
Ge). The grating period, thickness, and fill factor are sought in the intervals: 5.0 pm < A <
7.0 pm, 0.05 pm < d < 6.0 pm, and 0.1 < f < 0.9, respectively. A real encoding of the filter
parameters was employed with population of 1000, replacement of 400, crossover probability
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
of 0.8, and a Gaussian mutation type with mutation probability of 1.0 and standard deviation
of 0.1.
The higher reffactive-index materials available at these longer wavelengths enable the
design of bandpass filters with lower sidebands on a larger wavelength range than in the
previous filter example. Although the optimization was performed only over a 40 nm range,
the calculated filter o f figure 5.25 exhibits sideband transmittance below 10% over a
wavelength range exceeding 4 fim (corresponding to a relative spectral range (A.2 - \\)/A c =
37.7%), and under 1% over the range A.2 - A,i = 2.21 |im, (corresponding to a relative spectral
range (A,2 - X\)fkc = 20.86%). This is more clearly visible in figure 5.26 presenting the
transmittance of the same filter in dB. Lower sideband response may be obtained with
optimization performed over a wider wavelength range at the expense of increased
computational time.
5.5.3
Triple-layer/single-grating filters
In the two bandpass filter examples presented above, the only constraints imposed on
the search algorithm were the number o f layers (restricted to 1 or 2) and the discrete
refractive index values. Fabrication of bandpass filters may impose additional constraints on
the materials that can be utilized, as well as practical limitations in achieving the prescribed
geometrical parameters of the waveguide grating. Constraints that lead to realistic filter
designs can be incorporated in the genetic algorithm search procedure and one such example
is provided by the filter o f figure 5.27. The purpose of this bandpass filter design is to obtain
a filter centered at Ac = 1550 nm with a linewidth AA. = 0.5 nm (AA/Ac = 0.03%). and a low
sideband transmittance in the range 1530 - 1570 nm using only Si and Si02 as grating and
homogeneous layer materials. Thus, the optimization is performed only for grating period, fill
factor, and layer thicknesses in the ranges 0.7 pm < A < 1.06 pm, 0.05 pm < d < 2.5 pm, and
0.1 < f < 0.9, respectively. With these restrictions, the program finds single-layer and double­
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
148
layer devices with low sidebands but peak transmittance reaching only -85%. Peak responses
approaching 100% are obtained in 3-layer structures containing two homogeneous layers, Si
and SiC>2 , on silica substrate with a surface relief Si grating on top of the SiC>2 layer as shown
in figure 5.21(a). Repeatedly running the genetic algorithm routine with different random
number seeding of the chromosomes or for different genetic algorithm program parameters
leads to several filter structure parameters exhibiting very similar characteristics. Figure
5.21(b) illustrates two such examples with the geometrical parameters of the devices given in
the figure caption. These results were obtained with a real encoding of the filter parameters, a
population of 2000, a crossover probability of 0.8, and a Gaussian mutation type with
mutation probability of 0.5 and standard deviation o f 0.1. The number of replaced
chromosomes was 600 for filter 1, and 800 for filter 2 o f figure 5.27, respectively. Filter 1
exhibits a peak transmittance at Ac = 1.5501 jim with a linewidth AA, = 0.51 nm (AA/Ac =
0.033%) and sidebands below 2% over the 40 nm optimization range. Filter 2 has maximum
transmittance at Ac = 1.5489 (im with a linewidth AA. = 0.53 nm (AA/Ac = 0.034%) and
slightly higher sidebands than filter 1. The center wavelength of the second filter has been
shifted to lower wavelengths for comparison between the filter responses of the two devices.
In all guided-mode resonance filters, the central wavelength can be spectrally shifted by
appropriate changes in grating period [25]. The effect o f the high-modulation grating on the
transmittance o f the three-layer device is emphasized in figure 5.27 by comparison between
the spectral responses of the guided-mode resonance filter 1 and of a similar structure with
homogeneous layers only. The dashed line of figure 5.27 represents the transmittance of a
device identical to filter 1 except for the grating that is replaced by a homogeneous layer with
a refractive index equal to the average refractive index o f the grating (nav = 2.409).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
149
0.9
Generated
by GA
0.8
Reference
0.1
0
in T f i T- i—i t i ■ ■ i t i i i~i1
0.545
0.547
0.549
0.551
0.553
WAVELENGTH (p m )
* Il
0.555
Figure 5.15. TE-polarization transmittance at normal incidence of a bandpass
guided-mode resonance filter in the optical spectral region generated by genetic
algorithm optimization of the 2-grating structure of figure 5.3a (solid line) and the
Lorentzian reference response (dotted line). The parameters of the structure are: A
= 0.342 (im, nn.1 = 2.5, nm = 1.6, di = 0.332 pm, x y = 0.64, Xh,i = 1.0, n Hr2 =
2.5, nL,2 = 1.5, d2 = 0.093 pm, x l .2 = 0.2, xH^ = 0.54, nc = 1-0, ns = 1.52.
0.9
0.8
g o -7
H 0.6
E| 0.5
Z 0.4
0.5
0.52
0.54
0.58
0.56
WAVELENGTH (l011)
0.6
Figure 5.16. The filter response of figure 5.15 over a larger wavelength range.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
150
u— i i i i i i i i i i i i i i > i i i i
0.545
0.547
0.549
0.551
0.553
WAVELENGTH (p m )
0.555
Figure 5.17. Guided-mode resonance transmission filter response of the 9-layer
structure illustrated in the inset. The grating has the parameters A = 0.332 pm, n H ,5 =
2.5, nL,5 = 2.1, ds = 0.0595 pm. The homogeneous layers have the parameters n = 2.35
and d = 0.0585 fim (odd layers), and n = 1.38 and d = 0.0996 pm (even layers); nc =
1.0, ns = 1.52.
\J [ l I I I | L !! I | 1 1 ! I | l I II f l I I I1
0 .5 4 5
0 .5 4 7
0 .5 4 9
0 .5 5 1
0 .5 5 3
0 .5 5 5
WAVELENGTH (p m )
Figure 5.18. Guided-mode resonance transmission filter designed with the genetic
algorithm. With the notations of figure 5.3(a), the parameters o f the 2-layer/2-grating
structure are: A = 0.35 pm, nH,i = nn,2 = 2.35, nm = n y = 1.65, di = 0.1934 pm d 2 =
0.3404 pm, Xl,i = 0.0, Xh.i = 0.319, x ^ = 0.586, xh ,2 = 0.785; nc = 1.0, ns = 1.52.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
151
-
10
-
-60
-70
0.545
0.547
0.549
0.551
0.553
WAVELENGTH (pm )
0.555
Figure 5.19. Transmittance of the 9-layer/1-grating structure of figure 5.17
plotted in dB.
-10
-60
-70
0.545
0.547
0.549
0.551
0.553
WAVELENGTH (pm )
0.555
Figure 5.20. Transmittance of the 2-layer/2-grating structure of figure 5.18
plotted in dB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
Grating
0.9
Homogeneous
0.8
0.3
0.2
0.1
0.5
0.45
0.55
0.6
WAVELENGTH (pm)
0.65
Figure 5.21. Transmittance of the 9-layer/1-grating device of figure 5.17 over a
200 nm spectral range.
0.9
0.8
Grating
0.7
0.6
—
Homogeneous
0.5-
CO
0.4
0.3
0.2
0.1
0.45
0.5
0.55
0.6
WAVELENGTH (pm)
0.65
Figure 5.22. Transmittance of the 2-layer/2-grating structure of figure 5.18 over a
200 nm spectral range.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
153
U| >i i | l l i | i i i | I i l | i i l
0.545
0.547
0.549
0.551
0.553
WAVELENGTH (pm )
0.555
Figure 5.23. Guided-mode resonance transmission filter with the structure
generated by the genetic algorithm. W ith the notations of figure 5.3(a), the
parameters of the 2-layer/2-grating device are: A = 0.33 pm, nn,i = 2.5, qhj. = 2.4
hl,i = 1-75, nL,2 = 1-35, di = 0.1825 pm d 2 = 0.1875 pm, xm = 0.0, xh,i = 0.325,
Xl,2 = 0.424, Xh,2 = 0.945; nc —1.0, ns = 1.52.
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.45
0.5
0.55
0.6
W A V ELEN G TH ( p m )
0.65
Figure 5.24. The filter response of figure 5.23 illustrated over a 200 nm spectral
range (solid line) and transmittance of the corresponding equivalent homogeneous
structure with equal thicknesses and average refractive indices (dashed line).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
154
J
-------- Homogeneous
\
: \
X
V
\
✓
v
✓
o
✓
----- Grating
/
/
/
✓
n.
IIKB bNB
o-
8.5
9
1l-p11I M■1/,
9.5 10 10.5 11 11.5 12
WAVELENGTH (pm)
12.5
Figure 5.25. Infrared bandpass guided-mode resonance filter spectral response
centered at Ac = 10.6 pm (solid line). This response is generated by a single-layer
waveguide-grating with the cross-section illustrated in the inset and possessing the
following physical parameters: A = 6.91 pm, f = xh-xl = 0.42, d = 3.7 pm, nu = 4.0,
nL = 2.65, no = 1.0, and ns = 1.4. The dashed curve represents the transmittance of a
homogeneous layer with the same thickness and average refractive index (n = 3.285).
-
10
-
w -20
CJ
-60
-70
8.5
9
9.5
10 10.5 11 11.5 12
WAVELENGTH (p m )
12.5
Figure 5.26. Filter response of figure 5.25 expressed in dB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(a)
filter 1
homogeneous
® 0.7-
1 | I1I I 1
1.54
1.55
1.56
1.57
WAVELENGTH (pm)
(b)
Figure 5.27. Spectral responses of two infrared bandpass filters centered at Ac
-1.55 pm (b). The filters employ the same three-layer/single-grating structure (a)
and the same materials (nH = 3.2, nL = 1.0, n2 = 1.45, n3 = 3.2, no = 1.0, and ns =
1.45) but different grating periods, A, fill factors, f, and thicknesses, d. The
geometrical parameters of the two devices are as follows; filter 1: A = 0.985 pm, f
= 0.52, di = 0.228 pm , d2 = 0.958 pm, and d3 = 2.264 pm; filter 2: A = 1.06 pm, f
= 0.4, di = 0.087 pm , d2 = 1.48 pm, and d3 = 2.46 pm. The dashed line indicates
the calculated transmittance of a structure identical to filter 1 except for the
grating layer replaced with a homogeneous layer with a refractive index n = 2.409.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 6
GUIDED-MODE RESONANCE DEVICES ON
OPTICAL FIBER ENDFACES
6.1
Introduction
This chapter discusses a new type of guided-mode resonance device integrating a
waveguide grating with subwavelength period on the endface of an optical fiber. Such a
diffractive fiber-optic device can enable high-resolution biomedical or chemical sensors and
spectral filters for communications applications [80,81].
The fiber-endface guided-mode resonance filter may become a viable candidate for
fiber-optics communications, wavelength division multiplexing and demultiplexing,
provided that progress in device design and fabrication is made to enhance its performance.
A practical device for wavelength divison multiplexing would require narrow linewidths
(-0.1 nm), low sidebands (<-30 dB) over a specified wavelength range (-20-40 nm at 1.55
Jim), and a rectangular line shape [127]. Such filter features have been approximately
attained theoretically in separate devices (figures 3.8, 3.28, 3.32, 5.18, 5.25). Further
theoretical and experimental research is required to achieve these characteristics in a single
device.
The possibility of using the guided-mode resonance effect for sensing applications
has been proposed previously [24]. In order to perform a sensing function, the guided-mode
resonance device is not required to possess the demanding spectral characteristics
mentioned above for fiber-optic filters. This application relies on the sensitivity of the
resonance wavelength to the physical parameters of the waveguide grating (grating period,
fill factor, thicknesses, or refractive indices of the layers), or the refractive index of the
156
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
157
surrounding medium (i.e., cover region). Coupled with a wide-band source and a
spectroscopic detection system, a guided-mode resonance filter can be utilized to detect
changes in refractive index and/or thickness of the surface layer. The novelty of the
proposed device consists in its integration with an optical fiber, which combines the
advantageous features o f both these elements. The guided-mode resonance filter can exhibit
high sensitivity to parameter changes as well as a relatively large detection range. The
optical fiber enables remote access, increased spatial resolution, real-time operation, and
device miniaturization. Section 6.2 presents calculated examples o f thickness and
refractive-index sensors based on the guided-mode resonance effect. Section 6.3 discusses
experimental results obtained with gratings and waveguide gratings on optical-fiber
endfaces for future sensor development.
6.2
Guided-M ode Resonance Sensors
In general, a guided-mode resonance peak suffers a spectral shift whenever any one
physical parameter of the waveguide grating is varied. The magnitude of the resonance shift
depends on the specific physical parameter that is varied (e.g., grating period, refractive
index, thickness), as well as on the waveguide-grating structure employed (i.e., number of
grating and homogeneous layers and their order in a multilayer stack). Thus, it was found
that changes in grating period typically produce greater shifts of the resonance peak than
changes in any other physical parameter of a waveguide grating. It was also determined that,
for the same physical parameter variation, multilayer structures generally produce smaller
resonance shifts than single-layer devices. In spectral filtering applications, it is desirable
that the guided-mode resonance peak shifts less with changes in physical parameters such as
grating period, refractive indices, or thicknesses o f the layers. This ensures that fabrication
errors or slight changes in operating conditions (for instance temperature) do not affect the
properties o f the filter. On the contrary, in sensor applications a large spectral shift of the
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158
resonance is desirable for small variations in the physical properties of the device. The
amplitude o f the resonance spectral shift divided by the magnitude of the variation in a
physical parameter of the waveguide grating is used as a measure of the sensitivity of a
guided-mode resonance sensor [128]. Apart from sensitivity, other important sensor
characteristics are resolution and dynamic range [128]. The resolution is determined by the
sensor sensitivity in conjunction with detection equipment limitations, such as power
detector and monochromator sensitivities. The dynamic range of a sensor is defined, in the
case of guide-mode resonance sensors, as the maximum variation of a physical parameter
that can be detected by a shift in the resonance peak.
In this dissertation, waveguide-grating structures suitable for sensor applications
were sought. The optimum structure and materials for a sensor based on the guided-mode
resonance effect depends largely on the specific detection application. In some applications,
the refractive index of the cover region constitutes the physical parameter to be detected; in
others, the thickness of a deposited layer o f known material is of interest. Both these types
of detection are considered. The waveguide-grating examples provided here utilize
refractive indices corresponding to real materials. The materials employed in sensor design
are transparent in the spectral ranges of sensor operation and losses are thus neglected in
calculations.
In the first type of sensor function considered, the thickness of a material with
refractive index equal to 1.4 (typical of biological materials) is to be determined by
measuring the spectral shift of a guided-mode resonance. Figure 6.1 depicts a double-layer
structure formed with a SiC>2 (n = 1.45) surface-relief grating on a HfC>2 waveguide and its
spectral response. Uniform layers with thicknesses of 20 nm and 40 nm deposited on the
surface relief grating shift the resonance peak from 600.0 nm (for no material deposited) to
602.0 nm, and 603.5 nm, respectively. The spectral shift of the resonance is increased in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
159
structures that possess higher electric field amplitude on the surface of the grating where the
sensed material is deposited. In a double-layer device with the cross-section of figure 6.1a,
this can be accomplished by increasing the leakage of the grating and/or by decreasing the
mode confinement in the waveguide. Such an example is illustrated in figure 6.2. The
surface-relief grating layer is formed with a higher-index material, S i ^ (n = 2.0), and the
HfC>2 waveguide is quarter-wave thick instead of half-wave thick, as in the previous
example. The resonance peak shifts from 599.8 nm to 605.4 nm, and 610.2 nm as layers
with 20 nm and 40 nm are added, respectively. The larger leakage and reduced mode
confinement of this waveguide grating produces a larger width of the resonance peak. The
devices of figures 6.1 and 6.2 impose different requirements on the detection system in a
practical sensor. For equal layer thickness detection, the structure of figure 6.1 requires
higher wavelength resolution, while the device of figure 6.2 requires higher power detection
resolution.
The same sensing task can be performed with a single-layer device as illustrated by
the reflectance response of figure 6.3 in the wavelength range 740 - 770 nm. The materials
forming the grating are ZnSe (n = 2.55) and Si02 (n = 1.45). The resonance shifts from
750.4 nm to 752.1 nm, and 753.3 nm for a thickness of deposited material (n = 1.4) of 20
nm, and 40 nm, respectively. Larger shifts are obtained with gratings possessing higher
modulations. Figure 6.4 depicts a single-layer waveguide grating and the spectral sensitivity
of the resonance to adhesion of material to be sensed. This high-modulation guided-mode
resonance filter was generated with the genetic algorithm optimization routine. The device
of figure 6.4 is a surface relief grating formed with only one material (Si), and therefore is
simpler to fabricate than the structure of figure 6.3. It also possesses larger resonance shifts
than the other examples illustrated here (i.e., 10 nm shift for 10 nm of added material, and
20.8 nm shift for 20 nm of added material). A sensor utilizing this guided-mode resonance
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
160
structure and a monochromator with medium resolution of 0.1 nm, is capable of detecting
adhered material with thickness of ~1 A.
The sensor examples of figures 6.1 - 6.4 are designed to operate in air. The example
of figure 6.5 uses water as the operating environment. A single-layer surface relief grating
with ZnSe (n = 2.55) shifts its resonance by 1.9 nm and 4.5 nm, with 20 nm and 40 nm of
adhered material (n = 1.4), respectively. The resonance shift can be increased in gratings
with fill factors smaller or greater than 0.5. For instance, employing the structure o f figure
6.5 with a fill factor of 0.6 results in resonance shifts of 2.5 nm and 5.6 nm for additional
thickness layers of 20 nm and 40 nm thickness, respectively.
A different kind of sensing function results as the resonance peak shifts due to
changes in refractive index of the environment (i.e., cover medium of the waveguide
grating). Such an example is illustrated in figure 6.6. The structure is a silicon (n = 3.2)
surface-relief waveguide grating on a silica substrate (n = 1.45) with a cover medium o f
refractive index, nc = 1.33, operating in the infrared spectral range (-1.55 Jim). The
resonance peak shifts by 2.4 nm and 4.8 nm as the refractive index of the cover medium
changes from 1.33 to 1.34, and 1.35, respectively. Another example illustrating the
refractive-index sensing capabilities of guided-mode resonance devices is provided in figure
6.7. The single-layer device consisting of a surface-relief waveguide grating formed with
ZnSe is illustrated in figure 6.7 (a). The structure used in the example of figure 6.5 to detect
thickness variations is shown in figure 6.7 (a) to indicate changes in refractive index.
Variations in refractive index of 0.1 units induce resonance shifts of 2.6 nm, as
demonstrated in figure 6.7 (6). In surface-relief waveguide gratings, the resonance shift is
due to both changes in the low-refractive index of the grating, as well as changes in the
cover medium refractive index. The dependence of the resonance peak wavelength on the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
161
refractive index of the cover medium, indicating a linear characteristic in the range 1.331.34, is depicted in figure 6.8.
Transmitted
wave
A
t
Air
Deposited
material to
be detected
Grating
Waveguide
iSubstfaie::
;i jJncidebtj jif i i :ii^e)aected:
::: :: XVavie:::::::::::::: tvaVO: | |
(a)
R(0)
R(+20nm)
~ R(+40nm)
0.59 0.595 0.6 0.605 0.61 0.615 0.62
WAVELENGTH (Jim)
(b)
Figure 6.1. Spectral shift of a guided-mode resonance peak (b) exhibited by a
double-layer waveguide grating (a) as 20 nm (dotted line) and 40 nm (dashed line)
of material (n = 1.4) is deposited on top. The physical parameters of the structure
are as follows: grating period, A = 0.349 Jim, thicknesses, di = 0.12 (im, d2 = 0.15
|im, refractive indices of the grating, njtH = 1.45 (SiC>2 ) and n = 1.0 (air),
waveguide, n2 = 2.0 (HfC>2 ) and substrate, ns = 1.45 (silica).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
162
0.9
R(0)
0.8
R(+20nm)
0.70.6
U 03
m
0.4
0.3
0 . 1-
034
036
0.6
0.62
0.64
038
WAVELENGTH (pm )
0.66
Figure 6.2. Spectral shift of the guided-mode resonance generated with the
structure of figure 6.1a, for 20 nm and 40 nm of material (n = 1.4) deposited on
top. The parameters are: A = 0.38 pm, di = 0.09 pm, d2 = 0.075 pm, ni,H = 1.45
(Si3N4) and n = 1.0 (air), n2 = 2.0 (H f02), and ns = 1.45 (silica).
10.SH1
0 . 8"
W 0.7i
o
|
0.61
u 0 .5 i
BJ
5* 0 . 4 “
0 .3 i
0.2i
0 . 100 .7 4
n
t-
*
i
*
n
ll
II
ll
ll
R(0)
R(+20nm)
R(+40nm)
II
ll
il I
■?'
\«
i
i
i— |— i— i— —|— i— i—i—i—i
0.75
0.76
0.77
WAVELENGTH (pm)
Figure 6.3. Single-layer waveguide grating formed with ZnSe (n = 2.55) and S i0 2
(n = 1.45) performing the same sensing task as the examples o f figures 6.1 and
6.2. The grating period is A = 0.439 pm , the thickness is d = 0.362 pm, and the
substrate is silica (ns = 1.45).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
163
Transmitted
wave
Deposited
material to
be detected
Air
Waveguide
Grating
X
;wave
— R(0)
0.90.8 -
R(+20nm)
0.7-
R(+40nm)
0.30 .2 0 . 1“
1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6 1.61 1.62
WAVELENGTH (p m )
(b)
Figure 6.4. Resonance spectral shifts (b) in a single-layer surface-relief waveguide grating
(a) made with Si (n = 3.2) on silica substrate (n = 1.45). The grating period is A = 0.9076
pm and the thickness is d = 1.1 pm. The structure was generated with the genetic algorithm
optimization routine.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
164
i
r
[I]
j
R(0)
*i
■i
ii
■i
it
■i
it
■i
■i
h
u01
—
t:
R(+20nm)
R(+40nm)
—
-
i ll
Ti
1. _ _ J j M
‘ ‘ 1 ‘ 1 1 1 1 11
L
r 1 1 I ■1 1■1 r FW
0.74 0.745 0.75 0.755 0.76 0.765 0.77
WAVELENGTH (pm)
o-
Figure 6.5. Guided-mode resonance peaks generated with a single-layer surface-relief
grating (figure 6.4a) made with ZnSe (n = 2.55) and using water as cover medium.
The peaks correspond to thicknesses of adhered material with n = 1.4 of 0, 20 nm, and
40 nm, respectively. Other parameters are A = 0.454 pm , d = 0.371 pm, and ns =
1.45.
0.9
R (n = 1 .3 5 )
0.1 -
0. 6 U 0.5-:
0 .4 -i
0.3 r
0.2 r
0.1
1.54
1 .5 4 5
1.55
1.555
1.56
W AVELENGTH
1 .5 6 5
1.57
(p m )
Figure 6.6. Single-layer guided-mode resonance device (figure 6.7a) with a surfacerelief Si grating sensing changes in the refractive index o f the cover region. The
waveguide-grating parameters are A = 0.976 pm, d = 2.371 pm , ns = 1.45, nn = 3.2
(Si), nL = nc = 1.33 (solid line), 1.34 (dotted line), and 1.35 (dashed line).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
165
t
Transmitted
wave
-
■>(
Liquid to
be detected
Waveguide
Grating
dX
:Substrates::
:In cid en t x ::
:w ave'::
r:;B5efleiDbe^:
^a*ie:
( a)
0.9
R(n=1.33)
0.8
R(n=1.34)
o n -.
R(n=1.35)
0.6
E 0.40.3 "i
Q .2 - .
0.H
0.74
0.745
0.75 0.755 0.76 0.765
WAVELENGTH (|tm )
0.77
(b)
Figure 6.7. Single-layer guided-mode resonance device (a) indicating shifts in
resonant wavelength for different refractive indices o f the cover region (b). The
high-refractive index of the grating is nn = 2.55 and the low-refractive index
(equal to the cover index) is nL = nc = 1.33 (solid line), 1.34 (dotted line), and
1.35 (dashed line). Other parameters are A = 0.454 p.m, d = 0.371 pm, and ns =
1.45. The structure is the same as in figure 6.5.
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REFRACTIVE INDEX
Figure 6.8. Resonance wavelengths versus refractive index of the cover region for
the device of figure 6.7.
6.3
Experim ental Results
Compared to larger-area guided-mode resonance filters, devices fabricated on
optical fiber endfaces encounter additional fabrication challenges. Therefore, in this work
fabrication of a relatively simple structure with a photoresist grating on a Si3N4 waveguide
deposited on a fiber endface has been attempted [81]. Such a guided-mode resonance filter
with a typical response exhibiting a notch in transmission (corresponding to a peak in
reflectance) is illustrated in figure 6.9 (b) for a waveguide grating with the cross section
shown in figure 6.9 ( a). The calculations of figure 6.9 are performed with rigorous coupledwave analysis assuming plane waves at normal incidence on a structure with an infinite
number of grating periods. Nevertheless, this theory can be applied to accurately predict the
spectral locations and lineshapes of guided-mode resonance devices with finite sizes, as
demonstrated by recent experiments at microwave wavelengths [78,79]. The microwave
experiments [79] also showed that finite-size gratings with only -1 2 periods can yield
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167
guided-mode resonance notch filters with a decrease in the transmittance spectrum from
-81% outside resonance to -2% at resonance. It is therefore expected that waveguide
gratings on enfaces of multimode fibers (with diameters of -100 pm) encompassing -200
grating periods will yield deep transmittance notches in good agreement with the rigorous
coupled-wave calculations.
Practical implementation of fiber-endface waveguide gratings entails thin-film
deposition and diffraction grating fabrication. Homogeneous dielectric thin-film coatings
deposited on fiber endfaces are commercially available for applications such as
antireflection coatings or spectral filtering. However, fabrication o f fiber-endface diffraction
gratings has not been reported to date to the best of our knowledge. In this work, diffraction
gratings with submicron periods have been recorded in photoresist deposited on coated and
uncoated endfaces of single-mode and multimode fibers. The photoresist thickness was
approximately controlled by addition o f specific amounts of thinner. Reference silica
samples were deposited with photoresist in the same manner as the fibers and their
thickness was subsequently determined with an ellipsometer. A solution of 6 parts of
photoresist to 8 parts of thinner formed a layer thickness on fused silica of around 350 nm.
Holographic gratings were recorded on fiber endfaces with with an Ar+ laser (k = 365 nm)
using the Lloyd’s mirror interference set-up o f figure 3.34.
The grating diffraction efficiency is tested by coupling laser light into the uncoated
end o f the fiber, and measuring the power output of the transmitted diffraction orders on the
grating end. Figure 6.10 illustrates scanning electron micrographs o f diffraction gratings
with 800 nm period on optical fiber endfaces with 100 pm core diameters. This device
produces ±1 diffracted orders containing -50% o f the total output power, when tested with
a HeNe laser (X=633 nm). Gratings with a period of 530 nm were recorded on optical fiber
endfaces with 6.7 pm core diameters. The ±1 transmitted diffraction orders were measured
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168
to contain -10% of the total power coupled out of the fiber, at the wavelength of 442 nm
(HeCd laser).
Preliminary results have been obtained for resonant waveguide grating structures
integrated on optical fiber endfaces.
A thin-film layer of silicon nitride was deposited by
RF sputtering on multimode optical fiber endfaces with 100 |im core diameters.
A
photoresist grating (510 nm period) was subsequently recorded to yield resonant fiberendface waveguide gratings. Spectral measurements o f these devices were performed with a
tunable TirSapphire laser (X=735-880 nm) using the set-up of figure 6.11. Guided-mode
resonance notches were measured in the spectrum of the transmitted power, at the output of
the optical fiber. Figure 6.12 illustrates a transmittance notch of -18% at the wavelength of
760 nm. The transmittance in the plot of figure 6.12 was obtained by dividing all measured
data points to the highest value o f the transmitted power. The rapid oscillations in the
measured data are due to instabilities of the Ar+/Ti:sapphire laser system. The low
efficiency is partially attributed to the polarization sensitivity of the guided-mode resonance
effect, with TE and TM peaks occurring at different wavelengths and the polarization
scrambling induced by propagation through the optical fiber. However, similar devices that
are polarization independent can be achieved with two-dimensional gratings [35]. Scattering
due to imperfect fiber cleaves and rough silicon nitride films are also contributing factors to
a decrease in guided-mode resonance efficiency. Work is in progress to improve the
fabrication procedure of fiber-endface waveguide gratings, and thus achieve higher
efficiency guided-mode resonance devices on fiber endfaces.
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169
Transmitted
wave in air
fib e r
KeSe&ed
Saw
(a )
0.9
0.8
g 0.7
|
0.6
§ 0’5
TE
TM
Z 0.4
gs 0.3
0.2
0.1
0.75
0.8
0.85
WAVELENGTH (p m )
0.9
Figure 6.9. Calculated TE and TM-polarization spectral response (b) of a guidedmode resonance filter (a) with the following parameters: A = 0.51 pm, di = 0.4
pm, d2 = 0.18 pm, nH= 1-63 (photoresist), nL= 1.0 (air), n2 = 1.9 (Si3N4), and ns =
1.45 (fused silica optical fiber).
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170
w/
’v
Figure 6.10. Scanning electron micrographs of photoresist diffraction gratings
with a period of 0.8 p.m recorded on the endfaces o f cleaved optical fibers with
core diameter of 100 pm and cladding thickness of 125 pm. The gratings appear
as thin diagonal lines in the bottom picture and vertical lines in the top picture.
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171
Focusing lens
Grating on
tip of fiber
Detector
PC
Optical fiber
w aveguide g ratin g
Power
meter
Figure 6.11. Experimental set-up for measurement of the transmittance spectrum
o f a guided-mode resonance filter fabricated on the endface of an optical fiber. A
computer scans the wavelength of the Ti:Sapphire laser and simultaneously
measures the optical power transmitted through the optical fiber and the
waveguide grating on the fiber endface.
In
o
^
0 — ' ' ' I l - ' l' I I I I I I I I I I I I I I I I T - )—l- T I |
0.735 0.755 0.775 0.795 0.815 0.835 0.855 0.875
WAVELENGTH (pm)
Figure 6.12. Transmittance measurements with the set-up of figure 6.11 o f a
guided-mode resonance filter fabricated on an optical fiber endface. A
transmittance notch o f -18% is detected at the central wavelength of -0.76 J im .
The structure consists of a photoresist grating and a Si3N 4 waveguide on the
endface of a multimode silica fiber.
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CHAPTER 7
CONCLUSIONS
7.1
Contributions
The novel contributions to the study o f guided-mode resonances in waveguide
grating structures and development of spectral filtering devices based on these effects
presented in this dissertation are summarized below.
(1)
Two-layer and three-layer reflection filters in the optical spectral region
embedding gratings in classical anti-reflection thin-film coatings have been
characterized. These new calculated examples use practical materials commonly
used for thin-film coatings and combine the narrow resonance of the waveguide
grating with low sidebands extended over the whole visible spectrum.
(2)
A guided-mode resonance transmission filter with a single grating in the center
layer bordered by two dielectric thin-film mirrors has been invented. The
homogeneous layers on either side o f the grating have alternating high/low
refractive index values and quarter-wave thicknesses to satisfy high-reflectance
conditions, thus providing the low sideband transmittance of the filter. This single­
grating structure has a simpler design and a narrower linewidth than the previously
reported
double-grating
design
while
maintaining
comparable
sideband
transmittance and filter range.
(3)
A rigorous coupled-wave analysis has been developed for multilayer structures
with arbitrary number of homogeneous or grating layers, where the grating layers
can have one of two different periods, Ai or A 2 .
172
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173
(4)
The double-period rigorous coupled-wave analysis program has been used to
develop double-line filters at normal incidence. These filters are obtained with
structures containing two gratings in separate layers with a central homogeneous
layer. The two resonance wavelengths can be tuned independently by appropriate
choice of the grating periods. Calculated double-line filter examples are given in
the optical as well as in the microwave spectral region.
(5)
Multilayer structures with gratings having different phase-shifts have been studied
theoretically. In general, the lateral shift of half of a grating layer along the grating
vector, to obtain two identical phase-shifted gratings, produces a double-peak
response. It was shown that phase-shifted gratings can be used as design tools in
guided-mode resonance filter-line control. Linewidth broadening was obtained in
optical and microwave reflection filters using two adjacent gratings with a phase
shift value that depends on the layer thicknesses. The same phase shifts were
shown to narrow the linewidth in the case of single-grating transmission filters. In
optical reflection filters, a phase-shifts of 7t/2 has been shown to change the
triangular filter response to a more rectangular shape.
(6)
A comparison between guided-mode resonance filters and classical homogeneous
thin-film filters has been completed for the visible spectral region. Compared to
classical reflection filter designs (Bragg or quarter-wave notch filters), guidedmode resonance filters have a unique capability of achieving very small linewidths
with significantly fewer layers. Transmission filters with a single-grating structure
with adjacent thin-film dielectric mirrors also require fewer layers to achieve the
same linewidths as their classical counterparts (Fabry-Perot filters) but the
difference is not as great as in the case of reflection filters. Further reduction in
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174
total thickness and number of layers of the structure is achieved in guided-mode
resonance filters via genetic algorithm optimization.
(7)
A double-layer guided-mode resonance filter was fabricated and tested indicating a
TE-polarization peak reflectance of -98% at the central wavelength 0.86 fim with
-0.3 nm linewidth and sidebands below 5% in the range 0.8 — 0.9 }im. The
experimental measured reflectance spectrum was closely matched by the
theoretical calculations.
(8)
A guided-mode resonance filter with high efficiency was used as the output
coupling-mirror in a commercial dye laser. The laser power was -100 mW when
pumped with an Ar+ laser emitting a power of 5 W at the wavelength of 0.514 Jim.
The linewidth of the laser (-0.3 nm) was set by the guided-mode resonance filter
linewidth at the threshold reflectance for laser oscillation to occur.
(9)
A computer program that uses genetic algorithms as a search method and rigorous
coupled-wave analysis as a computational method for the merit function
optimization of diffractive optics devices has been developed. The program
functions as an inverse method of design. The desired reflectance or transmittance
spectrum constitutes the input data that the program uses to find the physical
parameters of a waveguide-grating structure that exhibits a response close to the
reference response. The program is modular and flexible; it can easily be adapted
to perform the optimization in terms of a restricted set of physical parameters (i.e.,
fixing some physical parameters) and to account for various design constraints
(e.g., use of selected refractive-index values).
(10)
A study of the convergence process of the genetic algorithm program for a variety
of input parameters has been conducted to determine the optimum starting
conditions in a guided-mode resonance optimization problem. Input parameters
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175
are of different kinds. There are input parameters related to the genetic algorithm
operations as for instance, type of encoding, population, replacement values,
mutation, and crossover probabilities. Other input parameters pertain to merit
function evaluation, such as the merit function power index, the number of
reference points, their spectral distribution and weight factors, or to the
optimization ranges for the physical parameters.
(11)
The genetic algorithm developed is versatile and can be applied for design of
multilayer structures with homogeneous and grating layers such as fan-out
gratings, anti-reflection coatings, high-reflectors, polarizing elements, beam­
splitters, and edge filters. In this work, the program is used to design a variety of
reflection, and transmission bandpass filters. Repeatedly running the genetic
algorithm program several solutions (i.e., structures with different physical
parameters) to the same problem (i.e., same reference response) are generally
found. This was demonstrated with both reflection and transmission filters. This
availability of alternative structures exhibiting similar filtering characteristics may
facilitate fabrication and implementation o f guided-mode resonance filters in
various applications.
(12)
The genetic algorithm program was able to find guided-mode resonance filters
possessing features not previously known to be possible. Thus, numerous
examples of transmission filters with peaks approaching 100% and low sidebands
have been found in structures containing only 1-3 layers. Filters with linewidths of
-0.2 nm (relative linewidth 0.036%) at a central wavelength of 0.55 pm are
demonstrated in 2-layer/2-grating structures with sideband below 0.5% over the 10
nm optimization range. The devices found by the genetic algorithm achieve, with
only two layers, slightly higher peaks, more symmetrical lineshapes, and lower
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176
sideband transmittances than the structures designed with the “direct” method. At
infrared wavelengths (10.6 pm), a filter consisting o f a single binary grating and
refractive indices corresponding to real materials is obtained featuring a linewidth
of 12.7 nm (relative linewidth 0.12%) and sideband transmittance below 1% over
a 2.2-|Xm spectral range. A 3-layer device with a surface relief Si grating and two
underlying homogeneous layers of SiC>2 and Si is shown to yield a high-efficiency
filter centered at 1.55 Jim with a linewidth of 0.5 nm (relative linewidth 0.032%).
A new feature discovered with the genetic algorithm approach is the low
transmittance realizable with high-modulation gratings. This has enabled
transmission filters that do not rely on classical high-reflectance stack to provide
the low filter sidebands as in previously known guided-mode resonance filters.
Therefore, with high-modulated gratings just one or two layers are utilized to
obtain the high transmittance peak as well as the low sideband response of these
filters.
(13)
Microwave reflection and transmission filters have been studied using single and
multilayer waveguide gratings. Using air as cover as well as substrate materials,
reflection filter characteristics with symmetrical lineshape and low sidebands have
been calculated at frequencies in the 4-20 GHz range. The available materials,
with dielectric constants larger than those in the optical region, provide new
design capabilities. Filters with larger linewidths and lower sidebands than in the
optical region can be designed with fewer layers.
(14)
Spectral measurements of the transmittance of single and multilayer waveguide
gratings in the 4-20 GHz range have been performed in an anechoic chamber with
microwave lenses for collimation of the incident beam on the devices and focusing
of the diffracted wave onto the receiver antenna. A single-layer reflection filter
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Ill
formed with Plexiglas strips in air exhibited notches in transmittance at spectral
locations in close agreement with the theoretical calculations for both TE and TM
polarizations.
(15)
Guided-mode resonance transmission peaks were measured experimentally for the
first time at microwave frequencies (4-20 GHz) in several 5-layer waveguide
gratings. A 5-layer device consisted o f a grating formed with G10 fiberglass
rectangular bars in air, and one fiberglass sheet at a distance in air, on each side of
the grating. It is important to note that these experimental results were found with
a finite-size microwave beam covering relatively few periods of the grating (-12
periods in the reflection filters and -18 periods in the transmission filters). Fivelayer Fabry-Perot bandpass filters with glass and air, or G10 fiberglass and air,
were also built and tested in the microwave spectral range. A comparison between
the Fabry-Perot filters and the guided-mode resonance filters built with the same
materials and number of layers indicated the narrower linewidths and higher
sensitivity to losses of the resonance filters. Excellent matching between
experimental transmittance spectra and the theoretical data was found for guidedmode resonance filters as well as for Fabry-Perot filters.
(16)
Diffraction gratings with submicron periods have been recorded for the first time
on coated and uncoated endfaces of single-mode and multimode fibers. The
diffraction efficiency of the holographic gratings was tested by coupling laser light
into the uncoated end of the fiber, and measuring the power output of the
transmitted diffraction orders on the grating end. Gratings with 0.8 pm period on
fiber endfaces with 100 pm core diameter, produced ±1 diffracted orders
containing -50% of the total output power, when tested with a HeNe laser (X =
0.633 pm).
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178
(17)
A new fiber optic sensor integrating dielectric diffraction gratings and thin film s
on optical fiber endfaces has been proposed for biomedical sensing applications.
An incident broad-spectrum signal is guided within an optical fiber and is filtered
to reflect or transmit a desired spectral band by the diffractive thin-film structure
on its endface.
Slight changes in one or more parameters o f the waveguide
grating, such as refractive index or thickness, can result in a responsive shift of the
reflected or transmitted spectral peak that can be detected with spectroscopic
instruments. Theoretical studies have been performed to find the waveguidegrating parameters that yield maximum sensitivity (i.e., maxim u m spectral shift of
the resonance) and detection range (i.e., maximum thickness or refractive index
variation that can be detected) of these sensors operating in different wavelength
ranges and in different environments.
(18)
Preliminary results have been obtained for resonant waveguide-grating structures
integrated on optical fiber endfaces.
A thin-film layer of silicon nitride was
deposited by RF sputtering on multimode optical fiber endfaces with 100-fim core
diameters. A photoresist grating (0.51 jim period) was subsequently recorded to
yield resonant fiber-endface waveguide gratings. Spectral measurements of these
devices performed with a tunable TiiSapphire laser (k = 0.73 —0.9 jim) exhibited
guided-mode resonance notches of -18% in the transmitted power, measured at
the output of the optical fiber.
7.2
Future Research
Future research in guided-mode resonance effects in waveguide gratings will be
oriented towards development of filters with improved characteristics, a more thorough
understanding o f the theoretically observed effects, and experimental validation of the
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179
theoretical predictions. During the past few years, guided-mode resonance filters operating
in reflection have been experimentally demonstrated in the optical field utilizing various
materials, layer configurations and fabrication techniques. Advantageous features exhibited
by resonance filters include peak reflectances exceeding 98% [74], linewidths up to 6.5%
(FWHM) [64], sidebands below 2% [42,62], filter ranges of 400 nm in the visible range
[17], and polarization sensitivity or independence [35]. However, these characteristics have
been found separately in different devices and much additional effort is needed on the
experimental side for fabrication of guided-mode resonance filters exhibiting all the
advantageous features predicted by the theoretical calculations. A variety of bandpass filters
discussed in this dissertation also await experimental demonstration. Future experimental
work in the optical and microwave spectral regions will focus on practical implementation
o f waveguide-grating filter designs with high efficiencies and low sidebands extended over
large spectral ranges, computed and illustrated in this dissertation.
For practical implementation of filter devices several important issues need to be
addressed. These include improving the filter line-shape, increasing the angular aperture,
decreasing the sidebands and extending the filter range (especially for transmission filters),
and studying the sensitivity of the filter response to parameter variations (thicknesses,
refractive indices, angle of incidence, grating profile, fill factor, losses).
In the microwave spectral range, bandpass filters with higher peak transmittance
than those reported in this dissertation can be achieved in waveguide gratings utilizing
materials with lower losses, higher degrees of homogeneity, and with structures possessing
improved uniformity in grating period, fill factor, and thickness. Such microwave materials
with low-loss and high dielectric constants are available and provide increased flexibility in
the realization of waveguide-grating filters. Examples of high-dielectric constant, low-loss
materials that are available commercially are TMM-10 (e = 9.8, tanS = 0.002) and E-10 (e =
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180
10.2, tan8 = 0.002). Fabrication of devices at microwave wavelengths is much easier than in
the optical range due to the physical dimensions of the gratings that scale with the
wavelengths. Therefore, waveguide-grating designs that are more difficult to implement at
optical wavelengths, such as phase-shifted or multi-grating structures, can be verified in the
microwave range.
A major objective of future theoretical research will be a more profound
understanding o f the resonance phenomena occurring in waveguide gratings and their
influence on the spectral response of guided-mode resonance filters. Promising future
research topics include resonances in phase-shifted gratings, high-modulation gratings,
multi-grating devices with one or multiple grating periods, non-rectangular grating profiles,
finite-size effects, and interaction between thin-film interference effects and resonance
effects in multilayer waveguide gratings.
The genetic algorithm program presented in this dissertation can be further
developed for higher versatility and applicability in design of practical devices by including
the following features:
(1)
The existing genetic algorithm program finds a structure with a response closest to
a given reflectance or transmittance spectrum, regardless of its angular
characteristics. The genetic algorithm code can be extended to include the angular
spectrum as well as the frequency spectrum in calculation of the merit function.
This would enable design of guided-mode resonance filters with large angular
apertures.
(2)
Currently the genetic algorithm code uses the same merit function and genetic
algorithm parameters (mutation and crossover probabilities, population replaced,
etc.) throughout the optimization process. A study of the influence of the genetic
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181
algorithm input parameter variation on the convergence during the optimization
process may indicate conditions for faster convergence.
(3)
The genetic algorithm could be expanded to include a rigorous coupled-wave code
for calculating the diffracted reflectance and transmittance of waveguide gratings
for arbitrary incident polarization. The inclusion of the arbitrary polarization
feature in the genetic algorithm program would enable design of devices with
user-specified spectral as well as polarization response.
(4)
A rigorous coupled-wave code for calculating diffraction efficiencies of arbitrary
grating profiles can be implemented in the genetic algorithm optimization code.
The grating shape could thus be used effectively as a parameter in guided-mode
resonance filter design.
(5)
A search and optimization routine for design of polarization-insensitive filters can
be realized by implementing a rigorous coupled-wave code for analyzing twodimensional structures into the existing genetic algorithm program.
(6)
The sensitivity of the waveguide gratings to slight changes in their physical
parameters could be calculated during the genetic-algorithm optimization and
included in the merit function evaluation of each device. Structures with desired
spectral response as well as low sensitivities to parameter variations can thus be
sought, as desired in filter applications. Conversely, the same program could be
used in sensor applications where a high sensitivity to parameter changes is
desirable.
(7)
The genetic algorithm library PGAPACK [110] utilized in the existing genetic
algorithm software allows parallel computing implementation on a variety of
computer systems. This would considerably decrease the computation time and
thus alleviate the main drawback of the genetic algorithm optimization programs.
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182
(8)
The possibility of increasing the convergence rate by developing a hybrid genetic
algorithm that combines a genetic algorithm with a hill-climbing (local
optimization) routine could be investigated. Genetic algorithms are good at finding
promising areas of the search space, but not as good at fine-tuning within those
areas. Hill-climbing heuristics, on the other hand, are good at fine-tuning, but lack
a global perspective. Therefore, a hybrid algorithm that combines genetic
algorithms with hill-climbing heuristics may results in an algorithm that can
outperform either one individually. Two hybrid algorithm schemes could be
experimented. The first is to run the genetic algorithm until it terminates and then
apply a hill-climbing heuristic to each (or just the best) solution. The second
approach is to integrate a hill-climbing heuristic with the genetic algorithm such
that local optimizations are performed on candidate solutions during the genetic
algorithm optimization process.
Research on fiber-endface grating structures has begun only recently and work is
currently in progress to develop the theoretical foundations as well as applications of such
devices. Existing software for simulation o f guided-mode resonance effects using rigorous
coupled-wave analysis assumes infinite lateral extent of the waveguide grating. This is a
valid approximation (confirmed by experiments) in the optical region when the incident and
diffracted waves are propagating in an unbounded medium. However, guided-mode
resonance devices with input and output waves propagating within a rectangular or
cylindrical waveguide require new computer codes that take into account the finite size of
the structures. These algorithms could be based on finite-difference, finite-element, or
finite-boundary element methods. The relationship between the number of grating periods
and the efficiency and linewidth of the waveguide-embedded guided-mode resonance
devices could thus be studied for structures with various physical parameters.
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183
Finite-size diffraction analysis codes are useful not only in describing waveguideembedded diffractive devices, but also in analysis of free-space grating structures with small
lateral dimensions and beam diameters (as for instance in the microwave and millimeter
waveguide-grating devices). Integration o f the finite-size code with a genetic algorithm
program for optimization of finite-size waveguide gratings could also be explored in the
future.
Experimental studies o f fiber-endface guided-mode resonance filters will focus on
achieving high-efficiency devices useful in spectral filtering and sensor applications. For
this purpose, new waveguide-grating structures, materials, and fabrication techniques will
be investigated. Sensor designs, such as the ones presented in this dissertation, will be
implemented to demonstrate the theoretically predicted attractive features o f the fiberendface resonant-grating devices.
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BIOGRAPHICAL INFORMATION
Sorin Tibuleac received his Master of Science degree in Physics from The University
o f Bucharest, Romania in 1988, and the Master of Science degree in Electrical Engineering
from The University of Texas at Arlington in 1996. From 1988 to 1993 he was a member of
the research staff of the Department of Lasers in the Institute of Atomic Physics Bucharest,
working in the Holography and Information Optics Laboratory, in the field of optical phase
conjugation, laser-induced dynamic gratings and holographic interferometry. Since 1993 he
has been a graduate research assistant in the Department of Electrical Engineering at The
University of Texas at Arlington, studying diffractive and waveguide optics, guided-mode
resonance phenomena in gratings, and applications to spectral filtering. He has also
developed and taught an undergraduate lab for the Optical Engineering course between 19951998. He is a member of the Optical Society of America, IEEE-Lasers and Electro-Optics
Society, and Sigma Xi.
194
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