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Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 20 — Oct. 7, 2002
  • pp: 1048–1059
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Electro-optical switching using coupled photonic crystal waveguides

Ahmed Sharkawy, Shouyuan Shi, Dennis W. Prather, and Richard A. Soref  »View Author Affiliations


Optics Express, Vol. 10, Issue 20, pp. 1048-1059 (2002)
http://dx.doi.org/10.1364/OE.10.001048


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Abstract

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

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

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.

]. 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.

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.

Lc=π(βeβo).
(1)

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,

Lc=π(2.5682.357)×106=14.88μm
=14.88μm0.5425μm=28a=9.6λ.
(2)

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.

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]

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,

Lc=π(3.5413.054)×106=6.44μm
=6.44μm0.4185μm=16a=4.0λ.
(3)

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

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]

] 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

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.

] 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

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

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]

]. 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

29. M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing with Photonic Crystal Waveguide couplers,” J. Lightwave Technol. , 19, 1970–1975, (2001). [CrossRef]

] 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

30. S. M. Sze, Physics of Semiconductor Devices, 2nd ed: John Wiley & Sons Inc., 1981).

], 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.
Fig. 5 Calculated switching characteristics of Fig. 1 crossbar switch (silicon pillars case).
Fig. 6. Dependence of σ upon N and P doping.

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.
Fig. 7 Calculated switching characteristics of Fig. 3 crossbar switch (perforated slab case).

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

31. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. , QE-23, 123–129, (1987). [CrossRef]

].

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

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]

, 32

32. A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B , 62, 4488–4492, (2000). [CrossRef]

], and narrowband [13

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]

, 33

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]

] 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.

Fig. 8 (171KB) Movie 2×2 cross bar PhC switch (silicon pillars) in OFF state.
Fig. 9 (253KB) Movie 2×2 cross bar PhC switch (silicon pillars) in ON state.

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]

OCIS Codes
(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers
(130.3120) Integrated optics : Integrated optics devices
(230.0230) Optical devices : Optical devices
(230.2090) Optical devices : Electro-optical devices
(230.4110) Optical devices : Modulators

ToC Category:
Research Papers

History
Original Manuscript: August 15, 2002
Revised Manuscript: September 20, 2002
Published: October 7, 2002

Citation
Ahmed Sharkawy, Shouyuan Shi, Dennis Prather, and Richard Soref, "Electro-optical switching using coupled photonic crystal waveguides," Opt. Express 10, 1048-1059 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-20-1048


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References

  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," Opt. Lett. 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 BetweenWaveguides 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: Awaveguiding 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 photoniccrystal waveguides," J. Opt. Soc. Am. B 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). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-1-167">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-1-167</a>. [CrossRef] [PubMed]
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