Factor 2 in the Equation for Cross-Phase Modulation

Posted on 2008-02-12. Permanent link: http://www.rp-photonics.com/spotlight_2008_02_12.html

Author: Dr. Rüdiger Paschotta, RP Photonics Consulting GmbH

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

Cross-phase modulation is often described with the equation

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 first look out for the terms describing self-phase modulation for beam 1. This must be related to components with a time dependence exp(±i ω₁), which can only result from such combinations of factors with ±ω₁, 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 ±ω₂, but also two terms with ±ω₁ 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 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.