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We present an electro-optical switch implemented in coupled photonic crystal waveguides. The switch is proposed and analyzed using both the FDTD and PWM methods. The device is designed in a square lattice of silicon posts in air as well as in a hexagonal lattice of air holes in a silicon slab. The switching mechanism is a change in the conductance in the coupling region between the waveguides and hence modulating the coupling coefficient and eventually switching is achieved. Conductance is induced electrically by carrier injection or is induced optically by electron-hole pair generation. Low insertion loss and optical crosstalk in both the cross and bar switching states are predicted.
©2002 Optical Society of America
1. Introduction
Optical components that permit the miniaturization of an application specific optical integrated circuit (ASPIC) to a scale comparable to the wavelength of light are a good candidate for next generation high density optical interconnects and integration. In recent years, there has been a growing effort in the realization of active and passive photonic crystals (PhCs) as optical components and circuits [1], which can be integrated monolithically on a single chip. In 1987 E. Yablonovitch [2] and S. John [3], proposed the idea that a periodic dielectric structure can posses the property of a band gap for a certain regions in the frequency spectrum, in much the same way as an electronic band gap exists in semiconductor materials. Two and three-dimensional photonic crystals have been explored both theoretically [4–8], and experimentally [9–11]. In particular two-dimensional photonic crystals have been studied extensively since they are relatively easier to fabricate for the optical regime. Two-dimensional photonic crystals were used to provide in-plane confinement for in-plane propagation. As an example, planar waveguides [12, 13] as well as coupled-cavity waveguides [14, 15] have been used to efficiently guide electromagnetic waves through a line defect or a chain of coupled cavities, respectively, in a PhC. To this end, a single line defect introduced to a photonic crystal resembled a waveguide, for which careful design of waveguide parameters such as width and thickness, number of propagating modes can be controlled [16].
Electro-optical switches are key components of such photonic integrated circuits, yet only one proposal for implementing such switches╍a resonator device╍has appeared in the literature [17]. In this paper we present the conception, numerical analysis of a PhC channel-waveguided 2 × 2 directional coupler switch that utilizes electrically or optically induced loss (conductivity) in the coupling region between two coupled waveguides. To our knowledge, this is the first PhC directional coupler switch that has been proposed and analyzed.
The organization of this paper is as follows. In section 2 we explain in detail the design procedure for two-single mode waveguides brought into close proximity to form a directional coupler, for both cases of silicon pillars in air background as well as perforated silicon slab, we also calculate the modal propagation constant of the eigenmodes of the coupled waveguide system, which we shall use to estimate the frequency dependant coupling coefficient and hence the coupling length necessary to design our switch. In section 3, we briefly discuss the switching approach we used, and in section 4 we present the numerical analysis of the electro-optical switch. In section 5 we present the FDTD results obtained for the proposed switch implementations, and finally in section 7 we present our concluding remarks.
2. Design procedure
When two PhC waveguides are brought in close proximity of each other they form what is known in the literature as a directional coupler, shown in Fig.1. Under suitable conditions, an electromagnetic lightwave launched into one of the waveguides can couple completely into the nearby waveguide. Once the wave has crossed over, the wave couples back into the first guide so that the power is exchanged continuously as often as the length between the two waveguides permits. However, complete exchange of optical power is only possible between modes that have equal phase velocities or, equal propagation constants. More specifically, the propagation constants must be equal for each guide in isolation. Equality of propagation constants, also known as phase synchronization, occurs naturally when the two waveguides are identical. In that case all the guided modes of both waveguides are in phase synchronism and can couple to each other at all wavelengths.
Fig. 1. Coupled Photonic Crystal Waveguided (CPhCW) system consisting of two closely coupled PBG waveguides separated by two PBG layers of length Lc. system formed using a periodic array of silicon pillars arranged in square lattice.
We begin with a structure composed of two single mode waveguides placed in close proximity of each other, as the one shown in Fig. 1. This structure is no longer a single mode device, which can be seen clearly if both waveguides were fused together into one wider waveguide that is no longer single mode waveguide. Instead, it now has two eigenmode solutions, an even (symmetric) and an odd (anti-symmetric) mode, which have slightly different propagation constants and hence they propagate at different velocities. In order to calculate the coupling length necessary for a certain wavelength to completely cross over from first waveguide to second waveguide, the frequency dependant propagation constant of the even and odd modes must be defined first, also known as the modal dispersion relation of the system of coupled waveguides. In order to determine this relation a computational unit cell shown in the bottom right corner of Fig. 2a is used since the structure is periodic. For our numerical experiments we consider first a directional coupler built using two single mode 2D-PhC waveguides, obtained by removing a row from a square lattice of infinitely long dielectric rods in air background. The design parameters for the photonic crystal are as follows: the dielectric rods have a dielectric constant εr =11.56 and a radius r = 0.2a. Where a is the lattice constant of the crystal. Using these values the structure was found to have a complete band gap in the spectral range 0.23 ≤ a/λ ≤ 0.41 for TM polarization (magnetic filed in plane).
The structure shown in Fig.2 (a) can be numerically analyzed using either plane wave expansion method [18] (PWM), or finite difference time domain (FDTD) method [19, 20] with periodic boundary conditions. The result of either method is a modal dispersion diagram for the eigenmodes of the structure, from which we will be able to extract the modal propagation constants and hence calculate the coupling length necessary for full transmission of the optical power from one waveguide to a nearby waveguide. Here we will briefly explain each method as well as discuss the results, starting with a super-cell shown in Fig. 2(a) by the dashed region, PWM was used to numerically compute the Bloch propagation constants for a plane wave propagating through the super-cell. The dispersion diagram obtained using PWM is shown in Fig. 2(a). On the other hand, if the FDTD method was to be used, a set of normalized propagation constants in the range (0<β2π/a<0.35) with an interval Δβ = 0.01×2π/a will be used. In order to categorize the odd and the even modes, excitations of a TM-even mode and a TM-odd mode were launched [21], from which it was found that eigenmodes with lower frequencies belongs to the even mode, while the higher frequencies belongs to the odd mode. The dispersion diagram is then plotted over the same one previously obtained from the PWM from which we can see that they overlap for a great extent, as shown in Fig. 2(a). We will only focus our attention to the modal dispersion curves within the band gap of the structure (0.23 ≤ a/λ ≤ 0.41), as shown in Fig. 2(b).
From the dispersion curves of both odd and even modes obtained previously we can calculate the length necessary for the signal launched in waveguide 1 to completely transfer to waveguide 2 using the following procedure; for a specific frequency, find the corresponding values of the normalized modal Bloch phase constants, for the even and the odd eigenmodes. The length required for full transmission can be then calculated using [22].
As an example for the device shown in Fig.2 (a) and for a wavelength of 1550 nm (a/λ = 0.35) a = 542.5 nm, r = 108.5 nm. From Fig.2 (b) we can find the propagation constant of the odd and even modes to be (β 0 =2π×0.1977/a=2.357×106 m -1 and (βe = 2π×0.2154/a = = 2.568×106 m -1) from which we can calculate the full transmission length as,
Which means that complete transmission from one waveguide to the other requires only ten wavelengths to occur, which makes such a theory viable for high density photonic integrated circuit applications.
Fig. 2 (a) Dispersion diagram for the structure shown in Fig. 1 obtained using both PWM and FDTD methods. Two solutions corresponding to the eigenmodes (odd and even) exists within the band gap (0.23<a/λ<0.41). Where dashed line corresponds to FDTD results and solid line solution corresponds to PWM results. (b) Modal dispersion curves of the eigenmodes of the system of CPhCW shown in (a). Where the odd mode is the high frequency mode and the even mode is the low frequency mode. A straight line drawn from a normalized frequency axis will intersect with the two curves from which modal propagation constants of the even and the odd modes can be determined and hence the coupling length Lc can be calculated. (c) Dispersion diagram for a system of CPhCW consisting of two waveguides created in a hexagonal array of air holes in high dielectric background. Three layers of air holes in the coupling region separate the two waveguides. Dispersion diagram was obtained using PWM, shows that two solutions (even and odd) modes exist within the band gap (0.24786<a/λ<0.3131). (d) Modal dispersion curves of the eigenmodes of the system of CPhCW shown in bottom right corner of plot (c), where the odd mode is the low frequency mode and the even mode is the low frequency mode. Similar to plot in (b) a straight line drawn from a normalized frequency axis will intersect with the two curves from which modal propagation constants of the odd and even modes can be extracted and used to calculate the frequency dependant coupling length Lc.
Next we consider the case of perforated silicon slab, for which we use effective index approximation to simplify 3D computational problem to 2D one. This method was previously used and proven to give similar results as full 3D while reducing computational time [23, 24]. We calculated neff = 2.88 for the slab by solving the transcendental equation in [25]. Air holes of radius r/a = 0.3 were arranged in a hexagonal lattice. Using these values the structure was found to have a band gap in the spectral range 0.24786 ≤ a\λ ≤ 0.3131 for TE polarization (electric field in plane). For a full 3D structure consisting of a perforated slab of air holes arranged in hexagonal lattice, the slab thickness t/a = 0.6 and the air holes radii of r/a = 0.3 was used in our numerical experiment. For such a structure the band gap was found in the spectral range 0.2475 ≤a/λ≤ 0.3125 for TE-like mode (even mode). Hence we can use effective index approximation to reduce computational time and space. Next we seek to design a single mode waveguide in the triangular lattice of air holes. Such a waveguide was previously designed in [26], here we use the results previously presented to form a coupled waveguide system of two single mode waveguides shown in Fig.3
Fig. 3 Coupled Photonic Crystal Waveguided (CPhCW) system consisting of two closely coupled PBG waveguides separated by two PBG layers of length Lc. system formed using a periodic array of air holes arranged in hexagonal lattice. Increasing the refractive index in the wave guiding direction to create an acceptor type waveguide created single mode waveguide. [26, 27]
To obtain the modal dispersion of the even and the odd mode we numerically solve for the eigenmodes within the super cell shown in Fig.2(c) using PWM. Again we only focus on the modal dispersion curves within the band gap of the structure (0.2475 ≤ a/λ ≤ 0.3125), as shown in Fig.2 (d). Once the modal dispersion curves were obtained for both the odd and the even modes, the frequency dependant coupling length can be then obtained following the same procedure presented above for the case of dielectric pillars.
As an example, for a wavelength 1550 nm (a/λ = 0.27) a = 418.5nm, r = 125.5nm. From Fig.2d we can find the propagation constant of the odd and even modes to be (β 0=2π×0.2034/a=3.054×106 m -1) and (βe =2π×0.2359/a=3.541×106 m -1), and the full transmission length is,
If we compare the modal dispersion curves of the even and odd modes in Fig. 2(d), we will notice that, unlike the silicon pillar case, Fig. 2(c) where higher frequency modes belong to the odd mode, and lower frequency modes belong to the even mode, for the perforated slab case higher frequency modes belong to the even mode, and lower frequency modes to the odd mode.
Once modal dispersion relation have been numerically extracted, next step will be utilize the frequency dependence of such relation to design an optical switch in PhC waveguides both for the cases of dielectric rods in air background as well as air holes in silicon background.
3. Switching approach
We propose that the “loss tangent” of dielectric material in the coupling region can be modified by external “commands” to spoil the coupling, thereby re-routing the light. This is a Δα switch (not the classical Δβ switch) in which the change in optical absorption coefficient Δα is employed (the change in conductance Δσ is proportional to Δα). We have found that the induced loss does not significantly attenuate the waves traveling in the straight-through channels. This behavior is analogous to that discussed in Soref and Little [28] where electro-absorption was assumed to reduce the Q of micro-ring resonators coupled to strip channel waveguides. To attain switching in 2D-PhC guides made from Si/air or Si/SiO2, the free-carrier absorption loss of Si can be controlled by (1) carrier injection from forward-biased PN junctions on the posts, (2) depletion of doped posts with MOS gates, (3) generation of electrons and holes by above-gap light shining upon the designated pillars, a contact-less process. If the PBG coupler is implemented in III-V semiconductor heterolayers, then the electro-absorption effect could be used. The present distributed-coupling device differs from the prior art PBG switching device of Fan et al [17] that relies upon a point-defect resonator, or two point defects, situated between two PBG channels. Fan et al assumed that the Q of those cavities would be spoiled by loss induced electrically at the defects.
4. Numerical analysis of switch
For the 1550 nm center wavelength, and for our first structure, we assumed a 2D photonic crystal of 217-nm-diamter silicon dielectric rods (εr = 11.6) arrayed in a square lattice (a = 542.5 nm) on an air background. Line defects and bent lines defined the channel waveguides. Our PBG waveguides are analogous to the practical 2D e-beam-etched silicon waveguide system developed by Loncar et al [9, 10]. For the perforated slab, we used a 2D-hexagonal-PhC lattice of air holes with 251-nm-diameter and lattice constant a = 418.5nm. The slab has an effective index of neff = 2.88. In this analysis, we used the finite-difference time-domain method with perfectly matched absorbing boundary conditions around the rectangle enclosing the 2 × 2 switch to truncate the computational domain and minimize reflections from the outer boundary. Our full wave solution for forward and backward traveling waves solved alternately for E and H fields at different spatial points (λ/20 sampling rate) as time progressed. Examination of several switching test structures at σ ~ 0, showed that the length l c = 28a for the square lattice and lc= 16a for the hexagonal lattice of the parallel-channel interaction region ensured that ~100% of the optical power launched into Port 1 was transferred to the other waveguide and output at Port 3. We analyzed the spectral transmission of this coupler and found a periodic response [29] whose first peak has a FWHM pass-band of about 20 nm.
5. Results
5.1 Silicon pillars case
Figure 4 presents a top view of the planar “crossbar” in it’s off and on conditions for the square lattice of silicon rods in air background. Top views of the corresponding infrared intensity distributions within the device are shown also. For a given value of σ, and assuming unity power input to Port 1, we determined the power emerging from Ports 2, 3, and 4, respectively. This switching response as a function of σ is shown in Fig. 5. The transmissions are: T(Port2) > 81 % for σ > 30 Ω-1cm-1 and T(Port 3) >88 % for σ < 0.0003 Ω-1cm-1. At σ = 10-4Ω-1cm-1, the predicted crosstalks are: Forward CT= P(2)/P(3) = -29.4 dB, Backward CT = P(4)/P(3) = -27.3 dB, while for σ = 100 Ω-1cm-1, Forward CT=P(3)/P(2) = -23.1 dB, Backward CT= P(4)/P(2) = -28.6 dB. Two accompanying movie files are included in Fig. 8 and Fig.9, respectively for the two extreme cases of the switch (OFF (σ = 0.001 Ω-1cm-1) and ON (σ = 10 Ω-1cm-1) states).
If we assume that the dielectric posts are undoped “intrinsic” silicon, then what concentrations of electrons or holes are required to be injected into those posts to obtain the desired increase in conductance? To answer this question, we assumed that the effect of injection is approximately the same as the effect of doping the silicon with n-type or p-type impurities. From Fig. 21 of Sze [30], the dependence of σ upon doping density is shown in Fig. 6.
Fig 4. Four snapshots for FDTD simulations of 2×2 electro-optical switch shown in Fig.1. the switch is formed in a square photonic crystal lattice of silicon pillars.
5.2 Perforated silicon slab case
For the crossbar switch in Fig.3 formed in a perforated slab of air holes arranged in a hexagonal lattice, we present the FDTD results in Fig. 6, for both the ON “bar” and OFF “cross” states. Where in this case only the conductivity of the center row was modulated, as shown in Fig. 3. The switching response for this device is shown in Fig. 7. The transmissions are: T(Port 2) > 85 % for σ > 105 Ω-1cm-1 and T(Port3) >90 % for σ < 102 Ω-1cm-1. At σ = 10 Ω-1cm-1, the predicted crosstalks are: Forward CT= P(2)/P(3) = -22.2 dB, Backward CT= P(4)/P(3) =-23 dB, while for σ = 3 × 105 Ω-1cm-1, Forward CT=P(3)/P(2) = -32.2 dB, Backward CT= P(4)/P(2) = -36.9 dB.
Fig. 6 Four snapshots for FDTD simulations of 2×2 electro-optical switch formed in a perforated slab of air holes arranged on a hexagonal photonic crystal lattice.
The switching response shown in Fig. 5 and Fig. 7 show that there is a minimum value for the output optical power at various ports for a specific value of conductivity (σ = 0.1 Ω-1cm-1) for the silicon pillars case and (σ = 104 Ω-1cm-1) for the perforated slab case, at this transient value the optical power launched at the input port will be absorbed in the coupling region between the two waveguides and the device suffer high attenuation coefficient α in the coupling region. An increase or decrease in the conductivity will redirect the optical power to either bar- or cross-states respectively. This behavior was previously discussed in by Soref and Bennett [31].
The Fig.-1 and 3 devices are intended to be interconnected and cascaded in the forward direction into an N × N optical cross-connect network. In this case further optimization to crosstalk can be achieved by minimizing the reflections at the waveguide bends. Techniques for enhancing transmission through waveguide bends and hence reducing reflections include, broadband [24, 32], and narrowband [13, 33] techniques.
6. Acknowledgments
The authors would like to thank the reviewers for their very constructive and insightful comments. The authors would also like to thank Emphotonics.com for providing the FDTD and PWM tools (EMP), which were used in the simulation and design of the work, presented in this article.
7. Summary
We have presented simulation results on a novel, compact, 2D-PhCwaveguided 2×2 directional coupler switch controlled by optical loss induced in the dielectric posts as well as perforated slab between the parallel line defects. Using the FDTD method on a 1.55 μm device, we predict low insertion loss and crosstalk below -23 dB in both switching states although the required change in conductance is large in this non-optimized switch. We are presently exploring improved switch designs that produce “complete” switching with smaller Δσ.
References and links
1. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic Crystals for Micro Lightwave Circuits Using Wavelength-Dependent Angular Beam Steering,” Appl. Phys. Lett. , 74, 1370–1372, (1999). [CrossRef]
2. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. , 58, 2059–2062, (1987). [CrossRef] [PubMed]
3. S. John, “Strong Localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. , 58, 2486, (1987). [CrossRef] [PubMed]
4. D. W. Prather, A. Sharkawy, and S. Shouyuan, “Photonic Crystals Design and Applications,” in Handbook of Nanoscience, Engineering, and Technology, Electrical Engineering Handbook, G. J. Iafrate, S. E. Lyshevski, D. W. Brenner, and W. A. Goddard III, Eds. (CRC Press, Boca Raton, FL.2002).
5. A. Adibi, R. K. Lee, Y. Xu, A. Yariv, and A. Scherer, “Design of photonic crystal optical waveguides with single mode propagation in the photonic bandgap,” Electron. Lett. , 36, 1376–1378, (2000). [CrossRef]
6. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals. (Princeton, New Jersey, 1995).
7. A. Sharkawy, S. Shi, and D. W. Prather, “Multichannel Wavelength Division Multiplexing Using Photonic Crystals,” Appl. Opt. , 40, 2247–2252, (2001). [CrossRef]
8. D. Pustai, A. Sharkawy, S. Shouyuan, and D. W. Prather, “Tunable Photonic Crystal Microcavities,” Appl. Opt. , 41, 5574–5579, (2002). [CrossRef] [PubMed]
9. M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in Planar Photonic Crystals,” Appl. Phys. Lett. , 77, 1937–1939, (2000). [CrossRef]
10. M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabrication of silicon photonic crystal optical waveguides,” J. Lightwave Technol. , 18, 1402–1411, (2000). [CrossRef]
11. D. W. Prather, J. Murakowski, S. Shouyuan, S. Venkataraman, A. Sharkawy, C. Chen, and D. Pustai, “High Efficiency Coupling Structure for a single Line-Defect Photonic Crystal Waveguide,” Optics Letters , 27, 1601–1603, (2002). [CrossRef]
12. R. Stoffer, H. J. W. M. Hoekstra, R. M. D. Ridder, E. V. Groesen, and F. P. H. V. Beckum, “Numerical Studies of 2D Photonic Crystals: Waveguides, Coupling Between Waveguides and Filters,” Opt. Quantum Electron. , 32, 947–961, (2000). [CrossRef]
13. C. J. M. Smith, H. Benisty, S. Olivier, M. Rattier, C. Weisbuch, T. F. Krauss, R. M. D. L. Rue, R. Houdre, and U. Oseterle, “Low-Loss Channel Waveguides with Two-Dimensional Photonic Crystal Boundaries,” Appl. Phys. Lett. , 77, 2813–2815, (2000). [CrossRef]
14. M. Bayindir, B. Temmelkuran, and E. Ozbay, “Propagation of Photons by Hopping: A Waveguiding Mechanism Through Localized Coupled Cavities in Three-Dimensional Photonic Crystals,” Phys. Rev. B , 61, R11855–R11858, (2000). [CrossRef]
15. M. Bayindir and E. Ozbay, “Heavy photons at coupled-cavity waveguide band edges in a three-dimensional photonic crystal,” Phys. Rev. B , 62, R2247–R2250, (2000). [CrossRef]
16. M. Loncar, J. Vuckovic, and A. Scherer, “Methods for controlling positions of guided modes of photonic-crystal waveguides,” J. Opt. Soc. Am. B-Optical Physics , 18, 1362–1368, (2001). [CrossRef]
17. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, B. E. Little, and H. A. Haus, “High Efficiency Channel drop filter with Absorption-Induced On/Off Switching and Modulation.” USA, 2000.
18. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B , 44, 8565–8571, (1991). [CrossRef]
19. D. Hermann, M. Frank, and K. Busch, “Photonic Band Structure Computations,” Opt. Express , 8, 167–172, (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-1-167. [CrossRef] [PubMed]
20. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second Edition. (Boston, MA: Artech House, 2000).
21. L. L. Liou and A. Crespo, “Dielectric Optical waveguide coupling analysis using two-dimensional finite difference in time-domain simulations,” Microwave and optical Technology Letters , 26, 234–237, (2000). [CrossRef]
22. S. Boscolo, M. Midiro, and C. G. Someda, “Coupling and Decoupling of Electromagnetic Waves in Parallel 2-D Photonic Crystal Waveguides,” IEEE J. Quant. Electron. , 38, 47–53, (2002). [CrossRef]
23. O. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically think dielectric slab,” J. opt. Soc. Am. B , 16, 275–285, (1999). [CrossRef]
24. A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. , 80, 1698–1700, (2002). [CrossRef]
25. A. Yariv and P. Yeh, Optical waves in Crystals. (New York: John Wiley & Sons, 1984).
26. M. L. Povinelli, S. G. Johnson, J. Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B , 64, 753131–753138, (2001). [CrossRef]
27. S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B , 60, 5751–5758, (1999). [CrossRef]
28. R. A. Soref and B. E. Little, “Proposed N-Wavelength M-Fiber WDM crossconnect switch using Active Microring Resonators,” IEEE Photon. Technol. Lett. , 10, 1121–1123, (1998). [CrossRef]
29. M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing with Photonic Crystal Waveguide couplers,” J. Lightwave Technol. , 19, 1970–1975, (2001). [CrossRef]
30. S. M. Sze, Physics of Semiconductor Devices, 2nd ed: John Wiley & Sons Inc., 1981).
31. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. , QE-23, 123–129, (1987). [CrossRef]
32. A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B , 62, 4488–4492, (2000). [CrossRef]
33. S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B , 18, 162–165, (2001). [CrossRef]
