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Cauchy's Equation for finding the refractive index for a light of given wavelength is:

$$n(\lambda)=A+\dfrac{B}{\lambda^2}+\dfrac{C}{\lambda^4}.....$$

This formula however is simplified to

$n(\lambda)=A+\dfrac{B}{\lambda^2}$

by neglecting higher order terms.

This is what I don' t understand. Wavelength of visible light is approximately $6\cdot10^{-7}m$ which is less than $1$. Shouldn't the contribution of higher order terms be more than lower order terms.

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The Cauchy equation is empirical relationship.

However, the refractive index can be obtained from the classical Lorentz model where a light wave creates oscillatory motion of the electrons and the electron displacements form dipole moments. This polarizes the medium , and the refractive index can be estimated theoretically. (https://www.phys.ksu.edu/personal/cdlin/class/class02a/s2-jing-li.ppt)

$$n=\sqrt{1+\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega} }$$

$\omega_p$ is the plasma frequency and $\omega_0$ are the resonance absorption edges. If we can neglect the absorption $\gamma=0$ .

$$n=\sqrt{1+\frac{p^2}{1-x^2}}\propto a+bx^2+cx^4...$$

Where $x=\frac{\omega}{\omega_0}$~$\frac{\lambda_0}{\lambda}$, $p=\omega_p/\omega_0$= constant

When we Taylor expand we see that in some approximation the empirical equation is justified. And we can expect that the Cauchy equation would fit the refractive index in limited spectral regions, for some materials.

Higher orders do not add much. More often is used the Sellmeier equation which describes better the behavior of the refractive index.

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