# Kerr Lens

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a lensing effect arising from the Kerr nonlinearity

When a short optical pulse propagates through a nonlinear medium, the Kerr effect leads to an optical 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

$${f^{ - 1}} = \frac{{8{n_2}d}}{{\pi {w^4}}}P$$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.

## More to Learn

Encyclopedia articles:

### Bibliography

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

## Questions and Comments from Users

2021-02-05

Could you provide the reference from which the formula is derived?

The author's answer:

That was my own calculation, although *surely* one could also find it elsewhere in the literature.

2024-01-26

What happens if the thickness in the calculator is set too long, for instance, longer than the focal length?

The author's answer:

As is indicated in the text, the formula works only for thin pieces. Outside that regime, calculations would be substantially more difficult.

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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?

The author's answer:

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