Abstract

We investigate the unidirectional transmission behavior of photonic crystal (PC) molecules consisting of defect pairs with Kerr nonlinearity and focus on how to enhance the transmission contrast and maximum transmission of the resulting optical diodes. Theoretical analyses in combination with the numerical simulations based on the finite-difference time-domain technique are employed to evaluate the designed optical diodes. It is found that by intentionally and properly misaligning the resonant frequencies of the constitutional PC atoms, the transmission contrast as well as the maximum transmission of the nonlinear PC molecules can be significantly improved. The figure of merit that characterizes the performance of optical diodes can be enhanced by a factor of 5 as compared with the optical diodes constructed by single asymmetrically confined PC atoms. In addition, the optimum performance of the optical diodes can be achieved only when the operating frequency is properly chosen.

© 2006 Optical Society of America

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  2. M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
    [CrossRef]
  3. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
    [CrossRef] [PubMed]
  4. S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Elimination of cross talk in waveguide intersections," Opt. Lett. 23, 1855-1857 (1998).
    [CrossRef]
  5. M. Bayindir, B. Temelkuran, and E. Ozbay, "Photonic-crystal-based beam splitters," Appl. Phys. Lett. 77, 3902-3904 (2000).
    [CrossRef]
  6. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
    [CrossRef]
  7. S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
    [CrossRef] [PubMed]
  8. S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses," J. Appl. Phys. 90, 4321-4327 (2001).
    [CrossRef]
  9. Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
    [CrossRef]
  10. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
    [CrossRef] [PubMed]
  11. P. R. Villeneuve, D. S. Abrams, S. Fan, and J. D. Joannopoulos, "Single-mode waveguide micro-cavity for fast optical switching," Opt. Lett. 21, 2017-2019 (1996).
    [CrossRef] [PubMed]
  12. P. Tran, "Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect," J. Opt. Soc. Am. B 14, 2589-2595 (1997).
    [CrossRef]
  13. S. Lan, S. Nishikawa, and O. Wada, "Leveraging deep photonic band gaps in photonic crystal impurity bands," Appl. Phys. Lett. 78, 2101-2103 (2001).
    [CrossRef]
  14. M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
    [CrossRef]
  15. M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "The photonic band edge optical diode," J. Appl. Phys. 76, 2023-2026 (1994).
    [CrossRef]
  16. K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, "All-optical diode in a periodically poled lithium niobate waveguide," Appl. Phys. Lett. 79, 314-316 (2001).
    [CrossRef]
  17. S. F. Mingaleev and Y. S. Kivshar, "Nonlinear transmission and light localization in photonic-crystal waveguides," J. Opt. Soc. Am. B 19, 2241-2249 (2002).
    [CrossRef]
  18. M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037602(1-4) (2005).
    [CrossRef]
  19. J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
    [CrossRef] [PubMed]
  20. X. S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
    [CrossRef]
  21. X. S. Lin, W. Q. Wu, H. Zhou, K. F. Zhou, and S. Lan, "Enhancement of unidirectional transmission through the coupling of nonlinear photonic crystal defects," Opt. Express 14, 2429-2439 (2006).
    [CrossRef] [PubMed]
  22. S. Lan and H. Ishikawa, "Coupling of defect pairs and generation of dynamical band gaps in the impurity bands of nonlinear photonic crystals for all-optical switching," J. Appl. Phys. 91, 2573-2577 (2002).
    [CrossRef]
  23. K. Yee, "Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). In this paper, a commercial software developed by Rsoft Design Group (http://www.rsoftdesign.com) is used for nonlinear FDTD simulation.
    [CrossRef]
  24. M. Qiu, "Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals," Appl. Phys. Lett. 81, 1163-1165 (2002).
    [CrossRef]
  25. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R)(1-4) (2002).
    [CrossRef]
  26. S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122(1-6) (2005).
    [CrossRef]
  27. X. S. Lin, X. W. Chen, and S. Lan, "Investigation and modification of coupling of photonic crystal defects," Chin. Phys. Lett. 22, 1698-1701 (2005).
    [CrossRef]

2006

2005

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122(1-6) (2005).
[CrossRef]

X. S. Lin, X. W. Chen, and S. Lan, "Investigation and modification of coupling of photonic crystal defects," Chin. Phys. Lett. 22, 1698-1701 (2005).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037602(1-4) (2005).
[CrossRef]

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

X. S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
[CrossRef]

2003

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

2002

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

S. Lan and H. Ishikawa, "Coupling of defect pairs and generation of dynamical band gaps in the impurity bands of nonlinear photonic crystals for all-optical switching," J. Appl. Phys. 91, 2573-2577 (2002).
[CrossRef]

M. Qiu, "Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals," Appl. Phys. Lett. 81, 1163-1165 (2002).
[CrossRef]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R)(1-4) (2002).
[CrossRef]

S. F. Mingaleev and Y. S. Kivshar, "Nonlinear transmission and light localization in photonic-crystal waveguides," J. Opt. Soc. Am. B 19, 2241-2249 (2002).
[CrossRef]

2001

S. Lan, S. Nishikawa, and O. Wada, "Leveraging deep photonic band gaps in photonic crystal impurity bands," Appl. Phys. Lett. 78, 2101-2103 (2001).
[CrossRef]

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, "All-optical diode in a periodically poled lithium niobate waveguide," Appl. Phys. Lett. 79, 314-316 (2001).
[CrossRef]

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses," J. Appl. Phys. 90, 4321-4327 (2001).
[CrossRef]

2000

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

M. Bayindir, B. Temelkuran, and E. Ozbay, "Photonic-crystal-based beam splitters," Appl. Phys. Lett. 77, 3902-3904 (2000).
[CrossRef]

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

1999

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

1998

1997

P. Tran, "Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect," J. Opt. Soc. Am. B 14, 2589-2595 (1997).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

1996

P. R. Villeneuve, D. S. Abrams, S. Fan, and J. D. Joannopoulos, "Single-mode waveguide micro-cavity for fast optical switching," Opt. Lett. 21, 2017-2019 (1996).
[CrossRef] [PubMed]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

1994

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "The photonic band edge optical diode," J. Appl. Phys. 76, 2023-2026 (1994).
[CrossRef]

1966

K. Yee, "Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). In this paper, a commercial software developed by Rsoft Design Group (http://www.rsoftdesign.com) is used for nonlinear FDTD simulation.
[CrossRef]

Abrams, D. S.

Asakawa, K.

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

Assanto, G.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, "All-optical diode in a periodically poled lithium niobate waveguide," Appl. Phys. Lett. 79, 314-316 (2001).
[CrossRef]

Bayindir, M.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Photonic-crystal-based beam splitters," Appl. Phys. Lett. 77, 3902-3904 (2000).
[CrossRef]

Bloemer, M. J.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "The photonic band edge optical diode," J. Appl. Phys. 76, 2023-2026 (1994).
[CrossRef]

Bowden, C. M.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "The photonic band edge optical diode," J. Appl. Phys. 76, 2023-2026 (1994).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Chen, J. D.

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122(1-6) (2005).
[CrossRef]

Chen, X. W.

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122(1-6) (2005).
[CrossRef]

X. S. Lin, X. W. Chen, and S. Lan, "Investigation and modification of coupling of photonic crystal defects," Chin. Phys. Lett. 22, 1698-1701 (2005).
[CrossRef]

Chutinan, A.

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Doll, T.

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

Dowling, J. P.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "The photonic band edge optical diode," J. Appl. Phys. 76, 2023-2026 (1994).
[CrossRef]

Fan, S.

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Elimination of cross talk in waveguide intersections," Opt. Lett. 23, 1855-1857 (1998).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

P. R. Villeneuve, D. S. Abrams, S. Fan, and J. D. Joannopoulos, "Single-mode waveguide micro-cavity for fast optical switching," Opt. Lett. 21, 2017-2019 (1996).
[CrossRef] [PubMed]

Feise, M. W.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037602(1-4) (2005).
[CrossRef]

Fejer, M. M.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, "All-optical diode in a periodically poled lithium niobate waveguide," Appl. Phys. Lett. 79, 314-316 (2001).
[CrossRef]

Ferrera, J.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Fink, Y.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R)(1-4) (2002).
[CrossRef]

Foresi, J. S.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Gallo, K.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, "All-optical diode in a periodically poled lithium niobate waveguide," Appl. Phys. Lett. 79, 314-316 (2001).
[CrossRef]

Haus, H. A.

Hwang, J.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Ibanescu, M.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R)(1-4) (2002).
[CrossRef]

Ikeda, N.

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

Imada, M.

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Ippen, E. P.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Ishikawa, H.

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

S. Lan and H. Ishikawa, "Coupling of defect pairs and generation of dynamical band gaps in the impurity bands of nonlinear photonic crystals for all-optical switching," J. Appl. Phys. 91, 2573-2577 (2002).
[CrossRef]

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses," J. Appl. Phys. 90, 4321-4327 (2001).
[CrossRef]

Ishikawa, K.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Joannopoulos, J. D.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R)(1-4) (2002).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Elimination of cross talk in waveguide intersections," Opt. Lett. 23, 1855-1857 (1998).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

P. R. Villeneuve, D. S. Abrams, S. Fan, and J. D. Joannopoulos, "Single-mode waveguide micro-cavity for fast optical switching," Opt. Lett. 21, 2017-2019 (1996).
[CrossRef] [PubMed]

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

Johnson, S. G.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R)(1-4) (2002).
[CrossRef]

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Elimination of cross talk in waveguide intersections," Opt. Lett. 23, 1855-1857 (1998).
[CrossRef]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Kimerling, L. C.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Kivshar, Y. S.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037602(1-4) (2005).
[CrossRef]

S. F. Mingaleev and Y. S. Kivshar, "Nonlinear transmission and light localization in photonic-crystal waveguides," J. Opt. Soc. Am. B 19, 2241-2249 (2002).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Lan, S.

X. S. Lin, W. Q. Wu, H. Zhou, K. F. Zhou, and S. Lan, "Enhancement of unidirectional transmission through the coupling of nonlinear photonic crystal defects," Opt. Express 14, 2429-2439 (2006).
[CrossRef] [PubMed]

X. S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
[CrossRef]

X. S. Lin, X. W. Chen, and S. Lan, "Investigation and modification of coupling of photonic crystal defects," Chin. Phys. Lett. 22, 1698-1701 (2005).
[CrossRef]

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122(1-6) (2005).
[CrossRef]

S. Lan and H. Ishikawa, "Coupling of defect pairs and generation of dynamical band gaps in the impurity bands of nonlinear photonic crystals for all-optical switching," J. Appl. Phys. 91, 2573-2577 (2002).
[CrossRef]

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

S. Lan, S. Nishikawa, and O. Wada, "Leveraging deep photonic band gaps in photonic crystal impurity bands," Appl. Phys. Lett. 78, 2101-2103 (2001).
[CrossRef]

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses," J. Appl. Phys. 90, 4321-4327 (2001).
[CrossRef]

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Lin, X. S.

X. S. Lin, W. Q. Wu, H. Zhou, K. F. Zhou, and S. Lan, "Enhancement of unidirectional transmission through the coupling of nonlinear photonic crystal defects," Opt. Express 14, 2429-2439 (2006).
[CrossRef] [PubMed]

X. S. Lin, X. W. Chen, and S. Lan, "Investigation and modification of coupling of photonic crystal defects," Chin. Phys. Lett. 22, 1698-1701 (2005).
[CrossRef]

X. S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
[CrossRef]

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122(1-6) (2005).
[CrossRef]

Loncar, M.

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

Manolatou, C.

Meade, R. D.

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

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Mingaleev, S. F.

Nedeljkovic, D.

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

Nishikawa, S.

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

S. Lan, S. Nishikawa, and O. Wada, "Leveraging deep photonic band gaps in photonic crystal impurity bands," Appl. Phys. Lett. 78, 2101-2103 (2001).
[CrossRef]

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses," J. Appl. Phys. 90, 4321-4327 (2001).
[CrossRef]

Nishimura, S.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Noda, S.

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

O'Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Ozbay, E.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Photonic-crystal-based beam splitters," Appl. Phys. Lett. 77, 3902-3904 (2000).
[CrossRef]

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Parameswaran, K. R.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, "All-optical diode in a periodically poled lithium niobate waveguide," Appl. Phys. Lett. 79, 314-316 (2001).
[CrossRef]

Park, B.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Pearsall, T. P.

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

Qiu, M.

M. Qiu, "Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals," Appl. Phys. Lett. 81, 1163-1165 (2002).
[CrossRef]

Scalora, M.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "The photonic band edge optical diode," J. Appl. Phys. 76, 2023-2026 (1994).
[CrossRef]

Scherer, A.

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Shadrivov, I. V.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037602(1-4) (2005).
[CrossRef]

Smith, H. I.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Soljacic, M.

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R)(1-4) (2002).
[CrossRef]

Song, M. H.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Steinmeyer, G.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Sugimoto, Y.

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

Takanishi, Y.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Takzoe, H.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Temelkuran, B.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Photonic-crystal-based beam splitters," Appl. Phys. Lett. 77, 3902-3904 (2000).
[CrossRef]

Thoen, E. R.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Toyooka, T.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Tran, P.

Villeneuve, P. R.

S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Elimination of cross talk in waveguide intersections," Opt. Lett. 23, 1855-1857 (1998).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

P. R. Villeneuve, D. S. Abrams, S. Fan, and J. D. Joannopoulos, "Single-mode waveguide micro-cavity for fast optical switching," Opt. Lett. 21, 2017-2019 (1996).
[CrossRef] [PubMed]

Vuckovic, J.

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

Wada, O.

S. Lan, S. Nishikawa, and O. Wada, "Leveraging deep photonic band gaps in photonic crystal impurity bands," Appl. Phys. Lett. 78, 2101-2103 (2001).
[CrossRef]

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses," J. Appl. Phys. 90, 4321-4327 (2001).
[CrossRef]

Winn, J. N.

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

Wu, J. W.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Wu, W. Q.

Yanik, M. F.

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Yee, K.

K. Yee, "Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). In this paper, a commercial software developed by Rsoft Design Group (http://www.rsoftdesign.com) is used for nonlinear FDTD simulation.
[CrossRef]

Zhou, H.

Zhou, K. F.

Appl. Phys. Lett.

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, "Waveguiding in planar photonic crystals," Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, "Photonic-crystal-based beam splitters," Appl. Phys. Lett. 77, 3902-3904 (2000).
[CrossRef]

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

S. Lan, S. Nishikawa, and O. Wada, "Leveraging deep photonic band gaps in photonic crystal impurity bands," Appl. Phys. Lett. 78, 2101-2103 (2001).
[CrossRef]

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, "All-optical diode in a periodically poled lithium niobate waveguide," Appl. Phys. Lett. 79, 314-316 (2001).
[CrossRef]

M. Qiu, "Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals," Appl. Phys. Lett. 81, 1163-1165 (2002).
[CrossRef]

Chin. Phys. Lett.

X. S. Lin, X. W. Chen, and S. Lan, "Investigation and modification of coupling of photonic crystal defects," Chin. Phys. Lett. 22, 1698-1701 (2005).
[CrossRef]

X. S. Lin and S. Lan, "Unidirectional transmission in asymmetrically confined photonic crystal defects with Kerr nonlinearity," Chin. Phys. Lett. 22, 2847-2850 (2005).
[CrossRef]

IEEE Trans. Antennas Propag.

K. Yee, "Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). In this paper, a commercial software developed by Rsoft Design Group (http://www.rsoftdesign.com) is used for nonlinear FDTD simulation.
[CrossRef]

J. Appl. Phys.

S. Lan and H. Ishikawa, "Coupling of defect pairs and generation of dynamical band gaps in the impurity bands of nonlinear photonic crystals for all-optical switching," J. Appl. Phys. 91, 2573-2577 (2002).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "The photonic band edge optical diode," J. Appl. Phys. 76, 2023-2026 (1994).
[CrossRef]

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses," J. Appl. Phys. 90, 4321-4327 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Mater.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takzoe, "Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions," Nat. Mater. 4, 383-387 (2005).
[CrossRef] [PubMed]

Nature

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122(1-6) (2005).
[CrossRef]

Phys. Rev. E

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R)(1-4) (2002).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, "Bistable diode action in left-handed periodic structures," Phys. Rev. E 71, 037602(1-4) (2005).
[CrossRef]

Phys. Rev. Lett.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Science

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Other

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

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Figures (7)

Fig. 1
Fig. 1

(a) Basic structure of the nonlinear PC molecules studied in this paper. (b) Linear transmission spectrum of the nonlinear PC molecule in which d A = 1.34 a and d B = 2.088 a (solid curve). The spectra of the two constitutional atoms (atom A, dashed curve, and atom B, dotted curve) are also provided for reference. The frequency of the input wave (or the pump frequency) ω p is indicated by an arrow.

Fig. 2
Fig. 2

Transmission behaviors (transmission as a function of the input power density) of the nonlinear PC molecule in which d A = 1.34 a and d B = 2.088 a at a pump frequency of 0.21796 ( 2 π c a ) for two launch directions. The transmission contrasts at the two thresholds ( C 1 and C 2 ) are indicated.

Fig. 3
Fig. 3

Linear transmission spectra of the nonlinear PC molecules (solid curves) in which (a) d A = 1.34 a and d B = 2.080 a , (b) d A = 1.34 a and d B = 2.075 a , (c) d A = 1.34 a and d B = 2.070 a . In each graph, the spectra of the two constitutional atoms (atom A, dashed curves; atom B, dotted curves) are also provided for reference.

Fig. 4
Fig. 4

Transmission behaviors (transmission as a function of the input power density) of the PC molecules in which (a) d A = 1.34 a and d B = 2.080 a , (b) d A = 1.34 a and d B = 2.075 a , (c) d A = 1.34 a and d B = 2.070 a at a pump frequency of 0.21796 ( 2 π c a ) for two launch directions. In each graph, the transmission contrasts at the two thresholds ( C 1 and C 2 ) are indicated.

Fig. 5
Fig. 5

(a) Linear transmission spectrum of the nonlinear PC molecule in which d A = 1.34 a and d B = 2.040 a (solid curve). The spectra of the two constitutional atoms (atom A, dashed curve; atom B, dotted curve) are also provided for reference. (b) Transmission behaviors (transmission as a function of the input power density) of the PC molecule at a pump frequency of 0.21796 ( 2 π c a ) for two launch directions. The transmission contrasts at the two thresholds ( C 1 and C 2 ) are indicated.

Fig. 6
Fig. 6

Transmission behaviors (transmission as a function of input power density) of the optimum nonlinear PC molecule ( d A = 1.34 a and d B = 2.070 a ) at different pump frequencies for two launch directions. (a) ω p = 0.21717 ( 2 π c a ) , (b) ω p = 0.21796 ( 2 π c a ) , (c) ω p = 0.21830 ( 2 π c a ) , (d) ω p = 0.21850 ( 2 π c a ) .

Fig. 7
Fig. 7

(a) Threshold transmission and transmission contrast of the optimum nonlinear PC molecule ( d A = 1.34 a and d B = 2.070 a ) as a function of the frequency of the input wave. (b) Dependence of the figure of merit of the resulting optical diode on the frequency of the input wave. The dashed line with arrows indicates the frequency range in which the figure of merit of the resulting optical diode is above 10.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

T ( ω ) = ( 4 τ 1 τ 2 τ 3 τ 4 sin 2 φ ) { [ 1 τ 1 τ 4 + 1 τ 2 τ 3 sin 2 φ i = 1 2 ( ω ω i ) ] 2 + ( ω ω 1 τ 4 + ω ω 2 τ 1 ) 2 } 1 ,
T ( ω ) = ( 4 τ 4 sin 2 φ ) { [ 1 τ 2 + 1 τ 2 sin 2 φ ( ω ω 1 ) ( ω ω 2 ) ] 2 + ( ω ω 1 τ + ω ω 2 τ ) 2 } 1 .
T peak = ( 1 + τ 2 ω 01 ω 02 2 4 ) 1 .

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