Modulational instability: Difference between revisions

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In the fieldfields of [[nonlinear optics]] and [[fluid dynamics]], '''modulational instability''' or '''sideband instability''' is a phenomenon whereby deviations from ana opticalperiodic 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="agrawalBenjaminFeir">{{cite bookjournal

| doi = 10.1017/S002211206700045X

| volume = 27

| issue = 3

| pages = 417–430

| last1 = Benjamin

| first1 = T. Brooke

| author1-link = T. Brooke Benjamin

| first2 = J.E.

| last2 = Feir

| title = The disintegration of wave trains on deep water. Part 1. Theory

| journal = Journal of Fluid Mechanics

| year = 1967

}}</ref><ref>{{Cite journal

| doi = 10.1098/rspa.1967.0123

| volume = 299

| issue = 1456

| pages = 59–76

| last = Benjamin

| first = T.B.

| author-link = T. Brooke Benjamin

| title = Instability of Periodic Wavetrains in Nonlinear Dispersive Systems

| journal = Proceedings of the Royal Society of London

| series = A. Mathematical and Physical Sciences

| year = 1967

}} Concluded with a discussion by [[Klaus Hasselmann]].</ref><ref name="agrawal">{{cite book

| last = Agrawal

| first = Govind P.

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| edition =2nd

| isbn = 0-12-045142-5

}}</ref>

The phenomenon was first discovered − and modelled − for periodic [[surface gravity 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 = 303−334

| last1 = Yuen

| first1 = H.C.

| first2 = B.M.

| lastr2 = Lake

| title = Instabilities of waves on deep water

| journal = Annual Review of Fluid Mechanics

| year = 1980

}}</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 = 863−884

| last = Janssen

| first = Peter A.E.M.

| title = Nonlinear four-wave interactions and freak waves

| journal = Journal of Physical Oceanography

| year = 2003

}}</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 wavelengths[[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.) 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, whilst at other frequencies, a perturbation will [[exponential growth|grow exponentially]]. The overall [[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.

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==References==

{{Reflist}}

<references />

{{DEFAULTSORT:Modulational Instability}}

[[Category:Nonlinear optics]]

[[Category:Photonics]]

[[Category:Water waves]]