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Optical -microwave interaction modeling in superconducting film for microwave /photonic applications

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O p tical-M icrow ave In tera ctio n M o d elin g in
S u p erco n d u ctin g F ilm for
M icr o w a v e /P h o to n ic A p p lic a tio n s
A . H am ed M ajedi
A thesis
presented to the University of Waterloo
in fulfillment of the
thesis requirement for the degree of
Doctor of Philosophy
in
Electrical and Computer Engineering
Waterloo. Ontario, Canada. 2001
© A . H am ed M ajedi 2001
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A. Hamed Majedi
(h.
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A. Helmed Majedi
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H- M fy z d i
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A bstract
The increasing demand for high quality
c o m m u n ic a tio n
systems and signal processing is
constantly pushing the researchers to find new device concepts in microwave and optical
domains. This continuous exploration has recently led to the possibility of using hightem perature superconductors (HTS) in optoelectronics and microwave/photonic devices
with low-noise/low-power and high-speed/high-frequency characteristics. However, anal­
ysis and design of the superconducting optoelectronic devices require a physical modeling
for the optical/microwave interaction in these materials. Toward this
a im ,
we propose
in this thesis the superconducting photoresponse mechanism-models in order to explore
the interaction of light with high-temperature superconducting film. Two-Fluid model
is used to study the electrical signal propagation along the photo-excited HTS struc­
ture. By employing the effective tem perature model into the heat transfer analysis, the
fast bolometric photoresponse is investigated for potential applications in optoelectronic
and microwave/photonic components. A multifunctional HTS Optoelectronic device con­
cept and optically-controlled passive HTS microwave devices are introduced based on
the lumped optical/microwave interaction model in the HTS films. We demonstrate the
possibility of microwave harmonic generation and signal
m ix in g
in a current-driven photo­
excited HTS film. The concept of optical control of HTS microwave devices is applied
to the HTS delay line and resonator, leading to fine tuning of their characteristics. Fi­
nally. the traveling-wave HTS microwave/photonic device concept is introduced
s t e m m in g
from the analytical investigation of the traveling-wave optical/microwave interaction in
the HTS transmission line. The microwave propagation along the spatio-temporal con­
ductivity grating in the HTS transmission line is rigorously solved by using Maxwell's
equations with Floquet's approach and time-varying coupled mode analysis. Numerical
results obtained from these analyses serve to demonstrate the potential applications of
these devices in periodic filtering of the microwave signal suited for high-performance
microwave/photonic communication systems and radio over optical fiber applications.
iv
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A cknowledgem ents
Looking back at the past three years. I realize that this research work would not have
been possible without the people or organizations mentioned herein.
I would first like to express my deep gratitude and appreciation to my advisor. Pro­
fessor Sujeet K. Chaudhuri. for his continuous guidance, encouragement and support
throughout the course of my graduate study. His wisdom, insight and gift of teaching
have always inspired me.
I am also deeply indebted to Professor S. Safavi-Naeini. my co-supervisor, for his valu­
able suggestions and novel ideas.
Special thanks also goes to Professor Tom P. Devereaux for his help and thorough
comments and also Professor Raafat R. Mansour for his motivative discussions. I am very
thankful to Professor John S. Preston of the University of McMaster. for taking his time
to review this thesis.
The financial supports of the Natural Sciences and Engineering Research Council of
Canada and the Material and Manufacture of Ontario are greatly appreciated.
Finally and most of all. I thank my beloved wife. Sheva Naahidi. for her patience,
unwavering support and encouragement.
v
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I dedicate this dissertation to my beloved wife, Sheva Naahidi.
Without her faith and support, I could not have done it.
vi
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C ontents
1
2
3
Introduction
1
1.1
Superconducting M icrow ave/Photonics.............................................................
3
1.2
Problem D e s c r ip tio n .............................................................................................
4
1.3
Thesis O verview .......................................................................................................
5
E lectrodynam ic T h e o r y o f Superconductivity
7
2.1
BCS T h e o ry ..............................................................................................................
8
2.2
Electrodynamics Based on the BCS Theory
...................................................
10
2.3
Two-Fluid M odel and Classical Electrodynam ics.............................................
12
2.4
Plane Wave in S u p erco n d u cto rs..........................................................................
15
2.5
Parallel Plate Superconducting Transmission Lines
......................................
16
2.6
Summary
.................................................................................................................
19
P hotoresponse M echanism -M odels in Su percond uctin g Film
20
3.1
Bolometric and Nonbolometric P h o to re sp o n se ................................................
22
3.2
Effective T em perature Model
.............................................................................
23
3.2.1
Photo-induced Changes in Electrical P a ra m e te rs................................
25
3.2.2
Simulation R e s u l t s ....................................................................................
27
Heat Transfer M odel of Superconducting P hotoresponse...............................
32
3.3.1
Continuous Uniform OpticalIrradiation
...............................................
36
3.3.2
Time Harm onic Optical I rr a d ia tio n .......................................................
37
3.3.3
Pulsed O ptical Irra d ia tio n .......................................................................
37
3.3
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3.3.4
3.4
4
Traveling-Wave Type Optical
..........................................
39
S u m m a r y ................................................................................................................
41
Illu m in a t i o n
Optical-M icrowave Interaction in Su percon du cting Film
42
4.1
Basic Theory and M o d e lin g .................................................................................
44
4.2
Photo-Induced Electrical Changes in HTS F ilm
4.3
Optoelectronic Device A p p licatio n s....................................................................
49
4.3.1
HTS Optoelectronic Device C haracteristics..........................................
57
Optical Control of Superconducting Microwave D e v ic e s ................................
59
4.4
.................................................45
4.4.1 Optically Tunable Propagation Delay Time in HTSTransmission
L i n e ..............................................................................................................
60
4.4.2 Optically Tunable HTS Microwave Resonator and F i l t e r ........................64
4.5
S u m m a r y ................................................................................................................
5 Traveling-W ave O ptical-M icrowave Interaction
in H T S Film
5.1
Laser-induced Dynamic Microwave Grating inHTS F i l m .............................
5.2
Microwave Propagation in Spatio-Temporal Grating in HTS Transmission
68
70
72
L i n e ..........................................................................................................................
77
5.2.1 Floquet's A pproach.....................................................................................
78
5.2.2 Simulation Result from Floquet’s Approach
........................................
84
...........................................................................
92
5.2.3 Coupled Mode Analysis
5.2.4 Numerical Simulation of Laser-Induced DFB Structure in HTS Trans­
mission L in e
5.3
97
S u m m a r y .................................................................................................................. 103
6 C onclusion
6.1
105
Summary of C o n trib u tio n s..................................................................................... 107
6.1.1
Superconducting Photoresponse M odels....................................................107
6 . 1.2
Superconducting Optoelectronics
6.1.3
Optically-Controlled Passive HTS D e v ic e s ............................................. I l l
............................................................. 109
viii
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6.1.4
6.2
Traveling-Wave HTS Microwave/Photonic Devices............................... 112
Future Research D irections...................................................................................... 115
A
B C S C om plex C on d u ctivity Below Gap Frequency
118
B
E nergy Gap C alculation U n d er O ptical Irradiation
119
C
Frequency A nalysis o f O ptoelectronic H T S M ixer
121
D
Three-Term R ecurrence R elation for Spatio-T em poral H arm onics
126
ix
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List of Tables
3.1 Parameters used in the analysis of Photo-excited YBa 2Cu 30 7 _i film [65].
[66]
28
4.1 Parameters used in the analysis of Photo-excited YBaoCuaOr-i film.
...
50
4.2 Parameters used for parallel plate resonator employing YBCO film on sap­
phire substrate
.......................................................................................................
67
4.3 Computed values of resonance frequency shift and quality factors for vari­
ous temperatures and relative absorbed optical powers
................................
68
5.1 Parameters used in the analysis of microwave propagation in travelingwave photo-excited YBasCusO?-,*- transmission line..........................................
x
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85
List o f Figures
2.1
Energy diagram for superconductor and superconducting gap energy . . . .
9
2.2
Superconducting energy gap and photon irra d ia tio n .......................................
10
2.3
Circuit representation of the two-fluid model for superconductors
.............
14
2.4
Parallel Plate Superconducting Transmission L i n e ..........................................
17
3.1
Photo-excited superconducting film operating at microwave frequency. . . .
27
3.2
Energy gap modification and effective temperature of photo-excited YBa2Cu 307_,5
thin films parameterized by u. the optical frequency............................................29
3.3
Relative change in th e film resistance and kinetic inductance of photo­
excited YBa 2Cu 30 7- i thin films parameterized by v. the optical frequency.
3.4
30
The photoresponse of the YBa2Cu307_<5 film superfluid density from our
model and measurements done by Y. Liu et. al. [68 ] .......................................
31
3.5
Superconducting film on substrate and its equivalent thermal model . . . .
34
3.6
Dynamic Laser Interference Grating in the Superconducting F i l m ................. 40
4.1
Photo-excited HTS film and its electrical c o n fig u ra tio n ................................
46
4.2
Circuit model of HTS microstrip neglecting edge e ffe c ts................................
47
4.3
Steady state photoresponse of HTS film effective tem perature
...................
50
4.4
Photo-induced steady-state voltage response of HTS b rid g e ..........................
51
4.5
Photo-induced steady-state voltage response of HTS bridge with 1 mA DC
bias c u r r e n t ..............................................................................................................
4.6
52
Photo-induced steady-state voltage response of HTS bridge with 300 MHz
LO current
..............................................................................................................
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53
4.7
Photo-induced steady-state voltage response of HTS bridge with 1 mA AC
c u rre n t.......................................................................................................................
4.8
Absorbed optical power (W ). followed by photo-induced transient temper­
ature (K)
4.9
54
55
Voltage response of HTS bridge to the pulsed optical irradiation with (a)
1 mA DC bias current (b) 300 MHz LO c u r r e n t .............................................
56
4.10 Delay time per unit length of the microwave signal in PPSTL for various
tem p eratu re..............................................................................................................
61
4.11 Propagation time for 50 mm long PPSTL made of YBCO for various rela­
tive absorbed optical power
................................................................................
63
4.12 Optically-timed parallel plate HTS re so n a to r..................................................
66
4.13 Shift in resonance frequency of PPSTL for YBa 2Cu 307_,5 film on sapphire
substrate at different temperatures. (*) indicates the measured value by
T. Sindlekht et. al....................................................................................................
5.1
TM-wave interaction with laser-induced dynamic grating in HTS transmis­
sion l i n e ....................................................................................................................
5.2
79
Dispersion Diagram for microwave signal along the uniformly irradiated
HTS film /su b stra te ................................................................................................
5.3
67
86
Dispersion Diagram for microwave signal along the laser-induced spatiotemporal conductivity g r a t i n g .............................................................................
87
5.4 Relative power of the second backward space-time m o d e ................................. 88
5.5 Relative power of the first backward space-time m o d e ...................................
89
5.6 Relative power of the first forward space-time m o d e ......................................
89
5.7 Relative power of the second forward space-time m o d e ................................
90
5.8
Dispersion diagram for low temporal frequency of dynamic g ra tin g ................ 91
5.9
Geometry of the Laser-induced distributed feedback dynamic grating in
HTS transmission line
..........................................................................................
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96
5.10 Relative power transmission (in dB) as a function of microwave angular
fre q u e n c y ................................................................................................................
98
5.11 Relative power transmission (in dB) as a function of microwave angular
f re q u e n c y ................................................................................................................... 100
5.12 Relative power transmission (in dB) as a function of microwave angular
f re q u e n c y ................................................................................................................... 101
5.13 Relative power transmission (in dB) as a function of absorbed optical ra­
diation p o w e r ............................................................................................................ 102
C .l
Even harmonicsvs.absorbed optical radiation power........................................... 124
C .2
Odd harmonics vs. absorbed optical radiation power.......................................... 124
xiii
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C hapter 1
Introduction
W ith the surprising discovery of superconductivity by H. K. Onnes in 1911. scientists
began dreaming about practical applications for this phenomenon. The alloy supercon­
ductors worked at tem perature below 23K. known as low tem perature superconductors
(LTSs) with liquid helium refrigeration have found many remarkable applications in mag­
netic levitation for vehicle transport, powerful magnets for electricity generation, medical
imaging and most importantly cryoelectronics, superconducting optoelectronics and meso­
scopic structures. The fastest integrated circuits (ICs) and single photon detectors in the
world today are made with a superconducting metal, niobium and niobium nitride [1], [2 j.
Superconducting IC’s are based on a new logic family so called rapid single flux quantum
(RSFQ) logic, operating on the storage and transmission of magnetic fluxons. working up
to 770 GHz and superconducting hot electron photodetectors detect infrared optical sig­
nals with 25 GHz frequency modulation with one photon sensitivity. The RSFQ IC's and
superconducting photodetectors will leverage superconductivity superiority in performing
low-power. high-speed and high-accuracy electronic functions while the miniaturization
and increased speed of semiconducting ICs are limited by heat loss and intrinsic response
time of semiconductors. Superconducting ICs are suitable candidates for some applica­
tions in high performance analog, digital and quantum com putation, communication and
signal processing. The recent development of single electron superconducting transistor
along with subsequent analog and digital electronic devices and the discovery of the first
1
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plastic LTS and superconducting DNA excite the field of low tem perature superconduc­
tivity for more futuristic applications [3]. [4].
A major breakthrough was made by A. Muller and G. Bednorz in 1986 by creating
a brittle ceramic compound th at was superconducting at 30K. Soon after that P. Chu
and his collaborators took the discovery one step further and announced the first hightemperature superconductor (HTS) based on Yttrium compounds at 92K. well above the
temperature of the liquid nitrogen. Applications currently being explored are mostly the
extensions of the technology used with LTSs including magnetic devices, medical imaging,
superconducting quantum interface devices (SQUTDs). infrared detectors, signal proces­
sors. microwave and millimeter components and systems. The field of microwave and
optoelectronic engineering hold great promise for practical applications of HTSs. Using
HTS for implementation of high performance microwave filters and multiplexers for cellu­
lar phone base station and satellite communication are amongst the most important HTS
applications [5]. [6]. The increasing demand of wireless and optical networking infras­
tructures with the interference and interface problems can significantly degrade capacity
and data rates which can be successfully improved by advanced HTS front-end technol­
ogy. Many compact microwave components such as delay lines, phase shifters, millime­
ter wave detectors, mixers and low power amplifiers and oscillators which have already
been used for applications in radio astronomy, remote sensing and military services, now
can be employed for real wireless and spread-spectrum communications and networks.
Superconducting optoelectronic devices could improve the quality and the speed of the
interface between electronic processing and optical transmission network. The ultrafast
data processing speeds that are achievable in RSFQ ICs can usher a transformation in
telecommunications networking. RSFQ switches and routers would enable wire-rate cir­
cuit switching and packet processing in the order of terabits per second in a single optical
wavelength channel. Crossing to the microwave/photonic applications, the microwave sig­
nal can be modulated onto an optical carrier and then the transmission, processing and
detection can be done in the optical domain and the recovered microwave signal would
2
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be delivered to the front-end cryoelectronics. The microwave signals in the supercon­
ducting devices and systems can be generated, manipulated or controlled externally by
optical radiation. The viability of the superconducting microwave/photonic applications
depends on the understanding of the photo-absorption phenomenon and its impact on
the electrical properties of the HTSs. and then introducing the device concepts, analyses,
simulations, designs, fabrications and implementation at the system level.
1.1
S u p erco n d u ctin g M icro w a v e/P h o to n ics
Recent developments in photonic technology and superconducting electronics is opening a
window of opportunity for superconducting optoelectronics [7] and microwave/photonics.
Superconducting optoelectronics refers to electronic functions performed by optical irradi­
ation via the photo-absorption phenomenon in superconductors in cryogenic environment,
superconducting microwave/photonics introduces a new technique for the distribution,
control and processing of microwave and millimeter-wave signals in the superconducting
transmission lines by optical radiation. Both disciplines take the advantage of unique com­
bination of high electrical conductivity and ultrafast photoresponse in superconductors for
a physical interaction to perm it coupling between the electrical and optical domains. The
photo-absorption phenomenon in superconductors, somewhat like a photoconductivity ef­
fect in semiconductors [8]. provides an opportunity for a novel class of cryo-optoelectronic
devices as well as a new technique for optical control of superconducting microwave de­
vices. These devices extend from integrated optoelectronic photodetectors, mixers, mod­
ulators [9]. [10]. [11] and optically tuned microwave devices for microwave/photonic ap­
plications to high current fast switches for power applications [12], [13], [14]. Since the
superconducting optoelectronic devices are intended to operate at cryogenic environment,
it is expected that they would exhibit low-noise/low-power and high-speed/high-frequency
characteristics. These characteristics are of critical importance for high performance radio
over optical fiber systems for wireless data and cellular radio systems, optically controlled
phased array antennas and microwave signal processing.
3
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1.2
P rob lem D escrip tio n
The main motivation and purpose of this thesis is to introduce a theoretical frame­
work for dealing with optical-microwave interaction in superconducting films for mi­
crowave/photonic device applications.
Central issue in this type of interaction is the
nonlinear variation of the complex conductivity of superconducting film in the microwave
and millimeter wave ranges under the application of proper optical radiation. Our ob­
jective is to analyze the microwave signal propagation along the superconducting trans­
mission line when the optical radiation changes the complex conductivity of the structure
both in the lumped and distributed forms.
In the lumped optical-microwave interaction, our focus will be placed on the photo-induced
changes in electrical properties of the superconducting film such as the kinetic inductance,
and the proposed analysis will be accomplished by combined heat transfer analysis and
the two fluid model. Because the length of the photoexcited region is much smaller than
the wavelength associated with the input electrical signal, the circuit and transmission
line theories are the best candidates for analysis. The result of this systematic approach is
demonstration of the novel superconducting optoelectronic devices such as optoelectronic
mixer and modulator.
Moving one stage further, the traveling-wave optical-microwave interaction will be inves­
tigated theoretically. The periodic changes of the complex conductivity of the supercon­
ducting film by the induced spatial and temporal optical radiation patterns can be used to
modify the propagation properties of the microwave signal. Our particular effort will be
set to find the condition for stable and optimum interaction and microwave propagation
characteristics control in distributed photoexcited structure. We will study the microwave
propagation along the photo-induced time-space periodic structure in superconducting
film by using rigorous electromagnetic analysis by Floquet's approach. This formula­
tion gives us the microwave propagation constant and the amplitude of the different
spatio-temporal harmonics in an infinite structure. The dispersion diagram reveals many
interesting potential applications for microwave signal processing by optical radiation.
4
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Crossing to the more practical case, approximate formalism based on the time-varying
coupled mode theory will help us to investigate the optical-microwave interaction in a fi­
nite structure. Photo-induced dynamic microwave distributed feedback (DFB) structure
can be achieved in finite length of superconducting film. Since this structure is frequency
selective in both tim e and space domains, the filtering functionality along with fine tuning
could be gained. This type of novel device can be employed for high-speed/high-frequency
microwave/photonic systems and high frequency microwave signal generation from laser
radiation.
1.3
T h esis O verview
This thesis is organized into six chapters.
The first chapter stated the research mo­
tivations. problem description and also introduced the new research direction in mi­
crowave/photonics.
A general background and mathematical formulation of the elec­
trodynamic theories of superconductivity is provided in Chapter 2 by an overview of BCS
theory and the two-fluid model. BCS theory will be used for developing an effective tem­
perature model of superconducting photoresponse and the two-fluid model will help us
to study the microwave signal propagation along the superconducting structures. Chap­
ter 3 will discuss the photoresponse mechanisms and associated models, with emphasis
on the heat transfer analysis for fast bolometric photoresponse. The lumped opticalmicrowave interaction will be considered in chapter 4. After basic theory and modeling,
the application will be split into two different categories. The first one is optoelectronic
device applications and the second one is optically controlled microwave devices. Each
device application will include the physical concept, simulation and some design issues.
One of our key contributions in this thesis will be presented in the Chapter 5. when the
traveling-wave optical-microwave interaction will be studied. First, the physical condition
for formation of photo-induced
d y n a m ic
microwave grating will be discussed in the con­
text of our previous photoresponse model. The microwave propagation is then analyzed
in a grating structure through Floquet's and time-varying coupled mode theories. At the
5
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end. the photo-induced DFB structure will be introduced for potential applications in mi­
crowave signal processing. Finally. Chapter 6 summarizes the thesis, provides conclusion
drawn from the research work, presents our contribution and outlines possible research
directions for future study.
6
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Chapter 2
E lectrodynam ic T heory of
Superconductivity
After the discovery of superconductivity, many theoretical studies have been made by
physicists to interpret this phenomenon. All the theories before 1956. attem pted to de­
scribe the behavior of superconductors most im portantly the zero resistivity and perfect
diamagnetism (Meissner effect), in terms of classical thermodynamics and electrodynam­
ics such as the Two-Fluid model. London and Ginzburg-Landau theories. These theories
did not however provide the physical understanding of resistanceless current and Meiss­
ner effect. They are however well-suited mathematically and phenomenologically for some
engineering applications.
The first widely-accepted theoretical understanding of superconductivity was advanced in
1957 by J. Bardeen. L. Cooper, and J. Schrieffer. the so-called BCS theory [15]. The quan­
tum mechanical-based BCS theory explained superconductivity for elements and simple
alloys. Despite the success of this theory which explained superconductivity phenomena
such as the electron pairing mechanism, the existence of the gap energy and superconduc­
tivity response to the electromagnetic radiation, it did not anticipate the development of
superconductivity at tem peratures above 77K. Now. 15 years after the discovery of HTSs.
the
m e c h a n is m
of superconductivity is still under debate and several models and theories
7
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have been proposed.
To set the proper context for our research work and establish the basic theoretical tools,
first the BCS theory will be discussed to provide a background for photoresponse studies
in both LTSs and HTSs. The focus will be placed on the existence of the energy gap and
the effect of electromagnetic radiation on the superconductors. After that, the Two-Fluid
model will be considered for dealing with microwave propagation along a superconducting
structure, since the electrodynamics based on this model can be justified with the BCS
theory and it is more mathematically tractable.
2.1
B C S T h eo ry
Superconductivity is a macroscopic quantum phenomenon. According to the BCS theory
[15]. when the tem perature of a material drops below its critical temperature. Tc. in the
absence of any external
s t im u l i ,
the normal electrons with opposite spins and momenta
begin to form pairs, the so-called Cooper pairs. The net spin of a Cooper pair is zero,
which means that Cooper pairs act like bosons and can occupy the same quantum state.
The condensation of normal electrons below Te is a consequence of the formation of a
tem perature dependence energy gap A (T) in the electron density of states above and
below the Fermi level, as depicted in Fig 2.1. The energy gap for temperatures near Tc
predicted by the BCS theory can be expressed in a more general case to consider both
LTSs and HTSs as:
A(D = A„(l- ( £ ) ’)
(2.1)
where Ao is energy gap at zero temperature, and 7 is an exponent which depends on the
material [16]. At any tem perature between zero and Tc the superconducting density of
state near the Fermi level is:
E
forE > A (T)
N,(E) = AT(0) ^ v/ ^ 2 " M T ) 2
0
(2 .2 )
for£ < A (T)
8
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Figure 2.1: Energy diagram for superconductor and superconducting gap energy
where 7V(0) is the density of states at the Fermi level. In an equilibrium state, the
distribution function of quasiparticles with respect to their energies has a form of wellknown Fermi-Dirac function [17]:
f ( B ) = ---------— g 1+
eXp(^ T )
(2.3)
where kg is Boltzmann's constant and E is the energy. Via electron-phonon-electron
attractive interaction Cooper pairs are bound together with an energy A (T). but in order
to overcome their energy gap to excite them above ground state. 2A(T) energy is needed.
The energy gap for most superconductors falls within a range corresponding to the infrared
region. Therefore, by applying photons with energies hu > 2A[T) the Cooper pairs can
be broken and electrons excited to the normal state, as shown in Fig 2.2. Historically, this
theoretical prediction was confirmed experimentally by Glover and Tinkham in 1957 [18].
The mean distance between paired electrons the so-called coherence length is quite large
in LTSs in the order of micrometer, and for HTSs in the order of nanometer so their
wavefunction overlap. Consequently, the bosonic behavior and large coherence length
9
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o
Normal electrons
Excited state
A(D
Ground state
Cooper pairs
Figure 2.2: Superconducting energy gap and photon irradiation
combine to lock the quantum phases of the pairs together, and their quantum state can
be described by a single wavefunction that obeys Schrodinger's equation. This makes the
dynamics of the Cooper pairs very different from quasi-classical behavior of the normal
electrons. The infinite conductivity and quantization of the magnetic flux are merely a
macroscopic consequences of the coherence of the Cooper pair condensate.
2.2
E lectro d y n a m ics B ased o n th e BC S T h e o r y
The electrodynamics based on BCS theory provides a justification for the phenomenologically accepted Two-Fluid model. Under the application of an external electric field
in the form Ee 7"4 the superconducting current consists of two non-interacting currents,
one is called a supercurrent representing the collisionless movement of Cooper pairs, and
the other one is a normal current representing the inertial and resistive movement of the
normal electrons. The superconductor can be described as having a complex conductivity
10
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(cr = <Ti —j(r2). which was first derived by M attis and Bardeen as [19]:
—
=
(Tnc
T~ T
[ f ( E ) - f ( E + Hu,)]g(E)dE
(2.4)
ftoJ J A [ T )
l
+
r-M T)
[ l - 2 f ( E + tku)]g(E)dE
t- /
n v JA(T)-hw
fL
<Tnc
=
' r
T)
[i-2/(g + M
y
*+*(T)+*»b
_
dE
y/& 2{T) - E*y/{E + M 2 - A 2(T)
« w i 4 |T ) - h ..4 ( I |l
(2.5)
where h is reduced Planck's constant,
uj
is angular frequency. f ( E ) is the energy dis­
tribution function for quasiparticles given by (2.3) in an equilibrium state, o v is the
conductivity at Te. and
E 2 + A 2{T) + HwE
y / E 2 - A 2( T ) y / ( E + M 2 - A 2( T ) '
While the first integral in equation (2.4) represents the effect of thermally excited normal
electrons, the second one accounts for the contribution of photo-excited quasiparticles
and is zero for hu> < 2A (T). The inertial term in equation (2.5) includes only the effect
of paired electrons and its lower limit is taken as —A (T) if tuu > 2 A (T). For microwave
2A(T)
frequency. / . below the gap frequency (ug = — -— ). the complex conductivity can be
expressed as [Appendix A]. [20]:
AhoT
hf
—
=
(Tnc
hf
f)f
} d E E=A{T)
[i - 2/(A (r )) ]
( 2 .6 )
(2.7)
The last two equations predict the thermal and frequency behavior of the superconducting
complex conductivity below its gap frequency, typically 100 GHz. We will use these results
in the next chapter when the photo-induced electrical properties of superconducting film
will be further studied.
11
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2.3
T w o -F lu id M od el an d C la ssica l E lectrod yn am ics
The classical electrodynamic theory of superconductors was originally established by F.
London. This macroscopic theory essentially incorporates the zero resistance and perfect
diamagnetism of superconductors into electromagnetic constitutive relations. The basic
idea of this phenomenological theory is the Two-Fluid model presented by C. Gorter
and H. Casimir in 1934.
According to this model, the electron fluid is composed of
superelectrons (similar to Cooper pairs in the BCS theory) at the lowest-energy state
and normal electrons in the excited state. The number of carriers is dependent on the
tem perature and can be given by the following empirical relations [16]:
».<T) =
„ ( l - ( |p )
(2.8)
T i-m
n (^ )’
(2.9)
=
where n„ and nn are the Cooper pair and normal electron number densities, n = n, +
Tin is
the totalelectron
number density and 7 is an exponent. For
the conventional
superconductors such as Pb and Nb. 7 Rs 4 and for mostHTSs such asYBaCuO. 7 s; 2
has been suggested [16]. Under the application of an external electromagnetic field, the
movement of the Cooper pairs is purely inertial but the motion of the normal electrons
includes the effects of both inertial and resistance due to their collision with the lattice.
W hen the applied electric field. E. and magnetic field. B . have a period of oscillation much
smaller than the electron-phonon relaxation time, re_pft. (or the associated frequency is
much less than the gap frequency in the context of the BCS theory) the hydrodynamical
equations for superelectrons and normal electrons, the so-called first London equations,
can be written as follows [21]. [22]:
D
r
m<!l i t Vs^r * ~ e [E (r ‘t ) + v *(r '
n
x
(2.10)
and
m e-^ -(v n( r .t) ) +
•t't
T~e—ph
= e [E (r.t) + (vn ( r .t)) x B ( r ,f )]
12
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(2.11)
where vs(r. £). and (vn(r. f )) are the velocity of the Cooper pairs and the average ve­
locity of normal electrons. me and e are mass and charge of the electron, respectively.
7 r-v(r.t) = -j-v(r.t) + (v(r.£).V)v(r.t) stands for total derivative, which includes the
Dt
dt
local variation of a vector v(r. t) at a specific point, r. as well as the change due to the
motion of the fluid [23]. Since the electromagnetic field. E. and B applied to the super­
conducting media are assumed to be far below their critical values, the carrier velocities
can be given by [16]:
f E(r. t)dt
(2.12)
(vn(r.£)) = - ^ ^ E ( r . £ )
me
(2.13)
vs(r.£) = — —
T7lc J
and
The total current density produced by the external electric field consists of two non­
interacting currents namely superfluid and normal current as [24]:
J(r. £) = J s(r.£) + Jn(r.£) = en,(r. £)vs(r. t) + en„(r.£)(vn(r.£))
(2.14)
We use the notions n ,(r. £). and nn(r. £) to generalize the definition of the electric current
according to the hydrodynamical basis for the Two-Fluid model. Physically speaking,
the total current might be varied through the carrier velocities or the number of carriers
by externally applied electromagnetic field or any other type of stimulus. This is the key
concept when the optical-microwave interaction will be considered. Since the optical ra­
diation changes the number of carriers via thermo-modulation effect, the carrier velocities
will be determined by the input microwave signal in a linear regime.
Combining equations ( 2 . 12 ). (2.13) and (2.14), the current-field relationship in the super­
conducting media can be written as:
J(r.£) = <r,(r.£)
J
E(r. t)dt + <rn(r. £)E(r, £)
(2.15)
where
<r,(r.£) =
<rn(r.£)
=
e2
— n,(r. t)
mc
2
Te. ph— nn( r . t )
7TO„
13
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(2.16)
(2.17)
cra(r.t) is the inverse London param eter, and <rn(r. t) is the normal conductivity l.
When the current-field relationship is considered under the lumped assumption, the equiv­
alent circuit model for the superconducting medium consists of the kinetic inductance.
Lit, indicating the presence of the Cooper pairs and the resistance. Rn. presenting the
effect of normal electrons, as illustrated in Figure 2.3 [25]. When the superfluid is much
Figure 2.3: Circuit representation of the two-fluid model for superconductors
greater than the normal current, the normal electrons are moving within the diffusion
skin depth and experiencing the lattice collision due to the skin effect, while the Cooper
pairs are flowing within the London penetration depth without any collision, due to the
screening effect [26]. According to the two-fluid model the London penetration depth is
frequency independent and can be w ritten as [25]:
mA im = — =
Ho
2e2fi0ns(T)
(2.18)
where Lk is the kinetic inductance and the diffusion skin depth can be expressed as:
K(T) =
2Rn
u)fi0
2
TTle
ijfi0 e2nn(T)re- ph
(2.19)
Since the London penetration depth is much smaller than the diffusion skin depth, this
implies that the conduction process and consequently the current flow is mainly focused
within the London penetration depth. This physical picture is very useful when the ap­
plied electromagnetic field does not cause any nonlinearity in the superconducting medium
and its frequency is well below the gap frequency.
1Note that the definition o f
<r3(r. t)
is slightly different from the com mon notion in the supercon­
ductivity textbooks. The inverse London parameter can be related to the complex conductivity in the
^
1
T71
frequency domain as <Ti —jcr-> = cr„ —j — = cr„ — j — where A = ~ ~ is London parameter.
o>
uA
e-n ,
14
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2.4
P la n e W ave in S u p erco n d u cto rs
Maxwell's equations form the basis of electromagnetism and can be easily adopted for use
with superconductors through the associated constitutive relations. The four fundamental
Maxwell's equations in superconducting media are:
3D
dB
dt
V xE =
V.D
=
(2 .20 )
(2 .21 )
0
( 2 .22 )
pP
(2.23)
where for most superconductors D = e0E. and B = /ioH.
The wave equations in superconductors can be derived from Maxwell's equations. Writing
down the general wave equation for the electric field as
v 2 E = /i0^
+ /i°e° W
(2'24)
and combing with the current-field relationship (2.15). the wave equation in superconduc­
tors can be given by:
V 2E ( r .t) =
J
E (r. t)dt + <r„(r. t)E(r. f)] + p QeQd
(2-25)
Similarly the wave equation for the magnetic field can be obtained. Recall that equation
(2.25) is the wave equation for superconductor using the Two-Fluid model and this equa­
tion might be considered with the results from the electrodynamics based on the BCS
theory, specifically relationships (2.6), and (2.7).
Now, we consider the case of sinusoidal time variation for the all field quantities in the
form of AeJ“‘ and furthermore we assume that the Cooper pair and normal electron num­
ber densities do not depend on the space and time variables. These assumptions lead to
the following wave equation:
V 2E = 7£E
15
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(2.26)
where j i is the complex propagation constant and can be given by:
7L = a +
jP= y //ioO-. -
w*
p- +
jwiWn
(2.27)
where c is the speed of light and a. and f3 are th e attenuation andpropagation constants.
respectively2. The intrinsic impedance of the superconducting medium can be obtained
as:
Z, = t ± ° = R a + j X .
(2.28)
1L
The solution of the wave equation (2.26) gives us a forward plane wave traveling in the
z direction
in a superconducting medium filling the half-space z > 0 . and backward
propagating plane wave in the —z direction in the material filling the half-space z < 0 . in
the form of:
E = E e - ^ a * + Ee7tZax
(2.29)
H = H e -^ a j, +
(2.30)
Equations (2.29) and (2.30) represent the TEM wave, which has a constant amplitude
and phase over the xy plane.
The intrinsic impedance and the velocity of plane wave in superconducting media can
be subsequently obtained through their definitions [29], [30]. The above results will be
extensively used in Chapter 4 and 5. when the microwave signal propagation is considered
in photo-excited superconducting film.
2.5
P a ra llel P la te S u p erco n d u ctin g T ransm ission Lines
Incorporating superconducting materials, especially HTSs, in planar transmission lines
is very promising for applications in microwave and millimeter-wave devices due to their
2Alternatively, the com plex propagation constant can be written as 7^ = -yju>; — u 2 + juTe- pi,ul.
- > 71, 6*
't
n„e"
iere w2
where
u~ = ------- . and w" = -------- are the plasma frequencies o f the Cooper pairs and normal electrons
mt e0
mee0
[27], [28]
16
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low surface resistance. Superconducting
tr a n s m is s io n
lines (STLs) are also attractive for
propagation of picosecond electrical pulses for RSFQ interconnects and high speed su­
perconducting optoelectronic devices [31], [32]. [33]. Ultrashort pulses or high frequency
analog signals can propagate down the STLs without suffering by resistive losses as long
as the bandwidth of the electrical pulses or frequencies of analog signals do not exceed
the gap frequency of the superconductor. Provided the transmission line propagates in
a TEM mode. STL is dispersionless. due to the penetration depth not varying with fre­
quency [section 2.3].
STLs have been used in different geometries ranging from parallel plate to coplanar struc­
tures to serve as components or interconnects [34]. The following subsection looks at the
parallel plate or wide microstrip structure because of its simple nature and widespread
applications. The parallel plate superconducting transmission line (PPSTL) consists of
two infinitely long superconducting plates with thickness d\ and do separated by a dis­
tance h filling by a dielectric material with relative dielectric constant er . as depicted in
Fig 2.4. The width of the plates is w such th a t w 3 > h. implying that no fringe fields
w
Figure 2.4: Parallel Plate Superconducting Transmission Line
17
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exist outside the plates. If the field penetration into the superconducting plates is not
important, namely h
A^. and d 12 2 > A^12. the lumped transmission line model is quite
sufficient and the PPSTL can be modeled by the kinetic and magnetic inductances, a
capacitance and a small resistance [35]. [36]. However, if the field penetration into the su­
perconducting plate is im portant, then the Maxwell's equations have to be solved directly
along with boundary conditions which completes the solution. The general solution of
wave equation (2.26) can be obtained for the TM case [Ex(x. z). Ez(x. z). Hy(x. z)]. and
Q
TE case [Hx(x. z ). Hz(x. z). Ey(x. z)\. when — = 0. For instance the Ez[x. z) component
oy
in the TM case for di = <i2 = d and A^ = \ L2 =
can be given by:
Ez(x. z) = (Bbe ^ +
)(Abe rLz + A f e ~ ^ )
(2.31)
where Bb. Bf. Ab. A f are the constants depend on the boundary conditions and
7 £ = 7 L + 7L
(2-32)
where 7 i x. and 7 lz are the complex propagation constants along x and z directions, re­
spectively. The rest of the field components can be easily obtained through the Maxwell's
equations and the undeterm ined coefficients can be found from matching the field at
the boundaries of the superconducting plate [37]. or the surface boundary condition
method [29]. It should be noted that if the PPSTL is operating far below critical tem­
perature and is not too wide. 7 ^ ~ 'yjjX gives a very good approximation. This helps us
to analyze the microwave propagation along the photo-excited PPSTL by keeping 7 x,r
unchanged, while the optical radiation mostly affects 7 l zA more interesting problem, and one of practical significance in the photo-excited P P ­
STL is the problem when the microstrip is formed of thin film superconductors to absorb
the optical radiation efficiently. For this case the complex propagation constant can be
expressed as [29]:
lLz=lL\j{l
w
+
2Xt
. 2R* \
-s r - 3 ---------- y
u}fi0h
uifioh
18
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.
.
(2.33)
where Z f = R f + j X f is th e complex surface impedance. According to the relationship
(2.28). the real and imaginary parts of the complex surface impedance of a superconduct­
ing film can be written as:
R f l .2 = /2 ,f c o th ( - ^ - ) + —
L
^ 1.2
K
1
(2.34)
lLl.2
X fl2
=
A ,coth( —
)
(2.35)
1L1.2
A very useful formula can be made by ignoring the superconducting loss, for a propagation
constant as:
,3 u
13 =
/
f i , ^ l i (T)
+ T
"
~
di
AL2{T) ~
d2
“ 'w
i 1 + ~ h ~ cotM k j T ) )i
( 2 ' 3 6 )
This formula represents th e dispersion-relation for an PPSTL. which is linear function
of angular frequency as it is expected. It should be noted that several analytical and
numerical solutions for multilayered PPSTL are also available in the literature for further
studies [38]. [39]. [40], [41]. [42].
2.6
S um m ary
In this chapter we introduced the electrodynamic theories of superconductivity which
will be used in this thesis. BCS theory was discussed in order to investigate the photo­
absorption phenomenon in superconductors and after th at the BCS based electrodynamics
presented for the future photoresponse models that will be introduced in the next chap­
ter. The Two-Fluid model along with the classical electromagnetic theory helped us to
investigate the microwave propagation characteristics including the complex propagation
constant and intrinsic impedance in the superconducting medium. Finally, the supercon­
ducting transmission line was introduced with the focus on the parallel plate structure
with thin superconducting film, which will be used as a basic structure for the future
optoelectronic device proposals.
19
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Chapter 3
P h otoresp on se M echanism -M odels in
Superconducting F ilm
The photoresponse of superconductors generally provides the fundamental insight into the
mechanism of superconductivity in both LTSs and HTSs and also the information needed
for potential optoelectronic device applications. Historically, the application of LTSs have
been limited to the far-infrared detectors and fast opening switches. Their performance
is superior to semiconducting counterparts in terms of low noise and high sensitivity
criteria, making them the best candidate for remote sensing and radio-astronomical ap­
plications.
The advent of high-Tc superconductors has opened new opportunities for
optoelectronic applications. Unlike the conventional low-Tc superconductors the copper
oxide HTS materials are very good light-absorbers due to their low plasma frequency
associated with the low carrier density [43]. in the range from ultraviolet to the near
infrared [44], Therefore in term s of external quantum efficiency. HTS materials are fa­
vorable material for potential optoelectronic applications. The HTS crystalline structure
is similar to optoelectronic materials, enabling their epitaxial growth on electro-optic
and magneto-optic substrates [7]. [45]. HTS m aterial can be made to exhibit interest­
ing nonlinear and contrasting electrical and optical properties by tuning their oxygen
content. This feature makes them attractive for design and fabricating monolithic super-
20
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conducting/semiconducting/insulating films for use in novel band gap structures, similar
to photonic band gap materials, and quantum devices. Keeping this type of applications
in mind, we investigate photoresponse
m e c h a n is m s
and models in superconducting films
with an engineering approach.
Two classes of photoresponse commonly referred to as bolometric and nonbolometric (nonequilibrium) have been identified in LTSs similar to other solid-state materi­
als [46]. [47]. These type of photoresponse can be understood within the BCS framework
and its extensions such as Eliashberg theory [48]. For the HTSs. theories describing the
photoresponse mechanisms are non-conclusive and have had limited success so far because
of the lack of clear understanding of the mechanism of superconductivity in in these ma­
terials.
In this thesis, we are interested in potential applications of superconductors in optoelec­
tronic devices, so the emphasis of our approach is to investigate the photoresponse in a
more applied rather than pure theoretical way. Therefore, after reviewing the generally
accepted picture of bolometric and nonbolometric photoresponse categories in supercon­
ductors. we will develop an effective temperature model within the BCS formalism. The
advantage of this model is that presents a systematic formulation that relates the optical
irradiation characteristics to the effective temperature of the superconductor from very
low level of optical radiation all the way to levels that destroy the superconductivity. The
heat transfer model will be then discussed for both classes of photoresponse in supercon­
ducting thin films on a dielectric substrate, and the concept of the thermo-modulation
will be presented. We will show that the heat transfer model agrees with the result of the
effective tem perature model. This general model will be then used as a basic model to
analyze the effect of optical radiation on the electrical properties of the superconducting
film in the next chapter.
21
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3.1
B olom etric an d N o n b o lo m etric P h o to resp o n se
As we previously pointed out in section 2.1. in the superconducting state, normal elec­
trons with opposite spins and momenta form Cooper pairs with an energy gap A(T) in
the electron density of states as depicted in Fig 2.1. Cooper pairs are bound together
with an energy A (IT), in the order of a few mili-electron Volts which falls within a range
corresponding to the infrared region. Thus, by applying photons with energy hv > 2A(T)
the Cooper pairs can be broken and the electrons excited to the normal state. These en­
ergetic electrons, so-called quasiparticles, have different ways to lose their excess energy
by exchanging it with the Cooper pairs, thermally-excited normal electrons and phonons.
The way in which the photo-excited electron energy gets distributed and subsequently
dissipated within the superconductors identify photoresponse mechanisms.
When optical radiation is applied in a time scale greater than the electron-phonon re­
laxation time, the photo-excited electrons reach thermodynamical equilibrium with the
phonons through electron-phonon collisions, and both electron and phonou subsystems
can be described by the same tem perature . resulting in a bolometric photoresponse [20 ].
Normally, the speed of the bolometric response is limited by the time required for trans­
ferring the excess heat from the photo-excited region to the phonon subsystem as a heat
sink [49] and the hot electron out-diffusion process [50]. If the superconducting film is
mounted on a dielectric substrate with a thickness comparable with the optical penetra­
tion depth, then the photogenerated heat might diffuse through the thermal boundary
leading to the exchange of the energy between quasiparticles and the lattice of the sub­
strate material. In this case, depending on the material, geometry and thickness of the
HTS film and its substrate as a heat sink the speed of the bolometric response may vary
from milliseconds to a few nanoseconds [51].
In the non-bolometric regime the quantum response of the electron and phonon subsys­
tems are manifested. Generally, the photon stream is applied in a very short time scale in
a form of pulsed radiation. The optical pulse duration is smaller than the electron-phonon
relaxation time, so the photo-excited electrons do not have any chance to reach equilib-
22
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ritun with phonons. More specifically, two quasiparticles with excess energy hv —2A(T)
are formed upon absorption of each photon. If the excess energy is greater than 2A(T).
then these quasiparticles can lose their excess energy by breaking more Cooper pairs in
an avalanche-like process, which is quite fast in a few picosecond time scale or by emit­
ting phonons at a rate dictated by electron-phonon relaxation time. re_p^. If the emitted
phonons have energies greater than 2A(T). then they can break more Cooper pair. Once
photo-excited electrons have lost most of their excess energy, they can then recombine
to form Cooper pairs again. For recombination to occur, an associated phonon must
be emitted which has an energy larger or at least equal to 2A (T). This recombination
phonon can then break more Cooper pairs or reach the equilibrium with phonons or can
escape out of the film and into the substrate. Note th at in an equilibrium state. Cooper
pair breaking and quasiparticle recombination are taking place with equal rates [52].
In both bolometric and nonbolometric photoresponses, most experimental works demon­
strate that the phonon escape through the substrate is the slowest process and imposes
a serious constraint on the speed of the photoresponse [53]. [54]. [55]. Thus, from an ap­
plication point of view, w ithout much more attention to the physics behind the exchange
of the energy between photo-excited electrons and other particles, we can still model the
manifestation of the excess energy as an effective tem perature along with the phonon es­
cape process as a bottle neck. This concept is the foundation for heat transfer models for
the photoresponse and will be treated mathematically in the context of the BCS theory
in the following section.
3.2
E ffective T em p eratu re M o d e l
This section presents a steady state photoresponse analysis in a superconducting film.
This model is developed for investigating the capability of superconductors for potential
optoelectronic application. We consider the thermodynamical variation under the appli­
cation of continuous optical radiation in the superconducting state, far below its critical
tem perature and in the absence of any other external stimulus. It is assumed that the su-
23
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perconducting film contains an electron gas including both paired and unpaired electrons.
Each absorbed photon with energy hu > 2A (T), breaks the Cooper pairs and creates
energetic quasiparticles. By ignoring the quantum effects that take place on a short time
scale of the thermalization process, we can calculate the thermodynamical parameters of
the superconducting film in terms of the photon characteristics [56]. This can be done
through the calculation of the chemical potential, /i. since the electron gas could be con­
sidered in thermal equilibrium with an effective temperature. T ' f f • although the paired
and unpaired electrons are not in chemical equilibrium. A. G. Aronov calculated the
chemical potential for a continuous optical pumping based on the Boltzmann transport
equation [57]. For a thin superconducting film, when the mean free path of all phonons is
larger than than the effective film thickness and photo-excited quasiparticles are scattered
by the equilibrium phonons. the energy distribution function can be represented by the
following quasi-equilibrium Boltzmann function well below Tc:
M E ) = exp ! *1
)
(3.1)
where kg is the Boltzmann s constant. The chemical potential \i is determined by the
conservation law for the quasiparticle num ber and is given by [58]:
, = A (r ) + ^
ta [
| ^
+ ^ ) ] .
(3.2,
where.
irkgT
, A (IY
Xo " V 2A (T) 6XP( kBT h
, 4
(3’3)
( 3 ' 4 )
In the above. x0 is a dimensionless concentration of quasiparticles inside the superconduct­
ing film. Ia is the absorbed radiation intensity.
is the energy gap relaxation time, A0 is
the skin depth for the optical wave, d is the film thickness. Z q is the intrinsic impedance of
the free space and D is quasiparticle diffusion coefficient. Using self-consistent equation
24
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of the energy gap. the energy gap modification. SA. by the optical irradiation can be
expressed as [Appendix B]:
6MT)
A(T)
n!,
, /x .. ttIcbT
,- A ( T ) .
eXp^kBT ^ ] j 2 A ( T ) eXp^ kBT *
( *
Equation (3.5) predicts that the energy gap would decrease as the chemical potential /x is
increased, since by Cooper pair breaking, normal electrons are created and consequently
the temperature is raised. This fact with equation (2.1) leads us to evaluate an effective
temperature. Te/ / . due to energy gap reduction as:
Tc
= [(— y
[[TC}
A0 1
(3.6)
K
’
S£.(T)
T
T
Since — ---- is negative. - *** > — Equation 3.6 indicates how the continuous optical
Ao
Tc
Tc
radiation can change the thermophysical properties of th e superconductor, more specif­
ically by decreasing the energy gap and raising the effective temperature. It is worth
noting that the effective temperature could be viewed as the electron temperature when
the quantum photoresponse is manifested as will be described later in the two-temperature
model, and as the macroscopic temperature when the bolometric response appears. The
superconducting to normal phase transition point is determ ined when the absorbed opti­
cal radiation intensity reaches its critical value. We can find the critical absorbed optical
intensity when S A ( T ) = —A ( T ) or Tef j = Tc. so the validity of discussed analysis is
restricted to the region when the effective tem perature is below the critical temperature.
This was confirmed by the experiments for conventional superconductors [46]. [59] and
YBaoCuaOT—i [60].
3.2.1
P h oto-in du ced Changes in E lectrical Param eters
According to the electrodynamic theories of superconductivity discussed in Chapter 2 .
below the gap frequency and under the application of an external electric field in the
form EeJU,£. the superconducting film can be considered as having a complex conductivity
25
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in the BCS theory. The effect of photo-excitation on the conduction process can be
considered by using the nonequilibrium distribution function, formula (3.1) and the energy
gap modification, formula (3.5). through the simplified Mattis-Bardeen equations (2.6)
and (2.7). Then, the change in the complex conductivity is as follows:
JS Il = 2L Wk-B— + c o t h ( ^ ^ ) ] ( e x p ( - ^ —) - 1) exp( —
<r2
y 2A(T)
y2kBT liK VKkBT }
}
kBT
(3. 8)
K ’
Equations (3.7) and (3.8) indicate that the relative change in the real part of the con­
ductivity is positive implying that the number of normal electrons contributed in the
conduction process is enhanced, while the relative change in the imaginary part of the
conductivity is negative implying th at the number of Cooper pairs contributed to the
conduction process is decreased by the optical radiation. Also these equations introduce
a modification on the Mattis-Bardeen formulas (2.4) and (2.5) for frequencies well below
the gap frequency and under the application of any external stimulation characterized by
chemical potential y. on the energy distribution function of quasiparticles.
Using the last two equations and the circuit representation for superconducting film [Fig
2.3]as a parallel combination of kinetic inductivity per length (Lk = —^—) and normal
UJCJn
resistivity per length (R = — ). the change in the film resistance and kinetic inductance
0"i
can be simply derived through the following relations:
SR
R
1 + fo ,
SLk
1
1
- 1
(3.9)
- 1
(3.10)
0"l
i +
^
02
The discussed analysis reveals th a t the conduction process and consequently the electrical
param eters of superconducting films can be affected by the external optical illumination,
appropriate for optoelectronic applications.
26
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3.2.2
Simulation R esults
At microwave frequencies / ( h f < 2A (T)) the superconducting film can be modeled by
a complex conductivity described by the two-fluid model or the BCS theory. Consider a
laser beam with optical frequency v (hi/ > 2A(T)) illuminating the structure uniformly.
In this configuration, the superconducting film acts as a transmission line for a microwave
signal while the laser illumination perturbs its electrical properties, as depicted in Figure
3.1. The fraction of the incident optical power absorbed within the optical penetration
Optical Radiation h v
Phouvexcited Region
Microwave Signal
Microwave Signal
Superconducting Film
Figure 3.1: Photo-excited superconducting film operating at microwave frequency.
depth drives the superconducting film out of the equilibrium and affects the conduction
process. This phenomenon can be traced through the thermodynamical and subsequent
electrical changes described in previous subsections. In order to observe a typical behavior
of photo-induced effects. YBa 2Cu 307 _,s thin film with parameters presented in Table 3.1
is considered. All of these parameters are sample-dependent and the typical values for
high quality thin films are used in our simulations. The thermodynamical parameters
and normal conductivity of YBa 2Cu307 _i are considered from [16], [61]. The optical
penetration depth is numerically calculated from Mattis-Bardeen formulas (2.4) and (2.5).
which is in agreement with the experimental results given in [62]. The quasiparticle
diffusion coefficient and energy gap relaxation time are also adapted from [63]. [64]. Based
27
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Value
Description
Energy Gap at T = 0 (Ao)
0.031 eV
E x p o n en t(7)
2
Critical Temperature (Tc)
91.5 K
Energy Gap R elaxation Tim e (r^ )
% 10-11s
Normal C onductivity at T = Tr (cr„)
1.8 x 106(Q m )-1
Thermal Diffusivity (D)
% 2 x 10- 6 m 2/ s
Optical Penetration Depth (Sa)
rs 9
x 10- 8 m
Table 3.1: Parameters used in the analysis of Photo-excited YBa 2Cu 307 _,j film [65]. [66 ].
on the Table 3.1. the changes in thermodynamical and electrical parameters of photo­
excited superconducting thin films made from YBa2Cu 307 _<j are shown in Figures 3.2 and
3.3. respectively. Figure 3.2 illustrates the alteration of the thermodynamical properties,
starting with the change in the energy gap followed by the reduced effective tem perature
as a function of absorbed radiation intensity for various optical frequencies. It is generally
observed that no particular interaction is seen at low absorbed optical intensities while
the optical radiation tends to affect on superconducting state properties after a certain
value of absorbed optical intensity, so-called effective absorbed optical intensity /<=//• until
the superconductivity completely disappears. The variation of effective temperature with
Ieff- justifies the bolometric photoresponse regime, when the absorbed optical intensity
is high enough to raise the tem perature of the superconducting film, regardless of the
origin of the photoresponse mechanism. The photo-induced changes in the film resistance
and the kinetic inductance are also shown in Figure 3.3. Within the effective and critical
absorbed optical intensity range the thermodynamical and electrical parameters are highly
dependent on the optical radiation characteristics in a nonlinear fashion. Our simulation
proves that the effective range of absorbed radiation intensity for controlling the properties
of superconducting films is narrow and the exact knowledge of this parameter is vital for
potential applications. At the superconducting phase transition, the change in the kinetic
inductance approaches infinity and the vertical asymptotes associated with each graph in
28
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-
0.2
v=3v \3Q v
5 A /A
-0.4
-
0.6
d=10Qnm
-
0.8
v : Gap Frequency
80v
0.9
v=3v . /30v
o 0.8
80v
0.7
©
0.6
0.5
0.4
Absorbed Optical Intensity, lQ, (W/m )
Figure 3.2:
Energy gap modification and effective temperature of photo-excited
YBa 2Cu 30 7-* thin films parameterized by v. the optical frequency.
29
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6R/R
-
0.2
-0.4
T/T =0-5
v=3v \ 30v.
-
0.6
d=100nm
-
0.8
v : Gap Frequency
80v
10
8
30v
6
. 80v
4
2
0
Absorbed Optical Intensity, lQ) (W/m2)
Figure 3.3: Relative change in the film resistance and kinetic inductance of photo-excited
YBa 2Cu 307_«5 thin films parameterized by v. the optical frequency.
30
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the second part of Figure 3.3 clearly indicate the amount of critical absorbed radiation
intensity for destruction of superconductivity in thin films, which is very suitable for
optical detection purposes in a bolometric regime [65].
In order to demonstrate the capability of our model, we compare our simulation results
with the photoresponse superfluid density measurement in YEa^C^C^-,} film done by Y.
Liu et. al. [66] in Fig 3.4. They used pulsed terahertz-beam spectroscopy technique to
probe the photoresponse from 90 nm YE^CuaO?-,* film under various optical radiation
intensities from CW argon laser with 2.3 eV photon energy at 24 K. We calculated the
0.9
0.8
’ Y. Liu et al
-Our model
0.7
0.6
T=24K
v=36.45v
!
0.4
d=90 nm
0.3
0.2
Absorbed Optical Intensity, I (W/m)
Figure 3.4: The photoresponse of the YBaoCuaOT-* film superfluid density from our
model and measurements done by Y. Liu e£. al. [68]
31
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71
T
superfluid fraction through the relation — = 1 —(
Y • It is seen that there is a good
n
Tc
agreement between our numerical calculation and their measurement, especially at the
relatively high optical intensities. A discrepancy seen at the first point with lower absorbed
optical intensity can be modified by using the exponential relationship between the energy
gap with the effective temperature appropriate for low initial temperature [67].
Since the n o n lin e a r photo-induced variations of electrical parameters in superconducting
films occur in a nanosecond to picosecond time scale, the photo-excited superconducting
film can intrinsically operate as a basic device for optoelectronic applications.
3.3
H ea t Transfer M o d e l o f S u p erco n d u ctin g P h o ­
to resp o n se
The use of an effective tem perature to characterize the steady-state photoresponse in
superconducting film may result in the application of a heat transfer approach. This
macroscopic approach enables us to consider the more practical case, when the supercon­
ducting film is mounted on a dielectric substrate, and where the optical radiation depends
on space and tim e variables. Also we are able to model the nonbolometric and bolometric
photoresponse types in a unique mathematical framework. Let us first consider a thin
superconducting film deposited on a dielectric substrate as the upper part of the parallel
plate superconducting transmission line discussed in section 2.5. When the optical radi­
ation is incident on the superconducting film part of the energy can penetrate into the
film. The absorbed optical radiation acts as an internal heat source within the optical
penetration depth, which creates heat diffusion in the superconducting film /substrate con­
figuration. As it is pointed out in section 3.1. the absorbed photons exchange their energy
through electron, phonon and substrate subsystems of the superconducting structure. As­
suming two different effective tem peratures assigned to the photo-excited electrons and
phonons, justified by our effective temperature model, the energy balance equations for
the coupled electron-phonon- substrate system may be constructed. When the absorbed
32
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optical power. Pa(r. t) is applied to the superconducting film, the space and time evolution
of the electron temperature. Te(r.t) and phonon tem perature. Tph(r.t). is governed by a
pair of coupled nonlinear differential equations, known as 2T-equations [68], [69]:
c £ T e{T.t) = K eV 2Te( r . t ) - - ^ [ T ' ( r . t ) - T ph(r.t)] +
at
Cph^ - T ph(r.t)
®
r e_pfc
(3.11)
V
= KphV 2Tph(r.t) + - ^ [ T e( r . t ) - T ph( r . t ) } - ^ [ T ph( r . t ) - T 0]
Te—ph
1~C3
(3.12)
where Ce and Cph are the electron and phonon heat capacities. K e and K ph are the elec­
tron and phonon thermal conductivities. rea is the phonon escape time to the substrate.
T0 is the initial temperature and V is the volume of photo-excited film.
The terms appeared in the left hand side of the 2T-equations represent the rate of energy
storage in electron and phonon subsystems and the first terms in right hand sides reflects
the diffusion process. The optical source term appears only in the first equation, since
the absorbed optical power is directly absorbed by cooper pairs and affects the electron
subsystem. The second terms then represent the interaction of electrons and phonons.
and finally the last term of the second equation indicates the phonon escape phenomenon
through the substrate. The photo-excited superconducting film and its equivalent ther­
mal model are depicted in Fig 3.5 [70]. In fact, the 2T-equations is a generalized form
of the heat conduction equation based on the Fourier law and the conservation of energy [71]. Note that the optical frequency does not show up in this picture, since this
phenomenological model does not take into account any microscopic phenomenon. It is
also possible to reconcile the 2T-equations with the Rothwarf-Taylor rate equations [72].
where the number of photo-excited electrons and phonons can be related to their thermal
conductivities and diffusivity through transport laws [73].
We now focus more on the some of parameters appearing in the 2T-equations, starting
with the absorbed optical radiation power as the most important term. By taking into
account the reflection and transmission of the normally irradiated optical wave from the
film and neglecting the reflection from the film-substrate interface and multiple refraction
33
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Incident
Light
‘e-ph
d
Superconducting FQm
es
Substrate
Figure 3.5: Superconducting film on substrate and its equivalent thermal model
within the film, the absorbed optical power can be expressed as [74]:
Pa = P i n ( l - R ) [ l - e x p ( - £ ) ]
(3.13)
where Pin is an incident optical power. R is the reflection coefficient. Sa is optical pene­
tration depth, and d is the HTS film thickness. Normally the optical penetration depth
for HTS materials is in the range of nanometers [62].
In the 2T model, the phonon escape time indicates the thermal decay time for heat loss out
of the photo-excited region through the substrate and is related to the thermal boundary
resistance. R bd by [75]:
Tet = C dR sD
The thermal boundary
(3-14)
resistance is expressed in K m 2/W and depends on the dielectric
substrate and the geometry of the film /substrate contact l. In order to increase the speed
of the photoresponse, it is im portant to reduce the thickness of the film and most impor­
tantly the boundary resistance. It is suggested to use the film thickness comparable to
1For com plete review o f thermal boundary resistance theories refer to reference [76].
34
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the optical penetration depth in the order of nanometer to reduce the heat diffusion effect
in the substrate. Also, a narrow strip contact geometry and a deposition of deoxygenated
superconducting layer on the substrate has been demonstrated to effectively reduce the
boundary resistance problem [49]. The choice of the substrate m aterial for a supercon­
ducting film is also a critical factor for having smaller thermal boundary resistance and
some research works have been carried out for
n o m in al
substrate materials such as MgO.
LaAiOs. and SrTiOs [77]. [78]. In its more complicated form, a comprehensive thermal
model should be applied for the superconducting film and substrate structure for more
accurate result [79]. [80].
Next we consider the nonbolometric and bolometric cases which corresponds to two types
of temporal measurements. First, the optical pulse with pulse width, r on the order of
Te_ph. or CW optical signal modulated by frequency f m on the order of T~_}ph is applied to
the superconducting film and the transient temporal evolution of the electron and phonon
temperatures are observed at the same time. The typical behavior of the electron temper­
ature indicates a very fast rise, reaching to a maximum corresponding to the maximum
optical power, while Tph has basically the same behavior with a slower rate. When Te
approaches Tph. further relaxation of Te follows relaxation of Tph and is determined by the
phonon escape time [49]. [64]. [81]. [82]. This type of photoresponse behavior corresponds
to the nonbolometric case and the discussed behavior is a manifestation of a quantum
process that takes place in a very short time scale. In the second case, the pulse width
r larger than re_p^. or CW optical signal modulated by frequency f m is chosen to be
smaller than r~^ph and also frequency gap. ug. In this case, in which the fast bolometric
photoresponse manifests itself over a time scale comparable with the phonon escape time,
the electron and phonon subsystems will remain in thermal equilibrium and the effective
tem perature of the system. T (r. t). can be described by a simplified 2T-equations as [20]:
C -^ A T (r.t) = ^ V 2A T (r.t) +
at
V
- ^ - A T ( r,t)
Tes
(3.15)
where AT'(r.f) = T ( r.t) — Ta and C = Ce + Cph. K = Ke + Kph are the heat capacity
and thermal conductivity of the superconducting film, respectively.
35
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In this thesis, the emphasis will be placed on the fast bolometric photoresponse, since we
are interested in microwave/photonic applications, where the microwave signal frequency
and optical modulation frequency do not exceed the gap frequency of the superconducting
film. Therefore, we seek the solution of equation (3.15) upon different optical illumination
schemes namely continuous, time harmonic, pulsed and travelling-wave type.
3.3.1
C ontinuous U niform O ptical Irradiation
We wish to find the temporal evolution of the temperature shift of the superconducting
film when a continuous power Pa is absorbed uniformly within the optical penetration
depth. A solution of equation (3.15) with boundary condition A T(t = 0) = 0 is:
m t) = ^ ( 1
- e x p (-^ i))
(3.16)
where Ta is the initial temperature, and the total temperature is T(t) = T0 + A T(t). The
thermal time constant of the bolometric response is
r* = ^
f-'ph
(3.17)
which determines the speed of photoresponse. In the steady state regime, the reduced
T
temperature. — can be given by:
*C
T
b
T
b
T P
^
W
i
(X 18)
where Pc is the critical absorbed optical power, given by:
Pc = S t l 2 k ( i _ *•)
(3.19)
c
This quantity indicates the absorbed optical power required for destruction of supercon­
ductivity. when
all Cooper pairs are broken by the absorbed photons.Equation
(3.18)
was experimentally confirmed for Y ^ a C ^ C ^ -* thin films[66].Comparing the behavior
of. Teff. in the effective temperature model developed in section 3.2, equation (3.6), with
the equation (3.18). clearly indicates qualitative agreement.
36
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3.3.2
Tim e Harm onic O ptical Irradiation
Consider a CW laser source, when the optical signal is modulated by a time harmonic
microwave signal with frequency f m. Since the optical frequency v represents an energy
level greater than the binding energy of the electron pairs. A (T). the modulation frequency
should be chosen in such a way th at h f m
2A(T). In this case the absorbed CW optical
power can be simply expressed as:
Pa(t) = Pm{ l + T n s m 2 n f mt)
(3.20)
where Pm is the maximum optical power absorbed in the superconducting film and m is
the modulation index. The tem perature shift after t = 0 would be:
AT(t)
where
=
^ ( 1 - e x p ( —-jfe-t)) + ■
,
2tt rnPmT?aCfm
V& 'j, + (
2
c rkv
C t„
m P"'T"
+
(2x/m)I,j c ,
sii.(2ir/„i +
, Cph
eXP( C r„ 1
1 1
= - t a n 'l (
Cph
It isseen that in the steady state regime, the temperature of the film can be modulated
by the optical radiation, so-called thermo-modulation effect. This fact wastheoretically
discussed by N. Perrin [69]. observed by C. Vanneste et al. for conventional superconduc­
tors [83] and M. Danerud. et al. for YBa 2Cu 307 _,j epitaxial films [84].
3.3.3
P ulsed Optical Irradiation
During the last decade, particular attention has been devoted to HTS photoresponse
subject to picosecond and nanosecond laser pulses. Most of the experiments have shown
non-bolometric response and the bolometric tail due to the time constant of the applied
optical pulse comparable with the electron-phonon relaxation time [85]. Here, we assume
that a laser source provides an optical pulse with the Gaussian temporal profile with, a
time constant r. In order to take advantage of the fast bolometric response, the tim e
37
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constant of the pulse must be greater than the electron-phonon relaxation time [49]. For
this condition, the absorbed optical power can be considered as:
Pa(t) = PmeXp ( J l ^ l )
(3.22)
where td is a time delay of a pulse.
Solution of equation (3.15). gives us the temperature shift as follows:
ATii)
V2t
\ / 2 7 r Pmr
\T m .\
Cph
£5
=
~ 2C V ~ e i f i l F - a) aC! , {- c Z i ~ 2^
+
—^ ^ - e r f ( a ) e x p ( —
2\
+ a)
+13)
(3.23)
u
\ ■ 4-u
e
CphT
,
Cph 2td
CphT2
where erf(.) is the error function, a = --- (-----b 77 — ). and p = -^77 (----- 1- _ , ).
2 t
C re,
2C Te s
Ct-s
As it will be seen later on. our analytic solution is in a complete agreement with the
experiments [74]. [60] and a quantitative analysis presented in [49]. [51].
Generally, a pulsed laser source consists of a train of the Gaussian pulse with the inter­
pulse time interval Tp or a repetition rate f p =
- 7.
If Pav is the average optical power in
p
the pulse train, then the instantaneous absorbed optical power in the film by the contri­
bution at a time. £. due to all of the pulses in the infinite train is given by [86 ]:
p .w =
y /s ir T
£
“ f ( - (t
til T+*n
T r))
“
(3-24>
If this is the case, when the interpulse interval time Tp is greater than the bolometric
response time Tth■the temperature shift can be simply written by means of the superpo­
sition of the tem perature shift presented in formula (3.23) shifted at time n T , and the
average power Pav is accountable for the average temperature shift in the film.
As most of the mode locked or Q-switched high power lasers are employed at the MHz
repetition rate, for the nanosecond photoresponse time the discussed analysis would be
applicable.
38
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3.3.4 Traveling-Wave T ype O ptical Illum ination
In the three previous subsections, it was assumed that the superconducting film was uni­
formly exposed to the light, preventing the heat diffusion along the structure. In this
subsection, we deliberately impose a periodic optical illumination into the film, not only
in time but also in space. This results in the creation of a dynamic temperature grating in
a superconducting film leading to a modulation of the Cooper pairs and normal electrons
in time and space dictated by an optical illumination pattern. The spatial nonuniformity
is a key feature for increasing the interaction length between the optical radiation and mi­
crowave signal with a required lower light intensity for traveling-wave type optoelectronic
devices. A similar technique has been widely used in a holographic process in photorefractive materials [87]. spatio-temporal probing of charge dynamics in semiconductors for
traveling-wave optoelectronic devices [88]. thermoelectric power conversion in pyroelectric
materials [89] and transient-grating measurements [90].
Optical interference patterns can be produced by two-beam laser interference tech­
niques. This technique provides a spatially modulated optical field, known as an interfer­
ence grating. The operation is schematically shown in Fig 3.6. Two coherent laser beams
with optical wavelength. A. impinge on the superconducting film under an angle S. The
two beams have a difference angular frequency. Q. thus generating an optical grating with
the propagation vector A that moves with the velocity Vp = ---------j- [91]. The absorbed
As in (- )
radiation power pattern in the z direction can be written as:
Pa{z, t) = P0{ l + 77icos(fit —Az)}
(3.25)
where m is the modulation index or power contrast. Using the equation (3.25) in the
heat transfer equation (3.15). gives us a steady-state tem perature grating equation in
superconducting film as:
T(z.t) = Ta +
1+
m
■ =
cos(to - Az + <£)}
39
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(3.26)
where <f>= tan 1(----- — ) and D = — is the thermal diffasivity of the film. This
Kl + DA.2Tth’
C
3
equation indicates that both amplitude and phase of the film tem perature can be modu­
lated by the optical radiation. It is seen that the amplitude of the temperature grating
can be adjusted by the optical power, difference frequency, and grating vector but its
phase does not depend on the optical power. The critical absorbed optical power for the
Laser 2
Laser i
Substrate
Figure 3.6: Dynamic Laser Interference Grating in the Superconducting Film
traveling-wave type illumination is different from continuous optical irradiation and can
be given by:
f
VKT‘m
, (!-£ )
(3-27)
T“ { 1+ V(1 + 0 A W + W rl j
When 77i= ft = A = 0. equation (3.27) reducesto
equation (3.19). as expected.
The
theoretical possibility of such a technique in superconducting film was predicted by N. E.
Glass and D. Rogovin in 1989 [92]. They discussed the formation of conductivity grating
based on the Rothwarf-Taylor model and pointed out some useful physics behind that.
First of all. again we impose the constraint on the difference angular frequency of the
two lasers namely Ml < 2A(T). in order to write down the relation (3.25). Secondly, the
40
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inverse of grating vector. A-1 , must be chosen larger than electron diffusion length, which
is given by. £ = y/Drth. This criteria ensures that the diffusion process does not wash
out the temperature grating imposed by laser irradiation and it will be mathematically
justified in Chapter 5. Normally the diffusion length of practical superconductors is in the
nanometer range, leading to the conductivity grating for microwave and millimeter-waves.
3 .4
Sum m ary
In this chapter a comprehensive discussion of photoresponse in superconducting
film
struc­
tures was presented. The generally accepted physical picture behind the two types of
photoresponse, bolometric and nonbolometric. was demonstrated. We then established a
theoretical formalism based on the BCS theory to develop the relationship between the
optical radiation characteristics and the effective tem perature of a single superconducting
film. Analysis of photo-induced changes in electrical properties of the superconducting
film followed by the numerical simulation and comparison with experimental results in the
literature show the capability of our model for optoelectronic applications. Crossing over
to a more practical case, the photoresponse of a superconducting film on the substrate
was investigated by incorporating a heat transfer model. This model provides a mathe­
matical framework to analyze the nonbolometric photoresponse using 2T-equations and
fast bolometric response using the modified heat transfer equation. The fast bolometric
photoresponse was then considered for different optical
illu m in a tio n
schemes. Thermo-
modulation effects resulting from bolometric photoresponse, shows the possibility of trans­
formation of temporal and spatial information of the optical irradiation into the effective
tem perature of superconducting film for potential device applications.
41
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C hapter 4
O ptical-M icrow ave Interaction in
Superconducting F ilm
Optical-microwave interaction studies in different materials and media has progressed
remarkably after the advent of masers and lasers and ground-breaking theories and ex­
periments in the 1970's to now. Central to any optical-microwave interaction type is the
presence of physical mechanisms for mediating the optical and electrical domains. There
is two type of optical-microwave interaction in different materials:
1- The electromagnetic field associated with microwave signal can alter the refractive
index of a material, leading to transferring information from electrical signal to optical
signal. Illustrative examples of such an interaction include electro-optic, magneto-optic
and acousto-optic effects.
2- The absorption of optical signals by a material can modulate the electrical properties
of the material such as conductivity, leading to th e manipulation of an electrical signal
while it passing through the illuminated region. Photoconductivity phenomenon in semi­
conductors and the photo-absorption in superconductors are two examples of such an
interaction.
These type of optical-microwave interaction studies are the theoretical foundation of wellestablished field of optoelectronics and rapidly growing microwave/photonics area.
42
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From a technological point of view, the distinct advantages of optical transmission systems
and the ever increasing use of microwave and millimeter-wave frequencies in communi­
cation systems have stim ulated an interest in the development of microwave/photonic
technology. Transmission and processing of analog microwave and millimeter-wave and
high-speed digital signals over optical fibers, known as RF-Photonics and using ultra­
fast photonics to generate, control and characterize high-speed/high-frequency electronic
signals, devices and systems are examples of such enabling technology [93]. Semicon­
ductors are the main m aterial to perform microwave/photonic functions, since all of the
main components of the system including laser, fiber, waveguides and detectors can be
made of them with incredible integration scale and high figure of merits. However, the
increasing demand for an advanced communication systems for achieving higher speed,
low consumption power and ultralow-noise characteristics push the researcher to seek
other enabling technologies. Superconducting electronics and optoelectronics are one of
the most competitive technologies finding applications in realization of high-speed IC's.
wireless and 3-G mobile communications and high-speed, long-haul fiber optic telecom­
munication links. In order to investigate the capability of these materials for p e rfo rm in g
optoelectronic functions and elaborating them in microwave/photonic systems, we will
study the optical-microwave interaction in high-temperature superconductors based on
the photo-absorption phenomenon in these materials.
In this chapter, the fast bolometric photoresponse model along with the thermo- modu­
lation concept developed in the previous chapter will be incorporated to the Two-Fluid
model discussed in section 2.3. This modeling provides a theoretical framework for dealing
with both lumped and traveling-wave param etric interaction between optical radiation
and electrical signal. In this chapter, we shall consider the case of lumped interaction
for novel optoelectronic and optically-controlled microwave devices based on the kinetic
inductance of superconducting film and simple circuit theory. The parametric opticalmicrowave interaction study and associated potential device investigation will then be
carried out in the next chapter.
43
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4.1
B a sic T heory and M o d elin g
One of the most significant properties of superconductors is the dependence of its char­
acteristics on external
s t im u li
such as temperature, magnetic field, electrical current, and
optical radiation. Phenomenologically. the Two-Fluid model directly shows the relation­
ship between the number of Cooper pairs and normal electrons with temperature, through
the formulas (2.8) and (2.9). Back to Chapter 3. the effect of optical radiation is modeled
by the effective temperature. Te/ / . using a heat transfer model. Therefore, the dependency
of the Cooper pairs and normal electron number densities with optical radiation charac­
teristics can be traced through the effective temperature models and thermo-modulation
concept developed in the previous chapter. One can rewrite the Two-Fluid equations as:
n ,(r.f)
=
n „(r.f)
=
n [l -
~T]
(4.1)
(4.2)
W ith the revised version of Two-Fluid equations, the current-field relationship (2.15) can
be applied for modeling of optical-microwave interaction in superconducting film. Phys­
ically speaking, the conduction process for electrical signal depends on both the velocity
and the number density of carriers according to the relationship (2.14). While the ve­
locity of carriers contributed to the super and normal currents are determined by the
electrical signal, through equations (2.12) and (2.13). the number of carriers contributed
to the electrical conduction are determined by the optical radiation by formulas (4.1)
and (4.2). From equivalent circuit point of view, the optical irradiation decreases the
number of Cooper pairs by raising the effective tem perature and correspondingly the ki­
netic inductance of the superconductor and normal resistance are increased. Thus often
the photo-absorption phenomenon in superconductors called photoresistivity in contrast
with its counterpart phenomenon in semiconductors known as photoconductivity [94].
The time required for photo-excited normal electrons to contribute to the normal cur­
rent identifies the speed of interaction between optical radiation and electrical signal. It
is worth mentioning that the supercurrent response time.
t j
.
can be estimated by the
44
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equivalent circuit model presented in Figure 2.3 [95]. When the number of Cooper pairs
decrease, the kinetic inductance will increase and the normal-fluid channel resistance will
shunt current away from the Lt.
Thus, the supercurrent will respond to the optical
irradiation in a time:
= re. ph
T
(— r
Tc
(4.3)
1 ~ {TCV
The comparison between the supercurrent response time and the thermalization time from
phonon escape phenomenon gives the interaction time bottleneck for the exchange of the
energy between optical and electrical signals. Most of the experiments confirms the theo­
retical prediction of phonon escape time as a bottleneck for HTS film structures [20]. [49].
Note that the discussed physical mechanism behind the optical-microwave interaction in
the framework of the Two-Fluid model combined with the effective tem perature concept
is correct under the following assumptions:
1- The applied electrical signal to the superconducting film does not cause any nonlinear­
ity and its associated frequency is well below the gap frequency.
2- Since the superfluid is much greater than the normal current for temperatures be­
low 0.9TC. the conduction process is mainly focused in the London penetration depth.
Consequently, to increase the internal quantum efficiency of the photo-absorption in the
film/substrate structure the optical penetration depth and the film thickness should be
comparable with the London penetration depth on the order of nanometers.
4.2
P h o to -In d u ced E le c tr ic a l C h an ges in H T S F ilm
In this section we consider a photo-excited, electrically biased, thin superconducting film
on a dielectric substrate, as a basic superconducting optoelectronic component. We delib­
erately choose the high-temperature superconductors to assure high light absorption, and
high optical penetration depth in the order of nanometers. The thickness of the film is
45
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much smaller than the substrate thickness, and the HTS film is employed in a microstrip
and microbridge configuration as depicted in Figure 4.1. This type of structure has been
used in many experiments related to the photoresponse measurements [74]. [96], [97].
This structure is electrically current-driven or injecting a local oscillator (LO) signal and
provide ultralow-loss and nearly nondispersive performances for electrical signals up to
terahertz frequencies, below the superconducting gap frequency. We also further assume
that this structure is intended to operate at:
a) temperature range 0.57’c < T < 0.9TC:
b) below critical magnetic field (Hcl) or in the Meissher state:
c) well below its critical current Ic:
The above operating conditions assure a fast bolometric photoresponse occurs and also
ensure that the thermophysical parameters of the HTS film do not possess a significant
temperature dependence. An equivalent distributed electrical circuit model, neglecting
Photo-induced Voltage
DC Bias C urrent o r LO Signal
Photo-excited
HTS Bridge
Substrate
Figure 4.1: Photo-excited HTS film and its electrical configuration
edge effects, is shown in Fig 4.2 [30]. The kinetic inductance (£*) and normal resistance
(Rn) are the manifestation of superconducting properties and magnetic inductance (Lm)
and capacitance (Cm) represent the the geometrical structure on the dielectric substrate.
46
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i, and in are the supercurrent and normal current, respectively. From the fact that the
magnetic inductance introduces nonlinearity in the current distribution especially near
the HTS film edges [98]. the width of the film, w should be chosen much greater than a
thickness of the dielectric spacer, h. Under this design criterion, the peak current density
is only about twice the average value, since the m in im u m current density takes place at
the center and is about 0.75 the average value for a thin HTS film [99]. It has been
recently shown that if the tem perature is changed or the optical radiation is absorbed,
the shape of the current distribution through the thickness of the film changes everywhere
uniformly except within \ i of the edges [100]. Therefore the assumption of uniform cur­
rent density is not an unreasonable approximation for the proposed configuration. The
v (d ' i
N
'
m
Figure 4.2: Circuit model of HTS microstrip neglecting edge effects
length of the HTS bridge is also chosen smaller than the input microwave wavelength
to guarantee the validity of the electrically lumped model and greater than out-diffusion
length. •y/12£)rf._p/l. to assure the validity of the heat transfer model for the HTS photore­
sponse. The main goal in achieving such a conditions is to alleviate the transmission line
effects on the propagation of microwave signals along the structure. This helps to verify
and observe the physical phenomenon that takes place in the lumped case of parametric
47
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optical-microwave interaction in the HTS films.
T he output of a laser source is focused on the HTS bridge so that it is illuminated uni­
formly. The microwave signal is applied to the bridge from the current source, and the
voltage across the bridge is m onitored by a fast, sensitive and high frequency oscilloscope.
According to the circuit model shown in Fig 4.2. if the electrical current through the
bridge. i(t) = i , ( t ) + i n{t). is m uch smaller than the critical current, then by ignoring the
normal current. i„(f) below tem perature range T < Tc. the photo-induced voltage. v(t)
across the bridge will be:
di(t)
d
v(t) = L { t ) - £ - + i(t)— L(t)
(4.4)
where L(t) denotes the time dependent inductance due to the optical irradiation and
according to formula (2.36) for a microstrip line can be expressed as [101]:
L(t) = Lm + L M
= ! ^ ( l + 2 M > c o t h ( J i 7 I))
,4.5)
where Lm and Lk are the magnetic and kinetic inductances of the HTS strip. It is seen that
the optical stimulation affects th e kinetic inductance part via the time dependent London
penetration depth and normal resistance, more specifically formulas (2.18) and (2.19). The
tim e dependent London penetration depth and the skin depth are then calculated through
the effective temperature Te/ f ( t ) obtained from the section 3.3.
From the electrical-
response-signal point of view, two different photoresponse might be appeared. Since the
voltage appearing in the photo-excited HTS configuration is mainly due to the presence
of the kinetic inductance for tem peratures below 0.9TC. this type of response is known
as “Kinetic Inductance" or “Superconducting-State" photoresponse. In contrast, near a
transition temperature Tc. when the optical radiation is able to drive HTS film to the
normal state, the photo-induced voltage will be due to the change in the resistivity rather
th an the kinetic inductance, and the voltage would be:
=
+
(4.6)
This kind of response is commonly referred to as “Resistive Bolometric” or "SwitchedState" Photoresponse.
48
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The photo-induced electrical changes in the HTS film can play a key role for potential
optoelectronic device applications and offer insight for optical control of HTS microwave
devices. Therefore in the following sections we shall investigate phenomenon further.
4 .3
O p to electro n ic D e v ic e A p p lica tio n s
In order to observe the capability of the photo-excited HTS film for optoelectronic ap­
plications. our discussion will be commenced by the numerical simulation of lumped
optical-microwave interaction in HTS film, when the kinetic inductance photoresponse
is manifested.
Consider a thin superconducting film made of oxygen-rich YBa 2Cu 30 7 _i deposited on
a substrate, normally LaA103 or MgO. The thickness of the film is much smaller than the
substrate thickness, and the YBa 2Cu 30 7 _,$ film is employed in a microstrip and micro­
bridge configuration as depicted in Figure 4.1. The geometrical and physical parameters
ofY B a 2Cu 307-,5 microstrip are taken from [74] and are listed in Table 4.1. As a first step,
we consider a periodic optical irradiation in the presence of either DC bias current IQ or
a pure sinusoidal microwave current with 1 mA amplitude. The absorbed optical power
with amplitude modulation frequency fm = 100 MHz. unity modulation index and peak
power 24 mW is considered, and the steady state photo-induced tem perature in the HTS
bridge is computed by formula (3.21). Under the application of such an optical signed,
the temperature is modulated periodically and controlled by the optical power when the
HTS bridge still remains in the superconducting state, as demonstrated in Fig 4.3.
Figure 4.4 illustrates the voltage response, when the HTS bridge is biased with 1mA
DC current, in the superconducting state. Further investigation of the photo-induced
voltage in the frequency domain reveals that, because of the nonlinear relationship be­
tween the kinetic inductance and the effective temperature, the photo-induced voltage
has harmonic frequencies of the modulation frequency of the optical signal, as depicted
in Figure 4.5 when the bridge is biased by DC current. Our simulation indicates that
49
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Description
Value
Initial Temperature (T0)
77 K
Critical Temperature (Tc)
86
Exponent (7 )
K
2
Critical Current (Ic)
5 mA
Bridge W idth (t»)
10
fan
Bridge Length (I)
20
fan.
Film Thickness (d)
30 nm
Dielectric Spacer ( h )
fan.
1
London Penetration Depth (Ao)
as 180 nm
Optical W avelength
532 nm
Optical Penetration D epth (60)
90 nm
Optical A bsorptivity (a)
0 .8
Heat Capacity ( C )
0.91 Jcm - 3 K - 1
Phonon Heat C apacity (Cph)
0.9 Jcm - 3 K - 1
Phonon Escape T im e (r „ )
1
ns
Table 4.1: Parameters used in th e analysis of Photo-excited YBaaCuaOr-,* film.
86
c
t—“
m
9o.
82
E
9
h9
>
O
9
UJ
Time (ns)
Figure 4.3: Steady state photoresponse of HTS film effective temperature
50
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there is a strong second harmonic component of the modulation frequency, due to the
second order nonlinearity of the penetration depth with temperature and time derivative
relationship between the kinetic inductance and voltage, which amplifies the harmonic
frequency in the output voltage.
This suggests a promising feasibility of "microwave
0.1
0.08
0.06
>■ 0.04
S’ 0 02
*D
•£
-
0.02
£ -0.04
-0.06
-0.08
-
0.1
Time (ns)
Figure 4.4: Photo-induced steady-state voltage response of HTS bridge
harmonic generation" in the HTS film by the optoelectronic technique.
Next, if a local oscillator (LO) current is fed to the HTS bridge, the photo-induced voltage
displays the mixing of the modulation frequency f m and LO frequency, as depicted in Fig
4.6. Figure 4.7 indicates the frequency spectrum of the output voltage in the absence
of main components. It is worth noting that because normally the first term in relation
(4.4) is greater than the second term, the amplitude of the voltage in sum and difference
frequencies (fco ± f m) are not the same.
This simulation demonstrates potential of an
optoelectronic HTS photodetector/m ixer in up and down conversion of R F signals in high
performance superconducting microwave/photonic systems [102], and feasible application
51
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1
---------- r
m 0-8
a>
Q
«w
O
©
w
©
so
a.
■o
a 0.4
"5
E
o
z
0.2
100
200
300
400
500
600
700
800
900
1000
1100
Frequency (MHz)
Figure 4.5: Photo-induced steady-state voltage response of HTS bridge with 1 mA DC
bias current
52
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0.3
0.2
□>
= 0.05
-0.05
-
0.1
-0.15
Time (ns)
Figure 4.6: Photo-induced steady-state voltage response of HTS bridge with 300 MHz
LO current
53
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0
0
»
100
200
^
300
400
500
600
700
Frequency (MHz)
Figure 4.7: Photo-induced steady-state voltage response of HTS bridge with 1 mA
current
54
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in microwave/optical subcarrier multiplexed system s [103].
The complete mathematical analysis of frequency response of the HTS optoelectronic
mixer is presented in Appendix C.
In the second step, the optical pulsed radiation with the absorbed peak power 100 mW
and time constant 2y/2 ns is incident on the HTS bridge. Figure 4.8 indicates the opti­
cal pulse followed by the* photo-induced transient temperature. Figure 4.9 indicates the
*0.08
0.06
0.04
1 0.02
-
30
Time (ns)
a 80 I—79
® 78
2 77
30
Time (ns)
Figure 4.8: Absorbed optical power (W). followed by photo-induced transient tem perature
(K)
photo-induced voltage, when the HTS bridge is biased with 1 mA DC current and LO
signal with frequency f i o = 300 MHz. respectively. The fast bipolar transient voltage
developed across the bridge, inherently acts as an HTS photodetector, optical-to-electrical
transducer or optoelectronic switch [104]. In case of DC bias, the structure is demand­
ing for employing in digital
c o m m u n ic a t io n s
using rapid single-flux-quantum (RSFQ)
circuits and optical fiber for high-speed data transmission into the cryogenic environ-
55
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ment [60]. [105]. When the LO signal is connected to the photo-excited HTS bridge
the amplitude of the voltage is modulated by th e optical pulse leading to optoelectronic
amplitude modulation.
25
(a) Time (ns)
3-10
i -2 0
.c -30
0
5
15
10
20
25
(b) Time (ns)
Figure 4.9: Voltage response of HTS bridge to the pulsed optical irradiation with (a) 1
mA DC bias current (b) 300 MHz LO current
Another interesting application of photo-excited HTS film is free-space electromag­
netic radiator. Consider an HTS film on a dielectric material with low relative dielectric
constant, such as MgO. in the configuration shown in Fig 4.1. In the presence of a DC bias
current, the optical radiation causes the thermo-modulation which in turns creates the
modulation of the supercurrent and normal current, according to the Two-Fluid model.
Following the classical electrodynamics, this device is similar to the dipole antenna, there­
fore the radiation field E rad is proportional to th e time derivative of the current density,
d'J
namely — . The electromagnetic field radiation can be in form of continuous wave or
pulsed radiation depending on the associated optical radiation waveform, detectable from
56
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back of the structure. THz pulses em itted from photo-excited HTS film by ultrashort
laser pulses was experimentally observed by two research groups. C. Jaekel et al. from
Germany [106] and M. Hangyo et al. from Japan in 1995 [107]. They both measured elec­
tromagnetic pulses with frequency spectrum extend up to few THz from photo-excited
YBa 2Cu 307 _,j film on MgO. The amplitude of the EM field was found to be proportional
to the bias current and absorbed optical power, proving the superradiant character of
emission. To improve the radiation efficiency, bow-tie antenna along with hemispherical
lens attached to the substrate can be used [108].
4.3.1
HTS O ptoelectronic D ev ice Characteristics
The performance of the HTS optoelectronic device can be verified and compared to its
semiconducting and superconducting counterparts based on the conversion gain band­
width. B q . intrinsic and extrinsic conversion gains and equivalent noise tem perature.
The conversion gain bandwidth of our proposed device can be found from the temporal
bottleneck of the fast bolometric photoresponse as:
Ba =
—
2n Tth
(4.7)
Effective reduction of phonon escape time results in higher bandwidth even up to THz
frequencies
sim ilar
to the superconducting hot-electron bolometric mixers (HEB) [109].
The intrinsic conversion gain of any optoelectronic mixer can be defined as the relative
output of the mixed signal with respect to the input LO power [110].
A ssu m in g
matching
network in the both input and output sides, the intrinsic conversion gain bandwidth. G,nt
would be:
✓~r
LO )
/A 0\
Gint = TloUloV
(4-8)
where P ( f m±f i, o) and Pio(fLo) are the electrical power at f m± f i o and / lo delivered to
a matched resistive load.The conversion gain of the optoelectronicmixer
can be defined
with respectto the optical signal as well. This param eter is calledextrinsic
conversion
57
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gain, presented by Gcxt. Taking the optical absorption phenomenon in the HTS film into
account, the extrinsic conversion gain is:
P{fm
i
fLo)
MO)
where Ptn is the input optical power. These two conversion gains can be easily calcu­
lated by the formula (C-16) in Appendix C along with the circuit configuration of the
input and output matching networks. In comparison between HTS optoelectronic mixer
and the others, it is generally found out that, the intrinsic conversion gain of the HTS
optoelectronic mixer is comparable with the HEB. b ut is less than optoelectronic mixer
based on semiconducting heterojunction bipolar transistors (HBT) [111]. Nearly perfect
isolation between the LO signal and RF carrier signal causes the lack of coupling loss,
which in turn increases the intrinsic conversion gain in comparison with the HEB mixers.
The extrinsic conversion gain is even higher than HBT. because of higher amount of light
absorption in YBa 2Cu307_<5 thin films and higher quantum efficiency [112].
Noise characteristic of our proposed device is expected to be significantly superior over
conventional optoelectronic devices. One of its advantages is that it operates exclusively
in the superconducting state, therefore the Johnson noise is nearly absent. Consequently,
the main source is the tem perature fluctuation during the photoresponse formation [20].
The mean square value of the spectral intensity of the tem perature fluctuations in a system
with specific heat capacity C. tem perature T . volume V and thermalization relaxation
time t? is [109]:
((A T f ) =
(4.10)
The equivalent output noise tem perature ( Tf i ) dissipated in a perfectly matched load R l
caused by the temperature fluctuations in the HTS optoelectronic mixer is given by [113]:
T
4tt2f l o i l o T 2TtH dLk 2
Tfl rl
c v ( H r ]1
(4'll)
where iio is the maximum LO current. As it is expected Tfl is quite comparable with
hf
HEB mixers and can approach to the ten times of the quantum noise limit (——), de2KB
pending on the initial temperature. R F carrier frequency and phonon escape time [111].
58
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4 .4
O ptical C on trol o f S u p erco n d u ctin g M icrow ave
D evices
Optical control of microwave devices has been an area of growing interest since the ar­
rival of compact, practical and cost effective lasers. Optical signals have been used to
control the parameters of microwave semiconductor devices based on photoconductivity
effect. Various microwave and millimeter-wave functions including optical injection lock­
ing. phase shifting, modulating, switching, limiting and mixing in passive devices and also
gain control and oscillator tuning in active microwave devices have already been demon­
strated [8]. [114]. [115]. The optical control of microwave devices offers unique advantages
in:
1) The inherent high DC and signal isolation between the optical and microwave signals.
2) High band-width operation associated with fast material photoresponse.
3) Immunity from electromagnetic interference.
4) Possibility of eliminating microwave feed for large array systems.
5) Possibilities for monolithic integration using photonic guided-wave structures.
The above-mentioned advantages make this approach attractive for use in active phased
array antennas, microwave signed processing and optical signal distribution in microwave
systems [116]. [117].
N. E. Glass and D. Rogovin pointed out th at the optical control of superconducting
microwave and millimeter-wave components is theoretically possible for LTSs using the
photo-absorption phenomenon in these materials, similar to the microwave semiconduc­
tor devices [118]. Since the incorporation of HTS materials in microwave devices exploits
the ultralow-loss and nearly dispersionless characteristics, it offers higher optical absorp­
tivity and penetration depth leading to higher quantum efficiency along with more fea­
sible and cheap cryocooler systems. This technique also provides a remarkable solution
for multi-input/m ulti-output superconducting microwave or RSFQ interconnection sys­
tems when the thermal runaway to the input/output modules or cryocoolant could be
problematic [119]. It is worth noting that the recent comprehensive comparison of fully
59
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three-dimensional optical, normally conducting and superconducting interconnections in
a system-level shows the superiority of optical approaches to the others [120]. Therefore,
in this section we investigate the possibility of optical control of HTS microwave devices
based on the parallel plate superconducting transmission line (PPSTL) using the devel­
oped photoresponse model in section 3.3.
4.4.1
Optically Tunable Propagation Delay T im e in HTS Trans­
m ission Line
Microwave and
m illim e t e r - wave
delay lines are essential components in signal processing
systems as memory elements in order to temporarily store signals while they are being
pre-processed in other parts of system. Typical delays fall in the range of nanoseconds to
picoseconds, therefore a delay line consists of a long
tra n sm iss io n
lines, typically in the
form of spiral strip or meander lines. Conventional electrical delay lines are formed by
the lengths of coaxial cable or metal strips along with amplifier sections to circumvent the
ohmic loss. They suffer from loss and dispersion. In contrast, superconductive delay lines
can provide an ultralow-loss and nearly dispersionless performance, without need of ampli­
fier in between, and they exhibit a distinctive advantage over the conventional technology.
Currently most of HTS-based delay lines play an important role in space-borne satellite
communication transponders [121]. high-performance cellular communication [122]. and
electronic warfare systems [123].
The microwave signal velocity along the superconducting delay line in the form of
parallel plate or microstrip line can reduce due to the presence of kinetic inductance. As
the kinetic inductance of the superconducting line is increased the microwave signals need
more time to traverse along the line.
A c c o r d in g
to equation (2.36). the propagation delay
time per unit length can be written as:
r = —
U)
60
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(4.12)
Fig 4.10 indicates the variation of delay tim e versus temperature for 30 nm thick YBa2Cus0 7 _
film on a LaAlOa substrate with eT = 24. It should be noted that this delay is not fre­
quency dependent provided the microwave signal frequency is less than D.\vg. It is seen
th at the delay can be enhanced by the tem perature shift. Therefore, using optical radi­
ation and employing the bolometric photoresponse the effective tem perature of the HTS
film can be increased and the delay tim e can be controlled through the absorbed optical
power. Due to the long length of the delay line, it is not practically appropriate to illu180
140 j-
r= 10 um
h = i jim
f =24
Q 100
lT
80 -
60 (-
40 -
20
20
30
40
50
60
70
80
Temperature (K)
Figure 4.10: Delay time per unit length of the microwave signal in PPSTL for various
temperature
ruinate all of the transmission line coherently, so the optical beam is ju st applied to some
parts of the transmission line locally. A heat transfer model for the resulting hotspots and
accurate voltage measurements at different positions along the superconducting line indi­
cates that the temperature rise in the hotspot regions is a local phenomenon [124], [125].
For a point source continuous optical illumination the photo-excited excess heat diffuses
over the “thermal healing length". 1/,. in a time of order r „ . as it also passes into the
61
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substrate [126]. This characteristic length can be expressed as:
lh = y/D ret
(4.13)
where D is the thermal diffusivity of the superconducting film. Having kept this fact in
mind, the whole heat transfer model developed in chapter 3.3 is valid under the substi­
tution of photo-illuminated length. I. approximately by / 4- 2Z&. It was suggested that to
perform the delay time tuning without disturbing the microwave propagation, it is more
convenient to apply time harmonic optical radiation with low modulation frequency to
avoid the phase modulation of the microwave signal in the irradiated spot [127]. This
fact can be mathematically confirmed through the model developed in subsection 3.3.2.
more specifically equation 3.21 which explicitly indicates that the phase modulation is
enhanced by increasing the modulation frequency.
The optical tuning of the propagation delay time is simulated for the 50 mm long YBa 2Cu 307
thin film on the LaAIO3 with er = 24. Taking into account the thermal healing length,
it is assumed that 2 mm of the PPSTL is illuminated by laser signal with low chopping
frequency or in the form of pulsed radiation with low repetition rate far less than the
input microwave frequency. Fig 4.11 demonstrates the variation of the propagation time
in terms of relative absorbed optical power. In accordance with temperature variation of
delay time, depicted in Fig 4.10. as the absorbed optical radiation is gradually increased, r
is increased due to the bolometric photoresponse. After a moderate applied optical power,
by approaching to the critical power, the propagation time is remarkably enhanced. This
promising technique can be also employed in realization of single and double transmission
line transversal delay line filters [128]. Optically modulated superconducting delay lines
are demonstrated for the first time by researchers at HYPRES Inc. in 1993 [129]. They
reported the microwave signal attenuation and phase modulation due to the optical irra­
diation. In 1997. S. Cho and H. R. Fetterm an reported an optoelectronic temporal and
frequency measurements of the optically tuned delay meander line made of YBa 2Cu 307 -<s
on LaAIOs [12]. They found th at the measured propagation time is well described by a
squared dependence on the applied optical pulse energy, and the bolometric photoresponse
62
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1=50 mm
L=2 mm
w=2 |im
h=1 |im
d=30 nm
e
=24
T=77K
T=55K
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 4.11: Propagation time for 50 mm long PPSTL made of YBCO for various relative
absorbed optical power
63
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is manifested [130]. Their measurement supports our simulation result shown in Fig 4.11.
but they gained less
t u n in g r a n g e
than it is expected from our structures according to our
models. The main reason behind this difference is that they used a relatively thick HTS
film deposited on the thick dielectric material in the form of microstrip line. This points
to two major drawbacks for achieving an optimum device. First, by using microstrip
line with thick spacer the magnetic inductance of the line becomes comparable to the
kinetic inductance, since the optical stimulation can only change the kinetic inductance
part resulting in small variation of propagation time. Second. HTS film thickness greater
than the optical penetration depth causes the kinetic inductance of the structure not to
change entirely along the film thickness. This comparison enlights the capability of our
developed model for optimum design of optically tuned HTS microwave devices.
4.4.2
Optically Tunable H TS Microwave R esonator and Filter
HTS microwave resonators and filters exhibit superior performances over its metallic-based
counterparts. Incorporating HTS into resonators significantly decreases the conduction
loss, resulting in a higher quality factor. Therefore. HTS can help to improve the perfor­
mance of the filters in two ways. First by decreasing the insertion loss, as well as the filter
roll off and reducing its bandwidth [131]. Secondly, filters can be miniaturized by a change
in filter geometry [132]. The increasing demand for high-quality HTS resonators and fil­
ters in satellite
com m u n ic a tio n
[133]. and mobile communication systems [134]. make the
design and the implementation of these devices crucial and their tuning becomes more
im portant. Some efforts have gone into resonance frequency and filter bandwidth tuning
by electrical technique [135]. by external optical radiation [136]. [137] and by magnetic
field [138]. The optically-tuned resonators and filters Eire again based on the variation of
the kinetic inductance of the structure suitable for applications when the large arrays of
these devices are existed. For a pmallei plate resonator, the resonance frequency, /o is as
64
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follows [139]:
f° =
I
r n *“ ( r )
t h(
d'
i , A“ ( r > t u
*
t [(5 i )S + (5 ^ )3]
(4.14)
where L is the length of the PPSTL. n and m are the mode numbers [for fundamental
mode (n.m)=(0.1) or (1.0)]. Note that the quality factor Q and bandwidth A/o are
directly related to the complex surface resistance stated in equation (2.34) through the
following relations for thick film limit when d > Al(T) [139]:
Q =
A /„
=
^
(4.15)
Kf
§■
(4.16)
Alternating the kinetic inductance of the HTS film results in the subsequent change in the
penetration depth follows by decreasing the resonance frequency and quality factor. To
observe how these quantities changed by optical irradiation, we compare our simulation
results based on the developed heat transfer model with the experiments done by M.
Tsindlekht et. al. [137]. [140]. They used relatively thick 300 nm YBa 2Cu 307 _,j film on
the 0.3 mm thick sapphire substrate in the form of parallel plate transmission line. This
structure shows the resonance frequency around 5.401 GHz depending on the operating
temperature. They used CW Ar laser with 514 nm wavelength guided by a multimode
optical
fiberto irradiate one of the HTS plate, with spotsizearound 1mm2, as depicted
in Fig
4.12. In order to
estim ate the thermal healing length
and the thermalization
time, there is a crucial concern. They illuminate a small part of the structure, while
the resonance frequency is determined by the total area of the PPSTL. Therefore we
should take a thermal healing length equal to the total length of the resonator, since
the bolometric photoresponse manifests itself over the entire structure. This makes a
thermalization time very slow, on the order of milliseconds. This thermalization time. Tgth
can be well estimated by the temperature penetration concept and thermal healing length
65
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Optical Radiation
Figure 4.12: Optically-tuned parallel plate HTS resonator
for time-varying temperature produced by CW optical illumination as following [141]:
r .* « ^
(4.17)
Substitution of lh by total length L. gives an estimated thermalization time 70 millisecond
which roughly agrees with their observed value.
Table 4.2 summarizes all of the geometrical and thermophysical parameters of the HTS
resonator and Fig 4.13 indicates our simulation results and their measurements, which are
in accord with each other well. The change in the resonance frequency is very significant
for the relative absorbed optical intensity greater than 0.5PC. This trend causes the
sadden decrease in the quality factor, thus we calculate the shift in resonance frequency
and change in Q for low absorbed optical power, presented in Table 4.3. The effect of low
absorbed optical illumination on the resonance frequency shift is quite significant while
the quality factor does not change considerably. So. it is clear to suggest that one should
use low optical radiation to achieve relatively high t u n i n g range with excellent sensitivity
and approximately fixed quality factor. This simulation gives us a very insightful guideline
66
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film thickness (d)
spacer thickness ( h )
0.30 fim
0.35 mm
penetration depth (Ao)
illuminated area ( A rad)
thermalization tim e (rt* )
0.27 p m
0.785 mm 2
0.07 sec
length (L )
width (ut)
9 mm
10
9.4
mm
Table 4.2: Parameters used for parallel plate resonator employing YBCO film on sapphire
substrate
5.45
5.4
r 5 .3 5
T=77K
T=55K
T=30 K
525
52
0.05
0,t
0.15
Relative Optical Radiation Intensity (I// y
Figure 4.13: Shift in resonance frequency of PPSTL for YBa 2Cu 307 _<s film on sapphire
substrate at different tem peratures. (*) indicates the measured value by T. Sindlekht et.
al.
67
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to design optically-tuned PPSTL resonator and filters as well as essential information to
optical tuning of readily available prototype components.
Pa
'
shift in resonance frequency A f„
change in quality factor A Q
0 . 1-0 . 2
38 MHz
%0 . 0 2
0.05-0.12
28.7 MHz
%0.03
0-0.05
21.5 MHz
%0.05
Pr
T = Z0K
T
=
55K
T = 77 K
Table 4.3: Computed values of resonance frequency shift and quality factors for various
temperatures and relative absorbed optical powers
4.5
Sum m ary
The Basic theory and modeling of the lumped interaction between optical radiation and
microwave signed was introduced in this chapter. The photo-induced effective temper­
ature eind consequent electrical changes in HTS film was studied in detail. All of the
discussion weis based on the HTS transmission line configuration with uniform optical il­
lumination for seeking potential optoelectronic applications and a technique for opticallytuned microwave devices. Our successful modeling was then proceeded with numerical
simulation of the HTS microbridge for various temporal forms of the optical radiation
and DC and time-harmonic microwave signal. Many basic optoelectronic functions such
as photo-detection, optical-to-electrical transducing, microwave harmonic generation and
mixing have been shown to be achievable with HTS materials. A complete mathematical
treatm ent of temporal analysis of the HTS optoelectronic mixer has been presented in
Appendix C based on the Fourier series approach. HTS optoelectronic device character­
istics including conversion gain bandwidth, external and internal conversion gains and
equivalent noise tem perature have been computed and compared with the other com­
petitive devices. The promising field of optical control of HTS passive microwave and
millimeter-wave devices along with its potential advantages has been introduced in the
last section. In order to extend the capability of our developed model to include this field
68
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as well, the optically tuned propagation HTS parallel plate delay line, resonators and
filters were discussed and the result of our simulation was compared with experimental
d ata available in the literature. Our theoretical discussion and simulation results not only
predicted the behavior of photo-induced changes in propagation delay tim e and shift in
the frequency resonance o f the structures fairly well but also provided some intuitive and
useful guidelines for design and implementation.
69
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C hapter 5
Traveling-W ave O ptical-M icrow ave
Interaction in H T S Film
Previous chapter dealt with lumped optical-microwave interaction in HTS material when
the optical radiation either illuminated the HTS microbridge uniformly to perform op­
toelectronic functions or excited a specific part of the HTS passive microwave devices
for tuning their electrical properties. In this chapter, we will study another attractive
method of interaction, when the optical illumination modifies the conductivity of a HTS
thin film in space and time periodically, while the microwave signal is passing through it.
As will be seen later, this configuration turns out to be a problem of wave propagation in
spatio-temporal conductivity grating.
Historically, the propagation of waves in periodically stratified media was discussed
as early as 1887 by Lord Rayleigh, who recognized that this problem was mathematically
characterized by the Hill and Mathieu differential equations [142]. This subject has been
applied to a number of physical problems ranging from atomic physics, theory of solids
and electrons in crystals [143], quantum mechanics [144] and optical multi-layers with
applications in filters, antireflection films and polarizers [145]. The field of slow wave
structures was stimulated by the development of the microwave tubes [146] and traveling-
70
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wave antennas [147] after 1940's. After that, the main emphasis was placed on the exact
solution of electromagnetic wave equation in sinusoidally modulated dielectric medium
with Floquet's theorem with application in propagation in plasma. In early 1970 s. the
technological achievements in the development of passive and active thin film solid state
devices ranging from acoustical to optical domains gave a strong impetus to the field
of wave propagation in periodic media. Periodic nonlinear optical materials, were pro­
posed and used for param etric interaction [148] and active optical materials were used to
demonstrate distributed feedback lasers [149]. Coupled wave analysis was widely applied
to the problem of optical wave propagation in periodic media with focus on perturbed
dielectric waveguides and semiconductors and also distributed feedback structures [150].
At the same time, the fabrication of traveling-wave thin film sources such as EMPATT
and Schottky diodes increased the progress of microwave and millimeter-wave communica­
tions significantly. Eventually the quest for high power and compact CW millimeter-wave
sources pushed the researchers to use laser-induced traveling-wave solid state devices for
generating millimeter waves. J. Soohoo et. al. presented a novel device concept based
on photo-induced current density in a semiconductor strip resulting in nmlti-watt output
power at GHz frequency range [91]. Following the first observation of picosecond photo­
conductivity in semiconductors and optical control of microwave propagation in dielectric
semiconductor waveguides by C. H. Lee et. al.. the periodically light-induced plasma re­
gion in semiconducting waveguide was analyzed for possible application in optically tuned
microwave filter by M. M atsum oto et. al. [151] and the implementation of such a device
was shown by W. P latte in 1990 [152]. The most significant feature of light-induced grat­
ing is dynamicity and the ability of its tuning for a desired characteristics. This helps to
form a read/write spatio-temporal information by modulating the electrical properties of
a material which is a centred idea for performing many optoelectronic functions such as
filtering and mixing. In 1991. N. E. Glass and D. Rogovin proved the theoretical possibil­
ity of having laser-induced microwave index grating in superconducting film [153]. Soon
after that, researchers at UCLA, utilized HTS materials to fabricate frequency selective
surfaces for potential applications in diffraction grating and quasi- optical microwave fil-
71
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ter applications [154]. To th e best of our knowledge, there is no theoretical studies of
microwave propagation in laser-induced conductivity grating in HTS materials in the lit­
erature. so based on the HTS photoresponse model developed in Chapter 3. for the first
time we will extend the optical-microwave interaction study in HTS transmission line for
the traveling-wave case.
In this chapter, after introducing a laser-induced conductivity grating in HTS materi­
als by two-beam laser interference techniques, mentioned in section 3.3.4. the propagation
of microwave signal with frequency well below the HTS gap frequency will be treated
rigorously in the infinite structure by solution of Maxwell s equations using Floquet's
approach. This analysis will give us the dispersion relationship and the amplitudes of dif­
ferent space-time harmonic modes presented in such a structure. Our theory supports the
well-specified propagation bands and stopbands well-suited for performing optoelectronic
functions. Crossing to the more practical case, the optical-microwave interaction will be
investigated by time-varying coupled mode theory in a finite length structure. In this
case, the distributed feedback characteristic of such a structure will be exploited for po­
tential microwave/photonic applications such as optically-controlled periodic microwave
signal filtering and measurement of thermophysical parameters of HTS thin films.
5.1
L aser-ind uced D y n a m ic M icrow ave G ra tin g in H TS
F ilm
Two degenerate laser beams th at coherently interfere with one another, can produce a
spatio-temporal grating within an optical penetration depth of a superconducting thin
film, as depicted in Fig 3.6. Within the framework of the heat transfer model devel­
oped in section 3.3. this optical power grating acts as am internal heat source along the
superconducting film, leading to establishment of temperature grating, given by relation­
ship (3.26). It is observed th a t by optical power interference, not only the amplitude of
72
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grating, but also the “phase" of the tem perature can be modulated. Since the angle be­
tween the two laser beam can be easily changed, the phase velocity of the “traveling-wave
tem perature distribution" can be adjusted for a desired value. Also, by changing the dif­
ference frequency of the two laser, namely 12. the characteristic of the tem perature wave
can be altered. Based on the Two-Fluid model, specifically equations (4.1) and (4.2).
the formation of dynamic tem perature grating in HTS film causes the num ber of Cooper
pairs and normal electrons to change in space and in time. The presence of light gives
rise to Cooper pair breaking that generates normal electrons, therefore the population of
carriers in HTS film vary in time and space in accordance with the optical power pattern.
These space-time population structure, in turn, establishes conductivity grating within
the HTS film, which are active for microwave and millimeter-wave frequency, below the
gap frequency of HTS material. By using th e temperature grating formula. (3.26) in (4.1)
and (4.2) and subsequent insertion in (2.16) and (2.17). the conductivity grating can be
w ritten as:
a a( z .t) = — { 1 - (
me I
\T C
{l +
TCV K X
^ —m ■—
cos(Qt — A z + <f>) }1 1(5.1)
y / ( l + DA2rth)2 + &T?h
r
cra{z.t) = re_ph- ( ^ - + ^ ^ { 1 +
.
m
m e \ T C TCV K X
y / { i + Dh?rthy- +
, cos {Sit - A z + <£)}) |[5.2)
V
It is seen the conductivity grating m athem atically contain the space-time harmonics of the
fundamental space and tim e frequencies 12 and A. but the amplitude of these harmonics
are rapidly decreasing, since the optical power can not exceed the critical power formulated
in (3.27). This condition is referred to as weak heating condition, since th e photo-induced
excess temperature does not cause the second phase transition to the normal state for
moderate bias tem peratures [155]. Most of HTS materials possess the exponent. 7 s: 2.
therefore we proceed our calculation based on 7 = 2. Later on, we will numerically justify
th at even the second time-space harmonic is at least one order of m agnitude smaller than
the first harmonic. Under this assumption, th e conductivity grating is modeled as follows:
<ra(z . t)
=
<Tn(z . t) =
<rao + <t,, cos(I21 — A z + <f>) + <r,, cos 2(f2£ — A.z + <f>)
(5.3)
(Tr^ + <7ni cos(I21 — Az + <f>) + <Tn2 cos 2(f2f —Az + <j>)
(5.4)
73
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where
'°
<T4'
=
e~n /-|
-l^ 7 1 “ ^¥e
VKTe '
(
OPaTthJTt
” 2 l T O v ^ T + DX2Tth)2 + W rfJ i
r \ r>
DPaTth
\ _________ Pn Pn aTthm_________
V K T C >V K T cy/ { 1 + DS?rthy + 9T-rfh
”
*>
rn
0 e2n
,T0
~ m e Tc
-
1 e 2 n / __________ P P a Tthm __________ ^2
^ ^
(5.6)
?)
2 me [ V K T cy/ ( 1 + PA?Tth)2 + f P r 2/
£~Tl f / To
ffn# =
PPa^th \ 2
vktJ
1/
PPjTthJ^
2 W r .y u T D A W
+
_
<r«, -
0r
e2n f Tp | PPgTth ^_________ P P aTthm _________
- r«-pfcTOel rc
; ^ 2; 7 ( 1 + PA.2Tthy +
<r„,
- T e. ph —
=
2
\ 2l
J
( ----------P P g T th m
)2
P m e V i r r cV/ ( l + ^ A 2rt/l)2 + n 2rtr
,—
(5.9)
(5.10)
The traveling-wave conductivity grating is composed of forward and backward waves dic­
tated by the laser interference grating. Following these mathematical expressions, we will
explore some of the im portant properties of such a grating structure deduced from the dis­
cussed analysis and mention some of the experimental issues and guidelines in more detail.
1-
These equations interestingly show th a t the conductivity grating along the HTS
film depends on the absorbed optical power. Pa. modulation index, m. difference laser
frequency. Q. and propagation vector A. Note that since the HTS film is employed in
moderate temperature, and applied microwave signal has frequency well below the gap
frequency, the Cooper pair conductivity. rra is much larger than normal conductivity (Tn.
Consequently the impact of the laser radiation is more observable in <rs. which collectively
represents the kinetic inductance of the HTS film.
2- The time and space frequency model of the conductivity grating includes the term
^
in all the expressions (5.5) to (5.10). For the difference frequency
V (1 + DAW
+ J P r 2*
up to nearly r ^ 1. the amplitude of the conductivity grating vs difference frequency Q is
approximately constant, and near and beyond that it decreases rapidly. Physically speak-
74
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ing. for the time frequency greater than r ^ 1. the electrical properties of the material do
not follow the optical signal, and the grating structure becomes less contrasting, and the
amplitude of the conductivity grating decreases. Therefore in order to increase the effec­
tive bandwidth, we should somehow decrease the thermalization time in the structure,
which was previously discussed in detail in section 3.3. The same scenario will occur
for the space frequency. A. around {DTth)~1^2- The effects of thermophysical parameters
of the HTS film /substrate to the laser-induced conductivity grating are clearly observed
through the discussed frequency response, which present basic physics underlying many
spectroscopic investigations for measurement of physical parameters of the superconduct­
ing materials [156], [157].
3-
The complete solution of the heat transfer equation (3.15) subject to the absorbed
power interference pattern given in (3.25) reveals that the steady state response should
be observed soon after the thermalization time. rth after the laser is turned on. After the
laser is switched off. the grating structure decays rapidly again by thermalization time
constant. In case of CW operation with on-off switching, the repetition period of the
laser radiation should be at least lO r^ to avoid cumulative thermal energy remains in the
medium.
4-
The presence of the dielectric substrate, mathematically modeled by the thermal­
ization time through the thermal boundary resistance, may affect the presence of con­
ductivity grating in the HTS film with direct absorption of the optical radiation. Heat
flowing in to the substrate takes place within the thermal penetration depth of the subj2D„
strate. and in this case given by At*. = y —— . where D, is the diffusivity of the substrate.
Considering the real values for diffusivity of the typical substrates, such as LaAlC>3 , MgO,
and SrTiOs. the therm al penetration depth into the substrate is in the order of nanome­
ter [78]. Choosing the substrate height, h in micrometers, justifies the validity of our
neglecting of such a thermal flow. Moreover, when we deal with microwave propagation
along the photo-excited PPSTL, it will be seen that microwave signed propagates along
75
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the stratified medium in which the substrate is represented by a dielectric constant. The
presence of the weak thermal grating in nanometer scale along the dielectric does not
have any impact on this parameter at all.
5-
It should be pointed out the conductivity grating model given in (5.3) and (5.4) is
the steady state response of the HTS film to the applied optical power. A laser source
may experience dynamic line broadening under CW operation especially when it is di­
rectly modulated, known as “Laser Chirping" [158]. In real experimental setup, the laser
interference pattern, due to a chirping effect might possess the nonstationary component
of the form cos 2(A vt —Az) corresponding to a grating velocity of ———. where Ai/ is the
laser linewidth around ft. Such an extra grating will have smaller amplitude, since most of
the laser radiation takes place at its center frequency, u . and consequently the amplitude
of the optical power at difference frequency, ft will be higher than the other frequency.
Also, this ex tra grating is still stationary as long as the A t' < r ^ 1. This linewidth criteria
is affordable well within microwave and millimeter-wave ranges [159].
The existence of the laser-induced spatio-temporal conductivity grating, can be used
to modify the propagation characteristics of the microwave propagation along the HTS
transmission line in both time and space domains. The unique properties of such grating
structure and its interaction with traveling microwave signals will eventually give rise to a
series of interesting applications in optically tunable filter and periodic filtering. Thus, we
will proceed our discussion by studying the microwave interaction with spatio-temporal
conductivity grating in HTS transmission line.
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5.2
M icrow ave P ro p a g a tio n in S p a tio -T em p o ra l G rat­
in g in H T S T ransm ission Line
Microwave propagation along the spatio-temporal conductivity grating is investigated by
solution of Maxwell's equations in this section. In the first treatm ent, we consider an
infinite length of a superconducting film having a conductivity grating given in (5.3) and
(5.4) mounted on a grounded dielectric substrate. The one dimensional rigorous solution
of Maxwell's equations with F loquets approach will be presented. The objective in this
development is to obtain analytical expressions for different spatio-temporal mode ampli­
tudes allowed to traverse along the grating structure and a dispersion relation expressing
the allowed wavenumber as a function of frequency and the parameters of the medium.
The analytical discussion will then continue with numerical simulation to explore the in­
teresting properties of such a grating in HTS film and potential applications for microwave
signal processing.
In the second treatment, we tu rn our attention to the microwave propagation along the
finite length of conductivity grating in HTS film by time-varying coupled mode analysis
deduced from Floquet's approach. In this scenario, by concentrating on the fundamental
forward and backward modes in the structure, the distributed feedback property of the
conductivity grating structure will be investigated by considering the Bragg condition in
both time and space domains. The cumulative reflection and transmission of a microwave
signal propagating along the conductivity grating presents the stopband and passband.
respectively. This implies the filtering functionality of such a structure, which is dynami­
cally tunable by changing the laser interference characteristics. The numerical simulation
again gives us a very illustrative situation for possibility of having tunable periodic filter
and also microwave signal generation in laser-induced grating in guided-wave HTS struc­
tures.
77
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5.2.1
Floquet’s Approach
Consider a thin superconducting
film
with thickness comparable with the optical pen­
etration depth on a grounded dielectric substrate forming an HTS transmission line as
depicted in Fig 5.1. We assume that the conductivity grating is well-defined through
the
film
thickness which is active for any electromagnetic wave with temporal frequency
below the gap frequency. According to the Two-Fluid model and Maxwell s equations,
the electric wave equation for a superconducting medium having a space and time varying
conductivity can be written in the following form as it has be already given in equation
(2.25):
V 2E (r.f) = fio^[o-a(z.t)
J
E (r.t)d t + q-n(z .t)E (r.t)] +
^—
(5-11)
<Te(z.t) and crn(z.t) possess the mathematical form expressed in relations (5.3) and (5.4).
The electric field equation in a lossless dielectric material. E d(r. t) with relative permit­
tivity
is as follows:
V 2E d( r.t) = ^ d2Eg ^ ' t)
(5-12)
The solutions of equations (5.11) and (5.12) can be sought for two different wave types,
namely TE and TM. Here the solution is accomplished for TM wave with components
E z. Ex and Hy since in this case the electric field has a longitudinal component and will
progressively change along the grating structure. Equations (5.11) and (5.12) must be
satisfied for both electric field components and the magnetic field Hy can be easily found
from the Maxwell's equations.
In a periodic structure, the shape, size or constitutive material relations vary periodically
along the longitudinal axis. The basis for the study of such periodic system is a theorem
ascribed to the French mathematician Floquet. which may be extended for time and space
periodicity as follows.
For any periodic structure, for a given mode of propagation at a specified time and space
frequencies, the field at one cross section differs from those one period (or an integer
multiple of periods) away by only a complex propagation constant. This theorem is valid
78
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Substrate
x=0
HTS Ground Plane
TM Wave
Figure 5.1: TM-wave interaction with laser-induced dynamic grating in HTS transmission
line
when a periodic structure of infinite length is replaced by one period. Suppose th at the
angular temporal and spatial frequencies of the grating be 12 and A. then according to
the Floquet's theorem, the electric field can be written as:
E [ x . y . z ± m ^ . t ± m ^ - ) = E ( x . y .z . t ),.**"*!84 ~ Az>
A
S2
(5.13)
where m is an integer number. The field quantity E ( x . y . z . t ) can be expressed in the
traveling-wave form E ( x . y j e ^ ^ e ^ 2. It is seen th at the field quantity is a periodic
function in both time and space domains which can be expanded into a Fourier series of
the following form:
E ( x .y .z .t ) =
E { x .y )e ii<UJ + n W t ~ t t P + nA) z
(5.14)
n=—00
The expansion (5.14) physically indicates that the field pattern at each cross section should
be fixed for a lossless case, and the field is changing periodically along the structure. This
expansion also represents the forward and backward waves with different phase velocity
79
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and the same group velocity that are continuously reflected and transm itted from each
grating cell. Using the plane wave solution for the transverse direction, the solution of
wave equation (2.25) for a TM-wave in HTS film and dielectric substrate can be sought
in the form of [160]:
E z(x.y. z .t) = <
e~ j k xx
+O0
Y , An( r i ^ + n ^ t - ^
+ n^ z
(5.15)
and
{
j kxdX
e j k =dX
| +00
\ J 2 Bnej ( u + n S l)t- j{ l3 + nA )z
(516j
J
where An and B n stand for the amplitude of the time-space harmonics of the electric
field, he and kxd the wavenumbers along x in HTS film and the dielectric substrate,
respectively. The mathematical representations of the electric field in (5.15) and (5.16)
indicate that the electric field is expanded in terms of infinite number of traveling-wave
modes specified with the space and time periodicity, where in each time and space periods
the propagation has a plane wave form with unknown )3 for a given frequency. u>. This
expansion can be also viewed as a Floquet's expansion of the field in a moving coordinate
system, representing the traveling-wave interaction of microwave signed with a progressive
sinusoidal grating imposed by laser interference pattern. The relationship between kx. and
j3 with constitutive parameters of the grating structure combined with continuity of the
tangential fields E, and Hy at the three boundaries, is known as “Dispersion relation".
Note that because the grating structure is formed in the HTS film, we should first find
the solution of the filed in this region and after th at by satisfying the boundary conditions
at the three boundaries, we are able to find the dispersion relation.
Introducing the solution format (5.15) in the wave equation (5.11) gives the following
relationship for the amplitude of spatio-temporal harmonics in the HTS film region:
D
n A n
~
^ - n .n — 1 A n _ 1 - |- K r i.n + 1 ■ ^■ n+ 1
d*
^ n . n —2 A n —2
^ n ,n + 2 ^ n + 2
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(5 - 1 7 )
where:
Dn =
k l-r n
(5.18)
7n =
(£ + nA )2 _ ^ + ” ^ - + fiocr,0 + jfi0(Tr^ (<jj + reft)
cr
(5.19)
and
M -+
- „ + (tf + iy S ) +
+ » + ( . ■- i m ) }
(5.20)
- B + ( , + 1 )0 )}
(5.21)
= - { f r p + M+ (f _ 2)ti) + y S l ( w + 20 H- ( , - 2)11)}
-
w+
( ”
2 ) f l >
+
~
2 S i
+
< "
+
2 > » ) }
(5.22)
< 5 -2 3 >
Note that, equations (5.19). will produce the complex propagation constant of the unper­
turbed superconducting medium given in equation (2.27) when n = 0.
The same procedure is simply applied for a dielectric substrate without existing any cou­
pling coefficients, resulted in a much simpler expressions. The recursive relationship in
(5.17) is similar to the relationships that Simons derived for the propagation of waves in
a progressive disturbance of a medium through its dielectric constant [115]. Our recursive
algebraic equation relates the different time and space harmonics to each other via the
conductivity grating parameters. A scrutiny of the right hand side of (5.19) reveals that
the first and second term individually account for the spatial and temporal evolution of
the field in the grating structure, while the third and fourth terms describe the station­
ary part of the material properties including the Cooper pair and normal conductivities.
The effect of traveling-wave grating can be seen through the coupling coefficients, k, j .
in (5.20) to (5.23) which represents the strength of coupling between different space-time
harmonics.
Satisfying the boundary conditions at the three boundaries, namely x = 0, x = h and
x = d + h. gives an equation for the propagation constant, (3 in the general matrix form
as follows:
D(uj.(3)A = 0.
81
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(5.24)
It should be noted th at the dispersion m atrix ID is a square matrix, in which the offdiagonal elements include the coupling coefficients. The solution of this equation will be
nontrivial if.
det D(w./?) = 0 .
(5.25)
This is the dispersion relation which gives the value of /? as a function of ui. the material
properties and grating parameters. The solution of (5.24) would then give the relative
values of the space harmonics in terms of the amplitude of the fundamental harmonic,
re = 0. The value of A 0 and B q itself is d e t e r m in ed from the source or boundary con­
ditions. In case of unperturbed structure, namely II = A = 0. the dispersion equation
(5.25) has the analytical form, which J. C. Swihart derived that in 1961 [37].
The behavior of the complex dispersion function 0(u;./?) for a given real frequency ui
indicates the presence of the passbands and stopbands for a traveling-wave field along
the periodic structure. As expected from an infinite grating structure, the stopbands are
A
centered at /I = re— in the space domain and at u; = re— in the tim e domain, according
2
2
to the zeroth order approximation. The transition between the passband to the stopband
can be used to determine the effective bandwidth of the inherent filtering functionality
of this medium. We will show this feature in much more detail by elaborating with the
time-varying coupled mode formulation in the next section.
It is necessary to mention that (5.25) will give us a pure real value for the allowed
wavenumber. (3 for a given real frequency u>. when the phase velocity of the dynamic grat12
Cl)
ing. Vp = —. is smaller than the phase velocity of the traveling-wave V„ — — [161]. [162].
Therefore the stability criterion for having a traveling-wave field along the spatio-temporal
grating can be stated as:
Q
U)
V p < V ' ==* A < 0
(5 *2 6 )
If this criterion is not satisfied, the time-growing field-wave can be established in the
grating structure1. The condition (5.26) can also be used as a check-point, when we will
1T he mathematical proof for the stable interaction in space-time periodic structure is beyond the
scope o f this thesis. For com plete study refer to the excellent appendix in reference [162], and discussion
82
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develop a numerical solution of equation (5.25).
A technique commonly employed in the solution of M athieu-type equations based on
the Floquet's expansion will eventually give rise to similar detenninental equation the so
called Hill determinant. Although, the numerical solution of equation (5.25) can be easily
obtained by typical numerical schemes for root finding, very intuitive evaluation of the
determinant presented by Morse and Feshbach. can also be applied to our problem [164].
In order to find the relative amplitude of space-time harmonics in the HTS film, we follow
the s im ila r procedure as was proposed for a solution of fourth-order, linear, homogeneous
difference equation by tra n s fo rm in g it to the three-term recursive equation [165]. The
detail procedure is presented in Appendix D. Following this procedure, the difference
equation (5.17) can be written as:
QnAn + R n -iA n -i +
— 0.
(5-27)
where.
r~. , ^n.n—2Kr»-l.n Kn+l.nKn.n+2
Qn = Dn H
------------- !----.
R*n—l.n—2
^n+l.n+2
n
Kn.n+1 Kn.n+2
^n.n-2 n
Rn-i = — ;---------------- «n.n-i Dn- X.(5.29)
Kn+l.n+2
^n—l.n—2
5
^n—l.n+l^n.n—2
^n,n+2 ^
n+1 =--;----------------- Kn.n+1 -^n+1*
Kn—l.n—2
KTl^_i>n_j>2
(5.28)
,c
/r on\
(5.JU)
A rigorous solution of three-term difference equation (5.27) can be obtained by the rapidly
convergent continued fraction as [166]:
An
Rn—\
-------------------------------
for n > 1
Q n -
*Sn+2^n+l
Qn+1 ~
Q r t+ 2
presented in reference [163].
83
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(5.31)
and
An
~ R”
A "+1
for n < 1
(5.32)
S n R a - 1
Q n
-----------------------------------------------------------------------------------
S n -1 R n
Q n -1 ~
Q n- 2
—
• * •
The convergence of the continued fractions can be verified by applying Poincare theorem
[166]. or Weyl theorem [147] in the limit when n —>oo. It can be easily shown that
,.
Rn-l
li m ———
n-yoo Qn
r
•?„+!
———
n-yoc Qn
= lim
c2
V~
= 1 ———
. where c is the speed of light.
Poincare theorem 2 asserts that for the convergence of the continued fraction (1 —
<?
1
) < -.
p
which can be satisfied by choosing Vp < y/2c.
It should be noted that the amplitudes B„ can be found via the boundary conditions.
The power of the discussed solution can be seen much more clearly when the conductivity
grating becomes very strong,
s im ilar
to pumping condition encountered in wave propa­
gation in plasma medium [160] and hypersonic waves in liquids [168]. This is not the
case for laser-induced conductivity grating in HTS film, since by applying the absorbed
optical power greater than the critical power, the thin film under irradiation will be no
longer superconductor. Although, our formulation will be valid if the conductivity of the
m aterial can be still presented by formulas (5.3) and (5.4).
5.2.2
Simulation R esult from F loq u et’s Approach
Consider a thin and long superconducting film made of YBa2Cu30 7 _i on LaAJ0 3 dielec­
tric which is illuminated by the dynamic laser interference pattern, described in section
3.3.4. Therefore, the steady-state conductivity grating is established along the HTS film.
We assume that the TM-plane wave with tem poral frequency smaller than the gap fre­
quency of the HTS film is
la u n c h e d
into the film, as depicted in Fig 5.1. The interaction
2For com plete discussion and m athem atical m anipulation, references [166] and [167] are recommended.
84
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of this wave with traveling-wave conductivity grating is then simulated by the Floquet s
approach discussed in previous section in an infinite length of the photo-excited trans­
mission line, where there are no physical boundaries present in the structure. All of the
geometrical and thermophysical parameters for a high-quality thin YBa 2Cu 30 7 _(} film
structure are presented in Table 5.1. The thermal conductivity and diffusivity are taken
from reference [156].
Initially we assess our numerical simulation with a very simple case when no conductivity
Description
Value
Initial Temperature (Tn)
77 K
C ritical Temperature (Tc)
92 K
100 nm
Film Thickness (d)
London Penetration Depth (Ao)
e~n
m*
% 180 nm
2.4693 x 1019
O ptical Penetration Depth (d0)
D iffusivity ( D )
90 nm
0.5 x 10"6 m 2/s
Therm al Conductivity (K)
0.5 W /(m K )
Phonon Escape Time [rr, )
0.1 ns
Dielectric Height (h)
300 nm
R elative Perm ittivity (ej)
24
Table 5.1: Parameters used in the analysis of microwave propagation in traveling- wave
photo-excited YBa 2Cu 30 7 _^ transmission line.
grating is presented, mathematically 12 = A = 0. but the line is uniformly irradiated by
the constant optical power. The dispersion diagram is showed in Fig 5.2. which confirms
the linear dispersion relation, leading to a constant phase velocity down the transmission
line. The effect of optical radiation is seen to increase the effective tem perature and con­
sequently the HTS penetration depth, which in turn decreases the speed of wave in the
transmission line. At the first stage, the laser interference pattern with A = 5 rad/m m
and 12 = 207T GHz is applied over a length of 15.7 mm. This length contains 125 period
2tt
of the space grating. — to assure the infinite interaction length assumption. Under this
A
85
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0 =0
A=0
P =0
40
P =0.9P
I
s
O 30
20
-
0.5
2.S
2
35
45
P (rad/mm)
Figure 5.2: Dispersion Diagram for microwave signal along the u n ifo rm ly irradiated HTS
film/substrate
condition the critical optical power given in (3.27) is 6.109 watt. Comparing the typical
optical power needed for creating periodic microwave structure in semiconductors, implies
that in case of HTS material, the required optical power is comparable with its semicon­
ducting counterparts such as silicon [169]. The dispersion relation for TM-microwave
signal obtained for frequencies near fl is sketched in Fig 5.3 for various absorbed opti­
cal power. It is interesting to note th a t by applying optical radiation, the propagation
constant plots produce two different regimes around a cross-over point around the first
order Bragg-condition. where two dispersion curves for different absorbed optical power
meet each other. For each absorbed optical power, there is a cross point in which the
propagation constant (3 can be reduced or increased for a fixed frequency. For any mi­
crowave signed with temporal frequency and propagation constant exactly equal to this
cross-over point, the optical radiation has no effect since the cumulative reflection from
grating structure is negligible. In case of Pa — 0.9Pc and Pa = 0.1 the cross over point is
occurred at (3 = 2.21 m m and u/ = 22.4Grad/s. This phenomenon potentially indicates
86
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x 10
A=5 rad/mm
P _= 0.9P
P = 0.1P
P =0.5P
CO
00
<5
3
0.5
2.5
3.5
4 .5
P(rad/mm)
Figure 5.3: Dispersion Diagram for microwave signal along the laser-induced spatiotemporal conductivity grating
87
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the ability of this structure for very fine microwave filtering and tuning. Note th a t this
feature is a consequence of space-time evolution of microwave signal with traveling-wave
modulated conductivity of the HTS film, and all of the values for laser interference param­
eters and thermophysical properties of the HTS film play crucial roles in this complicated
scenario. A relative power of the first four forward and backward modes are sketched in
Since a„x » crn , the first forward
Figures 5.4 to 5.7 for a fixed tem poral frequency.
0.12
x104
\=5 rad/mm
0.1
0.08
<
o
0.06
<
0.04
0.02
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
p ip
Figure 5.4: Relative power of the second backward space-time mode
mode is progressively traversed along the structure with greater amplitudes, and the first
backward mode resulting from cumulative reflection of the microwave signal from weak
and local conductivity grating is established along the infinite length of the photoexcited
HTS film. As the absorbed optical power is increased the relative amplitude of the four
(excluding the fundamental) forward harmonic becomes weaker and the rest of the ampli­
tudes become stronger. Comparison of the relative amplitude of the different space-time
harmonics clearly suggests th at the first forward mode is many orders higher than the
others, as the second order of the conductivity grating is far less manifested. This con­
clusion supports the use of the coupled mode approach in the next section where only the
88
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*10*
450
400
a=5
350
rad/mm
300
~p
250
< '
200
150
100
0.1
0.2
0.3
0.4
0.5
P
0.6
0.7
0.9
IP
Figure 5.5: Relative power of the first backward space-time mode
55
10k Grad/s
3.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
pip
Figure 5.6: Relative power of the first forward space-time mode
89
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80
60
<
o
CM
CM
<_
40
30
0.1
02
0.3
0.4
0.5
P
0.6
0.7
0.9
IP
Figure 5.7: Relative power of the secoud forward space-time mode
first mode will be taken into account.
The temporal frequency behavior of the dispersion diagram for a fixed space grating A = 5
rad/m m can be understood from Fig 5.8 for frequency 2ir Grad/s. By further lowering
the temporal frequency the dispersion digram does not change to much. Following the
same procedure, the spatial frequency behavior, below the spatial bandwidth of the struc­
ture must demonstrate a fixed trend, similar to Fig 5.2. Physically, by decreasing A, the
effect of the first term in equation (5.19) in comparison with the third and forth term
becomes negligible. In this case the dispersion diagram, gives the constant phase velocity
of the traveling-wave along the structure, when the optical irradiation decreases the phase
velocity of the wave. By raising A above the spatial bandwidth {Drcs)~1^2 the electron
diffusion counters the effect of laser-induced population grating and the similar curves
presented in 5.2 is observed. This feature potentially offers a novel technique to measure
the electron diffusion length of any superconducting film.
From numerical simulation presented in this subsection, it is concluded th at the contri-
90
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X 10
600
Q=2xx 1000 Mrad/s
500
P =0.1P
400
P .= 05P .
300
200
100
05
2.5
3.5
P (rad/mm)
Figure 5.8: Dispersion diagram for low temporal frequency of dynamic grating
bution of space and time evolutions o f the microwave signal along with optically-induced
variation of material properties manifested in the conductivity grating results in various
impact on dispersion diagram. The n o n lin e a r and effective interaction between the prop­
agating microwave signal and the laser-induced dynamic conductivity grating is critically
m a n ife s te d
around the spatial and tem poral bandwidth of the structure dictated by the
electron diffusion and phonon escape phenomena, around the first order Bragg condition.
In this nonlinear re g im e , the optical radiation enables us to change the propagation con­
stant for a fixed temporal frequency. At the cross over point, for instance seen in Fig 5.3.
a certain microwave signal can travel along the structure without being affected by the
optical radiation power level. Furtherm ore, the comparison of the relative amplitudes of
different spatio-temporal modes in th e grating structure, implies that the first forward
and backward modes are much more stronger than th e others, which leads us to focus on
the coupled mode approach.
91
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5.2.3
Coupled M ode A nalysis
The problem of microwave propagation in the laser-induced dynamic conductivity grating
is cast in the time-varying coupled mode formalism. The advantage of this novel treatm ent
is twofold. First, the microwave propagation is considered near the Bragg condition for
both time and space domains, leading to understanding of the filtering functionality of
the periodic structure more clearly. Second, the optical-microwave interaction can now
be analyzed for the finite length of the HTS transmission line.
Capturing these two
significant advantage, we will lose the accuracy of the result compared to the Floquet’s
expansion method discussed in the two previous subsections.
A physical boundary condition at the two ends of the laser-induced dynamic grating
can form a distributed feedback structure, which would be frequency selective in both
space
and time. As
it was concluded from the preceding in depth Floquet’s
analysis,
the second order of the conductivity grating results in the weaksecond and higher order
spatio-temporal harmonics with respect to the first forward and backward modes hence
the higher order coupling coefficients can be neglected. We refine our expressions for the
conductivity grating, formulas (5.3) and (5.4). as follows:
a„(z.t)
=
<
tSq + <7„ cos(ft< —Az)
(5.33)
an(z .t)
= (Tnv + crni cos(Clt — Az)
(5.34)
The phase difference (f>is om itted, as it does not affect our formulation. We focus our
attention on the TM-wave equation describing the variation of the electric field in the
longitudinal direction, z . by introducing (5.33) and (5.34) into the wave equation (5.11).
From Floquet’s analysis, one can guess that the spatio-temporal periodic structure can
create a non negligible backward wave if the phase delay between the small reflected wave
from each period are chosen correctly. Based on the zeroth order approximation such a
feedback system can generate reflected wave of significant amplitude, under the following
Bragg conditions:
p = n — and
lj
= n—
92
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(5.35)
where n corresponds to the various grating orders: in what follows, we pick the first order
grating n = 1. The Bragg conditions expressed in (5.35) indicate our first guess for the
space and time synchronization of the incoming microwave signal (wave characteristics
u; and /?) with moving conductivity grating parameters Q and A, for producing a strong
reflected or transmitted wave from laser-induced periodic structure. Thus, we deliberately
guess the field solution for the forward and backward electric field near Bragg conditions
as [170] and [171].i.e..:
_■ (
-
Ez(z.t) = f ( z ) e J 2 Z
7 -
- ~ t \
- ~ t \
+ r(z)e {2 Z
2
2 V H
(5.36)
The functions f ( z ) and r(z) represent the forward (+z) and backward ( —2 ) waves, respec­
tively and take care of boundary conditions at the ends of the structure. Substitution of
the solution (5.36) in wave equation (5.11) and collecting terms with common exponential
multipliers yields the following coupled differential equations:
+ [(/}(* + | ) ) 2 - ( | ) !] /( z )
^ r ( z ) - j \ 3 - r ( z ) + [(/3(u; -
^ ) ) ! - ( ^ ) s]r(z)
= * ,r(z |
= K ./(z)
(5.37)
(5.38)
where (3 is the propagation constant resulted from the previous Floquet’s analysis. The
coupling coefficients.
Kia and
and
k2
can be obtained from the modified coupling coefficients
ia the HTS film region which can be expressed as
« i.
+j(Tni(uj+ y ) }
=
U
Hi..
=
(5.39)
2
j j ) +jo-ni(u) + — )}
u + -
(5.40)
In the coupled differential equations (5.37) and (5.38). we neglect the terms involving
3
higher orders of the space and time variation, such
as
j ± »(Az —Qt)
e
2
. This coupled
differential equations model the exchange of the energy between the forward and backward
93
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electric fields in the spatio-temporal periodic structure. In order to find the unknown
functions f (z) and r(z) we make further intuitive approximations as follows:
1- The second derivative terms of f ( z) and r(z) are neglected, since they are additive to
the other terms with large multipliers in the order of A.
2- Based on the solution near Bragg condition, we do not distinguish between [3 and —
&
unless the difference occurs, thus by defining the detuning factors
(5.41)
and
Q.
s2 -i
2
(5.42)
2
we approximate the third terms of each differential equations as:
[0{w+
y ) ) 2 — t ^ ) ' = [/*(w +
(*» - | ) ) 2 - ( f )2 = [/#(- -
—( j ) ]
j) -
[fliw+
+ (^)] ~ asi
( f )] [/»<“ - § ) + ( £ > ] * « ,
3- We assume the functions / and r vary slowly with z [170].
The coupled differential equations (5.37) and (5.38) with the discussed approximations
reduces to a more manageable form:
f {z)
d z2 r ( z ) + j q Tz
f(z)
f(z)
r(z)
r{z)
= 0
(5.43)
where
2 A Ki K2
p
=
- jr -
q =
Si — S2
- .
-
(5.44)
(5.45)
Equation (5.43) is an extended version of the conventional coupled mode equation for
analyzing the permanent periodic structure [150] and [172], where the coupling coefficients
vary with input microwave signal and difference lasers frequencies. Our derived coupled
94
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mode differential equations also possess the extra term, which presents the damping of the
forward and backward fields due to its frequency detuning. In order to solve the differential
equation (5.43). the geometry shown in Fig 5.9 is considered. This geometry corresponds
to the distributed feedback structure which, has been widely used in grating assisted
integrated optical devices. The forward and backward field solutions are then sought in
L
the finite length. L of the dynamic grating with boundary condition r(z = —) = 0 as:
J(z) =
r(z)
^
r J 2(Z “ 2*{ - y/p* - <?)= cosh
- ( | )Hz - | ) ]
+ i ( | + <2)sinh [J p 1 - ( | ) 2(z - ^ ) j }
(5.46)
.q
L
________
= j A e ~ 3 T * ~ 2 1sinh [J p * - ( | ) 2(z - j ) ]
(5.47)
Relating the forward wave at z = —^ to incident field driving the structure and also
expressing all phases relative to that as [170]
AL
= - j ) = B t = Hz = - |) e J 4
(5.48)
yields the coefficient A and the reflected wave in the uniform HTS transmission line at
L .
z = —— is equal to:
L
1 4
_.A
B ; ( z = - | ) = r(z = - j ) e
95
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(5.49)
The reflection and transmission coefficients. T and 0 or S-parameters are then easily
calculated by equations (5.48) and (5.49). respectively as follows:
1-1
A
1
“
Ez (Z =
_ 2")
_
q
(r -
- T \
~
b l1 ~
J
^ J in c l2
—
_
n
*
2 '
________
_ .AZ
[ ^ P 2 ~ ( | ) 2^ ] e J 2
=
(5.50)
"J'
y^P2 - ( | ) 2cosh [ ^ p 2 - ( i f L ] - , - ( § + * ) Sinh [ ^ p » - ( | ) 2l]
g
a
0
-
^ ( Z—
o
T
^
q
_
q
2)
_
i )
- 5 ,2 - 5,1
-9 r
-A I
2„~J
n
~
3
~
f - q r - e
2 e
(5.51)
V
^ c o s h
[y/p2 - ( f ) 2^] - i ( | +
Laser 1
sinh [y/p2 - ( | ) 2£]
Laser 2
Uniform
Uniform
z=-U2
z= L/2
Figure 5.9: Geometry of the Laser-induced distributed feedback dynamic grating in HTS
transmission line
96
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Conservation of the total power in the structure yields the condition |5 n |2 + l-^isl2 = 1
for the lossless case, which is satisfied with equations (5.50) and (5.51). Having the
elements of the scattering matrix enables us to examine a variety of cases of practical
importance, such as distributed Bragg reflector when the power reflection coefficient |T|2
is quite high and also optically tunable filter based on quarter-wave bandpass filter [170].
5.2.4
N um erical Simulation o f Laser-Induced D F B Structure in
HTS Transm ission Line
Interaction of the TM-wave with the traveling-wave first order conductivity grating de­
scribed by equations (5.33) and (5.34) is simulated numerically in a finite structure, shown
in Fig 5.9. The geometrical and thermophysical parameters of the HTS film are chosen as
1507T
same as section 5.2.2 tabulated in Table 5.1. The length of the structure is held at —-—
A
providing 75 space periods typically in the order of millimeter. As a first step, we focus
our attention to a static conductivity grating where ft = 0. W ith the given geometrical
and physical parameters of the structure, and input microwave frequency we obtain the
wavenumber (3 as a function of absorbed optical power. Pa. so we can track the variation
of transmission and reflection of the microwave signal with various propagation vector A.
As Pa is increased the propagation constant is decreased, and for A > 2(3. the structure is
more transmissive, because it is getting far from the Bragg condition. This fact is graph­
ically shown in Fig 5.10 for A = 5.5 mm. The different behavior of the relative power
transmission for various amount of the absorbed optical radiation intensity and frequency
is a consequence of the nonlinearity of equation (5.51) with respect to absorbed optical
power and input frequency via the coupling coefficients. Fig 5.10 confirms the general
behavior of the structure when the static grating is made stronger by impinging more
optical radiation power, leading to creation of weak transmission grating.
At the second step we set ft = 2n G rad/s.
The relative power transmission is then
sketched for a broad frequency range around the first Bragg condition. Note that at
u} — ^ the first coupling coefficient K\t is a singular function of frequency as clearly
97
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P = 0.9P
-2
I, (dB)
Q=0
20log|S
A=5.5 rad/mm
-5
-7
P = 0.8P
-8
A ngular Frequency, to (Grad/s)
x 10
Figure 5.10: Relative power transmission (in dB) as a function of microwave angular
frequency
98
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seen in equation 5.39. This built-in singularity is a direct consequence of the temporal
interaction of the input electric field associated with microwave signal with the backward
laser-induced traveling-wave conductivity formed in a common period of time. This phe­
nomenon physically implies the synchronization between the photo-induced conductivity
traveling-wave and the input microwave signal. In our simulation the asymptotic behav­
ior of the coupling coefficient. k x, is taken into account with the polynomial expansion
around the singularity to obtain Ki. At this special frequency, which is indeed the first
order Bragg condition the transmission of the microwave signal approaches zero, as it can
be mathematically verified from equations (5.51). Fig 5.11 demonstrates the variation of
the relative power transmission of the microwave signal for A = 5.5
rad/ m m
after the
first order spatial Bragg condition and over the large frequency band after the first or­
der temporal Bragg condition. It is seen that for the frequencies less than the inherent
bandwidth of the structure r ^ 1. many frequency bands are established and after that
the transmission approaches to one. as expected. This behavior is observed for different
absorbed optical power radiation. Fig 5.12 illustrates the pass bands created at narrow
frequency intervals for different absorbed optical power. It is interesting to note that
under the intense optical radiation intensity, namely 0.9PC. narrow passbands are estab­
lished leading to a periodic filtering of microwave signal. Each band has a 3dB bandwidth
approximately equal to 1.25 M rad/s with the 5 M rad/s stopbands presented in between.
As the optical power is decreased, the bandwidths increase with associated broadening
of the stopbands. Further investigation over the frequency range from 4.70 Grad/s to
5.70 Grad/s reveals approximately 200 channels formed with 1.10 M rad/s to 1.25 Mrad/s
3dB bandwidth with 4.80 M rad/s to 5.00 M rad/s stopband in between with -8dB loss in
relative power transmission. The advantage of this scheme is a
d y n a m ic
tuning of the
filtering properties with the absorbed optical power, propagation vector and a difference
frequency of the laser interference patterns. Fig 5.13 indicates th e variation of the rel­
ative power transmission of the microwave signal near the Bragg-condition. It is seen
that the transmission coefficient is rapidly changing with the absorbed optical radiation
intensity, but the envelope of the variation is experienced its maximum just before Pc
99
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P = 0 .9 P
-2
(ap) f ls|6o|02
-4
Q =10 G ra d /s
A=5.5 rad/mm
-7
0
1
2
3
4
5
6
7
8
9
Angular Frequency, to (Grad/s)
Figure 5.11: Relative power transmission (in dB) as a function of microwave angular
frequency
100
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x 10
-2
P =0.1 P
(ap) f ls|6o|02
P =0.5P
-10
P =0.9P
-12
471
471.5
472
472.5
473
473.5
474
474.5
Angular Frequency, qj (Mrad/s)
Figure 5.12: Relative power transmission (in dB) as a function of microwave angular
frequency
101
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x 10
as expected. Although, the absorbed optical radiation in the range 0.5PC < Pa < 0.9PC
does not provide the deep band reject loss for the given propagation vector and difference
frequency between two lasers. It should be pointed out that due to the complexity of
the structure and its sensitivity to the chosen parameters the numerical simulation for
various combinations of the parameters would be the only way to yield the optimum result.
-2
20log|S12|, (dB)
-4
-6
Q = 20 tc G ra d /s
co=1 5ji G ra d /s
A =5.5 rad /m m
-1 0
-12
0
0.02
0.04
0.06
0.08
0.1
0.12
0 .1 4
0.16
Absorbed Optical Power, P (W)
Figure 5.13: Relative power transmission (in dB) as a function of absorbed optical radi­
ation power
The periodic filtering can potentially play an important role in many signal processing
102
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tasks in applications such as FDM (Frequency Devision Multiplexing) in microwave fre­
quencies. WDM (Wavelength Division Multiplexing) for optical domain [173] and analog
modulation/demodulation [174].
In su m m a r y the proposed HTS-optoelectronic tunable filter has potential applications in
microwave/photonic systems to build add/drop filters when the microwave signals carried
by optical signal. This device can also act as a photodetector and tunable microwave
filter simultaneously for a multi-channel system, suited for high quality radio over optical
fiber applications.
5.3
S u m m ary
The laser-induced traveling-wave tem perature grating described in section 3.3.4 has been
introduced in a generalized Two-Fluid model to derive the traveling-wave conductivity
grating along the HTS film. The super and normal conductivity gratings have been
analytically expressed in terms of the optical interference pattern characteristics and
thermophysical parameters of the HTS film. The microwave signal propagation along
the laser-induced conductivity grating in the HTS transmission line is then analytically
solved using Maxwell's equations. The solution of the Maxwell's equations was obtained
by Floquet expansion of the spatio-temporal harmonics of the electric field associated with
the input microwave signal in an infinite length of the HTS film. The immediate result
of this approach is to give dispersion diagram and the relative amplitudes of different
spatio-temporal modes as well as the stability criteria for having stable electromagnetic
fields along the dynamic grating. The numerical simulation of microwave propagation
including the dispersion diagram and relative amplitudes of different spatio-temporal har­
monics has been performed for the HTS transmission line made of YBajCuaOr-j. The
different dispersion diagram regimes observed under the variation of absorbed optical
power, laser-interference pattern characteristics and physical parameters were discussed
in detail. Detail numerical simulation revealed that the higher-order spatio- temporal
harmonics are many orders weaker than the first forward and backward modes, thus by
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re fin in g
our expressions of the super and normal conductivity gratings, the microwave
signal propagation was formulated by the time-varying coupled mode analysis. The main
advantage of this method is to include the physical boundary condition at the two ends of
the structure which results in the analysis of the traveling-wave optical-microwave inter­
action in a finite laser-induced distributed feedback structure. The Bragg-reflection and
transmission characteristic of such a device was investigated over a broad range of the
frequency. As expected, the traveling-wave nature of the conductivity grating resulted in
a periodic filtering functionality over a relatively wideband in the microwave region. This
demonstrated a potential applications of the traveling-wave HTS microwave/photonic
devices for high performance telecommunications and signal processing.
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Chapter 6
C onclusion
Optical-microwave interaction in superconducting film has been modeled in detail. This
interaction study was accomplished with two major steps. First, the effect of the optical
irradiation on the electrical properties of a superconducting film has been investigated in
the context of the microscopic BCS theory via the effective tem perature concept. That
model enabled us to calculate the effective tem perature in terms of optical radiation char­
acteristics namely absorbed optical power and frequency. The photo-induced effective
temperature was then used to modify the classical relationship of the super and normal
conductivities of a superconductor presented by M attis and Bardeen for any electrical
signal having the frequency well below the gap frequency. The photo-induced changes
in the kinetic inductance and normal resistance of a superconducting film were in turn
obtained analytically. The numerical simulation of a typical superconducting film led us
to use a heat transfer model to evaluate the effective tem perature for various type of
optical irradiation sources. By combining the heat transfer model with the Two-Fluid
model, our comprehensive thermo-electrical models provides a systematic way for dealing
with lumped interaction between the optical radiation and electrical signed. This lumped
interaction study resulted in two attractive potential applications in the field of optoelec­
tronics and optical control of the HTS microwave devices.
Many optoelectronic signal processing can be performed in a current driven photo-excited
HTS film when the optical signal illuminates the structure uniformly. Optoelectronic
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generation of microwave harmonics and signal mining were achievable when the optical
radiation is modulated with the time harmonic microwave signed and the HTS structure
was fed by DC current and a pure sinusoidal current, respectively. The photo-excited
HTS film can be a multifunctional optoelectronic device, since it inherently acts as a pho­
todetector and optical to electrical transducer. To assess the performance of our proposed
HTS optoelectronic device, we presented a detail simulation and discussion along with
the comparison with some experimental results reported in the literature.
The proposed thermo-electrical model could not only predict the behavior of the opticallycontrolled HTS microwave devices b u t also provided a guideline for their design criteria.
Optically-tunable delay line and parallel-plate resonator were successfully analyzed and
compared with the experimental works available in the literature.
Next, the distributed form of the optical-microwave interaction in an HTS film has
been modeled analytically. Laser-interference technique has been leveraged for creating
longitudinal traveling-wave conductivity grating in the HTS film. The microwave signal
propagation was then analyzed with Floquet's approach to obtain the dispersion diagram
and the field amplitudes of different spatio-temporal harmonics in an infinite length of
the HTS film. Numerical simulation of such an interaction revealed different regimes in
the dispersion diagram for various value of the spatial and temporal characteristics of
the dynamic conductivity grating versus the wave characteristics of the microwave signal.
Furthermore, the rigorous Floquet's analysis showed that the second and higher order
spatio-temporal modes were weak enough to be neglected and provided a justification
for applying the coupled mode analysis. A time-varying coupled mode analysis was then
developed in a finite length of a photo-excited HTS film to find the reflection and trans­
mission bands for the microwave signals traveling on the structure. The Bragg Reflection
characteristics of microwave signal was obtained, leading to a periodic filtering function­
ality of the structure. The analytical expressions for the relative power transmission and
reflection of the microwave signal in a laser-induced dynamic conductivity grating allowed
us to explore optimum ranges for periodic filtering of the microwave signal over a broad
106
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range of temporal and spatial frequencies, when the laser-induced conductivity grating
can be controlled by the laser-interference characteristics. Numerical simulations demon­
strated the potential applications of laser-induced DFB structure in the HTS film for
optically- controlled microwave signal processing.
In the next section, we highlight the major contribution of this thesis, followed by an
outline of suggested directions for future research work.
6.1
S u m m a ry o f C on trib u tion s
The major contributions of this thesis can be categorized into four areas: Superconduct­
ing photoresponse models, superconducting optoelectronics, optically-controlled passive
HTS devices, and traveling-wave HTS microwave/photonic devices. The contribution in
the first area is a consequence of our comprehensive studies in the superconducting pho­
toresponse from engineering application point of view. The second and third areas are
obtained by the lum ped optical-microwave interaction study . and finally the last area of
our contribution results from the traveling-wave optical-microwave interaction study in
the superconducting films. The contribution in each areas are discussed in more detail in
the following.
6.1.1
•
Superconducting P hotoresponse M odels
Effective T e m p e r a tu r e M odel. We presented a steady state photoresponse model
in a superconducting film. This model was developed for investigating the capability
of superconductors for potential optoelectronic applications. The effect of a continuous
optical radiation on the thermodynamical parameters of a superconducting film was in­
troduced into a energy distribution function for Cooper pairs and normal electrons via
a chemical potential. The energy gap modification was then calculated in terms of the
chemical potential which is a function of optical radiation characteristics and thermo-
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physical parameters of the material in the context of the well-known BCS theory. The
photo-induced effective tem perature of the superconducting film was calculated based on
the variation of the energy gap. This modeling provides a theoretical framework that
relates the optical irradiation characteristics to the superconducting properties from very
low levels of absorbed optical radiation intensity all the way to levels that destroy the
superconductivity for till optical frequencies greater than the gap frequency [175]. [176].
•
P h o to -In d u c e d C h a n g e s in E le c tric a l P a ra m e te rs . Based on the effective tem­
perature model, the photo-excited energy distribution function was introduced into an
electrodynamic formulation presented by M attis and Bardeen in the BCS framework.
Therefore, we reformulate the M attis and Bardeen equations in the presence of the optical
radiation. The change in the kinetic inductance and normal resistance of a superconduct­
ing film were then calculated easily. The discussed analysis revealed that the conduction
process and consequently the electrical parameters of the superconducting film can be
controlled by optical illumination, while the film remains in the superconducting state.
Numerical simulation of the photo-induced changes in the electrical parameters of the
film showed their nonlinear relationship with optical radiation characteristics for both
bolometric and nonbolometric cases. The result of our numerical simulation was finally
verified successfully with the experimental works available in the literature [176].
•
H e a t T ran sfer M o d el. The effective temperature model not only provided a steady
state superconducting photoresponse model but also justified a use of a heat transfer
model for the cases when the optical radiation varies with the time and space variables.
We reduced a two-temperature model which presents the evolution of the electron and
phonon effective temperatures to an electronic effective tem perature model, when the ef­
fect of the photo-excited phonons is considered via a thermal boundary conditions in a
film/substrate interface. Our simplified heat transfer equation is accountable for expla­
nation of fast bolometric photoresponse and can be analytically solved for many practiced
schemes of optical irradiation. The steady state photoresponse the effective temperature
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resulted in a definition of critical absorbed optical power. This quantity indicates the
absorbed optical power required for destruction of superconductivity, when all Cooper
pairs are broken by th e absorbed photons. The presence of this quantity in our effective
temperature model was in complete agreement w ith the heat transfer approach. The heat
transfer equation was also used to solve for time-harmonic and pulsed optical irradiations.
In both cases the photo-induced temperature followed the temporal variation of the opti­
cal sources after passing its transient response.
Finally, the solution of th e heat transfer equation was sought in the case of laser-induced
interference power p attern . This traveling-wave photo-excitation resulted in creation of
traveling-wave tem perature along the superconducting film, similar to acoustic wave prop­
agation in different m aterials. The advantage of this technique is th a t the characteristics
of the photo-induced traveling-wave temperature can be controlled by the laser interfer­
ence parameters, thus it provides a very attractive experimental benchmark for grating
based measurements of th e thermophysical properties of a superconducting film.
6.1.2
•
Superconducting O ptoelectronics
L u m p ed O p tical-M icro w av e In te ra c tio n in S u p e rc o n d u c tin g Film . We
introduced a comprehensive thermo-electrical modeling of the lumped optical-microwave
interaction in superconducting film based on the proposed heat transfer model and electro­
dynamic theory based on the Two-Fluid model. This model phenomenologically described
the effect of the optical radiation on the super and normal electron population numbers
via the photo-induced effective temperature to obtain the current density in terms of the
applied electric field associated with traveling microwave signal. The main assumption
in the developed current-field relationship was th a t the superconducting film was illu­
minated uniformly. T he temporal lumped optical-microwave interaction was manifested
in the superconducting film when the number of Cooper pairs and normal electrons are
temporally controlled by the optical irradiation and their velocities me temporally and
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locally determined by th e microwave signal. According to the equivalent circuit repre­
sentation of a superconducting film, the interaction time between the optical radiation
and microwave signal has been calculated. This interaction tim e is valid when the optical
penetration depth, film thickness and the London penetration depth are comparable with
each other [177]. [178].
•
O p to ele ctro n ic H T S M icrow ave H a rm o n ic G e n e ra to r. We used a basic multi­
functional device made of HTS film in a bridge/strip configuration to perform opto­
electronic functions. This device inherently acts as photodetector or optical-to-electrical
transducer. In a first configuration, the HTS bridge is illuminated with the time-harmonic
modulated optical signal with frequency / m. while the bridge is biased by DC current.
By the extension of the kinetic inductance photoresponse, we showed both analytically
and numerically that th e photo-induced voltage contains harmonics of the modulation
frequency f m. The am plitude of the first four harmonics were analytically derived with
the Fourier series expansion of the photo-induced voltage. We showed that proper design
of such a configuration m ay result in a relatively high second harmonic generation even
stronger than the main harmonic. This optoelectronic microwave harmonic generator
exhibits low power consumption and its bandwidth may be extended up to 100 GHz by
lowering its thermalization time [113].
•
O p to e lec tro n ic H T S M icrow ave M ix e r. In a second configuration of the pro­
posed HTS optoelectronic device, the HTS bridge is injected by a pure sinusoidal current
with local frequency fuo and optical signal with modulation frequency f m illuminates the
HTS bridge uniformly. We showed that the photo-induced voltage appeared on the bridge
contains mixing signal o f the local frequency and modulation frequency of optical signal.
This device not only detects the modulated optical radiation with any frequency less than
the energy gap of the HTS, typically below THz range, but also performs the heterodyne
receiving process. The am plitude of the mixed and intermodulated components can be
controlled via the initial tem perature, the absorbed optical power, and the LO current.
110
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Our simulation results demonstrated the potentials of the HTS optoelectronic mixer in
up and down conversion of microwave signals for high performance microwave/photonic
applications [177].
•
O p to e le c tro n ic H T S D evice C h a ra c te ristic s The performance of the multifunc­
tional HTS optoelectronic device was compared to its semiconducting counterparts. The
conversion gain bandwidth, intrinsic and extrinsic conversion gains, and the equivalent
noise temperature has been calculated analytically. The proposed device is generally ex­
pected to exhibit the low-noise/ low-power and high-speed/high-bandwidth characteris­
tics suited for applications in high-performance telecommunications and signal processing.
6.1.3
•
O ptically-Controlled Passive HTS D evices
O p tically -T u n ab le M icrow ave D elay Line. We presented the systematic analysis
of the optically-tunable HTS delay line, when the propagation time of the microwave signal
can be optically changed in partially-irradiated HTS parallel-plate waveguide. We showed
that in the relatively low absorbed optical power below 0.5PC- the delay time increases
approximately linearly with respect to the optical power, but after that it increases almost
quadratically. Our analytical discussion also provided some useful guidelines for optimum
design of such a device. The use of relatively thin film in the order of the penetration depth
of the HTS film and choosing thin dielectric spacer result in achieving wider tuning range.
•
O p tically -T u n ab le M icrow ave R e s o n a to r a n d F ilte r. The resonant frequency
of an HTS parallel-plate waveguide resonator can be tuned by optical illumination. The
change in the resonance frequency and the quality factor was successfully calculated in
terms of the absorbed optical power. We showed that under a proper design criteria,
the resonance frequency can be decreased while the quality factor has no significant drop.
Although the time required for changing the resonance frequency is relatively large, in the
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order of milliseconds, but the range of t im in g is quite significant. Since the change in the
resonance frequency is highly sensitive to the absorbed optical power, we suggested use
of the low optical illu m i n a tio n level below 0 .2 Pc. Our numerical simulation for an HTS
resonator employed in 5.5 GHz. showed an excellent agreement with the experimental
work done by the others. Furthermore, we calculated the relative change in the quality
factor and a shift in the resonance frequency in terms of various initial tem perature and
applied optical power level for an optimum design with a given resonator specifications.
6.1.4
•
Traveling-W ave HTS M icrow ave/Photonic D evices
Laser-Induced D yn am ic Microwave C on d u ctivity Grating in H T S Film.
It was shown that two degenerate laser beams with coherent interference can produce
a conductivity grating within an optical penetration depth of an HTS film. This laserinduced conductivity grating is active for any electrical signal with frequency smaller
than the gap frequency. The mathematical expressions describing the super and normal
conductivity gratings formed in the HTS film have been presented in detail in terms of
the optical interference characteristics and the thermophysical parameters. We showed
th at the laser-induced traveling-wave conductivity grating manifested itself as a combi­
nation of a forward and backward waves obtained from periodic change of the effective
temperature of the HTS film in space and time domains. The behavior of this dynamic
grating in both temporal and spatial frequency was discussed and it was shown that the
presence of spatio- tem poral conductivity grating in the HTS film can be used to modify
the propagation characteristics of any input microwave signal along the structure.
• Microwave P ropagation A long an Infinite L ength o f Laser-Induced Conduc­
tiv ity Grating in H T S Transm ission Line; F lo q u et’s Approach. The travelingwave super and normal conductivities of an HTS film has been used as a constitutive
relationship relating the super and normal currents in terms of the electric field associ-
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ated with input microwave signal in the Maxwell's equations in the HTS film. The solution
of Maxwell's equations has then been sought in the form of Floquet's expansion of dif­
ferent spatio-temporal harmonics. This formalism resulted in difference equations which
described the relationship between the amplitude of spatio-temporal modes in terms of
the thermophysical parameters of the HTS film/substrate and the conductivity grating
characteristics via the coupling coefficients. The solution of these recurrence relation­
ship not only gave the dispersion relation for the microwave signal but also enabled us
to find the relative amplitude of different spatio-temporal harmonics with respect to the
fundamental mode. It was shown th at the existence condition for the solution of the dif­
ference equation required that the phase velocity of the traveling-wave conductivity must
be smaller than the microwave signal velocity. This is the stability criterion for having a
nongrowing field along the structure in both time and space domains.
•
D ispersion D iagram s for M icrowave Signal p ropagatin g along the Spatio-
Tem poral C onductivity G ratin g. Numerical simulation of microwave propagation
along the laser-induced dynamic conductivity grating in a thin HTS transmission line
made of YBa2Cu307_,5 has been presented. Dispersion Diagram regimes for microwave
signal over a broad range of tem poral frequency and for different grating characteristics
have been investigated for various absorbed optical radiation levels. We presented a physi­
cal justification for different regimes observed in the dispersion diagrams. We showed that
at a specific points on the dispersion diagram the absorbed optical power has no effect
on the microwave signal propagation, the so-called cross-over points. The numerical sim­
ulation of the relative amplitude of different spatio-temporal modes indicated that the
second order conductivity grating cam be neglected without affecting the dispersion dia­
grams. Therefore, the discussed analysis supported the applicability of the coupled mode
treatm ent where only the forward and backward first order spatio-temporal modes are
considered.
•
Microwave P ropagation A lon g a Finite Length o f L aser-Induced C onduc-
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tiv ity G ra tin g in H T S T ran sm issio n Line; T im e-V ary in g C o u p led M o d e A n a l­
ysis. After refining the expressions describing the conductivity grating, the problem, of
microwave propagation along a finite length of the laser-induced dynamic grating was cast
into a generalized coupled mode formalism. A time-varying coupled mode equation when
the coupling coefficients are time-dependent was formulated successfully. The solution of
the derived coupled mode differential equations describing the forward and backward wave
functions based upon the zeroth-order Bragg condition have been obtained. Introducing
the physical boundary conditions at the two ends of the structure resulted in derivation of
the reflected and transm itted signal analytically. The scattering m atrix for the microwave
signal was then calculated which encompasses a useful information about the behavior of
the structure for periodic filtering functionality.
•
L a se r-In d u c e d T raveling-W ave D is trib u te d F eed b ack (D F B ) s tr u c tu r e in
H T S Film . Numerical simulation of the relative power transmission and reflection of
the microwave signal along the finite length of the laser-induced conductivity grating in
the HTS transmission line showed a periodic Bragg-reflection characteristic over a wide
range of microwave frequencies. This characteristic was demonstrated to be used for
optically-controlled microwave periodic filtering. The periodic filtering functionality can
be controlled by the absorbed optical radiation power and the laser-interference character­
istics. It was interestingly shown th at for a given microwave channels with specific 3-dB
bandwidth and channel separation, the laser-induced DFB structure can be dynamically
adjusted, since our analytical formulation along with the CAD simulation is able to find
the optimum design criteria. The traveling-wave optoelectronic HTS periodic filter can
potentially be used as a basic building block for many signal processing functions or in
receiver front-end in high-performance microwave/photonic communication systems and
radio over fiber applications.
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6.2
F uture R esea rch D irectio n s
The effective tem perature model developed in this thesis was obtained horn the conven­
tional s-wave BCS theory which is valid for low-temperature superconductors. Although,
this model provided an acceptable results for HTS materials, the extension of the model
to include d-wave superconductors would be desirable.
The heat transfer model presented in the thermo-electrical modeling of the HTS opto­
electronic devices might be improved to provide more accurate results by considering the
thermophysical boundary conditions presented in a two-dimensional or three-dimensional
structure. Since the HTS devices are expected to be placed in a cryogenic environment,
the thermal runaway phenomenon and the input/output electrical and optical intercon­
nection problems should be taken into account.
The concept of multifunctional HTS optoelectronic component resulted into a number
of individual devices such as optoelectronic mixer. The characteristics of these prototype
devices, such as noise, responsivity. and detectivity figure of merits need to be considered
in more detail. The next step is to experimentally verify the simulation result stemming
from our thermo-electrical modeling. This is accomplished by the fast electrical or the
electro-optic sampling measurement techniques on the current-driven photo-excited HTS
microstrip/microbridge configuration.
Many novel device concepts for optically-controlled microwave HTS structures may
be developed. For instance, the optically-controlled phased array HTS antennas or filter
structures is of interest. The extent of the thermo- electrical modeling of such a devices
or incorporating the proposed heat transfer analysis in the transmission line model will
provide more accurate model to predict the behavior of these devices as well as some
useful design guidelines.
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The proposed theoretical study of the traveling-wave optical-microwave interaction in
the HTS film can be extended to other materials such as semiconductors, plasma medium
and photonic bandgap structures by the establishment of a physical model describing the
effect of optical radiation on the material properties such as perm ittivity and conductiv­
ity. The solution of Maxwell's equation will then be used to describe the microwave and
millimeter-wave propagation along the photo-excited structure by either Floquet’s ap­
proach or time-varying coupled mode analysis. This theoretical investigation may result
in creation of a comprehensive models of the optical-microwave interaction in different
materials.
The solution of Maxwell's equations in the presence of the dynamic conductivity grat­
ing might be sought by a perturbative field expansion method. In this formalism, the
zeroth-order part of th e equation corresponds to an unperturbed structure and the higher
order parts take care of the spatio-temporal harmonic variations of the conductivity which
couples the zeroth-order mode with higher-order Floquet modes. The microwave disper­
sion diagram and the relative amplitudes of different spatio-temporal modes can be found
and the method can be extended to contain the transversal variation of the fields along
the structure.
A numerical m ethod for microwave propagation along the spatio-temporal conductiv­
ity grating along the HTS transmission line can be accomplished. FDTD (Finite Difference
Time-Domain) method seems the first choice, since the temporal and spatial variation of
the conductivity can be easily included into Maxwell's equation. The discrete equations
describing the field amplitudes can be found explicitly for any tim e step in terms of the
known field values at previous time step. The physical boundary conditions is included in
this scenario, and the m ethod can be extended to simulate the two and three
d im e n s io n a l
structures as well.
A laser-induced DFB structure in the HTS film and its associated Bragg-reflection
116
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characteristics might be experimentally verified to assess the results of the time-varying
coupled mode analysis and possible its limitation. This can be accomplished by perform­
ing microwave measurements and comparing the measured scattering parameters against
simulated results. Specifically, the validity of our physical modeling of the conductivity
grating needs to be experimentally verified. The success of our proposed model, theoreti­
cal treatment and simulated results might be useful for the measurement of the different
thermophysical parameters of the HTS films such as diffusivity and thermalization time.
117
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A ppendix A
BC S C om plex C onductivity B elow
Gap Frequency
We present here a brief calculation of — for low frequency limit and tem perature range
0-n
0 < T < Tc. Where h f —►0. equation (2.4) becomes:
_9 r
df
E 2 + A 2(T) + h f E
" A(T) d E ^ / e * - A2( T ) y / ( E + h f Y - A2(T)
„ 2A2(r) + h /A ( r )
r x df_____________ dE ______________
~ y / 2A( T) ( 2A( T) + hf ) J M T) d E ^ ( E - A ( T ) ) ( E + h f - A (T))
(A-l)
E
Using an estimation f { E ) = exp( —-——). and plugging in (A-l) yields:
kb T
fi =
crn
a .
hf-*o IcbT
2A(T)
Ub T
, A (T)
r_5Ev tzL ,
J0
\ / x {x + hf )
hf
hf
^
=
(A' 21
hf
4 kBT
where x = E — A (T). Since h / «C 2fcsr and if0( , , „ ) « ln(
- ) we can obtain
2kb T
hf
equation (2.6).
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A ppendix B
Energy Gap C alculation Under
O ptical Irradiation
In the absence of any external stimulation and in the thermal equilibrium, the supercon­
ducting energy gap. A (T). is related to the energy distribution function of the quasiparE
tides. f ( E) % exp( —■;——). through the BCS self-consistent gap equation [179]. [180]:
kBT
l n ( - ^ - ) = r 2f ( E ) —= = ^ = d E
A (7 y
J MT)
y / E ^ — A 2{T)
It has been shown
ductors subject
(B-l)
that the BCS self-consistent gap equation is still valid for supercon­
to a constant external stimulation if the nonequilibrium quasiparticle
distribution function and energy gap are used in equation (B-l) [56]. Using the normal
and photo-exdted distribution function, equation (3.1). BCS gap equation becomes [181]:
=
( B
- 2 )
where K a denotes the modified Bessd function of the second kind of order zero. In thermal
equilibrium, y. = 0 and under the influence of an external perturbation such as optical
radiation. \i ^ 0. From equation (B-2). we yidd:
( B
119
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 3 )
where A(fi.T ) = A (T) — S A (T ). Using an approximation ln (l + x) % x for 0 < x <S 1
and
~ 4/
VkBT 1
V 2 A (r )
exp( —
SA {T) _
A (T)
.
fc» r
for low temperatures, one can obtain:
n
1 7ckBT
-A (T )
eXp^kBT ^ \ l 2 A ( T ) eXp{ kBT *
which is the equation (3.5).
120
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppendix C
Frequency A nalysis o f
O ptoelectronic H T S M ixer
In order to analyze the frequency content of the periodic photo-induced voltage, a har­
monic analysis based on the Fourier series will be presented. The starting point is to
consider the time-dependent kinetic inductance of the HTS film upon the optical radia­
tion given in the relation (4.5). For sufficiently thin film, the kinetic inductance can be
approximated by the following relationship [74]:
Li-in(t) = ---- 7 —
e0u j - w d 1
7fqT\
_ (£(
0)2
(C-l)
Tc
•>
e~n
is the plasma frequency, n is the total number densities of Cooper pairs
Tne€o
and normal electrons, e and m e are the charge and the mass of the electron, respectively.
where uip =
Substituting the steady state part of the time harmonic tem perature T (t), relationship
(3.21). in (C-l) yields:
u* = -t-i—
jm
wd
(c-2)
= 1
1
- (p + qsm 2irfmt)2
(°-3)
where
M
121
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Parameters p and q are set up by initial tem perature, thermophysical quantities of the
HTS film and optical power characteristics and extracted from relationship (3.21) as
follows:
To
PaTei
P = ^T
+
Tc TCV Cph
(C-4)
mPaT.
(C-5)
It is worth noting that the optical radiation power. Pa, should be chosen in such a way
T it)
that the reduced temperature.
— < 1. This condition physically assures th at the kiC
netic inductance photoresponse is dominated over the resistive photoresponse near critical
temperature [182]. and also mathematically /(£) will remain nonsingular function during
optical illumination. Under this condition, f i t ) can be expanded in a Fourier series as
fit) = y + ^
(an cos 2irnfmt + bn sin 2 nnfmt )
(C-6)
n=l
where
ao
___
27T/m f
dt
7r J
1 — lp + q sin 2x f mt)2
2*/„
1
1
7^ 0
n is even
= 0
n is odd
0
sin 2irnfmt
dt <
IT J 2» 1 — Ip + qsin27r/m£)2
= 0
2w/m
n is odd
cos 2i:nfmt
2ir/m /*
-dt
an = -----/
;--7T J 2» 1 — lp + q sin 27r/m£)2
2*7m
(C-7)
(C-8)
and
6, = ^
/
n is even
122
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C-9)
The analytical expressions for the Fourier series coefficients are calculated for the first five
harmonics as
&! =
-\
1 P------------- r = + P
1
? V ( l- P ) 2- 9 2 V d+p)2- ^
, 4
2 ( 1 - p ) 2__________2(1 + p ) 2
°
=
«* -
a
ir
t f y / i l ~ P ) 2 ~ ?2
?2
(i-
93 *. 8 /,
■»+ ? ( 1 ?2V
p
-
v^
-
p )2 -
92v / ( 1 + P ) 2 “ ?2
? 2) 3
( i + p - V d + P)2 - g 2) 3j
(C. 12)
V (l-P)2- ? 2
Vd+P)2- ? 2
J
2
6p\
8 f
(1-p)2
(1 + p ) 2
? - i r ) - ? [ V ( 1 - | ,F
. +
92
<Z2 '
?2 V ( l - P ) 2 - 9 2 V/ ( 1 + p )2 - ? 2J
+
8 r
(1-p)4
(1+p)4
+,
1
f wVa( -i -p p) )22 - ?? 2 V
( i + p ) 2 - g? 2
2j
V(i+p)2-
_
l r ( i - p - V ( i ~ p )2 ~ g 2) 5
?5 «■
(C_1°)
(C-13)
(1 + p - \ / ( i + p ) 2 - ^ 2) 5-|
v 'd -P )2- ? 2
v/(i+p)2- r ’
J
The rest of coefficients can be evaluated by numerical integration schemes. Fig C .l and
C.2 show the dependency of the first five harmonics to the absorbed optical radiation
power, with parameters presented in Table 4.1. It is interestingly seen that the harmonics
become weaker when the absorbed optical power is increased. Once the frequency content
of the kinetic inductance is determined, the spectral response of the photo-induced voltage
can be easily calculated. Based on the relationship (4.4). when the HTS film is biased with
the DC current. I qc - the photo-induced voltage,
*„(£) is a periodic waveform containing
the harmonics of the carrier frequency f m as follows:
Vfcin(t) =
I
I
30
y '2 7 r n / TO(6n cos27r/mf - an sin 2 ttf m t )
€°u Pw d ^
j
(C-15)
As it can be seen from relation (C-15) the differentiation with respect to time inherently
increases the amplitude of higher harmonics presented in the output voltage but it is
obvious th at, the harmonic coefficients will rapidly decrease as n is increased.
In the more complete form, when the HTS bridge is properly biased with the pure sinu­
soidal current source with LO frequency
ii,o(t)
= ii,o sin(27r/x,ot). based on formula (4.4)
123
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xttr3
>
g
9o
■
3
a
E
<
oc
o
0
Z
e>
a
Z©
-1
*
Optical Power. P (mW)
Figure C .l: Even harmonics vs. absorbed optical radiation power.
B
o
* -3
-7
Optical Power, P (mW)
Figure C.2: Odd harmonics vs. absorbed optical radiation power.
124
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the photo-induced voltage contains the mixed version of the carrier and LO frequencies
as follows:
v(t)
=
(Lm + ^~)2TrfL0cos{2TrfLOt)
-I-
2n
{^
L°
^Qncos(27rf i o
± 2irnfm)t
-I-
sin(27r/ l o ± 27rre/m)£ j
n=l
(C-16)
The frequency components of the photo-induced voltage consists of the down and up
converted signals (/ m ± f i o ) and the intermodulated signals (n./m ± fio ) and their related
amplitude can be evaluated from formula (C-16).
125
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppendix D
Three-Term R ecurrence R elation for
Spatio-T em poral H arm onics
We present here a transformation of the linear forth-order difference equation (5.17) to the
second order difference equation (5.27). The starting point is to write the (n — l)th and
n + 1th equations given in (5.17). and calculating Kn.n_ 2-dn_2 from the n — 1th equation
and Kn,n+2 A n + 2 from n -f 1th equation as follows:
l.n—2
^n+l.n+2
{ ^ n —1-Ai—1
& n —l.n-^n
^n—l.n+3»dn_3
^n —l.n + l^n + l}
{^ n +
^ n -f - l.n ^ n
^ n + l , n —3 -^-n—1
^ n + l.n + 3 ^ n + 3
l -4 -n + l
}
^n.n—2-^n—2 (
— f * n .n + 2 - ^ - n + 2
1)
(D“2 )
By adding (D-l) and (D-2) and substituting the resulting expression in the right hand
side from (5.17). we yield the following equation:
^ ^n.n—2^n—l.n ^ ^n-t-l.n^n.n+2
+
.
+
^n—l.n—2
^n+l.n+2
r ^n+l.n—l^n.n+2
^W»,n—2 j-.
1 .
{ ------------------------ «n.n-1 - ------------y n- l ) A „ - i
^n+l.n+2
^n—l.n—2
r Kn -l.n + lKn,n-2
^n.n+2 n
1 A
{ -----;------------------- Kn.n+1 - ------------Dn+ij-An+i
^n —l.n—2
^n+l.n+2
^n.n—2^n—l.n —3 .
^n,n+ 2 ^n+l.n +3 *
«
^ n —l.n—2
■'mi—
3
^n+l.n+2
/ n Q\
(D- 3 )
^*n-f-3
From the last two term in (D-3) it is seen that the coefficients of -An-3 and A 1+3 are much
smaller than the rest, since they have involved with higher order coupling coefficients and
126
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the fact th at atx 2> <
t,2 and <7„, 3> crn,. Hence, the last two terms of the equation (D-3)
can be neglected and we obtain a linear second-order recurrence formula (D-3) as:
QnAn "(" IZn-l A i—1 "(■•Sn+l-^n+1 — 0
127
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