# Semiclassical description of EM waves reflection from metallic surfaces

Imagine an EM wave impinging on a metal. Fresnel's formulas tell us that no wave can propagate through the metal, or that the transmitted field is an evascent wave with some penetration depth dependent on the refraction index of the metal.

If we now would like to zoom into the microscopic physics of the reflection, we would have to take into account the crystalline structure of the metal, which imposes a certain electronic band-structure, which in turn determine the way the electrons respond to the external EM perturbation. We should calculate the reflected field from the back action of the electron motion on the incident field.

Do you know of a semiclassical way of solving this problem? Is it feasable to compute numerically a set of coupled Maxwell-Schroedinger equations?

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This was explained very successfully the work that Paul Drude began in 1900 and extended by Hendrik Lorentz in 1905, culminating in the Drude–Lorentz model which is a classical model. I will not go through the derivation but the main points. Inside a metal, the electrons do not feel a restoring force from the lattice when they interact with an impinging EM field. This is because they are considered free. I had a question about this recently and I got a satisfactory answer.

The model treats the electron as an oscillator being disturbed by light at a certain frequency. The equation is $$m_0 \frac{d^2 x}{dt^2} + m_0 \gamma\frac{d2 x}{dt} = -eE(t) =-e E_oe^{-i \omega t}$$ $m_0$ is the mass of the electorn and $\gamma$ encapsulates all of the friction the electron experiences (collision e.g.) and and the solution for the displacement is

$$x(t) =\frac{eE(t)}{m_o (\omega^2 - i \gamma \omega)}$$

Using the fact that $D = \epsilon_r \epsilon_0 E = \epsilon_0 E + P =\epsilon_0 E -Nex$

you can get the dielectric constant of the metal as $$\epsilon_r(\omega) =1 - \frac{\omega_p^2}{(\omega^2 - i \gamma \omega)}$$ where the plasma frequency is given by $$\omega_p^2 =\frac{Ne^2}{m_o \epsilon_0}$$

Then, using the fact that the complex refractive index $\hat n = \sqrt\epsilon_r$, you can calculate the reflectivity as $$R = |\frac{\hat n-1}{\hat n+1}|^2$$

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Yeah I know this, the Drude model that you described (the Drude-Lorentz has Lorentzians in the dielectric function due to interband transitions) is the way to derive electromagnetic optics, which is based on refraction index. However I'm more interested in the microscopic physics of the problem, namely what happens to the electronic wavefuntion at and inside the surface when perturbed by an external EM source, what is it the consequence on the solid bandstructure, how does the penetrating field look like down at this scale and so on.. –  Mattia Oct 19 '13 at 14:01