The law that only asks about one component

A boundary is a plane at which the material properties change and nothing else happens. It has no thickness, no clock and no structure along itself, and each of those absences forbids a wave from doing something.

The boundary does not move, so it cannot change the frequency: whatever oscillation arrives has to be matched on the far side, cycle for cycle, for ever, and a static object cannot supply a beat.

The boundary is uniform along itself, so it cannot change the wavenumber along itself: a field that repeats every so many nanometres as one walks along the interface must repeat at the same spacing on the other side, or the two solutions would agree at one point and disagree at the next.

Those two statements are the whole of refraction. Everything below is drawn with a compass.

Two symmetries, two conserved quantities #

The two statements above are not independent facts about boundaries. They are instances of one principle, and naming it makes clear why the list of conserved quantities is exactly two items long and not three.

A symmetry of a system implies a conserved quantity — that is Noether’s correspondence, and it applies to a wave meeting a boundary as readily as to a planet. The arrangement here is unchanged if the clock is reset, because none of the materials is doing anything that depends on when; the conserved quantity belonging to that invariance is the frequency. The arrangement is also unchanged if everything is slid along the interface, because the boundary is a plane and looks identical from every point of itself; the conserved quantity belonging to that invariance is the wavevector component along the slide.

Slide perpendicular to the boundary instead and the arrangement is not unchanged — one medium becomes the other. So there is no conservation law for the normal component, and its freedom to change is what makes refraction happen at all.

Counting the symmetries therefore counts the conserved quantities: one time translation and two independent translations within the plane give three, and for a wave in a plane of incidence the third is trivially zero. Everything else is free.

The construction #

Work in units of the vacuum wavenumber, so that a wave in a medium of index $n$ has a wavevector of length $n$, and every direction it can take lies on a circle of radius $n$. Draw the two circles concentrically with the interface along the horizontal.

An incident wave is an arrow from the centre to a point on the first circle. Its horizontal component — the one lying in the interface — is the quantity that must be preserved, so drop a vertical line through the arrow’s tip. Where that line cuts the second circle is the refracted wave.

Reading the two intersections off gives $n_1\sin\theta_1 = n_2\sin\theta_2$ immediately, because the horizontal component of an arrow of length $n$ at angle $\theta$ from the vertical is $n\sin\theta$. The ratio of sines is what the construction says when the two radii are called indices.

Nothing about least time, wavefronts or photons entered. The one physical input was that a uniform, static boundary preserves the component along itself, and the geometry did the rest.

The ray picture gives the same angle and has no way of saying where it came from. That is the point of running the argument the other way round: Snell’s law is usually presented as a fact about rays bending, and it is more honestly a consequence of the tangential wavevector being conserved across the boundary — which is a statement about matching phases along a surface, and which the ray picture cannot express.

The advantage over the usual statement is not economy but reach. The ratio of sines is a rule about two rays; the conserved component is a rule about a quantity, and quantities can be tracked into situations where rays cannot.

What crosses and what does not #

Put six quantities in a table and only two survive the crossing.

The frequency survives, for the reason above. The tangential wavenumber survives, for the other reason above. The normal wavenumber changes, the wavelength changes, the phase speed changes, and the direction changes — each by a factor the two indices set.

The wavelength is worth naming as the trap, because it is the quantity a drawing of wavefronts makes look conserved. Wavefronts crossing a boundary are drawn continuous, which is right, and the continuity is of the spacing along the boundary rather than of the spacing along the direction of travel. A helium-neon line at $633$ nm becomes $422$ nm inside glass of index $1.5$, and remains red, because what a detector responds to is the frequency.

Where the construction has no answer #

Send the light the other way, from glass into air, and the circle that has to be reached is the smaller one.

Beyond a certain incidence the conserved component exceeds the far circle’s radius entirely and the vertical line misses it. There is no direction available on the other side, and the incident light has nowhere to go except back.

That is total internal reflection, and the critical angle is where the line first fails to reach — $\sin\theta_c = n_2/n_1$, read off the picture rather than derived.

What makes the k-space version worth the trouble is what it says about the field beyond the surface. The normal component satisfies $k_z^2 = n_2^2 - k_x^2$, which is now negative, so $k_z$ is imaginary. An imaginary wavenumber is not an error: $\exp(ik_z z)$ with $k_z = i\kappa$ is $\exp(-\kappa z)$, a field that decays rather than travels.

The evanescent field’s reach past the surface diverges at the critical angle, where the wave is grazing and barely bound, and falls to a fraction of a wavelength well beyond it. That is where the construction has no answer in the ordinary sense: the tangential component demanded exceeds anything a propagating wave in the second medium can supply, so the solution that exists is one that decays instead of travelling.

So total reflection is not the absence of a field beyond the boundary; it is the absence of a direction beyond the boundary. The distinction is testable: bring a second piece of glass within the decay length and light crosses the gap, exponentially in its width, because the second medium restores a circle the line can reach. A picture in which light stops at the surface has nothing to say about that experiment.

What the decaying field does with its energy #

An evanescent field is a real field with a real energy density, and asking what it does with it is a good test of whether the picture is being taken seriously.

Time-averaged, it carries no power away from the surface. The reason is in the imaginary wavenumber: with $k_z$ purely imaginary the electric and magnetic fields are ninety degrees out of phase in the normal direction, and a product of two quantities in quadrature averages to nothing. Energy flows into the second medium during part of each cycle and back out during the rest, and the books balance exactly.

They balance exactly only for an infinite, lossless second medium. Put anything there that absorbs, or bring a third medium within the decay length, and a little of what flows in does not come back — which is the whole basis of two techniques. Total internal reflection microscopy illuminates a layer a hundred nanometres thick at the surface of a slide and nothing beyond it, because the evanescent field is the only illumination present. Attenuated total reflectance spectroscopy presses a sample against a prism and reads its absorption from the light that fails to return, sampling a depth of a fraction of a wavelength and requiring no thin section at all.

The sideways flow is a separate matter and does not vanish. The evanescent field carries power along the surface, and it has to: the reflected beam emerges laterally displaced from where geometry says it should, by a fraction of a wavelength, which is the Goos–Hänchen shift and is the bookkeeping of that sideways transport — the reflection that happens where the glass is not.

A grating is the same rule with an integer in it #

The second requirement was that the boundary be uniform along itself. Make it periodic instead, with period $d$, and it is no longer uniform — but it is not arbitrary either, and what a periodic boundary can do is add any whole multiple of $2\pi/d$ to the tangential component.

Each order is the same vertical line displaced sideways by an integer step. An order exists when its displaced line still meets the circle and does not exist when it misses. Reading off the angles gives

$$n_2\sin\theta_m - n_1\sin\theta_i = \frac{m\lambda}{d},$$

which is the grating equation, obtained as arithmetic on the shifts rather than as a separate law about path differences.

Two things fall out that the path-difference derivation makes work for. The number of propagating orders is the number of shifted lines that still reach the circle, so a grating whose period is less than about half a wavelength has only the zeroth: it stops diffracting altogether, which is why the fine gratings used as antireflection structures on solar cells look black rather than iridescent. And the orders that miss the circle are not absent — they are evanescent, bound to the surface, and they carry the energy that a rigorous grating calculation has to account for.

An object’s spatial frequencies appear as diffracted orders, and the same shift in the tangential wavevector describes a grating, a lens’s aperture and the resolution limit. They are one rule: a periodic structure adds a fixed increment to the tangential component, and whether the result can propagate on the far side decides whether that order exists.

The same construction in three dimensions is X-ray crystallography #

A grating is periodic in one direction. A crystal is periodic in three, and the argument survives the promotion with one change: the step that may be added is now any vector of the reciprocal lattice rather than any multiple of one number.

The condition for a diffracted beam is then that the incident wavevector plus a reciprocal-lattice vector lands on the sphere of allowed directions — the same sphere, with radius set by the wavelength, and the same requirement that the shifted point still reach it. Drawn out, that is the Ewald construction, and it is the hero figure of this essay with a lattice of dots in place of one vertical line and a sphere in place of a circle.

Bragg’s law follows from it in a line, and so does the fact that a stationary crystal illuminated by monochromatic X-rays usually shows no reflections at all: the sphere has to pass through a lattice point, and a discrete set of points generally misses a given sphere. Rotating the crystal sweeps the lattice through the sphere and each point crosses it in turn, which is why a single-crystal measurement is a rotation and not an exposure. Using a spread of wavelengths instead thickens the sphere into a shell and catches many points at once, which is the Laue method.

Nothing new was needed. A periodic structure adds a reciprocal step to the conserved component, and whether that produces a beam is a question about whether the shifted point is still on the shell.

The mode that nothing outside can reach #

There is a third case, and it is the one the construction makes obvious and every other treatment makes mysterious.

A wave guided along a film has a propagation constant $\beta$ along the film that lies between the cover’s index and the film’s — larger than $n_{\text{cover}}$, which is to say larger than the entire radius of the cover medium’s circle.

There is therefore no direction in the cover with the right tangential component, at any angle whatever. Shining light at a waveguide from outside does nothing, and that failure is not a matter of alignment or efficiency: it is the same impossibility as the missing intersection in total reflection, and it is exactly what “guided” means.

Two devices supply the difference and both are this construction read backwards. A prism of higher index has a bigger circle, so light inside it at the right angle has the necessary tangential component and can hand it across a thin gap through the evanescent field — the frustrated reflection of the previous section, used deliberately. A grating ruled on the surface adds an integer step, and choosing the period so that one step equals the shortfall couples light in from the normal direction.

Neither device changes the mode. Both change what can be phase-matched to it, which is the distinction the k-space picture makes visible and the ray picture cannot state.

The same idea where there is no boundary at all #

The word for the conserved quantity — phase matching — comes from a different subject, and it is the same condition.

A crystal that generates a second harmonic converts light at $\omega$ into light at $2\omega$. Each little volume of crystal radiates at the doubled frequency, and the contributions add up along the crystal only if the generated wave keeps in step with the polarisation driving it. That requires $k(2\omega) = 2k(\omega)$, which is to say $n(2\omega) = n(\omega)$ — and dispersion makes it false in every ordinary material.

The conversion efficiency is then a squared sinc of the accumulated mismatch: it rises for a coherence length, falls back to nothing, and oscillates. Getting a useful output means making the mismatch vanish, and the two standard methods are exactly the two devices above, moved inside the crystal. Birefringent phase matching uses a crystal whose index depends on polarisation and direction, and finds an angle at which the two indices coincide. Quasi-phase-matching gives up on that and instead flips the crystal’s sign periodically, adding a reciprocal step $2\pi/\Lambda$ to the mismatch — a grating, in the bulk rather than on a surface, doing the same arithmetic the shifted lines do above.

That the same phrase covers a refraction, a grating order and a nonlinear conversion is not a metaphor. In all three the requirement is that a phase relationship be maintained over an extended region, and the currency is the wavevector.

Where it stops #

The medium has to be isotropic for the loci to be circles. In a birefringent crystal the available directions lie on an ellipsoid and on a sphere — two surfaces, one per polarisation — so the vertical line generally cuts them at two different places and the light splits into two rays, each with its own polarisation. That is double refraction, and the k-space construction handles it with no new ideas and considerably more drawing.

The boundary has to be flat and stationary. A curved boundary is only locally uniform, so the conservation is only local and holds over a patch small compared with the radius of curvature. A moving boundary is not time-invariant, so the frequency is no longer conserved: light reflecting from a moving mirror is Doppler-shifted, which is the same theorem read the other way round: the symmetry that was giving the conservation has been removed, so the conserved quantity is not.

And the two media have to be linear. The whole argument assumes the response at each point is proportional to the field, so that a single frequency in gives a single frequency out. The nonlinear case above is what happens when it is not, and the appearance of phase matching there is the residue of the same bookkeeping under much weaker assumptions.

Fermat’s principle arrives at the same law from the other end — the path taken is the one whose travel time is stationary. The two descriptions are of one fact, and which is more fundamental is a question about taste rather than physics; the wavevector version generalises more easily, because it survives into situations where the notion of a ray has stopped being useful.

The history, and who had it first #

The ratio of sines is credited variously to Snell, who wrote it down about 1621 and did not publish, and to Descartes, who published it in 1637 with a derivation involving tennis balls that gets the answer right and the mechanism backwards. Both were anticipated by Ibn Sahl, who used the equivalent construction in Baghdad in 984 to design a lens that focuses without aberration.

Huygens’ construction builds the refracted front from wavelets of the right radius on each side, and that is the same phase-matching drawn rather than written. It is worth recording that he had it in 1690 — before there was any notion of a wavevector, and by a construction that needs no algebra at all.

Huygens’ construction of 1690 is the phase-matching argument in disguise and is worth recognising as such. Drawing wavelets of radius $c_1 t$ on one side and $c_2 t$ on the other and taking the common tangent is a geometrical way of insisting that the fronts meet along the boundary with the same spacing — which is the conserved tangential component, expressed as a picture in position rather than as a picture in wavenumber. The two constructions are Fourier transforms of one another, and it is a fair summary of nineteenth-century optics that everyone drew the first and calculated with the second. Which is preferable depends entirely on the question: a curved surface is easy in position and awkward in wavenumber, and a periodic surface is the reverse.

The ladder from here #

Later rungs on this anchor: the Fresnel coefficients, which say how much goes each way rather than where it goes, and which the phase-matching argument deliberately does not address; refraction into an absorbing medium, where the index is complex and the transmitted wavevector is too, so the surfaces of constant amplitude and constant phase are no longer parallel; negative refraction, where the available circle is traversed in the opposite sense and the refracted ray comes out on the same side of the normal; and refraction at a moving boundary, where the frequency is not conserved and the construction acquires a third dimension.

The neighbouring ladders are the bend at the boundary, which is this law met for the first time; the angle past which light cannot leave, which is the missing intersection; and what a thousand slits buy, which is the periodic boundary taken seriously.