# I Theoretical explanation for Dispersion? (Sellmeier)

1. Dec 1, 2015

### greswd

One of the most accurate formulas for dispersion is the Sellmeier equation:
$$n^2(\lambda) = 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i}$$

Dispersion does not arise with Huygen's Principle.

Is there a theoretical model that describes dispersion and explains why Sellmeier's equation takes the form that it does?

2. Dec 1, 2015

### Daz

As far as I’m aware, Sellmeier’s equation is purely empirical. It’s just a handy form that fits the measured data quite well. There are many similar expressions (Cauchy, normalised Cauchy, etc.) that also don’t have any physics behind them; they’re simply cooked up to describe the measured numbers.

Sellmeier, by the way, only works well for dielectrics in which the extinction coefficient is vanishingly small. It’s a poor fit for many other materials.

If you want a dispersion model which does have a physical foundation look up the Drude-Lorentz model. The physics behind it is very simple but it works remarkably well for a wide range of materials where Sellmeier fails.

So, to answer your question: Drude-Lorentz is one example of a theoretical model that describes dispersion very successfully, despite its simplicity.

3. Dec 2, 2015

### blue_leaf77

In fact, the expression of Sellmeier equation the OP posted can be derived from the classical harmonic oscillator mode that Drude and Lorentz proposed. Following this derivation, one can arrive at the expression for the complex dielectric function of the form
$$\epsilon(\omega) = 1 + \frac{Ne^2}{\epsilon_0 m}\frac{\omega_0^2-\omega^2 + i\gamma \omega}{(\omega_0^2-\omega^2)^2+(\gamma \omega)^2}$$
If the frequency plotted in the Sellmeier equation's curve is far from the resonance frequency, $\omega_0$, the above expression can be approximated as
$$\epsilon(\omega) = 1 + \frac{Ne^2}{\epsilon_0 m}\frac{1 }{\omega_0^2-\omega^2}$$
Changing the variable from frequency to wavelength, you will obtain the Sellmeier equation for only one term in the summation. In reality, there are many contributions (coming from the various atomic/molecular constituents of the matter) to the complex dielectric function and thus the summation over these contribution should be used.

4. Dec 2, 2015

### greswd

thanks guys. this is obscure enough that wikipedia doesn't cover it.

5. Dec 2, 2015

### blue_leaf77

Why do you seem to rely on Wikipedia? There are a couple of textbooks which discuss the issue of deriving the complex dielectric function, although through a classical approach.

6. Mar 2, 2016

### greswd

Wikipedia is a first resource. Can you recommend a textbook? thanks

7. Mar 2, 2016