## Circuit-based method for synthesizing of coupled-resonators bandpass photonic crystal filters |

Optics Express, Vol. 19, Issue 4, pp. 3667-3676 (2011)

http://dx.doi.org/10.1364/OE.19.003667

Acrobat PDF (1265 KB)

### Abstract

A method for synthesizing bandpass photonic crystal filters for wavelength division multiplexing (WDM) systems is presented. The proposed method permits the calculation of the physical dimensions of the crystalline structures given the desired frequency response of the filter in terms of bandwidth, in-band ripple, minimum out-of-band attenuation, and central frequency. The method, explained in detail for Chebyshev frequency responses, is equivalent circuit based. The resulting devices are very compact, have a high out-of-band attenuation, and are suitable for high density photonic integrated circuits. The validity of the proposed method is confirmed through contrasting the simulation concluded from the finite-difference time-domain (FDTD) method by the design of a third-order Chebyshev filter having a center frequency of 1THz, a flat bandwidth of 4GHz, and ripples of 0.5 dB in the passband.

© 2011 OSA

## 1. Introduction

5. M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. **28**(24), 2506–2508 (2003). [CrossRef] [PubMed]

6. X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B **81**, 779–781 (2005). [CrossRef]

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**(21), 4294–4296 (2003). [CrossRef]

9. J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. **14**(11), 2575–2580 (1996). [CrossRef]

10. M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. **20**(5), 873–878 (2002). [CrossRef]

11. R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. **15**(3), 401–403 (2003). [CrossRef]

12. D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. **23**(5), 1923–1928 (2005). [CrossRef]

13. X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. **89**(2), 327–332 (2007). [CrossRef]

15. S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express **3**(1), 4–11 (1998). [CrossRef] [PubMed]

18. Y. Akahane, T. Asano, H. Takano, B.-S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express **13**(7), 2512–2530 (2005). [CrossRef] [PubMed]

12. D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. **23**(5), 1923–1928 (2005). [CrossRef]

19. A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. **20**(2), 296–303 (2002). [CrossRef]

20. M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. **35**(10), 1451–1460 (1999). [CrossRef]

19. A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. **20**(2), 296–303 (2002). [CrossRef]

20. M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. **35**(10), 1451–1460 (1999). [CrossRef]

## 2. Filters synthesis

11. R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. **15**(3), 401–403 (2003). [CrossRef]

12. D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. **23**(5), 1923–1928 (2005). [CrossRef]

*ε*is a ripple constant,

*Ω*is a frequency variable. For our discussion here, it is convenient to let

*Ω*represent a radian frequency variable of a lowpass prototype filter which has a cutoff frequency at

*Ω*to that in the frequency domain

*ω*in which a practical bandpass filter response is expressed. The frequency transformation will have an effect on all the reactive elements accordingly, with no effect on the resistive elements. The immittance inverters have the ability to shift impedance or admittance levels depending on the choice of

*K*or

*J*parameters. Making use of these properties enables us to convert a filter circuit to an equivalent form that would be more convenient for implementation with microwave or optical structures. where

*Q*factors (or equivalently, the decay rates) can be determined from the standard LC filter circuit design theory, by treating the resonators as lumped elements [22

22. H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. **28**(1), 205–212 (1992). [CrossRef]

*J*inverter depicted in Fig. 2(c).

*β*is the propagation constant of the waveguide, and

*L*is the length between two reference planes as shown in Fig. 3(a).

*ϕ*is a critical parameter in the design of the multi-resonators reflection filter. It can be defined aswhere

22. H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. **28**(1), 205–212 (1992). [CrossRef]

*n*×

*n*matrix with all entries zero, except for

*q*

_{11}=

*q*

_{e1}and

*n*×

*n*unit or identity matrix,

*n*×

*n*reciprocal matrix (i.e.,

## 3. Examples

*a*’ is the lattice constant), length from resonator to waveguide

- • Center frequency 1.0THz
- • Fractional bandwidth 0.4%
- • Minimum stop-band attenuation 20dB at 1.006THz
- • Pass-band ripple 0.5dB

*n*=3 Chebyshev low-pass prototype. Then, based on the values calculated by the Eq. (6-9), the external quality factors and the coupling coefficients can be obtained by the Eq. (15-17), which are shown as following:

*a*(here

*a*. For the resonators with a

*a*. At first, we set the three resonators in a straight line with 4

*a*spacing, and extract the coupling coefficients of them. But the results are largely different from the desired ones. After extensive calculations, it has been found that

*a*. At last, the calculated result is plotted as real line and compared to the result obtained by 2-D FDTD method with dotted line in Fig. 6(b). It indicates that the method presented in this paper is valid for synthesizing of coupled-resonators bandpass photonic crystal filters.

## 4. Conclusion

## References and links

1. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

2. | M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. |

3. | M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. |

4. | M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. |

5. | M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. |

6. | X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B |

7. | M. Belotti, J. F. Galisteo Lòpez, S. De Angelis, M. Galli, I. Maksymov, L. C. Andreani, D. Peyrade, and Y. Chen, “All-optical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities,” Opt. Express |

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. | J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. |

10. | M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. |

11. | R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. |

12. | D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. |

13. | X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. |

14. | H. A. Haus, |

15. | S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express |

16. | C. Chen, X. Li, H. Li, K. Xu, J. Wu, and J. Lin, “Bandpass filters based on phase-shifted photonic crystal waveguide gratings,” Opt. Express |

17. | K. Fasihi and S. Mohammadnejad, “Highly efficient channel-drop filter with a coupled cavity-based wavelength-selective reflection feedback,” Opt. Express |

18. | Y. Akahane, T. Asano, H. Takano, B.-S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express |

19. | A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. |

20. | M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. |

21. | J. S. Hong and M. J. Lancaster, |

22. | H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: November 30, 2010

Revised Manuscript: January 12, 2011

Manuscript Accepted: January 21, 2011

Published: February 10, 2011

**Citation**

Zuoxing Dai, Jiali Wang, and Yan Heng, "Circuit-based method for synthesizing of coupled-resonators bandpass photonic crystal filters," Opt. Express **19**, 3667-3676 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3667

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### References

- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of Ligh, (Princeton Univ. Press, Princeton, 2008).
- M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004). [CrossRef]
- M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. 19(12), 1970–1975 (2001). [CrossRef]
- M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999). [CrossRef]
- M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28(24), 2506–2508 (2003). [CrossRef] [PubMed]
- X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005). [CrossRef]
- M. Belotti, J. F. Galisteo Lòpez, S. De Angelis, M. Galli, I. Maksymov, L. C. Andreani, D. Peyrade, and Y. Chen, “All-optical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities,” Opt. Express 16(15), 11624–11636 (2008). [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(21), 4294–4296 (2003). [CrossRef]
- J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996). [CrossRef]
- M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002). [CrossRef]
- R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15(3), 401–403 (2003). [CrossRef]
- D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. 23(5), 1923–1928 (2005). [CrossRef]
- X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007). [CrossRef]
- H. A. Haus, Wave and Fields in Optoelectronics (Englewood Cliffs, NJ: Prentice-Hall, 1984).
- S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3(1), 4–11 (1998). [CrossRef] [PubMed]
- C. Chen, X. Li, H. Li, K. Xu, J. Wu, and J. Lin, “Bandpass filters based on phase-shifted photonic crystal waveguide gratings,” Opt. Express 15(18), 11278–11284 (2007). [CrossRef] [PubMed]
- K. Fasihi and S. Mohammadnejad, “Highly efficient channel-drop filter with a coupled cavity-based wavelength-selective reflection feedback,” Opt. Express 17(11), 8983–8997 (2009). [CrossRef] [PubMed]
- Y. Akahane, T. Asano, H. Takano, B.-S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express 13(7), 2512–2530 (2005). [CrossRef] [PubMed]
- A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20(2), 296–303 (2002). [CrossRef]
- M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999). [CrossRef]
- J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications (John wiley & sons, INC. 2001).
- H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–212 (1992). [CrossRef]

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