Wave Optics Optical Bench (3D)

Interference and diffraction of light

A monochromatic wave passing through N narrow slits of width a and spacing d produces, far from the slits, the intensity pattern I(θ) = I0 sinc²(α) [ sin(Nβ) / (N sin β) ]² with α = π a sinθ / λ and β = π d sinθ / λ. For two slits the bracket is cos²β: the double-slit formula I = I₀ cos²(π d sinθ/λ) sinc²(π a sinθ/λ). The cosine (array) factor sets the fringes, the sinc factor (single-slit diffraction) is their envelope.

Key numbers

Principal maxima: d sinθ = mλ; on a screen at distance L the small-angle fringe spacing is Δy = λL/d. The central diffraction maximum has width 2λL/a and contains about 2d/a − 1 fringes. When d/a is an integer m, orders at multiples of d/a coincide with envelope zeros and are missing. With N slits there are N − 1 minima and N − 2 weak secondary maxima between principal maxima, whose width shrinks as ≈ 0.886 λ/(N d), which is why gratings resolve wavelengths (resolving power R = mN) and peak intensity grows as N².

Exact version used for the checks

The page also adds up Huygens wavelets exactly (scalar, two-dimensional): E(y) = ∑ exp(ik r)/√r over 40-64 sub-sources per slit with r = √(L² + (y − xs)²). This needs no far-field assumption; the Fresnel number a²/(λL) says which regime applies (below about 0.1 the Fraunhofer formula is accurate). White light is the sum over 16 wavelengths from 400 to 700 nm weighted by an approximate spectral colour, which is why the white-light fringes are coloured and wash out away from the centre.

The bench drawing

The bench is drawn compressed (a few decimetres of drawing per metre of screen distance), the slits on the card are enlarged, and the diffraction fan and the rays are drawn with the lateral scale of the screen (± the half-width slider maps to the screen edges); the pattern on the screen itself is the computed intensity at its true scale.

Honest simplifications

Scalar waves, slits infinitely long and perfectly opaque screen, no polarisation, no obliquity beyond a simple weight in the ripple view, a perfectly coherent monochromatic source, small-angle spacing formula; the bench schematic is not to scale; the colour rendering is a display approximation, not a colorimetric calculation.

Checks run in the page

(1) The fringe spacing measured from the peaks of the exact sum equals λL/d within 1%. (2) The Fraunhofer formula agrees with the exact sum over the pattern to within 3% RMS for a far-field reference geometry. (3) The first single-slit minimum is at λL/a and five slits give exactly four minima between principal maxima.