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# Cross-phase Modulation

Acronym: XPM

Definition: a nonlinear effect where the optical intensity of one beam influences the phase change of another beam

German: Kreuzphasenmodulation

Cross-phase modulation is the change in the optical phase of a light beam caused by the interaction with another beam in a nonlinear medium, specifically a Kerr medium. This can be described as a change in the refractive index:

where n2 is the nonlinear index. Here, the intensity I(1) of beam 1 causes a refractive index change for beam 2.

Compared with the corresponding equation for self-phase modulation, there is an additional factor of 2. This factor 2 is valid for beams with the same polarization; for cross-polarized beams in isotropic media (e.g. glasses), it must be replaced with 2/3.

A more fundamental description of cross-phase modulation effects refers to the nonlinear polarization caused in the medium, based on the χ(3) nonlinearity. For example, the above-mentioned factor 2 can be understood on that basis.

## Effects of Cross-phase Modulation

Cross-phase modulation can be relevant under different circumstances:

• It leads to an interaction of laser pulses in a medium, which allows e.g. the measurement of the optical intensity of one pulse by monitoring a phase change of the other one (without absorbing any photons of the first beam). This is basis of a scheme for quantum nondemolition (QND) measurements.
• The effect can also be used for synchronizing two mode-locked lasers using the same gain medium, in which the pulses overlap and experience cross-phase modulation.
• In optical fiber communications, cross-phase modulation in fibers can lead to problems with channel cross-talk.
• Cross-phase modulation is also sometimes mentioned as a mechanism for channel translation (wavelength conversion), but in this context the term typically refers to a kind of cross-phase modulation which is not based on the Kerr effect, but rather on changes in the refractive index via the carrier density in a semiconductor optical amplifier.

### Bibliography

 [1] M. N. Islam et al., “Cross-phase modulation in optical fibers”, Opt. Lett. 12 (8), 625 (1987) [2] M. Shtaif, “Analytical description of cross-phase modulation in dispersive optical fibers”, Opt. Lett. 23 (15), 1191 (1998) [3] A. Fellegara and S. Wabnitz, “Electrostrictive cross-phase modulation of periodic pulse trains in optical fibers”, Opt. Lett. 23 (17), 1357 (1998) [4] M. Margalit et al., “Cross phase modulation squeezing in optical fibers”, Opt. Express 2 (3), 72 (1998) [5] N. Matsuda et al., “Observation of optical-fibre Kerr nonlinearity at the single-photon levels”, Nature Photon. 3, 95 (2009) [6] G. P. Agrawal, Nonlinear Fiber Optics, 4th edn., Academic Press, New York (2007) [7] R. Paschotta, tutorial on "Passive Fiber Optics", Part 11: Nonlinearities of Fibers