Modulational instability explained

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 spectral-sidebands and the eventual breakup of the waveform into a train of pulses.^[1] ^[2] ^[3]

It is widely believed that the phenomenon was first discovered − and modeled − for periodic surface gravity waves (Stokes waves) on deep water by T. Brooke Benjamin and Jim E. Feir, in 1967.^[4] 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,^[5] and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966.^[6] Modulation instability is a possible mechanism for the generation of rogue waves.^[7] ^[8]

Initial instability and gain

Modulation instability only happens under certain circumstances. The most important condition is anomalous group velocity dispersion, whereby pulses with shorter wavelengths travel with higher group velocity than pulses with longer wavelength. (This condition assumes a focusing Kerr nonlinearity, whereby refractive index increases with optical intensity.)

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 grow exponentially. The overall gain spectrum can be derived 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 amplification. By tuning an input signal to a peak of the gain spectrum, it is possible to create an optical amplifier.

Mathematical derivation of gain spectrum

The gain spectrum can be derived by starting with a model of modulation instability based upon the nonlinear Schrödinger equation

	\partialA
	\partialz

i\beta

2	\partial^2A
	\partialt²

=i\gamma|A|^2A,

with time

and distance of propagation

. The imaginary unit

satisfies

i^2=-1.

The model includes group velocity dispersion described by the parameter

\beta₂

, and Kerr nonlinearity with magnitude

\gamma.

A periodic waveform of constant power

is assumed. This is given by the solution

A=\sqrt{P}e^i\gamma,

where the oscillatory

e^i\gamma

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

A=\left(\sqrt{P}+\varepsilon(t,z)\right)e^i\gamma,

where

\varepsilon(t,z)

is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as

). Substituting this back into the nonlinear Schrödinger equation gives a perturbation equation of the form

	\partial\varepsilon
	\partialz

+i\beta

2	\partial^2\varepsilon
	\partialt²

=i\gammaP\left(\varepsilon+\varepsilon^*\right),

where the perturbation has been assumed to be small, such that

|\varepsilon|^2\llP.

The complex conjugate of

\varepsilon

is denoted as

\varepsilon^*.

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

\varepsilon=c₁

	ik_mz-i\omega_mt
e

+c₂

	*
k
	m

z+i\omega_mt

where

k_m

and

\omega_m

are the wavenumber and (real-valued) angular frequency of a perturbation, and

c₁

and

c₂

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,

\omega_m

and

k_m

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

c_2=c

	*

	1

and subject to the condition

k_m=

	4
\pm\sqrt{\beta
	m

+2\gammaP\beta₂

	2}.
\omega
	m

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

	2
\beta
	m

+2\gammaP\beta₂<0,

that is for

	2
\omega
	m

<-2

	\gammaP
	\beta₂

This condition describes the requirement for anomalous dispersion (such that

\gamma\beta₂

is negative). The gain spectrum can be described by defining a gain parameter as

g\equiv2|\Im\{k_m\}|,

so that the power of a perturbing signal grows with distance as

Pe^g.

The gain is therefore given by

g=\begin{cases}

	4-2\gamma
2\sqrt{-\beta
	m

P\beta_2\omega

	2},

	m

&for\displaystyle

	2
\omega
	m

<-2

	\gammaP
	\beta₂

, \\[2ex] 0,&for\displaystyle

	2
\omega
	m

\ge-2

	\gammaP
	\beta₂

, \end{cases}

where as noted above,

\omega_m

is the difference between the frequency of the perturbation and the frequency of the initial light. The growth rate is maximum for

\omega^2=-\gammaP/\beta_2.

Modulation instability in soft systems

Modulation instability of optical fields has been observed in photo-chemical systems, namely, photopolymerizable medium.^[9] ^[10] ^[11] ^[12] Modulation instability occurs owing to inherent optical nonlinearity of the systems due to photoreaction-induced changes in the refractive index.^[13] 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.^[14]

Notes and References

10.1017/S002211206700045X. 27. 3. 417–430. Benjamin. T. Brooke. T. Brooke Benjamin. J.E. . Feir. The disintegration of wave trains on deep water. Part 1. Theory. Journal of Fluid Mechanics. 1967. 1967JFM....27..417B . 121996479.
10.1098/rspa.1967.0123. 299. 1456. 59–76. Benjamin. T.B.. T. Brooke Benjamin. Instability of Periodic Wavetrains in Nonlinear Dispersive Systems. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 1967. 1967RSPSA.299...59B . 121661209. Concluded with a discussion by Klaus Hasselmann.
Book: Agrawal , Govind P. . 1995 . Nonlinear fiber optics . San Diego (California) . Academic Press . 2nd . 978-0-12-045142-5.
10.1146/annurev.fl.12.010180.001511. 12. 303–334. Yuen. H.C.. B.M.. Lake. Instabilities of waves on deep water. Annual Review of Fluid Mechanics. 1980. 1980AnRFM..12..303Y .
Piliptetskii. N. F.. Rustamov. A. R.. 31 May 1965. Observation of Self-focusing of Light in Liquids. JETP Letters. 2. 2. 55–56.
Bespalov. V. I.. Talanov. V. I.. 15 June 1966. Filamentary Structure of Light Beams in Nonlinear Liquids. ZhETF Pisma Redaktsiiu. 3. 11. 471–476. 1966ZhPmR...3..471B. 17 February 2021. 31 July 2020. https://web.archive.org/web/20200731112029/http://www.jetpletters.ac.ru/ps/1621/article_24803.shtml. dead.
10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2. 33. 4. 863–884. Janssen. Peter A.E.M.. Nonlinear four-wave interactions and freak waves. Journal of Physical Oceanography. 2003. 2003JPO....33..863J . free.
10.1146/annurev.fluid.40.111406.102203. 40. 1. 287–310. Dysthe. Kristian. Harald E. . Krogstad . Peter . Müller. Oceanic rogue waves. Annual Review of Fluid Mechanics. 2008. 2008AnRFM..40..287D .
Burgess. Ian B.. Shimmell. Whitney E.. Saravanamuttu. Kalaichelvi. 2007-04-01. Spontaneous Pattern Formation Due to Modulation Instability of Incoherent White Light in a Photopolymerizable Medium. Journal of the American Chemical Society. 129. 15. 4738–4746. 10.1021/ja068967b. 17378567. 0002-7863.
Basker. Dinesh K.. Brook. Michael A.. Saravanamuttu. Kalaichelvi. Spontaneous Emergence of Nonlinear Light Waves and Self-Inscribed Waveguide Microstructure during the Cationic Polymerization of Epoxides. The Journal of Physical Chemistry C. en. 119. 35. 20606–20617. 10.1021/acs.jpcc.5b07117. 2015.
Biria. Saeid. Malley. Philip P. A.. Kahan. Tara F.. Hosein. Ian D.. 2016-03-03. Tunable Nonlinear Optical Pattern Formation and Microstructure in Cross-Linking Acrylate Systems during Free-Radical Polymerization. The Journal of Physical Chemistry C. 120. 8. 4517–4528. 10.1021/acs.jpcc.5b11377. 1932-7447.
Biria. Saeid. Malley. Phillip P. A.. Kahan. Tara F.. Hosein. Ian D.. 2016-11-15. Optical Autocatalysis Establishes Novel Spatial Dynamics in Phase Separation of Polymer Blends during Photocuring. ACS Macro Letters. 5. 11. 1237–1241. 10.1021/acsmacrolett.6b00659. 35614732 .
Kewitsch. Anthony S.. Yariv. Amnon. 1996-01-01. Self-focusing and self-trapping of optical beams upon photopolymerization. Optics Letters. EN. 21. 1. 24–6. 10.1364/ol.21.000024. 1539-4794. 1996OptL...21...24K. 19865292.
Book: Spatial Solitons Stefano Trillo Springer. en.

Modulational instability explained

Initial instability and gain

Mathematical derivation of gain spectrum

Modulation instability in soft systems

Further reading

Notes and References