Self-phase modulation

[2] Self-phase modulation has also been reported for nonlinear sound waves propagating in biological thin films, where the phase modulation results from varying elastic properties of the lipid films.

[3] The evolution along distance z of the equivalent lowpass electric field A(z) obeys the nonlinear Schrödinger equation which, in absence of dispersion, is:[4] with j the imaginary unit and γ the nonlinear coefficient of the medium.

The cubic nonlinear term on the right hand side is called Kerr effect, and is multiplied by -j according to the engineer's notation used in the definition of Fourier transform.

, it is: such that: The phase φ at coordinate z therefore is: Such a relation highlights that SPM is induced by the power of the electric field.

is called effective length [4] and is defined by: Hence, with attenuation the SPM does not grow indefinitely along distance in a homogeneous medium, but eventually saturates to: In presence of dispersion the Kerr effect manifests as a phase shift only over short distances, depending on the amount of dispersion.

In regions of anomalous dispersion, the opposite is true, and the pulse is compressed temporally and becomes shorter.

This effect can be exploited to some degree (until it digs holes into the spectrum) to produce ultrashort pulse compression.

If the pulse is of sufficient intensity, the spectral broadening process of SPM can balance with the temporal compression due to anomalous dispersion and reach an equilibrium state.

Self-phase modulation has stimulated many applications in the field of ultrashort pulse including to cite a few: The nonlinear properties of Kerr nonlinearity has also been beneficial for various optical pulse processing techniques such as optical regeneration[10] or wavelength conversion.

[11] In long-haul single-channel and DWDM (dense wavelength-division multiplexing) systems, SPM is one of the most important reach-limiting nonlinear effects.