# Kerr Lens

Definition: a lensing effect arising from the Kerr nonlinearity

German: Kerr-Linse

Categories: nonlinear optics, physical foundations

How to cite the article; suggest additional literature

Author: Dr. Rüdiger Paschotta

When a short optical pulse propagates through a nonlinear medium, the Kerr effect leads to a phase delay which is largest on the beam axis (where the optical intensity is highest) and smaller outside the axis.
This is similar to the action of a lens: the wavefronts are deformed, so that the pulse is focused (assuming a positive nonlinear index *n*_{2}).
This effect is called *self-focusing* and has important implications for passive mode locking of lasers (→ *Kerr lens mode locking*) and for optical damage of media (*catastrophic self-focusing*).
For negative *n*_{2}, the nonlinearity is *self-defocusing*.

When a Gaussian beam with optical power *P* and beam radius *w* propagates through a thin piece (thickness *d*) of a nonlinear medium with nonlinear index *n*_{2}, the dioptric power (inverse focal length) of the Kerr lens is

when considering only the phase changes near the beam axis in a parabolic approximation. This equation can be derived by calculating the radially dependent nonlinear phase change and comparing it with that of a lens.

The equation shows that for a given optical power Kerr lensing becomes more important for stronger beam focusing: this increases the optical intensities and even more so the intensity gradients.

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### Bibliography

[1] | P. A. Belanger and C. Pare, “Self-focusing of Gaussian beams: an alternate derivation”, Appl. Opt. 22 (9), 1293 (1983), doi:10.1364/AO.22.001293 |

[2] | F. Salin et al., “Modelocking of Ti:sapphire lasers and self-focusing: a Gaussian approximation”, Opt. Lett. 16 (21), 1674 (1991), doi:10.1364/OL.16.001674 |

[3] | V. Magni et al., “Astigmatism in Gaussian-beam self-focusing and in resonators for Kerr-lens mode locking”, J. Opt. Soc. Am. B 12 (3), 476 (1995), doi:10.1364/JOSAB.12.000476 |

[4] | J. H. Marburger, “Self-focusing: theory”, in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, Oxford, 1977), Vol. 4, pp. 35-110 (1977) |

[5] | Y. R. Shen, “Self-focusing: experimental”, in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, Oxford, 1977), Vol. 4, pp. 1-34 (1977) |

See also: dioptric power, focal length, Kerr effect, lenses, self-focusing, laser-induced damage, Kerr lens mode locking, self-phase modulation

and other articles in the categories nonlinear optics, physical foundations

2020-06-30

Regarding the formula above there is a difference by a factor of 2 between the printed book and the online Encyclopedia. Which formula is correct?

Answer from the author:

The formula here is correct. Sorry for the mistake in the printed book.