Abstract

We develop a systematic approach for simultaneous extraction of the dispersion relations and profiles of multiple modes in periodic waveguides though a special global optimization procedure applied to near-field electric field measurements in the waveguide plane. We apply this method to perform in-depth analysis of experimental data on wave propagation close to an interface between waveguide sections with different dispersion characteristics, and we successfully identify several modes contributing to the experimentally measured fields. We find clear evidence that when the group velocity is reduced across the interface, evanescent modes that facilitate the excitation of propagating slow-light waves appear, confirming previous theoretical predictions.

© 2011 Optical Society of America

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  1. R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007).
    [CrossRef]
  2. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
    [CrossRef] [PubMed]
  3. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
    [CrossRef] [PubMed]
  4. N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in k space,” Phys. Rev. B 78, 125301 (2008).
    [CrossRef]
  5. T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Lett. 33, 2644–2646(2008).
    [CrossRef] [PubMed]
  6. S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. 75, 3461–3463 (1999).
    [CrossRef]
  7. B. I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B 72, 165102 (2005).
    [CrossRef]
  8. A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B 80, 193101 (2009).
    [CrossRef]
  9. B. Dastmalchi, A. Mohtashami, K. Hingerl, and J. Zarbakhsh, “Method of calculating local dispersion in arbitrary photonic crystal waveguides,” Opt. Lett. 32, 2915–2917 (2007).
    [CrossRef] [PubMed]
  10. A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Yu. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express 17, 3716–3721(2009).
    [CrossRef] [PubMed]
  11. R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991).
    [CrossRef]
  12. V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Magn. Reson. Spectrosc. 38, 159–196 (2001).
    [CrossRef]
  13. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, C. M. de Sterke, and Yu. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. 34, 3776–3778 (2009).
    [CrossRef] [PubMed]
  14. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).
  15. 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]
  16. 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]
  17. L. H. Frandsen, A. V. Lavrinenko, J. Fage Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450(2006).
    [CrossRef] [PubMed]
  18. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef] [PubMed]
  19. M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. 85, 294–297 (2000).
    [CrossRef] [PubMed]
  20. M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
    [CrossRef] [PubMed]
  21. C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express 17, 17338–17343 (2009).
    [CrossRef]
  22. M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009).
    [CrossRef]

2009 (6)

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B 80, 193101 (2009).
[CrossRef]

A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Yu. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express 17, 3716–3721(2009).
[CrossRef] [PubMed]

S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, C. M. de Sterke, and Yu. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. 34, 3776–3778 (2009).
[CrossRef] [PubMed]

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express 17, 17338–17343 (2009).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

2008 (3)

2007 (3)

2006 (1)

2005 (3)

B. I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B 72, 165102 (2005).
[CrossRef]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

2001 (2)

V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Magn. Reson. Spectrosc. 38, 159–196 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

2000 (1)

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. 85, 294–297 (2000).
[CrossRef] [PubMed]

1999 (1)

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. 75, 3461–3463 (1999).
[CrossRef]

1991 (1)

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991).
[CrossRef]

Andryieuski, A.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B 80, 193101 (2009).
[CrossRef]

Appelbaum, I.

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. 75, 3461–3463 (1999).
[CrossRef]

Asakawa, K.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007).
[CrossRef]

Baba, T.

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

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]

Balistreri, M. L. M.

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. 85, 294–297 (2000).
[CrossRef] [PubMed]

Beggs, D. M.

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Borel, P. I.

Botten, L. C.

Burresi, M.

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Cummer, S. A.

B. I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B 72, 165102 (2005).
[CrossRef]

Dastmalchi, B.

de Sterke, C. M.

Dossou, K. B.

Engelen, R. J. P.

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007).
[CrossRef]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Fage Pedersen, J.

Fan, S. H.

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. 75, 3461–3463 (1999).
[CrossRef]

Frandsen, L. H.

Gersen, H.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007).
[CrossRef]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Gomez Iglesias, A.

Gray, S. K.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991).
[CrossRef]

Ha, S.

Hingerl, K.

Houdre, R.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in k space,” Phys. Rev. B 78, 125301 (2008).
[CrossRef]

Hughes, S.

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Ikeda, N.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. 75, 3461–3463 (1999).
[CrossRef]

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

Johnson, S. G.

Karle, T. J.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Kivshar, Yu. S.

Korterik, J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. 85, 294–297 (2000).
[CrossRef] [PubMed]

Kotlyar, M. V.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in k space,” Phys. Rev. B 78, 125301 (2008).
[CrossRef]

Krauss, T. F.

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in k space,” Phys. Rev. B 78, 125301 (2008).
[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]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

Kubo, S.

Kuipers, L.

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007).
[CrossRef]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. 85, 294–297 (2000).
[CrossRef] [PubMed]

Lavrinenko, A. V.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B 80, 193101 (2009).
[CrossRef]

L. H. Frandsen, A. V. Lavrinenko, J. Fage Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450(2006).
[CrossRef] [PubMed]

Le Thomas, N.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in k space,” Phys. Rev. B 78, 125301 (2008).
[CrossRef]

Li, J.

Malureanu, R.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B 80, 193101 (2009).
[CrossRef]

Mandelshtam, V. A.

V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Magn. Reson. Spectrosc. 38, 159–196 (2001).
[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, 1995).

Mohtashami, A.

Mori, D.

Noid, D. W.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991).
[CrossRef]

O’Faolain, L.

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (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]

Oosten, D.

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Opheij, A.

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

Patterson, M.

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Pfeffer, G. A.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991).
[CrossRef]

Popa, B. I.

B. I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B 72, 165102 (2005).
[CrossRef]

Powell, D. A.

Roy, R.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991).
[CrossRef]

Schulz, S.

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Shadrivov, I. V.

Sugimoto, Y.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007).
[CrossRef]

Sukhorukov, A. A.

Sumpter, B. G.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991).
[CrossRef]

van Hulst, N. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. 85, 294–297 (2000).
[CrossRef] [PubMed]

van Mori, D.

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

White, T. P.

Winn, J. N.

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

Zabelin, V.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in k space,” Phys. Rev. B 78, 125301 (2008).
[CrossRef]

Zarbakhsh, J.

Appl. Phys. Lett. (1)

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. 75, 3461–3463 (1999).
[CrossRef]

Nat. Phys. (1)

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

Phys. Rep. (1)

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991).
[CrossRef]

Phys. Rev. B (4)

B. I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B 72, 165102 (2005).
[CrossRef]

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B 80, 193101 (2009).
[CrossRef]

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in k space,” Phys. Rev. B 78, 125301 (2008).
[CrossRef]

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Phys. Rev. Lett. (4)

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. 85, 294–297 (2000).
[CrossRef] [PubMed]

M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[CrossRef] [PubMed]

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[CrossRef]

Other (1)

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

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Figures (8)

Fig. 1
Fig. 1

Procedure for determining optimal k m at a given frequency ω j is shown in steps. Alternatively, k m can be determined for all ω in a single global minimization by calculating W ( ω ) for every ω then minimizing the average of W ( ω ) .

Fig. 2
Fig. 2

Dispersion for the slow-light waveguide with p 2 = 0.375 d . (a) Bloch wavenumbers k m extracted from the measured field profiles: solid curves, Re ( k m ) ; dashed curves, Im ( k m ) . Propagating ( k 1 , k 2 ) and evanescent TE modes ( k 3 , k 4 , k 5 , k 6 ) are labeled “p” and “e”, respectively, and the label “TM” marks the TM-like mode. Dotted curves show the numerically calculated dispersion for propagating TE and TM waves based on 3D plane-wave simulations. (b) The value of mismatch W min between measured and reconstructed fields. (c) SFT of the measured fields. (d) Sketch of the waveguide geometry.

Fig. 3
Fig. 3

Dispersion for the slow-light waveguide with p 2 = 0.4 d . Notations correspond to Fig. 2.

Fig. 4
Fig. 4

Dispersion for the slow-light waveguide with p 2 = 0.425 d . Notations correspond to Fig. 2.

Fig. 5
Fig. 5

Extracted intensity profiles of (a) propagating and (b) evanescent modes for p 2 = 0.4 d away from the slow-light regime ( λ = 1537 nm ). The sum of the recovered modes and the measured field is shown in (c) and the difference between them is shown in (d).

Fig. 6
Fig. 6

Extracted intensity profiles of waveguide modes in the slow-light regime ( λ = 1534 nm ). Notations are the same as in Fig. 5.

Fig. 7
Fig. 7

Results of dispersion extraction when evanescent waves are excluded from consideration; parameters and notations correspond to Figs. 3a, 3b.

Fig. 8
Fig. 8

Results of local dispersion extraction procedure; parameters and notations correspond to Figs. 3a, 3b.

Equations (14)

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ψ m ( r ; ω ) exp ( i k m z / d ) ,
E ( r ; ω ) = m = 1 M a m ψ m ( r ; ω ) exp ( i k m z / d ) + w ( r ; ω ) .
U n ( r ; ω ) = E ( x , y , z + ( n 1 ) d ; ω ) ,
A m ( r ; ω ) = a m ψ m ( r ; ω ) exp ( i k m z / d ) ,
w n ( r ; ω ) = w ( x , y , z + ( n 1 ) d ; ω ) .
U n ( r ; ω ) = m = 1 M A m ( r ; ω ) exp [ i k m ( n 1 ) ] + w n ( r ; ω ) ,
W = r n = 1 N | w n | 2 r n = 1 N | U n | 2 ,
W A m = W A m * = 0 .
C H · C · A ˜ ( r ; ω ) = C H · U ˜ ( r ; ω ) ,
W A ( { k m } ) = W A = A ˜ = 1 r U ˜ H ( r ; ω ) · C · A ˜ ( r ; ω ) r U ˜ H ( r ; ω ) · U ˜ ( r ; ω ) .
ω m ( k ) j = 0 j m Q m ( j ) k j .
W min ( ω ) ω = min { Q } W A [ ω ; { k m ( ω , Q ) } ] ω ,
ω 1 ( k ) = Q 1 ( 0 ) + Q 1 ( 1 ) k + Q 1 ( 2 ) k 2 + Q 1 ( 3 ) k 3 .
ω 7 ( k ) = Q 7 ( 0 ) + Q 7 ( 1 ) k ,

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