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Factor 2 in the Equation for Cross-Phase Modulation

Posted on 2008-02-12 as a part of the Photonics Spotlight (available as e-mail newsletter!)

Permanent link: https://www.rp-photonics.com/spotlight_2008_02_12.html

Author: , RP Photonics Consulting GmbH

Abstract: It can be surprising that the refractive index change caused by cross-phase modulation is twice as large as that for self-phase modulation. The article explains the reason on the basis of the induced nonlinear polarization.

Dr. Rüdiger Paschotta

Ref.: encyclopedia articles on cross-phase modulation, self-phase modulation, and nonlinear polarization

Cross-phase modulation is often described with the equation

index change via XPM

for the change of the refractive index for a beam 2, as caused by an intense beam 1. It is assumed that both beams are linearly polarized in the same direction. The factor 2, which does not occur in the corresponding equation for self-phase modulation, should be rather surprising: how can the refractive index change for beam 2 be twice as large as that for beam 1 itself, even though the wavelengths of both beams can be very similar? Apparently it is not the wavelength making the difference, but what else?

The answer is obtained with a closer inspection of the physics behind cross-phase modulation and self-phase modulation. Both effects are caused by the nonlinear polarization in the medium as is related to the χ(3) tensor. Locally, this polarization is proportional to χ(3) (E1 exp(+i ω1) + E1 exp(-i ω1) + E2 exp(+i ω2) + E2 exp(-i ω2))3. This nonlinear polarization is responsible for a variety of effects. We have to expand the above equation, which leads to 43 = 64 terms. Now consider the two above mentioned nonlinear effects:

We see that although the nonlinear index n2 provides a convenient description of such nonlinear effects, in a way it hides the physics, and we should still be aware of the nonlinear polarization.

This article is a posting of the Photonics Spotlight, authored by Dr. Rüdiger Paschotta. You may link to this page and cite it, because its location is permanent. See also the RP Photonics Encyclopedia.

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