Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum. Self-phase modulation is an important effect in optical systems that use short, intense pulses of light, such as lasers and optical fiber communications systems. 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.
Theory with Kerr nonlinearity The evolution along distance z of the equivalent lowpass electric field A(z) obeys the nonlinear Schrödinger equation which, in absence of dispersion, is:
d A ( z ) d z = − j γ | A ( z ) | 2 A ( z ) {\displaystyle {\frac {dA(z)}{dz}}=-j\gamma \left|A(z)\right|^{2}A(z)}
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. The power of the electric field is invariant along z, since:
d | A | 2 d z = d A d z A ∗ + A d A ∗ d z = 0 {\displaystyle {\frac {d|A|^{2}}{dz}}={\frac {dA}{dz}}A^{*}+A{\frac {dA^{*}}{dz}}=0}
with * denoting conjugation. Since the power is invariant, the Kerr effect can manifest only as a phase rotation. In polar coordinates, with A = | A | e j φ {\displaystyle A=|A|e^{j\varphi }} , it is:
d | A | e j φ d z = d | A | d z ⏟ = 0 e j φ + j | A | e j φ d φ d z = − j γ | A ( z ) | 3 e j φ {\displaystyle {\frac {d|A|e^{j\varphi }}{dz}}=\underbrace {\frac {d|A|}{dz}} _{=0}e^{j\varphi }+j|A|e^{j\varphi }{\frac {d\varphi }{dz}}=-j\gamma \left|A(z)\right|^{3}e^{j\varphi }}
such that:
d φ d z = − γ | A | 2 . {\displaystyle {\frac {d\varphi }{dz}}=-\gamma |A|^{2}.}
The phase φ at coordinate z therefore is:
φ ( z ) = φ ( 0 ) − γ | A ( 0 ) | 2 z ⏟ S P M . {\displaystyle \varphi (z)=\varphi (0)-\underbrace {\gamma \left|A(0)\right|^{2}z} _{\mathrm {SPM} }.}
Such a relation highlights that SPM is induced by the power of the electric field. In presence of attenuation α the propagation equation is:
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