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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Grover Swartzlander
  • Vol. 30, Iss. 11 — Nov. 1, 2013
  • pp: 2849–2854

Average light velocities in periodic media

Peter Kaspar, Roman Kappeler, Daniel Erni, and Heinz Jäckel  »View Author Affiliations


JOSA B, Vol. 30, Issue 11, pp. 2849-2854 (2013)
http://dx.doi.org/10.1364/JOSAB.30.002849


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Abstract

Electromagnetic Bloch modes are used to describe the field distribution of light in periodic media that cannot be adequately approximated by effective macroscopic media. These modes explicitly take into account the spatial modulation of the medium and therefore contain the full physical information at any specific location in the medium. For instance, the propagation velocity of light can be determined locally, and it is not an invariant of space, as it is often implicitly assumed when definitions such as that of the group velocity vgr=dω/dk are used (where ω is the angular frequency and k is the Bloch index of a monochromatic mode). Spatially invariant light velocities can only be expected if the medium is assumed to show an effective behavior similar to a homogeneous material (where a plane-wave ansatz would be more appropriate). This inevitably leads to the question: what exactly is dω/dk of a Bloch mode, if it is not the group velocity? The answer is the average group velocity. This is not a trivial observation, and it has to be taken into account, for instance, when the enhancement of nonlinear effects induced by slow light is estimated. The example of a Kerr nonlinearity is studied, and we show formally that using the average group velocity can lead to an underestimation of the effect. Furthermore, this article critically reviews the concepts of energy and phase velocity. In particular, the different interpretations of phase velocity that exist in the literature are unified using a generic definition of the quantity.

© 2013 Optical Society of America

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(260.0260) Physical optics : Physical optics
(350.5030) Other areas of optics : Phase
(350.5500) Other areas of optics : Propagation
(160.5298) Materials : Photonic crystals

ToC Category:
Materials

History
Original Manuscript: February 25, 2013
Revised Manuscript: September 9, 2013
Manuscript Accepted: September 9, 2013
Published: October 10, 2013

Citation
Peter Kaspar, Roman Kappeler, Daniel Erni, and Heinz Jäckel, "Average light velocities in periodic media," J. Opt. Soc. Am. B 30, 2849-2854 (2013)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-11-2849


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References

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  5. A simple way of illustrating the difference in nature between wave vector and Bloch index is to consider the analogy to solid-state physics. The momentum of a given electronic Bloch state |ψ〉 is not simply given by ℏ multiplied by the Bloch index of the mode (as is the case for the plane-wave momentum of a free electron), but by 〈ψ|p^|ψ〉, which contains a weighted sum over all plane-wave components [6].
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