Abstract

We demonstrate that the spatial profiles of both propagating and evanescent Bloch modes in a periodic structure can be extracted from a single measurement of an electric field at the specified optical wavelength. We develop a systematic extraction procedure by extending the concepts of high-resolution spectral methods previously developed for temporal data series to take into account the symmetry properties of Bloch modes simultaneously at all spatial locations. We illustrate the application of our method to a photonic crystal waveguide interface and confirm its robustness in the presence of noise.

© 2009 Optical Society of America

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  1. R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (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, 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, Phys. Rev. Lett. 94, 073903 (2005).
    [CrossRef] [PubMed]
  4. N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, 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, Opt. Lett. 33, 2644 (2008).
    [CrossRef] [PubMed]
  6. S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, Appl. Phys. Lett. 75, 3461 (1999).
    [CrossRef]
  7. B. I. Popa and S. A. Cummer, Phys. Rev. B 72, 165102 (2005).
    [CrossRef]
  8. A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, in Theoretical and Computational Nanophotonics, D.Chigrin, ed., AIP Conference Proceedings (AIP, New York, 2009), pp. 37-39.
  9. B. Dastmalchi, A. Mohtashami, K. Hingerl, and J. Zarbakhsh, Opt. Lett. 32, 2915 (2007).
    [CrossRef] [PubMed]
  10. A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Yu. S. Kivshar, Opt. Express 17, 3716 (2009).
    [CrossRef] [PubMed]
  11. R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, Phys. Rep. 205, 109 (1991).
    [CrossRef]
  12. V. A. Mandelshtam, Prog. Nucl. Magn. Reson. Spectrosc. 38, 159 (2001).
    [CrossRef]
  13. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  14. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
    [CrossRef]
  15. K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
    [CrossRef]

2009 (1)

2008 (2)

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, Phys. Rev. B 78, 125301 (2008).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, Opt. Lett. 33, 2644 (2008).
[CrossRef] [PubMed]

2007 (2)

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (2007).
[CrossRef]

B. Dastmalchi, A. Mohtashami, K. Hingerl, and J. Zarbakhsh, Opt. Lett. 32, 2915 (2007).
[CrossRef] [PubMed]

2006 (1)

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

2005 (3)

B. I. Popa and S. A. Cummer, 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, 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, Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

2004 (1)

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

2001 (1)

V. A. Mandelshtam, Prog. Nucl. Magn. Reson. Spectrosc. 38, 159 (2001).
[CrossRef]

1999 (1)

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, Appl. Phys. Lett. 75, 3461 (1999).
[CrossRef]

1991 (1)

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, Phys. Rep. 205, 109 (1991).
[CrossRef]

Andryieuski, A.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, in Theoretical and Computational Nanophotonics, D.Chigrin, ed., AIP Conference Proceedings (AIP, New York, 2009), pp. 37-39.

Appelbaum, I.

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, Appl. Phys. Lett. 75, 3461 (1999).
[CrossRef]

Asakawa, K.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (2007).
[CrossRef]

Asatryan, A. A.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[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, 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, Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Botten, L. C.

T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, Opt. Lett. 33, 2644 (2008).
[CrossRef] [PubMed]

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Byrne, M. A.

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

Cummer, S. A.

B. I. Popa and S. A. Cummer, Phys. Rev. B 72, 165102 (2005).
[CrossRef]

Dastmalchi, B.

de Sterke, C. M.

T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, Opt. Lett. 33, 2644 (2008).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Dossou, K.

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

Dossou, K. B.

Engelen, R. J. P.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (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, 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, Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Fan, S. H.

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, Appl. Phys. Lett. 75, 3461 (1999).
[CrossRef]

Gersen, H.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (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, 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, Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Gray, S. K.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, Phys. Rep. 205, 109 (1991).
[CrossRef]

Ha, S.

Hingerl, K.

Houdre, R.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, Phys. Rev. B 78, 125301 (2008).
[CrossRef]

Ikeda, N.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (2007).
[CrossRef]

Joannopoulos, J. D.

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, Appl. Phys. Lett. 75, 3461 (1999).
[CrossRef]

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

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, 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, 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, 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, Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Kotlyar, M. V.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, Phys. Rev. B 78, 125301 (2008).
[CrossRef]

Krauss, T. F.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, Phys. Rev. B 78, 125301 (2008).
[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, 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, Phys. Rev. Lett. 94, 123901 (2005).
[CrossRef] [PubMed]

Kuipers, L.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (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, 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, Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Langtry, T. N.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Lavrinenko, A. V.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, in Theoretical and Computational Nanophotonics, D.Chigrin, ed., AIP Conference Proceedings (AIP, New York, 2009), pp. 37-39.

Le Thomas, N.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, Phys. Rev. B 78, 125301 (2008).
[CrossRef]

Malureanu, R.

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, in Theoretical and Computational Nanophotonics, D.Chigrin, ed., AIP Conference Proceedings (AIP, New York, 2009), pp. 37-39.

Mandelshtam, V. A.

V. A. Mandelshtam, Prog. Nucl. Magn. Reson. Spectrosc. 38, 159 (2001).
[CrossRef]

McPhedran, R. C.

T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, Opt. Lett. 33, 2644 (2008).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Meade, R. D.

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

Mohtashami, A.

Noid, D. W.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, Phys. Rep. 205, 109 (1991).
[CrossRef]

Pfeffer, G. A.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, Phys. Rep. 205, 109 (1991).
[CrossRef]

Popa, B. I.

B. I. Popa and S. A. Cummer, 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, Phys. Rep. 205, 109 (1991).
[CrossRef]

Shadrivov, I. V.

Sugimoto, Y.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (2007).
[CrossRef]

Sukhorukov, A. A.

Sumpter, B. G.

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, Phys. Rep. 205, 109 (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, 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, Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

White, T. P.

T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, Opt. Lett. 33, 2644 (2008).
[CrossRef] [PubMed]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Winn, J. N.

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

Zabelin, V.

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, Phys. Rev. B 78, 125301 (2008).
[CrossRef]

Zarbakhsh, J.

Appl. Phys. Lett. (1)

S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, Appl. Phys. Lett. 75, 3461 (1999).
[CrossRef]

J. Comput. Phys. (1)

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

Nat. Phys. (1)

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, Nat. Phys. 3, 401 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rep. (1)

R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, Phys. Rep. 205, 109 (1991).
[CrossRef]

Phys. Rev. B (2)

N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, Phys. Rev. B 78, 125301 (2008).
[CrossRef]

B. I. Popa and S. A. Cummer, Phys. Rev. B 72, 165102 (2005).
[CrossRef]

Phys. Rev. E (1)

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Phys. Rev. Lett. (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, 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, Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Prog. Nucl. Magn. Reson. Spectrosc. (1)

V. A. Mandelshtam, Prog. Nucl. Magn. Reson. Spectrosc. 38, 159 (2001).
[CrossRef]

Other (2)

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

A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, in Theoretical and Computational Nanophotonics, D.Chigrin, ed., AIP Conference Proceedings (AIP, New York, 2009), pp. 37-39.

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of a two-dimensional photonic crystal waveguide. The Bloch-wave extraction is performed in the PC2 section. (b) Numerically calculated amplitude (left) and phase (right) of a complex electric field profile at the normalized frequency d λ 0.2662 .

Fig. 2
Fig. 2

Complex wavenumbers k m extracted with spatial spectral analysis. Propagating and evanescent modes are marked by circles and triangles, respectively. Solid curves show exact dispersion obtained with scattering matrix approach [14, 15]. Real and imaginary parts of each mode are indicated above. Horizontal lines at d λ 0.2662 mark the frequency at the inflection point.

Fig. 3
Fig. 3

Extracted magnitudes and phases of mode profiles A m ( x , z ; ω ) at the slow-light frequency based on the original electric field data shown in Fig. 1b.

Equations (2)

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

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 ) = m = 1 M A m ( r ; ω ) exp [ i k m ( n 1 ) ] + w n ( r ; ω ) ,

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