Single-Slit Diffraction

Huygens's construction replaces a wavefront by sources sitting on it: every element of the wave filling the slit relaunches it, and what arrives at a distant point is the sum of all of them. Label a strip of width dy lying a distance y below the top edge of a slit of width a. Light from that strip travels y sinθ further than light from the top edge to reach a screen point at angle θ, so it arrives with a phase lag 2πy sinθ/λ. The lag runs smoothly from 0 at one edge to 2πa sinθ/λ at the other, and it is worth naming half of that total spread α = πa sinθ/λ now, because every result below is written in it. None of this is new physics — it is the same superposition that gives two slits their fringes — but the sum is an integral over a continuum instead of the addition of two terms, and that one change replaces a row of equal fringes with an envelope that has a bright middle.

Take the direction in which the edge-to-edge path difference is exactly one wavelength, a sinθ = λ, and split the slit down the middle. Every strip in the top half has a partner exactly a/2 below it, and the extra path between the members of a pair is (a/2) sinθ = λ/2. Half a wavelength: the pair cancels. The strips pair off one to one, so the aperture extinguishes itself completely. Nothing about that first λ was special. Split the slit into four equal strips and pair each with the one a/4 away, and the light dies whenever (a/4) sinθ = λ/2, that is a sinθ = 2λ. Repeating with 2m strips gives the general condition a sinθ = mλ with m = ±1, ±2, ±3 and so on. The argument needs at least two strips, so m = 0 is simply not available: on the axis every wavelet travels the same distance and they all add, which is why the centre of the pattern is its brightest point.

Pairing locates the zeros but says nothing about what lies between them. For that, add the contributions with their phases. Each strip dy contributes an equal amplitude with a phase that advances linearly in y, so the resultant is ∫₀^a cos(ωt − 2πy sinθ/λ) dy. Drawn as phasors — equal arrows, each turned a little further than the last — the sum is an arc of a circle. Its arc length is fixed at the on-axis amplitude E₀, and the resultant is the chord. An arc of length E₀ subtending an angle 2α has a chord E₀ sin α/α, and intensity goes as amplitude squared, so I(θ) = I₀ (sin α/α)² with α = πa sinθ/λ. Both earlier results fall out of it. As α → 0 the arc straightens, sin α/α → 1 and I → I₀. Whenever α = mπ the arc closes into a whole number of complete circles, the chord vanishes, and a sinθ = mλ reappears as the zeros of the curve.

The central maximum runs from the m = −1 minimum to the m = +1 minimum, so its angular half-width is θ₁ = arcsin(λ/a). On a screen at distance L, with angles small enough that sinθ ≈ tanθ, it is w = 2λL/a across — exactly twice the spacing of every fringe that follows it, which is the quickest way to recognise a single-slit pattern by eye. Beyond it the lobes are not equal. They peak where the derivative of (sin α/α)² vanishes, that is where tan α = α, at α = 4.4934, 7.7253 and 10.9041 rad, or 1.4303π, 2.4590π and 3.4709π — always a little inside the half-integer guess. Their heights are 4.72%, 1.65% and 0.83% of I₀. In practice the pattern is one bright band with a faint fringed skirt, which is why a single slit is measured from its dark fringes rather than its bright ones.

Slit width enters every result so far only through λ/a, so width and wavelength act inversely. With a 633 nm laser and a screen 2.00 m away, a 0.50 mm slit puts its first minimum 2.53 mm off axis, while a 0.050 mm slit — ten times narrower — puts it at 25.3 mm. Restricting light makes it spread. Push the trend both ways to find the edges of the model. For a ≫ λ the pattern shrinks onto the geometric image of the slit: a 1.0 mm slit gives θ₁ = 0.036°, which is why ray optics is safe for everyday openings. For a < λ the condition sinθ₁ = λ/a asks for a sine greater than one, so no minimum exists in any direction — a sub-wavelength slit produces no pattern at all, just a single Huygens wavelet flooding the forward half-space. Narrowing a slit past a wavelength does not give a narrower beam; it stops giving a beam.

The path difference y sinθ assumed that all the rays heading for one screen point leave the slit parallel. They do not, quite: the point sits at a finite distance L, and the exact path carries a correction of order a²/(8L) from the curvature of the wavefront across the aperture. Comparing that correction with λ gives the Fraunhofer condition, usually quoted through the Fresnel number F = a²/(λL) ≪ 1. For a = 0.10 mm at 633 nm, a²/λ = 1.6 cm, so a screen at 2.00 m has F = 0.0079 and the far field is reached with room to spare, while a screen 1 cm away, with F = 1.6, would show a quite different structured near-field pattern. A lens after the slit buys the same thing at any distance, since it brings genuinely parallel rays to one point of its focal plane. Once the condition holds, sinθₘ = mλ/a is exact, so measured sinθₘ plotted against m is a straight line through the origin of slope λ/a.