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Definition: a nonlinear interaction of light in a medium with an instantaneous response, related to the nonlinear electronic polarization
The Kerr effect is a nonlinear optical effect occurring when intense light propagates in crystals and glasses, but also in other media such as gases. Its physical origin is a nonlinear polarization generated in the medium, which itself modifies the propagation properties of the light. The Kerr effect is the effect of an instantaneously occurring nonlinear response, which can be described as modifying the refractive index. In particular, the refractive index for the high intensity light beam itself is modified according to
with the nonlinear index n2 and the optical intensity I. The n2 value of a medium can be measured e.g. with the z-scan technique. Note that in addition to the Kerr effect (a purely electronic nonlinearity), electrostriction can significantly contribute to the value of the nonlinear index [3, 4]. The electric field of light causes density variations (acoustic waves) which themselves influence the refractive index via the photoelastic effect. That mechanism, however, occurs on a much longer time scale and is thus relevant only for relatively slow power modulations, but not for ultrashort pulses.
Fused silica, as used e.g. for silica fibers, has a nonlinear index of ≈ 3 × 10−16 cm2/W. For soft glasses and particularly for semiconductors, it can be much higher, because it depends strongly on the bandgap energy. The nonlinearity is also often negative for photon energies above roughly 70% of the bandgap energy (self-defocusing nonlinearity).
The time- and frequency-dependent refractive index change leads to self-phase modulation and Kerr lensing, for different overlapping light beams also to cross-phase modulation. Note that the effective refractive index increase caused by some intense beam for other beams is twice as large as that according to the equation shown above, assuming that both beams are in the same polarization state.
The description of the Kerr effect via an intensity-dependent refractive index is actually based on a certain approximation, valid for light with a small optical bandwidth. For very short and broadband pulses, a deviation from this simple behavior can be observed, which is called self-steepening. It reduces the velocity with which the peak of the pulse propagates (i.e. it reduces the group velocity) and thus leads to an increasing slope of the trailing part of the pulse. This effect is relevant e.g. for supercontinuum generation. Furthermore, the strength of the Kerr effect is known to saturate at very high optical intensities.
At extremely high optical intensities, there may not be a further increase of refractive index in proportion to the intensity, but a saturation and even substantial decrease of refractive index . This can be understood as an effect of multiphoton ionization, leading to induced losses, which are related to additional phase changes via Kramers–Kronig relations [7, 8].
A nonlinear polarization with delayed (non-instantaneous) response cannot be simply described as a modification of the refractive index. Its effect is called Raman scattering, and is not considered to be part of the Kerr effect.
|||R. H. Stolen and A. Ashkin, “Optical Kerr effect in glass waveguide”, Appl. Phys. Lett. 22, 294 (1973)|
|||M. Sheik-Bahae et al., “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption”, Phys. Rev. Lett. 65 (1), 96 (1990)|
|||E. L. Buckland and R. W. Boyd, “Electrostrictive contribution to the intensity-dependent refractive index of optical fibers”, Opt. Lett. 21 (15), 1117 (1996)|
|||E. L. Buckland and R. W. Boyd, “Measurement of the frequency response of the electrostrictive nonlinearity in optical fibers”, Opt. Lett. 22 (10), 676 (1997)|
|||V. Loriot et al., “Measurement of high order Kerr refractive index of major air components”, Opt. Express 17 (16), 13429 (2009)|
|||D. N. Schimpf et al., “Circular versus linear polarization in laser-amplifiers with Kerr-nonlinearity”, Opt. Express 17 (21), 18774 (2009)|
|||C. Brée, A. Demircan and G. Steinmeyer, “Saturation of the all-optical Kerr effect”, Phys. Rev. Lett. 106 (18), 183902 (2011)|
|||B. Borchers et al., “Saturation of the all-optical Kerr effect in solids”, Opt. Lett. 37 (9), 1541 (2012)|
|||G. P. Agrawal, Nonlinear Fiber Optics, 4th edn., Academic Press, New York (2007)|