Self-focusing
Self-focusing is a non-linear optical process induced by the change in refractive index of materials exposed to intense electromagnetic radiation.
[1][2] A medium whose refractive index increases with the electric field intensity acts as a focusing lens for an electromagnetic wave characterized by an initial transverse intensity gradient, as in a laser beam.
Self-focusing is often observed when radiation generated by femtosecond lasers propagates through many solids, liquids and gases.
Kerr-induced self-focusing was first predicted in the 1960s[4][5][6] and experimentally verified by studying the interaction of ruby lasers with glasses and liquids.
[7][8] 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
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.
[9][10] Self-focusing beams have been found to naturally evolve into a Townes profile[5] regardless of their initial shape.
Although there is no general analytical expression for α, its value has been derived numerically for many beam profiles.
For air, n0 ≈ 1, n2 ≈ 4×10−23 m2/W for λ = 800 nm,[13] and the critical power is Pcr ≈ 2.4 GW, corresponding to an energy of about 0.3 mJ for a pulse duration of 100 fs.
Kerr-induced self-focusing is crucial for many applications in laser physics, both as a key ingredient and as a limiting factor.
On the other hand, self-focusing is a major mechanism behind Kerr-lens modelocking, laser filamentation in transparent media,[15][16] self-compression of ultrashort laser pulses,[17] parametric generation,[18] and many areas of laser-matter interaction in general.
Kelley[6] predicted that homogeneously broadened two-level atoms may focus or defocus light when carrier frequency
The small perturbations caused by roughnesses and medium defects are amplified in propagation.
[23] Thermal self-focusing is due to collisional heating of a plasma exposed to electromagnetic radiation: the rise in temperature induces a hydrodynamic expansion which leads to an increase of the index of refraction and further heating.
[24] Relativistic self-focusing is caused by the mass increase of electrons travelling at speed approaching the speed of light, which modifies the plasma refractive index nrel according to the equation where ω is the radiation angular frequency and ωp the relativistically corrected plasma frequency
[27][28][29] The evaluation of the contribution and interplay of these processes is a complex task,[30] but a reference threshold for plasma self-focusing is the relativistic critical power[2][31] where me is the electron mass, c the speed of light, ω the radiation angular frequency, e the electron charge and ωp the plasma frequency.
For an electron density of 1019 cm−3 and radiation at the wavelength of 800 nm, the critical power is about 3 TW.
Such values are realisable with modern lasers, which can exceed PW powers.
For example, a laser delivering 50 fs pulses with an energy of 1 J has a peak power of 20 TW.
Self-focusing in a plasma can balance the natural diffraction and channel a laser beam.
Such effect is beneficial for many applications, since it helps increasing the length of the interaction between laser and medium.
This is crucial, for example, in laser-driven particle acceleration,[32] laser-fusion schemes[33] and high harmonic generation.
[34] Self-focusing can be induced by a permanent refractive index change resulting from a multi-pulse exposure.
This effect has been observed in glasses which increase the refractive index during an exposure to ultraviolet laser radiation.
[35] Accumulated self-focusing develops as a wave guiding, rather than a lensing effect.
The scale of actively forming beam filaments is a function of the exposure dose.
Evolution of each beam filament towards a singularity is limited by the maximum induced refractive index change or by laser damage resistance of the glass.
Self-focusing can also been observed in a number of soft matter systems, such as solutions of polymers and particles as well as photo-polymers.
[36] Self-focusing was observed in photo-polymer systems with microscale laser beams of either UV[37] or visible light.
Self-focusing in photopolymerizable media is possible, owing to a photoreaction dependent refractive index,[37] and the fact that refractive index in polymers is proportional to molecular weight and crosslinking degree[44] which increases over the duration of photo-polymerization.
