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Larger complete photonic bandgaps in two-dimensional photonic crystal heterostructure
Solid State Communications 137 (2006) 74–77 www.elsevier.com/locate/ssc
Larger complete photonic bandgaps in two-dimensional photonic crystal heterostructure D.D. Wang, Y.S. Wang *, L.E. Deng, C.X. Zhang, X. Han Institute of Optoelectronic Technology, Beijing Jiaotong University, Beijing 100044, China Received 30 August 2005; received in revised form 12 October 2005; accepted 12 October 2005 by M.S. Skolnick Available online 2 November 2005
Abstract A two-dimensional photonic crystal heterostructure with a large complete photonic bandgap is presented. It consists of two photonic crystals, whose dielectric configurations are reverse, with triangular lattice of circular columns. The structure retains the ease of fabrication while increasing the maximum quality factor of the gap 2–3 times after optimization. Photonic bandgap properties are calculated using a plane-wave method and the transmission spectra are obtained. q 2005 Elsevier Ltd. All rights reserved. PACS: 42.70.Qs; 42.25.Bs Keywords: Photonic crystals; Photonic band gap; Heterostructure
In recent years, there is an increasing interest in fabricating periodic dielectric structures which give rise to photonic band gap (PBG)—a region of the frequency spectrum where propagating modes are forbidden [1,2]. Although threedimensional PBG structures suggest the most interesting potential in novel application, the fabrication of such structure with PBG at visible and near-infrared (IR) is quite difficult, despite progress in microfabrication technology. In contrast, two-dimensional (2D) lattice structures are much easier to fabricate in this regime and also have some promising applications, e.g. planar waveguide [3], photonic crystal fibers [4], and feedback mirrors [5]. For these reasons much attention has been drawn towards 2D photonic crystals (PCs). Since many potential applications of photonic crystals rely on their photonic band gaps, it is essential to design crystal structures with a complete band gap as large as possible. A 2D complete PBG was first demonstrated for a triangular lattice of air columns in a dielectric material [6] and other 2D PCs with different lattice structure, unit cell, or dielectric configuration were investigated later [7–11]. Recently, it was found that symmetry breaking is helpful to obtain a larger complete bandgaps, e.g. anisotropic inclusions [12] and two sets of air
* Corresponding author. Fax: C86 10 51683933. E-mail address: [email protected] (Y.S. Wang).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.10.006
inclusions were used [13]. Very recently, genetic algorithm was introduced to obtain a better structure [14]. In the 2D PCs, the electromagnetic wave can be decomposed into the transverse electric (TE) and transverse magnetic (TM) polarization modes. The complete band gap exists only when band gaps in both polarization modes are presented and they overlap each other. Physical arguments suggest that connectivity of high-3 regions is conducive to TE gaps, and isolated patches of high-3 regions lead to TM gaps [2], however, it seems difficult to arrange a photonic crystal with both isolated spots and connected region of high-3 dielectric material. In solid physics, it is well known that many novel features for electronic properties appear when two semiconductors form heterostructure. Recently, 1D and 2D photonic heterostructures are also proposed to widen the nontransmission frequency range [15,16]. In this letter, we will show theoretically that the enlargement of complete band gaps can be realized in a 2D photonic crystal heterostructure. The central idea is to obtain the TE and TM band gap independently with two PCs, in which the structure of one PC favors TE band gaps and the other favors TM bands gaps. The midgap frequency of TE and TM mode can be adjusted to overlap via carefully choosing the parameters of each PC, such as geometry and dielectric contrast. Using these two PCs to construct a 2D heterostructure, a large complete band gap will be expected. The proposed 2D photonic crystal heterostructure is shown in Fig. 1(c), which is composed of two photonic crystals with triangular lattice of circular columns. The first one (denoted as
D.D. Wang et al. / Solid State Communications 137 (2006) 74–77
75
Fig. 1. Schematic representation for (a) air columns in dielectric, (b) dielectric columns in air, (c) 2D photonic crystals heterostructure, (d) the irreducible Brillouin zone (light gray) of a triangular lattice with symmetry points, G, M, K.
PC1) is composed of air cylinders in dielectrics, and the second one (denoted as PC2) composed of dielectric cylinders in air. The PC1 is embedded into the center part of the PC2 as a defect. The dielectric constant of the constituent materials is 3airZ1, 3dieZ11.4 (3GaAs at lz1.5 mm). In the present paper, photonic band structures are calculated by the plane-wave expansion method [17]. Six hundred and twenty-five plane waves were used, and the difference in the band structures is less than 0.1% for the five lowest photonic bands of both TE and TM modes when these are compared with the results using 900 plane waves. All calculations were performed on Dell precision workstation 450. Fig. 2 shows how the bandgaps of a single PC vary as the radius of columns changes when aZb, where b is the lattice constant of PC1 and a is the lattice constant of PC2. The TE and TM mode bandgaps are indicated by the regions with different gray level. In the frequency range displayed, there is one noticeable TE gap for PC1 and two significant TM gaps for PC2. The TE bandgaps opens at rZ0.17a and disappear when rZ 0.5a, where r is the radius of the cylinder. The lowest TM bandgaps occurs at rZ0.05a and seals up when rZ0.46a. But for single PC1, the little TM gap limit the overlaping region and the resultant complete bandgap width. And for single PC2, there exist no TE gap for TM gap to overlap. After the heterostructure formed, the large overlapped frequency region from (0.20(2pc/ a) to 0.54(2pc/a)) of the bandgaps for TE and TM mode, which provided by PC1 and PC2 independently, will be able to offer a large complete two-dimensional photonic band gaps as desired. One important parameter determining the efficiency of the gap is the quality factor (QF), defined as the gap width divided by the midgap frequency. In Fig. 3, the QF is plotted with respect to the radius of cylinders for the TE gap of PC1 and the lowest TM gap of PC2 shown in Fig. 2. For TE mode, the maximum value occurs when rZ0.42a with a maximum QF Du/u0 Z49.3%, where DuZ0.175(2pc/a) and u 0Z 0.355(2pc/a). For TM mode, the QF reaches the maximum
Fig. 2. Photonic gap maps (a) for PC1 and (b) for PC2 when aZb.
value Du/u0Z47.6% when rZ0.18a, where DuZ0.186(2pc/ a) and u0Z0.391(2pc/a). When PC1 and PC2 have the same periodicity, the midgap frequencies of the two modes are different if both the TE and TM mode QF reach the maximum value. This means the overlapping of the two-mode bandgap is not optimal. Based on the fact that the bandgap frequency is directly related to the lattice constant and the QF is independent of the scale of PC, 2 we can improve the frequency overlapping condition simply by scaling down or up the lattice constant accordingly. In this
Fig. 3. Gap-midgap ratio as a function of the radius of the rods for TE mode of PC1 (dash dot line) and TM mode of PC2 (solid line) when aZb.
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D.D. Wang et al. / Solid State Communications 137 (2006) 74–77
Fig. 4. Photonic band structure for (a) PC1 and (b) for PC2 when b/aZ0.908 ((-) TM mode; (,) TE mode). The two gray areas represent the complete band gap.
work, the lattice constant of PC1 is chosen to be 0.908a, according to the midgap frequency ratio u0 =u00 Z 0:355=0:391, where u0 is the midgap frequency for TE mode, and u00 is the midgap frequency for TM mode. Fig. 4 gives the photonic band structure calculated for PC1 and for PC2 when bZ0.908a. The overlapping frequency region, from 0.298(2pc/a) to 0.484(2pc/a), represents the 2D complete PBGs of the heterostructure. The QF of our band gap reaches 47.6%. It has been shown that the maximum QF for a triangular lattice of air holes in GaAs is 18.6% [6], our bang gap is nearly 2.6 times larger than that maximum value, approximately 5.5 times larger than the maximum value (8.5%) of the chessboard lattice of square columns [10], approximately 2.8 times larger than the maximum value of the structure optimized by connecting rods and veins [11], and also approximately 2.4 times larger than the structure obtained by using genetic algorithm.[14] Furthermore, as far as practical fabrications are concerned, our structure has obvious advantage compared with Refs. [10–14]. The transmission spectra of the two-dimensional photonic crystal heterostructure calculated by the Translight program [18] are displayed in Fig. 5. The electromagnetic wave propagates through eights rows of air cylinders of PC1 and eight rows of dielectric cylinders of PC2 at the incident angel of 0, 30 and 898 (corresponding to the GM, GK and MK directions) in plane, respectively. In the heterostructure, the TE mode nontransmission frequency region reads from 0.292(2pc/a) to 0.480(2pc/a) while the TM mode nontransmission frequency region is from 0.272(2pc/a) to 0.484(2pc/a). The two regions have an overlap. The common nontransmission frequency region, from 0.292(2pc/a) to 0.480(2pc/a), is in good agreement with the results of the photonic band structure calculation, which proves that the large 2D complete PBG exists in the finite periods of photonic crystal heterostructure. Obviously, for the structure shown in Fig. 1(c), the electromagnetic waves transmitting inside PC1 in plane with frequency in the photonic bandgap, no matter their polarization modes and incident angles, will be localized by the heterostructure.
Fig. 5. Calculated optical transmission of the 2D photonic crystals heterostructure at different incident angles when b/aZ0.908. The dot, dash and solid lines stand for 0, 30 and 898 incident angels, (a) and (b) represent the transmission of TE and TM mode respectively. Computational program is available from Andrew L. Reynolds, University of Glasgow, UK.
It is worth noting that the complete photonic bandgap obtained with the composite structure discussed above differs from the conventional ones obtained with a single PC. Since each polarization bandgap was obtained with an independent PC, the inevitable compromise between TE and TM polarization bandgaps when requiring a single PC structure with large complete bandgaps is avoided. Thus, a much larger complete PBG will be expected by designing specific structures with adequately large bandgaps for one polarization and overlapping the bandgaps properly with heterostructure. Furthermore, with the capability to control TE and TM polarizations separately, some PC devices with especial properties will be innovated based on such a heterostructure, e.g., designing the shape of PC1 deliberately, or introducing some defects in PC1. In conclusion, a new method to obtain larger complete photonic bandgaps has been proposed. The two-dimensional photonic crystal heterostructure consists of two reverse dielectric configuration photonic crystals with triangular lattice of circular columns. This composite structure enlarges the complete bandgaps substantially, and holds the advantage of easy fabrication. Furthermore, the TM mode and TE mode bandgaps are obtained from two PCs independently, which bring in flexibility in the work to possess complete 2D PBGs. This structure, therefore, could have potential applications ranging from PBG fibers and feedback mirrors. Acknowledgement One of the authors (D.D. Wang) would like to thank Ph.D. Z. Wang sincerely for helpful discussions. This work was supported by the School Key Foundation of Beijing Jraotong University (DXJ02006), “973” national program (2003cB 314707).
D.D. Wang et al. / Solid State Communications 137 (2006) 74–77
References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic crystals, Princeton University Press, Princeton, 1995. [3] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Nature 386 (1997) 143. [4] P. Russell, Science 299 (2003) 358. [5] O. Painter, R.K. Lee, A. Scherer, A. Yariv, J.D. O’Brien, P.D. Dapkus, I. Kim, Science 284 (1999) 1819. [6] R.D. Mede, K.D. Brommer, A.M. Rappe, J.D. Joannopoulos, Appl. Phys. Lett. 61 (1992) 495. [7] P.R. Villeneuve, M. Piche´, Phys. Rev. B 46 (1992) 4973. [8] D. Cassagne, C. Jouanin, D. Bertho, Phys. Rev. B 52 (1995) 2217.
[9] [10] [11] [12] [13] [14] [15]
77
D. Cassagne, C. Jouanin, D. Bertho, Phys. Rev. B 53 (1996) 7134. M. Agio, L.C. Andreani, Phys. Rev. B 61 (2000) 15519. M. Qiu, S. He, J. Opt. Soc. Am. B 17 (2000) 1027. Z.Y. Lin, B.Y. Gu, G.Z. Yang, Phys. Rev. Lett. 81 (1998) 2574. C.M. Anderson, K.P. Giapis, Phys. Rev. Lett. 77 (1996) 2949. L.F. Shen, Z. Ye, S.L. He, Phys. Rev. B 68 (2003) 035109. X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, J. Zi, Appl. Phys. Lett. 80 (2002) 4291. [16] C. Zhang, F. Qiao, J. Wan, J. Zi, J. Appl. Phys. 87 (2000) 3174. [17] K.M. Ho, C.T. Chan, C.M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152. [18] The program is available from Andrew L. Reynolds, University of Glasgow, UK.
Larger complete photonic bandgaps in two-dimensional photonic crystal heterostructure D.D. Wang, Y.S. Wang *, L.E. Deng, C.X. Zhang, X. Han Institute of Optoelectronic Technology, Beijing Jiaotong University, Beijing 100044, China Received 30 August 2005; received in revised form 12 October 2005; accepted 12 October 2005 by M.S. Skolnick Available online 2 November 2005
Abstract A two-dimensional photonic crystal heterostructure with a large complete photonic bandgap is presented. It consists of two photonic crystals, whose dielectric configurations are reverse, with triangular lattice of circular columns. The structure retains the ease of fabrication while increasing the maximum quality factor of the gap 2–3 times after optimization. Photonic bandgap properties are calculated using a plane-wave method and the transmission spectra are obtained. q 2005 Elsevier Ltd. All rights reserved. PACS: 42.70.Qs; 42.25.Bs Keywords: Photonic crystals; Photonic band gap; Heterostructure
In recent years, there is an increasing interest in fabricating periodic dielectric structures which give rise to photonic band gap (PBG)—a region of the frequency spectrum where propagating modes are forbidden [1,2]. Although threedimensional PBG structures suggest the most interesting potential in novel application, the fabrication of such structure with PBG at visible and near-infrared (IR) is quite difficult, despite progress in microfabrication technology. In contrast, two-dimensional (2D) lattice structures are much easier to fabricate in this regime and also have some promising applications, e.g. planar waveguide [3], photonic crystal fibers [4], and feedback mirrors [5]. For these reasons much attention has been drawn towards 2D photonic crystals (PCs). Since many potential applications of photonic crystals rely on their photonic band gaps, it is essential to design crystal structures with a complete band gap as large as possible. A 2D complete PBG was first demonstrated for a triangular lattice of air columns in a dielectric material [6] and other 2D PCs with different lattice structure, unit cell, or dielectric configuration were investigated later [7–11]. Recently, it was found that symmetry breaking is helpful to obtain a larger complete bandgaps, e.g. anisotropic inclusions [12] and two sets of air
* Corresponding author. Fax: C86 10 51683933. E-mail address: [email protected] (Y.S. Wang).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.10.006
inclusions were used [13]. Very recently, genetic algorithm was introduced to obtain a better structure [14]. In the 2D PCs, the electromagnetic wave can be decomposed into the transverse electric (TE) and transverse magnetic (TM) polarization modes. The complete band gap exists only when band gaps in both polarization modes are presented and they overlap each other. Physical arguments suggest that connectivity of high-3 regions is conducive to TE gaps, and isolated patches of high-3 regions lead to TM gaps [2], however, it seems difficult to arrange a photonic crystal with both isolated spots and connected region of high-3 dielectric material. In solid physics, it is well known that many novel features for electronic properties appear when two semiconductors form heterostructure. Recently, 1D and 2D photonic heterostructures are also proposed to widen the nontransmission frequency range [15,16]. In this letter, we will show theoretically that the enlargement of complete band gaps can be realized in a 2D photonic crystal heterostructure. The central idea is to obtain the TE and TM band gap independently with two PCs, in which the structure of one PC favors TE band gaps and the other favors TM bands gaps. The midgap frequency of TE and TM mode can be adjusted to overlap via carefully choosing the parameters of each PC, such as geometry and dielectric contrast. Using these two PCs to construct a 2D heterostructure, a large complete band gap will be expected. The proposed 2D photonic crystal heterostructure is shown in Fig. 1(c), which is composed of two photonic crystals with triangular lattice of circular columns. The first one (denoted as
D.D. Wang et al. / Solid State Communications 137 (2006) 74–77
75
Fig. 1. Schematic representation for (a) air columns in dielectric, (b) dielectric columns in air, (c) 2D photonic crystals heterostructure, (d) the irreducible Brillouin zone (light gray) of a triangular lattice with symmetry points, G, M, K.
PC1) is composed of air cylinders in dielectrics, and the second one (denoted as PC2) composed of dielectric cylinders in air. The PC1 is embedded into the center part of the PC2 as a defect. The dielectric constant of the constituent materials is 3airZ1, 3dieZ11.4 (3GaAs at lz1.5 mm). In the present paper, photonic band structures are calculated by the plane-wave expansion method [17]. Six hundred and twenty-five plane waves were used, and the difference in the band structures is less than 0.1% for the five lowest photonic bands of both TE and TM modes when these are compared with the results using 900 plane waves. All calculations were performed on Dell precision workstation 450. Fig. 2 shows how the bandgaps of a single PC vary as the radius of columns changes when aZb, where b is the lattice constant of PC1 and a is the lattice constant of PC2. The TE and TM mode bandgaps are indicated by the regions with different gray level. In the frequency range displayed, there is one noticeable TE gap for PC1 and two significant TM gaps for PC2. The TE bandgaps opens at rZ0.17a and disappear when rZ 0.5a, where r is the radius of the cylinder. The lowest TM bandgaps occurs at rZ0.05a and seals up when rZ0.46a. But for single PC1, the little TM gap limit the overlaping region and the resultant complete bandgap width. And for single PC2, there exist no TE gap for TM gap to overlap. After the heterostructure formed, the large overlapped frequency region from (0.20(2pc/ a) to 0.54(2pc/a)) of the bandgaps for TE and TM mode, which provided by PC1 and PC2 independently, will be able to offer a large complete two-dimensional photonic band gaps as desired. One important parameter determining the efficiency of the gap is the quality factor (QF), defined as the gap width divided by the midgap frequency. In Fig. 3, the QF is plotted with respect to the radius of cylinders for the TE gap of PC1 and the lowest TM gap of PC2 shown in Fig. 2. For TE mode, the maximum value occurs when rZ0.42a with a maximum QF Du/u0 Z49.3%, where DuZ0.175(2pc/a) and u 0Z 0.355(2pc/a). For TM mode, the QF reaches the maximum
Fig. 2. Photonic gap maps (a) for PC1 and (b) for PC2 when aZb.
value Du/u0Z47.6% when rZ0.18a, where DuZ0.186(2pc/ a) and u0Z0.391(2pc/a). When PC1 and PC2 have the same periodicity, the midgap frequencies of the two modes are different if both the TE and TM mode QF reach the maximum value. This means the overlapping of the two-mode bandgap is not optimal. Based on the fact that the bandgap frequency is directly related to the lattice constant and the QF is independent of the scale of PC, 2 we can improve the frequency overlapping condition simply by scaling down or up the lattice constant accordingly. In this
Fig. 3. Gap-midgap ratio as a function of the radius of the rods for TE mode of PC1 (dash dot line) and TM mode of PC2 (solid line) when aZb.
76
D.D. Wang et al. / Solid State Communications 137 (2006) 74–77
Fig. 4. Photonic band structure for (a) PC1 and (b) for PC2 when b/aZ0.908 ((-) TM mode; (,) TE mode). The two gray areas represent the complete band gap.
work, the lattice constant of PC1 is chosen to be 0.908a, according to the midgap frequency ratio u0 =u00 Z 0:355=0:391, where u0 is the midgap frequency for TE mode, and u00 is the midgap frequency for TM mode. Fig. 4 gives the photonic band structure calculated for PC1 and for PC2 when bZ0.908a. The overlapping frequency region, from 0.298(2pc/a) to 0.484(2pc/a), represents the 2D complete PBGs of the heterostructure. The QF of our band gap reaches 47.6%. It has been shown that the maximum QF for a triangular lattice of air holes in GaAs is 18.6% [6], our bang gap is nearly 2.6 times larger than that maximum value, approximately 5.5 times larger than the maximum value (8.5%) of the chessboard lattice of square columns [10], approximately 2.8 times larger than the maximum value of the structure optimized by connecting rods and veins [11], and also approximately 2.4 times larger than the structure obtained by using genetic algorithm.[14] Furthermore, as far as practical fabrications are concerned, our structure has obvious advantage compared with Refs. [10–14]. The transmission spectra of the two-dimensional photonic crystal heterostructure calculated by the Translight program [18] are displayed in Fig. 5. The electromagnetic wave propagates through eights rows of air cylinders of PC1 and eight rows of dielectric cylinders of PC2 at the incident angel of 0, 30 and 898 (corresponding to the GM, GK and MK directions) in plane, respectively. In the heterostructure, the TE mode nontransmission frequency region reads from 0.292(2pc/a) to 0.480(2pc/a) while the TM mode nontransmission frequency region is from 0.272(2pc/a) to 0.484(2pc/a). The two regions have an overlap. The common nontransmission frequency region, from 0.292(2pc/a) to 0.480(2pc/a), is in good agreement with the results of the photonic band structure calculation, which proves that the large 2D complete PBG exists in the finite periods of photonic crystal heterostructure. Obviously, for the structure shown in Fig. 1(c), the electromagnetic waves transmitting inside PC1 in plane with frequency in the photonic bandgap, no matter their polarization modes and incident angles, will be localized by the heterostructure.
Fig. 5. Calculated optical transmission of the 2D photonic crystals heterostructure at different incident angles when b/aZ0.908. The dot, dash and solid lines stand for 0, 30 and 898 incident angels, (a) and (b) represent the transmission of TE and TM mode respectively. Computational program is available from Andrew L. Reynolds, University of Glasgow, UK.
It is worth noting that the complete photonic bandgap obtained with the composite structure discussed above differs from the conventional ones obtained with a single PC. Since each polarization bandgap was obtained with an independent PC, the inevitable compromise between TE and TM polarization bandgaps when requiring a single PC structure with large complete bandgaps is avoided. Thus, a much larger complete PBG will be expected by designing specific structures with adequately large bandgaps for one polarization and overlapping the bandgaps properly with heterostructure. Furthermore, with the capability to control TE and TM polarizations separately, some PC devices with especial properties will be innovated based on such a heterostructure, e.g., designing the shape of PC1 deliberately, or introducing some defects in PC1. In conclusion, a new method to obtain larger complete photonic bandgaps has been proposed. The two-dimensional photonic crystal heterostructure consists of two reverse dielectric configuration photonic crystals with triangular lattice of circular columns. This composite structure enlarges the complete bandgaps substantially, and holds the advantage of easy fabrication. Furthermore, the TM mode and TE mode bandgaps are obtained from two PCs independently, which bring in flexibility in the work to possess complete 2D PBGs. This structure, therefore, could have potential applications ranging from PBG fibers and feedback mirrors. Acknowledgement One of the authors (D.D. Wang) would like to thank Ph.D. Z. Wang sincerely for helpful discussions. This work was supported by the School Key Foundation of Beijing Jraotong University (DXJ02006), “973” national program (2003cB 314707).
D.D. Wang et al. / Solid State Communications 137 (2006) 74–77
References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic crystals, Princeton University Press, Princeton, 1995. [3] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Nature 386 (1997) 143. [4] P. Russell, Science 299 (2003) 358. [5] O. Painter, R.K. Lee, A. Scherer, A. Yariv, J.D. O’Brien, P.D. Dapkus, I. Kim, Science 284 (1999) 1819. [6] R.D. Mede, K.D. Brommer, A.M. Rappe, J.D. Joannopoulos, Appl. Phys. Lett. 61 (1992) 495. [7] P.R. Villeneuve, M. Piche´, Phys. Rev. B 46 (1992) 4973. [8] D. Cassagne, C. Jouanin, D. Bertho, Phys. Rev. B 52 (1995) 2217.
[9] [10] [11] [12] [13] [14] [15]
77
D. Cassagne, C. Jouanin, D. Bertho, Phys. Rev. B 53 (1996) 7134. M. Agio, L.C. Andreani, Phys. Rev. B 61 (2000) 15519. M. Qiu, S. He, J. Opt. Soc. Am. B 17 (2000) 1027. Z.Y. Lin, B.Y. Gu, G.Z. Yang, Phys. Rev. Lett. 81 (1998) 2574. C.M. Anderson, K.P. Giapis, Phys. Rev. Lett. 77 (1996) 2949. L.F. Shen, Z. Ye, S.L. He, Phys. Rev. B 68 (2003) 035109. X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, J. Zi, Appl. Phys. Lett. 80 (2002) 4291. [16] C. Zhang, F. Qiao, J. Wan, J. Zi, J. Appl. Phys. 87 (2000) 3174. [17] K.M. Ho, C.T. Chan, C.M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152. [18] The program is available from Andrew L. Reynolds, University of Glasgow, UK.