The Cauchy dispersion formula is valid only for visible light . What the general theory which is valid for a broad range of em frequencies ?
I dont think there is a general way to describe dispersion across all frequencies in matter. Generally you can describe the medium response using Maxwell's equations, but because of the vast range of wavelengths, the techniques used to derive dispersion relations and the index of refraction cannot be applied from frequency band one to another.
Additionally, the Cauchy dispersion relation is an empirical formula meaning that it was calculated using approximations, fitting, and experimental data. Dispersion relations in general are complicated to calculate as different materials have resonant absorption bands for various wavelengths. For instance, optical frequencies can use the Drude model for deriving dispersion relations to some degree of accuracy, but the same relations cannot be applied in the x-ray band or the radio wave band where the material response is very different.
If you consider high frequency radiation (x-ray or gamma-ray), most materials dont respond to that high a frequency as the electron plasma frequency in the material is much lower than the incident wavelength. In these cases, light is typically viewed as a particle and having some cross section of interaction in a given material. On the other hand, radio wavelengths are so long that they can penetrate most materials with ease with little attenuation.
If you ask about refractive lens theory, then I can say that lens calculations should be valid for any wavelength, if you use correct refraction index value and consider lens aberrations, and consider diffraction, when lens dimensions and the light wavelength are of the same order. Cauchi equation is a convenient approximation for practical use, in general, in visible range. But it is empirical formula, nor related to real physical model of optical media.
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