In 1887 Albert Michelson and Edward Morley used a table-top instrument to measure a distance, and got a null result so precise it demolished the idea of the luminiferous ether and cleared the ground for relativity. In 2015 the descendants of that same instrument — the LIGO detectors, each arm four kilometres long — measured a length change one ten-thousandth the width of a proton and heard two black holes collide a billion light-years away. Both machines are interferometers, and both work by the same trick: split a light wave in two, send the halves on different journeys, bring them back together, and read the tiny difference in the paths off the pattern of light and dark they make.

But splitting and recombining a wave only produces fringes if the two halves still "remember" each other's phase when they meet. That memory has a name — coherence — and it is the single idea of this page. Coherence is what separates a laser from a light bulb, what sets how far apart the two mirrors of an interferometer may sit, and what turns an interferometer into a spectrometer. We will pin down its two flavours, build the Michelson interferometer, and see how fringes let us measure lengths a fraction of a wavelength at a time.

Two kinds of coherence

A perfect wave would be an endless, unbroken sinusoid of a single frequency. Real light is not like that: an atom emits a short burst — a wave train — then stops, and the next burst starts with a random new phase. Coherence measures how close light comes to the perfect ideal, and it comes in two independent kinds.

Temporal (longitudinal) coherence asks: for how long does the wave stay a tidy sinusoid before the phase scrambles? That time is the coherence time \tau_c, and the distance the light travels in it is the coherence length L_c = c\,\tau_c. It is set entirely by how pure the colour is — a wave that is a single frequency lasts forever, and any spread of frequencies \Delta\nu limits the wave train through

A white-light bulb (\Delta\lambda \sim 300\ \text{nm}) has a coherence length of barely a micron — a couple of wavelengths. A good laser (\Delta\lambda vanishingly small) can stay coherent over kilometres. That single number decides how long the two arms of an interferometer may differ and still show fringes.

Spatial (transverse) coherence asks a different question: are two points across the beam in step with each other? This is set by the size of the source, not its colour. A point source is perfectly spatially coherent; a broad source (like the Sun's disc) is not, because light from its left edge and its right edge arrive with unrelated phases. The larger and closer the source looks, the smaller the patch over which the wave is coherent — a relationship (the van Cittert–Zernike theorem) that lets astronomers measure the diameters of stars from how their light loses coherence across a pair of telescopes.

The Michelson interferometer

Here is the machine. A beam of light meets a half-silvered mirror — the beam splitter — set at 45^\circ. Half the light passes straight through to a fixed mirror M_1; the other half reflects off to a second mirror M_2 mounted on a fine screw. Each beam bounces straight back, the two halves meet again at the splitter, and the recombined light travels to a detector. Because the two halves came from the same original wave, they can interfere — and whether they add up bright or cancel dark depends only on the difference in the two round-trip paths.

Let the two arms have lengths d_1 and d_2. Each beam travels its arm twice, so the path difference is \Delta = 2(d_1 - d_2). Constructive interference (bright) occurs whenever \Delta = m\lambda; destructive (dark) at \Delta = (m+\tfrac12)\lambda. Now the crucial consequence: nudge M_2 by just \lambda/2 and \Delta changes by a whole \lambda — the pattern advances by exactly one fringe. Count the fringes sliding past as you move the mirror and you have measured its displacement in units of half a wavelength: for green light, steps of 0.27\ \mu\text{m}.

Visibility: watching coherence run out

As you crank M_2 further and further from the balance point, the path difference \Delta grows. As long as \Delta stays well within the coherence length L_c, the two halves still remember their common phase and the fringes stay crisp. But once \Delta approaches L_c, the halves are drawn from different, unrelated wave trains — and the fringes fade away. We measure sharpness with the visibility

which is 1 for perfect black-and-white fringes and 0 when they have washed out to a uniform grey. Drag the slider to change the coherence length and watch the fringes — perfectly regular at the balance point — lose contrast as the path difference runs past L_c:

This fade is not a nuisance — it is a measurement. The width of the coherence envelope tells you L_c, and hence the spectral width \Delta\lambda of the light. Better still, the full shape of the visibility versus path difference is the Fourier transform of the source's spectrum — so scanning the mirror and recording the fringes lets you reconstruct the spectrum. That is Fourier-transform spectroscopy, the method behind the infrared spectrometers that identify molecules in chemistry labs and in the atmospheres of distant planets.

Worked examples

Example 1 — coherence length of a laser. A helium–neon laser emits at \lambda = 633\ \text{nm} with a linewidth \Delta\lambda = 0.002\ \text{nm}. Its coherence length is

Twenty centimetres — so this laser makes clean fringes even with the arms differing by many centimetres. A sodium lamp (\Delta\lambda \approx 0.6\ \text{nm}) manages only about 0.6\ \text{mm}; a white bulb, about a micron.

Example 2 — counting fringes. As mirror M_2 is moved smoothly through 0.10\ \text{mm}, how many fringes pass, using green light \lambda = 500\ \text{nm}? Each fringe is \lambda/2 of mirror travel, so

Four hundred fringes — and because you can comfortably estimate a fringe to a tenth, the instrument pins the mirror position to about 25\ \text{nm}.

Example 3 — reading visibility. A fringe pattern is measured to swing between I_{\max} = 9 and I_{\min} = 1 (arbitrary units). Its visibility is V = (9-1)/(9+1) = 0.8 — high-contrast, so the path difference is still comfortably inside the coherence length.