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Tunable full band gap in two-dimensional anisotropic photonic crystals infiltrated with liquid crystals
Optics Communications 282 (2009) 1584–1588
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Tunable full band gap in two-dimensional anisotropic photonic crystals infiltrated with liquid crystals B. Rezaei a,*, M. Kalafi a,b a b
Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran Physics Department, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 5 September 2008 Received in revised form 1 January 2009 Accepted 6 January 2009
a b s t r a c t We analyze the tunability of full band gap in two-dimensional photonic crystals created by square and triangular lattices of anisotropic tellurium rods in air background, considering that the rods are infiltrated with liquid crystal. Using the plane-wave expansion method, we study the variation of full band gap by changing the optical axis orientation of liquid crystal. Ó 2009 Elsevier B.V. All rights reserved.
PACS: 42.70.Qs 77.84.Nh 78.20.Bh Keywords: Tunable full band gap Anisotropic photonic crystal Liquid crystal
1. Introduction Photonic crystals (PCs) are periodic dielectric materials designed to affect the propagation of electromagnetic waves in the same way as the periodic potential in semiconductor crystals affects the electron motion by defining allowed and forbidden energy bands. Since first proposed by John [1] and Yablonovitch [2], their potential scientific and technological applications have inspired great interest among researchers. Photonic crystals offer an important opportunity to design new optical devices and hold a great potential for many significant applications, such as semiconductor lasers and solar cells, high quality resonator and filters, and optical fibers. In the last decade there has been an increasing emphasis on tuning the optical properties of photonic bandgap structures in order to design switchable or dynamical devices. Busch and John [3] predicted the tunability of band gap in three dimensional (3D) PCs by utilizing liquid crystals (LCs). Following this publication, some investigations of band gap tunability have been done by utilizing LCs in one-dimensional (1D) [4–7], twodimensional (2D) [8–17] and 3D [18–20] PCs. It is important to note that, in 2D PCs composed of LCs, the classification of the TE (transverse electric) and TM (transverse * Corresponding author. Tel.: +98 411 339 3027; fax: +98 411 334 7050. E-mail addresses: [email protected], [email protected] (B. Rezaei). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.01.007
magnetic) modes is possible when the directors of LCs are parallel and perpendicular to 2D plane. However, for other alignment of directors none of these two classifications of modes exist due to anisotropies of LCs, i.e. the mode coupling occurs. Materials play a significant role in determining the optical properties of a PC. The properties of PCs made of anisotropic materials differ largely from those of isotropic PCs. It has been reported by Zabel and Stroud that the anisotropy of materials can split degenerate bands and this will narrow the band gap of the PC [21]. In 1998, Li et al. have demonstrated that the band gap can be increased by using the anisotropic materials in a PC [22,23]. Recently, it has been reported that, the infiltration of nematic LC in 3D PC, made of anisotropic tellurium material, can affect the values and properties of full photonic band gaps [24]. They have found that, the band gaps can be controlled by rotating the directors of LCs under the influence of an applied electric field. Unlike the most potential applications of the 3D PCs, 2D ones are easier to fabricate, especially for the technologically important near-infrared or visible spectrum. Much attention has, therefore, been paid to 2D PCs. In this work, we consider 2D square and triangular PCs created by hollow anisotropic tellurium rods. Based on plane-wave expansion method (PWEM), we study the evolution of full band gap by changing the directors of LCs that are infiltrated throughout the hollow tellurium rods.
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2. Formulation To determine the photonic band gap in periodic dielectric structures, we study the propagation of electromagnetic waves from Maxwell’s equations using the plane-wave method as illustrated in several papers [25–27]. Following the discussion of Busch and John [3], the Maxwell’s equation for the magnetic field in 2D PCs utilizing LCs can be expressed as
r
1 x2 r Hð~ rÞ ¼ 2 Hð~ rÞ c eð~rÞ
ð1Þ
ei;j ð~rÞ ¼
i~ G:~ r
ei;j ð~ GÞe ; ði; j ¼ x; y; zÞ
ð2Þ
~ G
where eð~ GÞ is the Fourier transform of the eð~ rÞ, and plays a key role in determination of photonic band structure. Generally LCs possess two dielectric constants known as ordinary dielectric constant eo and extraordinary dielectric constant ee. The light waves with electric field perpendicular and parallel to the director of LC experience ordinary and extraordinary dielectric constants, respectively. When the director of LC rotates, the components of the dielectric tensor can be represented as [28]:
exx ð~rÞ ¼ eo þ ðee eo Þ sin2 ðhÞ cos2 ðuÞ eyy ð~rÞ ¼ eo þ ðee eo Þ sin2 ðhÞ sin2 ðuÞ exy ð~rÞ ¼ eyx ð~rÞ ¼ ðee eo Þ sin2 ðhÞ sinðuÞ cosðuÞ exz ð~rÞ ¼ ezx ð~rÞ ¼ ðee eo Þ sinðhÞ cosðhÞ cosðuÞ eyz ð~rÞ ¼ ezy ð~rÞ ¼ ðee eo Þ sinðhÞ cosðhÞ sinðuÞ exx ð~rÞ ¼ eo þ ðee eo Þ sin2 ðhÞ cos2 ðuÞ
ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ ð8Þ
where h is the zenith angle of the LC director (i.e. the angle between the LC director and the Z axis), u is the azimuth angle between the projection of the LC director on the X–Y plane and the X axis, and ~ n is the director of the LC, as shown in Fig. 1. Since light waves are transmitted in periodic structures, we can expand the magnetic field in terms of plane waves in the same way:
Hð~ rÞ ¼
2 XX ~ G
~ ~
h~G;k ^ek eiðkþGÞ:~r
k¼1
X
H~G;~G0
~ G0
h~G0 ;1
!
h~G0 ;2
¼
x2 h~G;1 c2
! ð10Þ
h~G;2
where
where, x is the frequency of light and c is the light velocity. The rÞ ¼ eð~ r þ~ RÞ is periodic with respect to the real dielectric tensor eð~ space lattice vector ~ R and we can use Bloch’s theorem to expand it as a sum of plane waves
X
where ~ k is a wave vector in the first Brillouin zone and ~ G is a 2D reciprocal lattice vector, ^ek ðk ¼ 1; 2Þ are orthogonal unit vectors perpendicular to ~ kþ~ G. In the case of dielectric rods parallel to the Z kþ~ Gj, direction, ^e1 and ^e2 are ð0; 0; 1Þ and ððky þ Gy Þ; kx þ Gx ; 0Þ=j~ respectively. Substituting Eqs. (2) and (9) into Eq. (1), we obtain the following linear matrix equation for the dispersion of electromagnetic waves:
ð9Þ
A H~G;~G0 ¼ j~ kþ~ Gjj~ kþ~ G0 j C
B
D
ð11Þ
With
A ¼ ^e2 e1 ð~ G~ G0 Þ ^e02 ¼ j~kþ~Gjj1~kþ~G0 j h 0 1 ~ ~ ~0 ~0 ðky þ Gy Þðky þ G0y Þe1 xx ðG G Þ ðky þ Gy Þðkx þ Gx Þexy ðG G Þ i 0 1 ~ ~ ~0 ~0 ðkx þ Gx Þðky þ G0y Þe1 yx ðG G Þ þ ðkx þ Gx Þðkx þ Gx Þeyy ðG G Þ B ¼ ^e2 e1 ð~ G~ G0 Þ ^e01 ¼
1 j~ kþ~ Gj
h i 1 ~ ~ ~0 ~0 ðky þ Gy Þe1 xz ðG G Þ ðkx þ Gx Þeyz ðG G Þ C ¼ ^e1 e1 ð~ G~ G0 Þ ^e02 ¼
1 j~ kþ~ G0 j
h i 0 1 ~ ~ ~0 ~0 ðky þ G0y Þe1 zx ðG G Þ ðkx þ Gx Þezy ðG G Þ ~ ~0 D ¼ e1 zz ðG G Þ ð12Þ The matrix H~G;~G0 is real and symmetric; thus the dispersion relation of electromagnetic waves can be solved by using standard diagonalization techniques for the linear system of Eq. (10). The main numerical problem in obtaining the eigenvalues is the evaluation of the Fourier coefficients of the inverse dielectric tensor in Eq. (12). The best method is to calculate the matrix of Fourier coefficients of real-space tensors and then take its inverse to obtain the required Fourier coefficients, which was shown by Ho et al. [29]. The structures under consideration and the corresponding first Brillouin zones are depicted in Fig. 2. We have considered (a) square and (b) triangular structures of circular shape anisotropic tellurium rods in air background, considering that the rods are infiltrated with LC. The anisotropic tellurium has two different principle refractive indices as ordinary-refractive index nTe o ¼ 4:8 and extraordinary refractive index nTe e ¼ 6:2, in which the extraordinary one is parallel to the Z-axis. We assumed the LC with ordinary index nLC o ¼ 1:59 and the extraordinary refractive index ¼ 2:223. This LC corresponds to the phenylacetylene type LC nLC e [30]. Parameters q1 and q2 denote the inner and outer radius of anisotropic tellurium rods, respectively. a is being the lattice constant. 3. Numerical results
Fig. 1. Schematic representation of rotation angles for the LC directors.
For this study, we consider 2D square and triangular PCs of circular shape anisotropic tellurium rods in a uniform air background with dielectric constant eb = 1, considering that the rods are infiltrated with LC as shown in Fig. 2. We choose the LCs as an anisotropic material for the convenient change of anisotropy by simply changing the orientation of LC molecules. The components of the dielectric tensor of the LC are given by Eqs. (3)–(8). The band structure of 2D PCs is obtained numerically by solving Eq. (10). A total of 441 plane waves were employed for both structures in
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Fig. 2. Photonic crystal structures and the corresponding first Brillouin zones for (a) square and (b) triangular lattices of LC-infiltrated anisotropic tellurium rods in air background.
these calculations. This number is known to be sufficient enough to obtain acceptable convergence for characterizing the band structures. Our main goal here is to study the modification of the band gap spectrum and the value of the full band gap by varying LC’s optical axis orientation as well as when the thickness of anisotropic tellurium outer shell varies. The band structures of these two kinds of lattices will be separately analyzed in the next two subsections. 3.1. Square Lattice We first consider a 2D PC of square lattice composed of LC-infiltrated anisotropic tellurium rods in air background, as illustrated in Fig. 2a. The corresponding first BZ is also depicted in Fig. 2a. For isotropic PCs, the full band structure can be obtained by simply considering all the wave vectors ~ k in the irreducible Brillouin zone (IBZ) which is enclosed by points C, X, and M as shown in Fig. 2a. The resulting full photonic band structure obtained from this IBZ for isotropic PCs based on the rotation and reflection symmetry of structure. In a PC made of anisotropic materials, the definition of the IBZ depends on the orientation of the principal axes of anisotropic material [31] and hence it maybe difficult to define the IBZ for each optic axis orientation. So, it is reasonable to consider the more symmetry directions in the first BZ for obtaining the complete photonic band structure. First we consider the case when there is no LC (solid tellurium rods), i.e. the inner rod radius is q1 = 0. Hence, the dielectric configuration is similar to that of an isotropic material. When the dielectric property is isotropic, the complete band structures will be calculated in the IBZ. Fig. 3 shows the full photonic band structures of this structure in the IBZ for optimum rod radius q2 = 0.36a, for which the full band gap reaches its maximum normalized width of Dx = 0.0341(2pc/a) between 0.2176 and 0.2517 frequencies in units of (2pc/a). At next step, for fixed optimum value of q2 = 0.36a, the inner spaces of rods are infiltrated with LC. The LC can be infiltrated in the air holes by a similar technique used for filling PCs made of macro-porous silicon [8]. For this purpose, the sample can be
Fig. 3. Full photonic band structures for square lattice of tellurium rods in air background at optimum rod radius of q2 = 0.36a.
placed in an evacuated flask. The LC is injected through shot needle. Because of strong capillarity forces, the air holes will be infiltrated easily. In this case, we study the variation of full band gap by changing the director of LC for different values of inner rod radius q1. We notice that, due to the symmetry of the underlying lattice, only the band shapes for / values between 0 and 45 are fundamentally unique. To obtain the maximum tunability of full band gap, we change the director of LC for different values of q1 by varying the h and / angles from 0° to 90° and from 0° to 45°, respectively. The numerical results show that the maximum tunability of full band gap is obtained at q1 = 0.3a and q2 = 0.36a. As shown in Fig. 4, the maximum tunability of 0.0098(2pc/a) is obtained for full band gap when h changes from 0° to 90° for all values of / except for the / = 45°. At h = 0° the full band gap with normalized width of 0.0098(2pc/a) is created between 4 and 5 bands and lies between 0.3145 and 0.3243 frequencies in units
B. Rezaei, M. Kalafi / Optics Communications 282 (2009) 1584–1588
Fig. 4. Tunability of full band gap for square lattice of LC-infiltrated anisotropic tellurium rods in air background at q1 = 0.3 and q2 = 0.36a. h and u are varied from 0° to 90° and from 0° to 0°, respectively.
1587
Fig. 5. Photonic band structures for triangular lattice of tellurium rods in air background at optimum rod radius of q2 = 0.355a.
of (2pc/a). When h = 90°, the photonic band gap is completely disappeared. It should be noted that, for / = 45°, the full band gap changes from 0.0098 to 0.0014 in units of (2pc/a). For a comprehensive investigation, we have also done such calculations for other values of q2. After extensive calculations, we have found that the maximum tunability of full band gap is obtained only for the above optimal values of q1 and q2. Taking into account the fabrication limit, for the lattice constant of a = 500nm, the square lattice of LC-infiltrated tellurium rods can be realized by inner radius q1 = 150 nm and outer radius q2 = 180 nm. This tunable PC can be used as a field-sensitive polarizer [32]. For example, the light of frequency xa/2pc = 0.315 ðk 1587 nmÞ cannot propagate through the PC if h = 0°, while at h = 90° it will be transmitted, because the full band gap has been disappeared. 3.2. Triangular lattice Fig. 2b shows the second structure to be analyzed; the 2D PC with triangular lattice composed of LC infiltrated anisotropic tellurium rods in air background. The corresponding first BZ of the reciprocal lattice is a hexagon, as depicted in Fig. 2b. The analysis of band structures for triangular lattice is similar to that of square lattice, as discussed in the last subsection. For a hexagonal BZ, it is necessary to consider the six distinct sub-zones as marked in Fig. 2b for obtaining the complete band structures of PCs made of anisotropic materials. Similar to the square lattice, we first consider the triangular lattice of solid tellurium rods (without LC material). Thus, the dielectric configuration, as explained in the last subsection, is similar to isotropic materials. Hence, the corresponding full band structures of this PC is shown in Fig. 5 in the IBZ which is enclosed by points C, M and K as shown in Fig. 2b. The maximum normalized width of Dx = 0.0414(2pc/a) is obtained for full band gap at optimum value of q2 = 0.355a and lies between 0.2198 and 0.2612 frequencies in units of (2pc/a). To get an insight into the effect of the orientation of LC directors on the full band gap, we first concentrate on optimum value of q2 = 0.355a as the inner spaces of rods are infiltrated with LC. Similar to the square lattice, because of the symmetry properties of the underlying lattice, the band structures for / values between 0° and 30° are unique. So, for different values of internal radius, q1, we study the tunability of full band gap as the LC molecules are
Fig. 6. Tunability of full band gap for triangular lattice of LC-infiltrated anisotropic tellurium rods in air background at q1 = 0.26 and q2 = 0.355a. h and u are varied from 0° to 90° and from 0° to 30°, respectively.
aligned in different orientations. The numerical results show that the maximum tunability of full band gap is obtained at q1 = 0.26a and q2 = 0.355a. Fig. 6 shows the variation of full band gap size as a function of h and / for these parameters. It can be seen that, the maximum tunability of 0.0252(2pc/a) is obtained for full band gap at / = 0°as h ranges from 0° to 90°. The tunable full band gap is created between 9 and 10 bands. The same calculations have been done for other values of q2. Extensive investigations show that the maximum tunability of full band gap is obtained for triangular lattice with optimal geometrical parameters q1 = 0.3a and q2 = 0.35a in the X–Z plane (u = 0°) when h ranges from 0° to 90° and in the X–Y plane (h = 90°) as u ranges from 0° to 30°. In Fig. 7a we plot the variation of full band gap size as a function of h at u = 0° for above values of q1 and q2. From this figure one sees that the tunability of full band gap, which is created between 7 and 8 bands, is 0.0328(2pc/a) and varies from 0 to 0.0328 in units of (2pc/a). The frequency range of the full band gap at h = 90° is 0.454–0.4868 (2pc/a). Also, as shown in Fig. 7b, the same tunability of full band gap, which exists between 7 and 8 bands, is obtained for these values of q1 and q2 in the X–Y plane (h = 90°) as u ranges from 0° to 30°. From the fabrication viewpoint, by setting a = 500 nm, the triangular lattice of LC-infiltrated tellu-
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Fig. 7. Variation of full band gap size as a function of (a) h in the X–Z plane (u = 0°) and (b) u in the X–Y plane (h = 90°) at q1 = 0.3a and q2 = 0.35a for triangular lattice of LCinfiltrated anisotropic tellurium rods in air background.
rium rods can be realized with q1 = 150 nm and q2 = 175 nm. As discussed in the last subsection, such tunability in PCs can be utilized as a field-sensitive polarizer. For instance, the incident light of frequency xa/2pc = 0.465 or k 1075 nm at h = 0° and u = 0° or h = 90° and u = 30° will be transmitted, because there is no full band gap, but it can not propagate if h = 90°and u = 0°. 4. Conclusion We have studied the band structure properties of 2D square and triangular PCs created by LC-infiltrated anisotropic tellurium rods in air background, using the plane-wave expansion method. The resulting effects on the full photonic band gap of PC have been analyzed as a function of optical axis orientations of anisotropic LC material. Numerical results show that, the tunability of full band gap is much larger in triangular lattices than in square lattice. The present results open up new scope for designing tunable devices in photonic integrated circuits. References [1] [2] [3] [4]
S. John, Phys. Rev. Lett. 58 (1987) 2486. E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. K. Busch, S. John, Phys. Rev. Lett. 83 (1999) 967. K.H. Young, Y.-C. Yang, J.-E. Kim, H.Y. Park, C.-S. Kee, H. Lim, J.-C. Lee, Appl. Phys. Lett. 79 (2001) 15. [5] R. Ozaki, T. Matsui, M. Ozaki, K. Yoshino, Appl. Phys. Lett. 82 (2003) 3593. [6] G. Pucker, A. Mezzetti, M. Crivellari, P. Bellutti, A. Lui, N. Daldosso, L. Pavesi, J. Appl. Phys. 95 (2004) 767.
[7] G. Alagappan, X.W. Sun, P. Shum, M.B. Yu, M.T. Doan, J. Opt. Soc. Am. B 23 (2006) 159. [8] S.W. Leonard, J.P. Mondia, H.M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, V. Lehmann, Phys. Rev. B 61 (2000) R2389. [9] C.-S. Kee, H. Lim, Y.-K. Ha, J.-E. Kim, H.Y. Park, Phys. Rev. B 64 (2001) 085114. [10] D.M. Pustai, A. Sharkawy, S. Shi, D.W. Prather, Appl. Opt. 41 (2002) 5574. [11] C. Schuller, F. Klopf, J.P. Reithmaier, M. Kamp, A. Forchel, Appl. Phys. Lett. 82 (2003) 2767. [12] T.T. Larsen, A. Bjarklev, D.S. Hermann, J. Broeng, Opt. Express 11 (2003) 2589. [13] B. Maune, M. Loncˇar, J. Witzens, M. Hochberg, T.B. Jones, D. Psaltis, A. Scherer, Y. Qiu, Appl. Phys. Lett. 85 (2004) 360. [14] C.-Y. Liu, L.-W. Chen, Phys. Rev. B 72 (2005) 045133. [15] Y.-Y. Wang, L.-W. Chen, Opt. Express 14 (2006) 10580. [16] S.-M. Hsu, M.-M. Chen, H.-Ch. Chang, Opt. Express 15 (2007) 5416. [17] J.A. Reyes, J.A. Reyes-Avendaño, P. Halevi, Optics Commun. 281 (2008) 2535. [18] K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, M. Ozaki, Appl. Phys. Lett. 75 (1999) 932. [19] D. Kang, J.E. Maclennan, N.A. Clark, A.A. Zakhidov, R.H. Baughman, Phys. Rev. Lett. 86 (2001) 4052. [20] H. Takeda, K. Yoshino, J. Appl. Phys. 92 (2002) 5658. [21] I.H.H. Zabel, D. Stroud, Phys. Rev. B 48 (1993) 5004. [22] Z.Y. Li, J. Wang, B.Y. Gu, Phys. Rev. B 58 (1998) 3721. [23] Z.-Y. Li, J. Wang, B.-Y. Gu, Phys. Rev. Lett. 81 (1998) 2574. [24] C.-Y. Liu, Y.-T. Peng, J.Z. Wang, L.-W. Chen, Physica B 388 (2007) 124. [25] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, second ed., Princeton University Press, Princeton, NJ, 1995. [26] M. Plihal, A.A. Maradudin, Phys. Rev. B 44 (1991) 8565. [27] K. Busch, S. John, Phys. Rev. E 58 (1998) 3896. [28] I.-C. Khoo, S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals, World Scientific, Singapore, 1993. [29] K.M. Ho, C.T. Chen, C.M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152. [30] H. Takeda, K. Yoshino, Phys. Rev. E 70 (2004) 026601-1. [31] G. Alagappan, X.W. Sun, P. Shum, M.B. Yu, D. den Engelsen, J. Opt. Soc. Am. A 23 (2006) 2002. [32] C.-Y. Liu, L.-W. Chen, Optics Commun. 256 (2005) 114.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Tunable full band gap in two-dimensional anisotropic photonic crystals infiltrated with liquid crystals B. Rezaei a,*, M. Kalafi a,b a b
Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran Physics Department, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 5 September 2008 Received in revised form 1 January 2009 Accepted 6 January 2009
a b s t r a c t We analyze the tunability of full band gap in two-dimensional photonic crystals created by square and triangular lattices of anisotropic tellurium rods in air background, considering that the rods are infiltrated with liquid crystal. Using the plane-wave expansion method, we study the variation of full band gap by changing the optical axis orientation of liquid crystal. Ó 2009 Elsevier B.V. All rights reserved.
PACS: 42.70.Qs 77.84.Nh 78.20.Bh Keywords: Tunable full band gap Anisotropic photonic crystal Liquid crystal
1. Introduction Photonic crystals (PCs) are periodic dielectric materials designed to affect the propagation of electromagnetic waves in the same way as the periodic potential in semiconductor crystals affects the electron motion by defining allowed and forbidden energy bands. Since first proposed by John [1] and Yablonovitch [2], their potential scientific and technological applications have inspired great interest among researchers. Photonic crystals offer an important opportunity to design new optical devices and hold a great potential for many significant applications, such as semiconductor lasers and solar cells, high quality resonator and filters, and optical fibers. In the last decade there has been an increasing emphasis on tuning the optical properties of photonic bandgap structures in order to design switchable or dynamical devices. Busch and John [3] predicted the tunability of band gap in three dimensional (3D) PCs by utilizing liquid crystals (LCs). Following this publication, some investigations of band gap tunability have been done by utilizing LCs in one-dimensional (1D) [4–7], twodimensional (2D) [8–17] and 3D [18–20] PCs. It is important to note that, in 2D PCs composed of LCs, the classification of the TE (transverse electric) and TM (transverse * Corresponding author. Tel.: +98 411 339 3027; fax: +98 411 334 7050. E-mail addresses: [email protected], [email protected] (B. Rezaei). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.01.007
magnetic) modes is possible when the directors of LCs are parallel and perpendicular to 2D plane. However, for other alignment of directors none of these two classifications of modes exist due to anisotropies of LCs, i.e. the mode coupling occurs. Materials play a significant role in determining the optical properties of a PC. The properties of PCs made of anisotropic materials differ largely from those of isotropic PCs. It has been reported by Zabel and Stroud that the anisotropy of materials can split degenerate bands and this will narrow the band gap of the PC [21]. In 1998, Li et al. have demonstrated that the band gap can be increased by using the anisotropic materials in a PC [22,23]. Recently, it has been reported that, the infiltration of nematic LC in 3D PC, made of anisotropic tellurium material, can affect the values and properties of full photonic band gaps [24]. They have found that, the band gaps can be controlled by rotating the directors of LCs under the influence of an applied electric field. Unlike the most potential applications of the 3D PCs, 2D ones are easier to fabricate, especially for the technologically important near-infrared or visible spectrum. Much attention has, therefore, been paid to 2D PCs. In this work, we consider 2D square and triangular PCs created by hollow anisotropic tellurium rods. Based on plane-wave expansion method (PWEM), we study the evolution of full band gap by changing the directors of LCs that are infiltrated throughout the hollow tellurium rods.
1585
B. Rezaei, M. Kalafi / Optics Communications 282 (2009) 1584–1588
2. Formulation To determine the photonic band gap in periodic dielectric structures, we study the propagation of electromagnetic waves from Maxwell’s equations using the plane-wave method as illustrated in several papers [25–27]. Following the discussion of Busch and John [3], the Maxwell’s equation for the magnetic field in 2D PCs utilizing LCs can be expressed as
r
1 x2 r Hð~ rÞ ¼ 2 Hð~ rÞ c eð~rÞ
ð1Þ
ei;j ð~rÞ ¼
i~ G:~ r
ei;j ð~ GÞe ; ði; j ¼ x; y; zÞ
ð2Þ
~ G
where eð~ GÞ is the Fourier transform of the eð~ rÞ, and plays a key role in determination of photonic band structure. Generally LCs possess two dielectric constants known as ordinary dielectric constant eo and extraordinary dielectric constant ee. The light waves with electric field perpendicular and parallel to the director of LC experience ordinary and extraordinary dielectric constants, respectively. When the director of LC rotates, the components of the dielectric tensor can be represented as [28]:
exx ð~rÞ ¼ eo þ ðee eo Þ sin2 ðhÞ cos2 ðuÞ eyy ð~rÞ ¼ eo þ ðee eo Þ sin2 ðhÞ sin2 ðuÞ exy ð~rÞ ¼ eyx ð~rÞ ¼ ðee eo Þ sin2 ðhÞ sinðuÞ cosðuÞ exz ð~rÞ ¼ ezx ð~rÞ ¼ ðee eo Þ sinðhÞ cosðhÞ cosðuÞ eyz ð~rÞ ¼ ezy ð~rÞ ¼ ðee eo Þ sinðhÞ cosðhÞ sinðuÞ exx ð~rÞ ¼ eo þ ðee eo Þ sin2 ðhÞ cos2 ðuÞ
ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ ð8Þ
where h is the zenith angle of the LC director (i.e. the angle between the LC director and the Z axis), u is the azimuth angle between the projection of the LC director on the X–Y plane and the X axis, and ~ n is the director of the LC, as shown in Fig. 1. Since light waves are transmitted in periodic structures, we can expand the magnetic field in terms of plane waves in the same way:
Hð~ rÞ ¼
2 XX ~ G
~ ~
h~G;k ^ek eiðkþGÞ:~r
k¼1
X
H~G;~G0
~ G0
h~G0 ;1
!
h~G0 ;2
¼
x2 h~G;1 c2
! ð10Þ
h~G;2
where
where, x is the frequency of light and c is the light velocity. The rÞ ¼ eð~ r þ~ RÞ is periodic with respect to the real dielectric tensor eð~ space lattice vector ~ R and we can use Bloch’s theorem to expand it as a sum of plane waves
X
where ~ k is a wave vector in the first Brillouin zone and ~ G is a 2D reciprocal lattice vector, ^ek ðk ¼ 1; 2Þ are orthogonal unit vectors perpendicular to ~ kþ~ G. In the case of dielectric rods parallel to the Z kþ~ Gj, direction, ^e1 and ^e2 are ð0; 0; 1Þ and ððky þ Gy Þ; kx þ Gx ; 0Þ=j~ respectively. Substituting Eqs. (2) and (9) into Eq. (1), we obtain the following linear matrix equation for the dispersion of electromagnetic waves:
ð9Þ
A H~G;~G0 ¼ j~ kþ~ Gjj~ kþ~ G0 j C
B
D
ð11Þ
With
A ¼ ^e2 e1 ð~ G~ G0 Þ ^e02 ¼ j~kþ~Gjj1~kþ~G0 j h 0 1 ~ ~ ~0 ~0 ðky þ Gy Þðky þ G0y Þe1 xx ðG G Þ ðky þ Gy Þðkx þ Gx Þexy ðG G Þ i 0 1 ~ ~ ~0 ~0 ðkx þ Gx Þðky þ G0y Þe1 yx ðG G Þ þ ðkx þ Gx Þðkx þ Gx Þeyy ðG G Þ B ¼ ^e2 e1 ð~ G~ G0 Þ ^e01 ¼
1 j~ kþ~ Gj
h i 1 ~ ~ ~0 ~0 ðky þ Gy Þe1 xz ðG G Þ ðkx þ Gx Þeyz ðG G Þ C ¼ ^e1 e1 ð~ G~ G0 Þ ^e02 ¼
1 j~ kþ~ G0 j
h i 0 1 ~ ~ ~0 ~0 ðky þ G0y Þe1 zx ðG G Þ ðkx þ Gx Þezy ðG G Þ ~ ~0 D ¼ e1 zz ðG G Þ ð12Þ The matrix H~G;~G0 is real and symmetric; thus the dispersion relation of electromagnetic waves can be solved by using standard diagonalization techniques for the linear system of Eq. (10). The main numerical problem in obtaining the eigenvalues is the evaluation of the Fourier coefficients of the inverse dielectric tensor in Eq. (12). The best method is to calculate the matrix of Fourier coefficients of real-space tensors and then take its inverse to obtain the required Fourier coefficients, which was shown by Ho et al. [29]. The structures under consideration and the corresponding first Brillouin zones are depicted in Fig. 2. We have considered (a) square and (b) triangular structures of circular shape anisotropic tellurium rods in air background, considering that the rods are infiltrated with LC. The anisotropic tellurium has two different principle refractive indices as ordinary-refractive index nTe o ¼ 4:8 and extraordinary refractive index nTe e ¼ 6:2, in which the extraordinary one is parallel to the Z-axis. We assumed the LC with ordinary index nLC o ¼ 1:59 and the extraordinary refractive index ¼ 2:223. This LC corresponds to the phenylacetylene type LC nLC e [30]. Parameters q1 and q2 denote the inner and outer radius of anisotropic tellurium rods, respectively. a is being the lattice constant. 3. Numerical results
Fig. 1. Schematic representation of rotation angles for the LC directors.
For this study, we consider 2D square and triangular PCs of circular shape anisotropic tellurium rods in a uniform air background with dielectric constant eb = 1, considering that the rods are infiltrated with LC as shown in Fig. 2. We choose the LCs as an anisotropic material for the convenient change of anisotropy by simply changing the orientation of LC molecules. The components of the dielectric tensor of the LC are given by Eqs. (3)–(8). The band structure of 2D PCs is obtained numerically by solving Eq. (10). A total of 441 plane waves were employed for both structures in
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Fig. 2. Photonic crystal structures and the corresponding first Brillouin zones for (a) square and (b) triangular lattices of LC-infiltrated anisotropic tellurium rods in air background.
these calculations. This number is known to be sufficient enough to obtain acceptable convergence for characterizing the band structures. Our main goal here is to study the modification of the band gap spectrum and the value of the full band gap by varying LC’s optical axis orientation as well as when the thickness of anisotropic tellurium outer shell varies. The band structures of these two kinds of lattices will be separately analyzed in the next two subsections. 3.1. Square Lattice We first consider a 2D PC of square lattice composed of LC-infiltrated anisotropic tellurium rods in air background, as illustrated in Fig. 2a. The corresponding first BZ is also depicted in Fig. 2a. For isotropic PCs, the full band structure can be obtained by simply considering all the wave vectors ~ k in the irreducible Brillouin zone (IBZ) which is enclosed by points C, X, and M as shown in Fig. 2a. The resulting full photonic band structure obtained from this IBZ for isotropic PCs based on the rotation and reflection symmetry of structure. In a PC made of anisotropic materials, the definition of the IBZ depends on the orientation of the principal axes of anisotropic material [31] and hence it maybe difficult to define the IBZ for each optic axis orientation. So, it is reasonable to consider the more symmetry directions in the first BZ for obtaining the complete photonic band structure. First we consider the case when there is no LC (solid tellurium rods), i.e. the inner rod radius is q1 = 0. Hence, the dielectric configuration is similar to that of an isotropic material. When the dielectric property is isotropic, the complete band structures will be calculated in the IBZ. Fig. 3 shows the full photonic band structures of this structure in the IBZ for optimum rod radius q2 = 0.36a, for which the full band gap reaches its maximum normalized width of Dx = 0.0341(2pc/a) between 0.2176 and 0.2517 frequencies in units of (2pc/a). At next step, for fixed optimum value of q2 = 0.36a, the inner spaces of rods are infiltrated with LC. The LC can be infiltrated in the air holes by a similar technique used for filling PCs made of macro-porous silicon [8]. For this purpose, the sample can be
Fig. 3. Full photonic band structures for square lattice of tellurium rods in air background at optimum rod radius of q2 = 0.36a.
placed in an evacuated flask. The LC is injected through shot needle. Because of strong capillarity forces, the air holes will be infiltrated easily. In this case, we study the variation of full band gap by changing the director of LC for different values of inner rod radius q1. We notice that, due to the symmetry of the underlying lattice, only the band shapes for / values between 0 and 45 are fundamentally unique. To obtain the maximum tunability of full band gap, we change the director of LC for different values of q1 by varying the h and / angles from 0° to 90° and from 0° to 45°, respectively. The numerical results show that the maximum tunability of full band gap is obtained at q1 = 0.3a and q2 = 0.36a. As shown in Fig. 4, the maximum tunability of 0.0098(2pc/a) is obtained for full band gap when h changes from 0° to 90° for all values of / except for the / = 45°. At h = 0° the full band gap with normalized width of 0.0098(2pc/a) is created between 4 and 5 bands and lies between 0.3145 and 0.3243 frequencies in units
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Fig. 4. Tunability of full band gap for square lattice of LC-infiltrated anisotropic tellurium rods in air background at q1 = 0.3 and q2 = 0.36a. h and u are varied from 0° to 90° and from 0° to 0°, respectively.
1587
Fig. 5. Photonic band structures for triangular lattice of tellurium rods in air background at optimum rod radius of q2 = 0.355a.
of (2pc/a). When h = 90°, the photonic band gap is completely disappeared. It should be noted that, for / = 45°, the full band gap changes from 0.0098 to 0.0014 in units of (2pc/a). For a comprehensive investigation, we have also done such calculations for other values of q2. After extensive calculations, we have found that the maximum tunability of full band gap is obtained only for the above optimal values of q1 and q2. Taking into account the fabrication limit, for the lattice constant of a = 500nm, the square lattice of LC-infiltrated tellurium rods can be realized by inner radius q1 = 150 nm and outer radius q2 = 180 nm. This tunable PC can be used as a field-sensitive polarizer [32]. For example, the light of frequency xa/2pc = 0.315 ðk 1587 nmÞ cannot propagate through the PC if h = 0°, while at h = 90° it will be transmitted, because the full band gap has been disappeared. 3.2. Triangular lattice Fig. 2b shows the second structure to be analyzed; the 2D PC with triangular lattice composed of LC infiltrated anisotropic tellurium rods in air background. The corresponding first BZ of the reciprocal lattice is a hexagon, as depicted in Fig. 2b. The analysis of band structures for triangular lattice is similar to that of square lattice, as discussed in the last subsection. For a hexagonal BZ, it is necessary to consider the six distinct sub-zones as marked in Fig. 2b for obtaining the complete band structures of PCs made of anisotropic materials. Similar to the square lattice, we first consider the triangular lattice of solid tellurium rods (without LC material). Thus, the dielectric configuration, as explained in the last subsection, is similar to isotropic materials. Hence, the corresponding full band structures of this PC is shown in Fig. 5 in the IBZ which is enclosed by points C, M and K as shown in Fig. 2b. The maximum normalized width of Dx = 0.0414(2pc/a) is obtained for full band gap at optimum value of q2 = 0.355a and lies between 0.2198 and 0.2612 frequencies in units of (2pc/a). To get an insight into the effect of the orientation of LC directors on the full band gap, we first concentrate on optimum value of q2 = 0.355a as the inner spaces of rods are infiltrated with LC. Similar to the square lattice, because of the symmetry properties of the underlying lattice, the band structures for / values between 0° and 30° are unique. So, for different values of internal radius, q1, we study the tunability of full band gap as the LC molecules are
Fig. 6. Tunability of full band gap for triangular lattice of LC-infiltrated anisotropic tellurium rods in air background at q1 = 0.26 and q2 = 0.355a. h and u are varied from 0° to 90° and from 0° to 30°, respectively.
aligned in different orientations. The numerical results show that the maximum tunability of full band gap is obtained at q1 = 0.26a and q2 = 0.355a. Fig. 6 shows the variation of full band gap size as a function of h and / for these parameters. It can be seen that, the maximum tunability of 0.0252(2pc/a) is obtained for full band gap at / = 0°as h ranges from 0° to 90°. The tunable full band gap is created between 9 and 10 bands. The same calculations have been done for other values of q2. Extensive investigations show that the maximum tunability of full band gap is obtained for triangular lattice with optimal geometrical parameters q1 = 0.3a and q2 = 0.35a in the X–Z plane (u = 0°) when h ranges from 0° to 90° and in the X–Y plane (h = 90°) as u ranges from 0° to 30°. In Fig. 7a we plot the variation of full band gap size as a function of h at u = 0° for above values of q1 and q2. From this figure one sees that the tunability of full band gap, which is created between 7 and 8 bands, is 0.0328(2pc/a) and varies from 0 to 0.0328 in units of (2pc/a). The frequency range of the full band gap at h = 90° is 0.454–0.4868 (2pc/a). Also, as shown in Fig. 7b, the same tunability of full band gap, which exists between 7 and 8 bands, is obtained for these values of q1 and q2 in the X–Y plane (h = 90°) as u ranges from 0° to 30°. From the fabrication viewpoint, by setting a = 500 nm, the triangular lattice of LC-infiltrated tellu-
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Fig. 7. Variation of full band gap size as a function of (a) h in the X–Z plane (u = 0°) and (b) u in the X–Y plane (h = 90°) at q1 = 0.3a and q2 = 0.35a for triangular lattice of LCinfiltrated anisotropic tellurium rods in air background.
rium rods can be realized with q1 = 150 nm and q2 = 175 nm. As discussed in the last subsection, such tunability in PCs can be utilized as a field-sensitive polarizer. For instance, the incident light of frequency xa/2pc = 0.465 or k 1075 nm at h = 0° and u = 0° or h = 90° and u = 30° will be transmitted, because there is no full band gap, but it can not propagate if h = 90°and u = 0°. 4. Conclusion We have studied the band structure properties of 2D square and triangular PCs created by LC-infiltrated anisotropic tellurium rods in air background, using the plane-wave expansion method. The resulting effects on the full photonic band gap of PC have been analyzed as a function of optical axis orientations of anisotropic LC material. Numerical results show that, the tunability of full band gap is much larger in triangular lattices than in square lattice. The present results open up new scope for designing tunable devices in photonic integrated circuits. References [1] [2] [3] [4]
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