# Factor 2 in the Equation for Cross-Phase Modulation

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

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

Author: Dr. Rüdiger Paschotta, RP Photonics AG, RP Photonics AG

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.

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

Cross-phase modulation is often described with the equation

$$\Delta {n^{(2)}} = 2\;{n_2}\;{I^{(1)}}$$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 <$\chi^{(3)}$> tensor. Locally, this polarization is proportional to <$\chi^{(3)} (E_1 \exp(+i \omega_1) + E_1 \exp(-i \omega_1) + E_2 \exp(+i \omega_2) + E_2 \exp(-i \omega_2))^3$>. This nonlinear polarization is responsible for a variety of effects. We have to expand the above equation, which leads to 4^{3} = 64 terms. Now consider the two above-mentioned nonlinear effects:

- We first look out for the terms describing self-phase modulation for beam 1. This must be related to components with a time dependence <$\exp(\pm i \omega_1)$>, which can only result from such combinations of factors with <$\pm \omega_1$>, where both signs occur: one sign twice, the other one just once. There are 6 terms of that kind.
- Next we look at terms for cross-phase modulation, where beam 1 influences beam 2. Such terms need to contain <$\pm\omega_2$>, but also two terms with <$\pm \omega_1$> which cancel each other via different signs. There are 12 of those – twice as many as for self-phase modulation. And this is the reason for the ominous factor of 2: the nonlinear polarization gets twice as large, compared with a situation where beam 2 influences itself.

We see that although the nonlinear index <$n_2$> 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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