# Modulational Instability

Acronym: MI

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

Alternative term: sideband instability

German: Modulationsinstabilität

Categories: nonlinear optics, physical foundations

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/o7s

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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 optical signal with a weak very high frequency modulation (4 THz) is injected into a passive single-mode fiber with anomalous dispersion. Only during a short time interval (e.g. 20 ps) of an ultrashort pulse, we assume the average power to be quite high (3.5 kW). Problems with stimulated Brillouin scattering are avoided in that way.

The optical spectrum of the original pulse (Figure 1) shows two weak side lobes around the central spectral component, and there is a noise background resulting from quantum fluctuations:

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.

### Bibliography

[1] | P. K. Shukla and J. Juul Rasmussen, “Modulational instability of short pulses in long optical fibers”, Opt. Lett. 11 (3), 171 (1986); https://doi.org/10.1364/OL.11.000171 |

[2] | M. Nakazawa, K. Suzuki and H. A. Haus, “Modulational instability oscillation in nonlinear dispersive ring cavity”, Phys. Rev. A 38, 5193 (1988); https://doi.org/10.1103/PhysRevA.38.5193 |

[3] | S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers”, Opt. Lett. 16 (13), 986 (1991); https://doi.org/10.1364/OL.16.000986 |

[4] | G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating light waves”, J. Opt. Soc. Am. B 9 (7), 1047 (1992); https://doi.org/10.1364/JOSAB.9.001047 |

[5] | E. A. Golovchenko and A. N. Pilipetskii, “Unified analysis of four-photon mixing, modulational instability, and stimulated Raman scattering under various polarization conditions in fibers”, J. Opt. Soc. Am. B 11 (1), 92 (1994); https://doi.org/10.1364/JOSAB.11.000092 |

[6] | C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings”, J. Opt. Soc. Am. B 15 (11), 2660 (1998); https://doi.org/10.1364/JOSAB.15.002660 |

[7] | P. M. Lushnikov, P. Lodahl and M. Saffman, “Transverse modulational instability of counterpropagating quasi-phase-matched beams in a quadratically nonlinear medium”, Opt. Lett. 23 (21), 1650 (1998); https://doi.org/10.1364/OL.23.001650 |

[8] | S. Pitois, M. Haelterman and G. Millot, “Bragg modulational instability induced by a dynamic grating in an optical fiber”, Opt. Lett. 26 (11), 780 (2001); https://doi.org/10.1364/OL.26.000780 |

[9] | G. Millot, “Multiple four-wave mixing–induced modulational instability in highly birefringent fibers”, Opt. Lett. 26 (18), 1391 (2001); https://doi.org/10.1364/OL.26.001391 |

[10] | T. Tanemura and K. Kikuchi, “Unified analysis of modulational instability induced by cross-phase modulation in optical fibers”, J. Opt. Soc. Am. B 20 (12), 2502 (2003); https://doi.org/10.1364/JOSAB.20.002502 |

[11] | S. Longhi, “Modulational instability and space–time dynamics in nonlinear parabolic-index optical fibers”, Opt. Lett. 28 (23), 2363 (2003); https://doi.org/10.1364/OL.28.002363 |

[12] | A. Armaroli and S. Trillo, “Modulational instability due to cross-phase modulation versus multiple four-wave mixing: the normal dispersion regime”, J. Opt. Soc. Am. B 31 (3), 551 (2014); https://doi.org/10.1364/JOSAB.31.000551 |

[13] | G. P. Agrawal, Nonlinear Fiber Optics, 4th edn., Academic Press, New York (2007) |

See also: nonlinearities, Kerr effect, supercontinuum generation

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