## Average light velocities in periodic media |

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 *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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- It is sometimes argued that phase fronts cannot be unambiguously defined because a Bloch mode is the result of multiple plane waves (some of them counterpropagating). Equation (10) clearly shows that this implication is incorrect.
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- Note that, due to the periodicity of uω(x) in Eq. (2), replacing k with k+m(2π/Γ) will result in a new function uω(x)e−im(2π/Γ)x with a fully periodic phase function φ(ω,x).
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