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In the fields of
nonlinear optics Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typic ...
and
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
, 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 In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the ...
. It is widely believed that the phenomenon was first discovered − and modeled − for periodic
surface gravity wave In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
s (
Stokes wave In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation serie ...
s) on deep water by
T. Brooke Benjamin Thomas Brooke Benjamin, FRS (15 April 1929 – 16 August 1995) was an English mathematical physicist and mathematician, best known for his work in mathematical analysis and fluid mechanics, especially in applications of nonlinear differential eq ...
and Jim E. Feir, in 1967. 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, and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966. Modulation instability is a possible mechanism for the generation of
rogue wave Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
s.


Initial instability and gain

Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity dispersion'', whereby pulses with shorter
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
s travel with higher group velocity than pulses with longer wavelength. (This condition assumes a ''focussing'' 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 Gain or GAIN may refer to: Science and technology * Gain (electronics), an electronics and signal processing term * Antenna gain * Gain (laser), the amplification involved in laser emission * Gain (projection screens) * Information gain in d ...
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 An optical amplifier is a device that amplifies an optical signal directly, without the need to first convert it to an electrical signal. An optical amplifier may be thought of as a laser without an optical cavity, or one in which feedback fr ...
.


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 : \frac + i\beta_2\frac = i\gamma, A, ^2A, which describes the evolution of a
complex-valued In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
slowly varying envelope A with time t and distance of propagation z. The
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...
i satisfies i^2=-1. The model includes group velocity dispersion described by the parameter \beta_2, and Kerr nonlinearity with magnitude \gamma. A periodic waveform of constant power P is assumed. This is given by the solution :A = \sqrt e^, where the oscillatory e^ phase factor accounts for the difference between the linear
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, o ...
, and the modified
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, o ...
, as raised by the Kerr effect. The beginning of instability can be investigated by perturbing this solution as :A = \left(\sqrt+\varepsilon(t,z)\right)e^, where \varepsilon(t,z) is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as A). Substituting this back into the nonlinear Schrödinger equation gives a perturbation equation of the form :\frac+i\beta_2\frac=i\gamma P \left(\varepsilon+\varepsilon^*\right), where the perturbation has been assumed to be small, such that , \varepsilon, ^2\ll P. The
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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=c_1 e^ + c_2 e^, where k_m and \omega_m are the
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
and (real-valued)
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
of a perturbation, and c_1 and c_2 are constants. The nonlinear Schrödinger equation is constructed by removing the
carrier wave In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has ...
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 = \pm\sqrt. 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
oscillation Oscillation is the repetitive or Periodic function, periodic variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. Familiar examples o ...
s around the unperturbed solution, whilst if negative, the wavenumber will become imaginary, corresponding to exponential growth and thus instability. Therefore, instability will occur when :\beta_2^2\omega_m^2 + 2 \gamma P \beta_2 < 0, that is for \omega_m^2 < -2 \frac. This condition describes the requirement for anomalous dispersion (such that \gamma\beta_2 is negative). The gain spectrum can be described by defining a gain parameter as g \equiv 2, \Im\, , so that the power of a perturbing signal grows with distance as P\, e^. The gain is therefore given by :g = \begin 2\sqrt, &\text \displaystyle \omega_m^2 < -2 \frac, \\ ex 0, &\text \displaystyle \omega_m^2 \ge - 2 \frac, \end 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=-\gamma P/\beta_2.


Modulation instability in soft systems

Modulation instability of optical fields has been observed in photo-chemical systems, namely, photopolymerizable medium. Modulation instability occurs owing to inherent optical nonlinearity of the systems due to photoreaction-induced changes in the refractive index. 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.


References


Further reading

* {{physical oceanography Nonlinear optics Photonics Water waves Fluid dynamic instabilities