Self-focusing is a non-linear optical process induced by the change in refractive index of materials exposed to intense electromagnetic radiation. [Cumberbatch, E. "Self-focusing in Non-linear optics", "J. Inst. Maths Applics" 6, 250 (1970)] Mourou, G. A. et al. "Optics in the relativistic regime", "Rev. Mod. Phys." 78, 309 (2006)] A medium whose refractive index increases with the electric field intensity acts as a focusing lens for an electromagnetic wave characterised by an initial transverse intensity gradient, as in a laser beam. The peak intensity of the self-focused region keeps increasing as the wave travels through the medium, until defocusing effects or medium damage interrupt this process.

Self-focusing is often observed when radiation generated by femtosecond lasers propagates through many solids, liquids and gases. Depending on the type of material and on the intensity of the radiation, several mechanisms produce variations in the refractive index which result in self-focusing: the main cases are Kerr-induced self-focusing and plasma self-focusing.

Kerr-induced self-focusing

Kerr-induced self-focusing was first predicted in the 1960s [Askar'yan, G. A. "Effects of the Gradient of Strong Electromagnetic Beam on Electrons and Atoms", "Soviet Phys. JETP" 15, 1088 (1962)] Chiao, R. Y. et al. "Self-trapping of optical beams", "Phys. Rev. Lett." 13, 479 (1964)] [Kelley, P. L. "Self-focusing of optical beams", "Phys. Rev. Lett." 15, 1005 (1965)] and experimentally verified by studying the interaction of ruby lasers with glasses and liquids. [Lallemand, P. and Bloembergen, N. "Self-focusing of laser beams and stimulated Raman gain in liquids", "Phys. Rev. Lett." 15, 1010 (1965)] [Garmire, E. et al. "Dynamics and characteristics of the self-trapping of intense light beams", "Phys. Rev. Lett." 16, 347 (1966)] 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 n as described by the formula n = n_0 + n_2 I, where "n"0 and "n"2 are the linear and non-linear components of the refractive index, and "I" is the intensity of the radiation. Since "n"2 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. [Gaeta, A. L. "Catastrophic Collapse of Ultrashort Pulses", "Phys. Rev. Lett." 84, 3582 (2000)] Self-focusing beams have been found to naturally evolve into a Townes profile regardless of their initial shape. [Moll, K. D. et al "Self-Similar Optical Wave Collapse: Observation of the Townes Profile", "Phys. Rev. Lett." 90, 203902-1 (2003)]

Self-focusing occurs if the radiation power is greater than the critical powerFibich, G. and Gaeta, A. L. "Critical power for self-focusing in bulk media and in hollow waveguides", "Opt. Lett." 25, 335 (2000)]

:P_{cr}= alpha frac{lambda^2}{4 pi n_0 n_2},

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, [Nibbering, E.T.J. et al. "Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses", "J. Opt. Soc. Am. B" 14, 650 (1997)] 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. For silica, n0 ≈ 1.453, n2 ≈ 2.4×10-20 m2/W, [Garcia, H. et al. "New approach to the measurement of the nonlinear refractive index of short (<25 m) lengths of silica and erbium-doped fibers", "Opt. Lett." 28 1796 (2003)] and the critical power is Pcr ≈ 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, [Kasparian, J. et al. "White-light filaments for atmospheric analysis", "Science" 301, 61 (2003)] [Couairon, A. and Mysyrowicz, A. "Femtosecond filamentation in transparent media", "Phys. Rep." 441, 47 (2007)] self-compression of ultrashort laser pulses, [Stibenz, G. et al. "Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament", "Opt. Lett." 31, 274 (2006)] parametric generation, [Cerullo, G. and Silverstri S. "Ultrafast optical parametric amplifiers", "Rev. Sci. Instrum." 74, 1 (2003)] and many areas of laser-matter interaction in general.

Plasma self-focusing

Advances in laser technology have recently enabled the observation of self-focusing in the interaction of intense laser pulses with plasmas. [Borisov, A. B. et al. "Observation of relativistic and charge-displacement self-channeling of intense subpicosecond ultraviolet (248 nm) radiation in plasmas", "Phys. Rev. Lett." 68, 2309 (1992)] [Monot, P. et al. "Experimental demonstration of relativistic self-channeling of a multiterawatt laser pulse in an underdense plasma", "Phys. Rev. Lett." 74, 2953 (1995)] Self-focusing in plasma can occur through thermal, relativistic and ponderomotive effects. [Mori, W. B. et al. "Evolution of self-focusing of intense electromagnetic waves in plasma", "Phys. Rev. Lett." 60, 1298 (1988)] 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. [Perkins, F. W. and Valeo, E. J. "Thermal self-focusing of electromagnetic waves in plasmas", "Phys. Rev. Lett." 32, 1234 (1974)] 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:n_{rel} = sqrt{1 - frac{omega_p^2}{omega^2,where ω is the radiation angular frequency and ωp the relativistically corrected plasma frequency omega_p= sqrt{frac{n e^{2{gamma mepsilon_0 . [Max, C. E. et al., "Self-modulation and self-focusing of electromagnetic waves in plasmas", "Phys. Rev. Lett." 33, 209 (1974)] [Pukhov, A. "Strong field interaction of laser radiation", "Rep. Prog. Phys." 66, 47 (2003)] Ponderomotive self-focusing is caused by the ponderomotive force, which pushes electrons away from the region where the laser beam is more intense, therefore increasing the refractive index and inducing a focusing effect. [Kaw, P. et al. "Filamentation and trapping of electromagnetic radiation in plasmas", "Phys. Fluids" 16, 1522 (1973)]

The evaluation of the contribution and interplay of these processes is a complex task, [Faure, J. et al. "Effects of pulse duration on self-focusing of ultra-short lasers in underdense plasmas", "Phys. Plasmas" 9, 756 (2002)] but a reference threshold for plasma self-focusing is the relativistic critical power [Sun, G. et al. "Self-focusing of short intense pulses in plasmas", "Phys. Fluids" 30, 526 (1987)]

:P_{cr}= frac{m_e c^5 omega^2}{e^2 omega_{p}^2} simeq 17 igg(frac{omega}{omega_{pigg)^2 extrm{GW},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, [Malka et al. "Laser–plasma accelerator: status and perspectives", "Phil. Trans. R. Soc. A" 364, 601 (2006)] laser-fusion schemes [Tabak W., et al. "Review of progress in Fast Ignition", "Phys. Plasmas" 12, 057305 (2005)] and high harmonic generation. [Umstadter, D. "Relativistic laser-plasma interactions", "J. Phys. D: Appl. Phys." 36, R151 (2003)]


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