- Email: [email protected]
Tunable lightwave propagation in two-dimensional hole-type photonic crystals infiltrated with nematic liquid crystal
Physica E 44 (2011) 313–316
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Tunable lightwave propagation in two-dimensional hole-type photonic crystals infiltrated with nematic liquid crystal Cheng-Yang Liu n Department of Mechanical and Electro-Mechanical Engineering, Tamkang University, No. 151, Ying-chuan Road, Tamsui District, New Taipei City, Taiwan
a r t i c l e i n f o
abstract
Article history: Received 10 June 2011 Accepted 30 August 2011 Available online 7 September 2011
Photonic crystals have many potential applications because of their ability to control lightwave propagation. We have investigated the tunability of lightwave propagation in two-dimensional holetype photonic crystal structures. The linear waveguides can be obtained by the infiltration of liquid crystals into air holes in hole-type photonic crystal with square lattices. The refractive indices of liquid crystals can be changed by rotating the directors of liquid crystals. Therefore, we can control the lightwave propagation in two-dimensional hole-type photonic crystal structures. Such a mechanism of lightwave adjustment should open up a new application for designing components in photonic integrated circuits. & 2011 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 regions where electromagnetic waves cannot propagate for both polarizations along any direction. This may bring about some peculiar physical phenomena. The electromagnetic wave is strongly confined in the defect channel, allowing low loss in linear waveguides and a sharp-bending [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 (2-D) PCs produced by insertion of linear defects into PC structures have been proposed [4] and experimentally proved [5]. 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 [6–8]. Recently the photonic bandgap of a 2-D PC has been continuously varied using the temperature dependent refractive index of a liquid crystal (LC) [9]. The propagation of tunable lightwave in Y-shaped waveguides in 2-D PCs by the use of LCs and semiconductors as linear defects is discussed [10,11]. A tunable PC waveguide coupler based on nematic LCs is presented by the authors [12], and then we have investigated the tunable bandgap in a 2-D photonic crystal modulated by a nematic liquid crystal [13]. These results can be used to obtain a tunable field-
n
Tel.: þ886 2 26215656x2061; fax: þ 886 2 26209745. E-mail address: [email protected]
1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.08.031
sensitive polarizer [14]. Electro-optical switches are the key components of such photonic integrated circuits. A PC channel waveguided 2 2 directional coupler switch that utilizes electrically or optically induced loss in the coupling region between coupled waveguides has been proposed and analyzed [15]. In this paper, we theoretically demonstrate the tunability of lightwave propagation in two-dimensional hole-type PC structures. We propose the infiltration of LCs in 2-D PC with square lattices composed of air holes, and then guided modes in PC waveguides can be changed based on the orientation of LCs by adjusting the applied field. It makes possible to control the lightwave propagation with a certain frequency in PC waveguides. Using a plane wave expansion method (PWE), we have solved Maxwell’s equations for the guided modes of electromagnetic waves in a 2-D hole-type PC with LC. The propagation properties of electromagnetic waves inside the hole-type PC waveguides are calculated by means of finite-difference time-domain simulations (FDTD). Details of the calculations and discussion of the results will be presented in the remainder of the paper.
2. Numerical method The PWE method is illustrated in several papers [2,16]. Here we summarize the theory very briefly. Following the discussion of previous literature, we can express the lightwave equation that is satisfied by the magnetic field in order to determine the photonic bandgaps of periodic structures as 2 1 o r r HðrÞ ¼ HðrÞ ð1Þ eðrÞ c
314
C.-Y. Liu / Physica E 44 (2011) 313–316
Since lightwaves are transmitted in periodic structures, Bloch’s theorem is used to expand the H field in plane waves: X hðGÞeG expfiðkþ GÞrg ð2Þ HðrÞ ¼ G
where k is the wave vector in the Brillouin zone, eG is the direction perpendicular to the 2-D plane, and G is the reciprocal lattice vector. The dielectric constant of photonic structure is periodic with respect to the lattice vector. Similarly, G can be expanded in the Fourier series: X ei,j ðrÞ ¼ ei,j ðGÞejGUr , i,j ¼ x,z ð3Þ G
Liquid crystals generally possess two dielectric constants. One is the ordinary dielectric constant eo and the other is the extraordinary dielectric constant ee. The lightwaves with electric fields perpendicular and parallel to the director of the LC have ordinary and extraordinary dielectric constants, respectively. When the director rotates on the 2-D plane, the components of the dielectric tensor can be represented as
ex,x ðrÞ ¼ eo ðrÞcos2 f þ ee ðrÞsin2 f, ez,z ðrÞ ¼ eo ðrÞsin2 f þ ee ðrÞcos2 f, ex,z ðrÞ ¼ ez,x ðrÞ ¼ ½ee ðrÞeo ðrÞcos f sin f
ð4Þ
where f is the rotation angle of the director. The director is presented by n ¼(cos f, sin f). By inserting Eqs. (2)–(4) in Eq. (1) and multiplying by eG, we can get the following infinite eigenvalue matrix problem: o 2 X HG,G0 hðG0 Þ ¼ hðGÞ ð5Þ c G0 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) [16]. The eigenvalues computed with the HCS method for 512 plane waves are estimated to be in less than 1% error. In our calculations, the convergence is quite fast for low-energy bands. We consider a square lattice of air holes in dielectric background and examine transverse electric (TE) mode where electric field is parallel to the axis of cylinders. PWE method is used to obtain photonic band structures and guided modes. The electromagnetic wave propagation is simulated using the FDTD method [17]. The FDTD method is a powerful, accurate numerical method that permits computer-aided design and simulation of PC structures. The photonic device is laid out in the x–y plane. The propagation is along the x direction. The space steps in the x and y directions are Dx and Dy, respectively. We assume that Dx ¼0.05 and Dy¼0.05. The sampling in time is selected to ensure numerical stability of the algorithm. A more detailed treatment of the FDTD method is given in Ref. [17].
velocity in the free space. In this structure, the linear waveguide is induced by the infiltration of LCs into air holes, as shown in Fig. 1. The ordinary and extraordinary refractive indices of liquid crystals LC (5CB type) are nLC o ¼ 1:5357 and ne ¼ 1:7128 at 25 1C, respectively. In Fig. 1, dark and light shaded regions indicate Si and liquid crystal, respectively. The insert indicates the director n of a liquid crystal and the rotation angle f of the director to the x-axis. The mesogenic temperature range of a single LC substance is usually quite limited [18]. 5CB is a nice material to work with because it exhibits a nematic phase at room temperature and its nematic range is more than 101 [19]. We assume that the operating temperature is at a constant room temperature and absorption loss is negligible. We examine the dispersion relations of guide modes in linear waveguide when the director of the LC is oriented at f. Fig. 2 shows the dispersion relations of guided modes in photonic crystal waveguide infiltrated with liquid crystal at f ¼01 and 901. The shaded regions correspond to the projected band structures of the perfect PC. Inside the photonic bandgap, there exist two kinds of dispersion relations at certain directors of LC and there exist cutoff frequencies of guide modes. The cutoff frequencies at f ¼01 and 901 are 0.2737 and 0.2851, respectively. For the 1550 nm center wavelength, we assumed a 2-D PC of 207.95 nm diameter air holes arrayed in a square lattice a¼433.23 nm in the dielectric background. Therefore, the cutoff wavelength lcutoff at f ¼01 and 901 are 1582.86 nm and 1519.57 nm, respectively. In order to let the incident lightwave propagate in the linear waveguide, incident wavelength linc%lcutoff must be satisfied. Fig. 3 shows transmission spectrum as a function of wavelength for the linear waveguide at rotation angle f ¼01 and 901. At f ¼ 01, the incident wavelength linc ¼1550 nm can propagate in the linear waveguide because linc%lcutoff is satisfied. At f ¼901, the incident wavelength linc ¼1550 nm cannot propagate in the linear waveguide because linc%lcutoff is not satisfied. Hence, the cutoff frequencies of PC waveguides depend on the directors of LCs and lightwave propagation in linear waveguides can be controlled by rotating the directors of LCs. Fig. 4 shows transmission spectrum as a function of rotation angle for the linear waveguide at incident wavelength linc ¼1550 nm and 1560 nm. For linc ¼1550 nm, the maximum of output power is 1.15 dB at f ¼251 and the minimum of output power is 30.28 dB at f ¼901. For linc ¼1560 nm, the maximum of output power is 1.06 dB at f ¼01 and the minimum of output power is 33.45 dB at f ¼901.
3. Hole-type PC waveguide infiltrated with LC We consider that the 2-D square lattice PC is composed of air holes utilizing LC as linear waveguide. The background 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 r¼0.48a. The refractive index of the dielectric background is nb ¼3.4 (Si). This structure has a photonic band gap for TE modes ranging from 0.2445 to 0.3102 in normalized frequency units (oa/2pc), where o is the angular frequency and c is the light
Fig. 1. Schematic view of the proposed 2-D hole-type photonic crystal with square lattices composed of air holes. Dark and light shaded regions indicate Si and liquid crystal, respectively. The inset indicates the director of a liquid crystal.
C.-Y. Liu / Physica E 44 (2011) 313–316
315
Fig. 4. Transmission spectrum as a function of rotation angle for the linear waveguide at incident wavelength linc ¼1550 nm (black line) and 1560 nm (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Dispersion relations of guided modes in photonic crystal waveguide infiltrated with liquid crystal at f ¼ 01 (red line) and 901 (blue line). Shaded regions correspond to the projected band structures of the perfect photonic crystals. The cutoff frequencies at f ¼ 01 and 901 are 0.2737 and 0.2851, respectively. The inset shows the supercell for this waveguide. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Transmission spectrum as a function of wavelength for the linear waveguide at rotation angle f ¼ 01 (black line) and 901 (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The FDTD method is used to solve the light propagation in 2-D PC linear waveguide with LC. Fig. 5 shows the field patterns observed in frequency domain of the linear waveguide with liquid crystals at f ¼01 and 901. The incident wavelength is linc ¼1550 nm. Fig. 5(a) shows that lightwave can propagate in the linear waveguide at f ¼01. Fig. 5(b) shows that lightwave cannot propagate in the linear waveguide at f ¼901. These results would provide a basis for the novel application of switching devices in photonic circuits. In the case of nematic LCs, the directors of LCs depend on the direction of the electric field. Indium tin oxide 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 z directions, which makes it possible to
Fig. 5. Field patterns observed in frequency domain of the linear waveguide with LCs at (a) f ¼ 01 and (b) f ¼ 901. The incident wavelength is linc ¼1550 nm.
rotate the directors of the LCs. In general, the response time of a LC is on the order of a millisecond. However, it has been reported that the response time on LCs in nanoscale voids becomes of the order
316
C.-Y. Liu / Physica E 44 (2011) 313–316
Fig. 6 shows a schematic view of the proposed T-shaped waveguide in a 2-D PC with square lattices composed of air holes. The radius of cylinders is r¼0.48a. The T-shaped waveguide is introduced by infiltrating LCs in the air holes. For the T-shaped waveguide shown in Fig. 6 the transmission spectrum thus obtained is shown in Fig. 7. The transmission coefficient is a function of the incident wavelength at f ¼01 and 901. At linc 1550 nm, the optical powers received in the two output ports are identical at f ¼01 and the transmittance is 6.21 dB per each output waveguide. When rotation angle is f ¼ 901, the output power is almost null and the transmittance is 37.75 dB per each output waveguide. The devices are intended to be interconnected and cascaded in the forward direction into an optical cross-connect network. In this case, further optimization to transmission can be achieved by minimizing the reflections at the waveguide region and waveguide bends. The broadband and narrowband techniques will be researched for enhancing transmission through waveguide region and reducing reflections.
4. Conclusion
Fig. 6. Schematic view of the proposed T-shaped waveguide in a 2-D PC with square lattices composed of air holes. Dark and light shaded regions indicate Si and liquid crystal, respectively. The inset indicates the director of a liquid crystal.
We have demonstrated numerically the tunability of lightwave propagation in linear waveguide in 2-D PC with square lattices composed of air holes utilizing LCs. The linear waveguide is obtained by the use of LCs as linear defects in 2-D hole-type PC structures. The refractive indices of LCs can be changed by rotating the directors of LCs. Then, we can control the lightwave propagation in 2-D linear PC waveguides. Such a switching mechanism should open up a new application for designing components in photonic integrated circuits. Further theoretical investigations and experimental efforts are needed to bring the devices into reality. References
Fig. 7. Transmission spectrum as a function of wavelength for the T-shaped PC waveguide at rotation angle f ¼ 01 (black line) and 901 (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
of 100 ms [20]. 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. [21] Therefore, 2-D hole-type PC utilizing LCs as linear waveguides may be appropriate for fast switching devices.
[1] E. Yablonovitch, Physical Review Letters 58 (1987) 2059. [2] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, New Jersey, 1995. [3] E. Centeno, D. Felbacq, Optics Communications 160 (1999) 57. [4] R.D. Meade, A. Devenyi, J.D. Joannopoulos, O.L. Alerhand, D.A. Smith, K. Kash, Journal of Applied Physics 75 (1994) 4753. [5] T. Baba, N. Fukaya, J. Yonekura, Electronics Letters 35 (1999) 654. [6] K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, M. Ozaki, Applied Physics Letters 75 (1999) 932. [7] H. Takeda, K. Yoshino, Journal of Applied Physics 92 (2002) 5658. [8] S. Kubo, Z.-Z. Gu, K. Takahashi, A. Fujishima, H. Segawa, O. Sato, Journal of the American Chemical Society 126 (2004) 8314. [9] S.W. Leonard, J.P. Mondia, H.M. van Driel, O. Toader, S. John, K. Busch, ¨ A. Birner, U. Gosele, V. Lehmann, Physical Review B 61 (2000) R2389. [10] H. Takeda, K. Yoshino, Physical Review B 67 (2003) 073106. [11] H. Takeda, K. Yoshino, Optics Communications 219 (2003) 177. [12] C.-Y. Liu, L.-W. Chen, IEEE Photonics Technology Letters 16 (2004) 1849. [13] C.-Y. Liu, L.-W. Chen, Physical Review B 72 (2005) 045133. [14] C.-Y. Liu, L.-W. Chen, Optics Communications 256 (2005) 114. [15] A. Sharkawy, S. Shi, D.W. Prather, R. Soref, Optics Express 10 (2002) 1048. [16] K.M. Ho, C.T. Chan, C.M. Soukoulis, Physical Review Letters 65 (1990) 3152. [17] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method, Artech House, Boston, 1998. [18] J. Li, S. Gauzia, S.-T. Wu, Optics Express 12 (2004) 2002. [19] I.-C. Khoo, S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals, World Scientific, Singapore, 1993. [20] Y. Shimoda, M. Ozaki, K. Yoshino, Applied Physics Letters 79 (2001) 3627. [21] A.V. Zakharov, L.V. Mirantsev, Physics of the Solid State 45 (2003) 183.
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Tunable lightwave propagation in two-dimensional hole-type photonic crystals infiltrated with nematic liquid crystal Cheng-Yang Liu n Department of Mechanical and Electro-Mechanical Engineering, Tamkang University, No. 151, Ying-chuan Road, Tamsui District, New Taipei City, Taiwan
a r t i c l e i n f o
abstract
Article history: Received 10 June 2011 Accepted 30 August 2011 Available online 7 September 2011
Photonic crystals have many potential applications because of their ability to control lightwave propagation. We have investigated the tunability of lightwave propagation in two-dimensional holetype photonic crystal structures. The linear waveguides can be obtained by the infiltration of liquid crystals into air holes in hole-type photonic crystal with square lattices. The refractive indices of liquid crystals can be changed by rotating the directors of liquid crystals. Therefore, we can control the lightwave propagation in two-dimensional hole-type photonic crystal structures. Such a mechanism of lightwave adjustment should open up a new application for designing components in photonic integrated circuits. & 2011 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 regions where electromagnetic waves cannot propagate for both polarizations along any direction. This may bring about some peculiar physical phenomena. The electromagnetic wave is strongly confined in the defect channel, allowing low loss in linear waveguides and a sharp-bending [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 (2-D) PCs produced by insertion of linear defects into PC structures have been proposed [4] and experimentally proved [5]. 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 [6–8]. Recently the photonic bandgap of a 2-D PC has been continuously varied using the temperature dependent refractive index of a liquid crystal (LC) [9]. The propagation of tunable lightwave in Y-shaped waveguides in 2-D PCs by the use of LCs and semiconductors as linear defects is discussed [10,11]. A tunable PC waveguide coupler based on nematic LCs is presented by the authors [12], and then we have investigated the tunable bandgap in a 2-D photonic crystal modulated by a nematic liquid crystal [13]. These results can be used to obtain a tunable field-
n
Tel.: þ886 2 26215656x2061; fax: þ 886 2 26209745. E-mail address: [email protected]
1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.08.031
sensitive polarizer [14]. Electro-optical switches are the key components of such photonic integrated circuits. A PC channel waveguided 2 2 directional coupler switch that utilizes electrically or optically induced loss in the coupling region between coupled waveguides has been proposed and analyzed [15]. In this paper, we theoretically demonstrate the tunability of lightwave propagation in two-dimensional hole-type PC structures. We propose the infiltration of LCs in 2-D PC with square lattices composed of air holes, and then guided modes in PC waveguides can be changed based on the orientation of LCs by adjusting the applied field. It makes possible to control the lightwave propagation with a certain frequency in PC waveguides. Using a plane wave expansion method (PWE), we have solved Maxwell’s equations for the guided modes of electromagnetic waves in a 2-D hole-type PC with LC. The propagation properties of electromagnetic waves inside the hole-type PC waveguides are calculated by means of finite-difference time-domain simulations (FDTD). Details of the calculations and discussion of the results will be presented in the remainder of the paper.
2. Numerical method The PWE method is illustrated in several papers [2,16]. Here we summarize the theory very briefly. Following the discussion of previous literature, we can express the lightwave equation that is satisfied by the magnetic field in order to determine the photonic bandgaps of periodic structures as 2 1 o r r HðrÞ ¼ HðrÞ ð1Þ eðrÞ c
314
C.-Y. Liu / Physica E 44 (2011) 313–316
Since lightwaves are transmitted in periodic structures, Bloch’s theorem is used to expand the H field in plane waves: X hðGÞeG expfiðkþ GÞrg ð2Þ HðrÞ ¼ G
where k is the wave vector in the Brillouin zone, eG is the direction perpendicular to the 2-D plane, and G is the reciprocal lattice vector. The dielectric constant of photonic structure is periodic with respect to the lattice vector. Similarly, G can be expanded in the Fourier series: X ei,j ðrÞ ¼ ei,j ðGÞejGUr , i,j ¼ x,z ð3Þ G
Liquid crystals generally possess two dielectric constants. One is the ordinary dielectric constant eo and the other is the extraordinary dielectric constant ee. The lightwaves with electric fields perpendicular and parallel to the director of the LC have ordinary and extraordinary dielectric constants, respectively. When the director rotates on the 2-D plane, the components of the dielectric tensor can be represented as
ex,x ðrÞ ¼ eo ðrÞcos2 f þ ee ðrÞsin2 f, ez,z ðrÞ ¼ eo ðrÞsin2 f þ ee ðrÞcos2 f, ex,z ðrÞ ¼ ez,x ðrÞ ¼ ½ee ðrÞeo ðrÞcos f sin f
ð4Þ
where f is the rotation angle of the director. The director is presented by n ¼(cos f, sin f). By inserting Eqs. (2)–(4) in Eq. (1) and multiplying by eG, we can get the following infinite eigenvalue matrix problem: o 2 X HG,G0 hðG0 Þ ¼ hðGÞ ð5Þ c G0 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) [16]. The eigenvalues computed with the HCS method for 512 plane waves are estimated to be in less than 1% error. In our calculations, the convergence is quite fast for low-energy bands. We consider a square lattice of air holes in dielectric background and examine transverse electric (TE) mode where electric field is parallel to the axis of cylinders. PWE method is used to obtain photonic band structures and guided modes. The electromagnetic wave propagation is simulated using the FDTD method [17]. The FDTD method is a powerful, accurate numerical method that permits computer-aided design and simulation of PC structures. The photonic device is laid out in the x–y plane. The propagation is along the x direction. The space steps in the x and y directions are Dx and Dy, respectively. We assume that Dx ¼0.05 and Dy¼0.05. The sampling in time is selected to ensure numerical stability of the algorithm. A more detailed treatment of the FDTD method is given in Ref. [17].
velocity in the free space. In this structure, the linear waveguide is induced by the infiltration of LCs into air holes, as shown in Fig. 1. The ordinary and extraordinary refractive indices of liquid crystals LC (5CB type) are nLC o ¼ 1:5357 and ne ¼ 1:7128 at 25 1C, respectively. In Fig. 1, dark and light shaded regions indicate Si and liquid crystal, respectively. The insert indicates the director n of a liquid crystal and the rotation angle f of the director to the x-axis. The mesogenic temperature range of a single LC substance is usually quite limited [18]. 5CB is a nice material to work with because it exhibits a nematic phase at room temperature and its nematic range is more than 101 [19]. We assume that the operating temperature is at a constant room temperature and absorption loss is negligible. We examine the dispersion relations of guide modes in linear waveguide when the director of the LC is oriented at f. Fig. 2 shows the dispersion relations of guided modes in photonic crystal waveguide infiltrated with liquid crystal at f ¼01 and 901. The shaded regions correspond to the projected band structures of the perfect PC. Inside the photonic bandgap, there exist two kinds of dispersion relations at certain directors of LC and there exist cutoff frequencies of guide modes. The cutoff frequencies at f ¼01 and 901 are 0.2737 and 0.2851, respectively. For the 1550 nm center wavelength, we assumed a 2-D PC of 207.95 nm diameter air holes arrayed in a square lattice a¼433.23 nm in the dielectric background. Therefore, the cutoff wavelength lcutoff at f ¼01 and 901 are 1582.86 nm and 1519.57 nm, respectively. In order to let the incident lightwave propagate in the linear waveguide, incident wavelength linc%lcutoff must be satisfied. Fig. 3 shows transmission spectrum as a function of wavelength for the linear waveguide at rotation angle f ¼01 and 901. At f ¼ 01, the incident wavelength linc ¼1550 nm can propagate in the linear waveguide because linc%lcutoff is satisfied. At f ¼901, the incident wavelength linc ¼1550 nm cannot propagate in the linear waveguide because linc%lcutoff is not satisfied. Hence, the cutoff frequencies of PC waveguides depend on the directors of LCs and lightwave propagation in linear waveguides can be controlled by rotating the directors of LCs. Fig. 4 shows transmission spectrum as a function of rotation angle for the linear waveguide at incident wavelength linc ¼1550 nm and 1560 nm. For linc ¼1550 nm, the maximum of output power is 1.15 dB at f ¼251 and the minimum of output power is 30.28 dB at f ¼901. For linc ¼1560 nm, the maximum of output power is 1.06 dB at f ¼01 and the minimum of output power is 33.45 dB at f ¼901.
3. Hole-type PC waveguide infiltrated with LC We consider that the 2-D square lattice PC is composed of air holes utilizing LC as linear waveguide. The background 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 r¼0.48a. The refractive index of the dielectric background is nb ¼3.4 (Si). This structure has a photonic band gap for TE modes ranging from 0.2445 to 0.3102 in normalized frequency units (oa/2pc), where o is the angular frequency and c is the light
Fig. 1. Schematic view of the proposed 2-D hole-type photonic crystal with square lattices composed of air holes. Dark and light shaded regions indicate Si and liquid crystal, respectively. The inset indicates the director of a liquid crystal.
C.-Y. Liu / Physica E 44 (2011) 313–316
315
Fig. 4. Transmission spectrum as a function of rotation angle for the linear waveguide at incident wavelength linc ¼1550 nm (black line) and 1560 nm (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Dispersion relations of guided modes in photonic crystal waveguide infiltrated with liquid crystal at f ¼ 01 (red line) and 901 (blue line). Shaded regions correspond to the projected band structures of the perfect photonic crystals. The cutoff frequencies at f ¼ 01 and 901 are 0.2737 and 0.2851, respectively. The inset shows the supercell for this waveguide. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Transmission spectrum as a function of wavelength for the linear waveguide at rotation angle f ¼ 01 (black line) and 901 (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The FDTD method is used to solve the light propagation in 2-D PC linear waveguide with LC. Fig. 5 shows the field patterns observed in frequency domain of the linear waveguide with liquid crystals at f ¼01 and 901. The incident wavelength is linc ¼1550 nm. Fig. 5(a) shows that lightwave can propagate in the linear waveguide at f ¼01. Fig. 5(b) shows that lightwave cannot propagate in the linear waveguide at f ¼901. These results would provide a basis for the novel application of switching devices in photonic circuits. In the case of nematic LCs, the directors of LCs depend on the direction of the electric field. Indium tin oxide 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 z directions, which makes it possible to
Fig. 5. Field patterns observed in frequency domain of the linear waveguide with LCs at (a) f ¼ 01 and (b) f ¼ 901. The incident wavelength is linc ¼1550 nm.
rotate the directors of the LCs. In general, the response time of a LC is on the order of a millisecond. However, it has been reported that the response time on LCs in nanoscale voids becomes of the order
316
C.-Y. Liu / Physica E 44 (2011) 313–316
Fig. 6 shows a schematic view of the proposed T-shaped waveguide in a 2-D PC with square lattices composed of air holes. The radius of cylinders is r¼0.48a. The T-shaped waveguide is introduced by infiltrating LCs in the air holes. For the T-shaped waveguide shown in Fig. 6 the transmission spectrum thus obtained is shown in Fig. 7. The transmission coefficient is a function of the incident wavelength at f ¼01 and 901. At linc 1550 nm, the optical powers received in the two output ports are identical at f ¼01 and the transmittance is 6.21 dB per each output waveguide. When rotation angle is f ¼ 901, the output power is almost null and the transmittance is 37.75 dB per each output waveguide. The devices are intended to be interconnected and cascaded in the forward direction into an optical cross-connect network. In this case, further optimization to transmission can be achieved by minimizing the reflections at the waveguide region and waveguide bends. The broadband and narrowband techniques will be researched for enhancing transmission through waveguide region and reducing reflections.
4. Conclusion
Fig. 6. Schematic view of the proposed T-shaped waveguide in a 2-D PC with square lattices composed of air holes. Dark and light shaded regions indicate Si and liquid crystal, respectively. The inset indicates the director of a liquid crystal.
We have demonstrated numerically the tunability of lightwave propagation in linear waveguide in 2-D PC with square lattices composed of air holes utilizing LCs. The linear waveguide is obtained by the use of LCs as linear defects in 2-D hole-type PC structures. The refractive indices of LCs can be changed by rotating the directors of LCs. Then, we can control the lightwave propagation in 2-D linear PC waveguides. Such a switching mechanism should open up a new application for designing components in photonic integrated circuits. Further theoretical investigations and experimental efforts are needed to bring the devices into reality. References
Fig. 7. Transmission spectrum as a function of wavelength for the T-shaped PC waveguide at rotation angle f ¼ 01 (black line) and 901 (red line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
of 100 ms [20]. 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. [21] Therefore, 2-D hole-type PC utilizing LCs as linear waveguides may be appropriate for fast switching devices.
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