Dispersion

The Speed of light in the Vide does not depend on its Wavelength. But in certain mediums, it can depend on it: it is the phenomenon of dispersion . One meets this phenomenon for all types of waves, like the his or the Vague S.

A medium causing a certain dispersion is then known as dispersive .

Waves in a dispersive medium

See also: Relation of dispersion

A sinusoidal wave is characterized by its Fréquence $\backslash \; nu$ or its Pulsation $\backslash \; Omega\; =\; 2\; \backslash \; pi\; \backslash \; nu$ (in rad/S), and by its Vecteur of wave of standard $k=2\; \backslash \; pi\; \backslash \; lambda$ (in rad m), where $\backslash \; lambda$ is the wavelength.

It is observed whereas there exist two different characteristic speeds:

• the Speed of phase $v\_\; \backslash \; phi=\; \backslash \; frac\; \{\backslash \; Omega\}\; \{K\}$ which corresponds to the displacement of the plane of wave;
• and the Speed of partial group $v\_g=\; \backslash \; frac\; \{\backslash \; \backslash \; Omega\}\; \{\backslash \; partial\; K\}$ which corresponds to the displacement of the envelope of the wave, or in other words, of energy.

These two speeds then have behaviors different according to the type of medium.

• If the medium is not-dispersive, i.e. $v\_\; \backslash \; phi$ does not depend on K , then $\backslash \; omega$ is inevitably proportional to K . It is thus obtained that two speeds are equal, and constant.
• If the medium is dispersive, these two speeds are not then equal any more, and depend on K .

These properties are remarkable in the study of the propagation of a Paquet of wave: it is, by definition, a superposition of several sinusoidal waves various wavelengths. One can show that its speed corresponds at the speed of group.

In a not-dispersive medium, like $v\_g=v\_\; \backslash \; phi$, the speed of the package of waves is the same one as the sinusoidal waves which compose it. Famous following animation this phenomenon:

In a dispersive medium, the package of wave does not go at the same speed as the waves which compose it, as shows it following animation:

In optics

Variation of the index of refraction

A medium is characterized by its Index of refraction N = C / v , where C is the celerity of the light in the vacuum and v its speed in the medium considered. A variation of this index on the way of a luminous ray will cause its deviation, or rather Réfraction according to the Lois of Snell-Descartes.

Thus, when the propagation medium are dispersive, the propagation velocity, and thus the index of refraction depend on the wavelength: the deviation of the rays depends on the wavelength, i.e. of the color. That is observed for materials like glass: the blue rays are deviated than the red rays and the colors are thus separated. This observation is known in the case of the prism S.

The variation of the index of refraction of a transparent medium in the visible light follows a law known as of Cauchy: $n\; (\backslash \; lambda)\; =\; has\; +\; B\; \{\backslash \; lambda\; ^2\}$.

Consequences

A luminous Onde is characterized by its spectrum, which is the distribution of the intensity emitted according to the Wavelength. In the case of the visible light, the wavelength is connected to the color perceived by the eye. In general, a light wave is polychromatic, i.e. it is made up several wavelengths. Thus, the sunlight contains the majority of the visible colors. Dispersion will make it possible to separate them and visualize thus the colors which compose the radiation, which in particular makes it possible to make Spectroscopie.

One of the visible examples in the everyday life is the Arc-en-ciel. The observable rainbow outside is the result of the dispersion of the sunlight by the water droplets in suspension in the Air.

But dispersion results also in to limit the performances of the optical systems. One can quote:

• chromatic dispersion in the fiberoptics limiting the Band-width of a transmission. In a dispersive medium, each wavelength is propagated at a different speed, from where a temporal widening of an impulse during its transmission. It is for this reason, inter alia, that one uses lasers diode whose spectral width is low.
• the chromatic aberrations in the optical systems involving a deviation different each wavelength. It follows a convergence point different each wavelength, thus causing an erroneous color application of the image. A solution consists in carrying out a achromatic Doublet (addition of two different material lenses with different Constringence S).

Characterization of the dispersive mediums

To measure an index of refraction in a dispersive medium, one needs a radiation Monochromatique of reference, as the line D of helium (wavelength 587,6 Nm), near to the medium of the visible spectrum, which is often used.

For radiation D, the absolute index $n\_D$ of the Eau to 20°C is of 1,333; that of a ordinary Verre lies between 1,511 to 1,535. The index of the Air is equal to 1,000  292  6 under the normal conditions of temperature and pressure.

In the field of visible (wavelengths ranging between 380 Nm and 780 Nm) dispersion is characterized by the Constringence. One classifies then glasses in type Crown or Flint according to whether the constringence is lower or higher than 50.

$\backslash \; naked\; =\; \backslash \; frac\; \{n\_D\; -\; 1\}\; \{n\_F\; -\; n\_C\}$,

F and C indicating two lines of hydrogen (wavelengths $\backslash \; lambda\_F$ = 486,1 Nm and $\backslash \; lambda\_C$ = 656,3 Nm)

Dispersion by a prism

Dispersion by materials such as glass was used to analyze the sunlight. The experiments of Newton using prisms are famous. In the experiments of demonstration, one always uses prisms which, when the light crosses the two diopters, allow a good dispersion of the colors.

In practice, one uses also “prisms with direct vision” which are actually optical systems made up of 3 prisms joined, chosen in order to minimize the deviation while optimizing dispersion.

Dispersion by a network

The word “dispersion” is also used to indicate the separation of the chromatic components by a network. However, the phenomenon is of very an other nature. The effect obtained does not come from the dispersive character of a medium but from the Interférence S between the rays transmitted by the network.

Dispersion in other fields of physics

Dispersion intervenes in all the types of waves in physics. One can quote for example the Vague S when the sea-bed is not flat, the waves of plasma, the his…

Related articles

• Constringence
• Refraction
• Speed of phase

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