## Optical filter based on contra-directional waveguide coupling in a 2D photonic crystal with square lattice of dielectric rods

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

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]

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]

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

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]

## 2. Theoretical analysis

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]

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

*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.

*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),

*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.

*δ*= 0 and

*γ*= |

*κ*|), the power transfer (drop) coefficient of Eq. (6) is simplified as

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

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]

*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]

*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.

*ω*≈0.3899(2

*πc/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 Δ

*ω*≈ 0.0016(2

*πc/a*). Thus the lattice constant

*a*= 600

*nm*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)≈120

*nm*. Calculated by Eq. (4), the coupling coefficient is about

*κ*≈ 0.00753

*a*

^{-1}, and the filtering bandwidth is about Δ

*λ*≈ 3.0

*nm*. Since

*κ*≪

*β*

_{+,-}, it is selfevident that the analysis based on CMT is valid.

*a*. A Gaussian pulse is excited at port1 with a center wavelength

*λ*

_{0}= 1550

*nm*, a full width at half maximum (FWHM) Δ

*t*= 150

*fs*, 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]

*a*

^{-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.

*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 (

*κ*≪

*β*

_{+- }, 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.

## 4. Conclusion

*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.

*a*to 0.23

*a*, and the filter will be more compact. Meanwhile, the filtering bandwidth will be double.

*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

## References and Links

1. | Turan Erdogan, “Optical add-drop multiplexer based on an asymmetric Bragg coupler,” Opt. Commun. |

2. | Jin Hong and WeiPing Huang, “Contra-directional Coupling in Grating-Assisted Guided-Wave Devices,” IEEE J. Lightwave Technol. |

3. | J. Joannopoulos, R. Meade, and J. Winn, |

4. | M. Tokushima and H. Yamada, “Photonic crystal line defect waveguide directional coupler,” Electron. Lett. |

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. |

6. | Min Qiu and Marcin Swillo, “Contra-directional coupling between two-dimensional photonic crystal waveguides,” Photonics and Nanostructures , |

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 |

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 |

9. | Steven G. Johnson, Pierre. R. Villeneuve, Shanhui Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B |

10. | Sergey Kuchinsky, Vladislav Y. Golyatin, Alexander Y. Kutikov, Thomas P. Pearsall, and Dusan Nedeljkovic, “Coupling Between Photonic Crystal Waveguides,” IEEE J. Quantum Electron. |

11. | Masanori Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Couplers,” IEEE J. Lightwave Technol. |

12. | H. A. Haus, |

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. |

14. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

15. | Min Qiu, “Effective method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. |

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 |

**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

- Turan Erdogan, �??Optical add�??drop multiplexer based on an asymmetric Bragg coupler,�?? Opt. Commun. 157, 249-264, (1998). [CrossRef]
- Jin Hong, and WeiPing Huang, �??Contra-directional Coupling in Grating-Assisted Guided-Wave Devices,�?? IEEE J. Lightwave Technol. 10, 873-881 (1992) [CrossRef]
- J. Joannopoulos, R, Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995)
- M. Tokushima, and H. Yamada, �??Photonic crystal line defect waveguide directional coupler,�?? Electron. Lett. 37, 1454-1455 (2001) [CrossRef]
- 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]
- Min Qiu, and Marcin Swillo, �??Contra-directional coupling between two-dimensional photonic crystal waveguides,�?? Photonics and Nanostructures, 1, 23-30, (2003). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- Masanori Koshiba, �??Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Couplers,�?? IEEE J. Lightwave Technol. 19, 1970-1975 (2001) [CrossRef]
- H. A. Haus, waves and fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, USA, 1985)
- 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]
- 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]
- Min Qiu, �??Effective method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,�?? Appl. Phys. Lett. 81, 1163-1165, (2002). [CrossRef]
- 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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