Single-Slit Diffraction

Single-slit diffraction

Concept: Light passing through a slit only a fraction of a millimetre wide does not make a thin stripe on the screen. It spreads out into a bright central band with dark fringes either side. The narrower the slit, the wider the pattern.

What the four steps show

• 1 · Setup — the three quantities that control the pattern (λ, a, L) and where the first dark fringe lands: y₁ = λL/a.
• 2 · Diffraction — geometric optics predicts a thin stripe (dashed); real waves spread into the shadow because every point of the opening acts as a new source (Huygens).
• 3 · Why dark? — a ray from the top edge and a ray from the centre are a/2 apart. At the first minimum the top ray travels λ/2 further, so they cancel. Every such pair across the slit cancels too.
• 4 · Pattern — intensity I(y) = I₀·sinc²(πay/λL): brightest at the centre, zeros at y = ±mλL/a, side bands only 4.7 % and 1.6 % as bright.

Key formulas

• Minima: a·sinθ = mλ (m = ±1, ±2, …), so y_m ≈ mλL/a
• Path difference between rays a/2 apart: Δ = (a/2)·sinθ
• Intensity: I = I₀ [sin β / β]², β = πa·sinθ/λ

Sources

Hecht, E. (2017). Optics (5th ed.), Pearson, ch. 10.
Halliday, Resnick & Walker, Fundamentals of Physics, “Diffraction” chapter.