The light that comes back bluer when it is warm

Why the sky is blue and the sunset is not treated scattering as an elastic process: a molecule driven by a light wave re-radiates at the same frequency, more strongly for blue than red, as the fourth power of the frequency. The sand that darkens when it is wet followed light through many such scatterings in a grainy material. In both, the scattered light kept its colour. That is true of nearly all of it, but not all. In 1928 C. V. Raman and K. S. Krishnan in Calcutta, and independently Grigory Landsberg and Leonid Mandelstam in Moscow working with quartz, found that a small fraction of light scattered by liquids and crystals comes back at slightly different colours, shifted from the original by amounts characteristic of the material. The shifts are the frequencies of the material’s own vibrations.

Raman scattering is now one of the standard ways of identifying materials, because the pattern of shifts is a fingerprint of a substance’s molecular vibrations. It has a second use that depends not on where the shifted lines are, but on how strong they are relative to each other. The two kinds of shift — towards the red and towards the blue — carry the temperature.

A discovery made with sunlight #

Raman’s discovery is a lesson in seeing a faint effect with crude means. Working without a laser, he and Krishnan focused sunlight through a telescope lens into a flask of purified liquid, passed the incident light through a violet filter, and looked at the scattered light through a second, complementary filter that blocked violet. Ordinary scattering, at the same colour, was stopped. A faint glow of a different colour got through. The Austrian physicist Adolf Smekal had predicted in 1923, from the quantum theory of light, that a molecule could take or give energy to a scattered photon, and Raman’s experiment showed that it did. The Nobel Prize came in 1930, two years after the discovery, one of the shortest delays on record.

The quantum view is what makes the two lines natural. In the classical picture a vibrating molecule modulates the light it scatters, as a vibrating mirror would, and the modulation puts equal sidebands above and below the original frequency, as light arrives in lumps found the classical wave picture getting the photoelectric effect half right. The inequality of the sidebands has no classical explanation at all. It appears only when the light and the vibration are both treated as coming in quanta, and the rates are counted as quanta exchanged.

Two ways to scatter with a vibration #

A molecule or crystal that vibrates at angular frequency $\Omega$ can exchange energy with light only in whole quanta $\hbar\Omega$. A photon scattered by it can leave a quantum behind, emerging with frequency $\omega_L - \Omega$: the Stokes line, named after George Stokes’s rule that fluorescent light is redder than the light that excites it. Or it can pick a quantum up, emerging at $\omega_L + \Omega$: the anti-Stokes line.

The two lines are not equally strong, and the reason is the most basic fact of quantum statistics. Picking up a quantum requires the vibration to have one to give. If the vibration holds on average $n$ quanta, the rate of anti-Stokes scattering is proportional to $n$. Leaving a quantum behind is possible even if the vibration holds none, and the rate of Stokes scattering is proportional to $n + 1$. In thermal equilibrium $n$ follows the Bose–Einstein distribution,

$$n = \frac{1}{e^{\hbar\Omega/kT} - 1},$$

the same that the share that is not half a kT used to explain why stiff vibrations freeze out of a solid’s heat capacity. The ratio of the two rates is then

$$\frac{n}{n+1} = e^{-\hbar\Omega/kT},$$

exactly. For silicon’s vibration, whose quantum corresponds to 748 kelvin, the anti-Stokes line at room temperature is a tenth of the Stokes line; at 100 kelvin, less than a thousandth. The Stokes line changes little because $n + 1$ stays close to one until the crystal is hot; nearly all the temperature dependence is in the anti-Stokes line, which needs a vibration already excited and so measures directly how excited the vibrations are.

A thermometer with nothing to calibrate #

The measured ratio of the two lines includes one more factor: the scattered power depends on the fourth power of the scattered light’s frequency, as Rayleigh’s does, so the bluer line is boosted by $((\omega_L+\Omega)/(\omega_L-\Omega))^4$ — about 1.2 for silicon with green light. That factor is known from the laser’s wavelength and the shift alone. Everything else cancels: the strength of the material’s response to light, the amount of material, the efficiency of the collecting optics, the brightness of the laser. The ratio depends on the shift, which is read off the spectrum, and on the temperature. It can be inverted to give the temperature directly, with no calibration against any other thermometer.

That makes it a primary thermometer in the same sense as a gas thermometer or the noise of a resistor: its relation to temperature comes from fundamental physics rather than from a material property measured once and trusted afterwards. It is also a non-contact one, reading the temperature of whatever the laser spot falls on — the channel of a transistor a micrometre across, a sample heated by another laser, a mineral squeezed in a diamond anvil cell at the pressure of the Earth’s core, where no wire can reach. The diamond anvils themselves, whose own vibration lies at 1332 cm⁻¹, are used to read their temperature in the same way.

The fourth-power factor connects the effect back to the blue sky. Raman scattering is the same induced radiation from an oscillating dipole that Rayleigh scattering is; the dipole is simply modulated by the molecule’s vibration, so it radiates at sum and difference frequencies as well as at the original one, in the way any modulated carrier does. Everything a scatterer removes added up all the light a scatterer takes out of a beam; the Raman lines are a minute part of it, typically between a thousandth and a millionth of the elastically scattered light, which is why they were not seen until 1928 and why a laser and a good filter are needed to see them now.

When the ratio is not a temperature #

The ratio measures the occupation of one particular vibration, and it is a temperature only when that vibration is in equilibrium with everything else. Often it is not. A laser pulse absorbed by graphene or a semiconductor creates hot electrons, which hand their energy first to a few optical vibrations faster than those vibrations can share it with the rest of the crystal; for a few picoseconds those modes are far hotter than the lattice, and a time-resolved anti-Stokes measurement reads their own, much higher, occupation. In a working transistor the vibrations that electrons excite most strongly can be out of equilibrium with the slower sound waves that carry heat away. In those cases the ratio is still exactly $n/(n+1)$ for the mode it sees, and that is useful — it is one of the few ways of measuring how energy flows between vibrations — but it is not the temperature a thermocouple would read.

The frequency factor can also mislead. If the laser’s colour lies close to an electronic absorption of the material, the scattering is resonantly enhanced, and the enhancement is different for the Stokes and anti-Stokes colours because they lie at different distances from the resonance. The simple fourth-power factor then fails, and the ratio has to be corrected with a measured profile of the resonance. A practical Raman thermometer chooses its laser well away from any absorption of the sample for exactly that reason.

How finely it can tell #

The sensitivity follows from differentiating the exponential: a fractional error $\delta r/r$ in the ratio is a temperature error $(kT^2/\hbar\Omega)\,\delta r/r$. It grows as the square of the temperature, so the method is sharpest when the material is cold and the vibration stiff. That is also exactly when the anti-Stokes line is faintest, which is the method’s practical limit. Diamond’s stiff vibration would give the sharpest thermometer at room temperature, but its anti-Stokes line there is three thousandths of the Stokes line, so a precise ratio needs long counting; silica’s softer band is less sensitive per photon, but gives ten times more of them. Choosing a vibration is a trade between how much the ratio changes per kelvin and how many photons there are to measure it with.

A fibre that is a thermometer along its whole length #

The most widespread use of the effect turns a single optical fibre into a line of thermometers kilometres long.

A short laser pulse sent into a glass fibre is scattered all along its length, and a little of the scattered light travels back to the input. Light scattered from a point $z$ along the fibre arrives back after the round-trip time $2z/v$, so timing the returning light locates where along the fibre it came from, as a radar times an echo. Filtering the returning light into its Stokes and anti-Stokes bands — for silica glass, a broad band centred about 440 cm⁻¹ from the laser — and taking their ratio at each delay gives the temperature at each point. The ratio has to be corrected for the slightly different attenuation of the two colours on their way back, which a calibration at one known temperature supplies.

The anti-Stokes signal rises about 0.8 per cent for each kelvin near room temperature, the Stokes signal hardly at all, and the instrument reads the difference. With a pulse ten nanoseconds long each reading covers about a metre of fibre, and a single unit can read thousands of points along tens of kilometres every few seconds. Such systems run along power cables to find where they are overheating, down oil and geothermal wells to map their temperature with depth, through tunnels as fire detectors, and along pipelines to find leaks by the cooling of escaping gas. The physics in every one is the ratio of $n$ to $n + 1$.

Sound instead of vibration, and light that amplifies itself #

A crystal or a glass has two kinds of vibration that scatter light. The ones drawn here are optical vibrations — atoms in neighbouring positions moving against each other — with frequencies of terahertz, shifting the light by hundreds of wavenumbers. The others are sound waves, in which neighbouring atoms move together, the collective motion that the gas of sound that carries a diamond’s heat found carrying heat through a crystal. Light scattered by a sound wave, Brillouin scattering, is shifted by only a few gigahertz — the Doppler shift from a sound wave moving at a few kilometres a second — and at room temperature its two lines are almost exactly equal, because a gigahertz vibration holds thousands of quanta and $n/(n+1)$ is indistinguishable from one. Brillouin thermometry in fibres therefore uses the shift’s frequency, which depends on the sound speed and so on temperature and strain, rather than the ratio of intensities. The choice between the two methods is the choice between a vibration stiff enough that its occupation is a sensitive Boltzmann factor and one soft enough that it is not.

Raman scattering can also be stimulated. In a long fibre carrying an intense beam, light at the Stokes frequency is amplified as it travels, because each Stokes photon present makes the scattering into its own mode more likely, as photons in a laser do. The gain of silica peaks about 13 terahertz below the pump — the same 440 cm⁻¹ band the fibre thermometer reads — and Raman amplifiers built on it boost signals in long-distance optical-fibre links without converting them to electronics. The process that makes a faint thermometer in one direction makes an amplifier in the other, and the asymmetry between the two is again $n + 1$ against $n$: stimulated Stokes scattering always wins.

What is left at absolute zero #

The “+1” in the Stokes rate deserves a closer look, because it is not classical.

A classical vibration with thermal energy $kT$ would scatter light into the two lines equally: it can be pushed or can push back with equal probability. The asymmetry exists only because the vibration is quantised, and at its root is the vibration’s zero-point motion. A quantum oscillator in its ground state still moves, and light can always lose a quantum to it; but there is nothing below the ground state, so light can never gain one from it. As the temperature falls, the anti-Stokes line fades away while the Stokes line stays, and the ratio measures how close the vibration has come to its ground state.

That is now used to show that a mechanical object is in its quantum ground state. In 2012 a group at Caltech cooled a silicon beam a few micrometres long, vibrating at a few gigahertz, to an average occupation of the order of one quantum, and measured the asymmetry between the light it scattered up and down in frequency: the motional sidebands were unequal, as the quantum rates predict and no classical motion could produce. The measurement is Raman thermometry carried to its limit, where what it reads is no longer a temperature but the zero-point motion itself — the same irreducible quantum noise that the noise every amplifier must add found setting a floor under any amplifier.

The reverse process runs too. If a solid absorbs light just below a transition and re-emits it on average at a slightly higher energy, taking the difference from its own vibrations, it cools — the anti-Stokes fluorescence that the glow that carries a voltage noted as a way of refrigerating a crystal with a laser. Its efficiency is limited by the same Boltzmann factor that makes the anti-Stokes Raman line faint: the colder the crystal, the fewer excited vibrations there are to take energy from.

A single vibration, a thin sample and no absorption #

The figures take a single vibrational mode with a sharp frequency and the Bose–Einstein occupation of a solid in equilibrium; real spectra have several modes, broadened lines whose position and width also shift with temperature, and — in glass — a broad band whose shape changes slightly with temperature too. The ratio is computed for a transparent sample in which the scattered light is not reabsorbed; an absorbing or a strongly heated sample needs corrections for the light’s path and for the laser’s own heating, which is often the largest error in practice. The fibre model has a single attenuation for each band, a pulse that does not broaden, and no noise; real instruments average many pulses and calibrate the differential attenuation against a reference coil. The domain is temperatures from tens of kelvin to about 1,500, where the materials keep their structure, and scattering weak enough that each photon scatters at most once.

Still open: how small a hot spot it can see #

Raman thermometry reads the temperature of whatever the laser spot covers, and the spot is limited by diffraction to about half a wavelength — a few hundred nanometres. The transistors in a modern processor are smaller than that, and their hottest points smaller still; heat in them is carried by vibrations that travel ballistically over tens of nanometres before they scatter, so the very idea of a local temperature becomes doubtful at that scale. Tip-enhanced Raman spectroscopy, which concentrates the light at the apex of a sharp metal tip, can read Raman signals from regions tens of nanometres across, and whether the anti-Stokes to Stokes ratio measured there is a temperature, a measure of particular vibrations driven out of equilibrium, or an artefact of the tip’s own enhancement is argued from one experiment to the next. How far the thermometer can be shrunk before what it measures stops being a temperature is not settled.

The thermometer itself is short. Light scattered by a vibration of quantum $\hbar\Omega$ loses a quantum at a rate proportional to $n + 1$ and gains one at a rate proportional to $n$, so their ratio is $e^{-\hbar\Omega/kT}$ times a known frequency factor — 0.103 for silicon at 300 K and 7 × 10⁻⁴ at 100 K — a temperature read without contact or calibration, along a fibre a metre at a time at 0.8 per cent per kelvin, and, in a resonator cold enough, down to the zero-point motion that keeps the Stokes line alive at absolute zero. A crystal can always take a quantum from light; it can give one back only if it is warm.