Abstract

We formulate and demonstrate experimentally the high-resolution spectral method based on Bloch-wave symmetry properties for extracting mode dispersion in periodic waveguides from measurements of near-field profiles. We characterize both the propagating and evanescent modes, and also determine the amplitudes of forward and backward waves in different waveguide configurations, with the estimated accuracy of several percent or less. Whereas the commonly employed spatial Fourier-transform (SFT) analysis provides the wavenumber resolution which is limited by the inverse length of the waveguide, we achieve precise dispersion extraction even for compact photonic structures.

© 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, "Ultrafast evolution of photonic eigenstates in k-space," Nature Phys. 3, 401-405 (2007).
    [CrossRef]
  2. H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
    [CrossRef] [PubMed]
  3. H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (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, 125301-8 (2008).
    [CrossRef]
  5. 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]
  6. V. A. Mandelshtam, "FDM: the filter diagonalization method for data processing in NMR experiments," Prog. Nucl. Magn. Reson. Spectrosc. 38, 159-196 (2001).
    [CrossRef]
  7. 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]
  8. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  9. S. L. Marple, Digital Spectral Analysis with Applications (Prentice-Hall, Englewood Cliffs, NJ, 1987).
  10. S. H. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, "Guided and defect modes in periodic dielectric wave-guides," J. Opt. Soc. Am. B 12, 1267-1272 (1995).
    [CrossRef]
  11. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173.
    [CrossRef] [PubMed]
  12. D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, "Multistability in nonlinear left-handed transmission lines," Appl. Phys. Lett. 92, 264104-3 (2008).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh ed. (Cambridge University Press, UK, 2002).
    [PubMed]

2008 (2)

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, 125301-8 (2008).
[CrossRef]

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, "Multistability in nonlinear left-handed transmission lines," Appl. Phys. Lett. 92, 264104-3 (2008).
[CrossRef]

2007 (2)

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

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]

2005 (2)

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (2005).
[CrossRef] [PubMed]

2001 (2)

1995 (1)

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]

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," Nature Phys. 3, 401-405 (2007).
[CrossRef]

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (2005).
[CrossRef] [PubMed]

Chen, J. C.

Dastmalchi, B.

Devenyi, A.

Engelen, R. J. P.

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

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
[CrossRef] [PubMed]

Fan, S. 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," Nature Phys. 3, 401-405 (2007).
[CrossRef]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
[CrossRef] [PubMed]

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]

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, 125301-8 (2008).
[CrossRef]

Hulst, N. F.

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (2005).
[CrossRef] [PubMed]

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," Nature Phys. 3, 401-405 (2007).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Karle, T. J.

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
[CrossRef] [PubMed]

Kivshar, Yu. S.

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, "Multistability in nonlinear left-handed transmission lines," Appl. Phys. Lett. 92, 264104-3 (2008).
[CrossRef]

Korterik, J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (2005).
[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, 125301-8 (2008).
[CrossRef]

Krauss, T. F.

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, 125301-8 (2008).
[CrossRef]

Kuipers, L.

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

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, 125301-8 (2008).
[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]

Meade, R. D.

Mohtashami, A.

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]

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]

Powell, D. A.

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, "Multistability in nonlinear left-handed transmission lines," Appl. Phys. Lett. 92, 264104-3 (2008).
[CrossRef]

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]

Shadrivov, I. V.

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, "Multistability in nonlinear left-handed transmission lines," Appl. Phys. Lett. 92, 264104-3 (2008).
[CrossRef]

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," Nature Phys. 3, 401-405 (2007).
[CrossRef]

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]

Winn, J. N.

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, 125301-8 (2008).
[CrossRef]

Zarbakhsh, J.

Appl. Phys. Lett. (1)

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, "Multistability in nonlinear left-handed transmission lines," Appl. Phys. Lett. 92, 264104-3 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature 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," Nature Phys. 3, 401-405 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

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 (1)

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, 125301-8 (2008).
[CrossRef]

Phys. Rev. Lett. (2)

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (2005).
[CrossRef] [PubMed]

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

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

Other (3)

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

S. L. Marple, Digital Spectral Analysis with Applications (Prentice-Hall, Englewood Cliffs, NJ, 1987).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh ed. (Cambridge University Press, UK, 2002).
[PubMed]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Sketch of the periodic waveguide made of dielectric pillars. (b) Dispersion retrieved from experimental measurements, shown are both the real and imaginary parts as indicated by labels. Grey shading marks the confidence interval. (c) Retrieved amplitudes of experimentally excited forward (A+) and backward (A-) waves. (d) Numerically calculated dispersion. (e) Normalized SFT of experimentally measured electric field profile at rod centers along the waveguide.

Fig. 2.
Fig. 2.

(a) Photograph of transmission line. (b) Dispersion retrieved from experimental measurements (dashed lines) and calculated numerically (solid lines). (c) Retrieved amplitudes of experimentally excited modes with positive (A+) and negative (A-) wavenumbers. (d) SFT of experimentally measured electric field profile along the line.

Fig. 3.
Fig. 3.

(a) Sketch of perspex dielectric slab waveguide. (b) Dispersion retrieved from experimental measurements (dashed lines) and calculated numerically (solid lines). Shaded are the regions above the free-space light line, and below dielectric light line, where guided modes do not exist. (c) SFT of experimentally measured electric field profile along the waveguide.

Equations (1)

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U n = m = 1 M A m exp ( i k m n ) + w n .

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