Single slit diffraction intensity distribution

Diffraction is the bending of light waves around the edges of obstacles/apertures of size comparable with λ. In Fraunhofer diffraction the source and screen are effectively at infinite distance (plane wavefronts, using lenses).

Intensity distribution at a single slit

 slit width a                 screen
   |  \  θ  |                 I
 --|-- \    |                /\
   |    \ ->|--------  ... _/  \_  ...

1. A parallel beam falls normally on a slit of width a. Divide the slit into N narrow strips; each sends a secondary wavelet of equal amplitude.
2. Path difference between the extreme rays at angle θ is a sin θ, so the total phase difference is 2α = (2π/λ) a sin θ, i.e. α = π a sin θ / λ.
3. The vector (phasor) sum of N small equal vectors with total phase 2α forms an arc of a circle. With arc length = A₀ (the amplitude at θ = 0) and radius r: R = 2r sin α, A₀ = 2rα.
4. So the resultant amplitude: R = A₀ (sin α)/α.
5. Intensity I = R²:

I = I₀ (sin α / α)² ,    α = π a sin θ / λ

Minima and maxima

• Central maximum at α → 0: sinα/α → 1, so I = I₀.
• Minima: sin α = 0, α = ±mπ (m = 1, 2 ...), i.e. a sin θ = mλ.
• Secondary maxima: dI/dα = 0 gives tan α = α, α ≈ ±3π/2, ±5π/2 ... (approximately, midway between minima).

Relative intensities

At α = (2m+1)π/2, sin²α = 1, so I_m = I₀ / α² = 4 I₀ / [(2m+1)² π²].

I₀ : I₁ : I₂ : ... = 1 : 4/(9π²) : 4/(25π²) : ...
                    = 1 : 0.045 : 0.016 : ...

So the first secondary maximum is only ~4.5% and the second ~1.6% of the central maximum: intensity falls rapidly, and most light lies in the central maximum, whose width is 2λ/a (angular).