Modulational instability

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]

The phenomenon was first discovered − and modelled − 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. It is a possible mechanism for the generation of rogue waves.[5][6]

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.[3] (This condition assumes a focussing Kerr nonlinearity, whereby refractive index increases with optical intensity.)[3]

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 [3] by starting with a model of modulation instability based upon the nonlinear Schrödinger equation

which describes the evolution of a complex-valued slowly varying envelope with time and distance of propagation . The imaginary unit satisfies The model includes group velocity dispersion described by the parameter , and Kerr nonlinearity with magnitude A periodic waveform of constant power is assumed. This is given by the solution

where the oscillatory 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

where 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

where the perturbation has been assumed to be small, such that The complex conjugate of is denoted as 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

where and are the wavenumber and (real-valued) angular frequency of a perturbation, and and 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, and 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 and subject to the condition

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

  that is for  

This condition describes the requirement for anomalous dispersion (such that is negative). The gain spectrum can be described by defining a gain parameter as so that the power of a perturbing signal grows with distance as The gain is therefore given by

where as noted above, is the difference between the frequency of the perturbation and the frequency of the initial light. The growth rate is maximum for

Modulation Instability in Soft Systems

Modulation instability of optical fields has been observed in photo-chemical systems, namely, photopolymerizable medium.[7][8][9][10] Modulation instability occurs owing to inherent optical nonlinearity of the systems due to photoreaction-induced changes in the refractive index.[11] 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.[12]


  1. ^ Benjamin, T. Brooke; Feir, J.E. (1967). "The disintegration of wave trains on deep water. Part 1. Theory". Journal of Fluid Mechanics. 27 (3): 417–430. Bibcode:1967JFM....27..417B. doi:10.1017/S002211206700045X.
  2. ^ Benjamin, T.B. (1967). "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems". Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 299 (1456): 59–76. Bibcode:1967RSPSA.299...59B. doi:10.1098/rspa.1967.0123. Concluded with a discussion by Klaus Hasselmann.
  3. ^ a b c d Agrawal, Govind P. (1995). Nonlinear fiber optics (2nd ed.). San Diego (California): Academic Press. ISBN 978-0-12-045142-5.
  4. ^ Yuen, H.C.; Lake, B.M. (1980). "Instabilities of waves on deep water". Annual Review of Fluid Mechanics. 12: 303–334. Bibcode:1980AnRFM..12..303Y. doi:10.1146/annurev.fl.12.010180.001511.
  5. ^ Janssen, Peter A.E.M. (2003). "Nonlinear four-wave interactions and freak waves". Journal of Physical Oceanography. 33 (4): 863–884. Bibcode:2003JPO....33..863J. doi:10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2.
  6. ^ Dysthe, Kristian; Krogstad, Harald E.; Müller, Peter (2008). "Oceanic rogue waves". Annual Review of Fluid Mechanics. 40 (1): 287–310. Bibcode:2008AnRFM..40..287D. doi:10.1146/annurev.fluid.40.111406.102203.
  7. ^ 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. doi:10.1021/ja068967b. ISSN 0002-7863. PMID 17378567.
  8. ^ Basker, Dinesh K.; Brook, Michael A.; Saravanamuttu, Kalaichelvi (2015). "Spontaneous Emergence of Nonlinear Light Waves and Self-Inscribed Waveguide Microstructure during the Cationic Polymerization of Epoxides". The Journal of Physical Chemistry C. 119 (35): 20606–20617. doi:10.1021/acs.jpcc.5b07117.
  9. ^ 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. doi:10.1021/acs.jpcc.5b11377. ISSN 1932-7447.
  10. ^ 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. doi:10.1021/acsmacrolett.6b00659.
  11. ^ Kewitsch, Anthony S.; Yariv, Amnon (1996-01-01). "Self-focusing and self-trapping of optical beams upon photopolymerization" (PDF). Optics Letters. 21 (1): 24. Bibcode:1996OptL...21...24K. doi:10.1364/ol.21.000024. ISSN 1539-4794.
  12. ^ Spatial Solitons | Stefano Trillo | Springer.

Further reading

Index of physics articles (M)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.


In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.

In structural engineering, a structure can become unstable when excessive load is applied. Beyond a certain threshold, structural deflections magnify stresses, which in turn increases deflections. This can take the form of buckling or crippling. The general field of study is called structural stability.

Atmospheric instability is a major component of all weather systems on Earth.

Kerr effect

The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric field instead of varying linearly with it. All materials show a Kerr effect, but certain liquids display it more strongly than others. The Kerr effect was discovered in 1875 by John Kerr, a Scottish physicist.Two special cases of the Kerr effect are normally considered, these being the Kerr electro-optic effect, or DC Kerr effect, and the optical Kerr effect, or AC Kerr effect.


In optics, a nematicon is a spatial soliton in nematic liquid crystals (NLC). The name was invented in 2003 by G. Assanto. and used thereafter Nematicons are generated by a special type of optical nonlinearity present in NLC: the light induced reorientation of the molecular director (i.e. the average molecular orientation). This nonlinearity arises from the fact that the molecular director (i.e., the optic axis of the corresponding uniaxial) tends to align along the electric field of light. Nematicons are easy to generate (with mW optical power or less ) because the NLC dielectric medium exhibits the following properties:

A very large nonlinear response : the effective nonlinearity is typically eight orders of magnitude larger than that of carbon disulfide. This means that much lower optical powers are necessary to obtain the same refractive index variation (increase) or self-focusing to balance out diffraction.

A nonlocal response : the nonlinear response is not limited to the location of the optical field. Instead the response profile is wider than the light beam. A high nonlocality allows for stable soliton propagation even in the case of two transverse dimensions. Higher or lower powers than the exact value required for a soliton to exist lead to breathing solitons.

A saturable all-optical response: the director of the liquid crystal tends to align along the electric field of the light beam. For powerful beams the molecular director becomes parallel to the field and no further reorientation is possible. Response saturation also stabilizes two-dimensional solitons.Since the reorientational optical nonlinearity of nematic liquid crystals is accompanied by an electro-optic response to low-frequency electric fields, i.e. applied voltages, nematicons and the associated waveguides can be steered in angle and routed in space by the application of an external bias, leading to reconfigurable interconnects.In waveguide arrays where discrete solitons are knows to form, discrete nematicons have also been demonstrated

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 typically observed only at very high light intensities (values of atomic electric fields, typically 108 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.

The first nonlinear optical effect to be predicted was two-photon absorption, by Maria Goeppert Mayer for her PhD in 1931, but it remained an unexplored theoretical curiosity until 1961 and the almost simultaneous observation of two-photon absorption at Bell Labs

and the discovery of second-harmonic generation by Peter Franken et al. at University of Michigan, both shortly after the construction of the first laser by Theodore Maiman. However, some nonlinear effects were discovered before the development of the laser. The theoretical basis for many nonlinear processes were first described in Bloembergen's monograph "Nonlinear Optics".

Optical rogue waves

Optical rogue waves are rare pulses of light analogous to rogue or freak ocean waves. The term optical rogue waves was coined to describe rare pulses of broadband light arising during the process of supercontinuum generation—a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input waveform—in nonlinear optical fiber. In this context, optical rogue waves are characterized by an anomalous surplus in energy at particular wavelengths (e.g., those shifted to the red of the input waveform) and/or an unexpected peak power. These anomalous events have been shown to follow heavy-tailed statistics, also known as L-shaped statistics, fat-tailed statistics, or extreme-value statistics. These probability distributions are characterized by long tails: large outliers occur rarely, yet much more frequently than expected from Gaussian statistics and intuition. Such distributions also describe the probabilities of freak ocean waves and various phenomena in both the man-made and natural worlds. Despite their infrequency, rare events wield significant influence in many systems. Aside from the statistical similarities, light waves traveling in optical fibers are known to obey the similar mathematics as water waves traveling in the open ocean (the nonlinear Schrödinger equation), supporting the analogy between oceanic rogue waves and their optical counterparts. More generally, research has exposed a number of different analogies between extreme events in optics and hydrodynamic systems. A key practical difference is that most optical experiments can be done with a table-top apparatus, offer a high degree of experimental control, and allow data to be acquired extremely rapidly. Consequently, optical rogue waves are attractive for experimental and theoretical research and have become a highly studied phenomenon. The particulars of the analogy between extreme waves in optics and hydrodynamics may vary depending on the context, but the existence of rare events and extreme statistics in wave-related phenomena are common ground.

Outline of oceanography

The following outline is provided as an overview of and introduction to Oceanography.

Padma Kant Shukla

Padma Kant Shukla (CorrFRSE, FInstP, FAPS, AFTWAS) (7 July 1950 – 26 January 2013) was a Distinguished Professor and first International Chair of the Physics and Astronomy Department of Ruhr-University Bochum (RUB) in Germany. He was also the Director of the International Centre for Advanced Studies in Physical Sciences at RUB. He held a Ph.D. in Physics from Banaras Hindu University (BHU) in Varanasi, India and a second doctorate in Theoretical Plasma Physics from Umeå University (UmU) in Sweden.


The photon is a type of elementary particle. It is the quantum of the electromagnetic field including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force (even when static via virtual particles). The invariant mass of the photon is zero; it always moves at the speed of light in a vacuum.

Like all elementary particles, photons are currently best explained by quantum mechanics and exhibit wave–particle duality, exhibiting properties of both waves and particles. For example, a single photon may be refracted by a lens and exhibit wave interference with itself, and it can behave as a particle with definite and finite measurable position or momentum, though not both at the same time as per Heisenberg's uncertainty principle. The photon's wave and quantum qualities are two observable aspects of a single phenomenon—they cannot be described by any mechanical model; a representation of this dual property of light that assumes certain points on the wavefront to be the seat of the energy is not possible. The quanta in a light wave are not spatially localized.

The modern concept of the photon was developed gradually by Albert Einstein in the early 20th century to explain experimental observations that did not fit the classical wave model of light. The benefit of the photon model is that it accounts for the frequency dependence of light's energy, and explains the ability of matter and electromagnetic radiation to be in thermal equilibrium. The photon model accounts for anomalous observations, including the properties of black-body radiation, that others (notably Max Planck) had tried to explain using semiclassical models. In that model, light is described by Maxwell's equations, but material objects emit and absorb light in quantized amounts (i.e., they change energy only by certain particular discrete amounts). Although these semiclassical models contributed to the development of quantum mechanics, many further experiments beginning with the phenomenon of Compton scattering of single photons by electrons, validated Einstein's hypothesis that light itself is quantized. In December 1926, American physical chemist Gilbert N. Lewis coined the widely adopted name "photon" for these particles in a letter to Nature. After Arthur H. Compton won the Nobel Prize in 1927 for his scattering studies, most scientists accepted that light quanta have an independent existence, and the term "photon" was accepted.

In the Standard Model of particle physics, photons and other elementary particles are described as a necessary consequence of physical laws having a certain symmetry at every point in spacetime. The intrinsic properties of particles, such as charge, mass, and spin, are determined by this gauge symmetry. The photon concept has led to momentous advances in experimental and theoretical physics, including lasers, Bose–Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics. It has been applied to photochemistry, high-resolution microscopy, and measurements of molecular distances. Recently, photons have been studied as elements of quantum computers, and for applications in optical imaging and optical communication such as quantum cryptography.

Plasma stability

The stability of a plasma is an important consideration in the study of plasma physics. When a system containing a plasma is at equilibrium, it is possible for certain parts of the plasma to be disturbed by small perturbative forces acting on it. The stability of the system determines if the perturbations will grow, oscillate, or be damped out.

In many cases, a plasma can be treated as a fluid and its stability analyzed with magnetohydrodynamics (MHD). MHD theory is the simplest representation of a plasma, so MHD stability is a necessity for stable devices to be used for nuclear fusion, specifically magnetic fusion energy. There are, however, other types of instabilities, such as velocity-space instabilities in magnetic mirrors and systems with beams. There are also rare cases of systems, e.g. the field-reversed configuration, predicted by MHD to be unstable, but which are observed to be stable, probably due to kinetic effects.

Rogue wave

Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unexpected and suddenly appearing surface waves that can be extremely dangerous, even to large ships such as ocean liners.Rogue waves present considerable danger for several reasons: they are rare, unpredictable, may appear suddenly or without warning, and can impact with tremendous force. A 12-metre (39 ft) wave in the usual "linear" wave model would have a breaking pressure of 6 metric tons per square metre [t/m2] (59 kPa; 8.5 psi). Although modern ships are designed to tolerate a breaking wave of 15 t/m2 (150 kPa; 21 psi), a rogue wave can dwarf both of these figures with a breaking pressure of 100 t/m2 (0.98 MPa; 140 psi).In oceanography, rogue waves are more precisely defined as waves whose height is more than twice the significant wave height (Hs or SWH), which is itself defined as the mean of the largest third of waves in a wave record. Therefore, rogue waves are not necessarily the biggest waves found on the water; they are, rather, unusually large waves for a given sea state. Rogue waves seem not to have a single distinct cause, but occur where physical factors such as high winds and strong currents cause waves to merge to create a single exceptionally large wave.Rogue waves can occur in media other than water. They appear to be ubiquitous in nature and have also been reported in liquid helium, in nonlinear optics and in microwave cavities. Recent research has focused on optical rogue waves which facilitate the study of the phenomenon in the laboratory. A 2015 paper studied the wave behavior around a rogue wave, including optical, and the Draupner wave, and concluded that "rogue events do not necessarily appear without a warning, but are often preceded by a short phase of relative order". A 2012 study confirmed the existence of oceanic rogue holes, the inverse of rogue waves, where the depth of the hole can reach more than twice the significant wave height.


Self-focusing is a non-linear optical process induced by the change in refractive index of materials exposed to intense electromagnetic radiation. A medium whose refractive index increases with the electric field intensity acts as a focusing lens for an electromagnetic wave characterized by an initial transverse intensity gradient, as in a laser beam. The peak intensity of the self-focused region keeps increasing as the wave travels through the medium, until defocusing effects or medium damage interrupt this process. Self-focusing of light was discovered by Gurgen Askaryan.

Self-focusing is often observed when radiation generated by femtosecond lasers propagates through many solids, liquids and gases. Depending on the type of material and on the intensity of the radiation, several mechanisms produce variations in the refractive index which result in self-focusing: the main cases are Kerr-induced self-focusing and plasma self-focusing.

Self-phase modulation

Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction.

An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum.

Self-phase modulation is an important effect in optical systems that use short, intense pulses of light, such as lasers and optical fibre communications systems. It has also been reported for nonlinear sound waves propagating in biological thin films, where the phase modulation results from varying elastic properties of the lipid films.

Silicon photonics

Silicon photonics is the study and application of photonic systems which use silicon as an optical medium. The silicon is usually patterned with sub-micrometre precision, into microphotonic components. These operate in the infrared, most commonly at the 1.55 micrometre wavelength used by most fiber optic telecommunication systems. The silicon typically lies on top of a layer of silica in what (by analogy with a similar construction in microelectronics) is known as silicon on insulator (SOI).

Silicon photonic devices can be made using existing semiconductor fabrication techniques, and because silicon is already used as the substrate for most integrated circuits, it is possible to create hybrid devices in which the optical and electronic components are integrated onto a single microchip. Consequently, silicon photonics is being actively researched by many electronics manufacturers including IBM and Intel, as well as by academic research groups, as a means for keeping on track with Moore's Law, by using optical interconnects to provide faster data transfer both between and within microchips.The propagation of light through silicon devices is governed by a range of nonlinear optical phenomena including the Kerr effect, the Raman effect, two-photon absorption and interactions between photons and free charge carriers. The presence of nonlinearity is of fundamental importance, as it enables light to interact with light, thus permitting applications such as wavelength conversion and all-optical signal routing, in addition to the passive transmission of light.

Silicon waveguides are also of great academic interest, due to their unique guiding properties, they can be used for communications, interconnects, biosensors, and they offer the possibility to support exotic nonlinear optical phenomena such as soliton propagation.

Stokes wave

In fluid dynamics, a Stokes wave is a non-linear 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 series approach, now known as the Stokes expansion – obtained approximate solutions for non-linear wave motion.

Stokes' wave theory is of direct practical use for waves on intermediate and deep water. It is used in the design of coastal and offshore structures, in order to determine the wave kinematics (free surface elevation and flow velocities). The wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. For long waves (as compared to depth) – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small amplitude. In such shallow water, a cnoidal wave theory often provides better periodic-wave approximations.

While, in the strict sense, Stokes wave refers to progressive periodic waves of permanent form, the term is also used in connection with standing waves and even for random waves.

Zero-dispersion wavelength

In a single-mode optical fiber, the zero-dispersion wavelength is the wavelength or wavelengths at which material dispersion and waveguide dispersion cancel one another. In all silica-based optical fibers, minimum material dispersion occurs naturally at a wavelength of approximately 1300 nm. Single-mode fibers may be made of silica-based glasses containing dopants that shift the material-dispersion wavelength, and thus, the zero-dispersion wavelength, toward the minimum-loss window at approximately 1550 nm. The engineering tradeoff is a slight increase in the minimum attenuation coefficient. Such fiber is called dispersion-shifted fiber.

Another way to alter the dispersion is changing the core size and the refractive indices of the material of core and cladding. Because fiber optic materials are already highly optimized for low scattering and high transparency alternative ways to change the refractive index were investigated. As a straightforward solution tapered fibers and holey fibers or photonic crystal fibers (PCF) were produced. Essentially they replace the cladding by air. This improves the contrast of refractive indices by a factor of 10. Therefore, the effective index is changed, especially for longer wavelengths. This type of refractive index change versus wavelength due to different geometry is called waveguide dispersion.

As these narrow waveguides (~1-3 μm core diameter) are combined with ultrashort pulses at the zero-dispersion wavelength pulses are not instantly destroyed by dispersion. After reaching a certain peak power within the pulse the non-linear refractive index starts to play an important role leading to frequency generation processes like self-phase modulation (SPM), modulational instability, soliton generation and soliton fission, cross phase modulation (XPM) and others. All these processes generate new frequency components, meaning that input light with narrow bandwidth expands into a wide range of new colours, through a process called supercontinuum generation.

The term is also used, more loosely, in multi-mode optical fiber. There, it refers to the wavelength at which the material dispersion is minimum, i.e. essentially zero. This is more accurately called the minimum-dispersion wavelength.

Zonal flow (plasma)

In toroidally confined fusion plasma experiments the term zonal flow means a plasma flow within a magnetic surface primarily in the poloidal direction. This usage is inspired by the analogy between the quasi-two-dimensional nature of large-scale atmospheric and oceanic flows, where zonal means latitudinal, and the similarly quasi-two-dimensional nature of low-frequency flows in a strongly magnetized plasma.

Zonal flows in the toroidal plasma context are further characterized by

being localized in their radial extent transverse to the magnetic surfaces (in contrast to global plasma rotation),

having little or no variation in either the poloidal or toroidal direction—they are m = n = 0 modes (where and m and n are the poloidal and toroidal mode numbers, respectively),

having zero real frequency when analyzed by linearization around an unperturbed toroidal equilibrium state (in contrast to the geodesic acoustic mode branch, which has finite frequency).

Arising via a self-organization phenomenon driven by low-frequency drift-type modes, in which energy is transferred to longer wavelengths by modulational instability or turbulent inverse cascade.

Ocean zones
Sea level


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