Abstract

The local dispersion relation of a photonic crystal waveguide is directly determined by phase-sensitive near-field microscopy. We readily demonstrate the propagation of Bloch waves by probing the band diagram also beyond the first Brillouin zone. Both TE and TM polarized modes were distinguished in the experimental band diagram. Only the TE polarized defect mode has a distinctive Bloch wave character. The anomalous dispersion of this defect guided mode is demonstrated by local measurements of the group velocity. The measured dispersion relation and measured group velocities are both in good agreement with theoretical calculations.

© 2005 Optical Society of America

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References

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  2. See for example, Photonic Crystals and Light Localization in the 21st Century, in NATO Science Series, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001)
  3. H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,�?? J. Appl. Phys. 79 7483�??7492 (1996)
    [CrossRef]
  4. S. G. Johnson, P. R. Villeneuve, S. Fan, J. D. Joannopoulos, �??Linear waveguides in photonic crystal slabs,�?? Phys. Rev. B 62 8212�??8222 (2000)
    [CrossRef]
  5. X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M.L. d�??Yerville, D. Cassagne, C. Jouanin, �??Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes�?? Appl. Phys. Lett. 79 2312�??2314 (2001)
    [CrossRef]
  6. S. Olivier, H. Benisty, C.J.M. Smith, M. Rattier, C. Weisbuch, T.F. Krauss, �??Transmission properties of two-dimensional photonic crystal channel waveguides,�?? Opt. Quantum Electron. 34 171�??181 (2002)
    [CrossRef]
  7. K. Inoue, N. Kawai, Y. Sugimoto, N. Ikeda, K. Asakawa, �??Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,�?? Phys. Rev. B 65 121-308 (2002)
    [CrossRef]
  8. T. Asano, K. Kiyota, D. Kumamoto, B.S. Song, S. Noda, �??Time-domain measurement of picosecond light-pulse propagation in a two-dimensional photonic crystal-slab waveguide,�?? Appl. Phys. Lett. 84 4690�??4692 (2004)
    [CrossRef]
  9. D. Mori, T. Baba, �??Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,�?? Appl. Phys. Lett. 85 1101�??1103 (2004)
    [CrossRef]
  10. A.Y. Petrov, M. Eich, �??Zero dispersion at small group velocities in photonic crystal waveguides�?? Appl. Phys. Lett. 85 4866�??4868 (2004)
    [CrossRef]
  11. P.St.J. Russell, �??Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,�?? J. Mod. Opt. 38 1599�??1619 (1991)
    [CrossRef]
  12. M. Lon�?ar, D. Nedeljkovi�?, T.P. Pearsall, J. Vu¡ckovi�?, A. Scherer, S. Kuchinsky, D.C. Allan, �??Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,�?? Appl. Phys. Lett. 80 1689�??1691 (2002)
    [CrossRef]
  13. P. E. Barclay, K. Srinivasan, M. Borselli, O. Painter, �??Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers,�?? Appl. Phys. Lett. 85 4�??6 (2004)
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  18. M.L.M. Balistreri, H. Gersen, J.P. Korterik, L. Kuipers, N.F. van Hulst, �??Tracking femtosecond laser pulses in space and time,�?? Science 294 1080�??1082 (2001)
    [CrossRef] [PubMed]
  19. H. Gersen, J.P. Korterik, N.F. van Hulst, L. Kuipers, �??Tracking ultrashort pulses through dispersive media: Experiment and theory�?? Phys. Rev. E 68 026604 (2003)
    [CrossRef]
  20. H. Gersen, T.J. Karle, R.J.P. Engelen, W. Bogaerts, J.P. Korterik, N.F. van Hulst, T.F. Krauss, L. Kuipers, �??Real space observation of ultraslow light in photonic crystal waveguides,�?? Phys. Rev. Lett. 94 073903 (2005)
    [CrossRef] [PubMed]
  21. M.L.M. Balistreri, A. Driessen, J.P. Korterik, L. Kuipers, N.F. van Hulst, �??Quasi interference of perpendicularly polarized guided modes observed with a photon scanning tunneling microscope,�?? Opt. Lett. 25 637�??639 (2000)
    [CrossRef]
  22. G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, Calif., 2001)

Adv. Mater. (1)

L. Vogelaar, W. Nijdam, H.A.G.M. van Wolferen, R.M. de Ridder, F.B. Segerink, E. Fl¨uck, L. Kuipers, N.F. van Hulst, �??Large Area Photonic Crystal Slabs for Visible Light with Waveguiding Defect Structures: Fabricated with Focussed Ion Beam Assisted Laser Interference Lithography�?? Adv. Mater. 13 1551�??1554 (2001)
[CrossRef]

Appl. Phys. Lett. (6)

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M.L. d�??Yerville, D. Cassagne, C. Jouanin, �??Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes�?? Appl. Phys. Lett. 79 2312�??2314 (2001)
[CrossRef]

M. Lon�?ar, D. Nedeljkovi�?, T.P. Pearsall, J. Vu¡ckovi�?, A. Scherer, S. Kuchinsky, D.C. Allan, �??Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides,�?? Appl. Phys. Lett. 80 1689�??1691 (2002)
[CrossRef]

P. E. Barclay, K. Srinivasan, M. Borselli, O. Painter, �??Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers,�?? Appl. Phys. Lett. 85 4�??6 (2004)
[CrossRef]

T. Asano, K. Kiyota, D. Kumamoto, B.S. Song, S. Noda, �??Time-domain measurement of picosecond light-pulse propagation in a two-dimensional photonic crystal-slab waveguide,�?? Appl. Phys. Lett. 84 4690�??4692 (2004)
[CrossRef]

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

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

J. Appl. Phys. (1)

H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,�?? J. Appl. Phys. 79 7483�??7492 (1996)
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

P.St.J. Russell, �??Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,�?? J. Mod. Opt. 38 1599�??1619 (1991)
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

S. Olivier, H. Benisty, C.J.M. Smith, M. Rattier, C. Weisbuch, T.F. Krauss, �??Transmission properties of two-dimensional photonic crystal channel waveguides,�?? Opt. Quantum Electron. 34 171�??181 (2002)
[CrossRef]

Phys. Rev. B (1)

S. G. Johnson, P. R. Villeneuve, S. Fan, J. D. Joannopoulos, �??Linear waveguides in photonic crystal slabs,�?? Phys. Rev. B 62 8212�??8222 (2000)
[CrossRef]

Phys. Rev. B (2)

K. Inoue, N. Kawai, Y. Sugimoto, N. Ikeda, K. Asakawa, �??Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,�?? Phys. Rev. B 65 121-308 (2002)
[CrossRef]

S.I. Bozhevolnyi, V.S. Volkov, T. Sondergaard, A. Boltasseva, P.I. Borel, M. Kristensen, �??Near-field imaging of light propagation in photonic crystal waveguides: Explicit role of Bloch harmonics,�?? Phys. Rev. B 66 235204 (2002)
[CrossRef]

Phys. Rev. E (1)

H. Gersen, J.P. Korterik, N.F. van Hulst, L. Kuipers, �??Tracking ultrashort pulses through dispersive media: Experiment and theory�?? Phys. Rev. E 68 026604 (2003)
[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, 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, L. Kuipers, �??Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,�?? Phys. Rev. Lett. 94 123901 (2005)
[CrossRef] [PubMed]

Science (1)

M.L.M. Balistreri, H. Gersen, J.P. Korterik, L. Kuipers, N.F. van Hulst, �??Tracking femtosecond laser pulses in space and time,�?? Science 294 1080�??1082 (2001)
[CrossRef] [PubMed]

Other (3)

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

See for example, Photonic Crystals and Light Localization in the 21st Century, in NATO Science Series, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001)

G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, Calif., 2001)

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Figures (4)

Fig. 1.
Fig. 1.

Calculation of the dispersion relation of the W1 waveguide by 3D FDTD simulation for both TE (a) and TM polarization (b). In both figures, the solid dots represent the simulation results. (a) For TE polarization, the continuous curves represent the suggested dispersive modes. In the TE crystal bandgap, between ω=0.425 and 0.510 (575–690nm), two modes are allowed to propagate. They are denoted even and odd by their in-plane symmetry. The shaded region depicts the crystal modes. (b) For TM polarization, the band diagram is dominated by crystal modes. The index guided waveguide mode and dominant crystal slab mode are drawn in red (solid and dashed lines, respectively). These bands are identified by mode solving and correspond to the modes with the lowest temporal decay in the FDTD simulation.

Fig. 2.
Fig. 2.

Image sizes: 41 µm×4 µm (a) SEM image of the W1 waveguide under investigation. (b,c) Result of a near-field measurement on a PhCW with λ=674 nm. (b) Measured distribution of Acosϕ. (c) Normalized distribution of the amplitude of the evanescent optical field.

Fig. 3.
Fig. 3.

Measured photonic band diagram of the W1 PhCW. Spatial Fourier Transforms (SFTs) of the measured A cosϕ of the optical field along the waveguide direction. The SFTs are represented by a linear false-color scale at their corresponding optical frequencies. Peaks represent the resonant wavevectors in the waveguide. The calculated dispersion relations of the TE modes are plotted in blue. The even and odd waveguide modes (TE polarization) are represented by the dashed blue lines. The index guided TM modes (waveguide and crystal) are both indicated by dashed green lines. Finally the light line (ω=ckx ) is drawn in solid black.

Fig. 4.
Fig. 4.

Group velocity determined by measurement of the propagation distance through the waveguide as a function of pulse delay. The squares and solid circles represent measurements at wavelengths 612 nm and 675 nm respectively. The straight lines are linear fits through the data. The group velocity found at λ=612 nm is c/(2.0±0.6). For the red-shifted pulse at 675 nm, the group velocity is c/(3.4±0.4).

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