Constructive interference

Constructive interference occurs when two or more coherent electromagnetic waves arrive at a point in phase — their peaks coincide with peaks, troughs with troughs — producing a resultant amplitude that is the sum of the individual amplitudes. For two equal-amplitude waves, the combined amplitude is 2A₀ and the intensity (proportional to amplitude squared) is 4 times that of a single wave, not 2 times — this is a key signature that distinguishes interference from incoherent addition, where intensities add.

The condition for constructive interference is that the path-length difference between the two waves is an integer number of wavelengths: ΔL = mλ for integer m. In Young's double-slit setup, bright fringes appear on a distant screen at angles satisfying d sin θ = mλ, where d is the slit spacing. In thin-film interference, constructive reflection from the top and bottom surfaces of an oil slick or soap bubble occurs at wavelengths where the round-trip optical path is an integer number of wavelengths (with a π phase shift accounted for if one reflection is off a denser medium and the other is not). Constructive interference is the physical basis of all lasers, holograms, interferometric telescopes (VLA, ALMA, VLBI), LIGO-class gravitational-wave detectors, and every spectroscopic instrument that relies on a diffraction grating.