Cauchy's equation

The Cauchy equation, also called Cauchy model is a mathematical description of the dispersion of electromagnetic waves in solids over a wide spectral range. It is normally within the range of visible light are used. The empirically determined relationship was published in 1830 by Augustin- Louis Cauchy.

Description

The Cauchy equation is a parametric description of the refractive index of a material as a function of the wavelength in the form:

For most materials but already rich from the first two terms of the series to describe the measured dispersion sufficiently well in a limited spectral range. For this reason, only the parameters A, B, C are given for the description often, this is also true for many optical simulation and analysis programs, such as are used for example in ellipsometry. The following applies:

However, the description is only valid for isotropic, almost perfectly transparent materials. That is, the extinction coefficient of the complex refractive index is very small. In order to describe the transition region to a spectral absorption with sufficiently good, the Cauchy equation to a wavelength- dependent term of the extinction coefficient can be expanded:

Which represent, and corresponding fitting parameters. For the simulation of birefringent so optically anisotropic materials, some analysis programs also offer additional model extensions.

Validity

As already described, the Cauchy equation is valid only in a limited spectral range. The material described in this section may no absorption bands, for example, caused by band transitions have. Therefore, only transparent materials can be sufficiently well described. Physical effects such as abnormal dispersion, as they occur in the area of absorption centers and the absorption behavior itself can not be described, hence no metals.

Sell ​​Wolfgang von Meier published in 1871 an expanded empirical model that is named after him sell Meier equation. It models the refractive index in the ultraviolet and infrared better. However, even this description is limited to wavelengths where the material is transparent. An improved description of the refractive indices for metals followed at the end of the 19th century with the Drude model of Paul Drude for metals. Hendrik A. Lorentz succeeded to the model of the Lorentz oscillator to combine the approaches of Drude and Sell Meier.