Self-focusing - Kerr-induced Self-focusing

Kerr-induced Self-focusing

Kerr-induced self-focusing was first predicted in the 1960s and experimentally verified by studying the interaction of ruby lasers with glasses and liquids. Its origin lies in the optical Kerr effect, a non-linear process which arises in media exposed to intense electromagnetic radiation, and which produces a variation of the refractive index as described by the formula, where n₀ and n₂ are the linear and non-linear components of the refractive index, and I is the intensity of the radiation. Since n₂ is positive in most materials, the refractive index becomes larger in the areas where the intensity is higher, usually at the centre of a beam, creating a focusing density profile which potentially leads to the collapse of a beam on itself. Self-focusing beams have been found to naturally evolve into a Townes profile regardless of their initial shape.

Self-focusing occurs if the radiation power is greater than the critical power

,

where λ is the radiation wavelength in vacuum and α is a constant which depends on the initial spatial distribution of the beam. Although there is no general analytical expression for α, its value has been derived numerically for many beam profiles. The lower limit is α ≈ 1.86225, which corresponds to Townes beams, whereas for a Gaussian beam α ≈ 1.8962. For air, n₀ ≈ 1, n₂ ≈ 4×10-23 m2/W for λ = 800 nm, and the critical power is P_cr ≈ 2.4 GW, corresponding to an energy of about 0.3 mJ for a pulse duration of 100 fs. For silica, n₀ ≈ 1.453, n₂ ≈ 2.4×10-20 m2/W, and the critical power is P_cr ≈ 1.6 MW.

Kerr induced self-focusing is crucial for many applications in laser physics, both as a key ingredient and as a limiting factor. For example, the technique of chirped pulse amplification was developed to overcome the nonlinearities and damage of optical components that self-focusing would produce in the amplification of femtosecond laser pulses. On the other hand, self-focusing is a major mechanism behind Kerr-lens modelocking, laser filamentation in transparent media, self-compression of ultrashort laser pulses, parametric generation, and many areas of laser-matter interaction in general.