The answer that cannot come first

Air removes the treble from a distant sound and glass removes almost nothing from visible light, and the coefficient that says how much is measured frequency by frequency. This essay is about what that measurement has already decided about everything else the material does.

A medium is a machine that answers #

Shine light on a piece of glass and the electrons in it are driven. They are bound, so they have a natural frequency; they are damped, so they lose energy; and they are being pushed at whatever frequency the light has. That is the driven oscillator, and nothing else is needed.

A medium is a machine that answers, and the two things it does to a wave are the two things any driven oscillator does: it responds with a size, and it responds with a lag. A material’s response to light is a driven-oscillator curve, one copy per kind of electron it contains, and everything that follows is a consequence of those two numbers being two components of one complex answer rather than two independent properties.

The response has a size and a phase. The size decides how much energy the wave gives up, which is absorption. The phase decides how much the re-radiated wave lags the driving one, and adding a lagged wave to the original one is what slows the light down — which is refraction. So the two are not two mechanisms. They are the real and imaginary parts of one complex number, and a complex number has two parts because a response has a size and a lag.

Written that way, the susceptibility of a single kind of oscillator is

$$\chi(\omega) = \frac{S}{\omega_0^2 - \omega^2 - i\gamma\omega}$$

whose real part is the refraction and whose imaginary part is the absorption. Both are drawn in the figure above, and the region where the dashed curve runs downhill is exactly as wide as the solid one is tall. That band — where the refractive index falls as the frequency rises — is called anomalous dispersion, which is a misleading name for something compulsory.

Converted into the quantities an instrument reports, the same response becomes a refractive index and an extinction coefficient. The conversion is a complex square root and it changes nothing about the argument — what it changes is which pair of numbers is being drawn. It is worth knowing that the tabulated pair a spectroscopist uses and the response this essay is about are the same object in different coordinates, because the constraint below applies to both.

That much is a model. What follows is not, and the difference matters: the constraint below holds for any linear medium whatever, whether or not its electrons behave like springs.

Delete half of it, and the material answers early #

Suppose absorption and refraction really were independent. Then a material could be specified by choosing them separately — an absorption line here, and a refractive index that does not vary at all. Materials are routinely quoted that way in a table: a loss figure, and one index.

The question is what such a material would do, and it can be asked without any argument at all. A response in frequency is a response in time; take the frequency response, transform it back, and out comes the material’s answer to a single sharp kick.

The material with only half a response answers before it is asked. It is not slightly acausal or acausal in some limit: the acausal part is the exact mirror image of the real one and is exactly half the size, 0.274 against a peak of 0.547 in the units drawn, because keeping only the odd part of a function keeps half of it on each side of zero.

The calculation assumes one thing beyond time order, and it is worth naming because it is the assumption that fails first in practice: the response is linear. Double the field and double the polarisation. That is an excellent approximation for ordinary light in ordinary matter and a poor one inside a pulsed laser, where the medium’s answer depends on how hard it was asked and the whole apparatus of a frequency response stops applying. Nonlinear optics has its own relations, of a more complicated kind, and they are derived the same way.

That is the whole argument, and it is worth stating in the abstract because it uses nothing about light. Any system whose output depends linearly on its input, and which does not respond before the input arrives, has a frequency response whose real and imaginary parts are tied together by an integral. The tie is the Kramers–Kronig relations, published independently in 1926 and 1927, and the reason they hold for such different objects — a dielectric, an amplifier, a scattering amplitude, a sheet of steel in a magnet — is that the only thing they use is time order.

The true impulse response of a damped oscillator has a shape worth looking at: silence, and then a decaying ring. Every material’s answer to a kick has that form, and everything in this essay is a consequence of the silence on the left-hand half. A response that began before the kick would be a material that knew what was coming, and the whole Kramers–Kronig apparatus is the arithmetic of forbidding it.

One curve computed from the other #

If the constraint is real then a measurement of absorption alone should be enough to produce the refraction, with nothing else supplied. It is, and the arithmetic can be done here.

The integral is a principal value: it runs over every frequency, and at the frequency of interest the integrand has a pole through which the value is taken symmetrically. That structure is not decoration. It says that the index here depends on the absorption there — on absorption at frequencies where nobody is looking, including frequencies far outside any band the instrument covers.

The practical consequence is large enough to have a name in the trade. The refractive index of an ordinary optical glass across the visible is set by absorption in the ultraviolet that the glass never shows in a visible measurement, plus a smaller contribution from the infrared. That is why glass is transparent and dispersive at once, and why the dispersion always has the same sign in the visible: everything visible is on the low-frequency tail of an ultraviolet line.

Two real glasses make the point with no absorption anywhere in sight. Their measured refractive indices across the visible, from the Sellmeier coefficients their makers publish, both slope — and neither glass absorbs anywhere on that chart. The slope is the far tail of an ultraviolet absorption each of them has, reaching in from outside the picture. The index at a frequency where nothing is absorbed is fixed by absorption at frequencies that are not being looked at.

The same statement runs the other way and is used that way in practice. A material’s absorption across a band that no spectrometer can reach is inferred from its index where it can be measured; a thin-film maker measures reflection at a range of angles and extracts both. The pair of relations is what makes the extraction possible, and two glasses can be combined to cancel a derivative only because that derivative is fixed by absorption neither of them displays.

An engineer meets the absorption as decibels per kilometre, which is a number at one frequency. The point of this essay is that the whole curve of such numbers has already fixed the material’s refractive index at every frequency, including all the ones where the coefficient is zero. There is no such thing as choosing a transparency and a dispersion independently, and a specification that asks for both is asking for a material that cannot exist.

The total, which cannot be exceeded #

There is a second consequence, and it constrains the absorption itself rather than relating it to anything.

Integrate the absorption over all frequencies, weighted by frequency, and the result depends only on how many electrons there are per unit volume. Not on where the absorption lines sit, not on how wide they are, not on the material’s structure. Broadening a line lowers it by exactly as much as it widens it.

The history ran through exactly this case. Kronig arrived at the relation in 1926 while working on X-ray dispersion, where the puzzle was that measured refractive indices came out below one and nobody could see how to reconcile that with the absorption edges the same materials showed. The relation resolved it by making the two the same measurement, and Kramers stated the general form the following year. It is one of the few results in optics that was found because a table of numbers refused to make sense.

This is the f-sum rule, and it is a statement about counting: a medium cannot absorb more than it has electrons to absorb with. It is what makes an absorption measurement into an inventory. It is also the reason for a fact that sounds impossible until the accounting is done.

Above every resonance a material has, the electrons are driven out of step with the field, so their contribution to the susceptibility is negative and the refractive index is less than one. For X-rays, which are above essentially every electronic resonance in every material, the index is below one for everything — by about one part in a million, and always with the same sign.

That is why an X-ray mirror is a grazing-incidence mirror. With an index below one, total internal reflection happens on the way in rather than on the way out, at angles within a fraction of a degree of the surface, and every X-ray telescope is built out of nested tubes for that reason. It is a piece of practical engineering that follows from the sign of a sum.

The cleanest case has no resonance at all. Free electrons in a plasma have nothing to resonate at, so the index is below one at every frequency above the plasma frequency and the relation between wavenumber and frequency bends away from the light line. The ionosphere is the ordinary example of it, and it is why short-wave radio comes back to the ground rather than escaping.

Nothing here goes faster than light #

An index below one means that crests travel faster than $c$, and in the anomalous band near a strong line the group velocity can exceed $c$ as well, or become negative. Both are measured facts and neither is a problem, but the reason is worth being precise about because it is usually waved at.

Nothing here goes faster than light, and it is worth being precise about the case that looks as though it does. A pulse crossing a medium with anomalous dispersion can have its peak emerge before the peak of a pulse that went the same distance through vacuum. The emerging pulse is built out of the leading edge of the incoming one, which arrived earlier; nothing has overtaken anything, and the front of the disturbance travels at exactly $c$. What moved early was a feature of a shape, not a piece of information.

What arrives first is the front — the first non-zero disturbance — and the front travels at $c$ in every medium, because the front is built from the highest frequencies and the index goes to one there. That is the sum rule again. The peak of a pulse is a feature of a shape that the medium is free to rearrange, and a peak carries no news: the information is in the discontinuity at the front, which cannot be moved.

The two statements — that a medium’s response is causal, and that no signal outruns light — are therefore the same statement, and this is where the argument closes. Causality in the microscopic sense produced the integral relation; the integral relation forces the index to one at high frequency; and that is what keeps the front at $c$ no matter what the medium does to the middle of the pulse.

Everything the material does slowly, from the lines it never shows #

Evaluate the relation at zero frequency and it says something unusually concrete: a material’s static response is a weighted total of its absorption at every frequency, with the weight one over the frequency. Not related to it, not constrained by it — equal to it.

$$\varepsilon(0) - 1 = \frac{2}{\pi}\int_0^\infty \frac{\varepsilon''(\omega)}{\omega}\,\mathrm{d}\omega.$$

So the number a capacitor manufacturer prints on a datasheet is an integral over a spectrum the manufacturer never measured. And because of the $1/\omega$, the low-frequency absorption is weighted far more heavily than the high — which is the opposite emphasis from the sum rule above, where the weight was $\omega$ and the ultraviolet dominated.

Water is the case that makes this vivid, because its two numbers disagree spectacularly. Its refractive index in the visible is 1.33, so the response it offers to light is about 1.77 in the units above. Its static dielectric constant is 78. A factor of forty-four separates what water does to a steady field from what it does to light, and the integral says exactly where the difference is: in absorption below the visible.

It is there. The water molecule has a permanent electric dipole, and in a steady field the molecules turn to line up with it — a mechanism that has nothing to do with distorting an electron cloud and everything to do with rotating a whole molecule against its neighbours. That rotation cannot keep up above about twenty gigahertz, so it contributes a broad absorption centred there and nothing at all in the visible. Weighted by $1/\omega$, a modest absorption at twenty gigahertz outweighs everything the ultraviolet does, and the static constant is forty-four times the optical one.

The same fact is why a microwave oven works, and why it works at the frequency it does. Driving water on the low-frequency shoulder of that relaxation is driving a response that lags — and a lagging response absorbs. The oven is exploiting the imaginary part of the number whose real part gives water its dielectric constant, and the two are the same measurement.

The general lesson is worth keeping separate from the arithmetic. A material’s response at any frequency is an inventory of everything it can do at every other, weighted by how far away in frequency it is. That is why a substance’s static properties and its spectrum are never independent facts, and why measuring one of them well is a constraint on the other.

Why slow light must be either lossy or narrow #

The most-discussed modern application of the relations is a prohibition, and it disappointed a great many people.

Light is slowed by making the refractive index vary steeply with frequency: the group index is $n + \omega\,\mathrm{d}n/\mathrm{d}\omega$, so a steep slope gives a large delay. Demonstrations in which a pulse crawls through a vapour at a few metres a second work exactly this way, by opening a very narrow transparent window inside an absorption line and using the steep dispersion the window’s edges create.

The relations then decide what such a window costs. A steep slope in the refraction over a band is tied, by the integral, to structure in the absorption over a band of the same width — that is the smallest form of the constraint, drawn in the last figure above. So the transparency window cannot be both narrow enough to be steep and wide enough to pass a short pulse: making it wider flattens the slope in proportion.

What comes out is a delay–bandwidth product of order one per resonance. A medium can hold a pulse for roughly one of the pulse’s own durations, and no arrangement of a single resonance improves it. Storing a nanosecond pulse for a microsecond needs something with a thousand resonances, or a structure with a thousand of them designed in, or a mechanism that is not a linear response at all.

That last escape is the one that works, and it is worth naming because it shows precisely which assumption was doing the forbidding. The demonstrations that stop light entirely, rather than merely slowing it, work by transferring the excitation into an atomic coherence and back — during which the medium’s properties are being changed by a control beam, so the response is no longer linear time-invariant and the relations do not apply to the process as a whole. They still apply at every instant to the medium as it is at that instant.

The habit generalises past optics. Whenever a device is to be built with a specified magnitude response, ask what phase response the relations have already assigned to it, because the answer is not negotiable and it is usually the thing that makes the design impossible. A filter with a sharp edge has a delay that peaks at that edge; an amplifier with a steep gain rolloff has a phase shift that goes with it, which is what sets the stability limit on any feedback loop wrapped around it. In each case what looks like a separate specification has already been written down by the first one.

Where the constraint bites elsewhere #

In every filter that has to be built. A filter with a specified amplitude response has its phase response already decided, up to a factor that all-pass networks supply. A designer who asks for a sharp cutoff and a flat delay is asking for something the relations forbid, and the phase distortion at the edge of a band is not a defect of the implementation.

In scattering. The optical theorem is the same relation for a scattering amplitude: the forward amplitude’s imaginary part is the total cross-section, which is why a large particle removes twice its own area from a beam. The forward amplitude is a response function, and it obeys the same time-ordering constraint that a dielectric does.

In every measurement of a solid’s optical constants. A reflectivity spectrum is easy to measure over a wide range and gives one number per frequency, where two are wanted. Running the Kramers–Kronig integral on the logarithm of the reflectivity returns the phase of the reflected wave, and from the pair the refractive index and the absorption follow separately. Half the tabulated optical constants of solids were obtained this way, which is a large practical debt to an argument about time order.

In magnets, in mechanics, in circuits. A viscoelastic material’s storage and loss moduli are a Kramers–Kronig pair, which is why a liquid that remembers cannot have a loss peak without a step in its stiffness. So are the real and imaginary parts of a magnetic permeability, and of an amplifier’s gain.

What the pictures cannot show #

The model is one oscillator and a real material has many. Everything computed here uses a Lorentz line because it has a closed form to check the integral against. The relations themselves use no model at all — they are properties of any causal linear response — and the figures would look different and mean the same for a real spectrum.

The integral needs every frequency and no measurement has them. Applying the relations to real data means extrapolating outside the measured band, and the extrapolation is where the error lives. Practitioners use subtractive forms, in which a single known value of the index anchors the result and the tails matter far less; the honest statement is that the relation is exact and its application is an estimate.

Nothing here is quantum mechanical. The oscillator is classical, the absorption is a damping coefficient, and the argument would have been available to Maxwell. What quantum mechanics supplies is the values — where the lines are and how strong each one is — and the sum rule is where the two pictures meet, because the classical count of electrons and the quantum sum over transitions give the same total.

And the relations do not say a medium is passive. An amplifying medium has negative absorption in a band, which the relations permit; what they forbid is response before excitation. A laser gain medium obeys them exactly, and the anomalous dispersion in its gain band is what the figure above draws.

The ladder from here #

Later rungs on this anchor: the subtractive form of the relations and how they are actually applied to reflectivity data; the optical theorem as the same statement for scattering; sum rules in general, including the ones that constrain a metal’s conductivity; and the Sokhotski–Plemelj formula, which is the piece of complex analysis that turns “no response before the kick” into a principal-value integral.

The neighbouring ladders are the frequency-dependent dielectric constant, which is this constraint applied to one material, and the driven oscillator, which is where the response function came from. The distance that takes the treble out is the measurement this essay says has already decided the rest.