The temperature dependence of the omni-directional reflection (ODR) band in a one-dimensional photonicbcrystal is proposed simultaneously considering thermal expansion effect and thermo-optic effect. The structure proposed in this study consists of a periodic arrangement of alternate layers of SiO2 as the material of low refractive index and Si as the material of high refractive index. As the refractive index and thickness of both materials used in this study are modulated by temperature, the ODR band can be tuned as a function of temperature. With the increase of temperature, it is noted that the ODR band shifts towards the longer wavelength region. Also, the ODR band broadens slightly. The ODR band can be tuned by variation in the operating temperature of the structure.

1. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.1, February 2014
9
THERMAL TUNING OF OMNI-DIRECTIONAL
REFLECTION BAND IN SI-BASED 1D PHOTONIC
CRYSTAL
B. Suthar1
, Vinay Kumar2
, Arun Kumar3
, Vipin Kumar4
* and A. Bhargva5
1
Department of Physics, Govt. College of Engineering & Technology, Bikaner 334004,
India
2
Department of Physics, Shri Venkteshwara University, Gajraula 244221, India
3
AITEM, Amity University, Noida 201313, India
4
Department of Physics, Digamber Jain (P.G.) College, Baraut 250611, India
5
Nanophysics Laboratory, Department of Physics, Govt. Dungar College, Bikaner
334001 India
ABSTRACT
The temperature dependence of the omni-directional reflection (ODR) band in a one-dimensional photonic
crystal is proposed simultaneously considering thermal expansion effect and thermo-optic effect. The
structure proposed in this study consists of a periodic arrangement of alternate layers of SiO2 as the
material of low refractive index and Si as the material of high refractive index. As the refractive index and
thickness of both materials used in this study are modulated by temperature, the ODR band can be tuned as
a function of temperature. With the increase of temperature, it is noted that the ODR band shifts towards
the longer wavelength region. Also, the ODR band broadens slightly. The ODR band can be tuned by
variation in the operating temperature of the structure.
KEYWORDS
Omnidirectional reflection, Photonic crystal, Thermal expansion, Transfer matrix method
1. INTRODUCTION
Recently, the study of various properties and potential applications of photonic crystals (PCs)
have become an area of intense research. Photonic crystals are structures of materials generally
with periodically modulated dielectric constants. Under some conditions, photonic crystals can
exhibit complete photonic band gaps (PBGs) i.e. some range(s) of the wavelengths of incident
wave that are reflected by the photonic crystal structure [1]. By introducing defect(s) in the
periodic structure of the PCs, one can obtain localized defect mode inside the forbidden band gap
[2]. This property of the photonic crystals guides the flow of light and manipulates the
propagation of photons within the PCs. This property can be used in many interesting applications
of optoelectronics [3-7]. One dimensional PC structures have many applications such as low-loss
optical waveguides, dielectric reflecting mirrors, optical switches, optical limiters, optical filters
etc. It has been demonstrated experimentally and theoretically that one-dimensional PCs have
complete omni-directional PBGs [8-13].
An omni-directional reflector is a mirror having 100% reflectivity at any angle of incidence for
TE polarized as well as TM polarized waves. In 1998, Fink et al. reported that one-dimensional
dielectric lattice displays total omni-directional reflection for incident light under certain
conditions [14, 15]. They constructed a stack of nine alternate layers of polystyrene/tellurium
having a thickness of the order of ~ μm and demonstrated omni-directional reflection over the
wavelength range from 10-15 μm. Gallas et al. [16] reported the annealing effect in the Si/SiO2

2. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.1, February 2014
10
PBG based omni-directional reflectors. Wang et al. [17] theoretically showed that overlapping
two photonic crystals could enlarge the range of total omni-directional reflection.
Mostly, photonic crystals have been made from III–V semiconductors. While the thermal effects
of these semiconductors are changed various optical properties of PCs. Previous reports on omni-
directional reflection were without taking the thermal effect. In this article, the design of an omni-
directional reflector using a one-dimensional photonic crystal has been proposed considering the
temperature modulation of thickness and refractive index on each alternate layer. The effect of
temperature on thickness is called thermal expansion effect and effect of temperature on
refractive index is called thermal-optical effect. The structure proposed in this study consists of a
periodic arrangement of alternate layers of SiO2 as the material of low refractive index and Si as
the material of high refractive index. As the refractive index and thickness of both materials used
in this study are modulated by temperature, the ODR band can be tuned as a function of
temperature. With the increase of temperature, it is noted that the ODR band shifts towards the
longer wavelength region. Many researchers have been worked on thermal tuning of photonic
band gaps and defect modes [18-24]. But to the best of our knowledge no other thermal tuning
ODR based theoretical/or experimental research has been worked out.
2. THEORETICAL MODEL
We consider a multi-layered structure [air/(n1n2)10
/air] which consists of alternate layers of
materials of high and low refractive indices along the x- axis, as shown in Figure 1.
Figure 1: Graphical representation of the proposed Structure
By employing the transfer matrix method (TMM), the matrices for both polarized waves (TE and
TM) for a bilayer system of such a structure is given by [25, 26]
.sin
cos
sin cos
j
j
j
j
j j j
i
p
A
ip


 
 
−
 
=  
 
−
 
(1)
where pj=njcosθj, ( j=1,2 ) for the TE mode of polarization and pj=cosθj/nj for the TM mode of
polarization, j=(2π/λ)njdjcosθj, θj is the angle of incidence of the propagated wave inside the
layer with refractive index nj and λ is the wavelength of the propagated wave in the medium of
incidence (air). The characteristics matrix for the proposed structure having N unit cells is given
by
11 12
21 22
( )N
j
A A
A A
A A
 
= =  
 
(2)

3. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.1, February 2014
11
The reflection coefficient of the proposed structure for TE and TM polarized waves are given by
11 12 21 22
11 12 21 22
( ) ( )
( ) ( )
f i f
f i f
A p A p A p A
r
A p A p A p A
+ − +
=
+ + +
(3)
where pi,f=ni,fcosθj,f for TE polarized wave and pi,f= (cosθj,f)/ ni,f for TM polarized wave, where the
subscripts i and f correspond to the mediums of incidence and the medium of emergence
respectively. The reflectivity of the proposed structure can be given as
2
r
R = (4)
There is no complete photonic band gaps (PBG) in one-dimensional PCs due to the two factors.
The first factor is that the edges of the directional PBGs (PBGs at a certain region) will shift
towards the longer frequency side with the increase in angle of incidence, usually leading to the
disappearance of the overall PBGs. The second factor is the Brewster angle; at which the TM
polarized wave cannot be reflected. However, the absence of a complete PBG does not mean that
there is no omni-directional reflection. The criterion for the existence of total omni-directional
reflection is that there are no waves propagate through the PC for both modes of polarization
[10,12,16]. From Snell’s law, we know nisinθi=n1sinθ1 and n1sinθ1=n2sinθ2 i.e.
)
/
sin
(
sin 1
1
1 n
n i
i 
 −
= and )
/
sin
(
sin 2
1
1
1
2 n
n 
 −
= , where n1 and n2 are the refractive indices of the
low and high refractive index mediums respectively, and ni is the refractive index of the incident
medium. The maximum refracted angle is defined as )
/
(
sin 2
1
max
2 n
ni
−
=
 and the Brewster angle is
given by )
/
sin
(
tan 2
1
1
n
n
B 
 −
= . If the Brewster’s angle is greater than the maximum angle of
refraction, then the propagated wave cannot couple to Brewster’s window which is the condition
of owing total reflection for all modes of polarization and angles of incidence. Thus, Brewster’s
angle must be equal to maximum angle of refraction for omni-directional reflection i.e.
max
2

 =
B
[10,12,27]. This condition must be satisfied by parameters used in our numerical
computations.
3. RESULTS AND DISCUSSION
From the computation of Equation (4), the reflection properties of one-dimensional photonic
crystals simultaneously considering thermal expansion effect and thermal-optic effect can be
represented graphically. For this purpose, we consider the PC structure having n1 and n2 as
refractive indices of SiO2 and Si respectively (n1<n2). Si is the good candidate for the photonic
crystals especially in infrared region because of its very low absorption in this region. Crystalline
Si has been taken in this study. The refractive indices are taken considering thermo-optic effect in
the following manner
n1(T)= n1(1+γ1ΔT) and n2(T)= n2(1+γ2ΔT) (5)
where n1=1.46, n2=3.4 and γ1, γ2 are called the thermo-optic coefficients for the SiO2 and Si
materials respectively. The values of these coefficients are taken as γ1=1.86 x 10-4
/o
C and γ2=6.8
x 10-6
/o
C [28].
The thickness of each layer is taken considering the effect of thermal expansion of each layer in
the following manner
a=a(1+ 1ΔT) and b=b(1+ 2ΔT) (6)
where 1, 2 are called the thermal expansion coefficients for the SiO2 and Si materials
respectively. The values of these coefficients are taken as 1=2.6 x 10-6
/o
C and 2=0.5 x 10-6
/o
C
[28].

4. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.1, February 2014
12
Figure 2: Reflection for both TE and TM modes in one dimensional PC with angle of incidence at room
temperature T=30o
C
The thickness of the layers of proposed structure are taken as a=300nm and b=129nm according
to the quarter wave stack condition a=λc/4n1 and b=λc/4n2, with the critical wavelength,
λc=1750nm. The critical wavelength is the mid wavelength of the spectrum considered in this
computation. The ODR band is found in the vicinity of this critical wavelength. So the optical
paths for the incident radiation of each material are equal. The reflectance spectra of the proposed
PC (for both TE and TM polarizations) at 30°
C, is shown in Figure 2. The reflectance spectra are
plotted in terms of wavelength and for incident angle θi. Figure 3, represents the complete
photonic band structure shown as a function of the angle incidence, which can be obtained by the
projections of Figures 2(a) and 2(b). In Figure 3, the shaded region gives the total omni-
directional reflection band. The data corresponding to nearly 100% reflectance is summarized in
Table 1.

5. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.1, February 2014
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TABLE 1: TE, TM AND ODR BAND GAP AT ROOM TEMPERATURE T =30O
C
Mode Upper (nm) Lower (nm) Band Gap (nm)
TM 1815.6 1404.2 411.4
TE 2243.0 1404.2 838.8
Complete ODR 1815.6 1404.2 411.4
TABLE 2: ODR BAND WITH THE TEMPERATURE
Temperature
(o
C)
Upper Edge of ODR
Band (nm)
Lower Edge of ODR
Band (nm)
Mid-Gap of
ODR Band
(nm)
ODR Band
Gap (nm)
50 1815.9 1404.4 1610.15 411.5
100 1816.7 1404.9 1610.80 411.8
150 1817.5 1405.4 1611.45 412.1
200 1818.3 1405.8 1612.05 412.5
250 1819.1 1406.3 1612.70 412.8
300 1819.9 1406.8 1613.35 413.1
350 1820.7 1407.3 1614.00 413.4
400 1821.5 1407.8 1614.65 413.7
450 1822.3 1408.3 1615.30 414.0
500 1823.1 1408.8 1615.95 414.3
Figure 3: Complete ODR Bandgap for both TE & TM mode of polarization and whole angle of incidence
at room temperature T=30o
C
From Table 1 it is observed that the TE polarized wave has its 100% reflection range from
1404.2nm to 1815.6nm and the 100% reflection range for the TM polarized wave is from
1404.2nm to 2243.0nm. Therefore, the range for which both the polarizations (TE and TM)
exhibit 100% reflection i.e. omni-directional reflection (ODR) has the bandwidth (∆λ=λH– λL) of
411.4nm. The upper edge of the complete band gap i.e. ODR band is λH=1815.6nm and the lower
wavelength edge is λL=1415.2nm. Our aim is to tune the ODR band due to the variation of

6. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.1, February 2014
14
temperature. The variations of ODR band gap with temperature (30-500 o
C) are shown in table 2.
Table 2 indicate that with the increase of temperature the ODR range increases slightly. The
average change in the ODR band is 0.006nm/ o
C. As we increase the temperature, the middle of
the ODR band shifts toward the longer wavelength region. This is approximately 0.013/ o
C, which
is just double of the broadening of ODR bands. This shifting behavior of ODR bands can be
explained by using the phase equation j=(2π/λ)nj(T)dj(T)cosθj where j=1,2. According to this
phase equation, as nj(T) and dj(T) increase with temperature, the wavelength must increase
accordingly to keep the phase j unchanged. Correspondingly, the value of wavelength must
increase.
Figure 4: Variation in Mid-gap with Temperature
Figure 5: Variation in Bandgap with Temperature
The variation of the middle of the ODR band with temperature is shown in Figure 4. From this
figure, it is clear that the change in the middle of the ODR band is linear approximately. Also, the
variation of the band width of ODR band with temperature is shown in Figure 5. This Figure
shows that the ODR band width also change linearly with temperature. So, we can tune the ODR
band width as well as middle of the ODR band to the desired wavelength region by choosing the
appropriate temperature of the medium.

7. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.1, February 2014
15
3. CONCLUSIONS
To summarize, we have investigated theoretically the ODR range of PC structure considering
simultaneously thermo-optic and thermal expansion effects. It is observed that the ODR range of
proposed photonic crystal structure can be tuned by increasing the operating temperature. These
types of optical reflectors are compact comparatively and may have many useful applications in
photonics and optoelectronics such as optical sensors, optical filters, etc.
ACKNOWLEDGEMENTS
The authors would like to thank Dr.Kh.S. Singh and Prof S.P.Ojha, for their valuable suggestions.
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