Optical theorem

Remarks

The proof is a balance of energy flux through a large sphere around the scatterer. The flux of the total field splits into the flux of $u_i$, which is zero, the flux of $u_s$, and a cross term between $u_i$ and $u_s$. For large $r$, the cross term is an oscillatory integral over the sphere, and the method of stationary phase shows that only the forward direction $\hat{x} = \hat{d}$ contributes. This gives the term $\frac{4\pi}{k} \operatorname{Im} f(\hat{d})$.

The same identity holds for electromagnetic scattering, with the vector forward amplitude projected on the incident polarization. It also holds in quantum potential scattering, where it is a consequence of the unitarity of the scattering operator.

An example is the extinction paradox. For a large opaque sphere of radius $a$ with $ka \to \infty$, the extinction cross-section tends to $2\pi a^2$, twice the geometric cross-section. Half of it comes from the light that hits the sphere, and the other half from the light diffracted into a narrow forward cone.

Going deeper

For a student

The theorem links the far-field pattern $f$ to the total cross-section, so it is a check on any numerical scattering code. In quantum mechanics it reads $\sigma_{\mathrm{tot}} = \frac{4\pi}{k} \operatorname{Im} f(0)$ and follows from conservation of probability. It also constrains approximations: the Born approximation gives a real forward amplitude for a real potential, so it violates the optical theorem at first order.