# Self-focusing

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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 characterised by an initial transverse intensity gradient, as in a laser beam. [3] 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 of light was discovered by Gurgen Askaryan.

Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (values of atomic electric fields, typically 108 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.

In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light propagates through the material. It is defined as

In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

## Contents

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 [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]

A ruby laser is a solid-state laser that uses a synthetic ruby crystal as its gain medium. The first working laser was a ruby laser made by Theodore H. "Ted" Maiman at Hughes Research Laboratories on May 16, 1960.

In physics, intensity is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy. In the SI system, it has units watts per square metre (W/m2). It is used most frequently with waves, in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.

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

In physics, power is the rate of doing work or transferring heat, the amount of energy transferred or converted per unit time. Having no direction, it is a scalar quantity. In the International System of Units, the unit of power is the joule per second (J/s), known as the watt in honour of James Watt, the eighteenth-century developer of the condenser steam engine. Another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written:

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

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.

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse gaussian mode describes the intended output of most (but not all) lasers, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a lens, the transverse phase dependence is altered; this results in a different Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam (for a given wavelength and polarization) are determined by a single parameter: the so-called waist w0. At any position z relative to the waist (focus) along a beam having a specified w0, the field amplitudes and phases are thereby determined as detailed below.

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. For silica, n0 ≈ 1.453, n2 ≈ 2.4×10−20 m2/W, [14] and the critical power is Pcr ≈ 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.

Chirped pulse amplification (CPA) is a technique for amplifying an ultrashort laser pulse up to the petawatt level with the laser pulse being stretched out temporally and spectrally prior to amplification.

Kerr-lens modelocking (KLM) is a method of modelocking lasers via a nonlinear optical process known as the optical Kerr effect. This method allows the generation of pulses of light with a duration as short as a few femtoseconds.

In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. They are commonly referred to as ultrafast events. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.

## Self-focusing and defocusing in gain medium

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 Nonlinear Schrodinger-Frantz-Nodvik equation. [19]

When ${\displaystyle \omega }$ is detuned downward or upward the ${\displaystyle \omega _{0}}$ the refractive index is changed. Noteworthy the red detuning leads to increase of index during saturation of 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 stimulated emission cross section, ${\displaystyle {N_{0}}({\vec {\mathbf {r} }})}$ is population inversion density before pulse arrival, ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ are longitudinal and transverse lifetimes of two-level medium, ${\displaystyle z}$ is propagation axis.

## Filamentation

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 Shrodinger 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] .

## Plasma self-focusing

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 nrel according to the equation

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

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

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 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]

## Accumulated self-focusing

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 in soft matter and polymer systems

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.

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