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Anomalous dispersion group velocity

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The chromatic dispersion of an optical medium is the phenomenon that the phase velocity and group velocity of light propagating in a transparent medium depend on the optical frequency ; that dependency results mostly from the interaction of light with electrons of the medium.

A related quantitative measure is the group velocity dispersion. Chromatic dispersion can also occur from geometrical effects, for example; see below the section on chromatic dispersion of optical components.

An early measure for the magnitude of chromatic dispersion was the Abbe number V Dintroduced by Ernst Abbe:. The denominator is also called the principal dispersion. The Abbe number depends on the refractive indices at only three different wavelengths:. Large values of the Abbe number indicate low chromatic dispersion and vice versa.

Such values can be used for the design of achromatic optical elementsbut can of course give only a quite rough indication. It is the derivative of the inverse group velocity with respect to angular frequency:.

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The dispersion of various orders for a medium can most conveniently be calculated if the refractive index is specified with a kind of Sellmeier formula. Tabulated index data are less suitable, since the numerical differentiation is sensitive to noise. Normal dispersion, where the group velocity decreases with increasing optical frequencyoccurs for most transparent media in the visible spectral region.

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Anomalous dispersion sometimes occurs at longer wavelengthse. Great care is recommended when the sign of dispersion is specified. The ultrafast optics community identifies that sign with the sign of k''. The opposite sign is usually used in optical fiber communicationswhere the dispersion is often specified with the dispersion parameter. The different signs result from using a frequency derivative in one case and a wavelength derivative in the other.The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. The expanding ring of waves is the wave groupwithin which one can discern individual wavelets of differing wavelengths traveling at different speeds. The shorter waves travel faster than the group as a whole, [ citation needed ] [ dubious — discuss ] but their amplitudes diminish as they approach the leading edge of the group.

The longer waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group.

The group velocity v g is defined by the equation: [2] [3] [4] [5]. One derivation of the formula for group velocity is as follows. By the superposition principlethe wavepacket at any time t is. There are two factors in this expression. Part of the previous derivation is the Taylor series approximation that:.

As a result, the envelope of the wave packet not only moves, but also distorts, in a manner that can be described by the material's group velocity dispersion. Loosely speaking, different frequency-components of the wavepacket travel at different speeds, with the faster components moving towards the front of the wavepacket and the slower moving towards the back.

Eventually, the wave packet gets stretched out. This is an important effect in the propagation of signals through optical fibers and in the design of high-power, short-pulse lasers.

The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.

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Hamilton inand the first full treatment was by Rayleigh in his "Theory of Sound" in For waves traveling through three dimensions, such as light waves, sound waves, and matter waves, the formulas for phase and group velocity are generalized in a straightforward way: [10]. If the waves are propagating through an anisotropic i.

The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform. However, if the wave is travelling through an absorptive or gainful medium, this does not always hold. In these cases the group velocity may not be a well-defined quantity, or may not be a meaningful quantity.In opticsdispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.

Sometimes the term chromatic dispersion is used for specificity. Although the term is used in the field of optics to describe light and other electromagnetic wavesdispersion in the same sense can apply to any sort of wave motion such as acoustic dispersion in the case of sound and seismic waves, in gravity waves ocean wavesand for telecommunication signals along transmission lines such as coaxial cable or optical fiber.

In optics, one important and familiar consequence of dispersion is the change in the angle of refraction of different colors of light, [2] as seen in the spectrum produced by a dispersive prism and in chromatic aberration of lenses.

Design of compound achromatic lensesin which chromatic aberration is largely cancelled, uses a quantification of a glass's dispersion given by its Abbe number Vwhere lower Abbe numbers correspond to greater dispersion over the visible spectrum. In some applications such as telecommunications, the absolute phase of a wave is often not important but only the propagation of wave packets or "pulses"; in that case one is interested only in variations of group velocity with frequency, so-called group-velocity dispersion.

The most familiar example of dispersion is probably a rainbowin which dispersion causes the spatial separation of a white light into components of different wavelengths different colors. However, dispersion also has an effect in many other circumstances: for example, group velocity dispersion GVD causes pulses to spread in optical fibersdegrading signals over long distances; also, a cancellation between group-velocity dispersion and nonlinear effects leads to soliton waves.

Most often, chromatic dispersion refers to bulk material dispersion, that is, the change in refractive index with optical frequency. However, in a waveguide there is also the phenomenon of waveguide dispersionin which case a wave's phase velocity in a structure depends on its frequency simply due to the structure's geometry.

More generally, "waveguide" dispersion can occur for waves propagating through any inhomogeneous structure e. Material dispersion can be a desirable or undesirable effect in optical applications. The dispersion of light by glass prisms is used to construct spectrometers and spectroradiometers. Holographic gratings are also used, as they allow more accurate discrimination of wavelengths.

However, in lenses, dispersion causes chromatic aberrationan undesired effect that may degrade images in microscopes, telescopes, and photographic objectives. The phase velocityvof a wave in a given uniform medium is given by. The wavelength dependence of a material's refractive index is usually quantified by its Abbe number or its coefficients in an empirical formula such as the Cauchy or Sellmeier equations. Because of the Kramers—Kronig relationsthe wavelength dependence of the real part of the refractive index is related to the material absorptiondescribed by the imaginary part of the refractive index also called the extinction coefficient.

The most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrum by a prism. From Snell's law it can be seen that the angle of refraction of light in a prism depends on the refractive index of the prism material.

Since that refractive index varies with wavelength, it follows that the angle that the light is refracted by will also vary with wavelength, causing an angular separation of the colors known as angular dispersion. For visible light, refraction indices n of most transparent materials e. In this case, the medium is said to have normal dispersion. Whereas, if the index increases with increasing wavelength which is typically the case in the ultraviolet [4]the medium is said to have anomalous dispersion.

Thus, blue light, with a higher refractive index, will be bent more strongly than red light, resulting in the well-known rainbow pattern. Beyond simply describing a change in the phase velocity over wavelength, a more serious consequence of dispersion in many applications is termed group velocity dispersion. When different frequency components are combined together, as when considering a signal or a pulse, one is often more interested in the group velocity which describes the speed at which a pulse or information superimposed on a wave modulation propagates.

In the accompanying animation, it can be seen that the wave itself orange-brown travels at a phase velocity which is much faster than the speed of the envelope black which corresponds to the group velocity. This pulse might be a communications signal, for instance, and its information only travels at the group velocity rate even though it consists of wavefronts advancing at a faster rate the phase velocity.

Or in terms of the phase velocity v p. When dispersion is present, not only will the group velocity not be equal to the phase velocity, but generally will itself vary with wavelength. This is known as group velocity dispersion GVD and causes a short pulse of light to be broadened, as the different frequency components within the pulse travel at different velocities. If a light pulse is propagated through a material with positive group-velocity dispersion, then the shorter wavelength components travel slower than the longer wavelength components.

The pulse therefore becomes positively chirpedor up-chirpedincreasing in frequency with time. On the other hand, if a pulse travels through a material with negative group-velocity dispersion, shorter wavelength components travel faster than the longer ones, and the pulse becomes negatively chirpedor down-chirpeddecreasing in frequency with time.The amplitude of the de Broglie wave is associated with the moving body represents the probability of finding a body at a particular time and space.

Thus, the de Broglie waves can be obtained by the superposition of the several waves. Let us find the velocity of the wave packet or de Broglie wave. We assume that the wave packet or group is formed due to combination of the two waves having equal amplitude and different their angular frequency by dw and wave velocity dk.

The waves are represented mathematically as. According to the superposition principle, the resultant displacement Y at any time t at any point x is the sum of two displacements. This equation represents a wave of angular frequency w and wave number k moving in the same direction superimposed by a modulated wave of angular frequency dw and wave vector dk.

Thus, the superposition of the two waves results a new waves and successively they form a new wave. The group velocity of the de Broglie wave depends upon the manner in which the phase velocity of the medium varies or constant.

This shows that the phase velocity of the de Broglie wave is same as the wave-velocity. Let us consider a de Broglie wave associated with a moving particle of a rest mass m 0 and velocity v.

The angular frequency of the de Broglie wave is given by. For electromagnetic waves in vacuum, the speed of light c is constant. Therefore group velocity vg and the phase velocity v, of the light radiations are same.

Physics Assignment. Phase and Group Velocities of the de Broglie Wave. The group of wave need not have the same velocity as the individual one.

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Equation 1 describes the de Broglie wave or wave packet or wave group.Note: this box searches only for keywords in the titles of encyclopedia articles. For full-text searches on the whole website, use our search page. Note: the article keyword search field and some other of the site's functionality would require Javascript, which however is turned off in your browser.

Group velocity dispersion is the phenomenon that the group velocity of light in a transparent medium depends on the optical frequency or wavelength. The term can also be used as a precisely defined quantity, namely the derivative of the inverse group velocity with respect to the angular frequency or sometimes the wavelength :. The group velocity dispersion is the group delay dispersion per unit length. Somewhere between these wavelengths at about 1.

For optical fibers e. This can be calculated from the above-mentioned GVD parameter:. Enter input values with units, where appropriate.

anomalous dispersion group velocity

After you have modified some values, click a "calc" button to recalculate the field left of it. Attention: The buttons do not work, as Javascript is turned off in your browser! In order to avoid confusion, the terms normal and anomalous dispersion can be used instead of positive and negative dispersion. Normal dispersion implies that the group velocity decreases for increasing optical frequency; this occurs in most cases.

I am wondering about the example numbers you are giving here. Why is there a difference or am I missing something here? Answer from the author :. The calculator is not calculating the chromatic dispersion of silica, but only converting chromatic dispersion values given with different units.

In order to get the shortest possible pulse duration from a mode-locked laser, should one minimize the total GDD inside the laser resonator? The shortest possible pulse duration is often achieved when the total group delay dispersion is negative, so that quasi- soliton pulses are formed. However, that depends on the type of laser.

To which extent spectral variations of the GDD matter, depends on the bandwidth of the pulses to be generated. Certainly, the GDD in spectral regions totally outside the pulse spectrum does not matter. On the other hand, it is not necessarily required that the GDD is hardly varying within the pulse bandwidth. A better criterion is that the round-trip phase shift should not very too much within the spectrum.

However, it again depends on the type of laser how critical that is. Modal dispersion, or more precisely intermodal dispersionis related to differences between different propagation modes, not within a single mode.

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Group Velocity Dispersion

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anomalous dispersion group velocity

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Group velocity

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Group Velocity / Phase Velocity Animation - Case 5: Positive Phase and Negative Group Velocity

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anomalous dispersion group velocity

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