Physical Optics

Interference is what happens when two or more waves meet and superpose. For a steady pattern to appear, the sources must be coherent: they must keep a constant phase difference over time. Two separate lamps never do this, because their emissions are independent. Thomas Young solved the problem by taking one source and passing it through two narrow slits. The two slits then behave as sources that begin in phase, since both are fed by the same wavefront, and their waves can interfere.

The path difference decides whether a point on the screen is bright or dark. In the double slit, let the slits be separated by $d$ and let the screen sit at an angle $\theta$ from the central line. The wave from the lower slit travels an extra distance

$$\Delta S = d\sin\theta$$

Constructive interference, a bright fringe, occurs when this extra distance is a whole number of wavelengths, $m\lambda = d\sin\theta$, with $m = 0, \pm 1, \pm 2, \dots$. Destructive interference, a dark fringe, occurs when the extra distance is a half-odd multiple, $(m + \tfrac{1}{2})\lambda = d\sin\theta$. The same two facts can be stated in phase: constructive interference corresponds to a phase difference of $2m\pi$ radians, destructive interference to $(m + \tfrac{1}{2})2\pi$ radians. A path difference of one wavelength is a phase shift of $360^\circ$, and a path difference of $\tfrac{1}{2}\lambda$ is a shift of $180^\circ$.

One more rule is needed before the fringe conditions can be trusted. When light reflects from a boundary where the refractive index is higher, the reflected wave is inverted, a $180^\circ$ phase reversal. When it reflects from a boundary where the index is lower, there is no inversion. This single rule explains the half-wavelength that appears in thin-film conditions.