The angle that is two angles

The rung below this one derived refraction from a single requirement: the phase along the boundary has to agree on both sides, at every point and at every instant. That fixes the component of the transmitted wavevector along the surface and nothing else, and the component across it is then whatever the second medium’s own relation between wavevector and frequency supplies.

For a transparent medium that relation is $|k| = n k_0$, a circle, and the construction gives one answer. For a medium that absorbs it is $|k| = N k_0$ with $N = n + ik$ complex, and the answer is complex too.

The usual response is to put the complex index into Snell’s law and call the result a complex angle. That is arithmetically fine and physically empty: an angle is a direction, directions are real, and a complex one is not the direction of anything. What has actually happened is that the transmitted wave has stopped having a single direction.

Where the two directions come from #

Write the transmitted wavevector as $(k_x, k_z)$. The boundary condition makes $k_x$ real and equal to $k_0\sin\theta_i$, because the incident wave’s tangential wavenumber is real and the two must match. The medium then supplies

$$k_z = k_0\sqrt{N^2 - \sin^2\theta_i},$$

which is complex. Split it: $k_z = \beta + i\alpha$.

The wave in the medium is $\exp[i(k_x x + \beta z)]\exp(-\alpha z)$. The first factor is an oscillation, and its surfaces of constant phase are perpendicular to the vector $(k_x, \beta)$. The second is a decay, and its surfaces of constant amplitude are perpendicular to $(0, \alpha)$ — level planes, parallel to the interface, because $k_x$ has no imaginary part to contribute.

So the two families are tilted with respect to each other by

$$\theta_{\text{phase}} = \arctan\frac{\sin\theta_i}{\beta},$$

and the wave is called inhomogeneous: its amplitude varies along its own wavefronts. That is the general transmitted solution. The familiar picture in which one set of parallel planes carries both the oscillation and the amplitude is the special case, and it requires the absorption to be exactly zero.

The check that the arithmetic is right is that it reduces. Setting $k = 0$ makes $k_z$ real, the tilt becomes $\arctan(\sin\theta_i/\sqrt{n^2-\sin^2\theta_i})$, and that is Snell’s law — verified in the figures to twelve decimal places at four angles in two transparent media, because a complex square root has a branch and a branch is exactly the kind of thing that is wrong without being visibly wrong.

Which side the answer falls on #

The natural expectation is that a strongly absorbing medium pulls the refracted wave toward the normal, and it is the expectation almost every textbook encourages. The construction says otherwise for half the cases.

The two extremes are worth separating because the difference is not a matter of degree.

Copper at ten micrometres has $n = 11$ and $k = 59$, so $\beta$ comes out at 11 and the tilt at sixty degrees of incidence is under five degrees. That is the standard picture, and it is a good description of a metal in the infrared and at radio frequencies, where the free-electron response is enormous.

Silver at 550 nanometres has $n = 0.055$ and $k = 3.32$. The real part of the index is smaller than one, which is legal — a phase velocity above $c$ carries no information, as a medium’s response to being asked quickly makes routine — and it makes $\beta$ come out at 0.053. The tilt is then 86 degrees. The phase fronts inside silver run within four degrees of the surface itself while the amplitude decays straight down into it.

That is not a curiosity. A wave whose phase travels along a metal surface and whose amplitude decays into it is the description of a surface plasmon — a channel with no walls, made of a boundary rather than of a guide, and the reason silver is the standard plasmonic metal at optical frequencies is exactly the small $\beta$ this figure computes. The rung’s construction and a whole applied field are looking at the same number.

A real part below one, which is allowed #

Silver’s index has a real part of 0.055, and the first reaction to that is usually that something has gone wrong: an index below one means a phase velocity above the speed of light.

It does, and nothing is wrong. A phase velocity is the speed at which a point of constant phase on an infinite monochromatic wave moves, and an infinite monochromatic wave carries no information — it has been going on for ever and will continue for ever, so nothing about it can be a signal. Information travels at the group velocity or, more carefully, at the front velocity, and neither exceeds $c$ in any material.

Where an index below one comes from is easier to see in the dielectric function than in the index. A medium’s response depends on how fast it is asked, and above a resonance the driven charges respond in antiphase, which subtracts from the applied field rather than adding to it. A metal is the extreme case: its free electrons have no restoring force at all, so they are in antiphase at every frequency below the plasma frequency, the permittivity is negative, and the index’s real part collapses.

X-rays make the same point without the absorption. Every material has an index slightly below one for X-rays — typically $1 - 10^{-5}$ — because the frequency is above every atomic resonance in the material. That is why X-ray mirrors work only at grazing incidence: with $n < 1$ the dense medium is the optically rarer one, so total external reflection is possible, and the critical angle is a few milliradians. An entire branch of optics rests on an index below one being real.

The practical statement for this essay is narrower. What matters to the geometry is $\beta$, the real part of the normal wavenumber, and it is small when the index’s real part is small. Absorption does not pull the phase fronts toward the normal; a large real index does, and in the visible a metal has not got one.

How far it gets #

The other half of the complex $k_z$ is the decay, and its size settles what kind of object the transmitted wave is.

Calling that a wave is generous. In silver at 550 nanometres the field falls by a factor of e in about twenty-six nanometres while the wavelength associated with $\beta$ is measured in micrometres, so what is inside the metal is a decaying disturbance with a phase gradient rather than a propagating wave with a small loss.

Three consequences follow directly and are usually taught separately.

The reflection is a surface property. All of the light that returns from a mirror interacted with a layer a few tens of nanometres thick, which is why a mirror’s performance depends on its surface finish and its top coating and not at all on the thickness of the metal beneath. An evaporated silver layer a hundred nanometres thick reflects exactly as well as a solid silver block.

Absorption in a metal is not a bulk process. The energy that does not come back was deposited within that same thin layer, which is why a metal surface heated by a laser has a temperature profile set by conduction away from a two-dimensional source rather than by where the light went.

And the same arithmetic is the skin effect. Run the frequency down to a megahertz and $\alpha$ falls, the depth grows to tens of micrometres — the same expression a plasma’s own free charges produce, and the identical expression is what tells a radio engineer to silver-plate a waveguide. Optical reflection and the skin effect are one calculation at two frequencies, and the only thing that changes is which term of the dielectric function dominates.

How the two numbers are measured #

Everything above takes $n$ and $k$ as given, and it is worth saying how they are got, because the method is the essay’s own construction run backwards.

A single reflectance measurement gives one number and there are two to find. The standard solution is ellipsometry: shine polarised light on the surface at a known angle and measure how the polarisation changes on reflection. The $s$ and $p$ components reflect with different complex coefficients, so their ratio has an amplitude and a phase — two numbers, from one measurement, at one angle.

Inverting the Fresnel expressions then gives $n$ and $k$ directly. Nothing is fitted and no model of the material is assumed, which is why ellipsometry is the reference method: it measures a ratio of two coefficients rather than an absolute intensity, so it needs no calibrated source and no calibrated detector, and it is insensitive to almost everything except the surface it is looking at.

That last property is also its difficulty. A metal’s optical constants are a property of the top few tens of nanometres — the depth the previous figure computed — so an oxide layer two nanometres thick is a significant fraction of the sample. Published values for silver differ between laboratories by more than the measurement uncertainty of any one of them, and the difference is real: they measured different films.

The technique’s sensitivity to thin layers is what turned it into an instrument for something else entirely. A monolayer of protein on a surface shifts the ellipsometric angles measurably, so the same apparatus that measures a metal’s index measures the thickness of an adsorbed film to a fraction of a nanometre — and does so in real time, in water, which is why it sits in biochemistry laboratories that have no interest in metals at all.

The branch that was chosen, and what the other one is #

The complex square root has two values and the figures take one of them. That choice is a physical statement and it deserves saying out loud.

The two roots differ in the sign of $\alpha$. The one taken makes the amplitude fall as the wave goes into the medium. The other makes it grow, exponentially, without limit.

That second root is not nonsense. It is the correct solution for a medium with gain — an amplifier, a laser medium above threshold, an inverted population — where the imaginary part of the index has the opposite sign because the material returns energy to the wave rather than taking it. Everything in this essay applies to such a medium with $k$ negative, and the geometry is the same: the constant-amplitude surfaces are still parallel to the interface, and the wave still has two directions in it.

What decides which root is physical is not the mathematics but the boundary condition at infinity — the same kind of statement, and the same kind of imposition, as the choice of the retarded solution in electromagnetism. A passive medium is one in which the field must not grow without bound far from the source, and that single requirement fixes the sign.

Where the transparent case sits #

The construction is more general than the transparent one and it contains it, which raises a question worth answering directly: is an inhomogeneous wave a strange object, or a common one?

It is common, and this collection has already met it twice under other names.

Beyond a totally internally reflecting surface the transmitted wave has real $k_x$ and purely imaginary $k_z$ — an inhomogeneous wave in a transparent medium, with constant-amplitude surfaces parallel to the interface and constant-phase surfaces perpendicular to them. That is the extreme case of this essay’s figure, with the tilt at exactly ninety degrees.

And a wave in a lossy transmission line, or in a waveguide below cutoff, has the same structure. The general solution of a wave equation with a boundary is a wave whose amplitude and phase vary in different directions, and the case where they do not is what happens when both the medium and the geometry decline to break the symmetry.

So the honest way to state the rung is that the rung below solved a special case without saying so. Phase matching does not produce a refracted ray. It produces a transmitted wavevector, real in one component and complex in the other, and a ray exists only when the second component happens to be real too.

What survives from the transparent case #

It is worth listing what the complex construction does not change, because the list is longer than the list of what it does.

The frequency is unchanged. The boundary is not moving, so nothing about it can shift a frequency, and every wave in the problem oscillates at the same rate. That is the assumption underneath the whole phase-matching argument and it survives absorption untouched.

The tangential wavenumber is unchanged. This is the same statement as the first, applied along the surface instead of in time, and it is what makes a grating a grating. An absorbing grating still diffracts into the same angles; it merely does so less brightly.

And the reflected wave is still a plane wave in the first medium, at the ordinary angle of reflection, with real everything. The complications live entirely on the far side, which is why the reflectance of a metal can be measured and modelled without ever confronting what the transmitted field is doing.

What changes is only the third component of the answer — the normal wavenumber in the second medium — and it changes from a number to a pair. That the change is so contained is the reason the standard treatment survives as long as it does before anybody notices what it has assumed.

Where this stops being right #

The optical constants are measurements, and they disagree. Published values of $n$ and $k$ for evaporated silver at 550 nanometres vary by tens of per cent between sources, because they depend on the deposition, the age of the film and the surface oxide. The qualitative result — $\beta$ small, tilt large — survives that; the number 86 degrees does not deserve three significant figures.

Nothing here computes how much goes each way. The Fresnel coefficients for a complex index are a separate calculation, and they are where the reflectance comes from. The construction says where the transmitted wave points and is silent on its size.

The material has been local and isotropic. A metal’s response at optical frequencies is neither, quite: spatial dispersion matters within a few nanometres of the surface, which is the same few nanometres the whole interaction happens in, and anisotropy matters in any crystalline film.

And a single frequency has been assumed throughout. A pulse entering an absorber has each of its components refracted differently and attenuated differently, so it changes shape as well as direction — which is the constraint tying absorption to refraction showing up as a practical effect.

What the pictures cannot show #

The hero figure draws two families of straight lines and calls one phase and one amplitude, and neither family is visible to any instrument. What could be measured is a field, and a field has one value at each point; the decomposition into an oscillation and an envelope is a way of writing it, chosen because the two factors have different geometries. A different decomposition would draw different lines through the same physics.

Nor can the figures show the reflected wave, which in every case here carries most of the energy. Silver at 550 nanometres returns about 98 per cent of what arrives, so the transmitted wave the whole essay is about is two per cent of the story by energy — and the essay’s subject is the small part, which is a choice the drawings do not disclose.

Where this ladder goes next #

Three rungs stand on refraction. The first met the law; the second derived it from phase matching alone; this one keeps the derivation and lets the index be complex, and finds that the refracted ray was an artefact of the special case.

The habit worth carrying away is about what a boundary condition constrains. Match what the boundary can see — the tangential components — and let the medium supply the rest, without assuming the result is real. Doing that here produced two directions where one was expected. The same discipline applied to a periodic boundary produced every order of a grating, and applied to a boundary between different symmetries it produces mode conversion.

What is left on this ladder is the other assumption the construction has been making without saying so. The circle in the second medium has two intersections with the conserved tangential wavenumber, and every figure so far has silently taken the upper one. Which of them is physical is decided by where the energy goes rather than by where the wavevector points — and there are materials in which those are opposite.