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The Cauchy’s dispersion formula is?

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Last updated date: 25th Jul 2024
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Answer
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Hint: Cauchy's transmission equation is an empirical relationship between a transparent material's refractive index and wavelength of light in optics. It was named after Augustin-Louis Cauchy, a mathematician who described it in 1836.


Complete step-by-step solution:
Cauchy's equation in its most general form is
$n(\lambda ) = A + \dfrac{B}{{{\lambda ^2}}} + \dfrac{C}{{{\lambda ^4}}} + ...$
where $n$ denotes the refractive index, $\lambda $ denotes the wavelength, and A, B, C, and so on are coefficients that can be calculated for a substance by fitting the equation to measured refractive indices at known wavelengths. The coefficients are normally expressed in micrometres as the vacuum wavelength.
In most cases, a two-term form of the equation suffices:
$n(\lambda ) = A + \dfrac{B}{{{\lambda ^2}}}$
Cauchy's theory of light-matter interaction, on which this equation was based, was later shown to be incorrect. The equation is only applicable in the visible wavelength region for regions of normal dispersion. The equation becomes unreliable in the infrared, and it is unable to reflect regions of anomalous dispersion. Despite this, it is useful in some applications due to its mathematical simplicity.
The Sellmeier equation is a later development of Cauchy's work that more precisely models a material's refractive index across the ultraviolet, visible, and infrared spectrum and manages anomalously dispersive regions.

 Note:The separation of components in any one spectrum increases continuously and almost uniformly with wavelength (as measured by an optical grating), with the separation being a monotonic property of the dispersion variable.