Self-focusing

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 ${\displaystyle n}$ as described by the formula ${\displaystyle n=n_{0}+n_{2}I}$, 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.[9][10] Self-focusing beams have been found to naturally evolve into a Townes profile[5] regardless of their initial shape.[11]

Self-focusing occurs if the radiation power is greater than the critical power[12]

${\displaystyle P_{cr}=\alpha {\frac {\lambda ^{2}}{4\pi n_{0}n_{2}}}}$,

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.[12] 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,[13] 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,[14] and the critical power is P_cr ≈ 2.8 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,[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 ${\displaystyle \omega }$ is detuned downward or upward the center of gain line ${\displaystyle \omega _{0}}$. Laser pulse propagation with slowly varying envelope ${\displaystyle E({\vec {\mathbf {r} }},t)}$ is governed in gain medium by the nonlinear Schrödinger-Frantz-Nodvik equation.[19]

When ${\displaystyle \omega }$ is detuned downward or upward from ${\displaystyle \omega _{0}}$ the refractive index is changed. "Red" detuning leads to an increased index of refraction during saturation of the resonant transition, i.e. to self-focusing, while for "blue" detuning the radiation is defocused during saturation:

${\displaystyle {\frac {\partial {{E}({\vec {\mathbf {r} }},t)}}{\partial z}}+{\frac {1}{c}}{\frac {\partial {{E}({\vec {\mathbf {r} }},t)}}{\partial t}}+{\frac {i}{2k}}\nabla _{\bot }^{2}E({\vec {\mathbf {r} }},t)=+ikn_{2}|E({\vec {\mathbf {r} }},t)|^{2}{{E}({\vec {\mathbf {r} }},t)}+}$

${\displaystyle {\frac {\sigma N({\vec {\mathbf {r} }},t)}{2}}[1+i(\omega _{0}-\omega )T_{2}]{{E}({\vec {\mathbf {r} }},t)},\nabla _{\bot }^{2}={\frac {\partial ^{2}}{{\partial x}^{2}}}+{\frac {\partial ^{2}}{{\partial y}^{2}}},}$

${\displaystyle {\frac {\partial {{N}({\vec {\mathbf {r} }},t)}}{\partial t}}=-{\frac {{N_{0}}({\vec {\mathbf {r} }})}{T_{1}}}-\sigma (\omega )N({\vec {\mathbf {r} }},t)|E({\vec {\mathbf {r} }},t)|^{2},}$

where ${\displaystyle \sigma (\omega )={\frac {\sigma _{0}}{1+T_{2}^{2}(\omega _{0}-\omega )^{2}}}}$ is the stimulated emission cross section, ${\displaystyle {N_{0}}({\vec {\mathbf {r} }})}$ is the population inversion density before pulse arrival, ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are longitudinal and transverse lifetimes of two-level medium and ${\displaystyle z}$ is the propagation axis.

The laser beam with a smooth spatial profile ${\displaystyle {E}({\vec {\mathbf {r} }},t)}$ is affected by modulational instability. The small perturbations caused by roughnesses and medium defects are amplified in propagation. This effect is referred to as Bespalov-Talanov instability.[20] In a framework of nonlinear Schrödinger equation : ${\displaystyle {\frac {\partial {{E}({\vec {\mathbf {r} }},t)}}{\partial z}}+{\frac {1}{c}}{\frac {\partial {{E}({\vec {\mathbf {r} }},t)}}{\partial t}}+{\frac {i}{2k}}\nabla _{\bot }^{2}E({\vec {\mathbf {r} }},t)=+ikn_{2}|E({\vec {\mathbf {r} }},t)|^{2}{{E}({\vec {\mathbf {r} }},t)}}$.

The rate of the perturbation growth or instability increment ${\displaystyle h}$ is linked with filament size ${\displaystyle \kappa ^{-1}}$ via simple equation: ${\displaystyle h^{2}=\kappa ^{2}(n_{2}|E({\vec {\mathbf {r} }},t)|^{2}-\kappa ^{2}/4k^{2})}$. Generalization of this link between Bespalov-Talanov increments and filament size in gain medium as a function of linear gain ${\displaystyle {\sigma N({\vec {\mathbf {r} }},t)}}$ and detuning ${\displaystyle \delta \omega =\omega _{0}-\omega }$ had been realized in .[19]

Advances in laser technology have recently enabled the observation of self-focusing in the interaction of intense laser pulses with plasmas.[21][22] Self-focusing in plasma can occur through thermal, relativistic and ponderomotive effects.[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 n_rel according to the equation

${\displaystyle n_{rel}={\sqrt {1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}}}}$,

where ω is the radiation angular frequency and ω_p the relativistically corrected plasma frequency ${\displaystyle \omega _{p}={\sqrt {\frac {ne^{2}}{\gamma m\epsilon _{0}}}}}$.[25][26]

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.[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]

${\displaystyle P_{cr}={\frac {m_{e}^{2}c^{5}\omega ^{2}}{e^{2}\omega _{p}^{2}}}\simeq 17{\bigg (}{\frac {\omega }{\omega _{p}}}{\bigg )}^{2}\ {\textrm {GW}}}$,

where m_e is the electron mass, c the speed of light, ω the radiation angular frequency, e the electron charge and ω_p 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.[38] The self-trapping of incoherent light was also later observed.[39] Self-focusing can also be observed in wide-area beams, wherein the beam undergoes filamentation, or Modulation Instability, spontaneous dividing into a multitude of microscale self-focused beams, or filaments.[40][41][39][42][43] The balance of self-focusing and natural beam divergence results in the beams propagating divergence-free. 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.

• List of plasma (physics) articles
• Filament propagation