Physical optics

Remarks

Because the equations are linear, fields add, and intensities do not: for $u = u_1 + u_2$ one gets $I = \abs{u_1}^2 + \abs{u_2}^2 + 2 \operatorname{Re}(u_1 \overline{u_2})$, and the last term produces interference fringes. Example: two plane waves $e^{i k x \sin\alpha}$ and $e^{-i k x \sin\alpha}$ in vacuum give $I = 4 \cos^2(k x \sin\alpha)$, a pattern of fringes with spacing $\lambda / (2 \sin\alpha)$. Geometrical optics is the limit $k \to \infty$, where $u \approx A e^{i k L}$ and $L$ solves the eikonal equation. The scalar model ignores polarisation; the vector theory uses Maxwell's equations for the electric and magnetic fields.