- Email: [email protected]
Electro-optical tenability properties of defective one-dimensional photonic crystal
Optik 145 (2017) 121–129
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Electro-optical tenability properties of defective one-dimensional photonic crystal Ashour M. Ahmed a,∗ , Mohamed Shaban a , Arafa H. Aly b a b
Nanophotonics and Applications (NPA) Lab, Physics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt Physics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt
a r t i c l e
i n f o
Article history: Received 13 April 2017 Accepted 6 July 2017 Keywords: Electro-optic effect Photonic crystal Defect mode Reflectance spectra Transfer matrix method
a b s t r a c t In this work, the optical transfer matrix formalism was applied to study the linear electrooptic effect on perfect and defective one-dimensional photonic crystal (1D-PC) composed of lithium niobate and polymer (polystyrene) multilayers. The effects of an external electrical field on the reflectance spectra of the proposed PCs were studied as a function of the angle of incidence and the types of polarization (TE and TM waves). The results show that, at a constant angle of the incident, the PBGs are shifted to shorter wavelengths for both TE and TM waves as the applied electric field increased. The position of defect mode decreases linearly from 1.401 to 1.379 m for TM wave and from 1.398 to 1.377 m for TE wave as the applied electric field increased from 0 to 100 v/m at an angle of incident 50◦ . By increasing the angle of the incident from 0 to 50 ◦ at constant applied electric field, the position of PBG for both TE and TM are shifted toward the shorter wavelengths. In addition, the width of the PBGs of TE wave is increased from 0.334 to 0.358 m and the width of the PBGs of TM wave is decreased from 0.334 to 0.248 m at 100 v/m. This study may be valuable for designing tunable photonic devices with driving electrical field, which has potential applications in the area of photonics and optoelectronics. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals (PCs) are periodic structures with alternating refractive indices in one, two, or three dimensions with periods comparable to the wavelength of the incident light [1]. In recent years, the studies of the electromagnetic propagation in the photonic crystals have become intense research area due to their novel applications in modern optical devices. PCs can be used to control the propagation of light by the appearance of a photonic band gap (PBG) and photon localization [2,3]. PBG is wavelength interval in which electromagnetic waves cannot propagate through the PC because the waves will undergo exponential decay (evanescent mode) and total reflection. The photon localization can be derived inside the PBG by adding a defect layer into the PC to break the periodic feature [4]. When the incident photon wavelength is equal to the defect-state wavelength, the photon will be localized in the place of the defect and a resonant transmission peak is generated within the PBG. The PBG and defect mode are strongly depending on some parameters such as index contrast, layer thickness, the state of polarization, filling fraction, and angle of incidence [5–7]. Hence, any variation in the refractive index of the PC can produce a significant modulation in the position of the defective mode. This modulation can be used as an indicator for measuring the index change.
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (A.M. Ahmed). http://dx.doi.org/10.1016/j.ijleo.2017.07.025 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
122
A.M. Ahmed et al. / Optik 145 (2017) 121–129
Fig. 1. Schematic diagram of the proposed 1D-PC with a defect layer under the effect of an external electric field.
Table 1 The values of the different parameters that used in the calculations. LiNbO3 (Material 1)
Refractive index ( in m)
n1 = −6
Electro-optic coefficient (10 Thickness (m)
m/V)
A+
B 2
− C2
␥1 = 30.9 [23] d1 = 0.1813
Polymer (Material 2) +
D 2
− H2
A = 5.35583; 2 − F C = 0.20692; H = 11.34927;
B = 0.100473; [20] D = 100; F = 153.34;
n2 = 1.6 [21,22] ␥2 = 150 [21,22] d2 = 0.2421
On the other side, certain materials can change their refractive indices under the exposure to external physical signals such as electrical, magnetic, thermal, acoustic, or mechanical signals. The electro-optic (EO) effect is a change in the optical properties of the material, especially its refractive index, in response to an external electric field. This effect occurs due electric field causes a redistribution of bond charges and possibly a slight deformation of the crystal lattice [8]. Therefore, the inverse dielectric constant (impermeability) tensor changes accordingly. Then, the EO effect can be used for the dynamic control of the refractive index. Also, the EO materials play a significant role in many practical applications such as optical communication systems, charge storage, optical sensors, electroluminescent devices, and optical computers [9–11]. Lithium Niobate (LN) and polymers (such as polystyrene) are typical electro-optic materials. The LN is a nonlinear ferroelectric material widely used in optoelectronics and surface acoustic wave devices [12,13]. Then, the design of LN photonic crystals can potentially decrease the size of the critical components in many optical systems [14]. Moreover, the EO polymers gained an increased interest for telecommunication applications due to their low optical losses in the 1.3 and 1.55 m telecommunication windows. Also, these polymers are characterized by high linear EO, low processing cost, and reliable mass production [15]. Therefore, it is imperative to study the effect of the external electric field on the PBG and photon localization of a PC from LN and a polymer such as a polystyrene. Most previous studies were focused on design and fabrication of 2D-PC structures for EO applications by etching hexagonal or square arrays of holes in LN or polymer film [16,17]. However, there are practical limitations for obtaining highly ordered 2D-PC because the difficulty of fabricating deep and cylindrical holes in bulk crystals. Also, PBG structures may be destroyed when a large number of defects or disorders are introduced in the PC [18]. Then 1D-PC structure (such as multilayer films) can be used as a real alternative for 2D-PC because it can be simulated and fabricated more easily than 2D-PC [19]. Moreover, it can be easily adapted for use in many applications. Here, an EO structure based on 1D-PC (LiNbO3 /polymer multilayers) with a defect layer is proposed. Using transfer matrix method (TMM), the reflection at different applied electric fields and incident angles for both transverse electric (TE) and transverse magnetic (TM) modes are calculated and discussed.
2. PC design To have an idea about the effect of the electric field on the optical spectra of PCs, a 1D-PC from alternately stacked LN and polymer layers with refractive indices n1 and n2 respectively is addressed. The LN layer is inserted as a defect layer. The PC is surrounded by air and glass substrate. The modeled structure was based on the following configuration: air/(n1 n2 )N /n2 /(n1 n2 )N /glass with N = 14 period. Also, two electrodes are placed on the sides of the PC to apply an electric field. Schematic diagram of the proposed structure is shown in Fig. 1. The physical thicknesses of LiNbO3 and polymer are taken according to the quarter wave arrangement at the operating wavelength in the infrared region 0 = 1.550 m, which is essential for the optical communication. The period of the lattice is d = d1 + d2 . The values of the different parameters that used in this study are given in Table 1.
A.M. Ahmed et al. / Optik 145 (2017) 121–129
123
3. Modeling of EO-PBG by using transfer matrix method Based on the Maxwell equations and the boundary conditions, the transfer matrix method (TMM) has been widely used to calculate the reflectance spectra of the propagated light waves in a 1D-PC [24–26]. An algorithm in MATLAB is implemented and used to determine the optical reflection of the proposed photonic structures. The characteristic matrix for the first layer, m1, and for the second layer, m2, are given by:
⎡
m1 = ⎣
⎤
⎡
⎤
−i sin1 p1 ⎦ m2 = ⎣
cosˇ1 −ip1 sin1
cosˇ2 −ip2 sin2
cosˇ1
−i sin2 p2 ⎦
(1)
cosˇ2
The phase differences at the first and second layer are given by 1 = 2n1 (E) d1 cos 1 /
2 = 2n2 (E) d2 cos 2 /
(2)
The values of p for the first and second layer are depends on the type of polarization and given by [27]: p1 = n1 (E) cos 1 and p2 = n2 (E) cos 2 for the TE
(3)
p1 = cos 1 /n1 (E) and p2 = cos 2 /n2 (E) for the TM
The angles of incidence 1 and 2 inside the first and second layer are related to the angle of incidence 0 by Snell’s law. n0 sin0 = n1 sin1 = n2 sin2
(4)
The characteristic matrix m for these two layers is given by
m = m1 m2 =
m11
m12
m21
m22
⎡ ⎢
cosˇ1 cosˇ2 −
=⎣
⎤
−i i sinˇ1 cosˇ2 − cosˇ1 sinˇ2 p1 p2 ⎥
p2 sinˇ1 sinˇ2 p1
−i p1 sinˇ1 cosˇ2 − i p2 cosˇ1 sinˇ2
cosˇ1 cosˇ2 −
p1 sinˇ1 sinˇ2 p2
⎦
(5)
The refractive index of EO materials under the applied electric field, n(E), is given by the relation [28]: n (E) = n − n3 ␥E/2
(6)
where, n is the refractive index of the material in the absence of the applied electric filed, ␥ is the linear electro-optic (Pockels) coefficient, and E is the applied electric filed in unit of v/m. The characteristic matrix M for a system of N periods can be obtained as follows:
N
N
M = m = (m1m2) =
M11 M12 M21
M22
=
m11TN−1 (A) − TN−2 (A)
m12TN−1 (A)
m21TN−1 (A)
m22TN−1 (A) − TN−2 (A)
(7)
Where TN (A) represenets the Chebyshev polynomials of the second kind and A is the Bloch phase for a single period. The TN (A) and A are given by
TN (A) = Sin (N + 1) cos−1 A /
1 + A2 and A = 0.5 (m11 + m22)
(8)
The total characteristic matrix, G, for the proposed structure is given by
G = Mm2M =
G11 G12 G21
G22
(9)
The reflectance coefficient of the multilayer is given by r=
(G11 + G12ps ) p0 − (G21 + G22ps ) (G11 + G12ps ) p0 + (G21 + G22ps )
(10)
Where, the values of p of air and glass substrate are given by p0 = n0 (E) cos 0 and ps = ns (E) cos s for the TE p0 = cos 0 /n0 (E) and ps = cos s /ns (E) for the TM
(11)
Then, the reflection for the defective 1D photonic crystal is given by R = |r2 |
(12)
124
A.M. Ahmed et al. / Optik 145 (2017) 121–129
Fig. 2. The reflectance spectrum of perfect 1D-PC and defective 1D-PC.
4. Results and discussion 4.1. Reflectance spectra of perfect 1D-PC and defective 1D-PC Fig. 2 shows the reflectance spectrum of a perfect 1D-PC and defective 1D-PC at normal incidence without the applied of an electric field. It can be seen that the PBG of defective PC is different compared to the perfect PC as shown in this Figure. The PBG of perfect 1D-PC exists in the NIR region with left and right band edges at L = 1.413 m and R = 1.716 m, respectively. The width of this PBG is = R − L = 0.303 m. The PBG of a 1D-PC structure arises due to the light beam divide into two parts at each layer when a light beam incident on the PC [29]. One part of the light will be transmitted, and another part will be reflected each time the light reaches the interface between the LN and polymer. There are multiple reflections arise from the periodicity of PC. The reflected waves can interfere constructively or destructively. Whether the multiple reflected beams are in phase or not, the phase difference between each succeeding reflection is given by Eq. (2). If the partially reflected waves are in phase and superimposed, then the incident waves are totally reflected, and zero transmission is obtained. The range of the wavelengths in which incident light cannot pass through the PC is called photonic band gap (PBG). On the other hand, the L of the PBG of a defective PC is slightly shifted to the short wavelength region whereas the R is shifted to the long wavelength region compared to that of the perfect PC. Then, the bandwidth of the defective PBG is increased to 0.329 m at the normal incidence. In addition, Fig. 2 shows a super narrow defective mode in the PBG at the selected wavelength (1.55 m). The defect mode divided the PBG into two parts. This may be ascribed to the constructive interference of incident and reflected waves in the defect layer [30]. The reflectivity of the defective mode is very low (R0 = 6.4%), so it can be considered as a resonant transmission mode in PBG at the selected wavelength. Also, the reflectance response outside the PBG shows some fluctuations due to the existence of the defect layer. The number of the ripples in the reflectance spectrum for the perfect 1D-PC is twice that of the defective 1D-PC. The reflection intensities of these ripples for the perfect 1D-PC are smaller than for that of the defective 1D-PC. The defect state can be significantly applied in many applications such as isolators, waveguides, filters, laser, a light-emitting diode, multiplex, PC fiber, splitters, sensors, and switches [31–33]. 4.2. Effect of an external electric field on optical properties of 1D-PC 4.2.1. Effect of the applied electric field on the refractive indices The effect of the applied electric field on the refractive indices of materials have been investigated at = 1.550 m. From Fig. 3, the refractive indices of the LN and polymer are deceased but the contrast between them is increased as the applied electric field increased due to the Pockels coefficient of the polymer is higher than that of LN. 4.2.2. Effect of applied external electric field for TE wave Fig. 4 shows the effect of applied electric field on the reflectance spectra for TE wave of the defective 1D-PC at two different incident angles, 0◦ and 50◦ . The analysis of the optical reflection was summarized in Fig. 5 for two angles of incidence (0◦ and 50◦ ). At a constant angle of incidence, the PBG edges are shifted to shorter wavelengths as the applied electric field increased accompanied by a slight linear increase of the PBG width, as shown in Fig. 5(a). Also, the bandwidth increases as the angle of incident increased. As the angle of incident increased, the slope of the linear relations between bandwidth and the applied electric field is increased. For the 1D-PC, the Bragg’s law is given by [34–37].
m = 2d
n2eff − sin2 , n2eff = n21 f1 + n22 f2
(13)
Where m is the diffraction order, d is the period, is the wavelength of the reflected light (or the so-called stop band), neff is the effective refractive index of the 1D-PCs, is the incident angle. n1 , f1, and n2 , f2 are the refractive indices and volume fractions of the LN and polymer, respectively. Then, the decrease in the effective refractive index of 1D-PC under the applied electric field leads to the shift of PBG edges to the shorter wavelengths, as shown in Fig. 4. Whereas the PBG enhancement
A.M. Ahmed et al. / Optik 145 (2017) 121–129
125
Fig. 3. The refractive indices of (a) LN and polymer, (b) the contrast between them as a function of applied electric field at = 1.550 m.
Fig. 4. Effect of applied electric field on the reflectance spectra of the defective 1D-PC for TE wave at two incident angle (a) 0◦ and (b) 50◦ .
may be ascribed to an increase in the index contrast n2 (E)/n1 (E) under the application of the external electric field. Also, the position of defect mode, 0 , decreases linearly with the external field, Fig. 5(c), whereas its reflected intensity, R0 , almost remains constant, Fig. 5(e). The decrease of the defect mode wavelength is ascribed to the reduction in the refractive indices n1 (E) and n2 (E). When electric field remains constant, the bandwidth is increased and the PBG is strongly shifted toward the shorter wavelengths as the angle of incident increased. This means that, the L and R band edges are shifted to the shorter wavelengths, but the movement speed of L is faster than that of R . Also, the position of defect mode, 0 , was shifted toward the shorter wavelengths with a small increase in the intensity of the defect mode reflectivity as the incident angle increased.
126
A.M. Ahmed et al. / Optik 145 (2017) 121–129
Fig. 5. Effect of the applied electric field and angle of incident, respectively, on the (a, b) PBG width, (c, d) position of defect mode and (e, f) reflectivity of the defect mode for TE wave.
The increase of the reflectivity may be ascribed to the decrease of the values of p as the angle of incident increased for TE waves (Eq. (3)). The shift of 0 may be due to the change of the phase difference according to Eq. (2). 4.2.3. Effect of applied external electric field for TM wave Fig. 6 shows the effect of applied electric field on the reflectance spectra of the defective 1D-PC at a different angle of incidence for TM wave. The important features are summarized and presented in Fig. 7.
A.M. Ahmed et al. / Optik 145 (2017) 121–129
127
Fig. 6. Effect of applied electric field on the reflectance spectra of the defective 1D-PC for TM wave at two incident angle (a) 0◦ and (b) 50◦ .
For normal incidence, there is no difference between TE and TM waves because both of them are equivalent at 0 = 0◦ . As in the case of TE wave, the PBG is shifted toward the shorter wavelengths, and the bandwidth is slightly increased for TM wave as the applied electric field increased, Fig. 7(a). For the defect mode, 0 decreases linearly, and R0 remains constant as the applied electric field increased, Fig. 7(c and e). As the incident angle increased from 0◦ to 50◦ , the PBG width decreases, Fig. 7(b), and the PBG strongly shifted to the shorter wavelengths for TM mode. This ascribed to the shift of the PBG edges, L, and R , to the short wavelengths, according to Bragg’s law. In contradicting, the results of the TE mode, the amount of shift in R is more pronounced than that in L leading to the decrease of the bandwidth for TE mode at the higher angles of incidence. Also, 0 of the defect mode is strongly shifted to the shorter wavelengths, Fig. 7(d), and R0 of defect mode is slightly decreased as the angle of incidence increased for TM wave, Fig. 7(f). It is remarkable to note that, when the angle of incidence is increased, PBG of the TE and TM waves are shifted to the shorter wavelength. However, the PBG of TE wave has comparatively wider bands than that of TM wave. Also, the PBG for TE wave is shifted more toward the shorter wavelength region than that for TM wave.
5. Conclusion Based on the method of the transfer matrix, the reflection spectra were calculated for a 1D perfect and defective PC designed from electro-optic materials (LN and polymer). The effect of the applied electric field and angle of incident light on the defect mode has been investigated. The results show that the refractive indices of the LN and polymer are deceased, but the contrast between them increases as the applied electric field increased. Also, the PBG edges are shifted to shorter wavelengths as the applied electric field increased accompanied with a slight linear increase of the PBG width at a constant angle of the incident for both TE and TM waves. Also, the position of defect mode decreases linearly with the external field. By increasing the angle of the incident from 0◦ to 50◦ at constant applied electric field of 100 v/m, the width of the PBGs of TE wave is increased from 0.334 to 0.358 m while the width of the PBGs of TM wave is decreased from 0.334 to 0.248 m This study may be valuable for designing tunable photonic devices with driving the electrical field, which has potential applications in the field of photonics and optoelectronics.
128
A.M. Ahmed et al. / Optik 145 (2017) 121–129
Fig. 7. Effect of the applied electric field and angle of incident, respectively, on the (a, b) PBG width, (c, d) position of defect mode and (e, f) reflectivity of the defect mode for TM wave.
References [1] J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed., Princeton University Press, Princeton, NJ, 2008. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (1987) 2486–2489. [3] E. Yablonovitch, Photonic band structure: the face- centered-cubic case employing nonspherical atoms, Phys. Rev. Lett. 67 (1991) 2295.
A.M. Ahmed et al. / Optik 145 (2017) 121–129 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
129
A.H. Aly, H.A. Elsayed, Defect mode properties in a one-dimensional photonic crystal, Physica B 407 (2012) 120. C.J. Wu, Y.H. Chung, B.J. Syu, Band gap extension in one-dimensional ternary metal-dielectric photonic crystal, Prog. Electromagn. Res. 102 (2010) 81. W. Xue, L. Ottaviano, E. Chen, Y. Yu, A. Lupi, J. Mork, K. Yvind, Thermal analysis of line-defect photonic crystal lasers, Opt. Express 23 (2015) 18277. J. Wang, C.W. Wang, Study of one-dimensional defect photonic crystal based on Ag-anodic aluminum oxide composites with super low refractive index, Eur. Phys. J. Plus. 130 (2015) 183. A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New York, 1984. B. Li, J. Vemagiri, R. Dinu, Design and modeling of traveling-wave electro-optic polymer modulator for ultrahigh speed applications, J. Lightw. Technol. 27 (2009) 606. K. Preston, S. Manipatruni, A. Gondarenko, C.B. Poitras, M. Lipson, Deposited silicon high-speed integrated electro-optic modulator, Opt. Express 17 (2009) 5118. A. Majumdar, C.M. Dodson, T.K. Fryett, A. Zhan, S. Buckley, D. Gerace, Hybrid 2D material nanophotonics: a scalable platform for low-power nonlinear and quantum optics, ACS Photonics 2 (2015) 1160. J.D. Witmer, J.T. Hill, A.H. Safavi-Naeini, Design of nanobeam photonic crystal resonators for a silicon-on-lithium-niobate platform, Opt. Express 24 (2016) 5876. R.I. Shostak, A.V. Yatsenko, Analysis of the refractive indices of lithium niobate at high temperatures, Opt. Spectrosc. 104 (2008) 457. H. Lu, B. Sadani, N. Courjal, G. Ulliac, N. Smith, V. Stenger, M. Collet, F.I. Baida, M.P. Bernal, Enhanced electro-optical lithium niobate photonic crystal wire waveguide on a smart-cut thin film, Opt. Express 20 (2012) 2974. D. Yang, X. Wang, H. Tian, J. Yuefeng, Electro-optic modulation property of slow light in coupled photonic crystal resonator arrays, Opt. Appl. 41 (2011) 753. J.E. Toney, V.E. Stenger, P. Pontius, N. Smith, J. Scholl, A. Pollick, B. Sadani, H. Lu, M.P. Bernal, S. Sriram, Photonic crystal electro-optic devices in engineered thin film lithium niobate substrates, Proc. SPIE 8376 (2012) (83760 H.). O. Yavuzcetin, H.P. Novikov, R.L. Dally, S.T. Malley, N.R. Perry, B. Ozturk, S. Sridhar, Photonic crystal fabrication in lithium niobate via pattern transfer through wet and dry etched chromium mask, J. Appl. Phys. 112 (2012) 074303. X.Y. Hu, Y. Li, Q.H. Gong, B.Y. Cheng, D.Z. Zhang, Fabrication of a two-dimensional organic photonic crystal, Chin. Phys. Lett. 22 (2005) 373. V. Kumar, K.S. Singh, S.P. Ojha, Enhanced Omni-directional reflection frequency range in Si-based one-dimensional photonic crystal with defect, Optik 122 (2011) 910. D.H. Jundt, Temperature-dependent Sellmeier equation for the index of refraction, ne , in congruent lithium niobate, Opt. Lett. 22 (1997) 1553. J.M. Brosi, C. Koos, L.C. Andreani, M. Waldow, J. Leuthold, W. Freude, High speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide, Opt. Express 16 (2008) 4177. J.H. Wulbern, A. Petrov, M. Eich, Electro-optical modulator in a polymer in filtrated silicon slotted photonic crystal waveguide heterostructure resonator, Opt. Express 17 (2009) 304. Y.Q. Lu, J.J. Zheng, Frequency tuning of optical parametric generator in periodically poled optical super-lattice LiNbO3 by electro-optical effect, Appl. Phys. Lett. 74 (1999) 123. P. Yeh, Optical Waves in Layered Media, Wiley, New York, 1988. M. Born, E. Wolf, Principles of Optics, 7th edition, Cambridge Press, London, 1999. J.B. Pendry, Photonic band structures, J. Mod. Opt. 41 (1994) 209. S.K. Awasthi, S.P. Ojha, Design of a tunable optical filter by using a one-dimensional ternary photonic band gap material, Prog. Electromagn. Res. M 4 (2008) 117. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, John Wiley & Sons, Inc, 1991. M. Shaban, A.M. Ahmed, E. Abdel-Rahman, H. Hamdy, Tunability and sensing properties of plasmonic/1D photonic crystal, Sci. Rep. 7 (2017) 41983. S.K. Srivastava, M. Upadhyay, S.K. Awasthi, S.P. Ojha, Tunable reflection bands and defect modes in one-dimensional tilted photonic crystal structure, Opt. Photon. J. 2 (2012) 230. W. Nakagawa, P.C. Sun, C.H. Chen, Y. Fainman, Wide field of view narrow band spectral filters based on photonic crystal nanocavities, Opt. Lett. 27 (2002) 191. M.W. Feise, I.V. Shadrivov, Y.S. Kivshar, Bistable diode action in left-handed periodic structures, Phys. Rev. E 71 (2005) 037602. Z. Qiaofen, Y. Zhang, Defect modes and wavelength tuning of one-dimensional photonic crystal with lithium niobate, Optik 120 (2009) 195. V. Morandi, F. Marabelli, V. Amendola, M. Meneghetti, D. Comoretto, Colloidal photonic crystals doped with gold nanoparticles: spectroscopy and optical switching properties, Adv. Funct. Mater. 17 (2007) 2779. B. Li, J. Zhou, L.Q. Li, Q. Li, S. Han, Z.B. Hao, One-dimensional photonic band-gap structure in abalone shell, Chin. Sci. Bull. 50 (2005) 1529. M. Li, Y. Song, High effective sensors based on photonic crystals, Front. Chem. China 5 (2010) 115. J.M. Lourtioz, H. Benisty, V. Berger, J.M. Gerard, D. Maystre, A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices, 2nd edition, Springer-Verlag, Berlin Heidelberg, New York, 2008.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Electro-optical tenability properties of defective one-dimensional photonic crystal Ashour M. Ahmed a,∗ , Mohamed Shaban a , Arafa H. Aly b a b
Nanophotonics and Applications (NPA) Lab, Physics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt Physics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt
a r t i c l e
i n f o
Article history: Received 13 April 2017 Accepted 6 July 2017 Keywords: Electro-optic effect Photonic crystal Defect mode Reflectance spectra Transfer matrix method
a b s t r a c t In this work, the optical transfer matrix formalism was applied to study the linear electrooptic effect on perfect and defective one-dimensional photonic crystal (1D-PC) composed of lithium niobate and polymer (polystyrene) multilayers. The effects of an external electrical field on the reflectance spectra of the proposed PCs were studied as a function of the angle of incidence and the types of polarization (TE and TM waves). The results show that, at a constant angle of the incident, the PBGs are shifted to shorter wavelengths for both TE and TM waves as the applied electric field increased. The position of defect mode decreases linearly from 1.401 to 1.379 m for TM wave and from 1.398 to 1.377 m for TE wave as the applied electric field increased from 0 to 100 v/m at an angle of incident 50◦ . By increasing the angle of the incident from 0 to 50 ◦ at constant applied electric field, the position of PBG for both TE and TM are shifted toward the shorter wavelengths. In addition, the width of the PBGs of TE wave is increased from 0.334 to 0.358 m and the width of the PBGs of TM wave is decreased from 0.334 to 0.248 m at 100 v/m. This study may be valuable for designing tunable photonic devices with driving electrical field, which has potential applications in the area of photonics and optoelectronics. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals (PCs) are periodic structures with alternating refractive indices in one, two, or three dimensions with periods comparable to the wavelength of the incident light [1]. In recent years, the studies of the electromagnetic propagation in the photonic crystals have become intense research area due to their novel applications in modern optical devices. PCs can be used to control the propagation of light by the appearance of a photonic band gap (PBG) and photon localization [2,3]. PBG is wavelength interval in which electromagnetic waves cannot propagate through the PC because the waves will undergo exponential decay (evanescent mode) and total reflection. The photon localization can be derived inside the PBG by adding a defect layer into the PC to break the periodic feature [4]. When the incident photon wavelength is equal to the defect-state wavelength, the photon will be localized in the place of the defect and a resonant transmission peak is generated within the PBG. The PBG and defect mode are strongly depending on some parameters such as index contrast, layer thickness, the state of polarization, filling fraction, and angle of incidence [5–7]. Hence, any variation in the refractive index of the PC can produce a significant modulation in the position of the defective mode. This modulation can be used as an indicator for measuring the index change.
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (A.M. Ahmed). http://dx.doi.org/10.1016/j.ijleo.2017.07.025 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
122
A.M. Ahmed et al. / Optik 145 (2017) 121–129
Fig. 1. Schematic diagram of the proposed 1D-PC with a defect layer under the effect of an external electric field.
Table 1 The values of the different parameters that used in the calculations. LiNbO3 (Material 1)
Refractive index ( in m)
n1 = −6
Electro-optic coefficient (10 Thickness (m)
m/V)
A+
B 2
− C2
␥1 = 30.9 [23] d1 = 0.1813
Polymer (Material 2) +
D 2
− H2
A = 5.35583; 2 − F C = 0.20692; H = 11.34927;
B = 0.100473; [20] D = 100; F = 153.34;
n2 = 1.6 [21,22] ␥2 = 150 [21,22] d2 = 0.2421
On the other side, certain materials can change their refractive indices under the exposure to external physical signals such as electrical, magnetic, thermal, acoustic, or mechanical signals. The electro-optic (EO) effect is a change in the optical properties of the material, especially its refractive index, in response to an external electric field. This effect occurs due electric field causes a redistribution of bond charges and possibly a slight deformation of the crystal lattice [8]. Therefore, the inverse dielectric constant (impermeability) tensor changes accordingly. Then, the EO effect can be used for the dynamic control of the refractive index. Also, the EO materials play a significant role in many practical applications such as optical communication systems, charge storage, optical sensors, electroluminescent devices, and optical computers [9–11]. Lithium Niobate (LN) and polymers (such as polystyrene) are typical electro-optic materials. The LN is a nonlinear ferroelectric material widely used in optoelectronics and surface acoustic wave devices [12,13]. Then, the design of LN photonic crystals can potentially decrease the size of the critical components in many optical systems [14]. Moreover, the EO polymers gained an increased interest for telecommunication applications due to their low optical losses in the 1.3 and 1.55 m telecommunication windows. Also, these polymers are characterized by high linear EO, low processing cost, and reliable mass production [15]. Therefore, it is imperative to study the effect of the external electric field on the PBG and photon localization of a PC from LN and a polymer such as a polystyrene. Most previous studies were focused on design and fabrication of 2D-PC structures for EO applications by etching hexagonal or square arrays of holes in LN or polymer film [16,17]. However, there are practical limitations for obtaining highly ordered 2D-PC because the difficulty of fabricating deep and cylindrical holes in bulk crystals. Also, PBG structures may be destroyed when a large number of defects or disorders are introduced in the PC [18]. Then 1D-PC structure (such as multilayer films) can be used as a real alternative for 2D-PC because it can be simulated and fabricated more easily than 2D-PC [19]. Moreover, it can be easily adapted for use in many applications. Here, an EO structure based on 1D-PC (LiNbO3 /polymer multilayers) with a defect layer is proposed. Using transfer matrix method (TMM), the reflection at different applied electric fields and incident angles for both transverse electric (TE) and transverse magnetic (TM) modes are calculated and discussed.
2. PC design To have an idea about the effect of the electric field on the optical spectra of PCs, a 1D-PC from alternately stacked LN and polymer layers with refractive indices n1 and n2 respectively is addressed. The LN layer is inserted as a defect layer. The PC is surrounded by air and glass substrate. The modeled structure was based on the following configuration: air/(n1 n2 )N /n2 /(n1 n2 )N /glass with N = 14 period. Also, two electrodes are placed on the sides of the PC to apply an electric field. Schematic diagram of the proposed structure is shown in Fig. 1. The physical thicknesses of LiNbO3 and polymer are taken according to the quarter wave arrangement at the operating wavelength in the infrared region 0 = 1.550 m, which is essential for the optical communication. The period of the lattice is d = d1 + d2 . The values of the different parameters that used in this study are given in Table 1.
A.M. Ahmed et al. / Optik 145 (2017) 121–129
123
3. Modeling of EO-PBG by using transfer matrix method Based on the Maxwell equations and the boundary conditions, the transfer matrix method (TMM) has been widely used to calculate the reflectance spectra of the propagated light waves in a 1D-PC [24–26]. An algorithm in MATLAB is implemented and used to determine the optical reflection of the proposed photonic structures. The characteristic matrix for the first layer, m1, and for the second layer, m2, are given by:
⎡
m1 = ⎣
⎤
⎡
⎤
−i sin1 p1 ⎦ m2 = ⎣
cosˇ1 −ip1 sin1
cosˇ2 −ip2 sin2
cosˇ1
−i sin2 p2 ⎦
(1)
cosˇ2
The phase differences at the first and second layer are given by 1 = 2n1 (E) d1 cos 1 /
2 = 2n2 (E) d2 cos 2 /
(2)
The values of p for the first and second layer are depends on the type of polarization and given by [27]: p1 = n1 (E) cos 1 and p2 = n2 (E) cos 2 for the TE
(3)
p1 = cos 1 /n1 (E) and p2 = cos 2 /n2 (E) for the TM
The angles of incidence 1 and 2 inside the first and second layer are related to the angle of incidence 0 by Snell’s law. n0 sin0 = n1 sin1 = n2 sin2
(4)
The characteristic matrix m for these two layers is given by
m = m1 m2 =
m11
m12
m21
m22
⎡ ⎢
cosˇ1 cosˇ2 −
=⎣
⎤
−i i sinˇ1 cosˇ2 − cosˇ1 sinˇ2 p1 p2 ⎥
p2 sinˇ1 sinˇ2 p1
−i p1 sinˇ1 cosˇ2 − i p2 cosˇ1 sinˇ2
cosˇ1 cosˇ2 −
p1 sinˇ1 sinˇ2 p2
⎦
(5)
The refractive index of EO materials under the applied electric field, n(E), is given by the relation [28]: n (E) = n − n3 ␥E/2
(6)
where, n is the refractive index of the material in the absence of the applied electric filed, ␥ is the linear electro-optic (Pockels) coefficient, and E is the applied electric filed in unit of v/m. The characteristic matrix M for a system of N periods can be obtained as follows:
N
N
M = m = (m1m2) =
M11 M12 M21
M22
=
m11TN−1 (A) − TN−2 (A)
m12TN−1 (A)
m21TN−1 (A)
m22TN−1 (A) − TN−2 (A)
(7)
Where TN (A) represenets the Chebyshev polynomials of the second kind and A is the Bloch phase for a single period. The TN (A) and A are given by
TN (A) = Sin (N + 1) cos−1 A /
1 + A2 and A = 0.5 (m11 + m22)
(8)
The total characteristic matrix, G, for the proposed structure is given by
G = Mm2M =
G11 G12 G21
G22
(9)
The reflectance coefficient of the multilayer is given by r=
(G11 + G12ps ) p0 − (G21 + G22ps ) (G11 + G12ps ) p0 + (G21 + G22ps )
(10)
Where, the values of p of air and glass substrate are given by p0 = n0 (E) cos 0 and ps = ns (E) cos s for the TE p0 = cos 0 /n0 (E) and ps = cos s /ns (E) for the TM
(11)
Then, the reflection for the defective 1D photonic crystal is given by R = |r2 |
(12)
124
A.M. Ahmed et al. / Optik 145 (2017) 121–129
Fig. 2. The reflectance spectrum of perfect 1D-PC and defective 1D-PC.
4. Results and discussion 4.1. Reflectance spectra of perfect 1D-PC and defective 1D-PC Fig. 2 shows the reflectance spectrum of a perfect 1D-PC and defective 1D-PC at normal incidence without the applied of an electric field. It can be seen that the PBG of defective PC is different compared to the perfect PC as shown in this Figure. The PBG of perfect 1D-PC exists in the NIR region with left and right band edges at L = 1.413 m and R = 1.716 m, respectively. The width of this PBG is = R − L = 0.303 m. The PBG of a 1D-PC structure arises due to the light beam divide into two parts at each layer when a light beam incident on the PC [29]. One part of the light will be transmitted, and another part will be reflected each time the light reaches the interface between the LN and polymer. There are multiple reflections arise from the periodicity of PC. The reflected waves can interfere constructively or destructively. Whether the multiple reflected beams are in phase or not, the phase difference between each succeeding reflection is given by Eq. (2). If the partially reflected waves are in phase and superimposed, then the incident waves are totally reflected, and zero transmission is obtained. The range of the wavelengths in which incident light cannot pass through the PC is called photonic band gap (PBG). On the other hand, the L of the PBG of a defective PC is slightly shifted to the short wavelength region whereas the R is shifted to the long wavelength region compared to that of the perfect PC. Then, the bandwidth of the defective PBG is increased to 0.329 m at the normal incidence. In addition, Fig. 2 shows a super narrow defective mode in the PBG at the selected wavelength (1.55 m). The defect mode divided the PBG into two parts. This may be ascribed to the constructive interference of incident and reflected waves in the defect layer [30]. The reflectivity of the defective mode is very low (R0 = 6.4%), so it can be considered as a resonant transmission mode in PBG at the selected wavelength. Also, the reflectance response outside the PBG shows some fluctuations due to the existence of the defect layer. The number of the ripples in the reflectance spectrum for the perfect 1D-PC is twice that of the defective 1D-PC. The reflection intensities of these ripples for the perfect 1D-PC are smaller than for that of the defective 1D-PC. The defect state can be significantly applied in many applications such as isolators, waveguides, filters, laser, a light-emitting diode, multiplex, PC fiber, splitters, sensors, and switches [31–33]. 4.2. Effect of an external electric field on optical properties of 1D-PC 4.2.1. Effect of the applied electric field on the refractive indices The effect of the applied electric field on the refractive indices of materials have been investigated at = 1.550 m. From Fig. 3, the refractive indices of the LN and polymer are deceased but the contrast between them is increased as the applied electric field increased due to the Pockels coefficient of the polymer is higher than that of LN. 4.2.2. Effect of applied external electric field for TE wave Fig. 4 shows the effect of applied electric field on the reflectance spectra for TE wave of the defective 1D-PC at two different incident angles, 0◦ and 50◦ . The analysis of the optical reflection was summarized in Fig. 5 for two angles of incidence (0◦ and 50◦ ). At a constant angle of incidence, the PBG edges are shifted to shorter wavelengths as the applied electric field increased accompanied by a slight linear increase of the PBG width, as shown in Fig. 5(a). Also, the bandwidth increases as the angle of incident increased. As the angle of incident increased, the slope of the linear relations between bandwidth and the applied electric field is increased. For the 1D-PC, the Bragg’s law is given by [34–37].
m = 2d
n2eff − sin2 , n2eff = n21 f1 + n22 f2
(13)
Where m is the diffraction order, d is the period, is the wavelength of the reflected light (or the so-called stop band), neff is the effective refractive index of the 1D-PCs, is the incident angle. n1 , f1, and n2 , f2 are the refractive indices and volume fractions of the LN and polymer, respectively. Then, the decrease in the effective refractive index of 1D-PC under the applied electric field leads to the shift of PBG edges to the shorter wavelengths, as shown in Fig. 4. Whereas the PBG enhancement
A.M. Ahmed et al. / Optik 145 (2017) 121–129
125
Fig. 3. The refractive indices of (a) LN and polymer, (b) the contrast between them as a function of applied electric field at = 1.550 m.
Fig. 4. Effect of applied electric field on the reflectance spectra of the defective 1D-PC for TE wave at two incident angle (a) 0◦ and (b) 50◦ .
may be ascribed to an increase in the index contrast n2 (E)/n1 (E) under the application of the external electric field. Also, the position of defect mode, 0 , decreases linearly with the external field, Fig. 5(c), whereas its reflected intensity, R0 , almost remains constant, Fig. 5(e). The decrease of the defect mode wavelength is ascribed to the reduction in the refractive indices n1 (E) and n2 (E). When electric field remains constant, the bandwidth is increased and the PBG is strongly shifted toward the shorter wavelengths as the angle of incident increased. This means that, the L and R band edges are shifted to the shorter wavelengths, but the movement speed of L is faster than that of R . Also, the position of defect mode, 0 , was shifted toward the shorter wavelengths with a small increase in the intensity of the defect mode reflectivity as the incident angle increased.
126
A.M. Ahmed et al. / Optik 145 (2017) 121–129
Fig. 5. Effect of the applied electric field and angle of incident, respectively, on the (a, b) PBG width, (c, d) position of defect mode and (e, f) reflectivity of the defect mode for TE wave.
The increase of the reflectivity may be ascribed to the decrease of the values of p as the angle of incident increased for TE waves (Eq. (3)). The shift of 0 may be due to the change of the phase difference according to Eq. (2). 4.2.3. Effect of applied external electric field for TM wave Fig. 6 shows the effect of applied electric field on the reflectance spectra of the defective 1D-PC at a different angle of incidence for TM wave. The important features are summarized and presented in Fig. 7.
A.M. Ahmed et al. / Optik 145 (2017) 121–129
127
Fig. 6. Effect of applied electric field on the reflectance spectra of the defective 1D-PC for TM wave at two incident angle (a) 0◦ and (b) 50◦ .
For normal incidence, there is no difference between TE and TM waves because both of them are equivalent at 0 = 0◦ . As in the case of TE wave, the PBG is shifted toward the shorter wavelengths, and the bandwidth is slightly increased for TM wave as the applied electric field increased, Fig. 7(a). For the defect mode, 0 decreases linearly, and R0 remains constant as the applied electric field increased, Fig. 7(c and e). As the incident angle increased from 0◦ to 50◦ , the PBG width decreases, Fig. 7(b), and the PBG strongly shifted to the shorter wavelengths for TM mode. This ascribed to the shift of the PBG edges, L, and R , to the short wavelengths, according to Bragg’s law. In contradicting, the results of the TE mode, the amount of shift in R is more pronounced than that in L leading to the decrease of the bandwidth for TE mode at the higher angles of incidence. Also, 0 of the defect mode is strongly shifted to the shorter wavelengths, Fig. 7(d), and R0 of defect mode is slightly decreased as the angle of incidence increased for TM wave, Fig. 7(f). It is remarkable to note that, when the angle of incidence is increased, PBG of the TE and TM waves are shifted to the shorter wavelength. However, the PBG of TE wave has comparatively wider bands than that of TM wave. Also, the PBG for TE wave is shifted more toward the shorter wavelength region than that for TM wave.
5. Conclusion Based on the method of the transfer matrix, the reflection spectra were calculated for a 1D perfect and defective PC designed from electro-optic materials (LN and polymer). The effect of the applied electric field and angle of incident light on the defect mode has been investigated. The results show that the refractive indices of the LN and polymer are deceased, but the contrast between them increases as the applied electric field increased. Also, the PBG edges are shifted to shorter wavelengths as the applied electric field increased accompanied with a slight linear increase of the PBG width at a constant angle of the incident for both TE and TM waves. Also, the position of defect mode decreases linearly with the external field. By increasing the angle of the incident from 0◦ to 50◦ at constant applied electric field of 100 v/m, the width of the PBGs of TE wave is increased from 0.334 to 0.358 m while the width of the PBGs of TM wave is decreased from 0.334 to 0.248 m This study may be valuable for designing tunable photonic devices with driving the electrical field, which has potential applications in the field of photonics and optoelectronics.
128
A.M. Ahmed et al. / Optik 145 (2017) 121–129
Fig. 7. Effect of the applied electric field and angle of incident, respectively, on the (a, b) PBG width, (c, d) position of defect mode and (e, f) reflectivity of the defect mode for TM wave.
References [1] J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed., Princeton University Press, Princeton, NJ, 2008. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (1987) 2486–2489. [3] E. Yablonovitch, Photonic band structure: the face- centered-cubic case employing nonspherical atoms, Phys. Rev. Lett. 67 (1991) 2295.
A.M. Ahmed et al. / Optik 145 (2017) 121–129 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
129
A.H. Aly, H.A. Elsayed, Defect mode properties in a one-dimensional photonic crystal, Physica B 407 (2012) 120. C.J. Wu, Y.H. Chung, B.J. Syu, Band gap extension in one-dimensional ternary metal-dielectric photonic crystal, Prog. Electromagn. Res. 102 (2010) 81. W. Xue, L. Ottaviano, E. Chen, Y. Yu, A. Lupi, J. Mork, K. Yvind, Thermal analysis of line-defect photonic crystal lasers, Opt. Express 23 (2015) 18277. J. Wang, C.W. Wang, Study of one-dimensional defect photonic crystal based on Ag-anodic aluminum oxide composites with super low refractive index, Eur. Phys. J. Plus. 130 (2015) 183. A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New York, 1984. B. Li, J. Vemagiri, R. Dinu, Design and modeling of traveling-wave electro-optic polymer modulator for ultrahigh speed applications, J. Lightw. Technol. 27 (2009) 606. K. Preston, S. Manipatruni, A. Gondarenko, C.B. Poitras, M. Lipson, Deposited silicon high-speed integrated electro-optic modulator, Opt. Express 17 (2009) 5118. A. Majumdar, C.M. Dodson, T.K. Fryett, A. Zhan, S. Buckley, D. Gerace, Hybrid 2D material nanophotonics: a scalable platform for low-power nonlinear and quantum optics, ACS Photonics 2 (2015) 1160. J.D. Witmer, J.T. Hill, A.H. Safavi-Naeini, Design of nanobeam photonic crystal resonators for a silicon-on-lithium-niobate platform, Opt. Express 24 (2016) 5876. R.I. Shostak, A.V. Yatsenko, Analysis of the refractive indices of lithium niobate at high temperatures, Opt. Spectrosc. 104 (2008) 457. H. Lu, B. Sadani, N. Courjal, G. Ulliac, N. Smith, V. Stenger, M. Collet, F.I. Baida, M.P. Bernal, Enhanced electro-optical lithium niobate photonic crystal wire waveguide on a smart-cut thin film, Opt. Express 20 (2012) 2974. D. Yang, X. Wang, H. Tian, J. Yuefeng, Electro-optic modulation property of slow light in coupled photonic crystal resonator arrays, Opt. Appl. 41 (2011) 753. J.E. Toney, V.E. Stenger, P. Pontius, N. Smith, J. Scholl, A. Pollick, B. Sadani, H. Lu, M.P. Bernal, S. Sriram, Photonic crystal electro-optic devices in engineered thin film lithium niobate substrates, Proc. SPIE 8376 (2012) (83760 H.). O. Yavuzcetin, H.P. Novikov, R.L. Dally, S.T. Malley, N.R. Perry, B. Ozturk, S. Sridhar, Photonic crystal fabrication in lithium niobate via pattern transfer through wet and dry etched chromium mask, J. Appl. Phys. 112 (2012) 074303. X.Y. Hu, Y. Li, Q.H. Gong, B.Y. Cheng, D.Z. Zhang, Fabrication of a two-dimensional organic photonic crystal, Chin. Phys. Lett. 22 (2005) 373. V. Kumar, K.S. Singh, S.P. Ojha, Enhanced Omni-directional reflection frequency range in Si-based one-dimensional photonic crystal with defect, Optik 122 (2011) 910. D.H. Jundt, Temperature-dependent Sellmeier equation for the index of refraction, ne , in congruent lithium niobate, Opt. Lett. 22 (1997) 1553. J.M. Brosi, C. Koos, L.C. Andreani, M. Waldow, J. Leuthold, W. Freude, High speed low-voltage electro-optic modulator with a polymer-infiltrated silicon photonic crystal waveguide, Opt. Express 16 (2008) 4177. J.H. Wulbern, A. Petrov, M. Eich, Electro-optical modulator in a polymer in filtrated silicon slotted photonic crystal waveguide heterostructure resonator, Opt. Express 17 (2009) 304. Y.Q. Lu, J.J. Zheng, Frequency tuning of optical parametric generator in periodically poled optical super-lattice LiNbO3 by electro-optical effect, Appl. Phys. Lett. 74 (1999) 123. P. Yeh, Optical Waves in Layered Media, Wiley, New York, 1988. M. Born, E. Wolf, Principles of Optics, 7th edition, Cambridge Press, London, 1999. J.B. Pendry, Photonic band structures, J. Mod. Opt. 41 (1994) 209. S.K. Awasthi, S.P. Ojha, Design of a tunable optical filter by using a one-dimensional ternary photonic band gap material, Prog. Electromagn. Res. M 4 (2008) 117. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, John Wiley & Sons, Inc, 1991. M. Shaban, A.M. Ahmed, E. Abdel-Rahman, H. Hamdy, Tunability and sensing properties of plasmonic/1D photonic crystal, Sci. Rep. 7 (2017) 41983. S.K. Srivastava, M. Upadhyay, S.K. Awasthi, S.P. Ojha, Tunable reflection bands and defect modes in one-dimensional tilted photonic crystal structure, Opt. Photon. J. 2 (2012) 230. W. Nakagawa, P.C. Sun, C.H. Chen, Y. Fainman, Wide field of view narrow band spectral filters based on photonic crystal nanocavities, Opt. Lett. 27 (2002) 191. M.W. Feise, I.V. Shadrivov, Y.S. Kivshar, Bistable diode action in left-handed periodic structures, Phys. Rev. E 71 (2005) 037602. Z. Qiaofen, Y. Zhang, Defect modes and wavelength tuning of one-dimensional photonic crystal with lithium niobate, Optik 120 (2009) 195. V. Morandi, F. Marabelli, V. Amendola, M. Meneghetti, D. Comoretto, Colloidal photonic crystals doped with gold nanoparticles: spectroscopy and optical switching properties, Adv. Funct. Mater. 17 (2007) 2779. B. Li, J. Zhou, L.Q. Li, Q. Li, S. Han, Z.B. Hao, One-dimensional photonic band-gap structure in abalone shell, Chin. Sci. Bull. 50 (2005) 1529. M. Li, Y. Song, High effective sensors based on photonic crystals, Front. Chem. China 5 (2010) 115. J.M. Lourtioz, H. Benisty, V. Berger, J.M. Gerard, D. Maystre, A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices, 2nd edition, Springer-Verlag, Berlin Heidelberg, New York, 2008.