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

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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

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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

| pages = 303−334303–334

| last1 = Yuen

| first1 = H.C.

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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

}}</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

| issue = 4

| pages = 863−884863–884

| last = Janssen

| first = Peter A.E.M.

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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

| pagesissue = 287−3101

| pages = 287–310

| last1 = Dysthe

| first1 = Kristian

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| journal = Annual Review of Fluid Mechanics

| year = 2008

|bibcode = 2008AnRFM..40..287D }}</ref>

==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.) There is also a threshold power, below which no instability will be seen.<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 [[Nonlinearnonlinear 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>

which describes the evolution of a [[Slowly varying envelope approximation|slowly varying envelope]] <math>A</math> with time <math>t</math> and distance of propagation <math>z</math>. The model includes group velocity dispersion described by the parameter <math>\beta_2</math>, and Kerr nonlinearity with magnitude <math>\gamma</math>. A waveform of constant power <math>P</math> is assumed. This is given by the solution

:<math>A = \sqrt{P}e^{i\gamma Pz}</math>

which describes the evolution of a [[complex number|complex-valued]] [[Slowly varying envelope approximation|slowly varying envelope]] <math>A</math> with time <math>t</math> and distance of propagation <math>z</math>. The [[imaginary unit]] <math>i</math> satisfies <math>i^2=-1.</math> The model includes [[group velocity]] dispersion described by the parameter <math>\beta_2</math>, and [[Kerr effect|Kerr nonlinearity]] with magnitude <math>\gamma.</math>. A [[periodic function|periodic]] waveform of constant power <math>P</math> is assumed. This is given by the solution

where the oscillatory <math>e^{i\gamma Pz}</math> phase factor accounts for the difference between the linear [[refractive index]], and the modified refractive index, as raised by the Kerr effect. The beginning of instability can be investigated by perturbing this solution as

:<math>A = \left(\sqrt{P}+\epsilon\left(t,z\right)\right)e^{i\gamma Pz}</math>

:<math>A = \sqrt{P} e^{i\gamma Pz},</math>

where <math>\epsilon\left(t,z\right)</math> is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as <math>A</math>). Substituting this back into the Nonlinear Schrödinger equation gives a [[perturbation theory|perturbation equation]] of the form

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

where the oscillatory <math>e^{i\gamma Pz}</math> [[wave phase|phase]] factor accounts for the difference between the linear [[refractive index]], and the modified [[refractive index]], as raised by the Kerr effect. The beginning of instability can be investigated by perturbing this solution as

where the perturbation has been assumed to be small, such that <math>\epsilon^2\approx 0</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>\epsilon=c_1 e^{i k_m z - i \omega_m t} + c_2 e^{- i k_m z + i \omega_m t}</math>

:<math>A = \left(\sqrt{P}+\epsilon\leftvarepsilon(t,z\right)\right)e^{i\gamma Pz},</math>

where <math>\omega_m</math> and <math>k_m</math> are the [[frequency]] and [[wavenumber]] of a perturbation, and <math>c_1</math> and <math>c_2</math> are constants. The Nonlinear Schrödinger equation is constructed by removing the [[carrier wave]] of the light being modelled, and so the frequency of the light being perturbed is formally zero. Therefore <math>\omega_m</math> and <math>k_m</math> don't represent absolute frequencies and wavenumbers, but the ''difference'' between these and those of the initial beam of light. It can be shown that the trial function is valid, subject to the condition

:<math>k_m = \pm\sqrt{\beta_2^2\omega_m^4 + 2 \gamma P \beta_2 \omega_m^2}</math>

where <math>\epsilon\leftvarepsilon(t,z\right)</math> is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as <math>A</math>). Substituting this back into the Nonlinearnonlinear Schrödinger equation gives a [[perturbation theory|perturbation equation]] of the form

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

where the perturbation has been assumed to be small, such that <math>|\epsilonvarepsilon|^2\approxll 0P.</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>\epsilonvarepsilon=c_1 e^{i k_m z - i \omega_m t} + c_2 e^{- i k_m^* z + i \omega_m t},</math>

where <math>\omega_mk_m</math> and <math>k_m\omega_m</math> are the [[frequencywavenumber]] and (real-valued) [[wavenumberangular frequency]] of a perturbation, and <math>c_1</math> and <math>c_2</math> are constants. The Nonlinearnonlinear Schrödinger equation is constructed by removing the [[carrier wave]] of the light being modelled, and so the frequency of the light being perturbed is formally zero. Therefore, <math>\omega_m</math> and <math>k_m</math> don't represent absolute frequencies and wavenumbers, but the ''difference'' between these and those of the initial beam of light. It can be shown that the trial function is valid, provided <math>c_2=c_1^*</math> and subject to the condition

:<math>k_m = \pm\sqrt{\beta_2^2\omega_m^4 + 2 \gamma P \beta_2 \omega_m^2}.</math>

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>

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

:<math>g = \begin{cases} 2\sqrt{-\beta_2^2\omega_m^4-2\gamma P \beta_2\omega_m^2} &;\, -\beta_2^2\omega_m^2 - 2 \gamma P \beta_2 > 0 \\ 0 &;\, -\beta_2^2\omega_m^2 - 2 \gamma P \beta_2 \leq 0\end{cases} </math>

where as noted above, <math>\omega_m</math> is the difference between the frequency of the perturbation and the frequency of the initial light.

:<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>

==Breakup==

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

The waveform will eventually breakup into a train of pulses.<ref name="agrawal" />

:<math>g = \begin{cases}

:<math>g = \begin{cases} 2\sqrt{-\beta_2^2\omega_m^4-2\gamma P \beta_2\omega_m^2}, &;\,text{for -\beta_2^2\omega_m^2 - 2} \gamma Pdisplaystyle \beta_2 > 0 \\ 0 &;\, -\beta_2^2\omega_m^2 < - 2 \frac{\gamma P }{\beta_2 \leq 0\end{cases} </math>,

\\[2ex]

0, &\text{for } \displaystyle \omega_m^2 \ge - 2 \frac{\gamma P}{\beta_2},

\end{cases} </math>

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==

{{Reflist}}

==Further reading==

* {{Cite journal

| doi = 10.1016/j.physd.2008.12.002

| volume = 238

| issue = 5

| pages = 540–548

| last1 = Zakharov

| first1 = V.E.

| author1-link = Vladimir E. Zakharov

| first2 = L.A.

| last2 = Ostrovsky

| title = Modulation instability: The beginning

| journal = Physica D: Nonlinear Phenomena

| 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}}

{{DEFAULTSORT:Modulational Instability}}

[[Category:Nonlinear optics]]

[[Category:Photonics]]

[[Category:Water waves]]

[[Category:Fluid dynamic instabilities]]