## Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs

Optics Express, Vol. 13, Issue 7, pp. 2596-2604 (2005)

http://dx.doi.org/10.1364/OPEX.13.002596

Acrobat PDF (416 KB)

### Abstract

A compact in-plane channel drop filter design in 2D hexagonal lattice photonic crystal slabs is presented in this paper. The system consists of two photonic crystal waveguides and a single cavity with two degenerate modes. Both modes are able to confine light strongly in the vertical dimension and prove to couple equally into the waveguides. Three dimensional finite difference time domain simulations show that the quality factor is around *3,000*. At resonance, power transferred to the drop waveguide is *78%* and only *1.6%* remains in the bus waveguide. We also show that by carefully tuning the drop waveguide boundary, light remaining in the bus can be further reduced to below *0.4%* and thus the channel isolation is larger than *22dB*.

© 2005 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059 (1987) [CrossRef] [PubMed]

6. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B , **60**, 5751 (1999) [CrossRef]

7. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944 (2003) [CrossRef] [PubMed]

10. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express **11**, 579 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579 [CrossRef] [PubMed]

11. M. Tokushima and H. Yamada, “Light propagation in a photonic-crystal-slab line-defect waveguide,” IEEE J. of Quantum Electron. **38**, 753 (2002) [CrossRef]

12. A. Sugitatsu, T. Asano, and S. Noda, “Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs,” Appl. Phys. Lett. **84**, 5395 (2004) [CrossRef]

13. M. Qiu and B. Jaskorzynska, “A design of a channel drop filter in a two-dimensional triangular photonic crystal,” Appl. Phys. Lett. **83**, 1074(2003) [CrossRef]

15. S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature **407**, 608 (2000) [CrossRef] [PubMed]

18. B. K. Min, J. E. Kim, and H. Y. Park, “High-efficiency Surface-emitting channel drop filters in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. **86**, 11106 (2005) [CrossRef]

18. B. K. Min, J. E. Kim, and H. Y. Park, “High-efficiency Surface-emitting channel drop filters in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. **86**, 11106 (2005) [CrossRef]

*50%*of light emitted upward. The in-plane design involves two waveguides (bus and drop) and a cavity system. Channel to be selected comes from the bus waveguide, tunnels into the cavity and is eventually transferred to the drop waveguide. Theoretical analysis of the in-plane channel drop filter has matured over the years [21

21. S. Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. **80**, 960 (1998) [CrossRef]

23. Y. Xu, Y. Li, E. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E. **62**, 7389 (2000) [CrossRef]

*100%*transfer between two waveguides can be realized if the cavity system provides at least a pair of degenerate modes of opposite symmetry with infinitely high vertical Q factor. Yet very few in-plane designs in 2D PCS’s have been reported [19

19. B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunnelling processes in two-dimensional triangular lattice photonic crystal slabs,” Optics Commun. **237**, 59 (2004) [CrossRef]

*|*

**T**^{2}and power dropped |

*|*

**D**^{2}at resonance can be expressed by [22

22. C. Manolatou, M. J. Khan, S. Fan, Pierre R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of Modes Analysis of Resonant Channel Add-Drop Filters,” IEEE J. of Quantum Electron. **35**, 1322 (1999) [CrossRef]

25. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propagation , **14**, 302 (1966) [CrossRef]

26. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185 (1994) [CrossRef]

9. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3988 [CrossRef] [PubMed]

27. M. Qiu and Z. Zhang, “High Q Microcavities in 2D Photonic Crystal Slabs Studied by FDTD Techniques and Padé Approximation,” Proc. SPIE.5733, (2005, to be published) [CrossRef]

28. W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of Resonant Frequencies and Quality Factors of Cavities by FDTD Technique and Padé Approximation,” IEEE Microwave Wireless Components Lett. **11**, 223 (2001) [CrossRef]

## 2. Cavity design and modes tuning

*Q*

_{⊥}larger than

*10*

^{6}[8

8. H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. **83**, 4294 (2003) [CrossRef]

9. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3988 [CrossRef] [PubMed]

*Q*

_{⊥}is achieved by carefully modifying the surrounding boundaries so that in the momentum (

*) space, most of the components are pushed away from the light cone [7*

**k**7. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944 (2003) [CrossRef] [PubMed]

9. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express **12**, 3988 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3988 [CrossRef] [PubMed]

24. J. Vučković, M. lončar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. **38**, 850 (2002) [CrossRef]

*k*space distribution of the cavity mode. As a result, more

*k*components are introduced into the light cone and thus

*Q*

_{⊥}is reduced greatly.

*Q*

_{⊥}of the cavity, this will reduce the coupling between the waveguide and the cavity,

*Q*

_{◃}is also increased and the ratio

*Q*

_{⊥}

*/Q*

_{◃}may not improve at all.

10. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express **11**, 579 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579 [CrossRef] [PubMed]

*R*

_{0}, surrounded by 4 periods of air holes with decreasing radii, along the outward direction denoted by

*R*

_{1}

*, R*

_{2}

*, R*

_{3}, and

*R*

_{4}respectively. The radius of the air holes on the same hexagon stays the same and

*R*

_{1}to

*R*

_{4}follows a parabolic pattern as shown in equation (3). The radius of air holes on the outmost hexagon

*R*

_{4}is the same as the regular air hole radius

*R*=

*0.30a*, where

*a*is the lattice constant and set to

*420nm*in our case. The slab thickness

*t*is

*0.6a*and the refractive index

*n*of the slab is

*3.4*, corresponding to silicon at

*1.55µm*. If

*R*

_{0}and

*R*

_{1}are given, the cavity structure is determined.

10. K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express **11**, 579 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579 [CrossRef] [PubMed]

*R*

_{0}) followed by a relatively large decrease in hole radius (

*R*

_{1}) for the nearest neighbor holes. The hole radii are then parabolically decreased in moving radially outwards down to

*R*

_{4}

*=R*at the edge of the cavity, forming the second level of confinement. Though increasing the number of hexagons surrounding the central large hole can enhance the vertical light confinement, it also reduces waveguide-cavity coupling. We choose the number of surrounding hexagons to be 4, which is sufficient to keep good waveguide-cavity coupling while maintaining high vertical Q. The hexagonal symmetry of the cavity itself makes it possible to support two modes of opposite symmetries (though only the mode with

*H*

_{z}field even in both in-plane directions was reported in [10

**11**, 579 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579 [CrossRef] [PubMed]

*H*

_{z}field even in both

*x*and

*y*directions and the second mode has

*H*

_{z}field even along

*y*direction but odd along

*x*direction. Their symmetry properties decide a forward drop when the two modes are degenerate [22

22. C. Manolatou, M. J. Khan, S. Fan, Pierre R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of Modes Analysis of Resonant Channel Add-Drop Filters,” IEEE J. of Quantum Electron. **35**, 1322 (1999) [CrossRef]

*R*

_{0}parameter will have more impact on the even mode and so it is with

*R*

_{1}on the odd mode. In order to maintain the vertical light confinement of the cavity, we keep

*R*

_{0}>

*R*

_{1}and follow Eq. (3) strictly during the tuning process. We start with

*R*

_{0}=

*0*.365

*a*and increase

*R*

_{0}by step of

*0.005a*to

*0*.395

*a*. For each value of

*R*

_{0}, we choose three values for

*R*

_{1}, i.e.,

*R*

_{1}=

*R*

_{0}

*-0.005a, R*

_{1}

*=R*

_{0}

*-0.01a*and

*R*

_{1}

*=R*

_{0}

*-0.015a*. The results are shown in Fig. 2.

*R*

_{0}and

*R*

_{1}. The central wavelength of the even mode

*λ*

_{1}is labeled with diamond maker and

*λ*

_{2}of the odd mode is labeled with circle maker. The Q factor variations of the two modes are shown in Fig. 2(b).

*Q*

_{1}is the total Q factor of the even mode and

*Q*

_{2}is the total Q factor of the odd mode. Fig. 2(c) and 2(d) show the central wavelength and Q factor difference respectively. Examine Fig. 2(c) and 2(d) in detail and we find the two modes have the closest central wavelengths and Q factors when

*R*

_{0}

*=0.38a*and

*R*

_{0}

*- 0.005a>R*

_{1}

*>R*

_{0}

*- 0.1a*. We set

*R*

_{0}

*=0.38a*and tune

*R*

_{1}in smaller steps. The result shows when

*R*

_{1}is

*0.372a*, the two modes have closest central wavelengths as well as Q factors, with

*λ*

_{1}

*=1554.51nm, λ*

_{2}

*=1554.52nm, Q*

_{1}

*=3,060*

*and Q*

_{2}

*=3,020*. Further computation separates

*Q*

_{⊥}and

*Q*

_{◃}from

*Q*total by

*Q*

^{-1}

*Q*

_{⊥1}

*=40,500*and

*Q*

_{◃1}

*=3,100*. For the odd mode,

*Q*

_{⊥2}

*=35,000*and

*Q*

_{◃2}=

*3,050*. The central wavelength difference is smaller than

*0.01nm*and is negligible compared to the line width.

*n*of the silicon slab by ±

*1%*and also the slab thickness

*t*by ±

*0.05a*, the two modes stay degenerate with Δ

*λ*<

*0.01nm*and Δ

*Q*<

*100*, despite the absolute shift in central wavelength and Q factors. This makes our design insensitive to some of the fabrication uncertainties, such as slab thickness and material index.

*0.267~0.28 (c/a)*, corresponding to

*1500~1573 (nm)*in wavelength assuming lattice constant

*a=420 nm*. The waveguide loss is avoided in our system as the central wavelengths of the cavity modes lie within this range.

*78%*of light is transferred along the forward direction of the drop waveguide,

*1.75%*is transferred along the backward direction of the drop waveguide, and

*1.6%*of light is still left in the bus waveguide. The Q factor measured as full width half maximum (FWHM) of the transmission spectrum is around

*3,000*, relevant to the previous

*Q*

_{◃}calculations.

## 3. Waveguide tuning to improve the drop effect

*CI*), defined as

*CI=10log(P*

_{1}

*/P*

_{2}

*)*, where

*P*

_{1}and

*P*

_{2}are the power of the selected channel transferred to the drop waveguide and the power remaining in the bus respectively. The channel isolation should be as large as possible to avoid the cross talk. The

*CI*value of the system we designed above is only

*16.9 dB*. We try to improve it by further reducing the

*1.6%*of light that remains in the bus while keeping the power transferred high. We re-write Eq. (1) in more detail. In equation (4) below,

*1/τ*

_{e1}and

*1/τ’*

_{e1}are the decay rate of the even mode into bus and drop waveguide respectively,

*1/τ*

_{o1}is the decay rate of the even mode due to loss and similarly defined are

*1/τ*

_{e2}

*, 1/τ’*

_{e2}and

*1/τ*

_{o2}for the odd mode.

*1/τ*

_{e1}

*=1/τ’*

_{e1}and

*1/τ*

_{e2}

*=1/τ’*

_{e2}. In this case, |

*|*

**T**^{2}will always be larger than zero according to equation (4) and the selected channel cannot be

*100%*dropped. However, if we modify the waveguide boundary and let

*τ’*

_{e1, 2}slightly larger than

*τ*

_{e1, 2}so that the nominator of Eq. (4) becomes zero, the input signal power at resonant wavelength is completely removed from the bus.

*R*

_{w}of the air holes at the upper boundary of the drop waveguide. This modification turns out to affect both modes equally. The in-plane Q factors of both modes are improved and thus

*τ’*

_{e1, 2}increased to the same extent. The two modes stay degenerate though their central wavelengths shift slightly by the same amount. There are other ways to change the decay rates, such as enlarging the air hole radius of the bus waveguide boundaries and increasing the drop waveguide width. These changes, however, are too abrupt and difficult to control, so we have skipped them in this paper.

*R*

_{w}

*=0.27a*, only 0.39% of signal power remains in the bus waveguide and the channel isolation is enhanced to

*22.7dB*. It is worth noting that due to the Lorentian line shape of the transmission spectra [22

22. C. Manolatou, M. J. Khan, S. Fan, Pierre R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of Modes Analysis of Resonant Channel Add-Drop Filters,” IEEE J. of Quantum Electron. **35**, 1322 (1999) [CrossRef]

*3,000*and if we use

*100GHz*channel spacing centered at the resonant frequency, the interchannel cross talk level is only

*13dB*below the desired channel. A higher Q factor is desired in the future channel drop filter designs using 2D PCS’s.

*H*

_{z}) at the resonant frequency for the case

*R*

_{w}

*=0.27a*. A silicon wire is connected to each end of the photonic crystal waveguides. Most of the field is transferred along the forward direction of the drop waveguide. The field along the forward direction of the bus waveguide and along the backward direction of the drop waveguide is negligible. Also note that the field is much stronger in the cavity region due to the high Q factor (~

*3,000*). The filter is compact and only occupies an area of a few wavelengths square.

## 4. Conclusions

## Acknowledgments

## References and Links

1. | E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. |

4. | A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B |

5. | T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature |

6. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B , |

7. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

8. | H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. |

9. | Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express |

10. | K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,” Opt. Express |

11. | M. Tokushima and H. Yamada, “Light propagation in a photonic-crystal-slab line-defect waveguide,” IEEE J. of Quantum Electron. |

12. | A. Sugitatsu, T. Asano, and S. Noda, “Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs,” Appl. Phys. Lett. |

13. | M. Qiu and B. Jaskorzynska, “A design of a channel drop filter in a two-dimensional triangular photonic crystal,” Appl. Phys. Lett. |

14. | M. Qiu, “Ultra-compact optical filter in two-dimensional photonic crystal,” Electron. Lett. |

15. | S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature |

16. | B. S. Song, S. Noda, and T. Asano, “Photonic Devices Based on In-Plane Hetero Photonic Crystals,” Science |

17. | A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, “Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. |

18. | B. K. Min, J. E. Kim, and H. Y. Park, “High-efficiency Surface-emitting channel drop filters in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. |

19. | B. K. Min, J. E. Kim, and H. Y. Park, “Channel drop filters using resonant tunnelling processes in two-dimensional triangular lattice photonic crystal slabs,” Optics Commun. |

20. | K. H. Hwang and G. H. Song, “Design of a Two-Dimensional Photonic-Crystal Channel-Drop Filter Based on the Triangular-Lattice Holes on the Slab Structure,” Proceedings of 30 |

21. | S. Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. |

22. | C. Manolatou, M. J. Khan, S. Fan, Pierre R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of Modes Analysis of Resonant Channel Add-Drop Filters,” IEEE J. of Quantum Electron. |

23. | Y. Xu, Y. Li, E. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E. |

24. | J. Vučković, M. lončar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. of Quantum Electron. |

25. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propagation , |

26. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

27. | M. Qiu and Z. Zhang, “High Q Microcavities in 2D Photonic Crystal Slabs Studied by FDTD Techniques and Padé Approximation,” Proc. SPIE.5733, (2005, to be published) [CrossRef] |

28. | W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of Resonant Frequencies and Quality Factors of Cavities by FDTD Technique and Padé Approximation,” IEEE Microwave Wireless Components Lett. |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 11, 2005

Revised Manuscript: March 21, 2005

Published: April 4, 2005

**Citation**

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.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2596

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### References

- E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059 (1987) [CrossRef] [PubMed]
- S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486 (1987) [CrossRef] [PubMed]
- H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,�?? J. Appl. Phys. 75, 4753 (1994)
- A. Mekis, S. Fan, and J. D. Joannopoulos, �??Bound states in photonic crystal waveguides and waveguide bends,�?? Phys. Rev. B 58, 4809 (1998) [CrossRef]
- T. F. Krauss, R. M. De La Rue, and S. Brand, �??Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,�?? Nature 383, 699 (1996) [CrossRef]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B, 60, 5751 (1999) [CrossRef]
- Y. Akahane, T. Asano, B. S. Song, and S. Noda, �??High-Q photonic nanocavity in a two-dimensional photonic crystal,�?? Nature 425, 944 (2003) [CrossRef] [PubMed]
- H. Y. Ryu, M. Notomi, and Y. H. Lee, �??High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,�?? Appl. Phys. Lett. 83, 4294 (2003) [CrossRef]
- Z. Zhang and M. Qiu, �??Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,�?? Opt. Express 12, 3988 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3988">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3988</a> [CrossRef] [PubMed]
- K. Srinivasan and O. Painter, �??Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals,�?? Opt. Express 11, 579 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-6-579</a> [CrossRef] [PubMed]
- M. Tokushima and H. Yamada, �??Light propagation in a photonic-crystal-slab line-defect waveguide,�?? IEEE J. of Quantum Electron. 38, 753 (2002) [CrossRef]
- A. Sugitatsu, T. Asano and S. Noda, �??Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs,�?? Appl. Phys. Lett. 84, 5395 (2004) [CrossRef]
- M. Qiu and B. Jaskorzynska, �??A design of a channel drop filter in a two-dimensional triangular photonic crystal,�?? Appl. Phys. Lett. 83, 1074 (2003) [CrossRef]
- M. Qiu, �??Ultra-compact optical filter in two-dimensional photonic crystal,�?? Electron. Lett. 40, 539 (2004) [CrossRef]
- S. Noda, A. Chutinan and M. Imada, �??Trapping and emission of photons by a single defect in a photonic bandgap structure,�?? Nature 407, 608 (2000) [CrossRef] [PubMed]
- B. S. Song, S. Noda and T. Asano, �??Photonic Devices Based on In-Plane Hetero Photonic Crystals,�?? Science 300, 1537 (2003) [CrossRef] [PubMed]
- A. Chutinan, M. Mochizuki, M. Imada and S. Noda, �??Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,�?? Appl. Phys. Lett. 79, 2690 (2001) [CrossRef]
- B. K. Min, J. E. Kim and H. Y. Park, �??High-efficiency Surface-emitting channel drop filters in two-dimensional photonic crystal slabs,�?? Appl. Phys. Lett. 86, 11106 (2005) [CrossRef]
- B. K. Min, J. E. Kim and H. Y. Park, �??Channel drop filters using resonant tunnelling processes in two-dimensional triangular lattice photonic crystal slabs,�?? Optics Commun. 237, 59 (2004) [CrossRef]
- K. H. Hwang and G. H. Song, �??Design of a Two-Dimensional Photonic-Crystal Channel-Drop Filter Based on the Triangular-Lattice Holes on the Slab Structure,�?? Proceedings of 30th European Conference on Optical Communication, 5, 76 (Stockholm, Sweden, 2004)
- S. Fan, Pierre R. Villeneuve, and J. D. Joannopoulos, �??Channel Drop Tunneling through Localized States,�?? Phys. Rev. Lett. 80, 960 (1998) [CrossRef]
- C. Manolatou, M. J. Khan, S. Fan, Pierre R. Villeneuve, H. A. Haus and J. D. Joannopoulos, �??Coupling of Modes Analysis of Resonant Channel Add-Drop Filters,�?? IEEE J. of Quantum Electron. 35, 1322 (1999) [CrossRef]
- Y. Xu, Y. Li, E. K. Lee and A. Yariv, �??Scattering-theory analysis of waveguide-resonator coupling,�?? Phys. Rev. E. 62, 7389 (2000) [CrossRef]
- J. Vuèkoviæ, M. lonèar, H. Mabuchi and A. Scherer, �??Optimization of the Q factor in photonic crystal microcavities,�?? IEEE J. of Quantum Electron. 38, 850 (2002) [CrossRef]
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