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Influence of graded index materials on the photonic localization in one-dimensional quasiperiodic (Thue–Mosre and Double-Periodic) photonic crystals
Optics Communications 333 (2014) 84–91
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Influence of graded index materials on the photonic localization in one-dimensional quasiperiodic (Thue–Mosre and Double-Periodic) photonic crystals Bipin K. Singh, Praveen C. Pandey n Department of Physics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, Uttar Pradesh, India
art ic l e i nf o
a b s t r a c t
Article history: Received 31 May 2014 Received in revised form 7 July 2014 Accepted 12 July 2014 Available online 25 July 2014
In this paper, we present the investigation on the photonic localization and band gaps in quasi-periodic photonic crystals containing graded index materials using a transfer matrix method in region 150–750 THz of the electromagnetic spectrum. The graded layers have a space dispersive refractive index, which vary in a linear and exponential fashion as a function of the depth of layer. The considered quasiperiodic structures are taken in the form of Thue–Morse and Double-Periodic sequences. The grading profile in the layers affects the position of reflection dips and forbidden bands, and frequency region of the bands. We observed that vast number of forbidden band gaps and dips are developed in its reflection spectra by increasing the number of quasi-periodic generation. Moreover, we compare the total forbidden bandwidths with increasing the generation of the quasi-periodic sequences for the structures with linear and exponential graded layer. Results show that the different graded profiles with same boundary refractive index can change the position of localization modes, number of photonic bands and change the frequency region of the bands. Therefore, we can achieve suitable photonic band gaps and modes by choosing the different gradation profiles of the refractive index and generation of the quasi-periodic sequences. & 2014 Elsevier B.V. All rights reserved.
Keywords: Photonic crystals Quasiperiodic sequence Photonic band gap Graded index material
1. Introduction In the past two decades, great efforts have been dedicated towards the investigation of the structure and physical properties of quasiperiodic systems after the discovery of the quasi-crystalline structure [1–3]. Quasi-periodic systems have long-range order but not in a repeating fashion yielding periodicity. These arrangements are fixed in a regular pattern and follow a simple deterministic recursion rule [4,5]. Quasi-periodic systems are one of the most interesting arrangements to obtain the suitable photonic band gaps because of several structural parameters available to tune as compared to the periodic and disordered systems [6–8]. Recently, some research groups have reported their works on electromagnetic (EM) wave propagation in quasi-periodic structures called photonic quasi-crystals. Due to a longrange order, such type of structures provide wide photonic band gap in photonic spectra as in periodic photonic crystals and simultaneously possess localized states as in disorder media [9,10]. Photonic quasicrystals exhibit unique influence on the optical properties such as optical transmission and reflectivity, photoluminescence, light transport, plasmonics and laser action, etc. Li et al. [11] and Luo et al. [12]
n
Corresponding author. Tel.: þ 91 5426702008; fax: þ 91 5422368428. E-mail address: [email protected] (P.C. Pandey).
http://dx.doi.org/10.1016/j.optcom.2014.07.043 0030-4018/& 2014 Elsevier B.V. All rights reserved.
proposed two-dimensional photonic crystals that achieve multimode lasing action, low pumping threshold and excellent linear polarization property as well as wide directional dependence. This opens a new field of research in photonics in view of their vast technical applications. Photonic band gap properties of quasi-periodic multi-layered structures have been extensively studied for different materials [13]. Specifically, one-dimensional (1-D) photonic quasi-crystals are very important because their formation is relatively easy and they may provide the description of light propagation in one direction [14–16]. One-dimensional photonic quasi-crystals are composed of layers according to substitutional sequences in form of the Fibonacci, Thue–Morse and Double-Periodic etc. Recently several researchers have been proposed the 1-D multilayer structures with gradual varying RI as a function of the depth of layer, and width of layers varies as a gradual fashion along the direction perpendicular to the surface of layer in the considered structures [17–21]. Such type of structures are called graded photonic crystals (GPCs). In two-dimensional GPC structures, gradual variations of the relative parameters can be distributed along the normal or perpendicular to the propagation of electromagnetic waves. Gradual variation of relative parameters of GPCs make them very different in the behaviour from the conventional PCs and enhance the ability to mold and control of the light wave propagation [22–24]. Such types of PCs play an important role to
B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
design spectral filters, beam aperture and deflector, high efficiency bending waveguides, high efficiency couplers, super bending and self-focusing media, lenses, artificial optical black holes and antireflection coating [25–30]. In our previous work [31], we have formulated the resonant Bragg condition for the quasiperiodic Fibonacci multilayer structures containing exponential graded material and shown that forbidden band gaps and omnidirectional band gaps to be obtained for periodicity of different generation Fibonacci sequence structures and their hetrostructures. In these structures, bandwidth of forbidden and omnidirectional band gaps can be tuned with graded profile parameter of exponential graded layer. Motivated by the ability to mold, confine and control of the electromagnetic waves by different types of GPCs, we now present the study of the photonic band gap characteristics of 1-D quasiperiodic GPCs constituted with an exponential/linear graded dielectric layer. The aims of this work are, we want to show the reflection spectra, which arise from the propagation of electromagnetic waves in quasi-periodic multilayer structures, comprised of alternating layers of both normal (SiO2) and graded index materials using a theoretical model based on a transfer matrix treatment. The quasi-periodic structures follow the Thue–Morse (TM) and Double-Periodic (DP) substitutional sequences and can be generated by the following inflation rules: A-AB, B-BA (TM); and A-AB, B-AA (DP),[4,5] where A and B are the building blocks modelling of the normal and graded index materials, respectively. Further, we intend to present a quantitative analysis of the results, pointing out the distribution of the forbidden band gaps and total bandwidths for up to the 6th generation, which gives a good insight about their photonic band gaps and localization. The plan of this paper is as follows. In Section 2, we present the method of calculation employed here, which is based on the transfer matrix approach. The reflection coefficient and dispersion relation is then determined. Section 3 is devoted to the discussion of this reflection spectra and dispersion relation for the Thue– Morse and Double-Periodic multilayer structures containing linear and exponential graded index material. Further, we present their total band gap with Thue–Morse and Double-Periodic generations for linear and exponential graded index material as one of the layer. The conclusions of this work are presented in Section 4.
2. Theoretical model and numerical analysis In this paper, we consider the system of multilayer that is composed of two layers and stacked alternatively along the x-direction. The stacks of two layers are arranged according to the recursion rule of the Thue–Morse (TM) and Double-Periodic (DP) sequence in different generations [4,5]. These sequences are based on the two-letter alphabet (A, B) and the substitution rule: σ (A) ¼AB, σ (B) ¼BA (TM); and σ (A)¼ AB, σ (B) ¼AA (DP). The substitution rule can be written in the form of the following equations: A 1 1 A AB σ: ¼ and B 1 1 B BA A 1 1 A AB σ: ¼ B 2 0 B AA The ratio of the frequencies of the letters A, B in the sequence is equal for TM and 2 to 1 for DP. The length of the sequence in both cases at the iteration n is 2n. On the basis of the above substitution rule, the first few words generated in this way are represented in Table 1 panels (i) and (ii). The proposed multilayer structures consist of two kinds of layers; one has a constant refractive index (1.5) and other has a
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linear or exponential varying refractive index (1.5–4.1) as the case may be. The variation of refractive index in the graded layers is taken along the direction of the thickness of the layer. The direction of wave propagation is considered along the x-axis i.e. the direction normal to the stacked layers and the considered materials assumed as non-magnetic, non-dispersive and isotropic. The refractive indices of the considered graded layers vary in linear and exponential fashion between the initial and final values as ½nL ðxÞ ¼ ni þ ðnf ni Þ=dL x and ½nE ðxÞ ¼ ni expðx=dE log nf =ni Þ, respectively. Here, ni and nf are the refractive indices at inward and outward boundaries of the graded layer, respectively [32]. The optical properties of the considered multi-layered structures are described by well-known theoretical model based on transfer matrix method (TMM). Transfer matrices are generated by applying boundary conditions in a plane wave solution of the Maxwell's wave equation at the interface boundary. The electric field distribution E(x) in different materials can be written as: (i) For normal layers: EN ðxÞ ¼ AN expð i UkN U xÞ þ BN expði U kN U xÞ………:
ð1Þ
where AN and BN are arbitrary constants and kN ¼ ω U nN =c represents the propagation wave vector at normal incidence with a constant refractive index nN . Subscript N represents the normal layer and ω and c are the angular frequency and velocity of light, respectively [30]. (ii) For linear graded layers the electric field equation can be written as " ! !# pffiffiffiffiffi ξ2 ξ2 EL ðxÞ ¼ ξL U AL J 1=4 L þ BL Y 1=4 L ð2Þ 2α 2α where, AL and BL are arbitrary constants and ξL ¼ ω UnL ðxÞ=c the propagation wave vector at normal incidence along xdirection for the linear graded layer with refractive index: nL ðxÞ ¼ ni þ ðnf ni Þ=dL Þ x, where ni and nf are the refractive indices at the initial and final boundary and dL is the layer thickness. Subscript L represents the linear graded layer and grading profile parameter of the linearly graded layer is α ¼ ðω=cÞðnf ni =dL Þ, ω and c are the angular frequency and velocity of light, respectively [32]. (iii) For exponential graded layers: ξ ξ ð3Þ EE ðxÞ ¼ AE J 0 E þ BE Y 0 E ………: γ γ where, AE and BE are arbitrary constants and ξE ¼ ω:nE ðxÞ=c represents the wave propagation vector at normal incidence along the x-direction for the exponential graded layer with refractive index nE ðxÞ ¼ ni expðγxÞ, where γ ¼ 1=dE log ðnf =ni Þ; is the grading profile parameter of the exponentially graded layer, ni and nf are same as defined in Eq. (2) and dE is the thickness of the exponentially graded layer. Subscript E represents the exponentially graded layer. The functions J and Y are the first and second kind Bessel's functions, respectively.
Using the transfer matrix approach, the amplitudes A0 and B0 of the electromagnetic field in the air medium at x o0 related to be the amplitudes An þ 1 and Bn þ 1 of the equivalent layer in the (n þ1) th region through the linear transformation. Therefore, for the multilayer structures, the total transfer matrix equation can be written as A0 B0
! ¼ M i;j
An þ 1 Bn þ 1
! ð4Þ
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B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
Table 1 Quasi-periodic multilayer arrangement for Thue–Morse and Double–Periodic sequences. (i) Generation of the words Tn ¼ σn (A) of the Thue–Morse sequence by the repeated action of the substitution rule σ (A)¼ AB, σ (B) ¼BA. Thue–Morse sequence
T n þ 1 ¼ T n T n ; T n þ 1 ¼ T n T n , with T 0 ¼ A and T 0 ¼ B
Generation number
T0 T1 T2 T3 T4 T5 … …
A AB ABBA ABBABAAB ABBABAABBAABABBA ABBABAABBAABABBABAABABBAABBABAAB …………………………………. ………………………………….
0 1 2 3 4 5 … So on.
(ii) Generation of the words Dn ¼ σn (A) of the Double-Periodic sequence by the repeated action of the substitution rule σ (A)¼ AB, σ (B) ¼AA. Double-periodic sequence
Dn þ 1 ¼ Dn Dn ; Dn þ 1 ¼ Dn Dn , with D0 ¼ A and D0 ¼ B
Generation number
D0 D1 D2 D3 D4 D5 … …
A AB ABAA ABAAABAB ABAAABABABAAABAA ABAAABABABAAABAAABAAABABABAAABAB …………………………………. ………………………………….
0 1 2 3 4 5 … So on.
where M i;j ði; j ¼ 1; 2Þ is the total characteristic matrix and if atmosphere around the system is air, M i;j ¼ M 0 1 ðM 1 M 2 M 3 … M n ÞM 0 for the n layer system and M n is the characteristic matrix of nth layer. These layers may be normal, linear or exponential graded material layer. After the simplification, the characteristic matrix for the different type of layer at the normal angle of incidence can be written as (i) For a normal (homogeneous) layer: " # sin ðkN dN Þ=nN cos ðkN dN Þ MN ¼ nN sin ðkN dN Þ cos ðkN dN Þ where M N and dN are the 2 2 characteristic matrix and thickness of normal layer, respectively. (ii) For a linear graded layer: 2 qffiffiffiffi 3 ni cα ω pffiffiffiffiffiffiffiffiffiffi n L2 n n L1 þ i f f 6 2ω nf 7 c 6 7
6 7 6 ωc pffiffiffiffiffiffiffiffiffiffiffi 7 n L1 n i f 6 7 6 7 q ffiffiffi ffi q ffiffiffi ffi M L ¼ Δ6 c3 α2 1 7 p ffiffiffiffiffiffiffiffiffiffi nf n cα cα ω i 6 4ω3 pffiffiffiffiffiffiffiffiL1 þ 2ωnf n L2 þ 2ωni n L3 þ c 3 ni nf L4 7 ni nf i f 6 7 6 7 6 7 qffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi 4 cα ni L1 ωn n n L3 5 i f 2ω nf c i
L3 ¼
L3 ¼
8 < :
0
J1
8 < :
4
0
J1 4
ξ2L 2α
!
ξ2L 2α
ξ2L 2α
Y1 x¼0
4
! 0
Y1 x¼0
4
!
ξ2L 2α
J1 x ¼ dL
ξ2L 2α
4
! 0
J1 x ¼ dL
4
!
ξ2L 2α
0
Y1 x ¼ dL
4
! 0
Y1 x ¼ dL
4
ξ2L 2α
9 =
!
ξ2L 2α
x¼0
;
! x¼0
9 = ; ;
M L and dL are the 2 2 characteristic matrix and thickness of the linearly graded layer, respectively, and prime mark represents the differentiation of the relative functions. (iii) For an exponential graded layer: 3 2 nf J 0 ξγE Y 1 ξγE Y 0 ξγE J 1 ξγE 7 6 x¼0 x ¼ dE x¼0 x ¼ dE 7 6 7 6 7 6 – J ξE ξE ξE ξE Y0 γ Y0 γ J0 γ 7 6 0 γ x¼0 x ¼ dE x¼0 x ¼ dE 7 6 7 6 M E ¼ Ω6 7 ξE ξE ξE 7 6 ni nf J ξE Y1 γ Y1 γ J1 γ 1 γ 7 6 x ¼ 0 x ¼ d x ¼ 0 x ¼ d E E 7 6 7 6 5 4 Y 0 ξγE Y 1 ξγE J 0 ξγE ni J 1 ξγE x¼0
where Ω ¼ 1=nf
x ¼ dE
x¼0
x ¼ dE
( ) ξ ξ ξ ξ J1 E Y0 E Y1 E J0 E ; γ x ¼ dE γ x ¼ dE γ x ¼ dE γ x ¼ dE
M L and dE are the 2 2 characteristic matrix and thickness of exponential graded layer, respectively.
where 8 ! ω < ξ2 Δ ¼ 1= n2f J 1 L c : 4 2α
8 ! < ξ2 L1 ¼ J 1 L : 4 2α
4
x ¼ dL
x¼0
8 ! < ξ2 L2 ¼ J 1 L : 4 2α
Y 01
x¼0
Y1 4
0
Y1 4
ξ2L 2α
ξ2L
! 4
x ¼ dL
! J1
2α ξ2L 2α
J 01
x ¼ dL
4
! 0
J1 x ¼ dL
4
ξ2L 2α
ξ2L 2α ξ2L 2α
!
ξ2L 2α
Y1 x ¼ dL
4
! Y1 x ¼ dL
4
! Y1 x ¼ dL
4
! x ¼ dL
ξ2L
!
2α ξ2L 2α
9 = ; ;
x¼0
! x¼0
9 = ; ;
9 = ;
;
Thus the transmittance (T) and reflectance (R) coefficients of electromagnetic wave can be written as T ¼ j1=M 11 j2 and R ¼ jM 21 =M 11 j2
;
ð5Þ
where M 11 and M 21 represent the elements of the optical transfer matrix M i;j ði; j ¼ 1; 2Þ. Naturally due to our consideration of lossless dielectric material, the transmittance here is just the reflectance's complement. According to Floquet's theorem, the electric field vector of a normal mode of propagation in a periodic medium is of the form
B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
as: EK ðx; zÞ ¼ EK ðxÞ U e i U β:z U e– i:K U x , where EK ðxÞ is a periodic function with period d, i.e. EK ðx þ dÞ ¼ EK ðxÞ and β is the z-component of the wave vector. The subscript K represents that the function EK ðxÞ depends on K. This optical wave is known as a Bloch mode and the parameter K, which satisfies the mode and its associated periodic function EK ðxÞ, is called the Bloch wave number. The dispersion is an equation relating the Bloch wave number K and the angular frequency ω and expressed in the form Kðβ; ωÞ ¼
1 1 cos 1 ðM 11 þ M 22 Þ d 2
ð6Þ
where d is the total thickness of a period of the periodic system, M 11 and M 22 is the elements of the optical transfer matrix M i;j ði; j ¼ 1; 2Þ.
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The dispersion relation exhibits multiple spectral bands classified into two regimes: first, where jðM 11 þ M 22 Þ=2j r 1 corresponds to real K and thus to propagating Bloch waves. Second, spectral bands within which K is complex corresponds to evanescent waves that are rapidly attenuated. Defined by the condition jðM 11 þ M 22 Þ=2j4 1, these bands correspond to the stop bands also called photonic band gaps/forbidden gaps since propagating modes do not exist for the systems [32].
3. Results and discussion In this section, we have presented the results obtained from the numerical calculations and the discussion on the reflection spectra of light for different 1-D quasi-periodic (Thue–Morse and
Fig. 1. Reflection coefficient (R) versus frequency for (a) 5th, (b) 6th and (c) 7th generation sequence of the Thue–Morse photonic crystals with linear and exponential graded layers.
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Double-Periodic) type multilayer structures. The unit cells of considered 1-D quasi-periodic photonic structures are stacked of two types of layers namely, A and B, where ‘A’ type layers represent constant refractive index (Normal) material of value 1.5 and ‘B’ type layers represent linear and exponential graded index material having refractive index variation from 1.5 to 4.5. The considered 1-D quasi-periodic multilayer structures using layers A and B are obtained according to the substitution rules of the Thue–Morse and Double–Periodic sequences, as tabulated in Table 1. We have calculated the reflectance and band spectrum of light incident normally through the air on the considered multilayer structures. These materials are considered lossless, non-
magnetic and their refractive index does not depend on frequencies. Also, we depicted the variation of total photonic band gap for 1-D quasi-periodic photonic structures as a function of the generation of Thue–Morse and Double-Periodic system with linear and exponential graded index material. We first present the reflection spectra of 1-D Thue–Morse quasi-periodic photonic structures for the fifth, sixth and seventh generation with linear and exponential graded index material as layer B. Thicknesses of the layer A and B are chosen in terms of quarter wave stacking i.e. nA dA ¼ nm dB ¼ λ0 =4 where dA and dB are the thickness of layer A and B, respectively, and nm is the mean value of the initial refractive index and final refractive index of the
Fig. 2. Reflection coefficient (R) versus frequency for (a) 5th, (b) 6th and (c) 7th generation sequence of the Double-Periodic photonic crystals with linear and exponential graded layers.
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graded layer. These types of stacking structures are very useful and suitable for designing various photonic devices because they provide large photonic band gap. The physical parameters used here are nA ¼ 1:5; nm ¼ 3:0 and wavelength λ0 is the mean value of the considered frequency region (150–750 THz). Analysing results shown in Fig. 1, we get number of localization modes and forbidden bands that increases with increasing the generation of Thue–Morse system for both linear and exponential graded index material. From all these figures, it is clearly visible that localization modes and forbidden bands exist in higher frequency region for exponential graded layer, while for the similar structures with linear graded layers position of localization modes and forbidden bands shifted towards lower frequency region. The influence of the different graded materials on the localization modes and band gaps are sighted on position and width of the band gaps whereas the numbers of localization modes and forbidden bands are same. Likewise, we have studied the influence of graded index materials on the localization modes and forbidden bands for 1-D Double-Periodic photonic structures for Fifth, sixth and seventh generation of quasi-periodic systems with linear and exponential graded index material under normal incidence of light. In these structures, composition of the layer arrangement has become different, but other parameters like thickness, total number of layer of the structures are same with respect to same generation of Thue–Morse quasi-periodic structures. Reflection spectra of these structures are shown in Fig. 2. The numbers of localization modes and forbidden band increases with the increase of the generation of Double-Periodic quasi-periodic system for both linear and exponential graded index material as layer B, and shift of the localization mode and forbidden bands is similar to that in 1-D quasi-periodic photonic structures with Thue–Morse system i.e. localization modes and forbidden bands exist at related higher and lower frequency regions for the structures with an exponential and linear graded layer, respectively. Conversely, one broader forbidden band always exists around the central frequency region for all these structures, while bands exist at lower and higher frequency region for the structures with Thue–Morse generations. Also, large number of modes and bands exist for these structures as compared to the same generation Thue–Morse based structures. In our configurations the Figs. 1 and 2, clearly demonstrated that the number of localization modes and forbidden bands of the considered PC structures depends on the generation of the quasiperiodic systems, and their position and bandwidth depend on the
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material properties and geometrical parameters. The results also show that, the position of modes and bands exist at higher frequency for structures with exponential graded layer as compared to the structures with the linear graded layer, because the rate of change of the refractive index contrast in the exponential graded layer is slightly higher in comparison to the linear graded layer. On the other hand, the refractive index boundary is same in both the type of graded layers and hence the change in the initial to final optical path length is approximately same; therefore due to this fact, structures have equivalent number of modes and bands. Furthermore, we have examined the photonic band gap spectrum of the periodic super lattice of different quasi-periodic (Thue–Morse and Double-Periodic) generations arising from competition between the linear and exponential graded layer, and depicted in panels (a) and (b) of Figs. 3 and 4, respectively. For this analysis, we have calculated the forbidden frequencies (stop bands), where jðM 11 þ M 22 Þ=2j 41, as a function of the generation of the quasi-periodic (Thue–Morse and Double-Periodic) sequence for a fixed value of nm dB ¼ λ0 =4, as shown in Fig. 3 (Thue–Morse sequence) and Fig. 4 (Double-Periodic sequence). The comparison of the photonic spectra of the Thue–Morse multilayer structures with linear and exponential graded layer for generation levels from 1 to 6 at normal incidence has been presented in Fig. 3. Fig. 3(a) and (b) shows the distribution of the forbidden (black region) and allowed (white region) frequencies, as a function of the generation of the Thue–Morse sequence of the structures with a linear and exponential graded layer, respectively up to the 6th generation of the Thue–Morse sequence, considering nm dB ¼ λ0 =4. As expected, for large number of generations, we get large number of stop bands and their bandwidth becomes narrower and narrower as an indication of more localized modes. We also observed that number of band gap in the structures with both linear and exponential graded layer presents a similar distribution as a function of the generation of the Thue– Morse sequence, but their bandwidth and frequency region are slightly different. This is illustrated in Fig. 3(a) and (b) for a fixed value nm dB ¼ λ0 =4 in the structures with linear and exponential graded layer, respectively. Likewise, the distribution of the bandwidths for the DoublePeriodic structures with linear and exponential graded layers, considering nm dB ¼ λ0 =4 and up to the fifth generation of the sequence, has been described in Fig. 4(a) and (b), respectively.
Fig. 3. The distribution of the photonic bandwidths as a function of the Thue–Morse generations for structures with (a) linear and (b) exponential graded layers.
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Fig. 4. The distribution of the photonic bandwidths as a function of the Double-Periodic generations for structures with (a) linear and (b) exponential graded layers.
Fig. 5. The total band gap variation with the generations of the sequence (a) Thue–Morse and (b) Double-Periodic photonic crystals with linear and exponential graded layers.
A similar behaviour like the Thue–Morse case i.e. the number of band gap in these structures with both linear and exponential graded layers also present a similar distribution as a function of the generation of the Double-Periodic sequence is found and like above their bandwidth and band region are slightly different. The frequency regions of the stop bands exist at higher frequency point for structures with an exponential graded layer as compared to the structures with a linear graded layer, but bandwidths in case of linear graded layer are slightly large as compared to the exponential graded layer. This fact is clearly demonstrated in Figs. 3 and 4, where we compare the photonic spectra of the Thue–Morse and Double-Periodic multilayer structures with linear and exponential graded layers considered for normal incidence. Finally, the distribution of the total bandwidth variation as a function of the generation of the sequence for the Thue–Morse and Double-Periodic structures with linear and exponential graded layers
is described in Fig. 5(a) and (b), respectively. This figure shows the total bandwidth of the forbidden energy regions increases with the generation of the sequences for the Thue–Morse and DoublePeriodic structures with linear and exponential graded layers, but the total bandwidths are higher for both types of structures with a linear graded layer in compare to the structures with an exponential graded layer for each generation level. In case of the Double-Periodic structures, the total bandwidths variation is in a linear way with generation of sequence, while in case of the Thue–Morse structures variations are not in linear way, and the values of the total bandwidths are high for the Double-Periodic structures with both types of graded layers in comparison to the Thue–Morse structures. These investigations clearly demonstrate that there is an influence of the linear and exponential graded index materials on the existence of localization modes, photonic bandwidth and their band region in the Thue–Morse and Double-Periodic multilayer structures.
B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
4. Conclusion We have investigated the influence of the graded index material on the photonic localization of the quasi-periodic (Thue–Morse and Double-Periodic) graded photonic crystals by using a transfer matrix approach in the region 150–750 THz of the electromagnetic spectrum. The graded layers have a space dispersive refractive index, which vary along the direction perpendicular to the surface of the layer in a linear and exponential fashion. We have calculated the reflection coefficients and band gaps of the quasi-periodic graded photonic crystals, and demonstrated how the photonic localization modes and band gaps in the structures affect the linear and exponential graded layers. We observed that more and more photonic modes and bands appear when the generations of the quasiperiodic sequences become sufficiently large. On the other hand, the position of the modes and bands affect with the graded layer, and the influence of the graded layer on the frequency region of the band gaps in the structures are slight, while refractive index at the initial and final boundary in the linear and exponential graded layers are same. Furthermore, we have compared the variations of the total band gaps with generations of the sequences for the Thue–Morse and Double-Periodic photonic crystals, in which one of the layers has a linear or exponential graded layer. We expect our results can provide the basic understanding of the influence of the graded profile on the photonic modes and band gaps in the Thue–Morse and Double-Periodic photonic crystals, and facilitate the design of the variation filters, sensors and other photonic devices. Acknowledgement The authors are thankful to Prof. S.P. Ojha and Dr. K.B. Thapa for their valuable suggestions and remarks in this work. This work has been partially supported by the Department of Science & Technology (DST), India in the form of Project Grant no. 100/IFD/2489/2011-12. References [1] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376 (2003) 225. [2] Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, Y. Silberberg, Phys. Rev. Lett. 103 (2009) 013901.
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[3] L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H.E. Beere, D.A. Ritchie, D.S. Wiersma, Nat. Photonics 4 (2010) 165. [4] W. Steurer, S. Widmer, J. Phys. D: Appl. Phys. 40 (2009) R229. [5] A.N. Poddubny, E.L. Lvchenko, Physica E 42 (2010) 1871. [6] L.D. Negro, J.H. Yi, V. Nguyen, Y. Yi, J. Michel, L.C. Kimerling, Appl. Phys. Lett. 86 (2005) 261905. [7] M. Florescu, S. Torquato, P.J. Steinhardt, Proc. Natl. Acad. Sci. 106 (2009) 20658. [8] Z.V. Vardeny, A. Nahata, A. Agrawal, Nat. Photonics 7 (2013) 177. [9] L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, M. Segev, Science 332 (2011) 1541. [10] L. Jia, I. Bita, E.L. Thomas, Adv. Funct. Mater. 22 (2012) 1150. [11] M.S. Li, A. Ying-Guey Fuh, Shing-Trong Wu, Opt. Lett. 37 (2012) 3249. [12] D. Luo, Q.G. Du, H.T. Dai, H.V. Demir, H.Z. Yang, W. Ji, X.W. Sun, Sci. Rep. 2 (2012) 627. [13] E. Macia, Rep. Prog. Phys. 75 (2012) 036502. [14] M. Ghulinyan, C.J. Oton, L.D. Negro, L. Pavesi, R. Sapienza, M. Colocci, D.S. Wiersma, Phys. Rev. B 71 (2005) 094204. [15] I.P. Coelho, M.S. Vasconcelos, C.G. Bezerra, Solid State Commun. 150 (2010) 1760. [16] Y. Trabelsi, N. Benali, Y. Bouazzi, M. Kanzari, Photonic Sens. 3 (2013) 246. [17] Z.F. Sang, Z. Ya Li, Opt. Commun. 259 (2006) 174. [18] P.C. Pandey, K.B. Thapa, S.P. Ojha, Opt. Commun. 281 (2008) 1607. [19] H. Rauh, G.I. Yampolskaya, S.V. Yampolskii, New J. Phys. 12 (2010) 073033. [20] H. Rauh, G.I. Yampolskaya, S.V. Yampolskii, J. Opt. 14 (2012) 015101. [21] B.K. Singh, P.C. Pandey, J. Mod. Opt. 61 (2014) 887. [22] E. Centeno, D. Cassagne, Opt. Lett. 30 (2005) 2278. [23] H. Kurt, D.S. Citrin, Opt. Express 15 (2007) 1240. [24] S.C.S. Lin, T.J. Huang, J.H. Sun, T.T. Wu, Phys. Rev. B 79 (2009) 094302. [25] M. Lu, B.K. Juluri, S.C.S. Lin, B. Kiraly, T. Gao, T.J. Huang, J. Appl. Phys. 108 (2010) 103505. [26] B.B. Oner, M. Turduev, H. Kurt, Opt. Lett. 38 (2013) 1688. [27] D. Luo, G. Alagappan, X.W. Sun, Z. Raszewski, J.P. Ning, Opt. Commun. 282 (2009) 329. [28] B. Vasic, R. Gajic, J. Appl. Phys. 110 (2011) 053103. [29] F. Gaufillet, E. Akmansoy, Opt. Commun. 285 (2012) 2638. [30] H.W. Wang, L.W. Chen, J. Appl. Phys. 109 (2011) 103104. [31] B.K. Singh, K.B. Thapa, P.C. Pandey, Opt. Commun. 297 (2013) 65. [32] P. Yeh, Optical Wave in Layered Media, John Wiley & Sons, Interscience, New Jersey, 1988.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Influence of graded index materials on the photonic localization in one-dimensional quasiperiodic (Thue–Mosre and Double-Periodic) photonic crystals Bipin K. Singh, Praveen C. Pandey n Department of Physics, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, Uttar Pradesh, India
art ic l e i nf o
a b s t r a c t
Article history: Received 31 May 2014 Received in revised form 7 July 2014 Accepted 12 July 2014 Available online 25 July 2014
In this paper, we present the investigation on the photonic localization and band gaps in quasi-periodic photonic crystals containing graded index materials using a transfer matrix method in region 150–750 THz of the electromagnetic spectrum. The graded layers have a space dispersive refractive index, which vary in a linear and exponential fashion as a function of the depth of layer. The considered quasiperiodic structures are taken in the form of Thue–Morse and Double-Periodic sequences. The grading profile in the layers affects the position of reflection dips and forbidden bands, and frequency region of the bands. We observed that vast number of forbidden band gaps and dips are developed in its reflection spectra by increasing the number of quasi-periodic generation. Moreover, we compare the total forbidden bandwidths with increasing the generation of the quasi-periodic sequences for the structures with linear and exponential graded layer. Results show that the different graded profiles with same boundary refractive index can change the position of localization modes, number of photonic bands and change the frequency region of the bands. Therefore, we can achieve suitable photonic band gaps and modes by choosing the different gradation profiles of the refractive index and generation of the quasi-periodic sequences. & 2014 Elsevier B.V. All rights reserved.
Keywords: Photonic crystals Quasiperiodic sequence Photonic band gap Graded index material
1. Introduction In the past two decades, great efforts have been dedicated towards the investigation of the structure and physical properties of quasiperiodic systems after the discovery of the quasi-crystalline structure [1–3]. Quasi-periodic systems have long-range order but not in a repeating fashion yielding periodicity. These arrangements are fixed in a regular pattern and follow a simple deterministic recursion rule [4,5]. Quasi-periodic systems are one of the most interesting arrangements to obtain the suitable photonic band gaps because of several structural parameters available to tune as compared to the periodic and disordered systems [6–8]. Recently, some research groups have reported their works on electromagnetic (EM) wave propagation in quasi-periodic structures called photonic quasi-crystals. Due to a longrange order, such type of structures provide wide photonic band gap in photonic spectra as in periodic photonic crystals and simultaneously possess localized states as in disorder media [9,10]. Photonic quasicrystals exhibit unique influence on the optical properties such as optical transmission and reflectivity, photoluminescence, light transport, plasmonics and laser action, etc. Li et al. [11] and Luo et al. [12]
n
Corresponding author. Tel.: þ 91 5426702008; fax: þ 91 5422368428. E-mail address: [email protected] (P.C. Pandey).
http://dx.doi.org/10.1016/j.optcom.2014.07.043 0030-4018/& 2014 Elsevier B.V. All rights reserved.
proposed two-dimensional photonic crystals that achieve multimode lasing action, low pumping threshold and excellent linear polarization property as well as wide directional dependence. This opens a new field of research in photonics in view of their vast technical applications. Photonic band gap properties of quasi-periodic multi-layered structures have been extensively studied for different materials [13]. Specifically, one-dimensional (1-D) photonic quasi-crystals are very important because their formation is relatively easy and they may provide the description of light propagation in one direction [14–16]. One-dimensional photonic quasi-crystals are composed of layers according to substitutional sequences in form of the Fibonacci, Thue–Morse and Double-Periodic etc. Recently several researchers have been proposed the 1-D multilayer structures with gradual varying RI as a function of the depth of layer, and width of layers varies as a gradual fashion along the direction perpendicular to the surface of layer in the considered structures [17–21]. Such type of structures are called graded photonic crystals (GPCs). In two-dimensional GPC structures, gradual variations of the relative parameters can be distributed along the normal or perpendicular to the propagation of electromagnetic waves. Gradual variation of relative parameters of GPCs make them very different in the behaviour from the conventional PCs and enhance the ability to mold and control of the light wave propagation [22–24]. Such types of PCs play an important role to
B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
design spectral filters, beam aperture and deflector, high efficiency bending waveguides, high efficiency couplers, super bending and self-focusing media, lenses, artificial optical black holes and antireflection coating [25–30]. In our previous work [31], we have formulated the resonant Bragg condition for the quasiperiodic Fibonacci multilayer structures containing exponential graded material and shown that forbidden band gaps and omnidirectional band gaps to be obtained for periodicity of different generation Fibonacci sequence structures and their hetrostructures. In these structures, bandwidth of forbidden and omnidirectional band gaps can be tuned with graded profile parameter of exponential graded layer. Motivated by the ability to mold, confine and control of the electromagnetic waves by different types of GPCs, we now present the study of the photonic band gap characteristics of 1-D quasiperiodic GPCs constituted with an exponential/linear graded dielectric layer. The aims of this work are, we want to show the reflection spectra, which arise from the propagation of electromagnetic waves in quasi-periodic multilayer structures, comprised of alternating layers of both normal (SiO2) and graded index materials using a theoretical model based on a transfer matrix treatment. The quasi-periodic structures follow the Thue–Morse (TM) and Double-Periodic (DP) substitutional sequences and can be generated by the following inflation rules: A-AB, B-BA (TM); and A-AB, B-AA (DP),[4,5] where A and B are the building blocks modelling of the normal and graded index materials, respectively. Further, we intend to present a quantitative analysis of the results, pointing out the distribution of the forbidden band gaps and total bandwidths for up to the 6th generation, which gives a good insight about their photonic band gaps and localization. The plan of this paper is as follows. In Section 2, we present the method of calculation employed here, which is based on the transfer matrix approach. The reflection coefficient and dispersion relation is then determined. Section 3 is devoted to the discussion of this reflection spectra and dispersion relation for the Thue– Morse and Double-Periodic multilayer structures containing linear and exponential graded index material. Further, we present their total band gap with Thue–Morse and Double-Periodic generations for linear and exponential graded index material as one of the layer. The conclusions of this work are presented in Section 4.
2. Theoretical model and numerical analysis In this paper, we consider the system of multilayer that is composed of two layers and stacked alternatively along the x-direction. The stacks of two layers are arranged according to the recursion rule of the Thue–Morse (TM) and Double-Periodic (DP) sequence in different generations [4,5]. These sequences are based on the two-letter alphabet (A, B) and the substitution rule: σ (A) ¼AB, σ (B) ¼BA (TM); and σ (A)¼ AB, σ (B) ¼AA (DP). The substitution rule can be written in the form of the following equations: A 1 1 A AB σ: ¼ and B 1 1 B BA A 1 1 A AB σ: ¼ B 2 0 B AA The ratio of the frequencies of the letters A, B in the sequence is equal for TM and 2 to 1 for DP. The length of the sequence in both cases at the iteration n is 2n. On the basis of the above substitution rule, the first few words generated in this way are represented in Table 1 panels (i) and (ii). The proposed multilayer structures consist of two kinds of layers; one has a constant refractive index (1.5) and other has a
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linear or exponential varying refractive index (1.5–4.1) as the case may be. The variation of refractive index in the graded layers is taken along the direction of the thickness of the layer. The direction of wave propagation is considered along the x-axis i.e. the direction normal to the stacked layers and the considered materials assumed as non-magnetic, non-dispersive and isotropic. The refractive indices of the considered graded layers vary in linear and exponential fashion between the initial and final values as ½nL ðxÞ ¼ ni þ ðnf ni Þ=dL x and ½nE ðxÞ ¼ ni expðx=dE log nf =ni Þ, respectively. Here, ni and nf are the refractive indices at inward and outward boundaries of the graded layer, respectively [32]. The optical properties of the considered multi-layered structures are described by well-known theoretical model based on transfer matrix method (TMM). Transfer matrices are generated by applying boundary conditions in a plane wave solution of the Maxwell's wave equation at the interface boundary. The electric field distribution E(x) in different materials can be written as: (i) For normal layers: EN ðxÞ ¼ AN expð i UkN U xÞ þ BN expði U kN U xÞ………:
ð1Þ
where AN and BN are arbitrary constants and kN ¼ ω U nN =c represents the propagation wave vector at normal incidence with a constant refractive index nN . Subscript N represents the normal layer and ω and c are the angular frequency and velocity of light, respectively [30]. (ii) For linear graded layers the electric field equation can be written as " ! !# pffiffiffiffiffi ξ2 ξ2 EL ðxÞ ¼ ξL U AL J 1=4 L þ BL Y 1=4 L ð2Þ 2α 2α where, AL and BL are arbitrary constants and ξL ¼ ω UnL ðxÞ=c the propagation wave vector at normal incidence along xdirection for the linear graded layer with refractive index: nL ðxÞ ¼ ni þ ðnf ni Þ=dL Þ x, where ni and nf are the refractive indices at the initial and final boundary and dL is the layer thickness. Subscript L represents the linear graded layer and grading profile parameter of the linearly graded layer is α ¼ ðω=cÞðnf ni =dL Þ, ω and c are the angular frequency and velocity of light, respectively [32]. (iii) For exponential graded layers: ξ ξ ð3Þ EE ðxÞ ¼ AE J 0 E þ BE Y 0 E ………: γ γ where, AE and BE are arbitrary constants and ξE ¼ ω:nE ðxÞ=c represents the wave propagation vector at normal incidence along the x-direction for the exponential graded layer with refractive index nE ðxÞ ¼ ni expðγxÞ, where γ ¼ 1=dE log ðnf =ni Þ; is the grading profile parameter of the exponentially graded layer, ni and nf are same as defined in Eq. (2) and dE is the thickness of the exponentially graded layer. Subscript E represents the exponentially graded layer. The functions J and Y are the first and second kind Bessel's functions, respectively.
Using the transfer matrix approach, the amplitudes A0 and B0 of the electromagnetic field in the air medium at x o0 related to be the amplitudes An þ 1 and Bn þ 1 of the equivalent layer in the (n þ1) th region through the linear transformation. Therefore, for the multilayer structures, the total transfer matrix equation can be written as A0 B0
! ¼ M i;j
An þ 1 Bn þ 1
! ð4Þ
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B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
Table 1 Quasi-periodic multilayer arrangement for Thue–Morse and Double–Periodic sequences. (i) Generation of the words Tn ¼ σn (A) of the Thue–Morse sequence by the repeated action of the substitution rule σ (A)¼ AB, σ (B) ¼BA. Thue–Morse sequence
T n þ 1 ¼ T n T n ; T n þ 1 ¼ T n T n , with T 0 ¼ A and T 0 ¼ B
Generation number
T0 T1 T2 T3 T4 T5 … …
A AB ABBA ABBABAAB ABBABAABBAABABBA ABBABAABBAABABBABAABABBAABBABAAB …………………………………. ………………………………….
0 1 2 3 4 5 … So on.
(ii) Generation of the words Dn ¼ σn (A) of the Double-Periodic sequence by the repeated action of the substitution rule σ (A)¼ AB, σ (B) ¼AA. Double-periodic sequence
Dn þ 1 ¼ Dn Dn ; Dn þ 1 ¼ Dn Dn , with D0 ¼ A and D0 ¼ B
Generation number
D0 D1 D2 D3 D4 D5 … …
A AB ABAA ABAAABAB ABAAABABABAAABAA ABAAABABABAAABAAABAAABABABAAABAB …………………………………. ………………………………….
0 1 2 3 4 5 … So on.
where M i;j ði; j ¼ 1; 2Þ is the total characteristic matrix and if atmosphere around the system is air, M i;j ¼ M 0 1 ðM 1 M 2 M 3 … M n ÞM 0 for the n layer system and M n is the characteristic matrix of nth layer. These layers may be normal, linear or exponential graded material layer. After the simplification, the characteristic matrix for the different type of layer at the normal angle of incidence can be written as (i) For a normal (homogeneous) layer: " # sin ðkN dN Þ=nN cos ðkN dN Þ MN ¼ nN sin ðkN dN Þ cos ðkN dN Þ where M N and dN are the 2 2 characteristic matrix and thickness of normal layer, respectively. (ii) For a linear graded layer: 2 qffiffiffiffi 3 ni cα ω pffiffiffiffiffiffiffiffiffiffi n L2 n n L1 þ i f f 6 2ω nf 7 c 6 7
6 7 6 ωc pffiffiffiffiffiffiffiffiffiffiffi 7 n L1 n i f 6 7 6 7 q ffiffiffi ffi q ffiffiffi ffi M L ¼ Δ6 c3 α2 1 7 p ffiffiffiffiffiffiffiffiffiffi nf n cα cα ω i 6 4ω3 pffiffiffiffiffiffiffiffiL1 þ 2ωnf n L2 þ 2ωni n L3 þ c 3 ni nf L4 7 ni nf i f 6 7 6 7 6 7 qffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi 4 cα ni L1 ωn n n L3 5 i f 2ω nf c i
L3 ¼
L3 ¼
8 < :
0
J1
8 < :
4
0
J1 4
ξ2L 2α
!
ξ2L 2α
ξ2L 2α
Y1 x¼0
4
! 0
Y1 x¼0
4
!
ξ2L 2α
J1 x ¼ dL
ξ2L 2α
4
! 0
J1 x ¼ dL
4
!
ξ2L 2α
0
Y1 x ¼ dL
4
! 0
Y1 x ¼ dL
4
ξ2L 2α
9 =
!
ξ2L 2α
x¼0
;
! x¼0
9 = ; ;
M L and dL are the 2 2 characteristic matrix and thickness of the linearly graded layer, respectively, and prime mark represents the differentiation of the relative functions. (iii) For an exponential graded layer: 3 2 nf J 0 ξγE Y 1 ξγE Y 0 ξγE J 1 ξγE 7 6 x¼0 x ¼ dE x¼0 x ¼ dE 7 6 7 6 7 6 – J ξE ξE ξE ξE Y0 γ Y0 γ J0 γ 7 6 0 γ x¼0 x ¼ dE x¼0 x ¼ dE 7 6 7 6 M E ¼ Ω6 7 ξE ξE ξE 7 6 ni nf J ξE Y1 γ Y1 γ J1 γ 1 γ 7 6 x ¼ 0 x ¼ d x ¼ 0 x ¼ d E E 7 6 7 6 5 4 Y 0 ξγE Y 1 ξγE J 0 ξγE ni J 1 ξγE x¼0
where Ω ¼ 1=nf
x ¼ dE
x¼0
x ¼ dE
( ) ξ ξ ξ ξ J1 E Y0 E Y1 E J0 E ; γ x ¼ dE γ x ¼ dE γ x ¼ dE γ x ¼ dE
M L and dE are the 2 2 characteristic matrix and thickness of exponential graded layer, respectively.
where 8 ! ω < ξ2 Δ ¼ 1= n2f J 1 L c : 4 2α
8 ! < ξ2 L1 ¼ J 1 L : 4 2α
4
x ¼ dL
x¼0
8 ! < ξ2 L2 ¼ J 1 L : 4 2α
Y 01
x¼0
Y1 4
0
Y1 4
ξ2L 2α
ξ2L
! 4
x ¼ dL
! J1
2α ξ2L 2α
J 01
x ¼ dL
4
! 0
J1 x ¼ dL
4
ξ2L 2α
ξ2L 2α ξ2L 2α
!
ξ2L 2α
Y1 x ¼ dL
4
! Y1 x ¼ dL
4
! Y1 x ¼ dL
4
! x ¼ dL
ξ2L
!
2α ξ2L 2α
9 = ; ;
x¼0
! x¼0
9 = ; ;
9 = ;
;
Thus the transmittance (T) and reflectance (R) coefficients of electromagnetic wave can be written as T ¼ j1=M 11 j2 and R ¼ jM 21 =M 11 j2
;
ð5Þ
where M 11 and M 21 represent the elements of the optical transfer matrix M i;j ði; j ¼ 1; 2Þ. Naturally due to our consideration of lossless dielectric material, the transmittance here is just the reflectance's complement. According to Floquet's theorem, the electric field vector of a normal mode of propagation in a periodic medium is of the form
B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
as: EK ðx; zÞ ¼ EK ðxÞ U e i U β:z U e– i:K U x , where EK ðxÞ is a periodic function with period d, i.e. EK ðx þ dÞ ¼ EK ðxÞ and β is the z-component of the wave vector. The subscript K represents that the function EK ðxÞ depends on K. This optical wave is known as a Bloch mode and the parameter K, which satisfies the mode and its associated periodic function EK ðxÞ, is called the Bloch wave number. The dispersion is an equation relating the Bloch wave number K and the angular frequency ω and expressed in the form Kðβ; ωÞ ¼
1 1 cos 1 ðM 11 þ M 22 Þ d 2
ð6Þ
where d is the total thickness of a period of the periodic system, M 11 and M 22 is the elements of the optical transfer matrix M i;j ði; j ¼ 1; 2Þ.
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The dispersion relation exhibits multiple spectral bands classified into two regimes: first, where jðM 11 þ M 22 Þ=2j r 1 corresponds to real K and thus to propagating Bloch waves. Second, spectral bands within which K is complex corresponds to evanescent waves that are rapidly attenuated. Defined by the condition jðM 11 þ M 22 Þ=2j4 1, these bands correspond to the stop bands also called photonic band gaps/forbidden gaps since propagating modes do not exist for the systems [32].
3. Results and discussion In this section, we have presented the results obtained from the numerical calculations and the discussion on the reflection spectra of light for different 1-D quasi-periodic (Thue–Morse and
Fig. 1. Reflection coefficient (R) versus frequency for (a) 5th, (b) 6th and (c) 7th generation sequence of the Thue–Morse photonic crystals with linear and exponential graded layers.
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B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
Double-Periodic) type multilayer structures. The unit cells of considered 1-D quasi-periodic photonic structures are stacked of two types of layers namely, A and B, where ‘A’ type layers represent constant refractive index (Normal) material of value 1.5 and ‘B’ type layers represent linear and exponential graded index material having refractive index variation from 1.5 to 4.5. The considered 1-D quasi-periodic multilayer structures using layers A and B are obtained according to the substitution rules of the Thue–Morse and Double–Periodic sequences, as tabulated in Table 1. We have calculated the reflectance and band spectrum of light incident normally through the air on the considered multilayer structures. These materials are considered lossless, non-
magnetic and their refractive index does not depend on frequencies. Also, we depicted the variation of total photonic band gap for 1-D quasi-periodic photonic structures as a function of the generation of Thue–Morse and Double-Periodic system with linear and exponential graded index material. We first present the reflection spectra of 1-D Thue–Morse quasi-periodic photonic structures for the fifth, sixth and seventh generation with linear and exponential graded index material as layer B. Thicknesses of the layer A and B are chosen in terms of quarter wave stacking i.e. nA dA ¼ nm dB ¼ λ0 =4 where dA and dB are the thickness of layer A and B, respectively, and nm is the mean value of the initial refractive index and final refractive index of the
Fig. 2. Reflection coefficient (R) versus frequency for (a) 5th, (b) 6th and (c) 7th generation sequence of the Double-Periodic photonic crystals with linear and exponential graded layers.
B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
graded layer. These types of stacking structures are very useful and suitable for designing various photonic devices because they provide large photonic band gap. The physical parameters used here are nA ¼ 1:5; nm ¼ 3:0 and wavelength λ0 is the mean value of the considered frequency region (150–750 THz). Analysing results shown in Fig. 1, we get number of localization modes and forbidden bands that increases with increasing the generation of Thue–Morse system for both linear and exponential graded index material. From all these figures, it is clearly visible that localization modes and forbidden bands exist in higher frequency region for exponential graded layer, while for the similar structures with linear graded layers position of localization modes and forbidden bands shifted towards lower frequency region. The influence of the different graded materials on the localization modes and band gaps are sighted on position and width of the band gaps whereas the numbers of localization modes and forbidden bands are same. Likewise, we have studied the influence of graded index materials on the localization modes and forbidden bands for 1-D Double-Periodic photonic structures for Fifth, sixth and seventh generation of quasi-periodic systems with linear and exponential graded index material under normal incidence of light. In these structures, composition of the layer arrangement has become different, but other parameters like thickness, total number of layer of the structures are same with respect to same generation of Thue–Morse quasi-periodic structures. Reflection spectra of these structures are shown in Fig. 2. The numbers of localization modes and forbidden band increases with the increase of the generation of Double-Periodic quasi-periodic system for both linear and exponential graded index material as layer B, and shift of the localization mode and forbidden bands is similar to that in 1-D quasi-periodic photonic structures with Thue–Morse system i.e. localization modes and forbidden bands exist at related higher and lower frequency regions for the structures with an exponential and linear graded layer, respectively. Conversely, one broader forbidden band always exists around the central frequency region for all these structures, while bands exist at lower and higher frequency region for the structures with Thue–Morse generations. Also, large number of modes and bands exist for these structures as compared to the same generation Thue–Morse based structures. In our configurations the Figs. 1 and 2, clearly demonstrated that the number of localization modes and forbidden bands of the considered PC structures depends on the generation of the quasiperiodic systems, and their position and bandwidth depend on the
89
material properties and geometrical parameters. The results also show that, the position of modes and bands exist at higher frequency for structures with exponential graded layer as compared to the structures with the linear graded layer, because the rate of change of the refractive index contrast in the exponential graded layer is slightly higher in comparison to the linear graded layer. On the other hand, the refractive index boundary is same in both the type of graded layers and hence the change in the initial to final optical path length is approximately same; therefore due to this fact, structures have equivalent number of modes and bands. Furthermore, we have examined the photonic band gap spectrum of the periodic super lattice of different quasi-periodic (Thue–Morse and Double-Periodic) generations arising from competition between the linear and exponential graded layer, and depicted in panels (a) and (b) of Figs. 3 and 4, respectively. For this analysis, we have calculated the forbidden frequencies (stop bands), where jðM 11 þ M 22 Þ=2j 41, as a function of the generation of the quasi-periodic (Thue–Morse and Double-Periodic) sequence for a fixed value of nm dB ¼ λ0 =4, as shown in Fig. 3 (Thue–Morse sequence) and Fig. 4 (Double-Periodic sequence). The comparison of the photonic spectra of the Thue–Morse multilayer structures with linear and exponential graded layer for generation levels from 1 to 6 at normal incidence has been presented in Fig. 3. Fig. 3(a) and (b) shows the distribution of the forbidden (black region) and allowed (white region) frequencies, as a function of the generation of the Thue–Morse sequence of the structures with a linear and exponential graded layer, respectively up to the 6th generation of the Thue–Morse sequence, considering nm dB ¼ λ0 =4. As expected, for large number of generations, we get large number of stop bands and their bandwidth becomes narrower and narrower as an indication of more localized modes. We also observed that number of band gap in the structures with both linear and exponential graded layer presents a similar distribution as a function of the generation of the Thue– Morse sequence, but their bandwidth and frequency region are slightly different. This is illustrated in Fig. 3(a) and (b) for a fixed value nm dB ¼ λ0 =4 in the structures with linear and exponential graded layer, respectively. Likewise, the distribution of the bandwidths for the DoublePeriodic structures with linear and exponential graded layers, considering nm dB ¼ λ0 =4 and up to the fifth generation of the sequence, has been described in Fig. 4(a) and (b), respectively.
Fig. 3. The distribution of the photonic bandwidths as a function of the Thue–Morse generations for structures with (a) linear and (b) exponential graded layers.
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B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
Fig. 4. The distribution of the photonic bandwidths as a function of the Double-Periodic generations for structures with (a) linear and (b) exponential graded layers.
Fig. 5. The total band gap variation with the generations of the sequence (a) Thue–Morse and (b) Double-Periodic photonic crystals with linear and exponential graded layers.
A similar behaviour like the Thue–Morse case i.e. the number of band gap in these structures with both linear and exponential graded layers also present a similar distribution as a function of the generation of the Double-Periodic sequence is found and like above their bandwidth and band region are slightly different. The frequency regions of the stop bands exist at higher frequency point for structures with an exponential graded layer as compared to the structures with a linear graded layer, but bandwidths in case of linear graded layer are slightly large as compared to the exponential graded layer. This fact is clearly demonstrated in Figs. 3 and 4, where we compare the photonic spectra of the Thue–Morse and Double-Periodic multilayer structures with linear and exponential graded layers considered for normal incidence. Finally, the distribution of the total bandwidth variation as a function of the generation of the sequence for the Thue–Morse and Double-Periodic structures with linear and exponential graded layers
is described in Fig. 5(a) and (b), respectively. This figure shows the total bandwidth of the forbidden energy regions increases with the generation of the sequences for the Thue–Morse and DoublePeriodic structures with linear and exponential graded layers, but the total bandwidths are higher for both types of structures with a linear graded layer in compare to the structures with an exponential graded layer for each generation level. In case of the Double-Periodic structures, the total bandwidths variation is in a linear way with generation of sequence, while in case of the Thue–Morse structures variations are not in linear way, and the values of the total bandwidths are high for the Double-Periodic structures with both types of graded layers in comparison to the Thue–Morse structures. These investigations clearly demonstrate that there is an influence of the linear and exponential graded index materials on the existence of localization modes, photonic bandwidth and their band region in the Thue–Morse and Double-Periodic multilayer structures.
B.K. Singh, P.C. Pandey / Optics Communications 333 (2014) 84–91
4. Conclusion We have investigated the influence of the graded index material on the photonic localization of the quasi-periodic (Thue–Morse and Double-Periodic) graded photonic crystals by using a transfer matrix approach in the region 150–750 THz of the electromagnetic spectrum. The graded layers have a space dispersive refractive index, which vary along the direction perpendicular to the surface of the layer in a linear and exponential fashion. We have calculated the reflection coefficients and band gaps of the quasi-periodic graded photonic crystals, and demonstrated how the photonic localization modes and band gaps in the structures affect the linear and exponential graded layers. We observed that more and more photonic modes and bands appear when the generations of the quasiperiodic sequences become sufficiently large. On the other hand, the position of the modes and bands affect with the graded layer, and the influence of the graded layer on the frequency region of the band gaps in the structures are slight, while refractive index at the initial and final boundary in the linear and exponential graded layers are same. Furthermore, we have compared the variations of the total band gaps with generations of the sequences for the Thue–Morse and Double-Periodic photonic crystals, in which one of the layers has a linear or exponential graded layer. We expect our results can provide the basic understanding of the influence of the graded profile on the photonic modes and band gaps in the Thue–Morse and Double-Periodic photonic crystals, and facilitate the design of the variation filters, sensors and other photonic devices. Acknowledgement The authors are thankful to Prof. S.P. Ojha and Dr. K.B. Thapa for their valuable suggestions and remarks in this work. This work has been partially supported by the Department of Science & Technology (DST), India in the form of Project Grant no. 100/IFD/2489/2011-12. References [1] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376 (2003) 225. [2] Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, Y. Silberberg, Phys. Rev. Lett. 103 (2009) 013901.
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