Diffraction

When a wave passes through an opening or around an obstacle whose size is comparable to its wavelength, it bends into the geometrical shadow — it does not travel in a straight line. This bending is called diffraction, and it is one of the clearest signatures that something is a wave.

Huygens's principle

The standard framework for understanding diffraction goes back to Christiaan Huygens (1678). His principle states that every point on a wavefront acts as the source of a new spherical (or circular, in two dimensions) wavelet. The actual wavefront at any later time is the envelope — the common tangent — of all these secondary wavelets. When a plane wave hits a narrow slit, only the wavelets that originate inside the slit can contribute to the field beyond it, and those wavelets spread in all directions, filling the shadow region. The narrower the slit relative to the wavelength, the more the wave spreads.

Single-slit diffraction

Consider a slit of width $a$ illuminated by a plane wave of wavelength $\lambda$. Divide the slit into infinitely many infinitesimal sources and add their contributions at a distant screen. When the path-length difference between a ray from the top of the slit and one from the middle is exactly $\lambda/2$, every source in the top half is paired with a source in the bottom half whose ray arrives exactly out of phase. The two contributions cancel, producing a dark fringe. This destructive condition generalises to

$a \sin\theta = m\lambda, \qquad m = \pm 1, \pm 2, \ldots,$

where $\theta$ is the angle from the slit normal and $m$ is the fringe order. Note the central maximum ($m = 0$) is broad and bright; the intensity falls to zero at the first dark fringe ($m = 1$), then rises to a weaker secondary maximum before falling again. The central peak has angular half-width $\theta_1 = \arcsin(\lambda/a)$, so the narrower the slit ($a \to \lambda$) the wider the central maximum — the wave spreads more when squeezed through a smaller opening.

The key parameter: $\lambda / a$

The ratio of wavelength to aperture size controls how much diffraction occurs.

• When $\lambda \ll a$ (e.g., visible light through a door-sized opening), diffraction is negligible and a sharp geometric shadow forms. This is the ray-optics limit.
• When $\lambda \sim a$ (e.g., red light through a slit a few micrometres wide, or sound waves through a doorway), diffraction is pronounced and the spreading is easily observed.
• When $\lambda \gg a$, the wave fans out nearly uniformly in all forward directions.

Sound diffracts strongly around corners because audible wavelengths (roughly 2 cm to 20 m) match everyday obstacle sizes. Visible light (400–700 nm) diffracts strongly only through microscopic slits, which is why diffraction was historically harder to observe for light than for sound.

Diffraction and the uncertainty principle

Diffraction has a direct quantum-mechanical interpretation. If a particle (say, an electron) passes through a slit of width $a$, its transverse position is known to within $\Delta y \approx a$. By Heisenberg's uncertainty principle, the transverse momentum is uncertain by at least $\Delta p_y \approx h/a$. This transverse momentum spread causes the particle beam to fan out after the slit — precisely the diffraction pattern. That matter waves diffract at all was first confirmed in the Davisson–Germer experiment (1927), in which electrons diffracting from a crystal lattice (Bragg-type reflection, not a single slit) produced intensity peaks matching de Broglie's hypothesis that particles have wavelength $\lambda = h/p$.

Why diffraction matters for quantum mechanics

Quantum mechanics inherits diffraction because every quantum particle has an associated de Broglie wavelength. Interference and diffraction of matter waves are not analogies — they are real, measurable phenomena. Modern double-slit experiments have observed diffraction patterns for electrons, neutrons, atoms, and even large molecules such as buckminsterfullerene ($\text{C}_{60}$). The inability to predict which specific detector pixel a single particle hits, while the long-run pattern follows the classical wave intensity, is one of the clearest demonstrations of quantum indeterminacy.

Sign in on the full site to ask questions and join the discussion.