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

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References

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    [Crossref]
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    [Crossref]
  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. 69(1), 111–114 (1999).
    [Crossref]
  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]
  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 16(15), 11624–11636 (2008).
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    [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]
  14. H. A. Haus, Wave and Fields in Optoelectronics (Englewood Cliffs, NJ: Prentice-Hall, 1984).
  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]
  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 15(18), 11278–11284 (2007).
    [Crossref] [PubMed]
  17. 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]
  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]
  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]
  21. J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications (John wiley & sons, INC. 2001).
  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]

2009 (1)

2008 (1)

2007 (2)

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]

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]

2005 (3)

2004 (1)

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]

2003 (3)

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]

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]

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

2002 (2)

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]

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

2001 (1)

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. 19(12), 1970–1975 (2001).
[Crossref]

1999 (2)

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. 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]

1998 (1)

1996 (1)

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]

1992 (1)

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]

Akahane, Y.

Altug, H.

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]

Andreani, L. C.

Asano, T.

Belotti, M.

Chen, C.

Chen, J. C.

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]

Chen, Y.

CHENG, B.

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]

Chutinan, A.

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]

Costa, R.

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

De Angelis, S.

Dodabalapur, A.

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]

Fan, S.

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. 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]

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]

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

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]

Fasihi, K.

Galisteo Lòpez, J. F.

Galli, M.

GONG, Q.

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]

Haus, H.

Haus, H. A.

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. 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]

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]

HU, X.

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]

Imada, M.

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]

Joannopoulos, J.

Joannopoulos, J. D.

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]

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]

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]

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]

Khan, M. J.

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]

Kim, S.

Koshiba, M.

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. 19(12), 1970–1975 (2001).
[Crossref]

Lai, Y.

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]

Lee, Y. H.

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]

Li, H.

Li, X.

Li, X. C.

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]

Lim, H.

Lin, J.

Lin, J. T.

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]

Liu, A. Q.

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]

LIU, Y.

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]

Maksymov, I.

Manolatou, C.

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]

Martinelli, M.

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

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

Meier, M.

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]

Mekis, M.

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]

Melloni, A.

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

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

Mochizuki, M.

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]

Mohammadnejad, S.

Noda, S.

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]

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]

Notomi, M.

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]

Park, D.

Park, I.

Peyrade, D.

Ryu, H. Y.

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]

Slusher, R. E.

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]

Soljacic, M.

Song, B.-S.

Takana, Y.

Takano, H.

Tanaka, T.

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]

Villeneuve, P.

Villeneuve, P. R.

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. 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]

Vuckovic, J.

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]

Wu, J.

Xu, J.

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]

Xu, K.

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]

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]

Yanik, M. F.

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. 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]

ZHANG, D.

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]

Appl. Phys. B (1)

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]

Appl. Phys. Lett. (1)

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]

Appl. Phys., A Mater. Sci. Process. (2)

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]

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]

IEEE J. Lightw. Technol. (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. 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]

IEEE J. Quantum Electron. (2)

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]

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]

IEEE Photon. Technol. Lett. (1)

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

J. Lightwave Technol. (4)

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]

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]

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

Opt. Express (5)

Opt. Lett. (1)

Other (3)

H. A. Haus, Wave and Fields in Optoelectronics (Englewood Cliffs, NJ: Prentice-Hall, 1984).

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).

J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications (John wiley & sons, INC. 2001).

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Figures (6)

Fig. 1
Fig. 1

Basic structure of filter for developing the theoretical method

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Fig. 2
Fig. 2

(a) Chebyshev low-pass response. (b) Low-pass prototype filters for Chebyshev filters. (c) Band-pass filters using frequency transformations and immittance inverters

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Fig. 3
Fig. 3

(a) Structure of the second-order coupled-resonator filter in a PC, where 1-D defect resonators are embedded in a 2-D PC waveguide. (b) Equivalent circuit of the structure in Fig. 3.(a), where the coupling between the resonators is treated as if it occurs through a waveguide

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Fig. 4
Fig. 4

Two states of two coupled resonators: (a) even mode. (b)odd mode

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Fig. 5
Fig. 5

(a) Schematic representation of the structure of the coupled-resonators. (b) Transmission characteristic of the filter obtained by coupled-resonators method and FDTD method

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Fig. 6
Fig. 6

(a) Schematic diagram of the proposed filter, which is composed of 3 resonators. (b) Transmission spectra of the desired filter by the synthesized method (real line) and the simulation result by the FDTD method (dotted line).

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Tables (1)

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Table 1 Coupling coefficients of two resonators a

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Equations (22)

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| S 21 ( j Ω ) | 2 = 1 1 + ε 2 F n 2 ( Ω )
L A ( Ω ) = 10 log 1 | S 21 ( j Ω ) | 2 d B
F n ( Ω ) = { cos ( n ⋅ cos − 1 Ω ) | Ω | ≤ 1 cos h ( n ⋅ cos h − 1 Ω ) | Ω | ≥ 1
ε = 10 L A r 10 − 1
n ≥ cos h − 1 10 0.1 L A s − 1 10 0.1 L A r − 1 cos h − 1 Ω s
g 0 = 1.0
g 1 = 2 γ sin ( π 2 n )
g i = 1 g i − 1 4 sin [ ( 2 i − 1 ) 2 n π ] ⋅ sin [ ( 2 i − 3 ) 2 n π ] γ 2 + sin 2 [ ( i − 1 ) n π ] i = 2 , 3 , ⋯ n
g n + 1 = { 1.0 for n odd cot h 2 ( β 4 ) for n even
J 0 , 1 = Y 0 ⋅ F B W ⋅ ω 0 ⋅ C p 1 Ω c ⋅ g 0 ⋅ g 1
J i , i + 1 = F B W ⋅ ω 0 Ω c ⋅ C p i ⋅ C p ( i + 1 ) g i ⋅ g i + 1 i = 1 to n − 1
J n , n + 1 = F B W ⋅ ω 0 ⋅ C p n ⋅ Y n + 1 Ω c ⋅ g n ⋅ g n + 1
{ C p i = ( Ω c F B W ⋅ ω 0 ) ⋅ g i γ 0 L p i = 1 / ( ω 0 2 ⋅ C p i ) i = 1 to n
ϕ = n e f f ω L c
Q e 1 = g 0 ⋅ g 1 ⋅ Ω c F B W
Q e n = g n ⋅ g n + 1 ⋅ Ω c F B W
k i , i + 1 = F B W Ω c ⋅ g i ⋅ g i + 1 | i = 1 ~ n − 1
Q e = 2 ω 0 Δ ω 3 d B
k i j = ± ( ω 0 j ω 0 i + ω 0 i ω 0 j ) ( ω j 2 − ω i 2 ω j 2 + ω i 2 ) 2 − ( ω 0 j 2 − ω 0 i 2 ω 0 j 2 + ω 0 i 2 ) 2
S 21 = | S + 2 S + 1 | = 2 1 q e 1 ⋅ q e n [ A ] n 1 − 1
[ A ] = [ q ] + p [ U ] − j [ k ]
q e i = Q e i ⋅ ( Δ ω / ω ) for i = 1 , n

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