Editing Sellmeier equation (section)

==Derivation==
Analytically, the Sellmeier equation models the refractive index as due to a series of optical resonances within the bulk material. Its derivation from the [[Kramers-Kronig relations]] requires a few assumptions about the material, from which any deviations will affect the model's accuracy:
*There exists a number of resonances, and the final refractive index can be calculated from the sum over the contributions from all resonances.
*All optical resonances are at wavelengths far away from the wavelengths of interest, where the model is applied.
*At these resonant frequencies, the imaginary component of the susceptibility (<math>{\chi_i}</math>) can be modeled as a [[delta function]].

From the last point, the complex refractive index (and the [[electric susceptibility]]) becomes:
:<math>\chi_i(\omega) = \sum_i A_i \delta(\omega-\omega_i)</math>

The real part of the refractive index comes from applying the Kramers-Kronig relations to the imaginary part:

:<math> n^2 = 1 + \chi_r(\omega) = 1 + \frac{2}{\pi}\int_0^\infty \frac{\omega \chi_i(\omega)}{\omega ^2 - \Omega ^2}d\omega</math>

Plugging in the first equation above for the imaginary component:

:<math> n^2 = 1 + \frac{2}{\pi}\int_0^\infty \sum_i A_i \delta(\omega-\omega_i) \frac{\omega}{\omega ^2 - \Omega ^2}d\omega</math>

The order of summation and integration can be swapped. When evaluated, this gives the following, where <math>H</math> is the [[Heaviside function]]:
:<math> n^2 = 1 + \frac{2}{\pi} \sum_i A_i \int_0^\infty \delta(\omega-\omega_i) \frac{\omega}{\omega ^2 - \Omega ^2}d\omega = 1 + \frac{2}{\pi} \sum_i A_i \frac{\omega_i H(\omega_i)}{\omega_i^2-\Omega^2}</math>

Since the domain is assumed to be far from any resonances (assumption 2 above), <math>H(\omega_i)</math> evaluates to 1 and a familiar form of the Sellmeier equation is obtained:
:<math> n^2 = 1 + \frac{2}{\pi} \sum_i A_i \frac{\omega_i}{\omega_i^2-\Omega^2}</math> 

By rearranging terms, the constants <math>B_i</math> and <math>C_i</math> can be substituted into the equation above to give the Sellmeier equation.<ref name="b841"></ref>