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# Modulational Instability

Acronym: MI

Definition: a nonlinear optical effect which amplifies modulations of optical power

Alternative term: sideband instability

German: Modulationsinstabilität

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Modulational instabilities (sometimes called sideband instabilities) can result from different kinds of nonlinearities. In the context of optics, and in particular in nonlinear fiber optics, they are usually caused by the Kerr nonlinearity of an optical fiber in conjunction with anomalous chromatic dispersion. Essentially, they imply the amplification of sidebands in the optical spectrum, and lead to increasing oscillations of the optical power.

## Mathematical Description

In the simplest case, one considers one-dimensional propagation of light, for example in a single-mode fiber. The light at a certain longitudinal position <$z$> and time <$t$> can then simply be described with a complex amplitude <$A(t,z)$>, the modulus squared is the optical power. If the only relevant physical effects are the Kerr nonlinearity, quantified with a nonlinear coefficient <$\gamma$> (with units of rad/(W m)) and a frequency-independent group velocity dispersion <$\beta_2$> (i.e., no higher-order dispersion, no propagation losses etc.), the propagation can be described with the nonlinear Schrödinger equation:

$$\frac{{\partial A}}{{\partial z}} + i{\beta _2}\frac{{{\partial ^2}A}}{{\partial {t^2}}} = i\gamma {\left| A \right|^2}A$$

Based on that equation, it can be shown relatively easily that a small sinusoidal amplitude modulation with frequency <$\omega_\textrm{m}$> added to a constant amplitude (with power <$P = |A|^2$>) can be amplified if the following condition is fulfilled:

$$\beta _2^2 \: \omega _{\rm{m}}^2 + 2\gamma \: P \: \beta_2 < 0$$

That is obviously possible if <$\beta_2 < 0$> (anomalous dispersion) and the optical power is high enough. The gain coefficient of the modulation can then be calculated as

$${g_{{\rm{mi}}}} = 2\sqrt { - 2\gamma \: P \: \beta_2 \: \omega _{\rm{m}}^2 - \beta _2^2 \: \omega _{\rm{m}}^4}$$

where the first term under the square root is positive because <$\beta_2 < 0$>. Note that the gain is zero if the above-mentioned condition is not fulfilled; there is then a purely oscillatory behavior.

The resulting nonlinear gain can be interpreted as resulting from parametric amplification by phase-matched four-wave mixing.

## Example Case

As an example, we consider a situation where an intense optical wave with a weak very high frequency modulation (4 THz) is injected into a passive single-mode fiber with anomalous dispersion. Normally, this would lead to strong stimulated Brillouin scattering, but that could be suppressed e.g. by applying the optical wave only for a quite limited time (e.g. 20 ps); therefore, we neglect that effect.

The optical spectrum of our input (Figure 1) shows two weak side lobes around the central spectral component, and there is a noise background resulting from quantum fluctuations, which we simulate semi-classically:

The following diagram, showing the situation after 0.2 m of fiber, exhibits increased side lobes, associated with a significantly amplified power oscillation (not shown). Also, the quantum noise background is amplified within a couple of terahertz around the central wavelength:

After 0.4 m of fiber, the sidelobes got stronger again, and the noise amplification also got more pronounced:

After 0.6 m of fiber:

The simulation has been done with the software RP Fiber Power.

Figure 5 shows the gain of the modulation or instability for three different levels of the optical power. One sees that increasing powers extend not only the magnitude but also the frequency range of the nonlinear gain.

The following diagram shows the resulting amplified modulation in the time domain. It is much stronger than the input modulation, is no longer sinusoidal, and it exhibits some influences of the random quantum noise:

## Variations of Modulation Instability

More complicated modulational instability phenomena can occur under various circumstances. For example, it can be observed in birefringent fibers even if the chromatic dispersion is in the normal dispersion regime; this is called the vector modulational instability. Also, the effect may involve additional fiber modes if the fiber does not have a single-mode design, and may occur in various forms by cross-phase modulation.

Modulational instability also often plays a role in the context of supercontinuum generation, where however the complexity of interacting effects is substantially larger. The modulation or instability is just one of many effects in such a situation. For example, it can be involved in the formation of Raman solitons.

Problems related to the modulation or instability are observed in some telecom systems. Even in the normal dispersion regime, such problems can occur in systems where the signal power is periodically raised with fiber amplifiers. That periodic variation can lead to a kind of quasi-phase matching.

## More to Learn

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### Bibliography

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