The Single-Slit Pattern in Detail

Bright bands between the dark ones

Between two dark bands of a single slit, the light rises again to a faint secondary maximum. It lies roughly midway between the two minima. The first dark bands are at \(a\sin\theta = \lambda\) and \(2\lambda\), so the first secondary maximum is near \(a\sin\theta = 3\lambda/2\). In general the secondary maxima are near \[ a\sin\theta \approx (2n + 1)\frac{\lambda}{2}, \qquad n = 1, 2, 3, \dots \]

Why is the first one so faint? Look again at the second chip of Fig 1.2 in the lesson before. At \(a\sin\theta = 3\lambda/2\), the slit splits into three zones, each with an edge-to-edge path difference of \(\lambda/2\). Two of the zones cancel each other pair by pair. Only the third zone is left to send light to the screen.

One third of the slit gives at most one third of the amplitude, and so about one ninth of the intensity. Even that third is not all in phase, which lowers it further. The exact calculation gives about \(1/22\), or \(4.5\,\%\), of the central intensity. The second secondary maximum has only \(1.6\,\%\).

The widths follow from the positions of the minima. The central maximum runs from \(\sin\theta = -\lambda/a\) to \(+\lambda/a\), an angular width of \(2\lambda/a\). Each secondary maximum runs from one minimum to the next, an angular width of \(\lambda/a\). So the central maximum is twice as wide as any other.

The slit that widens

The width of the central maximum, \(2\lambda D/a\), has the slit width \(a\) underneath. So the pattern does the opposite of what the slit does. Narrow the slit, and the light spreads wider. Widen it, and the pattern closes in towards a sharp image of the slit. Try it in the figure below.

In the figure above, a slit \(0.05\,\text{mm}\) wide spreads the light into a central band \(40\,\text{mm}\) wide. At \(0.2\,\text{mm}\) the band is \(10\,\text{mm}\), and at \(0.5\,\text{mm}\) only \(4\,\text{mm}\). Ten times the slit width gives a tenth of the band.

Too narrow for any dark band

Now make the slit very narrow, close to the wavelength. For a slit \(1\,\mu\text{m}\) wide and light of \(500\,\text{nm}\), \(\sin\theta_1 = 500/1000 = 0.5\). The first minimum is at \(30°\), and the small-angle formula no longer holds. Narrow the slit to \(0.4\,\mu\text{m}\). Then \(\lambda/a = 1.25\), and no angle has a sine that large. There is no minimum at all: the light spreads over the whole half space in front of the slit.

Diffraction under water

Put the whole apparatus under water, of refractive index \(\mu = \tfrac43\). The frequency of the light stays the same, but it slows to \(c/\mu\). So its wavelength shrinks to \(\lambda/\mu\). Every angle in the pattern shrinks with it. A central maximum \(10\,\text{mm}\) wide in air becomes \(10/\tfrac43 = 7.5\,\text{mm}\) wide in water.

When ray optics is enough

A wide slit still diffracts, only a little. Close to the slit, the beam is as wide as the slit itself, and ray optics describes it well. Far away, the spread \(D\lambda/a\) grows until it equals the slit width \(a\). That happens at the Fresnel distance \[ z_F = \frac{a^2}{\lambda}. \] Within \(z_F\), rays are a good picture; well beyond it, diffraction takes over.

An opening \(3\,\text{mm}\) wide, with light of \(500\,\text{nm}\), has \(z_F = (3 \times 10^{-3})^2/(500 \times 10^{-9}) = 18\,\text{m}\). A beam through it stays close to a ray for a few metres. This is why ray optics works so well on a laboratory bench.

Interference and diffraction compared

Both patterns come from the same principle of superposition. Waves add, and where they arrive in opposite phase they cancel. The difference lies in what is being added. In Young's double slit, two separate coherent sources are added. In a single slit, a continuous row of wavelets from one wavefront is added.

Two real slits

The two slits of Young's experiment are not mathematical lines. Each has a width \(a\), and each diffracts. The chapter on Young's double slit assumed that each slit lights the screen evenly. In fact each lights it with its own single-slit pattern. The two-slit fringes form inside that pattern, so their brightness follows it: the single-slit pattern is the diffraction envelope.

In Fig 2.1 each slit is \(0.1\,\text{mm}\) wide, so the envelope has its first minima at \(\lambda D/a = 5\,\text{mm}\). With \(d = 0.5\,\text{mm}\), the fringes are \(\beta = \lambda D/d = 1\,\text{mm}\) apart. Ten fringe widths fit across the central envelope, \(10\,\text{mm}\) wide.

Missing orders

The bright fringe of order \(n\) is at \(d\sin\theta = n\lambda\). The first minimum of the envelope is at \(a\sin\theta = \lambda\). Divide one by the other: the fringe that lands on that minimum has \(n = d/a\). Here \(d/a = 5\), so the fifth fringe falls where no light arrives from either slit. It is a missing order, and so are the tenth, the fifteenth and so on.

The fringes inside the central envelope are then the orders \(0, \pm1, \pm2, \pm3, \pm4\): nine bright fringes. With \(d = 0.3\,\text{mm}\), \(d/a = 3\). Only five fringes fit inside, and the third and sixth orders are missing.

This also explains a fact of every real double slit. The fringes far from the centre are fainter than the central ones. They are not fainter because of anything in the two-slit geometry. They are fainter because the envelope of each slit falls away from the centre.