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Creation of tunable absolute bandgaps in a two-dimensional anisotropic photonic crystal modulated by a nematic liquid crystal
Physics Letters A 372 (2008) 5198–5202
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Creation of tunable absolute bandgaps in a two-dimensional anisotropic photonic crystal modulated by a nematic liquid crystal Chen-Yang Liu ∗ Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu, Taiwan
a r t i c l e
i n f o
Article history: Received 8 May 2008 Accepted 12 June 2008 Available online 14 June 2008 Communicated by R. Wu PACS: 42.70.Qs 42.70.Df Keywords: Tunable absolute bandgap Anisotropic photonic crystal Liquid crystal
a b s t r a c t Photonic crystals (PCs) have many potential applications because of their ability to control light-wave propagation. We have investigated the tunable absolute bandgap in a two-dimensional anisotropic photonic crystal structures modulated by a nematic liquid crystal. The PC structure composed of an anisotropic-dielectric cylinder in the liquid crystal medium is studied by solving Maxwell’s equations using the plane wave expansion method. The photonic band structures are found to exhibit absolute bandgaps for the square and triangular lattices. Numerical simulations show that the absolute bandgaps can be continuously tuned in the square and triangular lattices consisting of anisotropic-dielectric cylinders by infiltrating nematic liquid crystals. Such a mechanism of bandgap adjustment should open up a new application for designing components in photonic integrated circuits. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Recently, there has been growing interest in studies of the propagation of electromagnetic waves in periodic dielectric structures [1]. These structures exhibit forbidden frequency region where electromagnetic waves cannot propagate for both polarizations along any direction. This may bring about some peculiar physical phenomena, many of which are based on the suppression of instantaneous emission in the presence of a photonic bandgap [2]. Furthermore, it has been shown that doped photonic crystals (PCs) permit the guiding of waves in two different geometric paths for two distinct wavelength ranges [3]. Such structures can be used to design highly efficient new optical devices. Optical waveguides in two-dimensional (2D) PCs produced by insertion of linear defects into PC structures had been proposed [4] and experimentally proved [5]. Planar PC circuits consist of devices, such as splitters [6], filters [7], and multichannel drop filters [8], by controlling the interaction between static devices, such as waveguides, cavities, or horns. It is also proposed that such PC structures may hold the key to continued progress towards photonic integrated circuits. It is well known that there exist only pseudogaps in an isotropic face-centered-cubic (fcc) lattice composed of a dielectric spheres in
*
Correspondence address: Room 216, Building 8, 321, Sec. 2, Kuang Fu Road, Hsinchu, Taiwan. Tel.: +886 3 5743765; fax: +886 3 5726445. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.06.019
a uniform background medium, because of symmetry-induced degeneracy between the conduction band and the valence band at the W point of the Brillouin zone [9,10]. The three-dimensional (3D) PC structure of the diamond lattice was then proposed to possess a full gap by theoretical simulations [11], as was soon verified by experiments [12]. The symmetry-induced degeneracy at the W point is broken by the nonspherical atom configuration, and a complete bandgap opens up between the second and third bands. Recently, it was found that the anisotropy in atom dielectricity can break the degeneracy of photonic bands such that partial band gaps can be created in fcc, body-centered-cubic (bcc), and simple-cubic (sc) lattices [13]. It was also demonstrated that an anisotropy in dielectricity can remarkably increase absolute bandgaps in 2D PC structures [14,15]. More recently, Kushwaha and Martinez treated a 2D periodic system of semiconductor cylinders embedded in a dielectric background [16]. It is important, however, to obtain tunable PC waveguides for applications in optical devices. Tunable PC structures that utilize synthetic opals and inverse opals infiltrated with functional materials have been proposed [17–19]. One can control the refractive indices of opals by adjusting various factors and fields. For example, one can change the refractive indices of conducting polymers and liquid crystals (LCs) by changing the temperature and the electric field of the polymer or crystal. Therefore one can change the optical properties of tunable PC waveguides composed of such materials by changing the temperature and the electric field in the same way. Recently the photonic bandgap of a 2D PC is continuously varied using the temperature dependent refractive index of
C.-Y. Liu / Physics Letters A 372 (2008) 5198–5202
an LC [20]. The propagation of tunable light in Y-shaped waveguides in 2D PCs by use of LCs as linear defects was discussed [21,22]. A tunable PC waveguide coupler based on nematic LCs was presented by the authors [23], and then we have investigated the tunable bandgap in a 2D photonic crystal modulated by a nematic liquid crystal [24]. These results can be used to obtain a tunable field-sensitive polarizer in photonic integrated circuits [25]. More recently, the effect of the optical axis orientation on the bandgap of a PC made of anisotropic materials was studied with examples of a one-dimensional PC infiltrated with LC and Pockel materials [26]. In this Letter, we theoretically demonstrate the tunable absolute bandgaps in 2D anisotropic PC structure with nematic LCs. Using a plane wave expansion method (PWE), we have solved Maxwell’s equations for the propagation of electromagnetic waves in a periodic arrangements of anisotropic dielectric cylinders by infiltrating nematic LCs. Photonic bandgaps can be improved in photonic crystals fabricated from anisotropic materials. We found that the PC structure with LC does possess a tunable photonic bandgap. The bandgaps can be controlled by rotating directors of LCs under the influence of an applied electric field. Such a mechanism of bandgap adjustment should open up a new application for designing components in photonic integrated circuits. Details of the calculations and discussion of the results will be presented in the remainder of the Letter.
5199
= where φ is the rotation angle of the director of the LCs, and n (cos φ, sin φ) is the director of the LC, as shown in Fig. 1. Since ε (r ) is periodic, we can use Bloch’s theorem to expand the H (r ) field in terms of plane waves, H (r ) = h(G )e G exp i (k + G ) · r , (7) G
where k is a wave vector in the Brillouin zone of the lattice, and we define e G as the direction which is perpendicular to the wave vector k + G because of the transverse character of the magnetic field H (r ) (i.e., ∇ · H (r ) = 0). We insert Eqs. (2)–(7) into Eq. (1) and multiply by e G . This results in the following infinite matrix eigenvalue problem:
G
H G ,G h (G ) =
ω c
2 h(G ).
(8)
where ∇ · H (r ) = 0. The dielectric tensor ε (r ) = ε (r + R ) is periodic with respect to the lattice vectors R generated by primitive translation, and it may be expanded in a Fourier series on G, the reciprocal lattice vector as
The main numerical problem in obtaining the eigenvalue is the evaluation of the Fourier coefficients of the inverse dielectric tensors. The best method is to calculate the matrix of Fourier coefficients of real space tensors and take its inverse in order to obtain the required Fourier coefficients. This method was shown by Ho, Chan, and Soukoulis (HCS) [11]. The eigenvalues computed with the HCS method for 512 plane waves are estimated to be in error less than 1%. In our calculations, the convergence is quite fast for low-energy bands. In particular, we study the photonic band structure of an anisotropic material Te (tellurium) which is a kind of positive uniaxial crystal with principal refractive indices ne = 6.2 and no = 4.8. The ordinary and extraordinary refractive indices of liquid crystals (5CB type) are noLC = 1.522 and neLC = 1.706, respectively. Fig. 1 in of a liquid crystal and the rotation angle φ dicates the director n of the director to the x-axis. The mesogenic temperature range of a single LC substance is usually quite limited [29]. For example, 5CB melts at 24 ◦ C and clears at 35.3 ◦ C. 5CB is a nice material to work with because it exhibits a nematic phase at room temperature and its nematic range is more than 10 degrees. We assume that the operating temperature is at a constant room temperature and that the absorption loss is negligible.
ε(r ) =
3. 2D anisotropic PC with LC
2. Numerical method The PWE method is illustrated in several papers [13,27,28]. Here we summarize the theory very briefly. Following the discussion of Busch and John [27], we can express the light-wave equation that is satisfied by the magnetic field in order to determine the photonic bandgaps of periodic structures utilizing nematic LCs,
∇×
1
ε(r )
2 ω H (r ), ∇ × H (r ) = c
ε(G ) exp(iG · r ).
(1)
(2)
G
The uniaxial material has two different principal refractive indices known as the ordinary refractive index no and the extraordinary refractive index ne . For such anisotropic materials, the dielectric tensor ε (r ) is a second rank tensor [13]. In the principal coordinates, the diagonal elements of ε (r ) are related to the principal refractive indices as
εxx = ne2 ,
ε y y = εzz = no2 ,
We consider that the 2D square-lattice PC is composed of isotropic and anisotropic dielectric cylinders surrounded by LC 5CB, as shown in Fig. 1. The material is homogeneous in the z direction, and periodic along x and y with lattice constant. The lattice constant is a and the radius of cylinders is r. The refractive index of the isotropic dielectric cylinders is n = 3.4 (Si), while for
(3)
while the other dyadic elements are all zero. Generally LCs possess also two kinds of dielectric index. One is the ordinary dielectric index εo , and the other is the extraordinary dielectric index ε e . Light waves with electric fields perpendicular and parallel to the director of the LC have ordinary and extraordinary refractive indices, respectively. The extended Jones matrix method [29] is a simple and powerful approach for dealing with the light transmission problem of a LC device at normal incidence. In the 2D materials, the components of the dielectric tensor of the nematic LC are represented as [29]
εxx (r ) = εo (r ) sin2 φ + εe (r ) cos2 φ, o
2
2
e
ε y y (r ) = ε (r ) cos φ + ε (r ) sin φ,
εxy (r ) = ε yx (r ) = ε (r ) − ε (r ) cos φ sin φ, e
o
(4) (5) (6)
Fig. 1. Photonic crystal structure with square lattices. The shaded region is infiltrated with liquid crystals. The left inset indicates the director of a liquid crystal.
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Fig. 2. Calculated photonic band structures of the 2D square array of isotropic dielectric cylinders in the air, anisotropic dielectric cylinders in the air, in the nematic liquid crystal at φ = 0◦ , and in the nematic liquid crystal at φ = 90◦ for TM mode (solid line) and TE mode (dotted lines). The radius of cylinders is taken as r = 0.35a. The gray areas represent absolute bandgaps.
anisotropic materials they are ne = 6.2 and no = 4.8 (Te). To investigate the existence of photonic bandgaps for such an anisotropic material, we calculate the photonic band structures along the edge of the irreducible Brillouin zone. The standard Brillouin zone of various lattices can be found in a textbook [2] and thus is not shown here. We show in Fig. 2 an example for the photonic band structure of an isotropic PC with a refractive index constant of n = 3.4 and r = 0.35a, where ω is the angular frequency, and c is the light velocity in the free space. This band structure is in excellent agreement with the result in Ref. [15]. The photonic properties of isotropic structures have been studied and shown to exhibit bandgaps for TM modes. It is clearly see that three bandgaps open for the TM mode (solid line), i.e., the 1–2 bandgap, 3–4 bandgap, and 6–7 bandgap. However, there are no significant TE gaps (dotted lines) in the frequency range displayed. The isotropic dielectric cylinders resulted in the absence of the absolute bandgap. The absolute bandgap can be created by introducing the anisotropy in the dielectric of cylinders. In Fig. 2, we also show the photonic band structures of 2D square lattice of anisotropic dielectric cylinders in the air, in the nematic liquid crystal at φ = 0◦ , and in the nematic liquid crystal at φ = 90◦ , respectively. The gray areas represent the absolute bandgap. We have chosen r = 0.35a, since the 2D square lattice of anisotropic cylinders in the air shows the largest absolute bandgap. The anisotropy-induced absolute bandgaps for the square lattice PC in air have a first bandgap (ωa/2π c = 0.2321–0.2701) and a second bandgap (ωa/2π c = 0.4631–0.4734). The photonic bandgaps can be improved in photonic crystals fabricated from anisotropic material. We have made a systematic examination of the photonic band structures for anisotropic dielectric cylinders in the LC background as a function of the director of the LC. The rotation angle φ was varied to change the dielectric tensor of LC medium. Fig. 2 also shows the photonic band structures of square lattice of anisotropic dielectric cylinders in the nematic LC background at φ = 0◦ and φ = 90◦ , respectively. The infiltration of LC can shift the first bandgap to 0.2279–0.2552, the second bandgap to 0.4317–0.443 and arise other two small bandgaps at φ = 0◦ . Rotation of the LC director causes the bandgap to disappear in the high frequency region, and the upper edge of the first bandgap moves down drastically with little decrease of bottom edge. The first bandgap was shifted to 0.2261–0.2335 at φ = 90◦ . The gap maps for a square lattice of anisotropic dielectric cylinders are shown in Fig. 3. At a
Fig. 3. Gap maps for a square lattice of anisotropic dielectric cylinders.
Fig. 4. Absolute gap maps for a square lattice of anisotropic dielectric cylinders.
glance, the gap map reveals some interesting regularities. First, the gaps all decrease in frequency as r /a increases. Second, the gaps all decrease in frequency as φ increases. The third, the gaps at higher frequencies disappear as φ increases. The fourth, all of the gaps
C.-Y. Liu / Physics Letters A 372 (2008) 5198–5202
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Fig. 5. Calculated photonic band structures of the 2D triangular array of isotropic dielectric cylinders in the air, anisotropic dielectric cylinders in the air, in the nematic liquid crystal at φ = 0◦ , and in the nematic liquid crystal at φ = 90◦ for TM mode (solid line) and TE mode (dotted lines). The radius of cylinders is taken as r = 0.35a. The gray areas represent absolute bandgaps.
seal up at r /a = 0.5. At that value, the dielectric cylinders begin to touch one another. The dielectric cylinders fill space at r /a = 0.7. The absolute gap maps for a square lattice of anisotropic dielectric cylinders are shown in Fig. 4. They all decrease in frequency as the rotation angle increases. The results mean that the absolute bandgaps of PC can be actively modulated by infiltrating nematic LCs. Such a concept is also applicable to other lattice types and atom configurations. We consider that a 2D PC composed of triangularlattice dielectric cylinders surrounded by LC 5CB. We calculated the photonic band structure of 2D triangular arrays using the same parameters as those employed in the square lattice. In Fig. 5, we also show the photonic band structures of 2D triangular lattice of anisotropic dielectric cylinders in the air, in the nematic liquid crystal at φ = 0◦ , and in the nematic liquid crystal at φ = 90◦ , respectively. The photonic properties of isotropic structures have been studied and shown to exhibit bandgaps for each of the two polarization modes. It is clearly see that several bandgaps open for the TM mode (solid line), i.e., the 1–2 bandgap, 3–4 bandgap, and 6–7 bandgap. For TE mode (dotted lines) a bandgap is opened between 1–2 bands. However, the bandgaps in the two polarization modes do not overlap with each other, resulting in the absence of the absolute bandgap. The absolute bandgap can be created by introducing the anisotropy in the dielectric of cylinders. The anisotropy-induced absolute bandgaps for the triangular lattice PC in air have a bandgap (ωa/2π c = 0.2311–0.2773). The infiltration of LC closes the first bandgap and produces other two bandgaps at φ = 0◦ . Rotation of the LC director causes the bandgaps to decrease in the high frequency region and opens the first bandgap to 0.2261–0.2358 at φ = 90◦ . The gap maps for a triangular lattice of anisotropic dielectric cylinders are shown in Fig. 6. The remarkable self-similarity of Fig. 3, which was for TM mode of the square lattice of dielectric cylinders, is mirrored here. The successive gaps are similar in shape and orientation, and stack regularly upon one another. The gaps decrease in frequency as φ increases. The dielectric cylinders begin touching one another at r /a = 0.5, and fill space at r /a = 0.58. The cutoff at r /a = 0.45 is once again near the cylinder-touching condition. The absolute gap maps for a triangular lattice of anisotropic dielectric cylinders are shown in Fig. 7. They all decrease in frequency as the rotation angle increases. The results show that the bandgap could be actively modulated after infiltrating nematic LCs. Such a mechanism of bandgap adjustment
Fig. 6. Gap maps for a triangular lattice of anisotropic dielectric cylinders.
Fig. 7. Absolute gap maps for a triangular lattice of anisotropic dielectric cylinders.
should open up new possibilities for designing components in photonic integrated circuits. Fig. 8 shows the variation of absolute bandgap due to the change of the rotation angle of LC. The colored areas represent tun-
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C.-Y. Liu / Physics Letters A 372 (2008) 5198–5202
of the square and triangular lattice of anisotropic dielectric cylinders. Photonic bandgaps can be improved in photonic crystals fabricated from anisotropic material. It is also demonstrated that the absolute bandgap can be controlled by rotating directors of LCs under the influence of an applied electric field. Due to large varieties of anisotropic materials in LC, the present results open up a new scope for designing tunable devices in photonic integrated circuits. Further theoretical investigations and experimental efforts are needed to bring the tunable PC into reality. References
Fig. 8. Absolute bandgaps as a function of rotation angle φ . The radius of cylinders is taken as r = 0.35a.
able absolute bandgaps. The fourth bandgap of the square lattice sealed up at φ = 50◦ . The first and fourth bandgaps of the triangular lattice opened up at φ = 30◦ and 60◦ . The absolute bandgaps could be actively modulated after infiltrating nematic LCs. This tunable PC can act as a tunable planar lightwave component. In the case of nematic LCs, the directors of LCs depend on the direction of the electric field. Indium tin oxide (ITO) layers can be attached to the top and the bottom of the PC structure. Then, the electric field can be applied by adjusting the magnitudes of the electric field in the x and y directions, which makes it possible to rotate the directors of the LCs. The director can be reoriented by an electric field, when the field strength exceeds the Fréedericksz transition threshold [29]. When the applied voltage V exceeds the Fréedericksz transition threshold (V th ), the directors begin to tilt. V th is the threshold voltage that is found to be 0.699V rms at 1 kHz sinusoidal frequency. In general, the response time of a LC is of the order of a millisecond. However, it has been reported that the response time of LCs in nanoscale voids becomes of the order of 100 μs [30]. The orientational relaxation times calculated by the molecular dynamics formalism and the experimental data determined by nuclear magnetic resonance spectroscopy for the nematic phase of a 5CB crystal at 300 K were presented in Ref. [31]. Therefore our novel tunable PC structure with LCs can be used as a fast switching component in photonic integrated circuits. 4. Conclusion We have demonstrated numerically, from the photonic band calculation, the effects of LC infiltration on the absolute bandgaps
[1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton Univ. Press, New Jersey, 1995. [3] E. Centeno, D. Felbacq, Opt. Commun. 160 (1999) 57. [4] R.D. Meade, A. Devenyi, J.D. Joannopoulos, O.L. Alerhand, D.A. Smith, K. Kash, J. Appl. Phys. 75 (1994) 4753. [5] T. Baba, N. Fukaya, J. Yonekura, Electron. Lett. 35 (1999) 654. [6] S. Fan, S.G. Johnson, J.D. Joannopoulos, C. Manolatou, H.A. Haus, J. Opt. Soc. Am. B 18 (2001) 162. [7] R. Costa, A. Melloni, M. Martinelli, Photon. Technol. Lett. 15 (2003) 401. [8] A. Sharkawy, S. Shi, D.W. Prather, Appl. Opt. 40 (2001) 2247. [9] K.M. Leung, Y.F. Liu, Phys. Rev. Lett. 65 (1990) 2646. [10] Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65 (1990) 2650. [11] K.M. Ho, C.T. Chan, C.M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152. [12] E. Yablonovitch, T.J. Gmitter, K.M. Leung, Phys. Rev. Lett. 67 (1991) 2295. [13] Z.-Y. Li, J. Wang, B.-Y. Gu, Phys. Rev. B 58 (1998) 3721. [14] Z.-Y. Li, B.-Y. Gu, G.-Z. Yang, Phys. Rev. Lett. 81 (1998) 2574. [15] Z.-Y. Li, B.-Y. Gu, G.-Z. Yang, Eur. Phys. J. B 11 (1999) 65. [16] M.S. Kushwaha, G. Martinez, Phys. Rev. B 65 (2002) 153202. [17] K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, M. Ozaki, Appl. Phys. Lett. 75 (1999) 932. [18] H. Takeda, K. Yoshino, J. Appl. Phys. 92 (2002) 5958. [19] S. Kubo, Z.-Z. Gu, K. Takahshi, A. Fujishima, H. Segawa, O. Sato, J. Am. Chem. Soc. 126 (2004) 8314. [20] 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. [21] H. Takeda, K. Yoshino, Phys. Rev. B 67 (2003) 073106. [22] H. Takeda, K. Yoshino, Opt. Commun. 219 (2003) 177. [23] C.-Y. Liu, L.-W. Chen, IEEE Photon. Technol. Lett. 16 (2004) 1849. [24] C.-Y. Liu, L.-W. Chen, Phys. Rev. B 72 (2005) 045133. [25] C.-Y. Liu, L.-W. Chen, Opt. Commun. 256 (2005) 114. [26] G. Alagappan, X.W. Sun, P. Shum, M.B. Yu, M.T. Doan, J. Opt. Soc. Am. B 23 (2006) 159. [27] K. Busch, S. John, Phys. Rev. E 58 (1998) 3896. [28] S. Guo, S. Albin, Opt. Express 11 (2003) 167. [29] I.-C. Khoo, S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals, World Scientific, Singapore, 1993. [30] Y. Shimoda, M. Ozaki, K. Yoshino, Appl. Phys. Lett. 79 (2001) 3627. [31] A.V. Zakharov, L.V. Mirantsev, Phys. Solid State 45 (2003) 183.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Creation of tunable absolute bandgaps in a two-dimensional anisotropic photonic crystal modulated by a nematic liquid crystal Chen-Yang Liu ∗ Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu, Taiwan
a r t i c l e
i n f o
Article history: Received 8 May 2008 Accepted 12 June 2008 Available online 14 June 2008 Communicated by R. Wu PACS: 42.70.Qs 42.70.Df Keywords: Tunable absolute bandgap Anisotropic photonic crystal Liquid crystal
a b s t r a c t Photonic crystals (PCs) have many potential applications because of their ability to control light-wave propagation. We have investigated the tunable absolute bandgap in a two-dimensional anisotropic photonic crystal structures modulated by a nematic liquid crystal. The PC structure composed of an anisotropic-dielectric cylinder in the liquid crystal medium is studied by solving Maxwell’s equations using the plane wave expansion method. The photonic band structures are found to exhibit absolute bandgaps for the square and triangular lattices. Numerical simulations show that the absolute bandgaps can be continuously tuned in the square and triangular lattices consisting of anisotropic-dielectric cylinders by infiltrating nematic liquid crystals. Such a mechanism of bandgap adjustment should open up a new application for designing components in photonic integrated circuits. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Recently, there has been growing interest in studies of the propagation of electromagnetic waves in periodic dielectric structures [1]. These structures exhibit forbidden frequency region where electromagnetic waves cannot propagate for both polarizations along any direction. This may bring about some peculiar physical phenomena, many of which are based on the suppression of instantaneous emission in the presence of a photonic bandgap [2]. Furthermore, it has been shown that doped photonic crystals (PCs) permit the guiding of waves in two different geometric paths for two distinct wavelength ranges [3]. Such structures can be used to design highly efficient new optical devices. Optical waveguides in two-dimensional (2D) PCs produced by insertion of linear defects into PC structures had been proposed [4] and experimentally proved [5]. Planar PC circuits consist of devices, such as splitters [6], filters [7], and multichannel drop filters [8], by controlling the interaction between static devices, such as waveguides, cavities, or horns. It is also proposed that such PC structures may hold the key to continued progress towards photonic integrated circuits. It is well known that there exist only pseudogaps in an isotropic face-centered-cubic (fcc) lattice composed of a dielectric spheres in
*
Correspondence address: Room 216, Building 8, 321, Sec. 2, Kuang Fu Road, Hsinchu, Taiwan. Tel.: +886 3 5743765; fax: +886 3 5726445. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.06.019
a uniform background medium, because of symmetry-induced degeneracy between the conduction band and the valence band at the W point of the Brillouin zone [9,10]. The three-dimensional (3D) PC structure of the diamond lattice was then proposed to possess a full gap by theoretical simulations [11], as was soon verified by experiments [12]. The symmetry-induced degeneracy at the W point is broken by the nonspherical atom configuration, and a complete bandgap opens up between the second and third bands. Recently, it was found that the anisotropy in atom dielectricity can break the degeneracy of photonic bands such that partial band gaps can be created in fcc, body-centered-cubic (bcc), and simple-cubic (sc) lattices [13]. It was also demonstrated that an anisotropy in dielectricity can remarkably increase absolute bandgaps in 2D PC structures [14,15]. More recently, Kushwaha and Martinez treated a 2D periodic system of semiconductor cylinders embedded in a dielectric background [16]. It is important, however, to obtain tunable PC waveguides for applications in optical devices. Tunable PC structures that utilize synthetic opals and inverse opals infiltrated with functional materials have been proposed [17–19]. One can control the refractive indices of opals by adjusting various factors and fields. For example, one can change the refractive indices of conducting polymers and liquid crystals (LCs) by changing the temperature and the electric field of the polymer or crystal. Therefore one can change the optical properties of tunable PC waveguides composed of such materials by changing the temperature and the electric field in the same way. Recently the photonic bandgap of a 2D PC is continuously varied using the temperature dependent refractive index of
C.-Y. Liu / Physics Letters A 372 (2008) 5198–5202
an LC [20]. The propagation of tunable light in Y-shaped waveguides in 2D PCs by use of LCs as linear defects was discussed [21,22]. A tunable PC waveguide coupler based on nematic LCs was presented by the authors [23], and then we have investigated the tunable bandgap in a 2D photonic crystal modulated by a nematic liquid crystal [24]. These results can be used to obtain a tunable field-sensitive polarizer in photonic integrated circuits [25]. More recently, the effect of the optical axis orientation on the bandgap of a PC made of anisotropic materials was studied with examples of a one-dimensional PC infiltrated with LC and Pockel materials [26]. In this Letter, we theoretically demonstrate the tunable absolute bandgaps in 2D anisotropic PC structure with nematic LCs. Using a plane wave expansion method (PWE), we have solved Maxwell’s equations for the propagation of electromagnetic waves in a periodic arrangements of anisotropic dielectric cylinders by infiltrating nematic LCs. Photonic bandgaps can be improved in photonic crystals fabricated from anisotropic materials. We found that the PC structure with LC does possess a tunable photonic bandgap. The bandgaps can be controlled by rotating directors of LCs under the influence of an applied electric field. Such a mechanism of bandgap adjustment should open up a new application for designing components in photonic integrated circuits. Details of the calculations and discussion of the results will be presented in the remainder of the Letter.
5199
= where φ is the rotation angle of the director of the LCs, and n (cos φ, sin φ) is the director of the LC, as shown in Fig. 1. Since ε (r ) is periodic, we can use Bloch’s theorem to expand the H (r ) field in terms of plane waves, H (r ) = h(G )e G exp i (k + G ) · r , (7) G
where k is a wave vector in the Brillouin zone of the lattice, and we define e G as the direction which is perpendicular to the wave vector k + G because of the transverse character of the magnetic field H (r ) (i.e., ∇ · H (r ) = 0). We insert Eqs. (2)–(7) into Eq. (1) and multiply by e G . This results in the following infinite matrix eigenvalue problem:
G
H G ,G h (G ) =
ω c
2 h(G ).
(8)
where ∇ · H (r ) = 0. The dielectric tensor ε (r ) = ε (r + R ) is periodic with respect to the lattice vectors R generated by primitive translation, and it may be expanded in a Fourier series on G, the reciprocal lattice vector as
The main numerical problem in obtaining the eigenvalue is the evaluation of the Fourier coefficients of the inverse dielectric tensors. The best method is to calculate the matrix of Fourier coefficients of real space tensors and take its inverse in order to obtain the required Fourier coefficients. This method was shown by Ho, Chan, and Soukoulis (HCS) [11]. The eigenvalues computed with the HCS method for 512 plane waves are estimated to be in error less than 1%. In our calculations, the convergence is quite fast for low-energy bands. In particular, we study the photonic band structure of an anisotropic material Te (tellurium) which is a kind of positive uniaxial crystal with principal refractive indices ne = 6.2 and no = 4.8. The ordinary and extraordinary refractive indices of liquid crystals (5CB type) are noLC = 1.522 and neLC = 1.706, respectively. Fig. 1 in of a liquid crystal and the rotation angle φ dicates the director n of the director to the x-axis. The mesogenic temperature range of a single LC substance is usually quite limited [29]. For example, 5CB melts at 24 ◦ C and clears at 35.3 ◦ C. 5CB is a nice material to work with because it exhibits a nematic phase at room temperature and its nematic range is more than 10 degrees. We assume that the operating temperature is at a constant room temperature and that the absorption loss is negligible.
ε(r ) =
3. 2D anisotropic PC with LC
2. Numerical method The PWE method is illustrated in several papers [13,27,28]. Here we summarize the theory very briefly. Following the discussion of Busch and John [27], we can express the light-wave equation that is satisfied by the magnetic field in order to determine the photonic bandgaps of periodic structures utilizing nematic LCs,
∇×
1
ε(r )
2 ω H (r ), ∇ × H (r ) = c
ε(G ) exp(iG · r ).
(1)
(2)
G
The uniaxial material has two different principal refractive indices known as the ordinary refractive index no and the extraordinary refractive index ne . For such anisotropic materials, the dielectric tensor ε (r ) is a second rank tensor [13]. In the principal coordinates, the diagonal elements of ε (r ) are related to the principal refractive indices as
εxx = ne2 ,
ε y y = εzz = no2 ,
We consider that the 2D square-lattice PC is composed of isotropic and anisotropic dielectric cylinders surrounded by LC 5CB, as shown in Fig. 1. The material is homogeneous in the z direction, and periodic along x and y with lattice constant. The lattice constant is a and the radius of cylinders is r. The refractive index of the isotropic dielectric cylinders is n = 3.4 (Si), while for
(3)
while the other dyadic elements are all zero. Generally LCs possess also two kinds of dielectric index. One is the ordinary dielectric index εo , and the other is the extraordinary dielectric index ε e . Light waves with electric fields perpendicular and parallel to the director of the LC have ordinary and extraordinary refractive indices, respectively. The extended Jones matrix method [29] is a simple and powerful approach for dealing with the light transmission problem of a LC device at normal incidence. In the 2D materials, the components of the dielectric tensor of the nematic LC are represented as [29]
εxx (r ) = εo (r ) sin2 φ + εe (r ) cos2 φ, o
2
2
e
ε y y (r ) = ε (r ) cos φ + ε (r ) sin φ,
εxy (r ) = ε yx (r ) = ε (r ) − ε (r ) cos φ sin φ, e
o
(4) (5) (6)
Fig. 1. Photonic crystal structure with square lattices. The shaded region is infiltrated with liquid crystals. The left inset indicates the director of a liquid crystal.
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Fig. 2. Calculated photonic band structures of the 2D square array of isotropic dielectric cylinders in the air, anisotropic dielectric cylinders in the air, in the nematic liquid crystal at φ = 0◦ , and in the nematic liquid crystal at φ = 90◦ for TM mode (solid line) and TE mode (dotted lines). The radius of cylinders is taken as r = 0.35a. The gray areas represent absolute bandgaps.
anisotropic materials they are ne = 6.2 and no = 4.8 (Te). To investigate the existence of photonic bandgaps for such an anisotropic material, we calculate the photonic band structures along the edge of the irreducible Brillouin zone. The standard Brillouin zone of various lattices can be found in a textbook [2] and thus is not shown here. We show in Fig. 2 an example for the photonic band structure of an isotropic PC with a refractive index constant of n = 3.4 and r = 0.35a, where ω is the angular frequency, and c is the light velocity in the free space. This band structure is in excellent agreement with the result in Ref. [15]. The photonic properties of isotropic structures have been studied and shown to exhibit bandgaps for TM modes. It is clearly see that three bandgaps open for the TM mode (solid line), i.e., the 1–2 bandgap, 3–4 bandgap, and 6–7 bandgap. However, there are no significant TE gaps (dotted lines) in the frequency range displayed. The isotropic dielectric cylinders resulted in the absence of the absolute bandgap. The absolute bandgap can be created by introducing the anisotropy in the dielectric of cylinders. In Fig. 2, we also show the photonic band structures of 2D square lattice of anisotropic dielectric cylinders in the air, in the nematic liquid crystal at φ = 0◦ , and in the nematic liquid crystal at φ = 90◦ , respectively. The gray areas represent the absolute bandgap. We have chosen r = 0.35a, since the 2D square lattice of anisotropic cylinders in the air shows the largest absolute bandgap. The anisotropy-induced absolute bandgaps for the square lattice PC in air have a first bandgap (ωa/2π c = 0.2321–0.2701) and a second bandgap (ωa/2π c = 0.4631–0.4734). The photonic bandgaps can be improved in photonic crystals fabricated from anisotropic material. We have made a systematic examination of the photonic band structures for anisotropic dielectric cylinders in the LC background as a function of the director of the LC. The rotation angle φ was varied to change the dielectric tensor of LC medium. Fig. 2 also shows the photonic band structures of square lattice of anisotropic dielectric cylinders in the nematic LC background at φ = 0◦ and φ = 90◦ , respectively. The infiltration of LC can shift the first bandgap to 0.2279–0.2552, the second bandgap to 0.4317–0.443 and arise other two small bandgaps at φ = 0◦ . Rotation of the LC director causes the bandgap to disappear in the high frequency region, and the upper edge of the first bandgap moves down drastically with little decrease of bottom edge. The first bandgap was shifted to 0.2261–0.2335 at φ = 90◦ . The gap maps for a square lattice of anisotropic dielectric cylinders are shown in Fig. 3. At a
Fig. 3. Gap maps for a square lattice of anisotropic dielectric cylinders.
Fig. 4. Absolute gap maps for a square lattice of anisotropic dielectric cylinders.
glance, the gap map reveals some interesting regularities. First, the gaps all decrease in frequency as r /a increases. Second, the gaps all decrease in frequency as φ increases. The third, the gaps at higher frequencies disappear as φ increases. The fourth, all of the gaps
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Fig. 5. Calculated photonic band structures of the 2D triangular array of isotropic dielectric cylinders in the air, anisotropic dielectric cylinders in the air, in the nematic liquid crystal at φ = 0◦ , and in the nematic liquid crystal at φ = 90◦ for TM mode (solid line) and TE mode (dotted lines). The radius of cylinders is taken as r = 0.35a. The gray areas represent absolute bandgaps.
seal up at r /a = 0.5. At that value, the dielectric cylinders begin to touch one another. The dielectric cylinders fill space at r /a = 0.7. The absolute gap maps for a square lattice of anisotropic dielectric cylinders are shown in Fig. 4. They all decrease in frequency as the rotation angle increases. The results mean that the absolute bandgaps of PC can be actively modulated by infiltrating nematic LCs. Such a concept is also applicable to other lattice types and atom configurations. We consider that a 2D PC composed of triangularlattice dielectric cylinders surrounded by LC 5CB. We calculated the photonic band structure of 2D triangular arrays using the same parameters as those employed in the square lattice. In Fig. 5, we also show the photonic band structures of 2D triangular lattice of anisotropic dielectric cylinders in the air, in the nematic liquid crystal at φ = 0◦ , and in the nematic liquid crystal at φ = 90◦ , respectively. The photonic properties of isotropic structures have been studied and shown to exhibit bandgaps for each of the two polarization modes. It is clearly see that several bandgaps open for the TM mode (solid line), i.e., the 1–2 bandgap, 3–4 bandgap, and 6–7 bandgap. For TE mode (dotted lines) a bandgap is opened between 1–2 bands. However, the bandgaps in the two polarization modes do not overlap with each other, resulting in the absence of the absolute bandgap. The absolute bandgap can be created by introducing the anisotropy in the dielectric of cylinders. The anisotropy-induced absolute bandgaps for the triangular lattice PC in air have a bandgap (ωa/2π c = 0.2311–0.2773). The infiltration of LC closes the first bandgap and produces other two bandgaps at φ = 0◦ . Rotation of the LC director causes the bandgaps to decrease in the high frequency region and opens the first bandgap to 0.2261–0.2358 at φ = 90◦ . The gap maps for a triangular lattice of anisotropic dielectric cylinders are shown in Fig. 6. The remarkable self-similarity of Fig. 3, which was for TM mode of the square lattice of dielectric cylinders, is mirrored here. The successive gaps are similar in shape and orientation, and stack regularly upon one another. The gaps decrease in frequency as φ increases. The dielectric cylinders begin touching one another at r /a = 0.5, and fill space at r /a = 0.58. The cutoff at r /a = 0.45 is once again near the cylinder-touching condition. The absolute gap maps for a triangular lattice of anisotropic dielectric cylinders are shown in Fig. 7. They all decrease in frequency as the rotation angle increases. The results show that the bandgap could be actively modulated after infiltrating nematic LCs. Such a mechanism of bandgap adjustment
Fig. 6. Gap maps for a triangular lattice of anisotropic dielectric cylinders.
Fig. 7. Absolute gap maps for a triangular lattice of anisotropic dielectric cylinders.
should open up new possibilities for designing components in photonic integrated circuits. Fig. 8 shows the variation of absolute bandgap due to the change of the rotation angle of LC. The colored areas represent tun-
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of the square and triangular lattice of anisotropic dielectric cylinders. Photonic bandgaps can be improved in photonic crystals fabricated from anisotropic material. It is also demonstrated that the absolute bandgap can be controlled by rotating directors of LCs under the influence of an applied electric field. Due to large varieties of anisotropic materials in LC, the present results open up a new scope for designing tunable devices in photonic integrated circuits. Further theoretical investigations and experimental efforts are needed to bring the tunable PC into reality. References
Fig. 8. Absolute bandgaps as a function of rotation angle φ . The radius of cylinders is taken as r = 0.35a.
able absolute bandgaps. The fourth bandgap of the square lattice sealed up at φ = 50◦ . The first and fourth bandgaps of the triangular lattice opened up at φ = 30◦ and 60◦ . The absolute bandgaps could be actively modulated after infiltrating nematic LCs. This tunable PC can act as a tunable planar lightwave component. In the case of nematic LCs, the directors of LCs depend on the direction of the electric field. Indium tin oxide (ITO) layers can be attached to the top and the bottom of the PC structure. Then, the electric field can be applied by adjusting the magnitudes of the electric field in the x and y directions, which makes it possible to rotate the directors of the LCs. The director can be reoriented by an electric field, when the field strength exceeds the Fréedericksz transition threshold [29]. When the applied voltage V exceeds the Fréedericksz transition threshold (V th ), the directors begin to tilt. V th is the threshold voltage that is found to be 0.699V rms at 1 kHz sinusoidal frequency. In general, the response time of a LC is of the order of a millisecond. However, it has been reported that the response time of LCs in nanoscale voids becomes of the order of 100 μs [30]. The orientational relaxation times calculated by the molecular dynamics formalism and the experimental data determined by nuclear magnetic resonance spectroscopy for the nematic phase of a 5CB crystal at 300 K were presented in Ref. [31]. Therefore our novel tunable PC structure with LCs can be used as a fast switching component in photonic integrated circuits. 4. Conclusion We have demonstrated numerically, from the photonic band calculation, the effects of LC infiltration on the absolute bandgaps
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