Abstract

We investigate the modes of double heterostructure cavities where the underlying photonic crystal waveguide has been dispersion engineered to have two band-edges inside the Brillouin zone. By deriving and using a perturbative method, we show that these structures possess two modes. For unapodized cavities, the relative detuning of the two modes can be controlled by changing the cavity length, and for particular lengths, a resonant-like effect makes the modes degenerate. For apodized cavities no such resonances exist and the modes are always non-degenerate.

© 2010 OSA

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  1. K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
    [Crossref]
  2. J. Vuĉković and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” Appl. Phys. Lett. 82, 2374–2376 (2003).
    [Crossref]
  3. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
    [Crossref] [PubMed]
  4. 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]
  5. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
    [Crossref]
  6. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
    [Crossref]
  7. Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with Q-factor of ∼109,” J. Lightwave Technol. 26, 1532–1539 (2008).
    [Crossref]
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    [Crossref]
  11. D. Englund, I. Fushman, and J. Vuĉković, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005).
    [Crossref] [PubMed]
  12. M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005).
    [Crossref] [PubMed]
  13. A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76, 053839 (2007).
    [Crossref]
  14. A. A. Chabanov, “Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media,” Phys. Rev. A 77, 033811 (2008).
    [Crossref]
  15. K. Y. Jung and F. L. Teixeira, “Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis,” Phys. Rev. A 78, 043826 (2008).
    [Crossref]
  16. S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010).
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  20. P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633.
  21. S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
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2010 (2)

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010).
[Crossref]

2009 (3)

2008 (5)

2007 (2)

A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76, 053839 (2007).
[Crossref]

S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. 32, 542–544 (2007).
[Crossref] [PubMed]

2006 (2)

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[Crossref]

T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

2005 (3)

2003 (1)

J. Vuĉković and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” Appl. Phys. Lett. 82, 2374–2376 (2003).
[Crossref]

1999 (1)

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]

1955 (1)

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
[Crossref]

Akahane, Y.

T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Asano, T.

Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with Q-factor of ∼109,” J. Lightwave Technol. 26, 1532–1539 (2008).
[Crossref]

T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Birks, T. A.

P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633.

Bordas, F.

Botten, L. C.

Bozio, R.

Bulla, D. A. P.

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

Chabanov, A. A.

A. A. Chabanov, “Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media,” Phys. Rev. A 77, 033811 (2008).
[Crossref]

Chan, J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

Chigrin, D. N.

Choi, D. Y.

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]

de Sterke, C. M.

Dossou, K. B.

Eggleton, B. J.

Eichenfield, M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

Englund, D.

Figotin, A.

A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76, 053839 (2007).
[Crossref]

Fink, Y.

Fushman, I.

Gai, X.

Gardin, S.

Grillet, C.

Ha, S.

S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010).
[Crossref]

Ha, S. W.

Ibanescu, M.

Joannopoulos, J. D.

M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. 30, 552–554 (2005).
[Crossref] [PubMed]

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

Johnson, S. G.

Jung, K. Y.

K. Y. Jung and F. L. Teixeira, “Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis,” Phys. Rev. A 78, 043826 (2008).
[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]

Kivshar, Yu. S.

S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010).
[Crossref]

S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
[Crossref] [PubMed]

Kohn, W.

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
[Crossref]

Kuramochi, E.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[Crossref]

Lavrinenko, A. V.

S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010).
[Crossref]

S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
[Crossref] [PubMed]

Lee, M. W.

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]

Letartre, X.

Lloyd Lucas, F. D.

P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633.

Luther-Davies, B.

Luttinger, J. M.

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
[Crossref]

Madden, S.

Magi, E. C.

Mahmoodian, S.

Matsuo, S.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

McPhedran, R. C.

Meade, R. D.

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

Mitsugi, S.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[Crossref]

Moss, D. J.

Noda, S.

Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with Q-factor of ∼109,” J. Lightwave Technol. 26, 1532–1539 (2008).
[Crossref]

T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Notomi, M.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[Crossref]

Nozaki, K.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

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]

Painter, O.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

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]

Poulton, C. G.

Rahmani, A.

Roundy, D.

Russell, P. St. J.

P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633.

Sato, T.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

Scherer, 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]

Seassal, C.

Shinya, A.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[Crossref]

Song, B. S.

T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Steel, M. J.

Sukhorukov, A. A.

S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010).
[Crossref]

S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
[Crossref] [PubMed]

Tanabe, T.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[Crossref]

Tanaka, Y.

Taniyama, H.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

Teixeira, F. L.

K. Y. Jung and F. L. Teixeira, “Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis,” Phys. Rev. A 78, 043826 (2008).
[Crossref]

Tomljenovic Hanic, S.

Vahala, K. J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

Viktorovitch, P.

Vitebskiy, I.

A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76, 053839 (2007).
[Crossref]

Vuckovic, J.

D. Englund, I. Fushman, and J. Vuĉković, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005).
[Crossref] [PubMed]

J. Vuĉković and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” Appl. Phys. Lett. 82, 2374–2376 (2003).
[Crossref]

Watanabe, T.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[Crossref]

Winn, J. N.

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

Yamamoto, Y.

J. Vuĉković and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” Appl. Phys. Lett. 82, 2374–2376 (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]

Appl. Phys. Lett. (2)

J. Vuĉković and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” Appl. Phys. Lett. 82, 2374–2376 (2003).
[Crossref]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

J. Lightwave Technol. (1)

Nat. Mater. (1)

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Nat. Photonics (1)

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[Crossref]

Nature (1)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (3)

Photonics Nanostruct.: Fundam. Appl. (1)

S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. 8, 310–317 (2010).
[Crossref]

Phys. Rev. (1)

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
[Crossref]

Phys. Rev. A (3)

A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A 76, 053839 (2007).
[Crossref]

A. A. Chabanov, “Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media,” Phys. Rev. A 77, 033811 (2008).
[Crossref]

K. Y. Jung and F. L. Teixeira, “Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis,” Phys. Rev. A 78, 043826 (2008).
[Crossref]

Science (1)

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 (2)

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

P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633.

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Figures (4)

Fig. 1
Fig. 1

Schematic of DHCs and their dispersion curves. (a) DHC based on a W1 waveguide. The highlighted region indicates the underlying PCW being perturbed to create a cavity. (b) Dispersion curve of the W1 PCW. Broken cyan curve shows the dispersion curve of a PCW with the background index changed to nb = 3.005. Green highlighting shows the mode gap. Vertical dashed line indicates BZ edge. (c) A DHC based on a PCW where the holes adjacent to the waveguide have had their radius increased to engineer the dispersion of the band minimum. (d) Dispersion curve of dispersion engineered PCW. The two band minima are at kxd/π = ±0.8034 (a reciprocal lattice vector is added in figure).

Fig. 2
Fig. 2

(a) Frequency of DHC modes versus cavity length for unapodized cavity in PCW with background index nb = 3.0, air hole radius 0.3d, where d is the period, and radius of holes adjacent to the waveguide are changed to 0.4d. The background index has been altered to nb = 3.005 to create the cavity. Broken red line corresponds to modes calculated using a fully numerical computation (COMSOL). The blue line corresponds to modes calculated using our perturbation theory. Top of figure corresponds to the band-edge frequency, d/λ = 0.26626, while the bottom is the frequency of the edge of the modegap, d/λ = 0.26584. (b) The detuning of the frequency of the two modes in (a). (c) Average of frequencies of the two modes in (a). (d)–(f) Same as (a)–(c) but for the Gaussian cavity with a maximum background index of nb = 3.005.

Fig. 3
Fig. 3

Normalised amplitudes (arbitrary units) of |Ey| for the DHC modes where the DHC has a length of 9d and is centred on x/d = 0. (a) |Ey| for the fundamental mode with frequency d/λ = 0.26607 computed using perturbation theory. (b) |Ey| for the higher order mode with frequency d/λ = 0.26611 computed using perturbation theory. (c) Cross section of (a) taken along the centre of the PCW (blue curve), but with an added comparison to fully numerical calculations of the fields using a finite element method (COMSOL) (broken red curve). Broken vertical lines indicate the edges of the cavity.

Fig. 4
Fig. 4

Sidelobes of F 1 ( k x k L ( 1 ) ) (arbitrary units) in the absence of coupling. (a) Broken green curve is for an unapodized cavity of length 8d. The blue curve is for an unapodized cavity with length 10.5d, i.e. near a point where the modes become degenerate. Although not shown, F 2 ( k x k L ( 1 ) ) has the same shape as F 1 ( k x k L ( 1 ) ), but is centred on k x = k L ( 2 ) indicated by the vertical dashed line. (b) Same as (a) but for a Gaussian cavity. The broken green curve is for a cavity with a 1/e length of 4d, while the blue curve is for a cavity with a 1/e length of 5.25d.

Equations (13)

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× × E k x ( r ) = ɛ ( r ) ω k x 2 c 2 E k x ( r ) ,
× × E ( r ) = [ ɛ ( r ) + δ ɛ ( r ) ] ω 2 c 2 E ( r ) ,
E ( r ) = d k x C ( k x ) E k x ( r ) .
C ( k x ) [ ω k x 2 ω 2 ] M = ω 2 d k x C ( k x ) 2 d r δ ɛ ( r ) E k x * ( r ) E k x ( r ) ,
k L ( 2 ) = k L ( 1 ) , ω k x = ω k x , E k x ( r ) = E k x * ( r ) .
F 1 ( k x k L ( 1 ) ) ( ω k x 2 ω 2 ) = ω 2 d k x [ F 1 ( k x k L ( 1 ) ) + F 2 ( k x k L ( 2 ) ) ] 2 d r δ ɛ ( r ) E k x * ( r ) E k x ( r ) .
M ( ω i d d x 2 ω 2 ) f 1 ( x )
L ^ f 1 ( x ) = 2 π ω 2 M [ f 1 ( x ) δ 11 + f 2 ( x ) δ 12 ] .
L ^ = ( ω 2 ω i d dx 2 ) , δ jm = d y δ ɛ ( r ) E k L ( j ) * ( r ) E k L ( m ) ( r ) .
[ L ^ + ω 2 d δ 11 ( x ) ω 2 d δ 12 ( x ) ω 2 d δ 12 * ( x ) L ^ + ω 2 d δ 11 ( x ) ] [ f 1 ( x ) f 2 ( x ) ] = [ 0 0 ] ,
E ( r ) = f 1 ( x ) E k L ( 1 ) ( r ) + f 2 ( x ) E k L ( 2 ) ( r ) ,
E ( r ) = f R ( x ) Re [ E k L ( 1 ) ( r ) ] + f I ( x ) Im [ E k L ( 1 ) ( r ) ] ,
[ L ^ + ω 2 d [ δ 11 ( x ) ] + Re ( δ 12 ( x ) ) ] ω 2 d Im [ δ 12 ( x ) ] ω 2 d Im [ δ 12 ( x ) ] L ^ + ω 2 d [ δ 11 ( x ) Re ( δ 12 ( x ) ) ] ] [ f R ( x ) f I ( x ) ] = [ 0 0 ] .

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