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Light passing through a gradient-index lens is focused as in a convex lens. In self-focusing, the refractive index gradient is induced by the light itself.

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.^[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.

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 $n$ 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.^[9]^[10] Self-focusing beams have been found to naturally evolve into a Townes profile^[5] regardless of their initial shape.^[11]

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^[12]

P_{\text{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.

Self-focusing and defocusing in gain medium

Kelley^[6] predicted that homogeneously broadened two-level atoms may focus or defocus light when carrier frequency $\omega$ is detuned downward or upward the center of gain line $\omega _{0}$ . Laser pulse propagation with slowly varying envelope $E({\vec {\mathbf {r} }},t)$ is governed in gain medium by the nonlinear Schrödinger-Frantz-Nodvik equation.^[19]

When $\omega$ is detuned downward or upward from $\omega _{0}$ 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:

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

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

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

where $\sigma (\omega )={\frac {\sigma _{0}}{1+T_{2}^{2}(\omega _{0}-\omega )^{2}}}$ is the stimulated emission cross section, ${N_{0}}({\vec {\mathbf {r} }})$ 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.

Filamentation

The laser beam with a smooth spatial profile ${E}({\vec {\mathbf {r} }},t)$ 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 : ${\frac {\partial {{E}({\vec {\mathbf {r} }},t)}}{\partial z}}+{\frac {1}{c}}{\frac {\partial {{E}({\vec {\mathbf {r} }},t)}}{\partial t}}+{\frac {i}{2k}}\nabla _{\bot }^{2}E({\vec {\mathbf {r} }},t)=+ikn_{2}|E({\vec {\mathbf {r} }},t)|^{2}{{E}({\vec {\mathbf {r} }},t)}$ .

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

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 $\omega _{p}={\sqrt {\frac {ne^{2}}{\gamma m\epsilon _{0}}}}$ .^[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]

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]

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.

References

^ Cumberbatch, E. (1970). "Self-focusing in Non-linear Optics". IMA Journal of Applied Mathematics. 6 (3): 250–62. doi:10.1093/imamat/6.3.250.
^ ^a ^b Mourou, Gerard A.; Tajima, Toshiki; Bulanov, Sergei V. (2006). "Optics in the relativistic regime". Reviews of Modern Physics. 78 (2): 309. Bibcode:2006RvMP...78..309M. doi:10.1103/RevModPhys.78.309.
^ Rashidian Vaziri, M.R. (2015). "Comment on 'Nonlinear refraction measurements of materials using the moiré deflectometry'". Optics Communications. 357: 200–1. Bibcode:2015OptCo.357..200R. doi:10.1016/j.optcom.2014.09.017.
^ Askar'yan, G. A. (1962). "Cerenkov Radiation and Transition Radiation from Electromagnetic Waves". Journal of Experimental and Theoretical Physics. 15 (5): 943–6.
^ ^a ^b Chiao, R. Y.; Garmire, E.; Townes, C. H. (1964). "Self-Trapping of Optical Beams". Physical Review Letters. 13 (15): 479. Bibcode:1964PhRvL..13..479C. doi:10.1103/PhysRevLett.13.479.
^ ^a ^b Kelley, P. L. (1965). "Self-Focusing of Optical Beams". Physical Review Letters. 15 (26): 1005–1008. Bibcode:1965PhRvL..15.1005K. doi:10.1103/PhysRevLett.15.1005.
^ Lallemand, P.; Bloembergen, N. (1965). "Self-Focusing of Laser Beams and Stimulated Raman Gain in Liquids". Physical Review Letters. 15 (26): 1010. Bibcode:1965PhRvL..15.1010L. doi:10.1103/PhysRevLett.15.1010.
^ Garmire, E.; Chiao, R. Y.; Townes, C. H. (1966). "Dynamics and Characteristics of the Self-Trapping of Intense Light Beams". Physical Review Letters. 16 (9): 347. Bibcode:1966PhRvL..16..347G. doi:10.1103/PhysRevLett.16.347. hdl:2060/19660014476.
^ Gaeta, Alexander L. (2000). "Catastrophic Collapse of Ultrashort Pulses". Physical Review Letters. 84 (16): 3582–5. Bibcode:2000PhRvL..84.3582G. doi:10.1103/PhysRevLett.84.3582. PMID 11019151.
^ Rashidian Vaziri, M R (2013). "Describing the propagation of intense laser pulses in nonlinear Kerr media using the ducting model". Laser Physics. 23 (10): 105401. Bibcode:2013LaPhy..23j5401R. doi:10.1088/1054-660X/23/10/105401. S2CID 250912159.
^ Moll, K. D.; Gaeta, Alexander L.; Fibich, Gadi (2003). "Self-Similar Optical Wave Collapse: Observation of the Townes Profile". Physical Review Letters. 90 (20): 203902. Bibcode:2003PhRvL..90t3902M. doi:10.1103/PhysRevLett.90.203902. PMID 12785895.
^ ^a ^b Fibich, Gadi; Gaeta, Alexander L. (2000). "Critical power for self-focusing in bulk media and in hollow waveguides". Optics Letters. 25 (5): 335–7. Bibcode:2000OptL...25..335F. doi:10.1364/OL.25.000335. PMID 18059872.
^ Nibbering, E. T. J.; Grillon, G.; Franco, M. A.; Prade, B. S.; Mysyrowicz, A. (1997). "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 of the Optical Society of America B. 14 (3): 650–60. Bibcode:1997JOSAB..14..650N. doi:10.1364/JOSAB.14.000650.
^ Garcia, Hernando; Johnson, Anthony M.; Oguama, Ferdinand A.; Trivedi, Sudhir (2003). "New approach to the measurement of the nonlinear refractive index of short (< 25 m) lengths of silica and erbium-doped fibers". Optics Letters. 28 (19): 1796–8. Bibcode:2003OptL...28.1796G. doi:10.1364/OL.28.001796. PMID 14514104.
^ Kasparian, J.; Rodriguez, M.; Méjean, G.; Yu, J.; Salmon, E.; Wille, H.; Bourayou, R.; Frey, S.; André, Y.-B.; Mysyrowicz, A.; Sauerbrey, R.; Wolf, J.-P.; Wöste, L. (2003). "White-Light Filaments for Atmospheric Analysis". Science. 301 (5629): 61–4. Bibcode:2003Sci...301...61K. CiteSeerX 10.1.1.1028.4581. doi:10.1126/science.1085020. PMID 12843384. S2CID 37270331.
^ Couairon, A; Mysyrowicz, A (2007). "Femtosecond filamentation in transparent media". Physics Reports. 441 (2–4): 47–189. Bibcode:2007PhR...441...47C. doi:10.1016/j.physrep.2006.12.005.
^ Stibenz, Gero; Zhavoronkov, Nickolai; Steinmeyer, Günter (2006). "Self-compression of millijoule pulses to 78 fs duration in a white-light filament". Optics Letters. 31 (2): 274–6. Bibcode:2006OptL...31..274S. doi:10.1364/OL.31.000274. PMID 16441054. S2CID 12957688.
^ Cerullo, Giulio; De Silvestri, Sandro (2003). "Ultrafast optical parametric amplifiers". Review of Scientific Instruments. 74 (1): 1. Bibcode:2003RScI...74....1C. doi:10.1063/1.1523642.
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^ Mori, W. B.; Joshi, C.; Dawson, J. M.; Forslund, D. W.; Kindel, J. M. (1988). "Evolution of self-focusing of intense electromagnetic waves in plasma". Physical Review Letters. 60 (13): 1298–1301. Bibcode:1988PhRvL..60.1298M. doi:10.1103/PhysRevLett.60.1298. PMID 10037999.
^ Perkins, F. W.; Valeo, E. J. (1974). "Thermal Self-Focusing of Electromagnetic Waves in Plasmas". Physical Review Letters. 32 (22): 1234. Bibcode:1974PhRvL..32.1234P. doi:10.1103/PhysRevLett.32.1234.
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^ Faure, J.; Malka, V.; Marquès, J.-R.; David, P.-G.; Amiranoff, F.; Ta Phuoc, K.; Rousse, A. (2002). "Effects of pulse duration on self-focusing of ultra-short lasers in underdense plasmas". Physics of Plasmas. 9 (3): 756. Bibcode:2002PhPl....9..756F. doi:10.1063/1.1447556.
^ Sun, Guo-Zheng; Ott, Edward; Lee, Y. C.; Guzdar, Parvez (1987). "Self-focusing of short intense pulses in plasmas". Physics of Fluids. 30 (2): 526. Bibcode:1987PhFl...30..526S. doi:10.1063/1.866349.
^ Malka, V; Faure, J; Glinec, Y; Lifschitz, A.F (2006). "Laser-plasma accelerator: Status and perspectives". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 364 (1840): 601–10. Bibcode:2006RSPTA.364..601M. doi:10.1098/rsta.2005.1725. PMID 16483951. S2CID 12223379.
^ Tabak, M.; Clark, D. S.; Hatchett, S. P.; Key, M. H.; Lasinski, B. F.; Snavely, R. A.; Wilks, S. C.; Town, R. P. J.; Stephens, R.; Campbell, E. M.; Kodama, R.; Mima, K.; Tanaka, K. A.; Atzeni, S.; Freeman, R. (2005). "Review of progress in Fast Ignition" (PDF). Physics of Plasmas. 12 (5): 057305. Bibcode:2005PhPl...12e7305T. doi:10.1063/1.1871246. hdl:11094/3277.
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^ Khrapko, Rostislav; Lai, Changyi; Casey, Julie; Wood, William A.; Borrelli, Nicholas F. (2014). "Accumulated self-focusing of ultraviolet light in silica glass". Applied Physics Letters. 105 (24): 244110. Bibcode:2014ApPhL.105x4110K. doi:10.1063/1.4904098.
^ Biria, Saeid (2017). "Coupling nonlinear optical waves to photoreactive and phase-separating soft matter: Current status and perspectives". Chaos. 27 (10): 104611. doi:10.1063/1.5001821. PMID 29092420.
^ ^a ^b Kewitsch, Anthony S.; Yariv, Amnon (1996). "Self-focusing and self-trapping of optical beams upon photopolymerization" (PDF). Optics Letters. 21 (1): 24–6. Bibcode:1996OptL...21...24K. doi:10.1364/ol.21.000024. PMID 19865292.
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^ ^a ^b Biria, Saeid; Malley, Philip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016). "Tunable Nonlinear Optical Pattern Formation and Microstructure in Cross-Linking Acrylate Systems during Free-Radical Polymerization". The Journal of Physical Chemistry C. 120 (8): 4517–28. doi:10.1021/acs.jpcc.5b11377.
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Bibliography

Carrigan, Richard A.; Ellison, James A., eds. (1987). Relativistic Channeling. NATO ASI Series. Vol. 165. doi:10.1007/978-1-4757-6394-2. ISBN 978-1-4419-3207-5.

[1] Cumberbatch, E. (1970). "Self-focusing in Non-linear Optics". IMA Journal of Applied Mathematics. 6 (3): 250–62. doi:10.1093/imamat/6.3.250.

[Mourou2006-2] Mourou, Gerard A.; Tajima, Toshiki; Bulanov, Sergei V. (2006). "Optics in the relativistic regime". Reviews of Modern Physics. 78 (2): 309. Bibcode:2006RvMP...78..309M. doi:10.1103/RevModPhys.78.309.

[3] Rashidian Vaziri, M.R. (2015). "Comment on 'Nonlinear refraction measurements of materials using the moiré deflectometry'". Optics Communications. 357: 200–1. Bibcode:2015OptCo.357..200R. doi:10.1016/j.optcom.2014.09.017.

[4] Askar'yan, G. A. (1962). "Cerenkov Radiation and Transition Radiation from Electromagnetic Waves". Journal of Experimental and Theoretical Physics. 15 (5): 943–6.

[Chiao1964-5] Chiao, R. Y.; Garmire, E.; Townes, C. H. (1964). "Self-Trapping of Optical Beams". Physical Review Letters. 13 (15): 479. Bibcode:1964PhRvL..13..479C. doi:10.1103/PhysRevLett.13.479.

[Kelley,_P._L_1965-6] Kelley, P. L. (1965). "Self-Focusing of Optical Beams". Physical Review Letters. 15 (26): 1005–1008. Bibcode:1965PhRvL..15.1005K. doi:10.1103/PhysRevLett.15.1005.

[7] Lallemand, P.; Bloembergen, N. (1965). "Self-Focusing of Laser Beams and Stimulated Raman Gain in Liquids". Physical Review Letters. 15 (26): 1010. Bibcode:1965PhRvL..15.1010L. doi:10.1103/PhysRevLett.15.1010.

[8] Garmire, E.; Chiao, R. Y.; Townes, C. H. (1966). "Dynamics and Characteristics of the Self-Trapping of Intense Light Beams". Physical Review Letters. 16 (9): 347. Bibcode:1966PhRvL..16..347G. doi:10.1103/PhysRevLett.16.347. hdl:2060/19660014476.

[9] Gaeta, Alexander L. (2000). "Catastrophic Collapse of Ultrashort Pulses". Physical Review Letters. 84 (16): 3582–5. Bibcode:2000PhRvL..84.3582G. doi:10.1103/PhysRevLett.84.3582. PMID 11019151.

[10] Rashidian Vaziri, M R (2013). "Describing the propagation of intense laser pulses in nonlinear Kerr media using the ducting model". Laser Physics. 23 (10): 105401. Bibcode:2013LaPhy..23j5401R. doi:10.1088/1054-660X/23/10/105401. S2CID 250912159.

[11] Moll, K. D.; Gaeta, Alexander L.; Fibich, Gadi (2003). "Self-Similar Optical Wave Collapse: Observation of the Townes Profile". Physical Review Letters. 90 (20): 203902. Bibcode:2003PhRvL..90t3902M. doi:10.1103/PhysRevLett.90.203902. PMID 12785895.

[Fibich2000-12] Fibich, Gadi; Gaeta, Alexander L. (2000). "Critical power for self-focusing in bulk media and in hollow waveguides". Optics Letters. 25 (5): 335–7. Bibcode:2000OptL...25..335F. doi:10.1364/OL.25.000335. PMID 18059872.

[13] Nibbering, E. T. J.; Grillon, G.; Franco, M. A.; Prade, B. S.; Mysyrowicz, A. (1997). "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 of the Optical Society of America B. 14 (3): 650–60. Bibcode:1997JOSAB..14..650N. doi:10.1364/JOSAB.14.000650.

[14] Garcia, Hernando; Johnson, Anthony M.; Oguama, Ferdinand A.; Trivedi, Sudhir (2003). "New approach to the measurement of the nonlinear refractive index of short (< 25 m) lengths of silica and erbium-doped fibers". Optics Letters. 28 (19): 1796–8. Bibcode:2003OptL...28.1796G. doi:10.1364/OL.28.001796. PMID 14514104.

[15] Kasparian, J.; Rodriguez, M.; Méjean, G.; Yu, J.; Salmon, E.; Wille, H.; Bourayou, R.; Frey, S.; André, Y.-B.; Mysyrowicz, A.; Sauerbrey, R.; Wolf, J.-P.; Wöste, L. (2003). "White-Light Filaments for Atmospheric Analysis". Science. 301 (5629): 61–4. Bibcode:2003Sci...301...61K. CiteSeerX 10.1.1.1028.4581. doi:10.1126/science.1085020. PMID 12843384. S2CID 37270331.

[16] Couairon, A; Mysyrowicz, A (2007). "Femtosecond filamentation in transparent media". Physics Reports. 441 (2–4): 47–189. Bibcode:2007PhR...441...47C. doi:10.1016/j.physrep.2006.12.005.

[17] Stibenz, Gero; Zhavoronkov, Nickolai; Steinmeyer, Günter (2006). "Self-compression of millijoule pulses to 78 fs duration in a white-light filament". Optics Letters. 31 (2): 274–6. Bibcode:2006OptL...31..274S. doi:10.1364/OL.31.000274. PMID 16441054. S2CID 12957688.

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