optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...

, Cauchy's transmission equation is an

empirical relationship In science, an empirical relationship or phenomenological relationship is a relationship or correlation that is supported by experiment and observation but not necessarily supported by theory. Analytical solutions without a theory An empirical rel ...

between the

refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, o ...

and

wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...

of light for a particular transparent material. It is named for the mathematician

Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...

, who defined it in 1837.

The equation

The most general form of Cauchy's equation is :

n(\lambda) = A + \frac  + \frac + \cdots,

where ''n'' is the refractive index, λ is the wavelength, ''A'', ''B'', ''C'', etc., are

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

s that can be determined for a material by fitting the equation to measured refractive indices at known wavelengths. The coefficients are usually quoted for λ as the vacuum wavelength in

micrometre The micrometre ( international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer ( American spelling), also commonly known as a micron, is a unit of length in the International System of ...

s. Usually, it is sufficient to use a two-term form of the equation: :

n(\lambda) = A + \frac,

where the coefficients ''A'' and ''B'' are determined specifically for this form of the equation. A table of coefficients for common optical materials is shown below: The theory of light-matter interaction on which Cauchy based this equation was later found to be incorrect. In particular, the equation is only valid for regions of normal

dispersion Dispersion may refer to: Economics and finance * Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item * Wage dispersion, the amount of variat ...

in the

visible wavelength The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. Electromagnetic radiation in this range of wavelengths is called ''visible light'' or simply light. A typical human eye will respond to wav ...

region. In the

infrared Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from aroun ...

, the equation becomes inaccurate, and it cannot represent regions of anomalous dispersion. Despite this, its mathematical simplicity makes it useful in some applications. The

Sellmeier equation The Sellmeier equation is an empirical relationship between refractive index and wavelength for a particular transparent medium. The equation is used to determine the dispersion of light in the medium. It was first proposed in 1872 by Wol ...

is a later development of Cauchy's work that handles anomalously dispersive regions, and more accurately models a material's refractive index across the

ultraviolet Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30 PHz) to 400 nm (750 THz), shorter than that of visible light, but longer than X-rays. UV radiati ...

, visible, and infrared spectrum.

Humidity dependence for air

Cauchy's two-term equation for air, expanded by Lorentz to account for humidity, is as follows: :

n_(\lambda, T, v, p) \approx 1 + \frac \left(1+ \frac \right)\left(p + 4810 \frac\right)

where ''p'' is the air pressure in millibar, ''T'' is the temperature in kelvin, and ''v'' is the

vapor pressure of water The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The saturation vapour pressure is the pressure at which water vapour is in thermodynamic ...

in millibar.

References

{{Reflist *F.A. Jenkins and H.E. White, ''Fundamentals of Optics'', 4th ed., McGraw-Hill, Inc. (1981). Augustin-Louis Cauchy Optics Equations Electric and magnetic fields in matter

The equation

Humidity dependence for air

See also

References