Abstract

We investigate the modes of coupled waveguides in a hexagonal photonic crystal. We find that for a substantial parameter range the coupled waveguide modes have dispersion relations exhibiting multiple intersections, which we explain both intuitively and using a rigorous tight-binding argument.

© 2010 Optical Society of America

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  1. S. Kubo, D. Mori, and T. Baba, "Low-group-velocity and low-dispersion slow light in photonic crystal waveguides," Opt. Lett. 32, 2981-2983 (2007).
    [CrossRef] [PubMed]
  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]
  3. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, "Slow-light dispersion in coupled periodic waveguides," J. Opt. Soc. Am. B 25, C65-C74 (2008).
    [CrossRef]
  4. D. Mori, and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004).
    [CrossRef]
  5. A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photon. Technol. Lett. 15, 694-696 (2003).
    [CrossRef]
  6. 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]
  7. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988). Chap. 11.
  8. K. Iizuka, Elements of Photonics, Volume II (Wiley, 2002). Chap. 9.
  9. 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]
  10. A. Locatelli, M. Conforti, D. Modotto, and C. D. Angelis, "Discrete negative refraction in photonic crystal waveguide arrays," Opt. Lett. 31, 1343-1345 (2006).
    [CrossRef] [PubMed]
  11. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, "Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers," Opt. Express 16, 1104-1114 (2008).
    [CrossRef] [PubMed]
  12. A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, "Slow-light vortices in periodic waveguides," J. Opt. A, Pure Appl. Opt. 11, 094016 (2009).
    [CrossRef]
  13. L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).
  14. F. S. Chien, Y. Hsu, W. Hsieh, and S. Cheng, "Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides," Opt. Express 12, 1119-1125 (2004).
    [CrossRef] [PubMed]
  15. S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009).
    [CrossRef] [PubMed]
  16. K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
    [CrossRef]
  17. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
  18. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).
    [CrossRef]
  19. L. C. Botten, R. A. Hansen, and C. M. de Sterke, "Supermodes in multiple coupled photonic crystal waveguides," Opt. Express 14, 387-396 (2006).
    [CrossRef] [PubMed]

2009

A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, "Slow-light vortices in periodic waveguides," J. Opt. A, Pure Appl. Opt. 11, 094016 (2009).
[CrossRef]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009).
[CrossRef] [PubMed]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

2008

2007

2006

A. Locatelli, M. Conforti, D. Modotto, and C. D. Angelis, "Discrete negative refraction in photonic crystal waveguide arrays," Opt. Lett. 31, 1343-1345 (2006).
[CrossRef] [PubMed]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

L. C. Botten, R. A. Hansen, and C. M. de Sterke, "Supermodes in multiple coupled photonic crystal waveguides," Opt. Express 14, 387-396 (2006).
[CrossRef] [PubMed]

2005

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

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]

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

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).
[CrossRef]

F. S. Chien, Y. Hsu, W. Hsieh, and S. Cheng, "Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides," Opt. Express 12, 1119-1125 (2004).
[CrossRef] [PubMed]

2003

A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

2001

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Angelis, C. D.

Asatryan, A. A.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (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]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Baba, T.

S. Kubo, D. Mori, and T. Baba, "Low-group-velocity and low-dispersion slow light in photonic crystal waveguides," Opt. Lett. 32, 2981-2983 (2007).
[CrossRef] [PubMed]

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

Botten, L. C.

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009).
[CrossRef] [PubMed]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

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

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

L. C. Botten, R. A. Hansen, and C. M. de Sterke, "Supermodes in multiple coupled photonic crystal waveguides," Opt. Express 14, 387-396 (2006).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (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]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Cheng, S.

Chien, F. S.

Chigrin, D. N.

Conforti, M.

Cuesta, F.

A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

de Sterke, C. M.

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009).
[CrossRef] [PubMed]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

L. C. Botten, R. A. Hansen, and C. M. de Sterke, "Supermodes in multiple coupled photonic crystal waveguides," Opt. Express 14, 387-396 (2006).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (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]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Desyatnikov, A. S.

A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, "Slow-light vortices in periodic waveguides," J. Opt. A, Pure Appl. Opt. 11, 094016 (2009).
[CrossRef]

Dossou, K. B.

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009).
[CrossRef] [PubMed]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

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

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

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]

Ha, S.

A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, "Slow-light vortices in periodic waveguides," J. Opt. A, Pure Appl. Opt. 11, 094016 (2009).
[CrossRef]

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

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]

Hansen, R. A.

Hsieh, W.

Hsu, Y.

Kivshar, Y. S.

Kubo, S.

Langtry, T. N.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Lavrinenko, A. V.

Locatelli, A.

Mahmoodian, S.

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009).
[CrossRef] [PubMed]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

Marti, J.

A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

Martijn de Sterke, C.

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

Martinez, A.

A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[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.

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009).
[CrossRef] [PubMed]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

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]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Modotto, D.

Mori, D.

S. Kubo, D. Mori, and T. Baba, "Low-group-velocity and low-dispersion slow light in photonic crystal waveguides," Opt. Lett. 32, 2981-2983 (2007).
[CrossRef] [PubMed]

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

Nicorovici, N. A.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

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]

Pelinovsky, D. E.

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]

Poulton, C. G.

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009).
[CrossRef] [PubMed]

Sukhorukov, A. A.

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.

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]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Wilcox, S.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

Appl. Phys. Lett.

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

IEEE Photon. Technol. Lett.

A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photon. Technol. Lett. 15, 694-696 (2003).
[CrossRef]

Int. J. Microwave Opt. Technol.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).

J. Opt. A, Pure Appl. Opt.

A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, "Slow-light vortices in periodic waveguides," J. Opt. A, Pure Appl. Opt. 11, 094016 (2009).
[CrossRef]

J. Opt. Soc. Am. B

Nature

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

Opt. Lett.

Phys. Rev. A

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

Phys. Rev. E

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Other

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988). Chap. 11.

K. Iizuka, Elements of Photonics, Volume II (Wiley, 2002). Chap. 9.

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Figures (6)

Fig. 1
Fig. 1

Dispersion curves of CWMs for (a) PCWs separated by four rows (staggered geometry) and (b) PCWs separated by five rows (inline geometry). The purple dashed curve and the solid green curves are for a single waveguide and for CPCWs, respectively. The red shaded area indicates the projected bands and the green area is the braided region, separated from the typical region by the blue dashed curve.

Fig. 2
Fig. 2

|Ey| fields of the even single PCW mode for a PCW centered at y = 0. The scale has been chosen so as to emphasize the weak fields which are involved in coupling the PCWs. The value of kxd for Fig. 2(b) corresponds to a crossing at spacing 4.

Fig. 5
Fig. 5

(a): Geometry of the PC unit cell, defined by the basis vectors e1 and e2. Points P1 and P2 are the phase origins of the plane wave expansion respectively at the upper (Π1) and lower (Π2) interfaces of a grating layer. (b): Schematic of a PC with a double waveguide, a composite of 5 elements characterized by their scattering matrices: RW, TW for the waveguides, RB, TB for the barrier between the waveguides and R for the semi-infinite PCs.

Fig. 3
Fig. 3

Complex band diagrams at different slices of kx for the bulk hexagonal lattice. Dark blue indicates real bands, while other colours indicate complex bands. (a) kxd is in the typical region; (b) kxd intersects the blue dashed curve in Fig. 1; (c) kxd is at the BZ edge.

Fig. 4
Fig. 4

Hz fields for two degenerate coupled waveguide modes separated by two rows at kxd = π. (a) The real part of both modes is identical. (b) The imaginary part of mode 1. (c) Imaginary part of mode 2. (d) Both modes have the same absolute value.

Fig. 6
Fig. 6

Roots of the correction term δν(kx, d/λ, ) when (kx d, d/λ) varies along the dispersion curve of a single waveguide (purple dashed curve in Fig. 1). The roots occurring at = 4 and = 5 are indicated by green and cyan dots respectively (see Table 1 for their coordinates). The blue and red curves correspond, respectively, to the upper and lower frequency modes in the braided region in Fig. 1. The former enters the braided region at kx d = 1.998, the latter at kx d = 2.048).

Tables (1)

Tables Icon

Table 1 Crossing points (kx d, d/λ) of the even and odd CWM dispersion curves in Fig. 1, according to the dispersion results in Fig. 1 (columns “converged results”) and the roots of δν(kx, d/λ, ) (columns “perturbation theory”)

Equations (54)

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J = δ ɛ E 1 ( r ) E 2 * ( r ) d 2 r = δ ɛ ( E x 1 ( r ) E x 2 * ( r ) + E y 1 ( r ) E y 2 * ( r ) ) d 2 r .
μ ± m = e i m k e 2 = e i k x m d / 2 e ± i k y m d 3 / 2 e κ m d 3 / 2 ,
ψ ( r 0 + m e 2 ) = c 1 φ 1 ( r 0 ) e i k x m d / 2 e i k y m d 3 / 2 e κ m d 3 / 2 + c 2 φ 2 ( r 0 ) e i k x m d / 2 e i k y m d 3 / 2 e κ m d 3 / 2 .
ψ ( r 0 + m e 2 ) = 2 e κ m d 3 / 2 e i k x m d / 2 Re [ c 1 φ 1 ( r 0 ) e i ( k y m d 3 / 2 ) + i ϑ / 2 ] .
| Ψ = | ψ 1 ± e i k x ( + 1 ) d / 2 | ψ 2
ψ ( r 0 + m e 2 ) = 2 e κ m d 3 / 2 e i k x m d / 2 c 1 φ 1 ( r 0 ) cos ( m π / 2 ) .
H z , s ( x , y ) = p = + χ p 1 2 [ f p , s e i χ p ( y y s ) + f p , s + e i χ p ( y y s ) ] e i α p ( x x s ) , for s = 1 , 2 ,
𝒯 [ f 1 f 1 + ] = μ [ f 1 f 1 + ]
= [ F F F + F + ] and = [ Λ 0 0 Λ ] .
Λ Λ = e i k x d I , [ F F + ] = [ Q 0 F + Q 0 F ] and Q 0 = diag ( e i p π ) p = diag ( ( 1 ) p ) p .
R = F + F 1 and R = F F + 1
R B = F + ( R 21 + Λ R 21 Λ ) ( I R 21 Λ R 21 Λ ) 1 F 1
T B = F ( I R 21 R 21 ) Λ ( I R 21 Λ R 21 Λ ) 1 F 1
R B = F ( R 21 + Λ R 21 Λ ) ( I R 21 Λ R 21 Λ ) 1 F + 1
T B = F + ( I R 21 R 21 ) Λ ( I R 21 Λ R 21 Λ ) 1 F + 1
{ f ^ 1 + = R W f ^ 1 + T W f 1 + f 1 = T W f ^ 1 + R W f 1 + f ^ 1 = R f ^ 1 + { f ^ 1 + = ( I R W R ) 1 T W f 1 + f ^ 1 = R ( I R W R ) 1 T W f 1 + f 1 = ( R W + T W R ( I R W R ) 1 T W ) f 1 +
f 2 + = ( R W + T W R ( I R W R ) 1 T W ) f 2 .
f 1 + = R B f 1 + T B f 2 + and f 2 = T B f 1 + R B f 2 + .
R B = Q 0 R B Q 0 , T B = e i k x d Q 0 T B Q 0 and R 21 = R 21 .
f ˜ 2 = diag ( e i α p d 2 ) f 2 = e i k x d 2 Q 0 f 2 .
( f 1 + ± Q 0 + 1 f ˜ 2 ) = ( R B ± e i k x d 2 Q 0 T B ) ( f 1 ± Q 0 + 1 f ˜ 2 + )
R = Q 0 R Q 0 ,
R W + T W R ( I R W R ) 1 T W = Q 0 ( R W + T W R ( I R W R ) 1 T W ) Q 0 .
f 1 = R ¯ ¯ f 1 + and Q 0 + 1 f ˜ 2 + = R ¯ ¯ Q 0 + 1 f ˜ 2 .
( f 1 ± Q 0 + 1 f ˜ 2 + ) = R ¯ ¯ ( f 1 + ± Q 0 + 1 f ˜ 2 + )
R ¯ ¯ = R W + T W R ( I R W R ) 1 T W .
A ( σ ) ( f 1 + + σ Q 0 + 1 f ˜ 2 + ) = 0 , with A ( σ ) = I ( R B + σ e i k x d 2 Q 0 T B ) R ¯ ¯
A 0 f 1 + = 0 with A 0 = I R R ¯ ¯ .
ν = ν 0 + δ ν , x = x 0 + δ x and A ( σ , ν ) = A 0 ( ν 0 + δ ν ) + δ A ( σ , ν 0 + δ ν ) .
A 0 ( ν 0 + δ ν ) + δ A ( σ , ν 0 + δ ν ) ) ( x 0 + δ x ) = 0 .
( A 0 ( ν 0 ) + A 0 ν δ ν + δ A ( σ , ν 0 ) ) ( x 0 + δ x ) = 0 .
x = x 0 + δ x = x 0 ( 1 ) + c ( 1 ) x 0 ( 1 ) + c ( 2 ) x 0 ( 2 ) with | c ( 1 ) | 1 and | c ( 2 ) | 1 .
δ ν A 0 ν x 0 ( 1 ) = δ A ( σ , ν 0 ) x 0 ( 1 ) c ( 2 ) A 0 ( ν 0 ) x 0 ( 2 ) .
δ ν = x 0 ( 1 ) H δ A ( σ , ν 0 ) x 0 ( 1 ) x 0 ( 1 ) H A 0 ν x 0 ( 1 )
R B = R + O ( || Λ || 2 )
T B = F ( I R 21 2 ) Λ F 1 + O ( || Λ || 2 ) = ( I ( Q 0 R ) 2 ) F Λ F 1 + O ( || Λ || 2 )
δ A ( σ , ν ) = A ( σ , ν ) A 0 ( ν ) = σ e i k x d 2 Q 0 ( I ( Q 0 R ) 2 ) F Λ F 1 R ¯ ¯ + O ( || Λ || 2 )
x 0 ( 1 ) H δ A ( σ , ν 0 ) x 0 ( 1 ) = σ x 0 ( 1 ) H Q 0 ( I ( Q 0 R ) 2 ) F ( e i k x d 2 Λ ) F 1 R ¯ ¯ x 0 ( 1 )
x 0 ( 1 ) H δ A ( σ , ν 0 ) x 0 ( 1 ) = σ x 0 ( 1 ) H Q 0 ( I ( Q 0 R ) 2 ) F ( e i k x d 2 Λ ) F 1 R 1 x 0 ( 1 )
x 0 ( 1 ) H A 0 ν x 0 ( 1 ) = x 0 ( 1 ) H ( R ( ν ) R ¯ ¯ ( ν ) ) ν x 0 ( 1 ) .
x 0 ( 1 ) H δ A ( σ , ν 0 ) x 0 ( 1 ) = σ x 0 ( 1 ) H ( Q 0 R H Q 0 R ) Q 0 F + ( e i k x d 2 Λ ) F + 1 x 0 ( 1 )
δ ν = σ Im ( R ) ( R ( ν ) R ¯ ¯ ( ν ) ) ν ( e i k x d 2 μ 1 )
x 0 ( 1 ) H δ A ( ν 0 ) x 0 ( 1 ) = σ ( a 1 ( e i k x d 2 μ 1 ) + a 2 ( e i k x d 2 μ 2 ) ) .
δ ν σ a 1 ( e i k x d 2 μ 1 ) .
e i k x d 2 μ 2 = ( e i k x d 2 μ 1 ) * e i k x d 2 μ 1 .
x 0 ( 1 ) H δ A ( ν 0 ) x 0 ( 1 ) = σ ( a 1 ( e i k x d 2 μ 1 ) a 1 * ( e i k x d 2 μ 1 * ) ) = 2 σ i | a 1 | | μ 1 | sin ( ( arg ( μ 1 ) + k x d 2 ) + arg ( a 1 ) )
x 0 ( 1 ) H δ A ( ν 0 ) x 0 ( 1 ) = σ a 1 ( ( i μ 1 ) ( i μ 1 ) ) = 2 σ i a 1 | μ 1 | sin ( π 2 )
μ n = e i k e 2 = e i k x d / 2 e i β n d 3 / 2 e κ n d 3 / 2
Ψ ( x , y ) ] { n = 1 c n + φ n + ( x , y ) , if y > h / 2 n = 1 c n φ n ( x , y ) , if y < h / 2
Ψ * ( x , y ) = γ ^ 2 Ψ ( x , y ) Ψ * ( x , y ) = γ ^ γ ^ * Ψ ( x , y ) ( γ ^ Ψ ( x , y ) ) * = γ ^ Ψ ( x , y ) .
μ 1 = e i k x d / 2 e i β 1 d 3 / 2 e κ 1 d 3 / 2 = ( i ) ( i ) e κ 1 d 3 / 2 = ( 1 ) e κ 1 d 3 / 2
μ 2 = e i k x d / 2 e i β 1 d 3 / 2 e κ 1 d 3 / 2 = ( i ) ( i ) e κ 1 d 3 / 2 = e κ 1 d 3 / 2
c 1 ( φ 1 ( ( x , y ) e 2 ) + φ 2 ( ( x , y ) e 2 ) ) = c 1 ( μ 1 φ 1 ( x , y ) + μ 2 φ 2 ( x , y ) )
= c 1 e κ 1 d 3 / 2 ( ( 1 ) φ 1 ( x , y ) + φ 1 ( x , y ) )

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