Why two lamps never interfere

Two lamps stand on a table and light the wall between them. Each sends out a wave, the two waves overlap across every square centimetre of that wall, and at every point their amplitudes add. The addition is not optional and does not wait for permission. The wall shows no fringes at all.

The fringes in that figure are on the wall. Everything below is an account of the two numbers that decide whether anything can record them, and of what those numbers cost.

Two waves always add, so what is the question #

Interference is taught as a property of light and is a property of a comparison.

That two overlapping waves add their displacements at every point is unconditional in any linear medium, and superposition is a consequence of that linearity rather than a special behaviour of some sources. The sum exists. What varies is whether the sum stays the same long enough for anything to measure it.

Two waves always add, and adding two waves 120° out of step of amplitudes 1 and 0.7 gives a resultant peak 0.88 times the larger of them — larger than either alone at some points on the screen, smaller at others. Held fixed, that is a fringe. Allowed to drift, it averages to exactly the same total as no interference at all, and the two situations differ in nothing but whether the phase relationship holds still.

A detector reports an average: the intensity integrated over however long it takes to respond, 40 ms for the eye and a nanosecond for a fast photomultiplier. If the phase difference is constant over that interval, the pattern stands still and is recorded. If it wanders through many cycles, the detector adds fringe patterns in every possible position and gets a uniform glow — the average of $\cos^2$ over all phases, which is one half at every point on the wall.

What “holds still” means can be put as a number. A source whose phase jumps at random every hundred femtoseconds delivers a different fringe pattern each time it jumps — the pattern shifted by whatever fraction of a cycle the jump was — and a detector integrating over a nanosecond sums ten thousand of them. A standing wave’s phase relationship is enforced by a boundary and holds indefinitely; a lamp’s is enforced by nothing.

The quantity that measures how well the pattern survives averaging is the visibility,

$$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$

which is 1 for perfect fringes and 0 for uniform illumination. For the figure above, with amplitudes 1 and 0.7, the geometrical limit is $2a_1a_2/(a_1^2+a_2^2) = 0.94$: unequal beams cannot quite reach unity even when the phase is nailed down.

The title’s claim is therefore a claim about detectors, and it has been tested by defeating one. In 1963 Magyar and Mandel combined the light of two independent ruby lasers and gated an image intensifier for about 20 ns, short compared with those lasers’ coherence time. Fringes appeared, from sources with no connection whatever. Two lamps produce the same fringes and no gate is fast enough.

The coherence time is one over the bandwidth #

A source’s phase drifts because the source is not emitting one frequency. A spread $\Delta\nu$ is a set of components whose relative phases advance at different rates, so a phase relationship established at one moment scrambles itself in a time of order $\tau_c \approx 1/\Delta\nu$. That is the coherence time, and it is a statement about Fourier analysis rather than about light. The figures below make it by construction.

Three real sources, with the arithmetic done. Bandwidth in frequency follows from bandwidth in wavelength by differentiating $\nu = c/\lambda$: $\Delta\nu = c\,\Delta\lambda/\lambda^2$.

A tungsten lamp behind a 10 nm filter at 550 nm. $\Delta\nu = (2.998\times10^8)(10\times10^{-9})/(550\times10^{-9})^2 = 9.9\times10^{12}$ Hz, so $\tau_c = 1.0\times10^{-13}$ s, or 100 fs.

Unfiltered white light, taken as 400 to 700 nm. The two edges sit at $7.49\times10^{14}$ and $4.28\times10^{14}$ Hz, so $\Delta\nu = 3.21\times10^{14}$ Hz and $\tau_c = 3.1$ fs — about one and a half optical cycles. Sunlight has almost no memory of its own phase.

A helium–neon laser with a 1 MHz linewidth. $\tau_c = 1$ µs, longer than the filtered lamp’s by a factor of ten million. Stabilised to 1 kHz it reaches a millisecond, within a factor of forty of the eye’s own integration time.

A filter is not the only thing that sets a bandwidth. An isolated line from a low-pressure discharge is narrower than any filter can cut, and its width comes from the thermal motion of the atoms emitting it — the Doppler shifts of the spread of speeds in a warm gas. For sodium at 300 K that width is $\nu\sqrt{8kT\ln 2/mc^2} = 1.3$ GHz and the coherence time 0.76 ns, better than a filtered continuum by six orders of magnitude. Hence the discharge lamp in every interference experiment before lasers.

The coherence length, and how thin a film has to be #

Light travels while the phase drifts, so the coherence time has a length attached to it:

$$L_c = c\,\tau_c = \frac{\lambda^2}{\Delta\lambda},$$

the second form following by substituting the bandwidth. This is the more useful of the two numbers, because a path difference is what an experiment arranges.

What is left when the medium adds nothing is the packet’s length, and that length is the coherence length. Two copies of a wavetrain, offset along their direction of travel by more than it, have no overlap in which to interfere: the leading edge of one arrives after the trailing edge of the other has gone. That is the whole of the longitudinal condition, and it is why a film has to be thin compared with a lamp’s coherence length before it shows colours.

The three sources become three lengths. The filtered lamp: $L_c = (550\ \text{nm})^2/(10\ \text{nm}) = 30\ \mu$m. White light: 0.93 µm, which is 1.7 wavelengths of green — the wavetrain barely outlasts its own crests. The helium–neon laser: 300 m, and 300 km stabilised.

Every familiar fact about thin films follows from the second number. Light reflected from the front and back of a film of thickness $d$ and index $n$ arrives with a path difference $2nd$, and fringes exist only where that is smaller than $L_c$. For white light and water, $d < L_c/2n = 0.93/(2\times1.33) = 0.35$ µm. A soap film shows colour where it is thinner than about 350 nm and nowhere else, which is exactly what a draining film looks like: bands at the top where it has thinned, colourless grey below. Put a 10 nm filter in front of the same film and the permitted thickness rises to 11 µm; Newton’s rings in an air gap go from visible over 0.47 µm of gap to 15 µm, which is the difference between a curiosity and an instrument, bought with a piece of coloured glass. The colours themselves come from the $\lambda$ in $2nd = m\lambda$, the same wavelength-dependent geometry that sets the rainbow’s angle.

Two closely spaced lines behave differently again, and audibly so.

Two frequencies one unit apart, added, give an envelope that pulses once per unit time — at the difference of the two — so the sum is large, then vanishes, then returns. A sodium lamp does that along a path rather than in time: its two D lines are separated by 5.1 × 10¹¹ Hz, so their beat period corresponds to a path difference of about 0.6 mm, and a Michelson interferometer fed by sodium light loses its fringes and regains them at exactly that interval as the mirror is walked out. The revival is the tell that the source has two lines rather than one broad one.

The sodium doublet sits at 588.995 nm and 589.592 nm, or $5.0899\times10^{14}$ and $5.0848\times10^{14}$ Hz — a difference of $5.1\times10^{11}$ Hz. Each line alone has a coherence length near 23 cm. Together they give fringes whose visibility falls to zero at a path difference of $c/2\Delta\nu = 0.29$ mm, then returns at 0.58 mm, and keeps returning with declining contrast. The vanishing is not a loss of coherence but the two line positions of a spectrum getting half a fringe out of step.

Spatial coherence, and the angular size of the source #

Bandwidth is a spread of frequencies. There is a second, independent spread — a spread of directions — and it is what the angular size of a source means.

Doubling the separation halves the angular spacing of the fringes: with the sources eight wavelengths apart instead of four, more whole wavelengths fit between them and the fan of reinforcement directions is twice as fine. Read backwards — which is the direction the second half of this essay reads it — a finer fan means a longer baseline, and measuring the fineness measures the separation.

Each point of an extended source runs its own fringe pattern, and patterns from points at different places are displaced with respect to one another. Where the displacement reaches half a fringe, the sum washes out. Adding across a source of angular diameter $\theta$ gives a transverse distance over which the field keeps a usable phase relationship,

$$\rho \approx \frac{1.22\,\lambda}{\theta},$$

the spatial coherence length, the 1.22 arriving for a uniform circular source from the same first zero of a Bessel function that sets the diffraction limit of a telescope. The two are one formula with the roles of aperture and source exchanged.

The Sun subtends 0.53°, or $9.25\times10^{-3}$ rad. At 550 nm that gives $\rho = 1.22\times550\times10^{-9}/9.25\times10^{-3} = 7.3\times10^{-5}$ m — 73 µm, rather less than the width of a human hair. A shadow cast in sunlight therefore has no fringes at its edge beyond the first fraction of a millimetre, because the diffraction pattern each point of the Sun makes is displaced from its neighbour’s.

Young’s arrangement of 1801 needed a pinhole before anything else, and the number says why. A lamp filament 1 mm across, viewed from 1 m, subtends $10^{-3}$ rad and gives $\rho = 0.67$ mm — workable. The same filament at 10 cm gives 67 µm, closer than any pair of slits can be cut. The pinhole reduces $\theta$ so the slits sample a field with a definite phase across both, which is also why each wavefront can be treated as a fresh set of sources only when it is one wavefront rather than many tilted ones superposed.

The two kinds of coherence are independent. Sunlight is broadband and spatially coherent across 73 µm; a large fluorescent panel behind a narrow filter is temporally coherent over centimetres and spatially coherent over microns.

The loss of visibility is the instrument #

Here the subject turns over. Everything so far has treated falling visibility as a nuisance. It is a measuring device, and for a century it was the finest one available.

Read the transverse formula backwards. If $\rho = 1.22\lambda/\theta$ is the separation at which fringes vanish, then measuring that separation gives $\theta$. Two mirrors feeding one telescope can be moved apart until the fringes disappear, and the baseline $B$ at which they do satisfies

$$\theta \approx \frac{1.22\,\lambda}{B}.$$

Michelson and Pease did exactly this on the 100-inch telescope at Mount Wilson in December 1920, with a 20-foot beam carrying two mirrors. The fringes from Betelgeuse vanished at a separation of 3.07 m. At 575 nm that gives $\theta = 1.22\times575\times10^{-9}/3.07 = 2.29\times10^{-7}$ rad, or 0.047 arcseconds — the first measurement of the diameter of any star other than the Sun. At the distance then adopted it corresponds to about $4\times10^{11}$ m across, a sphere that would enclose the orbit of the Earth.

No image was formed. The star was and remained a point in every telescope on Earth. What was measured was the baseline at which its light stopped being able to interfere with itself, and the diameter came out of a disappearance.

The comparison that makes the point is with imaging. The smallest angle two points can subtend and still be told apart is $1.22\lambda/D$ — 27.7 arcseconds for a 5 mm pupil at 550 nm, 0.058 for a 2.4 m telescope — and it falls exactly as the inverse of the aperture. Michelson’s stellar measurement reached 0.047 arcseconds with mirrors that could not have imaged anything of the kind. An interferometer beats what its own optics could resolve, because it measures a visibility rather than forming a picture.

That is the point that makes interferometry worth the trouble. An imaging telescope’s resolution is set by the diameter of its glass; an interferometer’s is set by the separation of its collectors, which can exceed any diameter that can be cast, polished and supported. Michelson’s 0.047″ is finer than the Hubble Space Telescope’s 0.058″, achieved in 1920 with two flat mirrors on a girder.

The same reading-backwards works in the temporal direction. Visibility against delay is the modulus of the field’s autocorrelation, and by the Wiener–Khinchin theorem its Fourier transform is the spectrum — so scanning one mirror and recording the fringe contrast measures a spectrum with no prism or grating in the apparatus. Every Fourier-transform infrared spectrometer works this way, at a resolution of $1/2L$ in wavenumbers for a mirror travel $L$: 0.5 cm⁻¹ for a centimetre.

Short coherence is an instrument too. Optical coherence tomography chooses a broadband source deliberately, because fringes then appear only where the two path lengths match, which locates a reflecting layer in depth. A superluminescent diode at 840 nm with $\Delta\lambda = 50$ nm has $L_c = \lambda^2/\Delta\lambda = 14$ µm and an axial resolution of about half that — seven micrometres inside a living retina, from a source chosen for its incoherence.

What it costs #

Filtering throws away almost all the light. A 10 nm slice of a 300 nm visible band keeps about 3% of it, so the experiment that needed the filter now needs thirty times the lamp. A pinhole is dearer still, discarding in proportion to solid angle: reducing $\theta$ tenfold to gain a factor of ten in $\rho$ costs a factor of a hundred in flux. Every gain in coherence is paid for in étendue at a fixed exchange rate.

The photon budget, which is the deeper reason two lamps fail. A fast enough detector would see a lamp’s fringes — except that there is nothing to see. The mean number of photons per mode of a thermal source is $1/(e^{h\nu/kT}-1)$, and for a tungsten filament at 2800 K at 550 nm the exponent is $h\nu/kT = 2.26\ \text{eV}/0.241\ \text{eV} = 9.34$, giving $8.8\times10^{-5}$. A detector fast enough to freeze the pattern would wait through eleven thousand independent fringe patterns to collect one photon. The blackbody spectrum makes the impossibility structural rather than technological, and a laser’s advantage is not that it is monochromatic but that it puts $10^{15}$ photons into one mode.

Coherence length becomes an engineering tolerance. LIGO’s laser is stabilised to the hertz because its coherence length must exceed the mismatch between two 4 km arms by a wide margin. Holography fails outright when the object is deeper than the coherence length: a 30 cm source cannot record a scene 50 cm deep.

Speckle is the price of success. A helium–neon beam on a rough wall produces a grainy pattern that shifts when the observer’s head moves. Every point on the wall scatters, the path differences are tens of micrometres, and 300 m of coherence length makes all of them interfere. Speckle is a perfectly stable interference pattern from an unprepared surface — the physics that made two lamps hopeless, made unavoidable. Laser projectors need a vibrating diffuser to destroy it.

Where the model stops #

The two-number account is an approximation with a well-defined domain, and three of its edges are worth naming with the size of the failure attached.

It assumes a quasi-monochromatic, stationary field. Both formulae treat $\Delta\nu$ as small compared with $\nu$ and the source’s statistics as unchanging. A 10 fs pulse at 800 nm violates the first badly: its bandwidth is $0.44/\tau = 4.4\times10^{13}$ Hz, or $\Delta\lambda = 94$ nm, 12% of the carrier. Its coherence time equals its duration by construction, so bandwidth and duration stop being separate ideas. This is the classical face of the relation that sharpness has to be paid for makes quantum: the Fourier bound the figures above measure as $\sigma_x\sigma_k = 0.500$ becomes $\Delta x\,\Delta p \ge \hbar/2$ on multiplication by $\hbar$, and nothing about the argument changes on the way.

It is entirely first-order coherence. Everything here concerns the correlation of amplitudes. The correlation of intensities is a second-order quantity, and it distinguishes sources this machinery cannot tell apart: a thermal source and a laser filtered to identical spectra have identical coherence times and identical visibilities, and yet the thermal source’s intensity correlation at zero delay is 2 while the laser’s is 1 — thermal photons arrive in bunches. Hanbury Brown and Twiss measured exactly this, on Sirius with radio dishes in 1952 and then optically at Narrabri, obtaining an angular diameter of 0.0068 arcseconds from intensity correlations alone, with no fringes anywhere in the apparatus. Their result was disbelieved for years because first-order thinking says it should not work.

The division into coherent and incoherent is a convenience. Visibility is a continuous number on $[0,1]$ and $\tau_c$ is an order of magnitude, not a threshold. At a path difference of $L_c$ the visibility of a Gaussian-spectrum source is about 0.5 rather than 0 or 1, so “coherence length” names where the fall happens and not where it finishes — and any statement of the form “coherent up to here” carries a factor-of-two ambiguity depending on which definition of bandwidth was used.

A fourth edge is narrower and still catches people. Coherence is necessary for fringes and not sufficient: two beams polarised at right angles produce none at any delay, however perfectly correlated, because their fields never share a component to add. The Fresnel–Arago laws state this, and the direction of the shaking is the missing variable.

The same two numbers in a radio telescope #

The formula that measured Betelgeuse is used every day at wavelengths a thousand times longer, and it works better there because the phase can be recorded.

Radio interferometry replaces mirrors with antennas and — the decisive difference — records each signal against a hydrogen maser clock, so the correlation can be computed hours later. Coherence is imposed rather than preserved. The Event Horizon Telescope’s baselines reach $10^7$ m, and at $\lambda = 1.3$ mm the fringe spacing $\lambda/B$ corresponds to 27 µas, with $1.22\lambda/B$ giving 33 µas. That is the number behind the image of M87’s shadow, from the same equation as the 1920 result on a girder.

None of it is about light. Radar range resolution is $c/2\Delta\nu$, so a 150 MHz chirp resolves 1 m — identical arithmetic, since a radar pulse is a wavetrain and cannot be located to better than its own length. Two loudspeakers driven by one amplifier are coherent and hold a fixed pattern of nulls in a room, while two playing different recordings hold none; and a membrane whose modes are not harmonically related has an autocorrelation with no clean recurrences, the acoustic counterpart of a visibility curve that never quite returns.

What the picture cannot show #

Coherence is a statistical property of a field over time, and every figure here is a single realisation at a single instant. That gap takes three forms.

No figure can show a phase that wanders. The travelling-wave figure draws one wave and one ghost at a definite offset. What a lamp does is redraw that offset at random $10^{13}$ times a second, and a static drawing cannot represent a distribution over offsets except by drawing one member of it. The fringe patterns in the two-source figures are all real; what cannot be drawn is the superposition of $4\times10^{11}$ of them that an eye receives.

No figure here shows a visibility curve. The quantity carrying the whole argument — $V$ against path difference, with its zero at 0.29 mm for sodium and its revivals afterwards — would need a horizontal axis that is a delay and a vertical axis that is the outcome of many measurements. The discrete-component packet comes closest, and only by analogy.

No figure here distinguishes first- from second-order coherence. All of them draw amplitudes, so a thermal source and a laser of the same spectrum would give identical figures. The bunching Hanbury Brown and Twiss measured would need a plot of coincidence counts against delay — a picture about the arrival statistics of light in lumps rather than about the shape of a wave, which is also the level at which one photon at a time still building a pattern becomes the relevant fact.

One further absence: the detector. Every figure draws the field, and the argument here is that the field is not the whole story — the integration time of whatever is watching is a parameter of the physics and appears in none of the drawings.

The ladder from here #

The rungs above this one: the coherence function $\gamma(\tau)$ and visibility as its modulus; the van Cittert–Zernike theorem, which makes the measured visibility the Fourier transform of the source’s brightness and turns an interferometer into an imaging device; partial coherence in imaging, where the illumination changes what a microscope resolves and by how much; second-order coherence, bunching and antibunching, and the experiment that separates a single atom from a lamp; and coherence in matter waves, where the coherence length is set by the temperature of an atom cloud.

The neighbouring ladders are close. Diffraction from a single slit is the phenomenon whose contrast these numbers control; what a thousand slits buy is the instrument whose resolving power the coherence length computed here limits; and the packet that moves at another speed is the same Fourier construction read for velocity rather than for phase memory.

Michelson’s fringes disappeared, and a star acquired a diameter. The most productive thing a fringe pattern can do is stop.