Abstract

We investigate the formation of photonic crystal waveguide (PCW) modes within the framework of perturbation theory. We derive a differential equation governing the envelope of PCW modes constructed from weak perturbations using an effective mass formulation based on the Luttinger-Kohn method from solid-state physics. The solution of this equation gives the frequency of the mode and its field. The differential equation lends itself to simple analytic approximations which reduce the problem to that of solving slab waveguide modes. By using this model, we demonstrate that the nature of the projected band structure and corresponding Bloch functions are central to the behaviour of PCW modes. With this understanding, we explain why the odd mode in a hexagonal PCW spans the entire Brillouin zone while the even mode is cut off.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. T. F. Krauss, "Slow light in photonic crystal waveguides," J. Phys. D: Appl. Phys. 40, 2666-2670 (2007).
    [CrossRef]
  4. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16, 6227-6232 (2008).
    [CrossRef] [PubMed]
  5. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
    [CrossRef] [PubMed]
  6. D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
    [CrossRef]
  7. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
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    [CrossRef]
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    [CrossRef]
  11. G. Bastard, "Superlattice band structure in the envelope-function approximation," Phys. Rev. B 24, 5693-5697 (1981).
    [CrossRef]
  12. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
    [CrossRef] [PubMed]
  13. K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. M. de Sterke, "Shallow defect states in two-dimensional photonic crystals," Phys. Rev. A 77, 063839 (2008).
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    [CrossRef]
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    [CrossRef]

2009 (2)

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

2008 (2)

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16, 6227-6232 (2008).
[CrossRef] [PubMed]

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. M. de Sterke, "Shallow defect states in two-dimensional photonic crystals," Phys. Rev. A 77, 063839 (2008).
[CrossRef]

2007 (1)

T. F. Krauss, "Slow light in photonic crystal waveguides," J. Phys. D: Appl. Phys. 40, 2666-2670 (2007).
[CrossRef]

2006 (1)

2005 (1)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

2004 (3)

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
[CrossRef]

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran "Modes of coupled photonic crystal waveguides," Opt. Lett. 29, 1384-1386 (2004).
[CrossRef]

A. Y. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

2002 (1)

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent. "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

2001 (1)

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

1999 (1)

1998 (1)

A. Mekis, S. Fan, and J. D. Joannopoulos, "Bound states in photonic crystal waveguides and waveguide bends," Phys. Rev. B 58, 4809-4817 (1998).
[CrossRef]

1981 (1)

G. Bastard, "Superlattice band structure in the envelope-function approximation," Phys. Rev. B 24, 5693-5697 (1981).
[CrossRef]

1955 (2)

J. M. Luttinger and W. Kohn, "Motion of electrons and holes in perturbed periodic fields," Phys. Rev. 97, 869-883 (1955).
[CrossRef]

W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915-922 (1955).
[CrossRef]

Allard, M.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent. "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Asatryan, A. A.

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. M. de Sterke, "Shallow defect states in two-dimensional photonic crystals," Phys. Rev. A 77, 063839 (2008).
[CrossRef]

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran "Modes of coupled photonic crystal waveguides," Opt. Lett. 29, 1384-1386 (2004).
[CrossRef]

Baba, T.

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
[CrossRef]

Bastard, G.

G. Bastard, "Superlattice band structure in the envelope-function approximation," Phys. Rev. B 24, 5693-5697 (1981).
[CrossRef]

Borel, P. I.

Botten, L. C.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. M. de Sterke, "Shallow defect states in two-dimensional photonic crystals," Phys. Rev. A 77, 063839 (2008).
[CrossRef]

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran "Modes of coupled photonic crystal waveguides," Opt. Lett. 29, 1384-1386 (2004).
[CrossRef]

Charbonneau-Lefort, M.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent. "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Corcoran, B.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

de Sterke, C. M.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. M. de Sterke, "Shallow defect states in two-dimensional photonic crystals," Phys. Rev. A 77, 063839 (2008).
[CrossRef]

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran "Modes of coupled photonic crystal waveguides," Opt. Lett. 29, 1384-1386 (2004).
[CrossRef]

Dossou, K. B.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. M. de Sterke, "Shallow defect states in two-dimensional photonic crystals," Phys. Rev. A 77, 063839 (2008).
[CrossRef]

Eggleton, B. J.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

Eich, M.

A. Y. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

Fage-Pedersen, J.

Fan, S.

A. Mekis, S. Fan, and J. D. Joannopoulos, "Bound states in photonic crystal waveguides and waveguide bends," Phys. Rev. B 58, 4809-4817 (1998).
[CrossRef]

Frandsen, L. H.

Gomez-Iglesias, A.

Grillet, C.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

Hamann, H. F.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Istrate, E.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent. "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Joannopoulos, J. D.

A. Mekis, S. Fan, and J. D. Joannopoulos, "Bound states in photonic crystal waveguides and waveguide bends," Phys. Rev. B 58, 4809-4817 (1998).
[CrossRef]

Kohn, W.

J. M. Luttinger and W. Kohn, "Motion of electrons and holes in perturbed periodic fields," Phys. Rev. 97, 869-883 (1955).
[CrossRef]

W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915-922 (1955).
[CrossRef]

Krauss, T. F.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16, 6227-6232 (2008).
[CrossRef] [PubMed]

T. F. Krauss, "Slow light in photonic crystal waveguides," J. Phys. D: Appl. Phys. 40, 2666-2670 (2007).
[CrossRef]

Lavrinenko, A. V.

Lee, R. K.

Li, J.

Luttinger, J.

W. Kohn and J. Luttinger, "Theory of donor states in silicon," Phys. Rev. 98, 915-922 (1955).
[CrossRef]

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]

Mahmoodian, S.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

McNab, S. J.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

McPhedran, R. C.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. M. de Sterke, "Shallow defect states in two-dimensional photonic crystals," Phys. Rev. A 77, 063839 (2008).
[CrossRef]

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran "Modes of coupled photonic crystal waveguides," Opt. Lett. 29, 1384-1386 (2004).
[CrossRef]

Mekis, A.

A. Mekis, S. Fan, and J. D. Joannopoulos, "Bound states in photonic crystal waveguides and waveguide bends," Phys. Rev. B 58, 4809-4817 (1998).
[CrossRef]

Monat, C.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

Mori, D.

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
[CrossRef]

Moss, D. J.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

Notomi, M.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

O’Boyle, M.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

O’Faolain, L.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16, 6227-6232 (2008).
[CrossRef] [PubMed]

Petrov, A. Y.

A. Y. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

Poon, J.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent. "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Poulton, C. G.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

K. B. Dossou, L. C. Botten, R. C. McPhedran, C. G. Poulton, A. A. Asatryan, and C. M. de Sterke, "Shallow defect states in two-dimensional photonic crystals," Phys. Rev. A 77, 063839 (2008).
[CrossRef]

Sargent, E. H.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent. "Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture," Phys. Rev. B 65, 125318 (2002).
[CrossRef]

Scherer, A.

Shinya, A.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Takahashi, C.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Takahashi, J.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

White, T. P.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16, 6227-6232 (2008).
[CrossRef] [PubMed]

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran "Modes of coupled photonic crystal waveguides," Opt. Lett. 29, 1384-1386 (2004).
[CrossRef]

Xu, Y.

Yamada, K.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yariv, A.

Yokohama, I.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

A. Y. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

D. Mori and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
[CrossRef]

J. Phys. D: Appl. Phys. (1)

T. F. Krauss, "Slow light in photonic crystal waveguides," J. Phys. D: Appl. Phys. 40, 2666-2670 (2007).
[CrossRef]

Nat. Photonics (1)

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, "Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides," Nat. Photonics 3, 206-210 (2009).
[CrossRef]

Nature (1)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (2)

J. M. Luttinger and W. Kohn, "Motion of electrons and holes in perturbed periodic fields," Phys. Rev. 97, 869-883 (1955).
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Supplementary Material (1)

» Media 1: AVI (964 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Contour plot of a quadrant of the second band in a hexagonal PC lattice with TE (Hz ) polarisation, background index nb =3.0, hole index, nc =1.0 and cylinder radius a=0.3d where d is the pitch. Dark shading indicates high frequencies while light shading indicates low frequencies. Red line is the BZ edge for the 2D PC while the broken black line indicates the edge of the BZ when the PCW is introduced. The projected band-edge trajectory (thick blue line) is obtained by finding the minimum frequency and its corresponding ky for each kx . The inset schematically illustrates the projected band trajectory over the entire BZ.

Fig. 2.
Fig. 2.

Schematic showing parts of infinite PCs with period d each with a PCW. We express the perturbation as a sum over columns for (a) square lattice PCW with period Λ=d and (b) hexagonal lattice PCW with period Λ=d. Filled circles indicate cylinders altered to construct a PCW. Region between black lines indicates a single period of the perturbation.

Fig. 3.
Fig. 3.

An illustration of replacing a perturbation with another which preserves the change in electric energy. Red curve shows C0∣∣Ekx,kLy(r)∣∣ dx for a Bloch mode at d/λ=0.2653 in a square PC with cylinder index nc =3 background index nb =1 and radius a=0.3d where d is the pitch. The box with broken blue lines corresponds to the area-preserving perturbation with height 1 (2a) C'0∣∣Ekx,kLy(r)∣∣2dxdy .

Fig. 4.
Fig. 4.

(a) Frequency of a PCW mode (dashed purple line) moving off the projected band-edge for PCW cylinder index nw =2.9. Blue shading indicates the band. (b) Frequency shift of (a) from the projected band-edge as a function of kxd. The dashed purple line is calculated from our theory, while red lines are numerical calculations. Broken vertical lines indicate the values of kxd for which field comparisons are made in Figs. 5(b) and 5(c)

Fig. 5.
Fig. 5.

(a) Contour plot of |Ez | field for a PCW mode constructed from a projected band-edge Bloch mode and an envelope function calculated from Eq. (14) and Eq. (15). Parameters as in text with nw =2.9, d/λ=0.245 and kxd=1.782 (b) Cross sectional comparison of (a) at x/d=0 with a numerical calculation. Here, the broken red line is the numerical result, while blue line is calculated using our asymptotic formulation. (c) Same as (b) but with d/λ=0.266 and kxd=2.962.

Fig. 6.
Fig. 6.

(a) Comparison of numerical (red lines) and asymptotic calculations (broken blue line) for PCW mode frequency splitting with respect to band-edge in region to the left of bifurcation point. Parameters as in Fig. 1 with PCW cylinder index nw =1.05 (b) Comparison of numerical (red lines) and asymptotic calculations (broken green and blue lines) for PCW mode frequency splitting with respect to band-edge in region to the right of bifurcation point. The red line below the green line cannot be distinguished.

Fig. 7.
Fig. 7.

Real part of Bloch modes (Hz field) along the projected band-edge, where black line indicates cylinder boundary (parameters as in text). Green shading indicates zero amplitude while red and blue indicate large positive and negative amplitudes respectively. (a) Shows Bloch mode at kxd=0, which is odd with respect to x. (b) and (c) show superposition of Bloch modes at kxd=π such that they are orthogonal over the perturbed cylinder. The modes then form (b) even and (c) odd superpositions with respect to x.

Fig. 8.
Fig. 8.

Comparison of numerics and analytic theory near the bifurcation point. Dashed curves indicate analytic theory, red lines indicate numerical calculations while the band is indicated by the orange shading. The blue line is the even mode, while purple and green lines are odd modes on the left and right of the bifurcation point respectively. The analytic theory for the odd mode diverges near the bifurcation point. Dashed vertical line indicates the bifurcation point.

Fig. 9.
Fig. 9.

(583 KB) Single frame excerpt of an animation (Media 1) showing dispersion relations of PCW modes as the index of PCW cylinders ranges from nw =1.1-3.0 for hexagonal lattice parameters as in Section 4. The red lines are numerical calculations while the green, purple and blue lines use our asymptotic treatment. The black line indicates the projected band-edge.

Equations (27)

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[(××)εPC(r)ωk2c2]Ek(r)=0 ,
[(××)(εPC(r)+pδεp(r))ωkx2c2]Ekx(r)=0 .
Ekx(r)= d ky C (ky) Ek (r) ,
M[ωkx,ky2ωkx2]C(ky)δ(kxkx)=ωkx2 d ky C (ky) d2 r pδεp(r)Ek*(r)·Ek(r) .
ωkx2 d ky d ky C (ky) d2 r pδεp(r)Ekx,kLy*(r)·Ekx,kLy(r)ei(kykLy)(yy)ei(kykLy)y .
f(y)= d ky C (ky) ei(kykLy)y ,
2 π ωkx2 f (y) dx pδεp(r)Ekx,kLY*(r)·Ekx,kLy(r) .
M [ωkx,iddy2ωkx2] f (y) δ (kxkx) .
ωkx2 f (y) dx δ ε (r) Ekx,kLy*(r)·Ekx,kLy(r)δ(kxkx)(2π)2Λ .
ωkx,iddy2f(y)=ωkx2 [1+Ω𝓔kxΛdxδε(r)Ekx,kLy(r)2] f (y) ,
ωkx,iddy=[ω(kx,kLy)12DLy(kx)d2dy2+]
ωkx,iddy2=[ω2(kx,kLy)ω(kx,kLy)DLy(kx)d2dy2+] .
[ω2(kx,kLy)ω(kx,kLy)DLy(kx)d2dy2]f(y)=ωkx2 [1+Ω𝓔ΛCdxδε∣∣Ekx,kLy(r)∣∣2] f (y) ,
[ω2(kx,kLy)ω(kx,kLy)DLy(kx)d2dy2]f(y)=ωkx2 [1+Ω2aΛδ𝓔kx𝓔] f (y) ,
[d2dy2α2]f(y)=0 , α2=DLy(kx)ω(kx,kLy) [ω2(kx,kLy)ωkx2]
[d2dy2+β2]f(y)=0, β2=α2+Ωωkx2DLy(kx)2aΛω(kx,kLy)δ𝓔kx𝓔
tan(βa)=αβ .
ωkx=ω(kx,kLy)DLy(kx)8(ω(kx,kLy)ΩΛ)2(δ𝓔kx𝓔)2 .
M [ωkx,ky2ωkx2] (C1(kykLy(1))+C2(kykLy(2))) δ (kxkx)
=ωkx2 d ky (C1(kykLy(1))+C2(kykLy(2))) pδεpEk*(r)·Ek(r)d2r.
M [ωkx,ky2ωkx2] C1 (kykLy(1)) δ (kxkx)
=ωkx2 (C1(kykLy(1))+C2(kykLy(2))) dky pδεpEk*(r).Ek(r)d2r ,
M [ωkx,ky2ωkx2] C2 (kykLy(2)) δ (kxkx)
=ωkx2 (C1(kykLy(1))+C2(kykLy(2))) dky pδεpEk*(r)·Ek(r)d2r .
[(ωkx2ωkx,iddy2)+ωkx2Ω2a𝓔Λδ𝓔11ωkx2Ω2a𝓔Λδ𝓔12ωkx2Ω2a𝓔Λδ𝓔21(ωkx2ωkx,iddy2)+ωkx2Ω2a𝓔Λδ𝓔22][f1(y)f2(y)]=[00] ,
[(ωkx2ωkx,iddy2)+ωkxΩ2𝓔Λδ𝓔AA]fA(y)=0,
[(ωkx2ωkx,iddy2)+ωkx2Ω𝓔Λδ𝓔BB]fB(y)=0 .

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