Modulational instability: Difference between revisions

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{{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

| doi = 10.1017/S002211206700045X

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| journal = Journal of Fluid Mechanics

| year = 1967

|bibcode = 1967JFM....27..417B | s2cid = 121996479
}}</ref><ref>{{Cite journal

| doi = 10.1098/rspa.1967.0123

| volume = 299

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| series = A. Mathematical and Physical Sciences

| year = 1967

|bibcode = 1967RSPSA.299...59B | s2cid = 121661209
}} Concluded with a discussion by [[Klaus Hasselmann]].</ref><ref name="agrawal">{{cite book

| last = Agrawal

| first = Govind P.

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| publisher = Academic Press

| edition =2nd

| isbn = 978-0-12-045142-5

}}</ref>

TheIt is widely believed that the phenomenon was first discovered − and modelledmodeled − for periodic [[surface gravity wave]]s ([[Stokes wave]]s) on deep water by [[T. Brooke Benjamin]] and Jim E. Feir, in 1967.<ref>{{Cite journal

| doi = 10.1146/annurev.fl.12.010180.001511

| volume = 12

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| journal = Annual Review of Fluid Mechanics

| 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|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

|bibcode = 1980AnRFM..12..303Y }}</ref> Therefore, it is also known as the '''Benjamin−Feir instability'''. It 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

| volume = 33

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| journal = Journal of Physical Oceanography

| year = 2003

|bibcode = 2003JPO....33..863J | doi-access = free
}}</ref><ref>{{Cite journal

| doi = 10.1146/annurev.fluid.40.111406.102203

| volume = 40

| issue = 1

| pages = 287–310

| last1 = Dysthe

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==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 ''focussingfocusing'' [[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, whilstwhile 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]].

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===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?|date=February 2024}}

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

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:<math>\frac{\partial \varepsilon}{\partial z}+i\beta_2\frac{\partial^2\varepsilon}{\partial t^2}=i\gamma P \left(\varepsilon+\varepsilon^*\right),</math>

where the perturbation has been assumed to be small, such that <math>|\varepsilon|^2\ll P.</math> The [[complex conjugate]] of <math>\varepsilon</math> is denoted as <math>\varepsilon^*.</math> Instability can now be discovered by searching for solutions of the perturbation equation which grow exponentially. This can be done using a trial function of the general form

:<math>\varepsilon=c_1 e^{i k_m z - i \omega_m t} + c_2 e^{- i k_m^* z + i \omega_m t},</math>

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This dispersion relation is vitally dependent on the sign of the term within the square root, as if positive, the wavenumber will be [[real number|real]], corresponding to mere [[oscillation]]s around the unperturbed solution, whilst if negative, the wavenumber will become [[imaginary number|imaginary]], corresponding to exponential growth and thus instability. Therefore, instability will occur when

:<math>\beta_2^2\omega_m^2 + 2 \gamma P \beta_2 < 0,</math> {{pad|2em}} that is for {{pad|2em}} <math>\omega_m^2 < -2 \frac{\gamma P}{\beta_2}.</math>

This condition describes the requirement for anomalous dispersion (such that <math>\gamma\beta_2</math> is negative). The gain spectrum can be described by defining a gain parameter as <math>g \equiv \Im [ 2|\Im\{k_m\}| ],</math> so that the power of a perturbing signal grows with distance as <math>P\, e^{g z}.</math> The gain is therefore given by

:<math>g = \begin{cases}

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where as noted above, <math>\omega_m</math> is the difference between the frequency of the perturbation and the frequency of the initial light. The growth rate is maximum for <math>\omega^2=-\gamma P/\beta_2.</math>

== Modulation instability in soft systems ==

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|bibcode=2007JAChS.129.4738B |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==

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| first1 = V.E.

| author1-link = Vladimir E. Zakharov

| first2 = L.A.

| last2 = Ostrovsky

| title = Modulation instability: The beginning

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| year = 2009

| url = http://www.math.umass.edu/~kevrekid/math697/sdarticle_ZO.pdf

| bibcode = 2009PhyD..238..540Z }}

}}{{Dead link|date=April 2020 |bot=InternetArchiveBot |fix-attempted=yes }}

{{physical oceanography}}

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[[Category:Photonics]]

[[Category:Water waves]]

[[Category:Fluid dynamic instabilities]]