The absorption measured with a stopwatch

The hooks that weigh an absorption line showed a beautiful way to measure an absorption line’s strength: not in terms of how much light it took, but by looking at the bend it put in the fringes of an interferometer, where its effect on the refractive index shows far from the line itself. That essay ended with how well oscillator strengths could be known in practice. An absorption measurement has always had this basic trouble. It is a comparison of a large quantity with itself. A sample absorbing one part in a million means the transmitted beam differs from the incident beam by one part in a million. Anything that makes the beam flicker by more than that will hide the signal.

The traditional fix is a longer path. If a gas absorbs one part in a million per metre, then a thousand metres gives a part in a thousand, which is measurable. That is the principle of the multipass cell, in which mirrors fold a beam many times through a sample, and also of atmospheric measurements taken across several kilometres of air. But folding needs enormous mirrors, and the source’s noise is still there.

Anthony O’Keefe and David Deacon, in 1988, introduced a different idea. They were interested in testing the reflectivity of high-grade mirrors, in which they had measured the leakage of light from a cavity as it died. Their insight was that the sample could be placed inside the cavity, and then what would matter was not how much light was transmitted but how fast it decayed. The method is called cavity ring-down spectroscopy. In converting a measurement of brightness into one of time, it removes the problem at its source.

An absorption that is a slope #

Take two mirrors, a distance $L$ apart, each with reflectivity $R$ close to 1. Send a short pulse of light in. Very little enters, because the first mirror reflects almost all of it. But what enters bounces between the mirrors and at each reflection a small fraction $1 - R$ leaks out. The leak is detected on the far side. It is proportional to the intensity inside, so the leak follows the internal decay exactly. Each transit takes $L/c$ and loses a fraction $1 - R$, along with whatever the gas in the cavity takes. If the gas absorbs a fraction $\alpha$ per unit length, the intensity falls exponentially:

$$I(t) = I_0e^{-t/\tau}, \qquad \tau = \frac{L}{c\left[(1 - R) + \alpha L\right]}.$$

The absorption coefficient comes out as

$$\alpha = \frac{1}{c}\left(\frac{1}{\tau} - \frac{1}{\tau_0}\right),$$

where $\tau_0$ is the decay time of the empty cavity. Nothing else enters: no length, no brightness and no reflectivity, once $\tau_0$ has been measured. The difference between the two lines in the figure is a difference of slopes on a log plot. A single trace, in about 100 µs, gives the measurement. Note what is missing here: $I_0$ has disappeared. A pulse twice as bright gives a line displaced upward and parallel to the first, and a slope does not depend on where a line starts. So one has a measurement that, in principle, is not affected by the source’s brightness or by its fluctuation from one shot to the next.

A long path, folded into a metre #

In one decay time the light travels a distance $c\tau_0 = L/(1 - R)$. That is the effective path length. A pair of mirrors with 99.99 per cent reflectivity, a metre apart, thus folds ten kilometres into a tube. Better mirrors with losses of a few parts per million give several hundred kilometres. No mirror made of silver or aluminium approaches this. Metals absorb a few per cent at best. These are dielectric mirrors, made of dozens of alternating quarter-wave layers, in the manner of the pile of glass that stops every colour, whose stack produces a reflectivity that depends on the absence of any absorbing material. The best examples, developed for gravitational-wave detectors and optical clocks, have total losses under 2 ppm, and the two mirrors that hold a beam deals with how to keep light confined between such mirrors.

The price is also easy to calculate, and it is surprising how little it is. Almost all of the light is reflected at the input mirror and never enters. With 99.99 per cent mirrors, only one part in ten thousand gets in, and only that fraction of the inside leaks out at each bounce. With a milliwatt continuous-wave laser this still leaves plenty of photons in the cavity for a sensitive detector. Moreover, when the laser’s frequency matches a resonance of the cavity, the build-up inside greatly boosts the power in the cavity by constructive interference. That resonance is the difficulty that needs managing in continuous-wave experiments, since the cavity accepts light only in narrow bands, with an acceptance spacing of $c/2L$ and a width of $(1 - R)c/2\pi L$. For a one-metre cavity at 99.99 per cent this is a width of about 5 kHz. A laser therefore has to be tuned to it, or the cavity’s length dithered across it, before each decay can be measured.

Lamp noise that a decay does not notice #

The simulation shows the most remarkable feature of the method. Both measurements use the same simulated source, whose brightness jumps by 5 per cent between shots. In the direct measurement, each transmission reading is the ratio of two noisy numbers, so the noise in the reading is of order $7 \times 10^{-2}$. The true absorption of $10^{-7}$ per metre is hopelessly below that. Averaging 400 shots gains a factor of 20, which is not nearly enough. In the ring-down measurement, each shot is a decay curve, and its slope is fitted, with noise only from a 0.1 per cent detector. It scatters by about $5 \times 10^{-8}$.

So the gain over a direct metre-long measurement is about a million. The long path accounts for ten thousand of that. The rest is from discarding the source’s noise. The slope of the logarithm of a decay has no information about how bright the pulse started out, and thus the source’s flicker never gets into the result in the first place. Comparable to this is the way a radioactive decay’s half-life can be measured with no knowledge of the sample’s size, a principle that a nucleus with no clock relied on. The result is a decay time, and time is what laboratory instruments can measure with the most ease and at the highest accuracy.

Real sources do have fluctuations at some level in other ways. A cavity’s excitation pattern can shift between shots if a laser’s spatial mode changes, so that high-order transverse modes are also excited, each with a slightly different decay. Fitting a sum of exponentials with one slope gives a biased result, and careful systems filter the excitation to one mode. The method’s main practical limits lie in things like that, along with mirror drifts and the small changes in the cavity with temperature and vibration, none of them to do with the light source’s brightness. In a well-engineered instrument a decay time can be reproduced to about 0.01 per cent or better, a figure that a direct absorption measurement cannot approach.

A line read from a stopwatch #

The laser is stepped in frequency across the line, and at each step a few thousand decays are measured. Turned into absorption, the result is the spectrum. Its baseline is the mirror’s loss. Mirrors do not have the same reflectivity at all wavelengths, and they can pick up material from the gas. So an absolute measurement needs a reference for the empty cavity’s decay, either by a separate measurement, or by fitting a baseline that varies slowly across the line. For narrow lines this is quite easy, because the line changes much faster with frequency than any baseline. The peak’s height, relative to the local baseline, needs almost no calibration at all. Since a decay time is measured against a quartz or atomic clock, which is extremely accurate, the method has become a standard for such things as the abundance of trace gases.

Commercial instruments based on this idea monitor atmospheric carbon dioxide and methane at parts per billion, on balloons and aircraft and at remote stations. A ring-down analyser that sits in a van can detect a methane leak from a natural gas main at several hundred metres, and regulators have come to rely on such surveys. In medicine, they can measure the amounts of certain isotopes of carbon in exhaled breath, which, after a tagged test meal, indicate an infection of the stomach. Water vapour can be measured to parts per trillion in semiconductor process gases. The strength of the method is that, once an instrument is calibrated, it stays calibrated, because it depends on nothing that drifts readily.

The two knobs that decide how little can be seen #

Differentiating the formula, the smallest change in $\alpha$ that can be detected is

$$\alpha_{\min} = \frac{1 - R}{L}\frac{\Delta\tau}{\tau}\frac{1}{\sqrt{N}},$$

where $\Delta\tau/\tau$ is the fractional precision of a decay measurement and $N$ the number of them averaged. Three separate things improve the result, and each of them is independent of the rest. A decay can be fitted well if it is much above the detector noise and if it has been sampled sufficiently. With an ordinary photodiode, a single shot at 0.02 per cent is reasonable. At $10^{-11}$ per metre, a line can be seen from a very small trace of a gas, or from a very weak overtone of a strong line. Weak overtone lines, which involve a molecule jumping by several vibrational quanta at once, can be measured by this method because the very long path brings them up to where they are visible. The spectrum that weighs a bond relied on these same kinds of overtones in its measurement of molecular anharmonicity.

In 2011, Iacopo Galli and colleagues in Florence used a variant of this idea to measure radiocarbon in carbon dioxide, at a concentration of 43 parts in a million billion, in an optical measurement that had until then been possible only with accelerators. Their method, called saturated-absorption cavity ring-down, uses the additional trick of saturating the molecules in the cavity, which the third process equilibrium demands gave the rate equations for, in order to separate a molecule’s absorption from the cavity’s own loss within a single decay. The implication is striking. The clock read by counting atoms is a story of accelerators that count individual carbon-14 atoms, and now a bench-top laser instrument can, in principle, reach some of the same abundances by reading the slope of a decay. There have been some reproducibility issues, and the claim has been debated, with several groups now working to confirm the detection limit.

A relation of the same width in another sense #

Inside the cavity, the decay time and the spectral width of its resonance are the same fact, by the relation in the width that is a lifetime. A 33 µs decay corresponds to a resonance about 5 kHz wide. Indeed, it would be possible, in principle, to measure the absorption by the width of a cavity resonance, in the frequency domain, instead of by the decay in the time domain. Since they are equivalent, why is the decay so much better in practice? Because a decay is a single trace with a shape, and a frequency scan is many separate measurements at different times, each with its own noise. It is the same reason that the fringe and the spectrum are one measurement gives a multiplex advantage for a Fourier spectrometer. Laying out the measurement in time has allowed one shot to contain all the information about the slope, and the source’s frequency noise does not matter during the decay either, because the laser is off.

A cavity is also the means by which the lowest losses in optics are known. Mirror reflectivities above 99.9 per cent cannot be measured by comparing input and output power with any precision, because the difference is too small, and ring-down was the first reliable method. That is the original reason for O’Keefe and Deacon’s work, and the technique was used for it even earlier: Herbelin and others used a phase-shift version in 1980, and Anderson, Frisch and Masser in 1984 used a decay for mirrors. Researchers looking to test the smallest losses in new mirror coatings, on which gravitational-wave detectors depend, continue to use it.

A comparison with the interferometer’s route #

The hook method that this story began with shares something with ring-down, in that neither measures how much light was removed. The hook method reads an absorption line through the refractive index beside it, since the two are tied together by causality, as the answer that cannot come first explains. Ring-down reads the same line through its effect on a rate. In each, a difficult comparison of two nearly equal intensities is replaced by an observable with a different nature: a fringe’s position in the first, a time in the second. It is a pattern that runs through precision measurement. The wait that sets the precision has an atomic clock reading a frequency from a phase accumulated over a time. Whenever it can be arranged, turning a quantity into a time or a frequency has been a good bargain, because those are what laboratory instruments measure best.

The bargain has a price, as usual. A ring-down instrument can only look where its mirrors reflect. A dielectric stack is highly reflective over a limited band, perhaps a tenth of its centre wavelength, so a mirror set covers one region of the spectrum and has to be replaced to reach another. Broadband sources and frequency combs have extended this, with a wide spectrum entering the cavity and leaving through a spectrometer, but they trade some sensitivity for the bandwidth. A traditional multipass cell, on the other hand, can look across an enormous range with the one set of metal mirrors, and for a survey of an unknown sample it may be the better choice.

A noise model with two sources #

The noise model is simple: a Gaussian jump in source brightness between shots, a uniform detector noise on the decay, and nothing else. A real instrument suffers from mode beating, in which several transverse modes decay together and give a non-exponential curve, and from a laser turning off too slowly for the first part of the decay to be usable. Fitting has to start some way after the laser has shut off, which removes some of the signal. Long-term drifts in mirror alignment and temperature matter, and these limit how far averaging can go, so that the $1/\sqrt{N}$ line in the last figure stops improving after a time, where a real noise floor sets in. Reported instruments typically reach a limit after minutes or hours of averaging.

The picture of a single exponential ignores saturation as well. If a laser is strong enough to saturate the gas, then the absorption itself changes during the decay as the intensity falls, giving a non-exponential curve. The method treats that as an error in simple use and as a signal in the saturated-absorption version. A line’s shape also depends on the gas’s pressure and temperature, so turning a line’s strength into a concentration requires knowing those to a matching accuracy. The figures for specific instruments, such as the ones monitoring methane or radiocarbon, are meant as orders of magnitude and vary from one instrument to another.

Still open: how low a cavity can go #

The technique’s lower limit depends on how far mirrors, fits and averaging can be pushed, and on how well the systematic effects can be controlled. Current best performances are near $10^{-13}$ per centimetre per root hertz, using techniques in which a frequency comb or a stabilised laser is locked to the cavity. This allows direct measurement of a molecular line’s shape at a level that tests quantum-mechanical calculations for hydrogen and other simple molecules. Measuring the spectrum of molecular hydrogen by ring-down, with its extremely weak quadrupole lines, has provided a test of the theory at parts per billion, and it could reveal new long-range forces between nuclei if there are any at nanometre scales.

Whether a truly optical method can replace accelerator mass spectrometry for radiocarbon dating is a practical question that has not been settled. Reports have made a case that this may eventually become possible for samples of a size relevant to dating, while others have raised concerns about how carbon dioxide’s strong lines overlap with those of other trace molecules, which need to be removed with great care. The stakes are large, since a bench-top radiocarbon instrument could change what is measured in fields from pharmacology to palaeoclimate, and it is a matter that will be settled by careful comparison between methods.

A pulse caught between two mirrors decays at a rate set by the mirrors and by what lies between them. That makes an absorption a change in a slope on a log plot, in which the pulse’s starting brightness does not appear. 99.99 per cent mirrors fold 10 km into a metre, and a noisy source that swamps a direct measurement by five orders of magnitude does not enter at all. It has turned one of optics’ hardest comparisons into a measurement of time.