Diffraction & Interference: The Double Slit Explained

Single-slit diffraction and the sinc envelope

Every point across an aperture of width a acts as a Huygens wavelet source; adding up their contributions at angle θ gives a sinc-squared intensity envelope:

β = πa·sinθ / λ
I(θ) = I₀ · [sin(β)/β]²         // single-slit diffraction envelope
// first dark fringe at a·sinθ = λ

Two slits: envelope times interference

Add a second slit a distance d away and the two beams interfere on top of the same diffraction envelope. Combined intensity is the product of the single-slit envelope and a cos² interference term:

δ = πd·sinθ / λ
I(θ) = I₀ · sinc²(β) · cos²(δ)
Bright fringes: d·sinθ = m·λ  (m = 0, ±1, ±2, …)
Fringe spacing on a screen at distance L: Δy = λL / d
Missing fringes: whenever d/a is an integer, an interference maximum lands
  exactly on a diffraction minimum and that fringe vanishes

Gratings: sharpening the peaks with N slits

Replace the two slits with N evenly spaced ones and the same peak positions survive (d·sinθ = mλ, the grating equation), but the peaks sharpen: intensity scales as N² while peak width shrinks as 1/N, giving resolving power R = λ/Δλ = m·N. A 600-line/mm grating illuminated across 500 lines at second order resolves R = 1000 — enough to separate the sodium doublet at 589.0/589.6 nm.

The Airy disk and the diffraction limit

A circular aperture of diameter D produces an Airy disk rather than a simple sinc pattern, described by a first-order Bessel function J₁. The Rayleigh criterion sets the angular resolution limit of any optical instrument:

θ_min = 1.22 · λ / D
  human eye (D≈4mm, λ=550nm):  θ ≈ 30 arcsec
  Hubble (D=2.4m, λ=500nm):    θ ≈ 0.05 arcsec

Fresnel versus Fraunhofer regimes

Whether you observe complicated near-field curved fringes or the simple far-field sinc² pattern above depends on the Fresnel number N_F = a²/(λ·z), where z is the observation distance: N_F ≫ 1 is the Fresnel (near-field) regime, N_F ≪ 1 is the Fraunhofer (far-field) regime the equations above assume. A 1 mm slit at 500 nm needs z > 2 m to reach Fraunhofer conditions unaided — or a lens can project the far-field pattern onto its focal plane regardless of distance, the founding principle of Fourier optics.

Where diffraction shows up in practice

X-ray crystallography treats crystal lattice planes as a 3D diffraction grating for X-rays via Bragg's law nλ = 2d·sinθ, reconstructing molecular structure from the diffraction pattern. Optical lithography for integrated circuits is fundamentally diffraction-limited (resolution = k₁λ/NA), which is exactly why EUV lithography at λ = 13.5 nm was needed to reach 3 nm feature sizes. Holography records the interference between a reference beam and light scattered from an object as a diffraction grating that reconstructs the object's full 3D wavefront when illuminated again.

Frequently asked questions

Try it live

Change the slit width, spacing and wavelength and watch the fringe pattern update in real time in Double-Slit Experiment.