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Dispersion (optics)

In optics, dispersion is a phenomenon that causes the separation of a wave into components with different frequency. It is most often observed in light waves, though it may happen to any kind of wave that interacts with a medium, such as sound waves.

Dispersion is caused by a variation of the speed of a wave with its frequency. In optics, the phase velocity of a wave v in a given medium is given by:

<math>v = \frac{c}{n}</math>

where c is the speed of light in a vacuum and n is the refractive index of the medium.

In general, the refractive index is some function of the frequency ν of the light, thus n = n(ν), or alternately, with respect to the wave's wavelength n = n(λ). The wavelength dependency of a material's refractive index is usually quantified by an empirical formula, the Sellmeier equation.

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 will also vary with wavelength, causing the angular separation of the colors.

For visible light, most transparent materials (e.g. glasses) have:

<math>n(\lambda_{\rm red})<n(\lambda_{\rm yellow})<n(\lambda_{\rm blue}),</math>

or alternatively:

<math>\frac{dn}{d\lambda} < 0</math>,

that is, refractive index n decreasing with increasing wavelength λ. This results in a stronger bending effect for blue light compared to red light, resulting in the well known angular spread of colors.

Group and phase velocity Another consequence of dispersion manifests itself as a temporal effect. The formula above, v = c / n calculates the phase velocity of a wave; this is the velocity at which the phase of any one frequency component of the wave will propagate. This is not the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will progagate. The group velocity vg is given by:

<math>v_g = c \left[ n - \lambda \frac{dn}{d\lambda} \right]^{-1}</math>.

The group velocity vg is often thought of as the velocity at which energy or information is conveyed along the wave. In most cases this is true, and the group velocity can be thought of as the signal velocity of the waveform. In some unusual circumstances, where the wavelength of the light is close to an absorption resonance of the medium, it is possible for the group velocity to exceed the speed of light (vg > c), leading to the conclusion that superluminal (faster than light) communication is possible. In practice, in such situations the distortion and absorption of the wave is such that the value of the group velocity essentially becomes meaningless, and does not represent the true signal velocity of the wave, which stays less than c.

The group velocity itself is usually a function of the wave's frequency. This results in group velocity dispersion (GVD), which causes a short pulse of light to spread in time as a result of different frequency components of the pulse travelling at different velocities. GVD is often quantified as the group delay dispersion parameter:

<math>D = - \frac{\lambda}{c} \left( \frac{d^2 n}{d \lambda^2} \right) </math>.

If D is less than zero, the medium is said to have normal, or positive dispersion. If D is greater than zero, the medium has anomalous, or negative dispersion. If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components travel slower than the lower frequency components. The pulse therefore becomes positively chirped, increasing in frequency with time. Conversely, if a pulse travels through an anomalously dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped (decreasing in frequency with time.)

The result of GVD, whether negative or positive, is temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge together, rendering the bit-stream unintelligible. This limits the length of fiber that a signal can be sent down without regeneration. One possible answer to this problem is to send signals down the optical fibre with a wavelength of ~1.3 μm; this is a wavelength at which GVD is zero in silica glass, so pulses at this wavelength suffer minimal spreading from dispersion. Another possible option is to use soliton pulses, a form of optical pulse which uses a nonlinear optical effect to self-maintain its shape.

Dispersion control is also important in lasers that produce short pulses. The overall dispersion of the optical resonator is a major factor in determining the duration of the pulses emitted by the laser. A pair of prisms can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium. Diffraction gratings[?] can also be used to produce dispersive effects; these are often used in high-power laser amplifier systems.

Dispersion in gemmology In the technical terminology of gemmology, dispersion is the difference in the refractive index of a material at the B and G Fraunhofer[?] wavelengths of 686.7 nm and 430.8 nm and is simple mean to express the degree to which the gemstone shows fire or color.

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