The faint light a bright one makes measurable

The fringe and the spectrum are one measurement ends by listing what comes after a scanning interferometer, and the second item is heterodyne detection, “where the transform is performed by mixing rather than by scanning”. The phrase undersells it. Heterodyne detection is how every radio receiver ever built works, how a wind lidar measures the speed of the air a kilometre up, how a coherent optical fibre link reads a signal of a few thousand photons per bit, and how a millimetre-wave telescope records a signal from a galaxy with its phase intact. It rests on a single fact about how light is detected, and the fact is one why two lamps never interfere already uses.

A photodetector cannot follow the oscillation of a light wave. Light at 1.55 micrometres oscillates 193 million million times a second, and a detector responds to the average over many cycles of the square of the field — to the power. Two fields falling on it together are added first and squared afterwards, and the square of a sum is not the sum of the squares:

$$|E_s + E_{LO}|^2 = |E_s|^2 + |E_{LO}|^2 + 2|E_s||E_{LO}|\cos(\Delta\omega\, t + \varphi).$$

The last term is the interference between the two, and if they differ in frequency by $\Delta\omega$ it is not a fixed fringe but a beat, oscillating at the difference frequency. When two waves meet describes beats between two tuning forks, heard as a slow throb at the difference of their pitches. Here the forks are a faint light and a bright laser, and the throb is an electrical signal at a radio frequency.

The same weak light, heard alone and heard against a strong one #

The drawing isolates the weak signal’s share. On its own, it adds its own power to the detector current: a steady unit, in the drawing’s units. With the strong beam — the local oscillator — added, it contributes a term that is its field multiplied by the local oscillator’s field, and with an oscillator a thousand times more powerful that term swings by sixty-three times the signal’s own contribution. The signal has been amplified, in a sense, before any electronics has touched it, and the amplification has come from the local oscillator’s power, not from the signal’s.

Three things are carried by the beat, and together they are why the arrangement is so useful. Its amplitude is proportional to the signal’s field, not its power, so a signal a hundred times weaker gives a beat only ten times smaller. Its frequency is the difference between the signal’s frequency and the oscillator’s, so an optical frequency is translated down to one an ordinary radio circuit can count. And its phase is the signal’s phase relative to the oscillator’s, so the measurement records not only how much light arrived but when its crests arrived — which direct detection, squaring the field, throws away.

The price of all three is coherence. The beat is only a clean oscillation if the signal and the oscillator hold their relative phase for longer than the detector takes to record it. Two independent lamps do not, which is why they never interfere; a signal and an oscillator that are both lasers with narrow lines do, for as long as a wave can remember its phase, and the beat’s spectrum is the two lines’ combined width.

A strong oscillator lifts a picowatt to the quantum limit #

Every real detector has an amplifier behind it, and the amplifier adds noise: random current fluctuations from the thermal motion of electrons in its input resistor. For a picowatt signal the detector current is a picoamp, and the amplifier’s noise over a megahertz bandwidth is thousands of times larger. Detected directly, the signal is buried sixty-six decibels deep — a factor of four million in power — and no averaging over any practical time will recover it.

Heterodyning lifts the signal’s current in proportion to the square root of the oscillator’s power, while the amplifier’s noise stays where it was. At some oscillator power the signal emerges; at higher power it keeps rising. But the oscillator brings noise of its own. Light arrives in photons, and photons arrive at random, so a steady beam produces a current with shot noise whose power grows in proportion to the beam’s. Once the oscillator is strong enough that its own shot noise dominates the amplifier’s — about twelve microwatts in the drawing — the signal and the noise both grow with the oscillator’s power, and the ratio levels off.

Where it levels off is the remarkable part. The limit is

$$\text{SNR} = \frac{\eta P_s}{h\nu B},$$

the number of signal photons the detector registers in one resolution time, $1/B$. Nothing about the oscillator, the detector’s amplifier or the electronics appears in it. A heterodyne receiver with a strong enough local oscillator detects a weak coherent signal as well as a perfect photon counter could, with the amplifier’s noise made irrelevant by being swamped rather than by being removed. That is why coherent optical communication links, which heterodyne or homodyne their incoming signal against a local laser, can work with a few photons per bit over thousands of kilometres of fibre, and why the technique was in use for decades before detectors quiet enough to count those photons directly existed.

Measuring both quadratures costs one extra vacuum #

The quantum limit carries a subtlety that decides how the method is used. A light wave’s field can be written as two amplitudes, the parts oscillating as a cosine and as a sine — its two quadratures — and in quantum mechanics these are conjugate quantities in the sense of the questions that can be asked together: their operators do not commute, and even a perfect coherent state has a minimum fluctuation in each, the vacuum’s noise. The noise pushed below the floor draws that fluctuation as a small circle round the tip of the field’s arrow in the plane of the two quadratures.

A homodyne detector — the local oscillator at exactly the signal’s frequency — measures one quadrature, the one in phase with the oscillator, and it does so with only the vacuum’s fluctuation: the left-hand histogram has variance ¼. A heterodyne detector, with its oscillator offset in frequency, sees the beat sweep through the signal’s phase and records both quadratures at once. The right-hand cloud shows what that costs: each quadrature comes out with twice the variance. The extra noise enters through the image frequency — the frequency on the other side of the oscillator, at the same offset, which beats down to the same radio frequency and contributes its own vacuum fluctuation whether or not anything is there.

This is not a defect of any particular receiver. Any measurement that tries to learn two conjugate quantities at once must pay it, a result proved in general by Edwin Arthurs and John Kelly in 1965, and heterodyne detection is the most familiar instance. The factor of two is three decibels, and it is the reason the limit in the previous drawing is stated for the signal’s photons rather than for twice as many: the oscillator has already paid half of it.

The noise a phase-sensitive receiver cannot avoid #

The extra vacuum can be expressed as a noise temperature — the temperature of a thermal source that would add the same noise — and the result is $h\nu/k$, which grows in proportion to frequency. Plotted across the electromagnetic spectrum, it sets the whole division of labour in astronomy.

At radio frequencies the floor is tiny. At the hydrogen line it is a twentieth of a kelvin, far below the thermal noise of the sky, the telescope and the receiver’s own electronics, so there is no penalty worth mentioning for recording the phase. Every radio telescope is therefore a heterodyne receiver, recording the signal’s amplitude and phase as numbers, and radio interferometers — from the Very Large Array to the Event Horizon Telescope — record each antenna’s signal separately, with an atomic clock’s timestamps, and combine them afterwards in a computer. The antennas can be on different continents.

At optical frequencies the floor is tens of thousands of kelvin, and the comparison is with the brightness of a star per spatial and spectral mode, which for all but the brightest is far lower. A heterodyne receiver would bury the starlight in its own quantum noise. Optical interferometers therefore bring the light itself together — through vacuum pipes and moving delay lines, as in the fringe that measures a star — and interfere it before detecting it, or give up the phase entirely and correlate intensities, as the correlation that survives what the phase does not describes. The one serious exception proves the rule: the Infrared Spatial Interferometer at Mount Wilson used heterodyne detection with carbon-dioxide laser oscillators at 11 micrometres, where the floor is about 1,300 kelvin, to measure the sizes of bright, cool, dusty stars that are luminous enough in the infrared to be seen above it.

The same curve explains why submillimetre astronomy is so hard. At 345 gigahertz the floor is about seventeen kelvin, comparable with the best receivers’ total noise, and the receivers at telescopes like ALMA operate within a few times of it — a rare case of an instrument working close to a quantum limit as a matter of routine.

Reading the wind off a beat note #

The frequency translation is what makes heterodyne detection the natural way to measure motion. Light scattered back from something moving along the beam is shifted in frequency by twice the speed over the wavelength — the moving-reflector shift of the shift a mirror gives twice — and at 1.55 micrometres that is 1.29 megahertz for every metre per second. The shift is a tiny fraction of the light’s frequency, about a hundred parts in a billion for a strong wind, far too small for any spectrometer that disperses the light directly. Beaten against a local oscillator taken from the same laser that sent the pulse out, it appears as a radio frequency of a few megahertz to a few tens, where counting it is trivial.

A coherent wind lidar sends pulses of laser light into the atmosphere and heterodynes the faint return from aerosols and dust. The return from each range — timed by the delay since the pulse left — is spread over a narrow band of frequencies by the turbulence within it, and in a single pulse it is weaker than the shot noise. The drawing simulates the standard remedy: average several hundred spectra and the random scatter of the noise floor shrinks as one over the square root of their number, while the signal’s peak does not. The wind at each range can then be read off the peak’s position to a fraction of a metre per second. Instruments of this kind are mounted beside runways and on wind turbines. The one wind lidar flown in orbit so far, the European Space Agency’s Aeolus, launched in 2018, worked at 355 nanometres in the ultraviolet and detected its returns directly, through interferometers that converted the Doppler shift into a change of brightness — for the reason the previous drawing gives. At ultraviolet frequencies the heterodyne floor is tens of thousands of kelvin, and a faint scattered return is better counted photon by photon than beaten against an oscillator.

Counting the frequency of light #

The same trick, pushed to its limit, is how the frequency of light is measured at all. No electronic counter can follow an oscillation of hundreds of terahertz. But a counter can follow a beat of a few megahertz, and if the local oscillator’s frequency is known, the beat gives the signal’s. The difficulty for most of the twentieth century was the known oscillator: optical frequencies were linked to the caesium clock’s microwave frequency through room-sized chains of lasers and mixers, each heterodyned against the next, and only a few national laboratories could run one.

The optical frequency comb, developed by the groups of John Hall and Theodor Hänsch around 2000, replaced the chain with a single laser that emits a train of ultrashort pulses. Its spectrum is a comb of hundreds of thousands of sharp lines spaced by exactly the pulse repetition rate, a radio frequency, and offset from zero by another radio frequency that can be measured by heterodyning the comb against a frequency-doubled copy of itself. Every line’s frequency is then known in terms of two radio frequencies. Beat any laser against its nearest comb line and count the beat, and the laser’s optical frequency is known with the precision of the radio clock that counts. Optical atomic clocks, which now reach parts in $10^{18}$, are read out this way, and the comparison of such clocks is heterodyne detection at its most exacting.

The radio in every pocket #

Heterodyning is older than any of this. Reginald Fessenden patented the idea of beating a received radio signal against a local one in 1901, and Edwin Armstrong’s superheterodyne receiver of 1918 made it the basis of practically every radio built since. A superheterodyne mixes the incoming signal with a tunable local oscillator so that whatever station is selected always emerges at the same intermediate frequency, where fixed, sharp filters and a well-behaved amplifier can do their work. Tuning a radio moves the local oscillator, not the filters.

Every such receiver faces the image problem that appeared in the quadrature drawing. A local oscillator at frequency $f_{LO}$ turns both $f_{LO} + f_{IF}$ and $f_{LO} - f_{IF}$ into the same intermediate frequency, so a station on the far side of the oscillator comes through as loudly as the wanted one unless it is filtered out beforehand. In a radio the image carries an unwanted station; in a quantum-limited optical receiver it carries the vacuum’s fluctuations, which cannot be filtered out because they are there even when nothing is. The three-decibel penalty is the image problem with no station left to remove.

Where the picture of mixing stops #

Perfect overlap. The beat’s full strength requires the signal and the oscillator to have the same shape and direction across the detector — the same spatial mode. A signal arriving from a slightly different direction beats with a phase that varies across the detector’s face, and the contributions cancel. A heterodyne receiver therefore sees only one spatial mode, which is the origin of the “antenna theorem”: its effective area times its field of view is one wavelength squared. That is the étendue limit counted in modes, from the other side.

Stable phases. The drawings assume the oscillator’s phase is steady relative to the signal’s. Laser phase noise broadens the beat, and when the combined linewidth exceeds the detection bandwidth the signal’s energy is spread over frequencies the receiver is not listening to.

Linear detectors. The derivation uses a detector whose current is proportional to the incident power. Real photodiodes saturate at high power, which limits how strong an oscillator can be used and is why receivers often split the oscillator between two detectors whose outputs are subtracted — a balanced receiver, which also cancels the oscillator’s own intensity fluctuations.

What the drawings cannot show #

None of the figures shows light. They show currents, noise and simulated measurement outcomes, which is appropriate — a heterodyne receiver’s entire output is an electrical signal — but it hides the part of the process that is optical: two beams, carefully matched in direction, focus and polarisation, combined on a beam splitter and focused onto a detector a few tens of micrometres across. Most of the engineering of a real heterodyne receiver is in making the two fields match, and none of that appears in a plot of signal-to-noise ratio against power.

Still open: phase-sensitive receivers below the quantum limit #

The three-decibel penalty of heterodyne detection belongs to measuring both quadratures. A phase-sensitive amplifier, which amplifies one quadrature and de-amplifies the other, can in principle avoid adding any noise at all to the quadrature it keeps, and squeezed light injected at the image frequency can cancel the vacuum noise that enters there. Both have been demonstrated in laboratories, and superconducting parametric amplifiers now operate near the quantum limit at microwave frequencies, where they read out superconducting qubits. Whether such techniques can improve a working astronomical receiver — where the signal is thermal, broadband and has no fixed phase to be sensitive to — is not settled, and the analysis depends on what exactly is being measured: a phase-insensitive quantity like a spectrum does not obviously benefit from amplifying one quadrature at the expense of the other.

The habit worth carrying away is to ask what a detector squares. A square-law detector cannot see a field, but it can see the product of two fields, and making one of them strong makes the product as large as the noise requires — which is how a picowatt is lifted to the quantum limit, how an optical frequency is counted with radio electronics, and why recording a light wave’s phase costs half a photon’s worth of noise per mode, a cost that is negligible for radio waves and ruinous for starlight.