self-focusing

Kerr-induced self-focusing was first predicted in the 1960s{{cite journal |last1=Askar'yan |first1=G. A. |title=Cerenkov Radiation and Transition Radiation from Electromagnetic Waves |journal=Journal of Experimental and Theoretical Physics |volume=15 |issue=5 |pages=943–6 |year=1962 |url=http://www.jetp.ac.ru/cgi-bin/e/index/e/15/5/p943?a=list }}{{cite journal |doi=10.1103/PhysRevLett.13.479 |title=Self-Trapping of Optical Beams |journal=Physical Review Letters |volume=13 |issue=15 |pages=479 |year=1964 |last1=Chiao |first1=R. Y. |last2=Garmire |first2=E. |last3=Townes |first3=C. H. |bibcode=1964PhRvL..13..479C }}{{cite journal |doi=10.1103/PhysRevLett.15.1005 |title=Self-Focusing of Optical Beams |journal=Physical Review Letters |volume=15 |issue=26 |pages=1005–1008 |year=1965 |last1=Kelley |first1=P. L. |bibcode=1965PhRvL..15.1005K }} and experimentally verified by studying the interaction of ruby lasers with glasses and liquids.{{cite journal |doi=10.1103/PhysRevLett.15.1010 |title=Self-Focusing of Laser Beams and Stimulated Raman Gain in Liquids |journal=Physical Review Letters |volume=15 |issue=26 |pages=1010 |year=1965 |last1=Lallemand |first1=P. |last2=Bloembergen |first2=N. |bibcode=1965PhRvL..15.1010L }}{{cite journal |doi=10.1103/PhysRevLett.16.347 |title=Dynamics and Characteristics of the Self-Trapping of Intense Light Beams |journal=Physical Review Letters |volume=16 |issue=9 |pages=347 |year=1966 |last1=Garmire |first1=E. |last2=Chiao |first2=R. Y. |last3=Townes |first3=C. H. |bibcode=1966PhRvL..16..347G |hdl=2060/19660014476 |hdl-access=free }} 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 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.{{cite journal |doi=10.1103/PhysRevLett.84.3582 |pmid=11019151 |title=Catastrophic Collapse of Ultrashort Pulses |journal=Physical Review Letters |volume=84 |issue=16 |pages=3582–5 |year=2000 |last1=Gaeta |first1=Alexander L. |bibcode=2000PhRvL..84.3582G }}{{cite journal |doi=10.1088/1054-660X/23/10/105401 |title=Describing the propagation of intense laser pulses in nonlinear Kerr media using the ducting model |journal=Laser Physics |volume=23 |issue=10 |pages=105401 |year=2013 |last1=Rashidian Vaziri |first1=M R |bibcode=2013LaPhy..23j5401R |s2cid=250912159 }} Self-focusing beams have been found to naturally evolve into a Townes profile regardless of their initial shape.{{cite journal |doi=10.1103/PhysRevLett.90.203902 |pmid=12785895 |title=Self-Similar Optical Wave Collapse: Observation of the Townes Profile |journal=Physical Review Letters |volume=90 |issue=20 |pages=203902 |year=2003 |last1=Moll |first1=K. D. |last2=Gaeta |first2=Alexander L. |last3=Fibich |first3=Gadi |bibcode=2003PhRvL..90t3902M }}

Self-focusing beyond a threshold of power can lead to laser collapse and damage to the medium, which occurs if the radiation power is greater than the critical power{{cite journal |doi=10.1364/OL.25.000335 |pmid=18059872 |title=Critical power for self-focusing in bulk media and in hollow waveguides |journal=Optics Letters |volume=25 |issue=5 |pages=335–7 |year=2000 |last1=Fibich |first1=Gadi |last2=Gaeta |first2=Alexander L. |bibcode=2000OptL...25..335F }}

:$P\_\{\backslash text\{cr\}\}=\; \backslash alpha\; \backslash frac\{\backslash lambda^2\}\{4\; \backslash pi\; n\_0\; n\_2\}$,

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,{{cite journal |doi=10.1364/JOSAB.14.000650 |title=Determination of the inertial contribution to the nonlinear refractive index of air, N₂, and O₂ by use of unfocused high-intensity femtosecond laser pulses |journal=Journal of the Optical Society of America B |volume=14 |issue=3 |pages=650–60 |year=1997 |last1=Nibbering |first1=E. T. J. |last2=Grillon |first2=G. |last3=Franco |first3=M. A. |last4=Prade |first4=B. S. |last5=Mysyrowicz |first5=A. |bibcode=1997JOSAB..14..650N }} 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,{{cite journal |doi=10.1364/OL.28.001796 |pmid=14514104 |title=New approach to the measurement of the nonlinear refractive index of short (< 25 m) lengths of silica and erbium-doped fibers |journal=Optics Letters |volume=28 |issue=19 |pages=1796–8 |year=2003 |last1=Garcia |first1=Hernando |last2=Johnson |first2=Anthony M. |last3=Oguama |first3=Ferdinand A. |last4=Trivedi |first4=Sudhir |bibcode=2003OptL...28.1796G }}

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,{{cite journal |doi=10.1126/science.1085020 |pmid=12843384 |title=White-Light Filaments for Atmospheric Analysis |journal=Science |volume=301 |issue=5629 |pages=61–4 |year=2003 |last1=Kasparian |first1=J. |last2=Rodriguez |first2=M. |last3=Méjean |first3=G. |last4=Yu |first4=J. |last5=Salmon |first5=E. |last6=Wille |first6=H. |last7=Bourayou |first7=R. |last8=Frey |first8=S. |last9=André |first9=Y.-B. |last10=Mysyrowicz |first10=A. |last11=Sauerbrey |first11=R. |last12=Wolf |first12=J.-P. |last13=Wöste |first13=L. |bibcode=2003Sci...301...61K |citeseerx=10.1.1.1028.4581 |s2cid=37270331 }}{{cite journal |doi=10.1016/j.physrep.2006.12.005 |title=Femtosecond filamentation in transparent media |journal=Physics Reports |volume=441 |issue=2–4 |pages=47–189 |year=2007 |last1=Couairon |first1=A |last2=Mysyrowicz |first2=A |bibcode=2007PhR...441...47C }} self-compression of ultrashort laser pulses,{{cite journal |doi=10.1364/OL.31.000274 |pmid=16441054 |title=Self-compression of millijoule pulses to 78 fs duration in a white-light filament |journal=Optics Letters |volume=31 |issue=2 |pages=274–6 |year=2006 |last1=Stibenz |first1=Gero |last2=Zhavoronkov |first2=Nickolai |last3=Steinmeyer |first3=Günter |bibcode=2006OptL...31..274S |s2cid=12957688 }} parametric generation,{{cite journal |doi=10.1063/1.1523642 |title=Ultrafast optical parametric amplifiers |journal=Review of Scientific Instruments |volume=74 |issue=1 |pages=1 |year=2003 |last1=Cerullo |first1=Giulio |last2=De Silvestri |first2=Sandro |bibcode=2003RScI...74....1C }} and many areas of laser-matter interaction in general.

Kelley predicted that homogeneously broadened two-level atoms may focus or defocus light when carrier frequency $\backslash omega$ is detuned downward or upward the center of gain line $\backslash omega\_0$. Laser pulse propagation with slowly varying envelope $E(\backslash vec\; \backslash mathbf\{r\},t)$ is governed in gain medium by the nonlinear Schrödinger-Frantz-Nodvik equation.{{cite journal |doi=10.1070/QE1988v018n02ABEH011482 |title=Compensation of self-focusing distortions in quasiresonant amplification of a light pulse |journal=Soviet Journal of Quantum Electronics |volume=18 |issue=2 |pages=233–7 |year=1988 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N |bibcode=1988QuEle..18..233O }}

When $\backslash omega$ is detuned downward or upward from $\backslash omega\_0$ 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:

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

=+ i k n_2 |E(\vec \mathbf{r},t)|^2 {{E}(\vec \mathbf{r},t)}+

$\backslash frac\; \{\backslash sigma\; N\; (\backslash vec\; \backslash mathbf\{r\},t)\}\{2\}\; [1\; +\; i\; (\; \backslash omega\_0\; -\; \backslash omega\; )T\_2]\; \{\{E\}(\backslash vec\; \backslash mathbf\{r\},t)\}\; ,\; \backslash nabla\_\{\backslash bot\}^2=\; \{\backslash frac\; \{\backslash partial^2\}\{\{\backslash partial\; x\; \}^2\}\}+\{\backslash frac\; \{\backslash partial^2\}\{\{\backslash partial\; y\; \}^2\}\},$

$\{\backslash frac\; \{\backslash partial\; \{\{N\}(\backslash vec\; \backslash mathbf\{r\},t)\}\}\; \{\backslash partial\; t\}\; \}\; =\; -\{\backslash frac\; \{\{\{N\_0\}(\backslash vec\; \backslash mathbf\{r\})\}\}\; \{T\_1\}\; \}-\; \backslash sigma\; (\backslash omega)\; N\; (\backslash vec\; \backslash mathbf\{r\},t)\; |E(\backslash vec\; \backslash mathbf\{r\},t)|^2\; ,$

where $\backslash sigma\; (\backslash omega)=\; \backslash frac\; \{\backslash sigma\_0\}\{1+T\_2^2\; (\; \backslash omega\_0\; -\; \backslash omega\; )^2\}$ is the stimulated emission cross section, $\{N\_0\}(\backslash vec\; \backslash mathbf\{r\})$ is the population inversion density before pulse arrival, $T\_1$ and $T\_2$ are longitudinal and transverse lifetimes of two-level medium and $z$ is the propagation axis.

The laser beam with a smooth spatial profile $\{E\}(\backslash vec\; \backslash mathbf\{r\},t)$ 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.{{cite journal |title=Filamentary Structure of Light Beams in Nonlinear Liquids |journal=JETP Letters |volume=3 |issue=12 |pages=307–310 |year=1966 |last1= Bespalov |first1=VI |last2=Talanov |first2=VI|

url=http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml }}

In a framework of nonlinear Schrödinger equation :

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

=+ i k n_2 |E(\vec \mathbf{r},t)|^2 {{E}(\vec \mathbf{r},t)}.

The rate of the perturbation growth or instability increment

$h$ is linked with filament size $\backslash kappa^\{-1\}$ via simple equation:

$h^2=\backslash kappa^2(n\_2\; |E(\backslash vec\; \backslash mathbf\{r\},t)|^2-\backslash kappa^2/4k^2)$. Generalization of this link between Bespalov-Talanov increments and filament size in gain medium as a function of linear gain $$

{\sigma N (\vec \mathbf{r},t)} and detuning $\backslash delta\; \backslash omega=\backslash omega\_0\; -\; \backslash omega$

had been realized in

.

Advances in laser technology have recently enabled the observation of self-focusing in the interaction of intense laser pulses with plasmas.{{cite journal |doi=10.1103/PhysRevLett.68.2309 |pmid=10045362 |title=Observation of relativistic and charge-displacement self-channeling of intense subpicosecond ultraviolet (248 nm) radiation in plasmas |journal=Physical Review Letters |volume=68 |issue=15 |pages=2309–2312 |year=1992 |last1=Borisov |first1=A. B. |last2=Borovskiy |first2=A. V. |last3=Korobkin |first3=V. V. |last4=Prokhorov |first4=A. M. |last5=Shiryaev |first5=O. B. |last6=Shi |first6=X. M. |last7=Luk |first7=T. S. |last8=McPherson |first8=A. |last9=Solem |first9=J. C. |last10=Boyer |first10=K. |last11=Rhodes |first11=C. K. |bibcode=1992PhRvL..68.2309B }}{{cite journal |doi=10.1103/PhysRevLett.74.2953 |pmid=10058066 |title=Experimental Demonstration of Relativistic Self-Channeling of a Multiterawatt Laser Pulse in an Underdense Plasma |journal=Physical Review Letters |volume=74 |issue=15 |pages=2953–2956 |year=1995 |last1=Monot |first1=P. |last2=Auguste |first2=T. |last3=Gibbon |first3=P. |last4=Jakober |first4=F. |last5=Mainfray |first5=G. |last6=Dulieu |first6=A. |last7=Louis-Jacquet |first7=M. |last8=Malka |first8=G. |last9=Miquel |first9=J. L. |bibcode=1995PhRvL..74.2953M }} Self-focusing in plasma can occur through thermal, relativistic and ponderomotive effects.{{cite journal |doi=10.1103/PhysRevLett.60.1298 |pmid=10037999 |title=Evolution of self-focusing of intense electromagnetic waves in plasma |journal=Physical Review Letters |volume=60 |issue=13 |pages=1298–1301 |year=1988 |last1=Mori |first1=W. B. |last2=Joshi |first2=C. |last3=Dawson |first3=J. M. |last4=Forslund |first4=D. W. |last5=Kindel |first5=J. M. |bibcode=1988PhRvL..60.1298M |url=https://zenodo.org/record/1233864 }} 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.{{cite journal |doi=10.1103/PhysRevLett.32.1234 |title=Thermal Self-Focusing of Electromagnetic Waves in Plasmas |journal=Physical Review Letters |volume=32 |issue=22 |pages=1234 |year=1974 |last1=Perkins |first1=F. W. |last2=Valeo |first2=E. J. |bibcode=1974PhRvL..32.1234P }}

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

:$n\_\{rel\}\; =\; \backslash sqrt\{1\; -\; \backslash frac\{\backslash omega\_p^2\}\{\backslash omega^2\}\}$,

where ω is the radiation angular frequency and ω_p the relativistically corrected plasma frequency $\backslash omega\_p=\; \backslash sqrt\{\backslash frac\{n\; e^\{2\}\}\{\backslash gamma\; m\backslash epsilon\_0\}\}$.{{cite journal |doi=10.1103/PhysRevLett.33.209 |title=Self-Modulation and Self-Focusing of Electromagnetic Waves in Plasmas |journal=Physical Review Letters |volume=33 |issue=4 |pages=209 |year=1974 |last1=Max |first1=Claire Ellen |last2=Arons |first2=Jonathan |last3=Langdon |first3=A. Bruce |bibcode=1974PhRvL..33..209M }}{{cite journal |doi=10.1088/0034-4885/66/1/202 |title=Strong field interaction of laser radiation |journal=Reports on Progress in Physics |volume=66 |issue=1 |pages=47–101 |year=2003 |last1=Pukhov |first1=Alexander |bibcode=2003RPPh...66...47P |s2cid=250909633 }}

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.{{cite journal |doi=10.1063/1.1694552 |title=Filamentation and trapping of electromagnetic radiation in plasmas |journal=Physics of Fluids |volume=16 |issue=9 |pages=1522 |year=1973 |last1=Kaw |first1=P. |last2=Schmidt |first2=G. |last3=Wilcox |first3=T. |bibcode=1973PhFl...16.1522K }}{{cite journal |doi=10.1088/0022-3727/12/8/005 |title=Evidence of filamentation (self-focusing) of a laser beam propagating in a laser-produced aluminium plasma |journal=Journal of Physics D: Applied Physics |volume=12 |issue=8 |pages=1261–73 |year=1979 |last1=Pizzo |first1=V Del |last2=Luther-Davies |first2=B |bibcode=1979JPhD...12.1261D |s2cid=250749005 }}{{cite journal |doi=10.1007/BF00934416 |title=Self-focussing of a laser beam in a multiply ionized, absorbing plasma |journal=Applied Physics |volume=18 |issue=2 |pages=199–204 |year=1979 |last1=Del Pizzo |first1=V. |last2=Luther-Davies |first2=B. |last3=Siegrist |first3=M. R. |bibcode=1979ApPhy..18..199D |s2cid=122912958 }}

The evaluation of the contribution and interplay of these processes is a complex task,{{cite journal |doi=10.1063/1.1447556 |title=Effects of pulse duration on self-focusing of ultra-short lasers in underdense plasmas |journal=Physics of Plasmas |volume=9 |issue=3 |pages=756 |year=2002 |last1=Faure |first1=J. |last2=Malka |first2=V. |last3=Marquès |first3=J.-R. |last4=David |first4=P.-G. |last5=Amiranoff |first5=F. |last6=Ta Phuoc |first6=K. |last7=Rousse |first7=A. |bibcode=2002PhPl....9..756F }} but a reference threshold for plasma self-focusing is the relativistic critical power{{cite journal |doi=10.1063/1.866349 |title=Self-focusing of short intense pulses in plasmas |journal=Physics of Fluids |volume=30 |issue=2 |pages=526 |year=1987 |last1=Sun |first1=Guo-Zheng |last2=Ott |first2=Edward |last3=Lee |first3=Y. C. |last4=Guzdar |first4=Parvez |bibcode=1987PhFl...30..526S }}

:$P\_\{cr\}=\; \backslash frac\{m\_e^2\; c^5\; \backslash omega^2\}\{e^2\; \backslash omega\_\{p\}^2\}\; \backslash simeq\; 17\; \backslash bigg(\backslash frac\{\backslash omega\}\{\backslash omega\_\{p\}\}\backslash bigg)^2\backslash \; \backslash textrm\{GW\}$,

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,{{cite journal |doi=10.1098/rsta.2005.1725 |pmid=16483951 |title=Laser-plasma accelerator: Status and perspectives |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=364 |issue=1840 |pages=601–10 |year=2006 |last1=Malka |first1=V |last2=Faure |first2=J |last3=Glinec |first3=Y |last4=Lifschitz |first4=A.F |bibcode=2006RSPTA.364..601M |s2cid=12223379 }} laser-fusion schemes{{cite journal |doi=10.1063/1.1871246 |title=Review of progress in Fast Ignition |journal=Physics of Plasmas |volume=12 |issue=5 |pages=057305 |year=2005 |last1=Tabak |first1=M. |last2=Clark |first2=D. S. |last3=Hatchett |first3=S. P. |last4=Key |first4=M. H. |last5=Lasinski |first5=B. F. |author5-link=Barbara Lasinski|last6=Snavely |first6=R. A. |last7=Wilks |first7=S. C. |last8=Town |first8=R. P. J. |last9=Stephens |first9=R. |last10=Campbell |first10=E. M. |last11=Kodama |first11=R. |last12=Mima |first12=K. |last13=Tanaka |first13=K. A. |last14=Atzeni |first14=S. |last15=Freeman |first15=R. |bibcode=2005PhPl...12e7305T |hdl=11094/3277 |url=https://ir.library.osaka-u.ac.jp/repo/ouka/all/3277/PP12-5-057305.pdf |hdl-access=free }} and high harmonic generation.{{cite journal |doi=10.1088/0022-3727/36/8/202 |title=Relativistic laser plasma interactions |journal=Journal of Physics D: Applied Physics |volume=36 |issue=8 |pages=R151–65 |year=2003 |last1=Umstadter |first1=Donald |url=https://deepblue.lib.umich.edu/bitstream/2027.42/48918/2/d308r2.pdf |hdl=2027.42/48918 |s2cid=10185064 |hdl-access=free }}

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.{{cite journal |doi=10.1063/1.4904098 |title=Accumulated self-focusing of ultraviolet light in silica glass |journal=Applied Physics Letters |volume=105 |issue=24 |pages=244110 |year=2014 |last1=Khrapko |first1=Rostislav |last2=Lai |first2=Changyi |last3=Casey |first3=Julie |last4=Wood |first4=William A. |last5=Borrelli |first5=Nicholas F. |bibcode=2014ApPhL.105x4110K |doi-access=free }} 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.{{Cite journal|last=Biria|first=Saeid|title=Coupling nonlinear optical waves to photoreactive and phase-separating soft matter: Current status and perspectives|journal=Chaos|volume=27|issue=10|pages=104611|doi=10.1063/1.5001821|pmid=29092420|year=2017|bibcode=2017Chaos..27j4611B }} Self-focusing was observed in photo-polymer systems with microscale laser beams of either UV{{cite journal |doi=10.1364/ol.21.000024 |pmid=19865292 |title=Self-focusing and self-trapping of optical beams upon photopolymerization |journal=Optics Letters |volume=21 |issue=1 |pages=24–6 |year=1996 |last1=Kewitsch |first1=Anthony S. |last2=Yariv |first2=Amnon |bibcode=1996OptL...21...24K |url=https://authors.library.caltech.edu/2845/1/KEWol96.pdf }} or visible light.{{cite journal |doi=10.1109/JLT.2005.850783 |title=Fabrication of light-induced self-written waveguides with a W-shaped refractive index profile |journal=Journal of Lightwave Technology |volume=23 |issue=8 |pages=2542–8 |year=2005 |last1=Yamashita |first1=T. |last2=Kagami |first2=M. |bibcode=2005JLwT...23.2542Y |s2cid=36961681 }} The self-trapping of incoherent light was also later observed.{{cite journal |doi=10.1021/acs.jpcc.5b11377 |title=Tunable Nonlinear Optical Pattern Formation and Microstructure in Cross-Linking Acrylate Systems during Free-Radical Polymerization |journal=The Journal of Physical Chemistry C |volume=120 |issue=8 |pages=4517–28 |year=2016 |last1=Biria |first1=Saeid |last2=Malley |first2=Philip P. A. |last3=Kahan |first3=Tara F. |last4=Hosein |first4=Ian D. }} 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.{{cite journal |doi=10.1021/ja068967b |pmid=17378567 |title=Spontaneous Pattern Formation Due to Modulation Instability of Incoherent White Light in a Photopolymerizable Medium |journal=Journal of the American Chemical Society |volume=129 |issue=15 |pages=4738–46 |year=2007 |last1=Burgess |first1=Ian B. |last2=Shimmell |first2=Whitney E. |last3=Saravanamuttu |first3=Kalaichelvi |bibcode=2007JAChS.129.4738B }}{{cite journal |doi=10.1021/acs.jpcc.5b07117 |title=Spontaneous Emergence of Nonlinear Light Waves and Self-Inscribed Waveguide Microstructure during the Cationic Polymerization of Epoxides |journal=The Journal of Physical Chemistry C |volume=119 |issue=35 |pages=20606 |year=2015 |last1=Basker |first1=Dinesh K. |last2=Brook |first2=Michael A. |last3=Saravanamuttu |first3=Kalaichelvi }}{{cite journal |doi=10.1021/acsmacrolett.6b00659 |title=Optical Autocatalysis Establishes Novel Spatial Dynamics in Phase Separation of Polymer Blends during Photocuring |journal=ACS Macro Letters |volume=5 |issue=11 |pages=1237–41 |year=2016 |last1=Biria |first1=Saeid |last2=Malley |first2=Phillip P. A. |last3=Kahan |first3=Tara F. |last4=Hosein |first4=Ian D. |pmid=35614732 }}{{Cite journal|last1=Biria|first1=Saeid|last2=Hosein|first2=Ian D.|date=2017-05-09|title=Control of Morphology in Polymer Blends through Light Self-Trapping: An in Situ Study of Structure Evolution, Reaction Kinetics, and Phase Separation|journal=Macromolecules|volume=50|issue=9|pages=3617–3626|doi=10.1021/acs.macromol.7b00484|issn=0024-9297|bibcode=2017MaMol..50.3617B}} 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, and the fact that refractive index in polymers is proportional to molecular weight and crosslinking degree{{cite journal |doi=10.1016/0032-3950(90)90361-9 |title=Influence of crosslinking density on the properties of polymer networks |journal=Polymer Science U.S.S.R |volume=32 |issue=10 |pages=2061–9 |year=1990 |last1=Askadskii |first1=A.A }} which increases over the duration of photo-polymerization.

• Filament propagation

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• {{cite book |editor1-first=Richard A. |editor1-last=Carrigan |editor2-first=James A. |editor2-last=Ellison |doi=10.1007/978-1-4757-6394-2 |title=Relativistic Channeling |volume=165 |series=NATO ASI Series |year=1987 |isbn=978-1-4419-3207-5 }}

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Category:Nonlinear optics

Category:Plasma phenomena

Category:Laser science