The Doppler shift with nothing coming closer

The note that changes on approach began the Doppler effect where everyone meets it, with a siren passing in the street, and the essays that followed carried it into relativity: the shift that survives at right angles, the sky that crowds into a cone, the brightness that depends on who is watching, the shift a mirror gives twice, and finally everything from an exchange of pulses, which rebuilt special relativity from nothing but the ratio of the rates at which two observers send and receive flashes. In every one of them, the shift came from a change in distance. A source approaching packs its wave crests closer; one receding spreads them out. The rate at which the crests arrive is the rate at which they are sent, corrected for how fast the path between is shortening.

This essay finds a frequency shift with no change in distance at all. Light can come back from a surface spinning in place, neither approaching nor receding at any point the light touches, with its frequency shifted by an exact amount. Light can pass straight through a turning plate of crystal and come out at a different frequency. The shift is a Doppler shift in every sense that matters, but its ingredients are not a speed and a wavelength. They are a rotation rate and an angular momentum.

Crests that wind round the axis #

An ordinary beam of light has flat wavefronts: at any instant the crests are planes, one wavelength apart, perpendicular to the beam. Light can also be made with wavefronts shaped like a corkscrew. Its phase advances by $2\pi\ell$ on going once round the axis, so the field varies as $e^{i\ell\varphi}$ with $\varphi$ the angle round the axis and $\ell$ a whole number, and the crests form $\ell$ interleaved helices running along the beam.

In cross-section, at one instant, the crests are $\ell$ spokes radiating from the axis. As time passes the spokes turn: at a fixed plane the pattern rotates once every $\ell$ optical periods, because a full turn of the helix passes by in that time. Such beams are made by passing an ordinary laser beam through a spiral phase plate, whose thickness rises by $\ell$ wavelengths in one turn, or through a hologram designed to imprint the spiral. Each photon in such a beam carries an angular momentum of $\ell\hbar$ about the beam’s axis, quite separate from the $\pm\hbar$ of its circular polarisation. The first is called orbital angular momentum and the second spin; the angular momentum that is in nothing at all found angular momentum stored in static fields, and these beams carry it in a travelling one.

Looking at the pattern from a turning frame #

Now look at the beam from a frame that turns about its axis at angular speed $\Omega$. In that frame the angle round the axis is $\varphi' = \varphi - \Omega t$, so the field $e^{i\ell\varphi - i\omega t}$ becomes $e^{i\ell\varphi' - i(\omega - \ell\Omega)t}$: its frequency is shifted by $\ell\Omega$.

That is the rotational Doppler shift, and its form is the ordinary Doppler shift with every quantity replaced by its rotational counterpart. A plane wave $e^{ikx - i\omega t}$, seen by an observer moving at speed $v$, has its frequency shifted by $kv$: wavenumber times speed. A helical wave $e^{i\ell\varphi - i\omega t}$, seen by an observer turning at $\Omega$, has its frequency shifted by $\ell\Omega$: winding number times angular speed. The winding number is to angle what the wavenumber is to position, and the angle the uncertainty principle cannot be written for found the same pairing in quantum mechanics, where angular momentum is to angle what momentum is to position.

A surface turning at $\Omega$ is such an observer. A rough disc spinning at $\Omega$, lit along its axis by a helical beam, scatters light into many winding numbers, and the light it scatters into winding number $\ell$ is shifted by $\ell\Omega$ relative to the light it would scatter at rest. The disc is not approaching anything. Every point of it moves sideways, across the beam.

The shifts are small. A disc turning a hundred times a second — six thousand revolutions a minute — shifts light of winding number one by a hundred hertz and winding number twenty by two thousand, against an optical frequency of about $4.7 \times 10^{14}$ hertz: a few parts in $10^{12}$. No spectrometer could see that as a change of colour. An interferometer sees it easily, as a beat between the shifted light and an unshifted reference, at the difference frequency.

Why the radius cancels #

The rotational shift has a property that the ordinary Doppler shift does not seem to share, and examining it shows they are the same thing.

A helical wavefront is tilted. At a distance $r$ from the axis, going once round the circumference $2\pi r$ advances the phase by $2\pi\ell$, so the wavefront has a sideways wavenumber of $\ell/r$: a beam that looks straight overall is, locally, travelling very slightly askew, steeply near the axis and nearly straight far out. A surface spinning at $\Omega$ moves sideways at $\Omega r$, slowly near the axis and fast far out. The ordinary Doppler shift at that point is the sideways wavenumber times the sideways speed, $(\ell/r)(\Omega r) = \ell\Omega$, and the radius cancels. Every part of the illuminated surface shifts the light by the same amount, which is why the scattered light from the whole disc comes back at one frequency rather than smeared over many.

So the rotational Doppler shift is not a new effect. It is the ordinary one, read locally, in a beam whose local direction of travel varies round the axis in just the way that makes the tilt and the speed scale oppositely. What is new is the bookkeeping: the shift is quantised by the beam’s winding number and set by the rotation rate, with no speed or wavelength anywhere in the answer.

Hearing a spin along its own axis #

That makes the effect an instrument. In 2013 Lavery, Padgett and colleagues lit a spinning rough surface along its axis with ordinary laser light, collected the light it scattered into winding numbers $+\ell$ and $-\ell$, and let the two interfere on a detector. One was shifted up by $\ell\Omega$, the other down, and their beat came out at $2\ell\Omega$, from which the rotation rate could be read directly. The measurement looks straight down the axis of the spinning object, where every point of it moves across the line of sight and an ordinary Doppler measurement records nothing at all.

The same idea has been proposed for measuring the rotation of objects that cannot be approached — a turbine, a tumbling satellite, perhaps the rotation of astronomical sources, whose light might carry a twist from the spinning matter it left. Whether that last application will work is debated, because turbulence and the vast distances scramble the twist; on the laboratory bench it is a standard technique.

Winding numbers in the thousands #

Because the shift is proportional to the winding number, the measurement becomes more sensitive the more the wavefronts wind. Nothing limits $\ell$ in principle; in practice the beam’s dark core grows with $\ell$, its bright ring moves outward, and making a clean high-$\ell$ beam takes a finely made phase plate. Beams with winding numbers in the hundreds are routine, and in 2016 photons with winding numbers above ten thousand were made and even entangled with one another. At a winding number of a thousand, a disc turning a hundred times a second shifts the light by a hundred kilohertz, and a disc turning once an hour by a quarter of a hertz — a rotation slow enough to be invisible to the eye, measured from along its own axis by a beat on a detector.

A plate that turns and a photon that pays #

There is a second rotational shift, and it comes with an energy argument that makes its size inevitable. Pass circularly polarised light through a half-wave plate. The plate reverses the light’s handedness: right-circular in, left-circular out. A photon of right-circular light carries spin angular momentum $+\hbar$ along its direction of travel; the same photon, left-circular, carries $-\hbar$. The plate has taken $2\hbar$ from each photon, and it feels the torque — Beth measured exactly that torque in 1936, by hanging a half-wave plate from a fine fibre and shining circularly polarised light through it.

Now let the plate turn, at angular speed $\Omega$, in the direction of the torque or against it. A torque acting on something turning does work, at a rate of torque times angular speed. Per photon, the angular momentum transferred is $2\hbar$ and the work is $2\hbar\Omega$, and that energy must come from or go into the light. A photon’s energy is $\hbar\omega$, so its frequency changes by exactly $2\Omega$. Garetz and Arnold demonstrated this in 1979 with a spinning half-wave plate as a frequency shifter, and the same principle underlies some electro-optic frequency shifters in which a rotating birefringence is made electronically rather than mechanically.

The argument does not care what the plate is made of or how thick it is, only that it reverses the spin. It also says what the orbital shift must be: a rotating element that reverses the winding number from $+\ell$ to $-\ell$, such as a spinning Dove prism, takes $2\ell\hbar$ per photon and shifts the light by $2\ell\Omega$. Angular momentum transferred times angular speed is energy transferred, and the frequency follows.

The geometric phase that runs #

The phase that is only a shape found that light taken round a closed loop of polarisation states picks up a phase that depends only on the area the loop encloses on the sphere of polarisations, the geometric phase of Pancharatnam. A half-wave plate turned through an angle $\theta$ sends circularly polarised light round such a loop and gives it a geometric phase of $2\theta$. Turn the plate steadily, so that $\theta = \Omega t$, and the phase grows steadily, at $2\Omega$. A phase that grows steadily in time is a frequency shift. The spinning wave plate’s $2\Omega$ is that geometric phase made to run: what was a fixed offset for a plate at rest is, for a plate in motion, a change of colour.

Light that turns what it touches #

The energy argument has a mechanical mirror image. If a rotating element shifts the light by giving or taking angular momentum, then light absorbed by a small object must give it angular momentum and set it turning. In 1995 a group in Brisbane trapped a microscopic absorbing particle in a laser beam with winding number three and watched it spin, each absorbed photon handing it $3\hbar$: an optical spanner. A transparent birefringent particle in circularly polarised light spins the same way, by taking the spin angular momentum that Beth’s plate took. Such particles reach thousands of revolutions per second, driven by milliwatts.

The two effects are one ledger. A particle spun up by twisted light is doing work on nothing; it is being worked on, and the energy comes from the light, whose scattered part is shifted down in frequency by the rotational Doppler shift, by just enough to pay for the particle’s growing kinetic energy and the drag of the liquid around it. Light has a pressure found the same ledger for linear motion: a mirror pushed by light recedes, and the reflected light comes back redder by exactly the energy the mirror gained. Turning replaces receding, and angular momentum replaces momentum.

Not the Sagnac effect #

Rotation also changes the light that goes round a ring, and the two effects are often confused. The ring where the two beams disagree found that light sent both ways round a rotating loop comes back at different times, by $4A\Omega/c^2$, which is how ring-laser and fibre gyroscopes in aircraft measure rotation. That effect needs light to travel round the rotating thing, and it measures the rotation of the frame the loop is built in. The rotational Doppler shift needs light to strike or pass through the rotating thing along its axis, and it measures the rotation of the object relative to the beam. A Sagnac gyroscope works with no rotating part and ordinary flat wavefronts; a rotational Doppler measurement works only with helical wavefronts or a spin-reversing element. One is about paths, the other about phases wound round an axis.

The same winding works for any wave with a phase. Acoustic beams with helical wavefronts have been made with arrays of loudspeakers, and they carry angular momentum and show the rotational shift when they scatter from a spinning target; radio beams with winding numbers have been made with spiral antennas. In each case the shift is the winding number times the rotation rate, whatever the wave and whatever its speed, because the argument never used the speed of the wave at all.

The cycle a turning antenna adds #

The same shift has to be corrected every day in the most precise uses of satellite navigation.

Navigation satellites transmit circularly polarised radio waves, and receivers measure the phase of the carrier to a small fraction of a cycle, which at the GPS L1 frequency is a fraction of nineteen centimetres. A receiving antenna is a polarisation analyser with an orientation. Turn it about the line of sight to the satellite and the phase it measures shifts by one cycle per turn — the spin rotational shift, $\Omega$ per antenna rotation, integrated over the rotation — exactly as if the satellite had moved nineteen centimetres further away. The satellites themselves turn slowly to keep their solar panels facing the Sun, and their antennas turn with them. The effect is called phase wind-up, it was worked out for precise positioning in 1993, and every processing package that reaches centimetre accuracy corrects for it from the known orientations of the satellite and the receiver.

What the pictures cannot show #

The figures treat beams of pure winding number and ideal spinning surfaces. A real rough surface scatters into many winding numbers at once and the detector selects some of them with a mask or hologram, losing most of the light; the beat in the figure is drawn clean, with a little noise added, where a real signal sits on speckle that fluctuates as the surface turns. The shift calculations ignore any wobble of the axis, which adds an ordinary Doppler shift from the parts of the surface moving towards and away from the detector, and the radius figure assumes the beam’s tilt is exactly $\ell/r$, which holds for the helical phase but not for the beam’s intensity profile, which has a dark core on the axis. The phase wind-up figure assumes a perfectly circular signal and an antenna turning exactly about the line of sight.

Still open: twisted light from rotating sources in the sky #

If the light from a rotating astronomical object — a spinning accretion disc around a black hole, a rotating star — picks up a rotational shift or a twist in its phase from the motion of the matter that emitted or scattered it, then the winding numbers in the received light would carry information about the rotation that no ordinary spectrum contains. Proposals to look for it have been made for radio waves from black holes, where the phase structure of the field can in principle be measured across an array of telescopes. Whether such a twist survives the journey through turbulent plasma between the source and the Earth, and whether it can be separated from the twist the instruments themselves introduce, has not been settled, and claims of detection have been disputed.

The habit worth carrying away is to ask what plays the role of a wavenumber in a given geometry. A field that winds ℓ times round an axis, seen from a frame turning at Ω, is shifted by ℓΩ — the angular counterpart of k·v — and a half-wave plate turning at Ω shifts circular light by exactly 2Ω, because it takes 2ħ of angular momentum per photon and does work 2ħΩ. The Doppler effect needs a change of phase per unit time, and turning can supply one as well as approaching.