Explore how multiple coherent sources create interference patterns from double-slit to diffraction grating
When multiple coherent wave sources emit waves simultaneously, the waves overlap and create an interference pattern. Constructive interference occurs where waves arrive in phase (phase difference = 2nπ), producing bright fringes. Destructive interference occurs where waves arrive out of phase (phase difference = (2n+1)π), producing dark regions. The resulting pattern depends on the number of sources, their spacing, and the wavelength.
With 2 sources (double-slit), the interference pattern shows broad, equally-spaced fringes with sinusoidal intensity variation. As the number of sources increases, the principal maxima become sharper and more intense, while secondary maxima appear between them. With many sources (e.g., 12), the pattern resembles a diffraction grating with very narrow, sharply defined peaks. The angular width of principal maxima decreases as 1/N, producing extremely narrow bright lines.
Circular waves radiate outward from each point source, creating near-field interference patterns that depend on the exact geometry. The amplitude decreases as 1/√r in 2D. Plane waves represent the far-field (Fraunhofer) approximation, where wavefronts are approximately flat. The far-field intensity follows I(θ) = I₀·[sin(Nβ)/(N·sin(β))]² where β = πd·sin(θ)/λ.
Multi-source interference principles are used in many fields. Diffraction gratings separate light into precise wavelengths for spectroscopy. Phased array antennas steer beams electronically without moving parts, used in radar and 5G communications. Acoustic beamforming focuses sound in specific directions for sonar and ultrasound imaging. X-ray crystallography uses interference from crystal lattice planes to determine atomic structures. Optical phased arrays enable LiDAR for autonomous vehicles.