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Dispersion (optics)

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Dispersion (optics)

In a dispersive prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a rainbow.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.[1] Media having this common property may be termed dispersive media. 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 waves, dispersion 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 waves), and for telecommunication signals propagating 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 lenses, in which chromatic aberration is largely cancelled, uses a quantification of a glass's dispersion given by its Abbe number V, where 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 (GVD).

Contents

  • Examples of dispersion 1
  • Material and waveguide dispersion 2
  • Material dispersion in optics 3
  • Group and phase velocity 4
  • Dispersion in waveguides 5
  • Higher-order dispersion over broad bandwidths 6
  • Dispersion in gemology 7
  • Dispersion in imaging 8
  • Dispersion in pulsar timing 9
  • See also 10
  • References 11
  • External links 12

Examples of dispersion

The most familiar example of dispersion is probably a rainbow, in 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, GVD causes pulses to spread in optical fibers, degrading signals over long distances; also, a cancellation between group-velocity dispersion and nonlinear effects leads to soliton waves.

Material and waveguide dispersion

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 dispersion, in 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.g., a photonic crystal), whether or not the waves are confined to some region. In a waveguide, both types of dispersion will generally be present, although they are not strictly additive.

Material dispersion in optics

The variation of refractive index vs. vacuum wavelength for various glasses. The wavelengths of visible light are shaded in red.
Influences of selected glass component additions on the mean dispersion of a specific base glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm (red))[3]

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 aberration, an undesired effect that may degrade images in microscopes, telescopes and photographic objectives.

The phase velocity, v, of a wave in a given uniform medium is given by

v = \frac{c}{n}

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 f of the light, thus n = n(f), or alternatively, with respect to the wave's wavelength n = n(λ). 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 relations, the wavelength dependence of the real part of the refractive index is related to the material absorption, described by the imaginary part of the refractive index (also called the extinction coefficient). In particular, for non-magnetic materials (μ = μ0), the susceptibility \chi that appears in the Kramers–Kronig relations is the electric susceptibility \chi_e = n^2 - 1.

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.g., air, glasses) decrease with increasing wavelength λ:

1 < n(\lambda_{\rm red}) < n(\lambda_{\rm yellow}) < n(\lambda_{\rm blue})\ ,

or alternatively:

\frac{\nu^2}\right)

where the dispersion constant k_\mathrm{DM} is given by

k_\mathrm{DM} = \frac{e^2}{2 \pi m_\mathrm{e}c} \simeq 4.149 \mathrm{GHz}^2\mathrm{pc}^{-1}\mathrm{cm}^3\mathrm{ms},[11]

and the dispersion measure (\mathrm{DM}) is the column density of free electrons (total electron content) — i.e. the number density of electrons n_e (electrons/cm3) integrated along the path traveled by the photon from the pulsar to the Earth — and is given by

\mathrm{DM} = \int_0^d{n_e\;dl}

with units of parsecs per cubic centimetre (1pc/cm3 = 30.857×1021 m−2).[12]

Typically for astronomical observations, this delay cannot be measured directly, since the emission time is unknown. What can be measured is the difference in arrival times at two different frequencies. The delay \Delta t between a high frequency \nu_{hi} and a low frequency \nu_{lo} component of a pulse will be

\Delta t = k_\mathrm{DM} \times \mathrm{DM} \times \left( \frac{1}{\nu_{\mathrm{lo}}^2} - \frac{1}{\nu_{\mathrm{hi}}^2} \right)

Re-writing the above equation in terms of \Delta t allows one to determine the \mathrm{DM} by measuring pulse arrival times at multiple frequencies. This in turn can be used to study the interstellar medium, as well as allow for observations of pulsars at different frequencies to be combined.

See also

References

  1. ^  
  2. ^ Dispersion Compensation Retrieved 25-08-2015.
  3. ^ Calculation of the Mean Dispersion of Glasses
  4. ^ Brillouin, Léon. Wave Propagation and Group Velocity. (Academic Press: San Diego, 1960). See esp. Ch. 2 by A. Sommerfeld.
  5. ^ Wang, L.J., Kuzmich, A., and Dogariu, A. (2000). "Gain-assisted superluminal light propagation". Nature 406 (6793): 277.  
  6. ^ Stenner, M. D., Gauthier, D. J., and Neifeld, M. A. (2003). "The speed of information in a 'fast-light' optical medium". Nature 425 (6959): 695–8.  
  7. ^ Rajiv Ramaswami and Kumar N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press: London 1998).
  8. ^ Chromatic Dispersion, Encyclopedia of Laser Physics and Technology (Wiley, 2008).
  9. ^ a b Walter Schumann (2009). Gemstones of the World: Newly Revised & Expanded Fourth Edition. Sterling Publishing Company, Inc. pp. 41–2.  
  10. ^ What is Gemstone Dispersion? by International Gem Society (IGS). Retrieved 03-09-2015
  11. ^ Single-Dish Radio Astronomy: Techniques and Applications, ASP Conference Proceedings, Vol. 278. Edited by Snezana Stanimirovic, Daniel Altschuler, Paul Goldsmith, and Chris Salter. ISBN 1-58381-120-6. San Francisco: Astronomical Society of the Pacific, 2002, p. 251-269
  12. ^ Lorimer, D.R., and Kramer, M., Handbook of Pulsar Astronomy, vol. 4 of Cambridge Observing Handbooks for Research Astronomers, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A, 2005), 1st edition.

External links

  • Dispersive Wiki – discussing the mathematical aspects of dispersion.
  • Dispersion – Encyclopedia of Laser Physics and Technology
  • Animations demonstrating optical dispersion by QED
  • Interactive webdemo for chromatic dispersion Institute of Telecommunications, University of Stuttgart
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