Wave Optics

The decisive episode happened in France, and it is the best-known example in physics of a prediction intended as a refutation. The Academy of Sciences set diffraction as the subject of its 1819 prize, expecting the particle theory to be vindicated. Augustin Fresnel, a road engineer, submitted a wave theory in which the field at any point is obtained by adding contributions from every element of the wavefront, each with its own phase — an integral, in modern terms.

Siméon Poisson, on the jury and a convinced Newtonian, examined Fresnel's integrals and extracted an absurdity: behind an opaque circular disc, at the exact centre of the shadow, all the contributions from the rim arrive in phase, so there must be a bright spot. François Arago set up the experiment. The spot was there, and it is there in every undergraduate laboratory now. Fresnel won.

Fresnel and Arago then found something less convenient. Two beams polarised at right angles produce no fringes at all, at any path difference. Young supplied the interpretation and Fresnel made it quantitative: light's oscillation is perpendicular to its travel, and perpendicular polarisations have no common component to add. This was a problem, because a transverse wave requires a medium that resists shear — a solid — and the planets move through it without resistance. The ether grew steadily more preposterous until Maxwell's equations showed in 1865 that the oscillating quantities are the electric and magnetic fields themselves, which is the subject of electromagnetism.

The fringe geometry is the whole of Young's measurement. With slits separated by $d$, a screen at distance $L \gg d$, and wavelength $\lambda$, the $m$th bright fringe sits where the path difference is $m\lambda$, which puts it at $x_m = m \lambda L / d$. So the spacing between neighbouring fringes is

$$\Delta x = \frac{\lambda L}{d}.$$

Everything on the right except $\lambda$ is measurable with a ruler. Take a modern demonstration: $d = 0.25$ mm, $L = 1$ m, and fringes measured 2.4 mm apart. Then

$$\lambda = \frac{\Delta x \, d}{L} = \frac{(2.4 \times 10^{-3})(2.5 \times 10^{-4})}{1} = 6.0 \times 10^{-7} \text{ m} = 600 \text{ nm}.$$

Young's own numbers, in his units, were 0.0000266 inch for the extreme red and 0.0000167 inch for the violet. Converting at 25.4 mm to the inch:

$$0.0000266 \text{ in} \times 25.4 = 6.76 \times 10^{-4} \text{ mm} = 676 \text{ nm}, \qquad 0.0000167 \text{ in} \to 424 \text{ nm}.$$

The accepted range for visible light is about 400 to 700 nm. In 1803, with sunlight, a slit and a screen, he was within a few per cent at both ends of the spectrum — and he had measured a length ten thousand times smaller than anything he could see, by counting something he could.

The same wave behaviour that makes this measurement possible imposes a ceiling. Ernst Abbe showed in 1873 that a microscope forms an image by collecting the light a specimen diffracts, so it can only reconstruct detail whose diffracted orders fall inside the objective's cone. The smallest resolvable separation is about

$$d_{\min} \approx \frac{\lambda}{2\,\mathrm{NA}},$$

where $\mathrm{NA} = n\sin\alpha$ is the numerical aperture. The best oil-immersion objectives reach $\mathrm{NA} \approx 1.4$, so at $\lambda = 550$ nm,

$$d_{\min} \approx \frac{550}{2 \times 1.4} = 196 \text{ nm}.$$

Two hundred nanometres is a hard line drawn across biology. A mitochondrion, at 0.5 to 1 µm, is resolvable; a ribosome at 25 nm, a virus at 100 nm, the 20-nm gap at a synapse are not, and no improvement in glass or grinding changes that. Everything below the line had to be inferred, or stained, or killed and examined with electrons — a constraint that shaped cell biology until fluorescence tricks found a way round it in the 1990s.

The same formula pointed the other way gives the telescope version, the Rayleigh criterion $\theta \approx 1.22\lambda/D$. For the Hubble Space Telescope's 2.4-m mirror at 550 nm,

$$\theta = \frac{1.22 \times 550 \times 10^{-9}}{2.4} = 2.8 \times 10^{-7} \text{ rad} = 0.058 \text{ arcsec}.$$

Which is why large telescopes are large: resolution is bought by aperture, and nothing else.

By 1893 the wave theory was complete enough to be used as a ruler. Albert Michelson compared the prototype metre with the red line of cadmium and found it to be 1,553,163.5 wavelengths long, tying a unit of length to a property of an atom instead of a bar of platinum-iridium. The idea became the SI definition, first through krypton and now through the fixed speed of light, and the instrument became the standard way to measure small distances: count fringes, each worth half a wavelength.

Classical wave optics was, by then, apparently finished. It had a complete theory of propagation, a quantitative limit on imaging, and an instrument that measured length to a fraction of a wavelength. What it did not have was any account of how light is emitted or absorbed, and that is where it broke. A blackbody's spectrum, the photoelectric effect and the sharp lines of atoms all involve light arriving in lumps, and the wave description says nothing about them — the opening of old quantum theory. Re-describing the interference of this chapter in terms of individual quanta, and asking what it means for one photon to interfere with itself, is the business of quantum optics.