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Indian J Phys
DOI 10.1007/s12648-017-1130-z
ORIGINAL PAPER
Binary photonic crystal for refractometric applications (TE case)
S A Taya* and S A Shaheen
Physics Department, Islamic University of Gaza, P.O.Box 108, Gaza, Palestine
Received: 19 November 2016 / Accepted: 24 August 2017
Abstract: In this work, a binary photonic crystal is proposed as a refractometric sensor. The dispersion relation and the
sensitivity are derived for transverse electric (TE) mode. In our analysis, the first layer is considered to be the analyte layer
and the second layer is assumed to be left-handed material (LHM), dielectric or metal. It is found that the sensitivity of the
LHM structure is the highest among other structures. It is possible for LHM photonic crystal to achieve a sensitivity
improvement of 412% compared to conventional slab waveguide sensor.
Keywords: Photonic crystal; Refractometric application; Sensitivity
PACS No.: 42.70.Qs; 87.85.fk; 84.40.Az
1. Introduction
In the past few years, extensive research and development
activities have been devoted to evanescent- field- based
optical waveguide sensors [1–8]. They play an important
function in a variety of sensing applications. By means of
measuring small changes in optical phase or intensity of the
guided light, these sensors present excellent properties such as
high sensitivity, fast response, immunity to electromagnetic
fields, and safety in the detection of combustible and explosive
materials [1]. An optical sensor is a device in which light
interacts with the substance to be detected (measurand) and
converts light affected by the measurand substance into
electrical signal which gives information about the analyte. In
slab waveguides, light is confined within the guiding layer,
with a small part of the guided mode called the evanescent
field that extends to the surrounding media. The evanescent
field detects any refractive index variation of the covering
medium in homogeneous sensing. The interaction of the
evanescent field with the measurand causes a change in the
effective refractive index of the guided mode. The change in
the effective refractive index due to the change in the analyte
index is the sensing criteria.
Artificial materials with simultaneously negative electric
permittivity (e) and magnetic permeability (l) have received
an increasing interest [9–19]. These metamaterials were called
left-handed materials (LHMs) or negative-index materials
(NIMs). They were named LHMs because the electric field,
magnetic field and wave vector of an electromagnetic wave
form a left-handed set. Veselago [9] showed that such materials exhibit a number of unusual properties such as negative
refractive index, reversal of Snell’s law, and reversal of
Doppler shift of radiation. Slab waveguides comprising LHM
have been suggested as optical sensors [20–27]. A multilayer
waveguide configuration comprising a metal layer and a LHM
as a guiding layer was proposed as a metal-clad waveguide
sensor [10]. Reflection and transmission coefficients were
derived and utilized to investigate the resonance dips at which
the reflectance vanishes. The results showed that the proposed
configuration has an advantage over the well-known surfaceplasmon resonance (SPR) structure since it gives a much
sharper reflectance dip and can result in a considerable sensitivity enhancement [10]. The propagation of transverse
magnetic (TM) waves in a four-layer waveguide configuration
was investigated for refractometric applications [13]. The
waveguide configuration consists of a substrate, a metal layer,
a material of LHM as a core layer, and a cladding. The metal
layer was very thin so that a peak was obtained in the reflectance profile. The angular position of the reflectance peak was
observed to sense any changes in the index of refraction of the
cladding. The results showed that for aluminum metal layer, a
thickness of about 9 nm represents the optimum metal
thickness which corresponds to the highest and sharpest peak.
The negative parameters of the LHM were found to have a
considerable effect on the performance of the proposed sensor
*Corresponding author, E-mail: staya@iugaza.edu.ps
Ó 2017 IACS
S A Taya and S A Shaheen
[13]. A planar waveguide with air core and anisotropic LHM
cladding was investigated for refractometric applications [16].
Different from the SPR structures in which the measurand is
placed in the evanescent field layer, the proposed sensor
contains the measurand in the guiding layer that supports the
oscillating field. As a result of the strong concentration of the
field in the measurand layer, the proposed structure showed a
dramatic sensitivity improvement [16].
Photonic crystals are artificial periodic structures having
energy bands for photons which either allow or forbid the
propagation of electromagnetic waves of certain frequency
ranges in the same manner as the periodic potentials do for
electrons in atomic crystals [12]. Depending on the
geometry of the structure, photonic crystals can be divided
into one-dimensional (1D), two-dimensional (2D) and
three-dimensional (3D) structures. The most important
property which determines practical significance of the
photonic crystals is the presence of the photonic band gap.
When a radiation with a frequency inside the periodic band
gap is incident on the structure, it is completely reflected.
Traditional waveguides operate by total internal reflection. In a purely two-dimensional photonic-crystal linear
waveguide, a linear one-dimensionally periodic defect is
introduced into the crystal, creating a localized band that
falls within and is guided by the photonic band gap [15].
Light is therefore prohibited from escaping the waveguide.
Photonic crystals were proposed as sensing elements for
detection small changes in the refractive index of a medium [2]. The suggested sensor was operated in reflection
mode. Ternary photonic crystals are found more sensitive
than binary photonic crystal. The fabrication of a refractometric of a quasi-one-dimensional waveguide photonic
crystal was reported [3]. The transmission stopband was
found to shift by 0.8-nm wavelength for either a cladding
refractive index change of 0.05.
A novel way to improve the temperature sensitivity in
photonic crystals by using a ternary periodic structure was
demonstrated [7]. The design of a two-dimensional photonic crystal coupled resonating optical waveguide based
integrated-optic sensor platform was proposed [8].
In this work, two-layer slab photonic crystal is investigated
for sensing applications. The dispersion relation, sensitivity to
any change in an analyte index, and power flow relations are
derived, plotted and analyzed for TE mode.
2. Basic equations
The geometry of two-layer slab photonic crystal structure
is shown in Fig. 1. It consists of two different layers in
periodic arrangement which have refractive indices n1 and
n2 and thicknesses d1 and d2 , respectively.
The refractive index profile of the structure is written as.
X
n
d
1
1
n n
2
d
2
d
n n
2
1
1
d
2
d
1
1
n
2
d
2
...
...
0
n
d
i
...
i
z
Fig. 1 Two-layer photonic crystal consisting of two different layers
in periodic arrangement
nð z Þ ¼
n1
n2
0\z\d1
d1 \z\d
ð1Þ
where nðzÞ ¼ nðz þ dÞ and d ¼ d1 þ d2 .
Consider TE polarized light in which the waves are
travelling along the x-axis and the electric field is polarized
along the y- axis, then there’s a non-zero y-component of
the electric field.
Helmholtz equation for TE mode can be written as
o 2 E y o 2 Ey
þ 2 þ k2 Ey ¼ 0;
ox2
oz
ð2Þ
Since the propagation is along x-axis, Eyðx; zÞ can be
written as
Ey ðx; zÞ ¼ Ey ðzÞeibx :
ð3Þ
Equation (2) becomes
d 2 Ey
þ k02 ei li N 2 Ey ðzÞ ¼ 0;
2
dz
ð4Þ
where b ¼ k0 N and k2 ¼ k02 ei li ;
The solution of Eq. (4) in the l-th cell can be written as
Ey1 ðzÞ ¼ al eiq1 ðzldÞ þ bl eiq1 ðzldÞ ; ðl 1Þd þ d2 \z\ld
ð5Þ
Ey2 ðzÞ ¼ cl eiq2 ðzldÞ þ fl eiq2 ðzldÞ ; ðl 1Þd\z\ðl 1Þd
þ d2
ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where q1 ¼ k0 e1 l1 N 2 and q2 ¼ k0 e2 l2 N 2 .
The nonzero components of the magnetic field
(Hx and Hz Þ can be calculated using
Hx ¼
1 oEy
;
ilx oz
from which
q1 iq1 ðzldÞ
Hx1 ¼
al e
bl eiq1 ðzldÞ ;
l1 x
q2 iq2 ðzldÞ
cl e
fl eiq2 ðzldÞ ;
Hx2 ¼
l2 x
and
ð7Þ
ð8Þ
ð9Þ
Binary photonic crystal for refractometric applications
i oHx
;
b
q2 oz
b iq1 ðzldÞ
al e
Hz1 ¼
þ bl eiq1 ðzldÞ ;
l1 x
b iq2 ðzldÞ
cl e
þ fl eiq2 ðzldÞ :
Hz2 ¼
l2 x
Hz ¼
ð10Þ
a0
b0
¼
A
C
B
D
l al
;
bl
ð21Þ
ð11Þ
The electric field vector in a periodic layered medium can
be written using Bloch theorem as [17]
ð12Þ
EKB ðx; zÞ ¼ EKB ðzÞeiKB z eiðxtbxÞ ;
The boundary conditions require that the tangential components of E and H to be continuous at z ¼ ðl 1Þd and
z ¼ ðl 1Þd þ d2 , yielding a set of homogeneous linear
equations for the coefficients ai, bi, ci, and fi.
The continuity of Ey and Hx at z ¼ ðl 1Þd gives
iq2 d
eiq2 d
e
1 1
al1
cl
¼ q2 iq2 d q2 iq2 d
;
1 1
bl1
fl
qTE e
qTE e
ð22Þ
where
EKB ðz þ dÞ ¼ EKB ðzÞ;
ð23Þ
and KB is the Bloch wave number
In terms of column-vector representation, the periodic
condition in Eq. (23) for Bloch wave is
al
a
¼ eikB d l1 :
ð24Þ
bl
bl1
ð13Þ
al1
Substituting for
from Eq. (16) into (24), we get
l2
bl1 where qTE ¼ l q1 .
1
a
A B
al
The continuity of Ey and Hx at z ¼ ½ðl 1Þd þ d2 gives
¼ eikB d l ;
ð25Þ
b
bl
C
D
l
iq d
iq1 d1
iq1 d1
iq2 d1
2 1
e
e
cl
al
e
e
¼ qTE iq2 d qTE iq1 d1
: A eikB d
al
B
e
e
fl
b
eiq2 d1 eiq2 d1
l
q2
q2
¼ 0:
ð26Þ
bl
C
D eikB d
ð14Þ
By taking the determinant of the matrix equal to zero, then
which can be written as
iq d
iq1 d1
A eikB d
B
iq1 d1
iq
d
2
1
2
1
ð27Þ
1
e
e
cl
al e
e
ikB d ¼ 0;
C
De
¼
:
qTE iq2 d
qTE iq1 d1
iq
d
iq
d
2
1
2
1
e
e
fl
bl e
2 e
q2
q2
A eikB d D eikB d BC ¼ 0:
ð28Þ
ð15Þ
Substituting from Eq. (15) into (13), we obtain
al1
A B
al
a
¼
¼M l ;
bl1
bl
bl
C D
But AD BC ¼ 1; which imply that
ð16Þ
where M is called the transfer matrix with the elements
1
1
qTE 1 e2iq2 d2
A ¼ cosðq2 d2 Þeiq1 d1 eiq2 d1
2
4
q2
1 iq2
1
sinðq2 d2 Þeiq1 d1 þ eiq2 d1 1 þ e2iq2 d2 ;
þ
2 qTE
4
ð17Þ
1
qTE
q2
B ¼ eiq1 d1 isinðq2 d2 Þ
;
ð18Þ
2
q2
qTE
1
1
qTE 1 e2iq2 d2
C ¼ cosðq2 d2 Þeiq1 d1 eiq2 d1
2
4
q2
1 iq2
1 iq2 d1 iq1 d1
sinðq2 d2 Þe
e
1 þ e2iq2 d2 ;
þ
2 qTE
4
ð19Þ
D¼e
iq1 d1
1 q2
qTE
cosðq2 d2 Þ þ i
þ
sinðq2 d2 Þ :
2 qTE
q2
ð20Þ
The transfer matrix of the structure shown in Fig. 1 can be
written as
e2ikB d ðA þ DÞeikB d þ 1 ¼ 0:
ð29Þ
The solution of the above equation for eikB d is given by
!12
AþD
AþD 2
ikB d
¼
1 :
ð30Þ
e
2
2
The eigenvectors corresponding to the eigenvalues are
obtained from Eq. (26) as
a0
B
;
ð31Þ
¼
b0
eikB d A
Then the corresponding column vectors for the l-th unit cell
are given by
al
B
lkB d
:
ð32Þ
¼e
eikB d A
bl
Multiplying Eq. (29) by eikB d , we get
eikB d þ eikB d ¼ A þ D:
Using the identity cos x ¼ e
ð33Þ
ix
þe
2
ix
, we get
S A Taya and S A Shaheen
2 cosðkB d Þ ¼ A þ D:
ð34Þ
Substituting for A and D from Eqs. (17) and (20) into (34),
we get
cosðkB dÞ ¼ cosð
q2 d2 Þ cosðq1d1 Þ
1 q2
qTE
þ
sinðq2 d2 Þsinðq1 d1 Þ
2 qTE
q2
1
ð35Þ
Equation (35) represents the dispersion relation of TE
wave.
The sensitivity of the effective refractive index to any
change in an analyte refractive index is calculated as the
change of the effective index (N) with respect to the
change in the analyte index (n1); i.e.,
S¼
oN
:
on1
sensitivity of the waveguide sensors and the power flowing
in the analyte layer. The power is given by
2
Z1 Ey ðzÞ
b
dz
ð40Þ
Ptotal ¼
2x
li ðzÞ
ð36Þ
Differentiating the dispersion relation given by Eq. (35)
with respect to N, then calculating S as
1
on1
S¼
ð37Þ
oN
k2 N
2d2 cosðq1 d1 Þ sinðq2 d2 Þ 0
q2
1
k2 n1 on
oN N
þ 2d1 cosðq2 d2 Þ sinðq1 d1 Þ 0
q1
2
2
l K0 N q2 k0
on1
N
sinðq1 d1 Þ sinðq2 d2 Þ
þ 1
3 n1
l2 q1 q2
oN
q1
l1 q2
k02
on1
N
cosðq1 d1 Þ sinðq2 d2 Þ
n1
þ d1
l2 q1
q1
oN
l q2
k2 N
d2 1 cosðq2 d2 Þ sinðq1 d1 Þ 0
q2
l2 q1
!
on
l2 K02 n1 oN1 N
q1 2
þ
þ 3 k0 N sinðq1 d1 Þ sinðq2 d2 Þ
q1 d2
l1
q2
l q1
k2
on1
N
þ d1 2 cosðq1 d1 Þ sinðq2 d2 Þ 0 n1
l1 q2
q1
oN
2
l q1
k N
d2 2 cosðq2 d2 Þ sinðq1 d1 Þ 0 ¼ 0
q2
l1 q2
ð38Þ
After some arrangement, we obtain
h
i
n1 2 dq11 u dq11 sTE v þ q12 rTE w
1
2 2
i;
S¼ h q2 q1
d1
d2
N 2 q1 u þ q2 v þ q2 q2 rTE w dq11 v þ dq22 u sTE
1 2
ð39Þ
where u = cosðq2 d2 Þ sinðq1 d1Þ; w ¼ sinðq1 d1 Þ sinðq2 d2 Þ;
sTE ¼ ll1 qq21 þ ll2 qq12 ; and rTE ¼ ll1 qq21 ll1 qq21 :It is useful to
2
1
2
2
find the total time-average power carried by the photonic
crystal layers. There is a close connection between the
For l
th
cell, the power for the first layer can be written as
2
b
al 2iq1 d2
pl ¼
e
e2iq1 d
2xll 2iq1
ð41Þ
b2l 2iq1 d2
2iq1 d
e
e
þ 2al bl d1 :
2iq1
and for the second layer as
2
2iq2 d2
b
cl 0
2iq2 d2
2 1e
pl ¼
1e
þ 2cl fl d2 :
fl
2xl2 2iq2
2iq2
ð42Þ
3. Results and discussion
The dispersion relation given by Eq. (35) was solved
numerically using Maple 17 software and the sensitivity
was calculated using Eq. (39). In the following computations, a two layer photonic crystal was assumed in which
the first layer was considered to be the analyte layer [2, 3]
and the second layer was assumed to be LHM, dielectric or
metal. This section was divided into three subsections. In
the first subsection, the second layer was assumed to be
LHM. In the second and third subsections, the second layer
was considered metal and dielectric, respectively. In all
subsections, the sensitivity was plotted with different
parameters of the structure such as layer thickness and
wave frequency.
3.1. Left-handed material (LHM)
The LHM is characterized by e2 and l2 which can be
written as [20]
eeff ðxÞ ¼ 1 x2p
;
x2 þ icx
ð43Þ
leff ðxÞ ¼ 1 Fx2
;
x2 x2 ixc
ð44Þ
where x° is the resonance frequency,c is the electron
scattering rate, xp is the plasma frequency and F is the
fractional area occupied by the split ring. We consider
x0 = 4.0 GHz,
xp = 10.0 GHz,
F = 0.56
and
c = 0.012 xp. These values are experimental values as
given in many references [26]. The frequency range was
taken from 4.0 to 6.0 GHz in which e2 and l2 are
Binary photonic crystal for refractometric applications
1.4
1.0
0.8
0.6
0.4
0.2
8
10
12
14
16
18
20
22
24
d2(mm)
Fig. 3 Sensitivity of the proposed sensor versus the thickness of
LHM layer for different values of the analyte thickness for e1 = 1.77,
n1 = 1.33, F = 0.56, c = 0.012xp, x = 5.0 GHz, x0 = 4.0 GHz,
xp = 10.0 GHz and kB = p/(5d)
sensitivity reaches a maximum value of 1.28 at
d1 = 7.2 mm and d2 = 13.36 mm, whereas it has a value
of 1.24 at d2 = 12.3 mm for d1 = 7.1 mm. But at
d1 = 7.0 mm, the sensitivity reaches 1.18 at an optimum
LHM thickness of 12.12 mm. It can be noted from Fig. 3
that when d1 increases from 7.0 to 7.1 mm, the maximum
sensitivity enhances by 3.2%. On the other hand, the
maximum sensitivity improves by 5.1% as d1 increases
from 7.1 to 7.2 mm.
The dependence of the sensitivity of the proposed sensor
on the thickness of the LHM layer for different values of
electron scattering rate (c) is plotted in Fig. 4. The sensitivity can be enhanced with decreasing the electron scattering rate (c). As can be seen from the figure, as the
electron scattering rate changes, the maximum sensitivity
occurs at the same value of an optimum LHM thickness.
The maximum sensitivity obtained has a value of 1.03 at
1.1
1.4
=5.0 GHz
1.2
1.0
=5.1 GHz
0.9
=5.2 GHz
1.0
Sensitivity
Sensitivity
d1=7.0mm
d1=7.1mm
d1=7.2mm
1.2
Sensitivity
simultaneously negative according to Eqs. (43) and (44),
the analyte layer is assumed to be water with n1 = 1.33.
Figure 2 shows the sensitivity of the proposed photonic
crystal sensor versus the thickness of the LHM layer for
different values of the wave frequency. The sensitivity can
be dramatically enhanced with decreasing the guided wave
frequency. As the frequency increases, the wavelength
decreases and the wave confinement in the LHM layer
increases. Consequently the evanescent field in the analyte
medium decreases and as a result the sensitivity also
decreases. It is also clear that as the wave frequency
increases, the sensitivity peak shifts toward higher value of
optimum thickness of the LHM layer. For x = 5.0 GHz,
the maximum sensitivity is 1.28 obtained at d2 = 13.4 mm
whereas it is 1.057 at d2 = 14.4 mm for x = 5.1 GHz. On
the other hand, for x = 5.2 GHz, the sensitivity reaches a
peak of 0.87 at an optimum thickness of 16.7 mm. It is
worth comparing our results for the sensitivity with those
of the conventional slab waveguide sensor comprising
lossless dielectric media which was proposed by Tiefenthaler [24]. The maximum sensitivity obtained by Tiefenthaler and his co-workers was 0.25. Our results show that it
is possible for LHM photonic crystal to achieve a sensitivity improvement of 412% compared to conventional slab
waveguide sensor. This improvement was calculated as
ð1:280:25Þ100%
.
0:25
Figure 3 shows the sensitivity versus the LHM layer
thickness for some different values of the analyte thickness.
It is clear that the maximum sensitivity can be enhanced
with increasing the analyte layer thickness. As the analyte
layer thickness increases, the evanescent field in the analyte medium is enhanced and as a result the maximum
sensitivity also increases. It is also observed that as the
analyte layer thickness increases, the sensitivity shifts
toward higher value of optimum LHM thickness. The
0.8
0.6
0.8
0.7
0.6
0.5
0.4
0.4
0.2
8
10
12
14
16
18
20
22
24
d 2 (mm)
Fig. 2 Sensitivity of the proposed sensor versus the thickness of
LHM layer for different values of frequency for e1 = 1.77, n1 = 1.33,
d1 = 7.13 mm,
x0 = 4.0 GHz,
F = 0.56,
c = 0.012xp,
xp = 10.0 GHz and kB = p/(5d)
14
15
16
17
18
19
20
21
22
23
d2(mm)
Fig. 4 Sensitivity of the proposed sensor versus the thickness of
LHM layer for different values of electron scattering rate for
e1 = 1.77, n1 = 1.33, F = 0.56, kB = p/(3d), x = 5.0 GHz,
x0 = 4.0 GHz, xp = 10.0 GHz and d1 = 7.1 mm
S A Taya and S A Shaheen
c = 0.012 xp. For c = 0.013 xp, the sensitivity has a peak
value of 0.99 whereas it is 0.95 at c = 0.014 xp. As c
reduces from 0.014 xp to 0.013 xp, the maximum sensitivity can be improved by 4.2%, whereas when c reduces
from 0.013 xp to 0.012 xp, the maximum sensitivity can be
improved by 4.0%.
In Figs. 2, 3 and 4, the sensitivity increases with
increasing the LHM layer thickness and peaks at an optimum value of the thickness. At this value of the thickness,
the evanescent field in the analyte is maximum. For
thicknesses beyond the optimum one, the sensitivity
decreases towards extremely low values and this can be
attributed to the high confinement of the electromagnetic
wave in the LHM layer. Increasing the LHM layer thickness beyond the optimum value enhances the electric field
in the LHM layer and the sensitivity decreases.
The variation of the sensitivity of the proposed sensor
with the analyte thickness for different values of the
thickness of LHM layer is illustrated in Fig. 5. As is clearly
seen from the figure, the sensitivity increases slowly with
d1 until it reaches 0.87 at d1 = 6.7 mm, then it shows a
sharp increase with further increasing of d1. This behavior
can be attributed to the part of the propagating field in the
analyte layer. As the thickness of the analyte layer
increases, this part increases.
One interesting feature can be seen from the figure, the
sensitivity can reach 2.17 for d1 = 8.82 mm and
d2 = 14.1 mm which is extremely high value. The sensitivity exhibits a very slight dependence on d2 as it changes
from 13.8 to 14.1 mm.
3.2. Metal material
In this subsection, we assume the layer of thickness d2 to be
metal. The proposed structure can be operated using many
2.5
metals such as Au, Ag, Al, Ni, Cr, etc. We considered two
metals: nickel (Ni) and chromium (Cr). The thickness of
the metal is taken in the nanometer range since the
wavelength of the propagating light and thickness of layers
should be comparable.
Figures 6 and 7 show the sensitivity of the proposed
sensor versus the thickness of metals (Ni) and (Cr) layer,
respectively, for different values of the wavelength (k).
The sensitivity increases with increasing d2 until it reaches
a maximum value at optimum d2, then decreases very
slightly with further increasing of d2 until it has a fixed
value. Increasing the metal layer thickness beyond the
optimum value does not show a significant effect on the
sensitivity. As can be seen from Figs. 6 to 7, increasing the
wavelength decreases the sensitivity of the proposed sensor. It is also obvious that as the wavelength decreases, the
sensitivity peak shifts toward lower value of optimum
thickness of the metal layer. We mean that the sensitivity
peak occurs at lower values of the metal layer thickness as
the wavelength decreases.
For k = 632.8 nm, the sensitivity has a maximum value
of 0.385 at Ni thickness of 148.7 nm whereas it has a
maximum value of 1.27 at Cr thickness of 156.49 nm. For
k = 650 nm, the sensitivity has a peak of 0.375 at Ni
thickness of 150.2 nm, whereas it has a peak of 1.15 at Cr
thickness of 161.95 nm. On the other hand, the sensitivity
of Ni metal structure has a peak of 0.348 at a thickness of
170 nm for k = 700 nm, but for Cr metal structure it has a
peak of 0.86 at a thickness of 200 nm for the same
wavelength. It can be seen from Fig. 6 that as k decreases
from 700 to 650 nm, the maximum sensitivity improves by
7.75% for Ni metal whereas it enhances by 33.7% for Cr
metal as shown in Fig. 7. As k decreases from 650 to
632.8 nm, the maximum sensitivity enhances by 2.7% for
0.40
d2=13.8 mm
d2=13.9 mm
2.0
0.38
d2=14.1 mm
0.34
1.5
Sensitivity
Sensitivty
0.36
1.0
0.5
0.32
0.30
0.28
0.26
0.24
0.0
0
2
4
6
8
10
d1(mm)
Fig. 5 Sensitivity of the proposed sensor versus analyte thickness for
different values of the thickness of LHM layer for e1 = 1.77,
n1 = 1.33, F = 0.56, c = 0.012xp, x = 5.0 GHz, x0 = 4.0 GHz,
xp = 10.0 GHz and kB = p/(5d)
0.22
50
100
150
200
250
300
350
400
450
d2(nm)
Fig. 6 Sensitivity of the proposed sensor versus the thickness of
metal (Ni) layer for different values of wavelength (k) for e1 = 1.77,
l2 = 1, kB = p/(5d) and
l1 = 1, e2 = - 9.96 ? 14.66i,
d1 = 550 nm
Binary photonic crystal for refractometric applications
1.3
1.3
1.2
1.1
Sensitivity
Sensitivity
1.2
1.0
0.9
0.8
1.1
1.0
0.9
0.7
d1=650nm
d1=660nm
d1=670nm
0.8
0.6
50
100
150
200
250
0.7
300
50
100
150
d2
200
250
300
d2(nm)
Fig. 7 Sensitivity of the proposed sensor versus the thickness of
metal (Cr) layer for different values of wavelength (k) for e1 = 1.77,
l2 = 1, kB = p/(5d) and d1 =
l1 = 1, e2 = - 1.1 ? 20.79i,
670 nm
Fig. 9 Sensitivity of the proposed sensor versus the thickness of
metal layer (Cr) for different values of the analyte thickness for
k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = - 1.1 ? 20.79i, l2 = 1 and
kB = p/(5d)
Ni metal as shown in Fig. 6 whereas it improves by 10.4%
for Cr metal as shown in Fig. 7.
It is worth mentioning that d1 = 550 nm in case of
using Ni metal and d1 = 670 nm in case of using Cr metal
are found numerically to correspond to the maximum
sensitivity and the following figures will confirm that.
The sensitivity of the proposed sensor as a function of
the metal layer thickness is plotted in Fig. 8 for (Ni) and in
Fig. 9 for (Cr) for different values of the analyte thicknesses. Figures 8 and 9 show that the sensitivity can be
increased by increasing the analyte thickness. It is observed
from Fig. 8 that the sensitivity has maximum values of
0.370, 0.378, and 0.385 at Ni thickness of 145.54 nm for
d1 = 530 nm, 540 and 550 nm, respectively, but Fig. 9
shows that it has maximum values of 1.14, 1.22 and 1.27 at
Cr thickness of 163.37 nm for d1 = 650, 660 and 670 nm,
respectively. As d1 increases from 530 to 540 nm, the
sensitivity improves by 2.16% whereas it enhances by
1.85% as d1 increases from 540 to 550 nm for Ni metal
structure. On the other hand, the sensitivity enhances by
7.0% when d1 increases from 650 to 660 nm, but it
improves by 5.0% as d1 increases from 660 to 670 nm for
Cr metal structure.
The sensitivity of the proposed sensor as a function of
the analyte thickness is illustrated in Fig. 10 for (Ni) and in
Fig. 11 for (Cr) for different values of the metal thicknesses. It can be shown from Fig. 10 that the sensitivity of
the proposed sensor with (Ni) metal can be increased by
increasing the thickness of analyte layer whereas Fig. 11
shows that the sensitivity of the proposed sensor with (Cr)
metal can be increased by increasing the thickness of
analyte layer until it reaches a maximum value at an
optimum d1 = 670 nm, then decreases with further
increasing of d1. In a similar to Fig. 5, this behavior can be
0.39
0.40
0.38
0.38
0.36
0.36
Sensitivity
Sensitivity
0.37
0.35
0.34
0.33
d1=530nm
0.32
d1=550nm
d1=540nm
50
100
150
200
250
300
350
400
d2(nm)
Fig. 8 Sensitivity of the proposed sensor versus the thickness of
metal layer (Ni) for different values of the analyte thickness for
k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = - 9.96 ? 14.66i, l2 = 1
and kB = p/(5d)
d2=180nm
d2=200nm
d2=220nm
0.34
0.32
0.30
0.28
0.26
400
420
440
460
480
500
520
540
560
d1(nm)
Fig. 10 Sensitivity of the proposed sensor versus the analyte
thickness for different values of thickness of the metal layer (Ni)
for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = - 9.96 ? 14.66i,
l2 = 1 and kB = p/(5d)
S A Taya and S A Shaheen
1.4
1.2
1.2
1.0
Sensitivity
1.0
Sensitivity
1.4
d2=150nm
d2=170nm
d2=190nm
0.8
0.6
0.8
0.6
0.4
0.2
0.4
d1=100nm
d1=105nm
0.0
0.2
d1=110nm
-0.2
210
600
650
700
750
215
220
800
3.3. Dielectric material
In this subsection, we assume the layer of thickness d2 to be
dielectric, silicon dioxide, of parameters e2 = 4.0 and
l2 = 1.
The dependence of sensitivity on the thickness of the
dielectric layer for various values of the analyte layer is
shown in Fig. 12. It is found that the sensitivity increases
very sharply with increasing the thickness of dielectric
layer and reaches a maximum value at an optimum thickness, then decreases very slowly with further increasing of
the thickness of dielectric layer beyond the optimum value.
The behavior of the sensitivity with the analyte thickness is
also illustrated in Fig. 12. It is obvious from the figure that
as d1 decreases, the maximum sensitivity shifts toward
higher values of dielectric layer. For d1 = 100 nm, the
sensitivity has a maximum value of 1.31 at
d2 = 218.77 nm whereas it has a peak of 1.23 at
d2 = 215.82 nm for d1 = 105 nm. The sensitivity has a
peak of 1.16 at d2 = 212.72 nm for d1 = 110 nm. An
enhancement of 6.5% can be obtained when the analyte
thickness is reduced from 105 to 100 nm whereas the
sensitivity increases by 6.0% as the analyte thickness is
reduced from 110 to 105 nm.
235
240
Fig. 12 Sensitivity of the proposed sensor versus the thickness of
dielectric layer for different values of thickness of the analyte layer
for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = 4, l2 = 1 and kB = p/
(4d)
Figure 13 is plotted to study the variation of the sensitivity with the thickness of dielectric layer for different
wavelengths. The Figure shows that the maximum sensitivity increases with increasing the wavelength. It is also
clear that the maximum sensitivity at the optimal thickness
of dielectric layer shifts toward larger values of dielectric
thickness as the wavelength increases. For k = 632.8 nm,
the sensitivity has a maximum value of 1.16 at
d2 = 212.81 nm and d1 = 110 nm whereas it has a maximum value of 1.30 at d2 = 218.67 nm and d1 = 100 nm.
On the other hand, for k = 650 nm, the sensitivity reaches
a peak of 1.22 at an optimum dielectric thickness of
219.68 nm and d1 = 110 nm whereas it has a peak of 1.29
at an optimum thickness 227.74 nm and d1 = 100 nm. As
can be seen from Fig. 13 when k increases from 632.8 to
650 nm, the sensitivity enhances by 0.8% for d1 = 100 nm
whereas it enhances by 5.2% for d1 = 110 nm.
1.4
1.2
1.0
Sensitivity
attributed to the part of the propagating field in the analyte
layer. As the thickness of the analyte layer increases, this
part increases until it reaches a maximum value. After that
it decays again towards lower values.
For 180 B d2 B 220 nm, changing the thickness of
metal layer has an ignorable effect on the sensitivity of Ni
metal structure as noted from Fig. 10. Also changing the
thickness of Cr layer has a slight effect on the sensitivity in
the range of 150 B d2 B 190 nm.
230
d2(nm)
d1(nm)
Fig. 11 Sensitivity of the proposed sensor versus the analyte
thickness for different values of the thickness of metal layer (Cr)
for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = - 1.1 ? 20.79i, l2 = 1
and kB = p/(5d)
225
d1=110nm
0.8
d1=100nm
0.6
0.4
0.2
0.0
-0.2
210
215
220
225
230
235
240
d2(nm)
Fig. 13 Sensitivity of the proposed sensor versus the thickness of
dielectric layer for different values of wavelength (k), for
d1 = (100,110)nm, e1 = 1.77, l1 = 1, e2 = 4, l2 = 1 and kB = p/
(4d)
Binary photonic crystal for refractometric applications
d2=218nm
d2=220nm
1.40
1.35
d2=225nm
Sensitivity
1.30
1.25
1.20
sensor. When the results of the structures comprising LHM,
metal, and dielectric are compared, it can be concluded that
the highest sensitivity was achieved with LHM structures.
It is worth mentioning that the proposed sensor in the
current work can be fabricated in a similar manner to that
proposed in reference 3.
1.15
1.10
References
1.05
96
98
100 102 104 106 108 110 112 114
d1(nm)
Fig. 14 Sensitivity of the proposed sensor versus the thickness of
analyte layer for different values of thickness of the dielectric layer
for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = 4, l2 = 1 and kB = p/
(4d)
Figure 14 is plotted to illustrate the effect of the thickness of the analyte layer on the sensitivity at some values of
the thickness of the dielectric layer. It is clear that the
sensitivity of the proposed sensor increases with decreasing
the thickness of the analyte. The sensitivity has the range of
1.06 B S B 1.33 at d2 = 220 nm. For d2 = 225 nm, the
sensitivity has the range of 1.05 B S B 1.22 whereas it has
the range of 1.06 B S B 1.39 for d2 = 218 nm. It is also
obvious that the maximum sensitivity enhances by 4.5% as
d2 decreases from 225 to 220 nm, but it improves by 9.0%
as d2 decreases from 220 to 218 nm.
4. Conclusion
In this work, a two-layer photonic crystal was assumed in
which the first layer was considered to be the analyte layer
and the second layer was considered to be LHM, dielectric
or metal. It is possible for LHM photonic crystal structure
to achieve a sensitivity of 2.17 which is extremely high
value whereas a metal photonic crystal structure can
achieve a sensitivity of 1.27. The dielectric photonic crystal
structure can reach a sensitivity of 1.39. The sensitivity
improvement with LHM photonic crystal was 412% for TE
mode when compared to conventional slab waveguide
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