Kerr Lens (Self-Focusing)
Thin Kerr lens of a Gaussian beam, f = πw⁴/(8n₂PL), with the on-axis nonlinear phase, the critical power for self-focusing and the Rayleigh range in the crystal.
Kerr focal length f
49.70 mm
πw⁴/(8n₂PL); lens power 20.12 m⁻¹
On-axis nonlinear phase
0.03556 rad
k₀n₂I₀L, peak intensity I₀ = 2P/(πw²) = 7.074e9 W/cm²
Critical power P_cr
1.705 MW
3.77 λ²/(8π n₀ n₂); P/P_cr = 0.05867
Rayleigh range in the crystal
6.220 mm
π n₀ w²/λ; L/z_R = 0.3215
The intensity raises the index, n = n₀ + n₂I. Near the axis a Gaussian beam's I(r) = I₀ e^(−2r²/w²) is a parabola, I₀(1 − 2r²/w²), so a crystal of length L adds the phase of a thin lens with 1/f = 4n₂I₀L/w² = 8n₂PL/(πw⁴). In a Kerr-lens mode-locked laser the pulse peak sees this lens and the low-power continuous light doesn't. An aperture or the gain spot then favours the pulse. The lens holds for L below z_R and f, and for P well below P_cr. The plot shows how the real Gaussian phase leaves the parabola beyond r ≈ w/2. The lens comes from the on-axis curvature of the Kerr phase, so it overstates how strongly the whole beam focuses: this parabolic lens alone would trap a Gaussian at λ²/(8πn₀n₂) = P_cr/3.77.
Nonlinear phase across the beam
- Kerr phase k₀n₂I(r)L
- Parabolic (thin-lens) fit
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Parabolic approximation of the Gaussian intensity near the axis, I ≈ I₀(1 − 2r²/w²), in a thin crystal: 1/f = 4n₂I₀L/w² with I₀ = 2P/(πw²). Holds for L below z_R and f and for P well below P_cr = 3.77 λ²/(8π n₀ n₂); the page warns outside that range. The on-axis lens overstates whole-beam focusing: on its own it would trap a Gaussian at λ²/(8π n₀ n₂) = P_cr/3.77. Cavity design (the stability-limit shift) isn't modelled.
- Magni V., Cerullo G., De Silvestri S. (1993). ABCD matrix analysis of propagation of gaussian beams through Kerr media. Opt. Commun. 96, 348–355. doi:10.1016/0030-4018(93)90284-C
- Fibich G., Gaeta A. L. (2000). Critical power for self-focusing in bulk media and in hollow waveguides. Opt. Lett. 25, 335–337. doi:10.1364/OL.25.000335
- Boyd R. W. (2020). Nonlinear Optics, 4th ed. Academic Press. Self-focusing and the nonlinear index n₂. doi:10.1016/C2015-0-05510-1