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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 15 — Jul. 25, 2005
  • pp: 5608–5613
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Optical filter based on contra-directional waveguide coupling in a 2D photonic crystal with square lattice of dielectric rods

Zhenfeng Xu, Jiangang Wang, Qingsheng He, Liangcai Cao, Ping Su, and Guofan Jin  »View Author Affiliations


Optics Express, Vol. 13, Issue 15, pp. 5608-5613 (2005)
http://dx.doi.org/10.1364/OPEX.13.005608


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Abstract

A coupler-type optical filter in 2D photonic crystal (PhC) with square lattice of dielectric rods in air is presented. The reduced-index and increased-index waveguides of filter have dispersion curves with opposite slopes to realize contra-directional coupling, and the point of anti-crossing is designed below the light line to avoid vertical radiation. The filter has a broad operable bandwidth due to the absence of mini stop bands. The transmission properties are analyzed using coupled modes theory (CMT) and simulated using the finite-difference time-domain (FDTD) method. The results show that a filtering bandwidth of 4 nm can be achieved in the range of 1500∼1600 nm, and over 83% drop coefficient is obtained.

© 2005 Optical Society of America

1. Introduction

In today’s dense wavelength division multiplexing (DWDM) system, optical filters are building blocks to multiplex and de-multiplex wavelengths. Various designs of optical filters have been proposed. Among these designs, waveguide-grating-based coupler-type filters are of great interest, due to their advantages of simple structure and nearly ideal spectral filtering characteristics [1

1. Turan Erdogan, “Optical add-drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. 157, 249–264, (1998). [CrossRef]

, 2

2. Jin Hong and WeiPing Huang, “Contra-directional Coupling in Grating-Assisted Guided-Wave Devices,” IEEE J. Lightwave Technol. 10, 873–881 (1992) [CrossRef]

]. However, the coupler-type filters usually require length in the range of several millimeters, which is far from the demands of compact integrated photonic devices.

Photonic crystals (PhCs) open new possibilities for ultra compact, highly wavelength selective optical devices, owing to their photonic band gaps (PBGs) [3

3. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995)

, 4

4. M. Tokushima and H. Yamada, “Photonic crystal line defect waveguide directional coupler,” Electron. Lett. 37, 1454–1455 (2001) [CrossRef]

]. Line defects in otherwise perfect PhCs can be utilized as waveguides for light. Due to the periodicity, PhCs waveguide couplers are natural grating-assisted waveguide couplers, so contra-directional coupling between two PhC nearby waveguides can be easily achieved [5

5. Min Qiu, Mikael Mulot, Marcin Swillo, Srinivasan Anand, Bozena Jaskorzynska, and Anders Karisson, “Photonic crystal optical filter based on contra-directional waveguide coupling,” Appl. Phys. Lett. 83, 5121–5123, (2003). [CrossRef]

, 6

6. Min Qiu and Marcin Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photonics and Nanostructures , 1, 23–30, (2003). [CrossRef]

]. Although the optical filter based on contra-directional PhC waveguide coupler is not as compact as other filtering devices in photonic crystals [7

7. Kyu H. Hwang and G. hugh Song, “Design of high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948–1957 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-1948 [CrossRef] [PubMed]

, 8

8. Ziyang Zhang and Min Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express 13, 2596–2604 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2596. [CrossRef] [PubMed]

], it is not so sensitive to loss and fabrication errors [6

6. Min Qiu and Marcin Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photonics and Nanostructures , 1, 23–30, (2003). [CrossRef]

]. The wavelength-selective operation of this type of filter has been demonstrated. However, that filter is based on two-dimensional (2D) PhC with triangular lattice of air holes in dielectric medium, where the operating wavelength is in the light cone. So the out-of-plane losses may become a big issue since the guided modes will couple vertically with the radiating modes in free space [9

9. Steven G. Johnson, Pierre. R. Villeneuve, Shanhui Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000). [CrossRef]

]. Besides, the total operable bandwidth (e.g ≈ 60 nm) of the couplertype filter is narrowed by the mini stop bands, which limits the number of multiplexed channels (e.g <20, which is probably inappropriate for a 32-wavelength DWDM system).

In this paper, we propose an optical filter based on waveguide contra-directional coupling in 2D PhC with square lattice of dielectric rods in air. The reduced-index and increased-index waveguides of the filter have band curves with opposite slopes to achieve contra-directional coupling, and the point of anti-crossing is designed below the light line to avoid vertical radiation, which is important for the low loss photonic integrated circuit. Furthermore, the filter has a broad total operable bandwidth due to the absence of mini stop bands, which ensures the filter appropriate for a 32-wavelength DWDM system. Some relevant radii are adjusted to reduce the coupling length and filtering bandwidth of filter. The transmission properties of the filter are analyzed using coupled modes theory (CMT) and simulated using the finite-difference time-domain (FDTD) method.

2. Theoretical analysis

The design layout of the filter is shown in Fig. 1. The waveguide W1 (W2) of filter is formed by reducing (increasing) the radius (equivalently index) of one row of rods respectively. The guided mode in the W1 PhC waveguide has a positive group velocity while that in the W2 PhC waveguide has a negative one. The periodicity of PhC ensures the contra-directional coupling between the two modes, like in the distributed feedback (DFB) structure.

Fig. 1. The structure of optical filter based on contra-directional waveguide coupling in 2D PhC with square lattice of dielectric rods. The two waveguides are formed by reducing and increasing the radius of one row of rods respectively.

In the PhC couplers based on two nearby PhC waveguides, if the coupling coefficient is much smaller than the propagation constants of the guided modes in the absence of coupling, the CMT is still valid to estimate some coupling characteristics [10

10. Sergey Kuchinsky, Vladislav Y. Golyatin, Alexander Y. Kutikov, Thomas P. Pearsall, and Dusan Nedeljkovic, “Coupling Between Photonic Crystal Waveguides,” IEEE J. Quantum Electron. 38, 1349–1352 (2002) [CrossRef]

, 11

11. Masanori Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Couplers,” IEEE J. Lightwave Technol. 19, 1970–1975 (2001) [CrossRef]

]. This condition is satisfied in our design, which is shown in section 3. According to the CMT, the coupling can be described as the following equations [2

2. Jin Hong and WeiPing Huang, “Contra-directional Coupling in Grating-Assisted Guided-Wave Devices,” IEEE J. Lightwave Technol. 10, 873–881 (1992) [CrossRef]

, 12

12. H. A. Haus, waves and fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, USA, 1985)

].

dα+dz=jβ+α++κα,
(1)
dαdz=jβα+κ*α+,
(2)

where α + and α - represent the amplitude of the forward and the backward waves respectively; β + and β - represent the corresponding propagation constants; κ represents the coupling coefficient and κ * is the conjugate value, and z is the propagation direction.

For contra-directional coupling, the general solutions with arbitrary constants for Eq. (1) and (2) are

α+=(A+eγz+Ae+γz)ejβ++β2z,
(3)
α=(B+eγz+Be+γz)ejβ++β2z,
(4)

where

γκ2(β+β2)2=κ2((dβ+dωdβdω)(ωω0))2κ2δ2,
(5)

and γ is real, ω 0 represents the crossover frequency without coupling. The power transfers from W1 waveguide to W2 waveguide, and if α - is negligible at z = 0 , the power transfer (drop) coefficient at z = -l can be obtained from Eq. (1) ∼ (5),

D(l)=[sinhγl(γ/κ*)coshγl+(jδ/κ*)sinhγl]2,
(6)

From Eq. (6), the filtering bandwidth of the filter can be expressed approximately as

Δλ=2λ02κπ(β+ωβω)=2λ02κπ(cvg+cvg),
(7)

where λ 0 represents the center wavelength, v g+ and v g- represent the group velocities of the relevant guided modes of each waveguide respectively. For the frequencies near ω 0, v g+ and v g- can be considered as constants.

At the center frequency (δ = 0 and γ = | κ|), the power transfer (drop) coefficient of Eq. (6) is simplified as

D(l)=tanh2(κl)=(eκleκleκl+eκl)2.
(8)

As shown in Eq. (7), the filtering bandwidth of filter is determined by the coupling coefficient and the group velocities of the guided modes. The coupling coefficient can be adjusted by changing the rows of dielectric rods between two waveguide or the radius of those rods. Note that a tradeoff should be made between the coupling coefficient and the coupling length according to Eq. (8). On the other hand, we can get small filtering bandwidth and coupling length at the same time by only reducing the group velocities of the guided modes. In the current design, we can tune the radius of defect rods in W2 (or W1) to set the point of anti-crossing near the edge of Brillouin zone [13

13. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely Large Group-Velocity Dispersion of Line-defect Waveguides in Photonic Crystal Slabs,” Phys. Rev. Lett. 87, 253902, (2001). [CrossRef] [PubMed]

].

3. Design and simulation

Based on the theoretical analysis in the preceding section, a contra-directional coupler-type optical filter is designed in a 2D PhC slab with square lattice of dielectric rods in air. This 2D PhC slab has a photonic band gap (PBG) only for the TM-like modes, so only the TM-like modes are concerned in this work. For the calculations of dispersion relation, a block-iterative frequency-domain method [14

14. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

] is carried out; and for the simulations, we use the FDTD code with perfect matched layer (PML) boundary condition. In order to simplify the calculation, the effective index method [15

15. Min Qiu, “Effective method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165, (2002). [CrossRef]

] is also used because of the relatively small frequency range.

In our design shown in Fig. 1, the radius of rods in PhC bulk is 0.20 a, where a is the lattice constant, and the height of rods is set as 2.0 a in order to get a large gap size [9

9. Steven G. Johnson, Pierre. R. Villeneuve, Shanhui Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000). [CrossRef]

]. For simplicity, we assume that the substrate and cladding of PhC are both air, then the corresponding effective dielectric constant of PhC slab is about 11.09. The waveguide W1 is form by reducing the radius of a row of dielectric rods to 0.10 a. The dispersion curve projected onto ⌜ -X direction is plotted in Fig. 2(a), which shows that only one guided mode with even symmetry occurs in the PBG. The waveguide W2 is formed by increasing the radius of a row of dielectric rods to 0.25 a. The projected dispersion curve is plotted in Fig. 2(b), and only one guided mode with even symmetry occurs in the PBG too. The effective indices are higher in the two waveguides than in the region above and below the slab, so the modes can be guided vertically by index confinement. The propagation loss along the waveguide direction mainly originates from extrinsic scattering due to structural disorder. The two guided modes have opposite group velocities and a cross point, which satisfy the criteria to achieve contra-directional coupling. If we just change the radius of defect rods, the dispersion curves in Fig. 2 will move up and down and cross at different points, then we can easily design the center frequency of filter. When the cross point closes to the edge of Brillouin zone, the low group velocity will decrease the bandwidth of filter drastically according to Eq. (4). The filter is composed of W1 and W2 PhC waveguides, and there is one row of dielectric rods between the two waveguides.

Fig. 2. The band structures of (a) W1 waveguide and (b) W2 waveguide. The PhC is a square lattice of dielectric rods in air. The radius of rods in bulk is 0.20 a, and the waveguide W1 (W2) is form by reducing (increasing) the radius of a row of dielectric rods to 0.10 a (0.25 a).
Fig. 3. (a) The band structure of the optical filter, or the band structure for the coupled W1 and W2 waveguides. (b) The band structure near the anti-crossing point (encircled region of part a).

The band structure of this coupler-type filter is shown in Fig. 3. The anti-crossing occurs below the light line and the normalized center frequency is ω≈0.3899(2πc/a) (the crossover frequency of the two dashed lines in Fig. 2(b)), where c is the velocity of light in free space. The frequency gap for the filter is Δω ≈ 0.0016(2πc/a). Thus the lattice constant a = 600nm is used to ensure the operating wavelength at around 1550 nm. The radius of the rods is then 120 nm. The total operable bandwidth of filter (or W1 PhC waveguide) is about 600×(1/0.3700-1/0.4004)≈120nm. Calculated by Eq. (4), the coupling coefficient is about κ ≈ 0.00753a -1, and the filtering bandwidth is about Δλ ≈ 3.0nm. Since κβ +,-, it is selfevident that the analysis based on CMT is valid.

Then the transmission characteristic of the filter is simulated. The length of the filter is 200 a. A Gaussian pulse is excited at port1 with a center wavelength λ 0 = 1550nm, a full width at half maximum (FWHM) Δt = 150fs, and polarization parallel to the rods. In order to avoid Fabry-Perot reflections, a gradual taper transition should be added to couple light efficiently from conventional waveguide into PhC waveguide [16

16. Steven G. Johnson, Peter Bienstman, M. A. Skorobogatiy, Mihai Ibanescu, Elefterios Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608–5758 (2002). [CrossRef]

].

Fig. 4. Normalized transmission spectra at different output ports for the 200 a long coupler-type filter. The port numbers correspond to those in Fig. 1.

The transmission spectra at different output ports are shown in Fig. 4. Nearly 83% drop coefficient (the transfer coefficient in port 3) is obtained in the range of 1500∼1600 nm, the center wavelength is around 1553 nm, and the filtering bandwidth is around 4 nm. All the parameters are identical with the theoretical analysis.

Fig. 5. (a) The coupling coefficient as a function of the radius of the rods between W1 and W2 PhC waveguide. The other parameters of filter maintain unchanged. (b) Normalized transmission spectra at different output ports for the 100 a long coupler-type filter, and the radius of the rods between W1 and W2 PhC waveguides is tuned to 0.23 a.

If the radius of the rods between W1 and W2 PhC waveguides is tuned slightly, the coupling coefficient will increase with the increasing of radius. It is shown in Fig. 5(a) that the coupling coefficient reaches 0.0167a -1 when the radius is 0.23 a. This implies that the coupling length will be less than half of the one mentioned above. Meanwhile, the bandwidth of filter will be double. Figure. 5(b) shows the simulation result. The length of the filter is reduced to 100 a, and nearly 80% drop coefficient is achieved around the center wavelength of 1548 nm with the filtering bandwidth of about 10 nm. When the radius is tuned > 0.24a, the coupling coefficient increases rapidly, and the condition of weak coupling is no longer satisfied, so it should not be analyzed with the CMT.

We can also tune the radius of defect rods in W2 (or W1) PhC waveguide to set the point of anti-crossing near the edge of Brillouin zone, then the low group velocities will decrease the bandwidth of filter and conserve the coupling length. When the radius of W2 decreases from 0.25 a to 0.24 a, the dispersion curve of guided mode moves up and the point of anti-crossing occurs close to the band edge where the group velocities of the two guided modes are around c/15, which is shown in Fig. 6 (κβ +- , so the CMT is still valid). Then the filtering bandwidth will reduce to 0.8 nm, which can be possibly used in today’s 128-wavelength DWDM system. But the filtering bandwidth is very sensitive to parameter error due to the flat slope of ω-κ curve, and this subject is now under consideration.

Fig. 6. (a) The band structure when the radius of W2 PhC waveguide decreases from 0.25 a to 0.24 a. The point of anti-crossing occurs close to the band edge. (b) The band structure near the anti-crossing point (encircled region of part a).

4. Conclusion

In summary, we have proposed an optical filter based on waveguide contra-directional coupling in 2D PhC with square lattice of dielectric rods in air. The reduced-index and increased-index PhC waveguides of the filter have band curves with opposite slopes, which ensures the contra-directional coupling between the two waveguides, and the point of anticrossing is designed below the light line to avoid the vertical radiation. These waveguides have broad total operable bandwidth because there is no mini stop band. For a 200 a long coupler-type filter, nearly 83% drop coefficient is obtained in the range of 1500∼1600 nm with the center wavelength of 1553 nm, and the filtering bandwidth is around 4 nm. All the parameters agree well with the theoretical analysis.

Moreover, we can tune the radius of the rods between W1 and W2 PhC waveguides slightly in order to change the coupling coefficient (length). The coupling length will be reduced to half when the radius is increased from 0.20 a to 0.23 a, and the filter will be more compact. Meanwhile, the filtering bandwidth will be double.

Additionally, we can also adjust the radius of defect rods in W2 (or W1) PhC waveguide to set the point of anti-crossing near the edge of Brillouin zone, then the low group velocities will decrease the filtering bandwidth of filter and keep the coupling length conserved. When the radius decreases from 0.25 a to 0.24 a, the filtering bandwidth of filter will reduce to 0.8nm, which has the potential of using in today’s 128-wavelength DWDM system.

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 60277011) and National Research Fund for Fundamental Key Projects NO.973 (G19990330).

References and Links

1.

Turan Erdogan, “Optical add-drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. 157, 249–264, (1998). [CrossRef]

2.

Jin Hong and WeiPing Huang, “Contra-directional Coupling in Grating-Assisted Guided-Wave Devices,” IEEE J. Lightwave Technol. 10, 873–881 (1992) [CrossRef]

3.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995)

4.

M. Tokushima and H. Yamada, “Photonic crystal line defect waveguide directional coupler,” Electron. Lett. 37, 1454–1455 (2001) [CrossRef]

5.

Min Qiu, Mikael Mulot, Marcin Swillo, Srinivasan Anand, Bozena Jaskorzynska, and Anders Karisson, “Photonic crystal optical filter based on contra-directional waveguide coupling,” Appl. Phys. Lett. 83, 5121–5123, (2003). [CrossRef]

6.

Min Qiu and Marcin Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photonics and Nanostructures , 1, 23–30, (2003). [CrossRef]

7.

Kyu H. Hwang and G. hugh Song, “Design of high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948–1957 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-1948 [CrossRef] [PubMed]

8.

Ziyang Zhang and Min Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,” Opt. Express 13, 2596–2604 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2596. [CrossRef] [PubMed]

9.

Steven G. Johnson, Pierre. R. Villeneuve, Shanhui Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000). [CrossRef]

10.

Sergey Kuchinsky, Vladislav Y. Golyatin, Alexander Y. Kutikov, Thomas P. Pearsall, and Dusan Nedeljkovic, “Coupling Between Photonic Crystal Waveguides,” IEEE J. Quantum Electron. 38, 1349–1352 (2002) [CrossRef]

11.

Masanori Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Couplers,” IEEE J. Lightwave Technol. 19, 1970–1975 (2001) [CrossRef]

12.

H. A. Haus, waves and fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, USA, 1985)

13.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely Large Group-Velocity Dispersion of Line-defect Waveguides in Photonic Crystal Slabs,” Phys. Rev. Lett. 87, 253902, (2001). [CrossRef] [PubMed]

14.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

15.

Min Qiu, “Effective method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165, (2002). [CrossRef]

16.

Steven G. Johnson, Peter Bienstman, M. A. Skorobogatiy, Mihai Ibanescu, Elefterios Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608–5758 (2002). [CrossRef]

OCIS Codes
(230.3990) Optical devices : Micro-optical devices
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Research Papers

History
Original Manuscript: June 6, 2005
Revised Manuscript: July 1, 2005
Published: July 25, 2005

Citation
Zhenfeng Xu, Jiangang Wang, Qingsheng He, Liangcai Cao, Ping Su, and Guofan Jin, "Optical filter based on contra-directional waveguide coupling in a 2D photonic crystal with square lattice of dielectric rods," Opt. Express 13, 5608-5613 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-15-5608


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References

  1. Turan Erdogan, �??Optical add�??drop multiplexer based on an asymmetric Bragg coupler,�?? Opt. Commun. 157, 249-264, (1998). [CrossRef]
  2. Jin Hong, and WeiPing Huang, �??Contra-directional Coupling in Grating-Assisted Guided-Wave Devices,�?? IEEE J. Lightwave Technol. 10, 873-881 (1992) [CrossRef]
  3. J. Joannopoulos, R, Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995)
  4. M. Tokushima, and H. Yamada, �??Photonic crystal line defect waveguide directional coupler,�?? Electron. Lett. 37, 1454-1455 (2001) [CrossRef]
  5. Min Qiu, Mikael Mulot, Marcin Swillo, Srinivasan Anand, Bozena Jaskorzynska, and Anders Karisson, �??Photonic crystal optical filter based on contra-directional waveguide coupling,�?? Appl. Phys. Lett. 83, 5121-5123, (2003). [CrossRef]
  6. Min Qiu, and Marcin Swillo, �??Contra-directional coupling between two-dimensional photonic crystal waveguides,�?? Photonics and Nanostructures, 1, 23-30, (2003). [CrossRef]
  7. Kyu H. Hwang and G. hugh Song, �??Design of high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,�?? Opt. Express 13, 1948-1957 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-1948">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-6-1948</a> [CrossRef] [PubMed]
  8. Ziyang Zhang and Min Qiu, �??Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs,�?? Opt. Express 13, 2596-2604 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2596.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2596</a> [CrossRef] [PubMed]
  9. Steven G. Johnson, Pierre. R. Villeneuve, Shanhui Fan, and J. D. Joannopoulos, �??Linear waveguides in photonic-crystal slabs,�?? Phys. Rev. B 62, 8212-8222 (2000). [CrossRef]
  10. Sergey Kuchinsky, Vladislav Y. Golyatin, Alexander Y. Kutikov, Thomas P. Pearsall, and Dusan Nedeljkovic, �??Coupling Between Photonic Crystal Waveguides,�?? IEEE J. Quantum Electron. 38, 1349-1352 (2002) [CrossRef]
  11. Masanori Koshiba, �??Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Couplers,�?? IEEE J. Lightwave Technol. 19, 1970-1975 (2001) [CrossRef]
  12. H. A. Haus, waves and fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, USA, 1985)
  13. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, �??Extremely Large Group-Velocity Dispersion of Line-defect Waveguides in Photonic Crystal Slabs,�?? Phys. Rev. Lett. 87, 253902, (2001). [CrossRef] [PubMed]
  14. S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis,�?? Opt. Express 8, 173-190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef] [PubMed]
  15. Min Qiu, �??Effective method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,�?? Appl. Phys. Lett. 81, 1163-1165, (2002). [CrossRef]
  16. Steven G. Johnson, Peter Bienstman, M. A. Skorobogatiy, Mihai Ibanescu, Elefterios Lidorikis, and J. D. Joannopoulos, �??Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,�?? Phys. Rev. E 66, 066608-5758 (2002). [CrossRef]

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