Diffraction: single slit, gratings and resolution

Waves bend around obstacles and spread out after passing through openings. Sound turns corners easily because its wavelength is comparable to doorways; light seems to travel in straight lines because its wavelength is a million times smaller than everyday openings. But look closely at the edge of a shadow or at a bright point through a small aperture and the spreading is there, as fringes and rings. This is diffraction. It is the same physics as interference, with a continuum of sources instead of two, and this lesson develops it from Huygens's principle: the single slit, the diffraction grating that sorts light by wavelength, and the circular aperture whose pattern sets the sharpness of every camera, microscope and telescope. We end with X-ray diffraction by crystals, which revealed the arrangement of atoms in matter.

Huygens's principle and the two regimes

Huygens proposed in 1678 that every point on a wavefront acts as a source of secondary wavelets, and the new wavefront is their envelope. Fresnel added in 1818 that the wavelets interfere, so the field at any point is the sum of the contributions from every point of the aperture with their phases. That Huygens–Fresnel principle, later put on a firm footing by Gustav Kirchhoff's integral theorem of 1882, explains diffraction quantitatively. Two regimes are distinguished by the Fresnel number \(N_F = a^2/L\lambda\) for an aperture of half-width \(a\) viewed from distance \(L\). For \(N_F \gtrsim 1\), near the aperture, the pattern is Fresnel diffraction, a blurred image of the aperture edged with fringes. For \(N_F \ll 1\), far away or in the focal plane of a lens, it is Fraunhofer diffraction, which depends only on angle and is the Fourier transform of the aperture. We work in the Fraunhofer regime.

The single slit

A slit of width \(a\) is lit by a plane wave at normal incidence. Divide it into strips of width \(dy\) at position \(y\) from its centre. In direction \(\theta\), the strip at \(y\) is ahead of the centre by path \(y\sin\theta\), so it contributes a phasor proportional to \(e^{-iky\sin\theta}\,dy\). Summing,

The intensity has a bright central maximum and zeros where \(\beta = m\pi\), that is

There is a neat way to see the first zero: pair each strip in the top half of the slit with the strip \(a/2\) below it. When \(a\sin\theta = \lambda\), each pair differs in path by \(\lambda/2\) and cancels, so the whole slit gives nothing. The central maximum has angular half-width \(\lambda/a\), so a narrower slit spreads the light more, the bandwidth theorem again, now relating aperture size to angular spread. The secondary maxima fall off quickly: the first is only 4.7 parts in a hundred of the central peak. A slit 0.10 mm wide lit with 633 nm light makes a central band \(2L\lambda/a = 25\ \mathrm{mm}\) wide on a screen 2.0 m away.

Two slits and many slits

Two slits of width \(a\) and separation \(d\) produce Young's \(\cos^2\) fringes inside the single-slit envelope:

The product structure is general: the pattern of an array of identical apertures is the pattern of one aperture times the pattern of the array, a consequence of the Fourier-transform nature of Fraunhofer diffraction. For \(N\) equally spaced slits, adding \(N\) phasors each turned by \(2\alpha\) from the last gives a geometric series whose sum is

The array factor has principal maxima, of height \(N^2\), where \(\alpha = m\pi\), that is

the same angles as for two slits. Between them are \(N - 1\) zeros and \(N - 2\) weak subsidiary maxima. The principal maxima get narrower as \(N\) grows: the first zero beside the peak of order \(m\) is at \(\alpha = m\pi + \pi/N\). With thousands of slits the maxima become sharp lines, and that is a diffraction grating.

Diffraction gratings

Joseph von Fraunhofer made the first precise gratings in 1821 by winding fine wire between two screws, after he had mapped the dark absorption lines of the solar spectrum with a prism in 1814. Modern gratings have up to a few thousand grooves per millimetre ruled or holographically written on glass or metal. A grating with spacing \(d\) sends wavelength \(\lambda\) to angles given by \(d\sin\theta = m\lambda\), so it disperses light into spectra, one per order \(m\), and unlike a prism it does so with a dispersion fixed by geometry, which makes it ideal for measuring wavelengths. For light at incidence angle \(\theta_i\) the grating equation becomes \(d(\sin\theta_m - \sin\theta_i) = m\lambda\), with signs depending on convention.

How close can two wavelengths be and still be separated? Two lines are resolved when the peak of one falls on the first zero of the other. The peak of order \(m\) for \(\lambda + \Delta\lambda\) is at \(\alpha = m\pi(1 + \Delta\lambda/\lambda)\), and the first zero of \(\lambda\) is at \(m\pi + \pi/N\). Equating gives the resolving power

The sodium D lines at 589.00 and 589.59 nm need \(\mathcal R \approx 1000\): a grating with 1000 grooves illuminated in first order just separates them, while a 5 cm grating with 600 grooves per millimetre, 30,000 grooves in all, separates them thirty times over. Astronomical spectrographs reach resolving powers of 100,000 and more, enough to measure the Doppler wobble of a star of a few metres per second caused by an orbiting planet, which is how most of the first exoplanets were found after 1995. Gratings also show why compact discs and the wings of some insects shimmer: their periodic structures diffract white light into spectra.

The circular aperture and the Rayleigh criterion

Lenses, mirrors and pupils are circular. Fraunhofer diffraction by a circular aperture of diameter \(D\) gives a bright central disc surrounded by faint rings, the Airy pattern, calculated by George Airy in 1835:

where \(J_1\) is the Bessel function of order one. Its first zero is at \(x = 3.8317\), so the first dark ring is at

About 84 parts in a hundred of the light falls inside this ring. Even a perfect lens therefore images a point as a small disc, and two points closer than the disc size blur together. Lord Rayleigh proposed in 1879 that two equal point sources are just resolved when the centre of one Airy pattern falls on the first dark ring of the other, giving the Rayleigh criterion

At that separation the combined pattern dips to about 73 parts in a hundred of the peaks between them, a dip the eye can see. The criterion is a convention rather than a law; modern image processing can do somewhat better with high signal-to-noise ratio, but it sets the scale. A human pupil of 3 mm at 550 nm gives \(2.2\times10^{-4}\ \mathrm{rad}\), about 0.8 minutes of arc, close to the measured acuity of the eye, whose receptor spacing evolved to match. The 2.4 m Hubble Space Telescope resolves 0.06 seconds of arc in visible light. The 6.5 m James Webb Space Telescope, launched on 25 December 2021, works in the infrared where wavelengths are longer, and its resolution at \(2\ \mathrm{\mu m}\) is similar to Hubble's in the visible.

X-ray diffraction by crystals

A crystal is a three-dimensional grating with spacings of a few tenths of a nanometre, matched to X-ray wavelengths. In 1912 Max von Laue proposed that crystals should diffract X-rays, and Walter Friedrich and Paul Knipping confirmed it. William Lawrence Bragg and his father William Henry Bragg then showed in 1913 that the strong reflections can be understood as mirror reflection from planes of atoms spaced \(d\) apart, reinforced when the path difference between successive planes is a whole number of wavelengths:

with \(\theta\) measured from the planes. Bragg's law turned X-ray diffraction into a tool for finding atomic structures, from rock salt in 1913 to the double helix of DNA, whose X-ray photograph by Rosalind Franklin and Raymond Gosling in 1952 showed the cross-shaped pattern of a helix, to the thousands of protein structures solved each year. The Braggs shared the 1915 Nobel Prize in Physics; the younger, at 25, remains the youngest physics laureate.

Computing diffraction patterns

The program sums Huygens wavelets across a slit numerically and compares with the \(\operatorname{sinc}^2\) formula, evaluates the \(N\)-slit pattern, finds the first Airy zero from a series for \(J_1\), computes the dip at the Rayleigh separation, and checks the grating worked example.

import math, cmath

lam, a = 633e-9, 0.10e-3
k = 2 * math.pi / lam
def slit_numeric(theta, n=2000):
    s = sum(cmath.exp(-1j * k * (-a/2 + (j + 0.5) * a / n) * math.sin(theta)) for j in range(n))
    return abs(s / n) ** 2
def slit_formula(theta):
    b = math.pi * a * math.sin(theta) / lam
    return 1.0 if b == 0 else (math.sin(b) / b) ** 2
print("single slit, a = 0.10 mm, 633 nm:")
for mrad in (0.0, 3.0, 6.33, 9.0, 12.66):
    th = mrad * 1e-3
    print(f"  theta = {mrad:5.2f} mrad: numeric {slit_numeric(th):.5f}, formula {slit_formula(th):.5f}")
b1 = 1.4303 * math.pi                     # first side maximum, tan b = b
print(f"first side maximum / central peak = {(math.sin(b1)/b1)**2:.4f}")

def nslit(s, N, d, w):
    al = math.pi * d * s / lam; be = math.pi * w * s / lam
    env = 1.0 if be == 0 else (math.sin(be) / be) ** 2
    arr = N * N if abs(math.sin(al)) < 1e-12 else (math.sin(N * al) / math.sin(al)) ** 2
    return env * arr
print("\nfive slits, d = 3a: principal maxima at sin(theta) = m lam/d")
for m in range(0, 4):
    s = m * lam / (3 * a)
    print(f"  m = {m}: I/I0 = {nslit(s, 5, 3*a, a):7.3f}  (N^2 = 25 times envelope {slit_formula(math.asin(s)):.3f})")

def J1(x, terms=40):
    return sum((-1)**m * (x/2)**(2*m + 1) / (math.factorial(m) * math.factorial(m + 1)) for m in range(terms))
lo, hi = 3.0, 4.5
for _ in range(60):
    mid = (lo + hi) / 2
    if J1(lo) * J1(mid) <= 0: hi = mid
    else: lo = mid
x1 = (lo + hi) / 2
print(f"\nfirst zero of J1 at x = {x1:.4f}; x/pi = {x1/math.pi:.4f}")
airy = lambda x: 1.0 if x == 0 else (2 * J1(x) / x) ** 2
dip = 2 * airy(x1 / 2)                    # midpoint between two images separated by x1
print(f"Rayleigh separation: midpoint intensity / peak = {dip / (1 + airy(x1)):.3f}")

d = 1e-3 / 600
for L in (486e-9, 656e-9):
    print(f"grating 600/mm, {L*1e9:.0f} nm: first order at {math.degrees(math.asin(L/d)):.2f} deg, max order {int(d // L)}")
print(f"resolving power with 15000 lines in first order: smallest resolvable step at 656 nm = {656/15000:.4f} nm")

single slit, a = 0.10 mm, 633 nm:
  theta =  0.00 mrad: numeric 1.00000, formula 1.00000
  theta =  3.00 mrad: numeric 0.44808, formula 0.44807
  theta =  6.33 mrad: numeric 0.00000, formula 0.00000
  theta =  9.00 mrad: numeric 0.04716, formula 0.04716
  theta = 12.66 mrad: numeric 0.00000, formula 0.00000
first side maximum / central peak = 0.0472

five slits, d = 3a: principal maxima at sin(theta) = m lam/d
  m = 0: I/I0 =  25.000  (N^2 = 25 times envelope 1.000)
  m = 1: I/I0 =  17.098  (N^2 = 25 times envelope 0.684)
  m = 2: I/I0 =   4.274  (N^2 = 25 times envelope 0.171)
  m = 3: I/I0 =   0.000  (N^2 = 25 times envelope 0.000)

first zero of J1 at x = 3.8317; x/pi = 1.2197
Rayleigh separation: midpoint intensity / peak = 0.735
grating 600/mm, 486 nm: first order at 16.95 deg, max order 3
grating 600/mm, 656 nm: first order at 23.18 deg, max order 2
resolving power with 15000 lines in first order: smallest resolvable step at 656 nm = 0.0437 nm

The numerical Huygens sum matches the formula, with zeros at 6.33 and 12.66 mrad, which are \(\lambda/a\) and \(2\lambda/a\). The first side maximum is 0.047 of the centre. For five slits the principal maxima reach \(25\) times the envelope, and the third order vanishes because it lands on an envelope zero. The bisection finds the first zero of \(J_1\) at 3.8317, which is \(1.2197\pi\), the origin of the factor 1.22, and the Rayleigh dip comes out at 0.735 of the peaks.