Three-visible-light wave combiner based on photonic crystal waveguides Dingwen Liu, Yiling Sun,* and Zhengbiao Ouyang Shenzhen Key Laboratory of Micro-Nano Photonic Information Technology, College of Electronic Science and Technology, Shenzhen University, Shenzhen, Guangdong 518060, China *Corresponding author: [email protected] Received 18 April 2014; revised 27 June 2014; accepted 27 June 2014; posted 30 June 2014 (Doc. ID 210345); published 18 July 2014
We present a three-visible-light wave combiner based on two-dimensional photonic crystal waveguides whose widths are not integral multiples of the lattice period. The proposed device consists of two cascaded directional couplers. It combines three visible light waves with different wavelengths from three input ports into a single output port. As an example, a combiner for combining light waves of 635, 532, and 488 nm, which are commonly used as the three primary colors in laser display systems, is designed and demonstrated through the finite-difference time-domain method. The results show that the proposed device can perform efficient synthesis for three visible light waves with transmittance exceeding 89% for each wavelength and high ability in preventing the backward coupling of waves from different waveguides. The method for designing the combiner is useful for designing other waveguide couplers based on photonic crystals made of dispersion materials. © 2014 Optical Society of America OCIS codes: (250.5300) Photonic integrated circuits; (300.6550) Spectroscopy, visible; (060.4230) Multiplexing. http://dx.doi.org/10.1364/AO.53.004791
1. Introduction
Photonic crystals (PhCs) are microstructured material whose refractive index changes periodically. Their main feature is the presence of photonic bandgaps (PBGs) [1–3]. Photonic crystal waveguides (PCWs) are formed by removing one row of rods in perfect PhCs. The PCWs can guide light waves with frequencies that fall into the PBGs. When there are two or more PCWs, under certain conditions, coupling will occur between them [4]. Through coupling of PCWs, wavelength selection can be achieved that can be used for wavelength division multiplexing or demultiplexing [5,6]. However, to our knowledge, most wavelength combiners based on PhCs operate in the communication wavelength band [7,8] while less attention has been paid to PhC wave combiners operating in visible 1559-128X/14/214791-04$15.00/0 © 2014 Optical Society of America
bands. In particular, the material dispersion characteristics of PhCs have not been explored in the design of wave combiners. However, with the development of laser display technology and laser illumination, especially the development of portable laser TV or projectors, compact visible band combiners are needed. Thus, we focus on the combination of the three basic colors, which are generally 635, 532, and 488 nm, for laser display, laser TV, and laser illumination. In this paper, a three-wavelength combiner based on two-dimensional PhCs in the visible wavelength region is presented. A combiner for combining light waves of 635, 532, and 488 nm, which are commonly used as three primary colors in laser display systems, is designed and demonstrated through finite-difference time-domain (FDTD) method. The results show that the proposed device can perform efficient synthesis for the three primary colors with transmittance exceeding 89% for each wavelength and high ability in preventing the backward coupling of waves from different waveguides. The method for 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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designing the combiner is useful for designing other PCW couplers based on PhCs made of dispersion materials. 2. Design of Three-Wavelength Combiners Based on PCWs
For combiners of light waves with different wavelengths, a large PBG is often required. As is known, dielectric pillars with large refractive index can be used to build PhCs with large PBGs. It is also known that GaP has a large refractive index and a low extinction coefficient in the visible band. Therefore, we consider a PhC composed of circular pillars or rods of GaP, arranged in air in a square lattice with a lattice constant a. The largest TM bandgap occurs when the rods have a radius r 0.18a. For such a structure, confinement in the vertical direction is necessary. We can use two periodic woodpile three-dimensional PhC structures or 5–8 periodic one-dimensional PhC films for this confinement. We consider PhC wave combiners through directional coupling PCWs, in which a wave of a certain frequency is to be coupled from one PCW to another. To realize this function, we consider the PCW directional couplers indicated in Fig. 1. The directional couplers shown in Fig. 1 consist of two parallel PCWs with one row of dielectric rods in the interaction region. In Fig. 1(a), the PCW directional couple has a regular PCW width, which is twice the lattice constant. In Fig. 1(b), the PCW directional couple has an irregular waveguide width, which is different from 2a, to realize coupling of waves with different wavelengths. As an example, we choose the three wavelengths in the visible band to be 488, 532, and 635 nm, which are commonly used as the three primary colors in optical display systems. The refractive index of GaP rods is varied with operating wavelength. The refractive indices are 3.63613, 3.47749, and 3.31119 for the three wavelengths of 488, 532, and 635 nm, respectively [9]. Through a plane wave expansion (PWE) method, we can obtain the dispersion curves of the coupled PCWs in Fig. 1. Figure 2(a) shows the dispersion curve for the refractive index of the rods n 3.63613. When the two same PCWs are close, a single defect mode is split into two eigenmodes (the even- and odd-modes). It can be seen from Fig. 2(a) that there
are two modes in the structure indicated in Fig. 1(a). These two modes can be regarded as the defect modes in the PhC with a line defect. The two modes can couple from one PCW to the other in the structure with a coupling length of Lc
a ; 2k1 − k2
(1)
where k1 and k2 are the normalized wave vectors of the even and odd modes at a given normalized frequency [10]. Equation (1) implies the coupling length will be infinitely long when the two modes have the same wave number (k1 k2 ), which means that there would be no coupling between the PCWs with finite length. From Fig. 2(a) we can see that there is a decoupling point at which the two modes degenerate as one mode, with a normalized frequency of 0.433a∕λ. We design the main PCW for transmitting this mode as one of the three waves to be combined. In such a way, this mode can only transmit in the main PCW, preventing losses that may be caused by coupling of this mode to the branch PCWs. Setting this normalized frequency 0.433 corresponding to the wavelength 488 nm, we can get the lattice constant a 211.3 nm. With this lattice constant, we have the normalized frequencies corresponding to 532 and 635 nm as 0.397 and 0.333, respectively. The dispersion curve of the structure shown in Fig. 1(a) corresponding to n 3.31119 is shown in Fig. 2(b). As we can see from Fig. 2(b), the structure can only support one mode for the normalized frequency of 0.333. This means that the wave of 635 nm cannot be coupled between the two PCWs with different wave numbers. This problem is caused by the limitation of the PBG width and the material dispersion. Thus, conventional design methods for wavelength division multiplexing/demultiplexing in the communication band cannot work for cases where PhCs adopt practical materials, with dispersion and limited PBG width in the visible wavelength band. We present a new method to solve this problem as follows. We increase the width of the line defect by 55 nm, which changes the corresponding dispersion curve [11]. The structure with an increased width of
Fig. 1. Schematic of the PCW directional couplers. (a) Structure with regular waveguide width and (b) structure with irregular waveguide width. 4792
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Fig. 3. Configuration of the three-wavelength combiner.
Fig. 2. Dispersion curve for the structure indicated in Fig. 1(a) with the refractive index n 3.63613 (a), in Fig. 1(a) with the refractive index n 3.31119 (b), and in Fig. 1(b) with the refractive index n 3.31119 (c).
PCW is shown in Fig. 1(b), which, in this paper, is also called the irregular PCW. The dispersion curve for this structure is shown in Fig. 2(c). The normalized frequency of the decoupling point shifts from 0.433 to 0.428. We recalculate the lattice constant and get a 208.9 nm by setting the normalized frequency 0.428 corresponding to the wavelength 488 nm; then, we have: 55 nm 0.26a. As shown in Fig. 1(b), the width of the PCW is increased from 2a to 2.26a. Furthermore, the normalized frequencies corresponding to the wavelengths of 635 and 532 nm become 0.329 and 0.393, respectively. From Fig. 2(c), we can see that the structure can support two modes at the normalized frequency of 0.329. This means that the waves of 635 nm can be coupled between the branch PCW and the main PCW. Using Eq. (1), we can obtain the coupling lengths as Lc1 29a and Lc2 7a for the waves of 532 and 635 nm, respectively.
With the above consideration and analysis, we can obtain a design for the three-wavelength combiner, as illustrated in Fig. 3. The central PCW is the main one while the PCWs marked with 1# and 2# are the branch PCWs. The light waves of wavelengths 635, 488, and 532 nm are input from port A of PCW 1#, port B of the main PCW, and port C of PCW 2#, respectively. The coupling lengths between the main PCW and the branch PCWs 1# and 2# are 7a and 29a, respectively, and the PCW widths of 1# and 2# increase by 55 nm, as shown in Fig. 1(b). Light waves of 635 nm and 532 nm are coupled into the main PCW through the directional coupling between the PCWs. It should be noted that the light wave with the wavelength 488 nm propagates only in the main PCW in our design. In addition, the wave from the PCW 1# will not be coupled to the PCW 2# because the waves of 635 and 532 nm have different coupling lengths between the main PCW and the branch PCWs. Therefore, the structure has a high isolation among the three PCWs. Accordingly, the transmission of the waves from the input to the output is enhanced. To improve the transmittance further, some dielectric rods were added at the input ports A, B, and C, as indicated in Fig. 3. 3. Simulation and Demonstration
We use the FDTD method to simulate the wave propagation in the three-wavelength wave combiner shown in Fig. 3 using the commercial software module, Rsoft FullWave. The field distribution obtained in the combiner is shown in Fig. 4. The transmittances for the three wavelengths 635, 488, and 532 nm are 93.6%, 89.6%, and 91.8%, respectively. The transmittances for the waves of 635 and 532 nm are all above 90% while that for the wavelength 488 nm is approximately 90%. The reason is that the normalized frequency corresponding to 488 nm is at the edge of the band, so the confinement effect of the PhC is weaker than that of the other two wavelengths. The ratios of the reflected power to the input power are 2%, 7%, and 5% for the wavelengths of 635, 488, and 532 nm, respectively. The power reflections are below 10% for three-wavelength channels. For applications, we generally need a circulator to block the reflected light from going to the laser source. We have also calculated the insertion losses and operating bandwidths for the three waves, as shown in Fig. 5. From Fig. 5, we can see that the insertion 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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Fig. 4. Field distribution in the combiner shown in Fig. 3.
for designing other PCW couplers based on PhCs made of dispersive materials. This work was supported by the National Natural Science Foundation of China (No. 61275043) and the Fund of Shenzhen Key Laboratory of Micro-Nano Photonic Information Technology (No. MN201113). References
Fig. 5. Insertion loss as a function of Δλ.
losses corresponding to 635, 488, and 532 nm are 0.29, 0.48, and 0.37 dB, respectively. The 1 dB bandwidths of the three light waves are 9, 13, and 20 nm, respectively. 4. Conclusions
In this paper, we propose a new method for designing visible light wave combiners that overcomes the limitations and difficulties caused by PBG width and material dispersion. We adopt PCWs of widths in nonintegral multiples of the PhC lattice period, such that the dispersion curve can be adjusted and the desired coupling properties obtained. The transmittances for the three wavelengths 635, 488, and 532 nm are 93.6%, 89.6%, and 91.8%, respectively. The PhC wave combiners for three wavelengths in the visible region can miniaturize the light source in laser display systems, and may have potential applications in laser imaging, laser illumination, and other fields. The method for designing the combiner is useful
4794
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1. R. P. Zaccaria, P. Verma, and S. Kawaguchi, “Manipulating full photonic band gap in two dimensional birefringent photonic crystal,” Opt. Express 16, 14812–14820 (2008). 2. F. Wen, S. David, X. Checoury, M. E. Kurdi, and P. Boucaud, “Two-dimensional photonic crystals with large complete photonic band gaps in both TE and TM polarizations,” Opt. Express 16, 12278–12289 (2008). 3. X. Cheng and J. Yang, “Maximizing band gaps in twodimensional photonic crystals in square lattices,” J. Opt. Soc. Am. A 30, 2314–2319 (2013). 4. S. Kuchinsky, V. Golyatin, A. Kutikov, T. Pearsall, and D. Nedeljkovic, “Coupling between photonic crystal waveguides,” IEEE J. Quantum Electron. 38, 1349–1352 (2002). 5. M. Koshiba, “Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. 19, 1970–1975 (2001). 6. S. Boscolo, M. Midrio, and C. G. Someda, “Coupling and decoupling of electromagnetic waves in parallel 2-D photonic crystal waveguides,” IEEE J. Quantum Electron. 38, 47–53 (2002). 7. Y. Wu, K. Hsu, and T. Shih, “Thirty-two-channel dense-wavelength-division multiplexer based on cascade two dimensional photonic crystal waveguide structure,” J. Opt. Soc. Am. B 24, 2075–2080 (2007). 8. Y. Wu, K. Hsu, and T. Shih, “New design of four-channel add–drop filters based on double-resonant cavity photonic crystals,” J. Opt. Soc. Am. B 26, 640–644 (2009). 9. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983). 10. F. S.-S. Chien, Y.-J. Hsu, W.-F. Hsieh, and S.-C. Cheng, “Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides,” Opt. Express 12, 1119–1125 (2004). 11. N. Yamamoto, T. Ogawa, and K. Komori, “Photonic crystal directional coupler switch with small switching length and wide bandwidth,” Opt. Express 14, 1223–1229 (2006).
We present a three-visible-light wave combiner based on two-dimensional photonic crystal waveguides whose widths are not integral multiples of the lattice period. The proposed device consists of two cascaded directional couplers. It combines three visible light waves with different wavelengths from three input ports into a single output port. As an example, a combiner for combining light waves of 635, 532, and 488 nm, which are commonly used as the three primary colors in laser display systems, is designed and demonstrated through the finite-difference time-domain method. The results show that the proposed device can perform efficient synthesis for three visible light waves with transmittance exceeding 89% for each wavelength and high ability in preventing the backward coupling of waves from different waveguides. The method for designing the combiner is useful for designing other waveguide couplers based on photonic crystals made of dispersion materials. © 2014 Optical Society of America OCIS codes: (250.5300) Photonic integrated circuits; (300.6550) Spectroscopy, visible; (060.4230) Multiplexing. http://dx.doi.org/10.1364/AO.53.004791
1. Introduction
Photonic crystals (PhCs) are microstructured material whose refractive index changes periodically. Their main feature is the presence of photonic bandgaps (PBGs) [1–3]. Photonic crystal waveguides (PCWs) are formed by removing one row of rods in perfect PhCs. The PCWs can guide light waves with frequencies that fall into the PBGs. When there are two or more PCWs, under certain conditions, coupling will occur between them [4]. Through coupling of PCWs, wavelength selection can be achieved that can be used for wavelength division multiplexing or demultiplexing [5,6]. However, to our knowledge, most wavelength combiners based on PhCs operate in the communication wavelength band [7,8] while less attention has been paid to PhC wave combiners operating in visible 1559-128X/14/214791-04$15.00/0 © 2014 Optical Society of America
bands. In particular, the material dispersion characteristics of PhCs have not been explored in the design of wave combiners. However, with the development of laser display technology and laser illumination, especially the development of portable laser TV or projectors, compact visible band combiners are needed. Thus, we focus on the combination of the three basic colors, which are generally 635, 532, and 488 nm, for laser display, laser TV, and laser illumination. In this paper, a three-wavelength combiner based on two-dimensional PhCs in the visible wavelength region is presented. A combiner for combining light waves of 635, 532, and 488 nm, which are commonly used as three primary colors in laser display systems, is designed and demonstrated through finite-difference time-domain (FDTD) method. The results show that the proposed device can perform efficient synthesis for the three primary colors with transmittance exceeding 89% for each wavelength and high ability in preventing the backward coupling of waves from different waveguides. The method for 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
4791
designing the combiner is useful for designing other PCW couplers based on PhCs made of dispersion materials. 2. Design of Three-Wavelength Combiners Based on PCWs
For combiners of light waves with different wavelengths, a large PBG is often required. As is known, dielectric pillars with large refractive index can be used to build PhCs with large PBGs. It is also known that GaP has a large refractive index and a low extinction coefficient in the visible band. Therefore, we consider a PhC composed of circular pillars or rods of GaP, arranged in air in a square lattice with a lattice constant a. The largest TM bandgap occurs when the rods have a radius r 0.18a. For such a structure, confinement in the vertical direction is necessary. We can use two periodic woodpile three-dimensional PhC structures or 5–8 periodic one-dimensional PhC films for this confinement. We consider PhC wave combiners through directional coupling PCWs, in which a wave of a certain frequency is to be coupled from one PCW to another. To realize this function, we consider the PCW directional couplers indicated in Fig. 1. The directional couplers shown in Fig. 1 consist of two parallel PCWs with one row of dielectric rods in the interaction region. In Fig. 1(a), the PCW directional couple has a regular PCW width, which is twice the lattice constant. In Fig. 1(b), the PCW directional couple has an irregular waveguide width, which is different from 2a, to realize coupling of waves with different wavelengths. As an example, we choose the three wavelengths in the visible band to be 488, 532, and 635 nm, which are commonly used as the three primary colors in optical display systems. The refractive index of GaP rods is varied with operating wavelength. The refractive indices are 3.63613, 3.47749, and 3.31119 for the three wavelengths of 488, 532, and 635 nm, respectively [9]. Through a plane wave expansion (PWE) method, we can obtain the dispersion curves of the coupled PCWs in Fig. 1. Figure 2(a) shows the dispersion curve for the refractive index of the rods n 3.63613. When the two same PCWs are close, a single defect mode is split into two eigenmodes (the even- and odd-modes). It can be seen from Fig. 2(a) that there
are two modes in the structure indicated in Fig. 1(a). These two modes can be regarded as the defect modes in the PhC with a line defect. The two modes can couple from one PCW to the other in the structure with a coupling length of Lc
a ; 2k1 − k2
(1)
where k1 and k2 are the normalized wave vectors of the even and odd modes at a given normalized frequency [10]. Equation (1) implies the coupling length will be infinitely long when the two modes have the same wave number (k1 k2 ), which means that there would be no coupling between the PCWs with finite length. From Fig. 2(a) we can see that there is a decoupling point at which the two modes degenerate as one mode, with a normalized frequency of 0.433a∕λ. We design the main PCW for transmitting this mode as one of the three waves to be combined. In such a way, this mode can only transmit in the main PCW, preventing losses that may be caused by coupling of this mode to the branch PCWs. Setting this normalized frequency 0.433 corresponding to the wavelength 488 nm, we can get the lattice constant a 211.3 nm. With this lattice constant, we have the normalized frequencies corresponding to 532 and 635 nm as 0.397 and 0.333, respectively. The dispersion curve of the structure shown in Fig. 1(a) corresponding to n 3.31119 is shown in Fig. 2(b). As we can see from Fig. 2(b), the structure can only support one mode for the normalized frequency of 0.333. This means that the wave of 635 nm cannot be coupled between the two PCWs with different wave numbers. This problem is caused by the limitation of the PBG width and the material dispersion. Thus, conventional design methods for wavelength division multiplexing/demultiplexing in the communication band cannot work for cases where PhCs adopt practical materials, with dispersion and limited PBG width in the visible wavelength band. We present a new method to solve this problem as follows. We increase the width of the line defect by 55 nm, which changes the corresponding dispersion curve [11]. The structure with an increased width of
Fig. 1. Schematic of the PCW directional couplers. (a) Structure with regular waveguide width and (b) structure with irregular waveguide width. 4792
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Fig. 3. Configuration of the three-wavelength combiner.
Fig. 2. Dispersion curve for the structure indicated in Fig. 1(a) with the refractive index n 3.63613 (a), in Fig. 1(a) with the refractive index n 3.31119 (b), and in Fig. 1(b) with the refractive index n 3.31119 (c).
PCW is shown in Fig. 1(b), which, in this paper, is also called the irregular PCW. The dispersion curve for this structure is shown in Fig. 2(c). The normalized frequency of the decoupling point shifts from 0.433 to 0.428. We recalculate the lattice constant and get a 208.9 nm by setting the normalized frequency 0.428 corresponding to the wavelength 488 nm; then, we have: 55 nm 0.26a. As shown in Fig. 1(b), the width of the PCW is increased from 2a to 2.26a. Furthermore, the normalized frequencies corresponding to the wavelengths of 635 and 532 nm become 0.329 and 0.393, respectively. From Fig. 2(c), we can see that the structure can support two modes at the normalized frequency of 0.329. This means that the waves of 635 nm can be coupled between the branch PCW and the main PCW. Using Eq. (1), we can obtain the coupling lengths as Lc1 29a and Lc2 7a for the waves of 532 and 635 nm, respectively.
With the above consideration and analysis, we can obtain a design for the three-wavelength combiner, as illustrated in Fig. 3. The central PCW is the main one while the PCWs marked with 1# and 2# are the branch PCWs. The light waves of wavelengths 635, 488, and 532 nm are input from port A of PCW 1#, port B of the main PCW, and port C of PCW 2#, respectively. The coupling lengths between the main PCW and the branch PCWs 1# and 2# are 7a and 29a, respectively, and the PCW widths of 1# and 2# increase by 55 nm, as shown in Fig. 1(b). Light waves of 635 nm and 532 nm are coupled into the main PCW through the directional coupling between the PCWs. It should be noted that the light wave with the wavelength 488 nm propagates only in the main PCW in our design. In addition, the wave from the PCW 1# will not be coupled to the PCW 2# because the waves of 635 and 532 nm have different coupling lengths between the main PCW and the branch PCWs. Therefore, the structure has a high isolation among the three PCWs. Accordingly, the transmission of the waves from the input to the output is enhanced. To improve the transmittance further, some dielectric rods were added at the input ports A, B, and C, as indicated in Fig. 3. 3. Simulation and Demonstration
We use the FDTD method to simulate the wave propagation in the three-wavelength wave combiner shown in Fig. 3 using the commercial software module, Rsoft FullWave. The field distribution obtained in the combiner is shown in Fig. 4. The transmittances for the three wavelengths 635, 488, and 532 nm are 93.6%, 89.6%, and 91.8%, respectively. The transmittances for the waves of 635 and 532 nm are all above 90% while that for the wavelength 488 nm is approximately 90%. The reason is that the normalized frequency corresponding to 488 nm is at the edge of the band, so the confinement effect of the PhC is weaker than that of the other two wavelengths. The ratios of the reflected power to the input power are 2%, 7%, and 5% for the wavelengths of 635, 488, and 532 nm, respectively. The power reflections are below 10% for three-wavelength channels. For applications, we generally need a circulator to block the reflected light from going to the laser source. We have also calculated the insertion losses and operating bandwidths for the three waves, as shown in Fig. 5. From Fig. 5, we can see that the insertion 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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Fig. 4. Field distribution in the combiner shown in Fig. 3.
for designing other PCW couplers based on PhCs made of dispersive materials. This work was supported by the National Natural Science Foundation of China (No. 61275043) and the Fund of Shenzhen Key Laboratory of Micro-Nano Photonic Information Technology (No. MN201113). References
Fig. 5. Insertion loss as a function of Δλ.
losses corresponding to 635, 488, and 532 nm are 0.29, 0.48, and 0.37 dB, respectively. The 1 dB bandwidths of the three light waves are 9, 13, and 20 nm, respectively. 4. Conclusions
In this paper, we propose a new method for designing visible light wave combiners that overcomes the limitations and difficulties caused by PBG width and material dispersion. We adopt PCWs of widths in nonintegral multiples of the PhC lattice period, such that the dispersion curve can be adjusted and the desired coupling properties obtained. The transmittances for the three wavelengths 635, 488, and 532 nm are 93.6%, 89.6%, and 91.8%, respectively. The PhC wave combiners for three wavelengths in the visible region can miniaturize the light source in laser display systems, and may have potential applications in laser imaging, laser illumination, and other fields. The method for designing the combiner is useful
4794
APPLIED OPTICS / Vol. 53, No. 21 / 20 July 2014
1. R. P. Zaccaria, P. Verma, and S. Kawaguchi, “Manipulating full photonic band gap in two dimensional birefringent photonic crystal,” Opt. Express 16, 14812–14820 (2008). 2. F. Wen, S. David, X. Checoury, M. E. Kurdi, and P. Boucaud, “Two-dimensional photonic crystals with large complete photonic band gaps in both TE and TM polarizations,” Opt. Express 16, 12278–12289 (2008). 3. X. Cheng and J. Yang, “Maximizing band gaps in twodimensional photonic crystals in square lattices,” J. Opt. Soc. Am. A 30, 2314–2319 (2013). 4. S. Kuchinsky, V. Golyatin, A. Kutikov, T. Pearsall, and D. Nedeljkovic, “Coupling between photonic crystal waveguides,” IEEE J. Quantum Electron. 38, 1349–1352 (2002). 5. M. Koshiba, “Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. 19, 1970–1975 (2001). 6. S. Boscolo, M. Midrio, and C. G. Someda, “Coupling and decoupling of electromagnetic waves in parallel 2-D photonic crystal waveguides,” IEEE J. Quantum Electron. 38, 47–53 (2002). 7. Y. Wu, K. Hsu, and T. Shih, “Thirty-two-channel dense-wavelength-division multiplexer based on cascade two dimensional photonic crystal waveguide structure,” J. Opt. Soc. Am. B 24, 2075–2080 (2007). 8. Y. Wu, K. Hsu, and T. Shih, “New design of four-channel add–drop filters based on double-resonant cavity photonic crystals,” J. Opt. Soc. Am. B 26, 640–644 (2009). 9. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983). 10. F. S.-S. Chien, Y.-J. Hsu, W.-F. Hsieh, and S.-C. Cheng, “Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides,” Opt. Express 12, 1119–1125 (2004). 11. N. Yamamoto, T. Ogawa, and K. Komori, “Photonic crystal directional coupler switch with small switching length and wide bandwidth,” Opt. Express 14, 1223–1229 (2006).
