Modulational instability - Revision history

186.169.134.10: /* Initial instability and gain */

2024-12-28T04:40:29Z

Initial instability and gain

← Previous revision		Revision as of 04:40, 28 December 2024
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	Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity [[dispersion relation\|dispersion]]'', whereby pulses with shorter [[wavelength]]s travel with higher [[group velocity]] than pulses with longer wavelength.<ref name="agrawal" /> (This condition assumes a ''focusing'' [[Kerr nonlinearity]], whereby refractive index increases with optical intensity.)<ref name="agrawal" />		Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity [[dispersion relation\|dispersion]]'', whereby pulses with shorter [[wavelength]]s travel with higher [[group velocity]] than pulses with longer wavelength.<ref name="agrawal" /> (This condition assumes a ''focusing'' [[Kerr nonlinearity]], whereby refractive index increases with optical intensity.)<ref name="agrawal" />

	The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, ~~whilst~~ at other frequencies, a perturbation will [[exponential growth\|grow exponentially]]. The overall [[Gain (electronics)\|gain]] spectrum can be derived [[Analytical expression\|analytically]], as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.		The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, while at other frequencies, a perturbation will [[exponential growth\|grow exponentially]]. The overall [[Gain (electronics)\|gain]] spectrum can be derived [[Analytical expression\|analytically]], as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.

	The tendency of a perturbing signal to grow makes modulation instability a form of [[amplifier\|amplification]]. By tuning an input signal to a peak of the gain spectrum, it is possible to create an [[optical amplifier]].		The tendency of a perturbing signal to grow makes modulation instability a form of [[amplifier\|amplification]]. By tuning an input signal to a peak of the gain spectrum, it is possible to create an [[optical amplifier]].

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← Previous revision		Revision as of 18:28, 12 May 2024
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	\| journal = Annual Review of Fluid Mechanics		\| journal = Annual Review of Fluid Mechanics
	\| year = 1980		\| year = 1980
	\|bibcode = 1980AnRFM..12..303Y }}</ref> Therefore, it is also known as the '''Benjamin−Feir instability'''. However, spatial modulation instability of high-power lasers in organic solvents was observed by Russian scientists N. F. Piliptetskii and A. R. Rustamov in 1965,<ref>{{Cite journal\|last1=Piliptetskii\|first1=N. F.\|last2=Rustamov\|first2=A. R.\|date=31 May 1965\|title=Observation of Self-focusing of Light in Liquids\|url=http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml\|journal=JETP Letters\|volume=2\|issue=2\|pages=55–56}}</ref> and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966.<ref>{{Cite journal\|last1=Bespalov\|first1=V. I.\|last2=Talanov\|first2=V. I.\|date=15 June 1966\|title=Filamentary Structure of Light Beams in Nonlinear Liquids\|url=http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml\|journal=ZhETF Pisma Redaktsiiu\|volume=3\|issue=11\|pages=471–476\|bibcode=1966ZhPmR...3..471B }}</ref> Modulation instability is a possible mechanism for the generation of [[rogue wave]]s.<ref>{{Cite journal		\|bibcode = 1980AnRFM..12..303Y }}</ref> Therefore, it is also known as the '''Benjamin−Feir instability'''. However, spatial modulation instability of high-power lasers in organic solvents was observed by Russian scientists N. F. Piliptetskii and A. R. Rustamov in 1965,<ref>{{Cite journal\|last1=Piliptetskii\|first1=N. F.\|last2=Rustamov\|first2=A. R.\|date=31 May 1965\|title=Observation of Self-focusing of Light in Liquids\|url=http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml\|journal=JETP Letters\|volume=2\|issue=2\|pages=55–56}}</ref> and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966.<ref>{{Cite journal\|last1=Bespalov\|first1=V. I.\|last2=Talanov\|first2=V. I.\|date=15 June 1966\|title=Filamentary Structure of Light Beams in Nonlinear Liquids\|url=http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml\|journal=ZhETF Pisma Redaktsiiu\|volume=3\|issue=11\|pages=471–476\|bibcode=1966ZhPmR...3..471B\|access-date=17 February 2021\|archive-date=31 July 2020\|archive-url=https://web.archive.org/web/20200731112029/http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml\|url-status=dead}}</ref> Modulation instability is a possible mechanism for the generation of [[rogue wave]]s.<ref>{{Cite journal
	\| doi = 10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2		\| doi = 10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2
	\| volume = 33		\| volume = 33

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2024-02-01T13:40:55Z

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← Previous revision		Revision as of 13:40, 1 February 2024
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	===Mathematical derivation of gain spectrum===		===Mathematical derivation of gain spectrum===

	The gain spectrum can be derived <ref name="agrawal" /> by starting with a model of modulation instability based upon the [[nonlinear Schrödinger equation]]{{what\|reason=Time and space reversed?}}		The gain spectrum can be derived <ref name="agrawal" /> by starting with a model of modulation instability based upon the [[nonlinear Schrödinger equation]]{{what\|reason=Time and space reversed?\|date=February 2024}}

	: <math>\frac{\partial A}{\partial z} + i\beta_2\frac{\partial^2A}{\partial t^2} = i\gamma\|A\|^2A,</math>		: <math>\frac{\partial A}{\partial z} + i\beta_2\frac{\partial^2A}{\partial t^2} = i\gamma\|A\|^2A,</math>

2A02:1210:2642:4A00:B019:594A:E4BC:B46B: /* Mathematical derivation of gain spectrum */

2024-02-01T11:40:52Z

Mathematical derivation of gain spectrum

← Previous revision		Revision as of 11:40, 1 February 2024
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	===Mathematical derivation of gain spectrum===		===Mathematical derivation of gain spectrum===

	The gain spectrum can be derived <ref name="agrawal" /> by starting with a model of modulation instability based upon the [[nonlinear Schrödinger equation]]		The gain spectrum can be derived <ref name="agrawal" /> by starting with a model of modulation instability based upon the [[nonlinear Schrödinger equation]]{{what\|reason=Time and space reversed?}}

	: <math>\frac{\partial A}{\partial z} + i\beta_2\frac{\partial^2A}{\partial t^2} = i\gamma\|A\|^2A,</math>		: <math>\frac{\partial A}{\partial z} + i\beta_2\frac{\partial^2A}{\partial t^2} = i\gamma\|A\|^2A,</math>

129.119.235.10: /* Initial instability and gain */

2023-10-08T15:59:33Z

Initial instability and gain

← Previous revision		Revision as of 15:59, 8 October 2023
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	==Initial instability and gain==		==Initial instability and gain==

	Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity [[dispersion relation\|dispersion]]'', whereby pulses with shorter [[wavelength]]s travel with higher [[group velocity]] than pulses with longer wavelength.<ref name="agrawal" /> (This condition assumes a ''~~focussing~~'' [[Kerr nonlinearity]], whereby refractive index increases with optical intensity.)<ref name="agrawal" />		Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity [[dispersion relation\|dispersion]]'', whereby pulses with shorter [[wavelength]]s travel with higher [[group velocity]] than pulses with longer wavelength.<ref name="agrawal" /> (This condition assumes a ''focusing'' [[Kerr nonlinearity]], whereby refractive index increases with optical intensity.)<ref name="agrawal" />

	The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, whilst at other frequencies, a perturbation will [[exponential growth\|grow exponentially]]. The overall [[Gain (electronics)\|gain]] spectrum can be derived [[Analytical expression\|analytically]], as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.		The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, whilst at other frequencies, a perturbation will [[exponential growth\|grow exponentially]]. The overall [[Gain (electronics)\|gain]] spectrum can be derived [[Analytical expression\|analytically]], as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.

OAbot: Open access bot: doi added to citation with #oabot.

2023-08-12T14:41:38Z

Open access bot: doi added to citation with #oabot.

← Previous revision		Revision as of 14:41, 12 August 2023
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	\| journal = Journal of Physical Oceanography		\| journal = Journal of Physical Oceanography
	\| year = 2003		\| year = 2003
	\|bibcode = 2003JPO....33..863J }}</ref><ref>{{Cite journal		\|bibcode = 2003JPO....33..863J \| doi-access = free
			}}</ref><ref>{{Cite journal
	\| doi = 10.1146/annurev.fluid.40.111406.102203		\| doi = 10.1146/annurev.fluid.40.111406.102203
	\| volume = 40		\| volume = 40

JJMC89 bot III: Moving :Category:Fluid dynamic instability to :Category:Fluid dynamic instabilities per Wikipedia:Categories for discussion/Speedy

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Moving Category:Fluid dynamic instability to Category:Fluid dynamic instabilities per Wikipedia:Categories for discussion/Speedy

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	[[Category:Photonics]]		[[Category:Photonics]]
	[[Category:Water waves]]		[[Category:Water waves]]
	[[Category:Fluid dynamic ~~instability~~]]		[[Category:Fluid dynamic instabilities]]

Citation bot: Alter: pages. Add: pmid, bibcode, s2cid, authors 1-1. Removed parameters. Formatted dashes. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Whoop whoop pull up | Category:Water waves | #UCB_Category 9/72

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Alter: pages. Add: pmid, bibcode, s2cid, authors 1-1. Removed parameters. Formatted dashes. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Whoop whoop pull up | Category:Water waves | #UCB_Category 9/72

← Previous revision		Revision as of 23:06, 14 August 2022
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	\| journal = Journal of Fluid Mechanics		\| journal = Journal of Fluid Mechanics
	\| year = 1967		\| year = 1967
	\|bibcode = 1967JFM....27..417B }}</ref><ref>{{Cite journal		\|bibcode = 1967JFM....27..417B \| s2cid = 121996479
			}}</ref><ref>{{Cite journal
	\| doi = 10.1098/rspa.1967.0123		\| doi = 10.1098/rspa.1967.0123
	\| volume = 299		\| volume = 299
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	\| series = A. Mathematical and Physical Sciences		\| series = A. Mathematical and Physical Sciences
	\| year = 1967		\| year = 1967
	\|bibcode = 1967RSPSA.299...59B }} Concluded with a discussion by [[Klaus Hasselmann]].</ref><ref name="agrawal">{{cite book		\|bibcode = 1967RSPSA.299...59B \| s2cid = 121661209
			}} Concluded with a discussion by [[Klaus Hasselmann]].</ref><ref name="agrawal">{{cite book
	\| last = Agrawal		\| last = Agrawal
	\| first = Govind P.		\| first = Govind P.
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	\| journal = Annual Review of Fluid Mechanics		\| journal = Annual Review of Fluid Mechanics
	\| year = 1980		\| year = 1980
	\|bibcode = 1980AnRFM..12..303Y }}</ref> Therefore, it is also known as the '''Benjamin−Feir instability'''. However, spatial modulation instability of high-power lasers in organic solvents was observed by Russian scientists N. F. Piliptetskii and A. R. Rustamov in 1965,<ref>{{Cite journal\|~~last~~=Piliptetskii\|~~first~~=N. F.\|last2=Rustamov\|first2=A. R.\|date=31 May 1965\|title=Observation of Self-focusing of Light in Liquids\|url=http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml\|journal=JETP Letters\|volume=2\|issue=2\|pages=~~55-56~~}}</ref> and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966.<ref>{{Cite journal\|~~last~~=Bespalov\|~~first~~=V. I.\|last2=Talanov\|first2=V. I.\|date=15 June 1966\|title=Filamentary Structure of Light Beams in Nonlinear Liquids\|url=http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml\|journal=ZhETF Pisma Redaktsiiu\|volume=3\|issue=11\|pages=~~471-476~~}}</ref> Modulation instability is a possible mechanism for the generation of [[rogue wave]]s.<ref>{{Cite journal		\|bibcode = 1980AnRFM..12..303Y }}</ref> Therefore, it is also known as the '''Benjamin−Feir instability'''. However, spatial modulation instability of high-power lasers in organic solvents was observed by Russian scientists N. F. Piliptetskii and A. R. Rustamov in 1965,<ref>{{Cite journal\|last1=Piliptetskii\|first1=N. F.\|last2=Rustamov\|first2=A. R.\|date=31 May 1965\|title=Observation of Self-focusing of Light in Liquids\|url=http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml\|journal=JETP Letters\|volume=2\|issue=2\|pages=55–56}}</ref> and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966.<ref>{{Cite journal\|last1=Bespalov\|first1=V. I.\|last2=Talanov\|first2=V. I.\|date=15 June 1966\|title=Filamentary Structure of Light Beams in Nonlinear Liquids\|url=http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml\|journal=ZhETF Pisma Redaktsiiu\|volume=3\|issue=11\|pages=471–476\|bibcode=1966ZhPmR...3..471B }}</ref> Modulation instability is a possible mechanism for the generation of [[rogue wave]]s.<ref>{{Cite journal
	\| doi = 10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2		\| doi = 10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2
	\| volume = 33		\| volume = 33
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	== Modulation instability in soft systems ==		== Modulation instability in soft systems ==
	Modulation instability of optical fields has been observed in photo-chemical systems, namely, photopolymerizable medium.<ref>{{Cite journal\|~~last~~=Burgess\|~~first~~=Ian B.\|last2=Shimmell\|first2=Whitney E.\|last3=Saravanamuttu\|first3=Kalaichelvi\|date=2007-04-01\|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=4738–4746\|doi=10.1021/ja068967b\|pmid=17378567\|issn=0002-7863}}</ref><ref>{{Cite journal\|~~last~~=Basker\|~~first~~=Dinesh K.\|last2=Brook\|first2=Michael A.\|last3=Saravanamuttu\|first3=Kalaichelvi\|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\|language=en\|volume=119\|issue=35\|pages=20606–20617\|doi=10.1021/acs.jpcc.5b07117\|year=2015}}</ref><ref>{{Cite journal\|~~last~~=Biria\|~~first~~=Saeid\|last2=Malley\|first2=Philip P. A.\|last3=Kahan\|first3=Tara F.\|last4=Hosein\|first4=Ian D.\|date=2016-03-03\|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=4517–4528\|doi=10.1021/acs.jpcc.5b11377\|issn=1932-7447}}</ref><ref>{{Cite journal\|~~last~~=Biria\|~~first~~=Saeid\|last2=Malley\|first2=Phillip P. A.\|last3=Kahan\|first3=Tara F.\|last4=Hosein\|first4=Ian D.\|date=2016-11-15\|title=Optical Autocatalysis Establishes Novel Spatial Dynamics in Phase Separation of Polymer Blends during Photocuring\|journal=ACS Macro Letters\|volume=5\|issue=11\|pages=1237–1241\|doi=10.1021/acsmacrolett.6b00659}}</ref> Modulation instability occurs owing to inherent optical nonlinearity of the systems due to photoreaction-induced changes in the refractive index.<ref>{{Cite journal\|~~last~~=Kewitsch\|~~first~~=Anthony S.\|last2=Yariv\|first2=Amnon\|date=1996-01-01\|title=Self-focusing and self-trapping of optical beams upon photopolymerization\|journal=Optics Letters\|language=EN\|volume=21\|issue=1\|pages=24–6\|doi=10.1364/ol.21.000024\|issn=1539-4794\|bibcode=1996OptL...21...24K\|url=https://authors.library.caltech.edu/2845/1/KEWol96.pdf\|pmid=19865292}}</ref> Modulation instability of spatially and temporally incoherent light is possible owing to the non-instantaneous response of photoreactive systems, which consequently responds to the time-average intensity of light, in which the femto-second fluctuations cancel out.<ref>{{Cite book\|url=https://www.springer.com/us/book/9783540416531\|title=Spatial Solitons {{!}} Stefano Trillo {{!}} Springer\|language=en}}</ref>		Modulation instability of optical fields has been observed in photo-chemical systems, namely, photopolymerizable medium.<ref>{{Cite journal\|last1=Burgess\|first1=Ian B.\|last2=Shimmell\|first2=Whitney E.\|last3=Saravanamuttu\|first3=Kalaichelvi\|date=2007-04-01\|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=4738–4746\|doi=10.1021/ja068967b\|pmid=17378567\|issn=0002-7863}}</ref><ref>{{Cite journal\|last1=Basker\|first1=Dinesh K.\|last2=Brook\|first2=Michael A.\|last3=Saravanamuttu\|first3=Kalaichelvi\|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\|language=en\|volume=119\|issue=35\|pages=20606–20617\|doi=10.1021/acs.jpcc.5b07117\|year=2015}}</ref><ref>{{Cite journal\|last1=Biria\|first1=Saeid\|last2=Malley\|first2=Philip P. A.\|last3=Kahan\|first3=Tara F.\|last4=Hosein\|first4=Ian D.\|date=2016-03-03\|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=4517–4528\|doi=10.1021/acs.jpcc.5b11377\|issn=1932-7447}}</ref><ref>{{Cite journal\|last1=Biria\|first1=Saeid\|last2=Malley\|first2=Phillip P. A.\|last3=Kahan\|first3=Tara F.\|last4=Hosein\|first4=Ian D.\|date=2016-11-15\|title=Optical Autocatalysis Establishes Novel Spatial Dynamics in Phase Separation of Polymer Blends during Photocuring\|journal=ACS Macro Letters\|volume=5\|issue=11\|pages=1237–1241\|doi=10.1021/acsmacrolett.6b00659\|pmid=35614732 }}</ref> Modulation instability occurs owing to inherent optical nonlinearity of the systems due to photoreaction-induced changes in the refractive index.<ref>{{Cite journal\|last1=Kewitsch\|first1=Anthony S.\|last2=Yariv\|first2=Amnon\|date=1996-01-01\|title=Self-focusing and self-trapping of optical beams upon photopolymerization\|journal=Optics Letters\|language=EN\|volume=21\|issue=1\|pages=24–6\|doi=10.1364/ol.21.000024\|issn=1539-4794\|bibcode=1996OptL...21...24K\|url=https://authors.library.caltech.edu/2845/1/KEWol96.pdf\|pmid=19865292}}</ref> Modulation instability of spatially and temporally incoherent light is possible owing to the non-instantaneous response of photoreactive systems, which consequently responds to the time-average intensity of light, in which the femto-second fluctuations cancel out.<ref>{{Cite book\|url=https://www.springer.com/us/book/9783540416531\|title=Spatial Solitons {{!}} Stefano Trillo {{!}} Springer\|language=en}}</ref>

	==References==		==References==

Headbomb: ZhETF Pisma Redaktsiiu (

2021-10-30T18:29:14Z

ZhETF Pisma Redaktsiiu (

← Previous revision		Revision as of 18:29, 30 October 2021
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	\| journal = Annual Review of Fluid Mechanics		\| journal = Annual Review of Fluid Mechanics
	\| year = 1980		\| year = 1980
	\|bibcode = 1980AnRFM..12..303Y }}</ref> Therefore, it is also known as the '''Benjamin−Feir instability'''. However, spatial modulation instability of high-power lasers in organic solvents was observed by Russian scientists N. F. Piliptetskii and A. R. Rustamov in 1965,<ref>{{Cite journal\|last=Piliptetskii\|first=N. F.\|last2=Rustamov\|first2=A. R.\|date=31 May 1965\|title=Observation of Self-focusing of Light in Liquids\|url=http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml\|journal=JETP Letters\|volume=2\|issue=2\|pages=55-56}}</ref> and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966.<ref>{{Cite journal\|last=Bespalov\|first=V. I.\|last2=Talanov\|first2=V. I.\|date=15 June 1966\|title=Filamentary Structure of Light Beams in Nonlinear Liquids\|url=http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml\|journal=ZhETF ~~Pis ma~~ Redaktsiiu\|volume=3\|issue=11\|pages=471-476}}</ref> Modulation instability is a possible mechanism for the generation of [[rogue wave]]s.<ref>{{Cite journal		\|bibcode = 1980AnRFM..12..303Y }}</ref> Therefore, it is also known as the '''Benjamin−Feir instability'''. However, spatial modulation instability of high-power lasers in organic solvents was observed by Russian scientists N. F. Piliptetskii and A. R. Rustamov in 1965,<ref>{{Cite journal\|last=Piliptetskii\|first=N. F.\|last2=Rustamov\|first2=A. R.\|date=31 May 1965\|title=Observation of Self-focusing of Light in Liquids\|url=http://www.jetpletters.ac.ru/ps/1596/article_24469.shtml\|journal=JETP Letters\|volume=2\|issue=2\|pages=55-56}}</ref> and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966.<ref>{{Cite journal\|last=Bespalov\|first=V. I.\|last2=Talanov\|first2=V. I.\|date=15 June 1966\|title=Filamentary Structure of Light Beams in Nonlinear Liquids\|url=http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml\|journal=ZhETF Pisma Redaktsiiu\|volume=3\|issue=11\|pages=471-476}}</ref> Modulation instability is a possible mechanism for the generation of [[rogue wave]]s.<ref>{{Cite journal
	\| doi = 10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2		\| doi = 10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2
	\| volume = 33		\| volume = 33

Qwerfjkl: Removing 'a(n)' from the beginning of short descriptions, per WP:SHORTDESC, from request at WP:Reward board. (via WP:JWB)

2021-06-27T11:31:32Z

Removing 'a(n)' from the beginning of short descriptions, per WP:SHORTDESC, from request at WP:Reward board. (via WP:JWB)

← Previous revision		Revision as of 11:31, 27 June 2021
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	{{short description\|~~A phenomenon~~ whereby deviations from a periodic waveform are reinforced by nonlinearity}}		{{short description\|Phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity}}
	In the fields of [[nonlinear optics]] and [[fluid dynamics]], '''modulational instability''' or '''sideband instability''' is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of [[Frequency spectrum\|spectral]]-sidebands and the eventual breakup of the waveform into a train of [[wave packet\|pulses]].<ref name="BenjaminFeir">{{cite journal		In the fields of [[nonlinear optics]] and [[fluid dynamics]], '''modulational instability''' or '''sideband instability''' is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of [[Frequency spectrum\|spectral]]-sidebands and the eventual breakup of the waveform into a train of [[wave packet\|pulses]].<ref name="BenjaminFeir">{{cite journal
	\| doi = 10.1017/S002211206700045X		\| doi = 10.1017/S002211206700045X

← Previous revision		Revision as of 04:40, 28 December 2024
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	Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity [[dispersion relation\|dispersion]]'', whereby pulses with shorter [[wavelength]]s travel with higher [[group velocity]] than pulses with longer wavelength.<ref name="agrawal" /> (This condition assumes a ''focusing'' [[Kerr nonlinearity]], whereby refractive index increases with optical intensity.)<ref name="agrawal" />		Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity [[dispersion relation\|dispersion]]'', whereby pulses with shorter [[wavelength]]s travel with higher [[group velocity]] than pulses with longer wavelength.<ref name="agrawal" /> (This condition assumes a ''focusing'' [[Kerr nonlinearity]], whereby refractive index increases with optical intensity.)<ref name="agrawal" />

	The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, ~~whilst~~ at other frequencies, a perturbation will [[exponential growth\|grow exponentially]]. The overall [[Gain (electronics)\|gain]] spectrum can be derived [[Analytical expression\|analytically]], as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.		The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, while at other frequencies, a perturbation will [[exponential growth\|grow exponentially]]. The overall [[Gain (electronics)\|gain]] spectrum can be derived [[Analytical expression\|analytically]], as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.

	The tendency of a perturbing signal to grow makes modulation instability a form of [[amplifier\|amplification]]. By tuning an input signal to a peak of the gain spectrum, it is possible to create an [[optical amplifier]].		The tendency of a perturbing signal to grow makes modulation instability a form of [[amplifier\|amplification]]. By tuning an input signal to a peak of the gain spectrum, it is possible to create an [[optical amplifier]].