Interference

When two light waves overlap, their electric fields add. If the waves have a fixed phase relationship, the intensity of the sum depends on their relative phase $\varphi$:

Where the waves arrive in phase the intensity exceeds the sum of the two; where they arrive half a wavelength apart it falls below it, to zero if $I_1 = I_2$. Energy is not lost, only redistributed: the bright fringes gain what the dark fringes lose. The phase difference comes from the optical path difference, $\varphi = 2\pi\,\Delta\mathrm{OPL}/\lambda$.

Visibility and coherence

The contrast of the fringes is described by the visibility, $V = (I_{max} - I_{min})/(I_{max} + I_{min})$. For perfectly coherent beams it is $2\sqrt{I_1 I_2}/(I_1 + I_2)$: 1 for equal beams, 0.58 for a 10:1 intensity ratio and still 0.20 for 100:1, which is why weak reflections produce visible fringes and parasitic etalons in optical systems. Partial coherence multiplies the visibility by the magnitude of the degree of coherence. Temporal coherence limits how large a path difference still gives fringes, set by the coherence length; spatial coherence limits how far apart two points of a beam can be and still interfere. Orthogonal polarizations do not interfere at all.

Two-beam and multiple-beam interference

Two plane waves crossing at a full angle $\theta$ make straight fringes with period $\lambda/[2\sin(\theta/2)]$: 36 µm for 632.8 nm beams crossing at 1°. This is the basis of holography, interference lithography and fiber Bragg grating writing, and the double-slit experiment is its classic demonstration. Many-beam interference, as in a Fabry-Perot resonator, thin-film coatings or a grating, produces much sharper maxima, because many waves must all be in phase at once.

Where it is used

Interferometers turn optical path differences into intensity changes and measure length, surface shape, refractive index and vibration to a small fraction of a wavelength. Mach-Zehnder modulators and switches use it to turn phase into amplitude. Anti-reflection coatings cancel reflections by it, and ring resonators, arrayed waveguide gratings and directional couplers are interference devices. Unwanted interference, from stray reflections, appears as fringes and noise, and in laser light scattered from rough surfaces as speckle.

References: M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 7 and 10; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 9.