Swap the ends and nothing changes

Build the most awkward thing anybody can think of out of layers of glass, foam, metal and air. Put a source at one end and a detector at the other, and note the reading. Then swap them.

The reading does not change. It does not approximately not change, or not change on average over frequency, or not change for symmetrical arrangements; it does not change, in modulus and in phase, for any arrangement at all.

This is one of the few statements in wave physics with essentially no conditions attached, and the conditions it does have are worth more than the statement. They rule out an entire category of device that people continue to try to invent, and the two loopholes they leave are the basis of two real instruments.

What is being claimed, exactly #

The precise version is about a Green’s function: the disturbance at $\mathbf{r}_2$ produced by a point source at $\mathbf{r}_1$ equals the disturbance at $\mathbf{r}_1$ produced by the same source at $\mathbf{r}_2$, for any medium in between.

$$G(\mathbf{r}_2, \mathbf{r}_1) = G(\mathbf{r}_1, \mathbf{r}_2).$$

The medium may vary from point to point, may have sharp boundaries, may absorb, may be shaped into a horn or a maze. None of that appears in the statement.

The figure is the one-dimensional case computed rather than argued. Nine slabs are drawn at random; the transfer matrices of the interfaces and the propagations are multiplied in one order to get the transmission from the left, and in the other order for the transmission from the right. The two matrix products share no intermediate step, and they agree to the last bit.

What is not claimed is that the two sides look alike. The reflections in the same figure have equal moduli — but only because nothing in this stack absorbs, so what is not transmitted must be reflected either way — and their phases differ by $1.6$ radians, because a wave coming from the left meets a different first interface from one coming from the right. An interferometer would tell the two sides apart at once. Reciprocity is a statement about a pair of ends, not about symmetry.

Where it comes from #

The shortest honest account is a symmetry of the equation rather than of the apparatus.

The wave equation for a medium with position-dependent properties can be written so that the spatial operator is self-adjoint: $\nabla \cdot (p\nabla u) + \omega^2\rho u$ with $p$ and $\rho$ functions of position. Self-adjointness means that for any two fields $u$ and $v$ vanishing on the boundary,

$$\int u\,\mathcal{L}v = \int v\,\mathcal{L}u,$$

and putting a point source into each of them turns that identity into the symmetry of $G$ directly. Nothing about $p$ and $\rho$ is used except that they are functions of position and not of anything else.

The physical reading is time reversal. The equation contains a second derivative in time, so running a solution backwards gives another solution; a wave that went from $A$ to $B$ through the maze, reversed, goes from $B$ to $A$ through the same maze along the same path. Every reflection, refraction and scattering event is reversible individually, so the whole journey is.

That reading also says immediately what could break it. A first derivative in time — friction, absorption — does not break it, because absorption is described by a complex $p$ that is still symmetric, and an absorbing maze is still reciprocal. What breaks it is a term that changes sign under time reversal while the others do not, and there are exactly two ordinary ways to get one: a medium that is itself moving, so that reversing time reverses the flow; and a magnetic field acting on the medium’s charges, since reversing time reverses currents and hence the field.

The object the theorem is about is the medium’s response to a point disturbance — a Green’s function. Reciprocity is the statement that this object is symmetric in its two arguments: the response at B to a source at A is the response at A to a source at B, so the picture would be identical with the source and the observation point exchanged. Everything else in this essay is a consequence of that one symmetry, and every exception is a case in which the symmetry fails.

It is worth being careful about one word. “Passive” in the statement of the theorem does not mean “lossless” and does not mean “not powered”; it means that the medium’s response at a point depends on the field at that point through a symmetric relation, with no preferred direction built into the material itself. An amplifier is not passive in that sense — its gain medium is being driven by something with a direction in it — and a chain of amplifiers is a perfectly good one-way path. That is not a counterexample to the theorem; it is a system with an energy source and a designed asymmetry, and the theorem was never about those.

The distinction matters because it is where most attempted counterexamples land. A device with a battery in it can be one-way. A device made only of shaped, layered, absorbing, refracting matter cannot, and no amount of ingenuity in the shaping changes that, because the shaping never enters the proof.

What follows, in places that look unrelated #

The theorem is used constantly by people who never name it.

An antenna radiates in the pattern it listens in. A dish with a narrow beam transmits a narrow beam and receives from a narrow cone, with the same shape, the same sidelobes and the same nulls — and the width of both is the diffraction limit of its aperture. Nobody measures both: an antenna range measures the receiving pattern because that is easier, and the transmitting pattern is then known.

A loudspeaker is a microphone. Drive it and it moves air; move the air and it produces a voltage. The efficiency in the two directions is related by exactly this theorem, which is why a dynamic microphone and a small speaker are the same object and why one can be used as the other.

Seismic sources and receivers can be exchanged. A survey shot at one point and recorded at another gives the same trace as the reverse arrangement, through the whole complicated Earth, refractions and all. That is used as a data-quality check and as a way of building surveys that would otherwise be impossible — one expensive source and many cheap receivers, processed as though it were the other way round.

And if a room lets sound from a stage reach a seat, it lets a shout from that seat reach the stage. Acoustic design cannot make an auditorium loud in one direction and quiet in the other, and the many devices sold on that promise are either doing something else or doing nothing.

A window that transmits light also transmits it back. The same theorem covers the coating that makes a reflection vanish: an antireflection stack designed for light entering a lens works identically for light leaving it, which is convenient, and which is also why a coating cannot be used to suppress an internal reflection while passing the wanted light through the same surface.

A stack of layers — an antireflection coating, an acoustic matching section, an impedance transformer — transmits the same fraction in both directions however the layers are ordered. That is not obvious from the arithmetic, which multiplies the layers’ matrices in one order going forward and in the other going back, and it comes out because the determinant of each layer’s matrix is one. So a coating designed for light going in works for light coming out, and a designer never has to say which way the beam runs.

The last consequence is the one that most often surprises: the theorem forbids the passive one-way window, and it forbids it for waves of every kind at once. A structure that let light through from outside and not from inside would violate the same identity as one that let sound through from a stage and not toward it. What a real one-way mirror does is reflect most of the light in both directions and rely on the two sides being lit differently, which is a statement about the lamps and not about the glass.

A use that runs the theorem backwards #

There is a technique that treats reciprocity as a tool rather than as a constraint, and it is worth describing because it makes the abstract symmetry concrete.

Record the field arriving at an array of transducers from a source somewhere in a complicated, unknown medium — a body, a block of rock, a room. Then play the recording back through the same array, reversed in time, so that what arrived last is emitted first. The reversed field is a solution of the same equation, so it retraces every scattering event and converges on the original source, focusing there far more tightly than the array’s own aperture would suggest.

Nothing about the medium needs to be known, and nothing about how the wave spread on the way out has to be corrected for. The scattering that would ordinarily blur a focus is what sharpens this one, because a more complicated medium gives the array more independent paths to work with — which is the opposite of the usual relation between complexity and resolution. The technique is used to focus ultrasound through the skull, to concentrate shock waves on kidney stones, and to locate the source of a seismic event by refocusing its own recorded wavefield.

It works because the wave equation is reversible and because the array can be at either end of the journey, which are the two halves of the theorem stated as an operation rather than as an identity.

The two exceptions, and one of them is an instrument #

A moving medium breaks the symmetry because reversing the sign of time reverses the flow, so the reversed journey is through a different medium. The size of the effect is small and easy to compute: a pulse sent downstream over a path $L$ arrives sooner than one sent upstream by

$$\Delta t = L\left(\frac{1}{c-v} - \frac{1}{c+v}\right) \approx \frac{2vL}{c^2},$$

which for water at a metre per second over ten centimetres is $91$ nanoseconds — a fractional difference of $9\times10^{-7}$, so nothing about the measurement is easy except that everything it does not depend on has cancelled. That is a comfortable measurement for modern electronics and an impossible one for anything else, which is why the transit-time flowmeter is a recent instrument for an old idea. It is also why the instrument has no moving parts and does not touch the fluid: the entire measurement is a difference of two arrival times, and everything else about the pipe cancels.

The magnetic exception is the one optics uses. A magnetised medium rotates the plane of polarisation of light passing through it, and — this is the whole point — it rotates in the same absolute sense for light going either way, so a return trip doubles the rotation instead of undoing it.

A rotator followed by a polariser at the rotated angle is the classic isolator, and it is worth following in both directions. Light entering is rotated by 45°, meets a polariser at 45°, and passes. Light returning is rotated by a further 45° rather than back — because Faraday rotation is non-reciprocal — meets the input polariser at 90°, and is blocked. The device transmits one way and not the other, and it works only because one of its parts breaks the symmetry the rest of this essay relies on.

An isolator is that fact plus a polariser. Light enters through a polariser, is rotated $45°$, and leaves through a second polariser set at $45°$. Light returning is rotated another $45°$ in the same absolute sense, arrives at the input polariser at $90°$ to it, and is extinguished. The magnet supplies the non-reciprocity; the polariser converts a phase relationship into an attenuation. Neither alone would do it, and that division of labour is worth noticing: the magnetised medium by itself is lossless and passes light equally in both directions, and inserting it into a reciprocal system does not by itself make anything one-way. That is the same division of labour a quarter-wave plate and a polariser make between them, with the crucial difference that the plate’s effect is undone by a return trip and the magnet’s is not.

Where it stops #

Linearity is required and is not usually mentioned. The theorem is about a linear operator, and a nonlinear medium is not covered by it at all. A frequency-doubling crystal, an acoustic device whose stiffness depends on amplitude, a saturable absorber — all of these can be arranged to behave differently in the two directions, and several practical isolators work this way. The catch is that they are non-reciprocal only at the intensity they were designed for, which is why they are useless for the job an isolator usually has: protecting a laser from a weak reflection.

Stationarity is required too. A medium whose properties are being modulated in time — a stiffness switched at a frequency comparable with the wave’s — is outside the theorem, and time-modulated structures are a current way of building non-reciprocal devices without a magnet. Reversing time in such a system does not give back the same modulation, which is the loophole in a form nobody exploited until the last decade or so.

And the source has to be a source. The theorem relates a point source of one kind to a point detector of the matching kind — a force to a displacement, a current to a voltage — and exchanging a source with a detector of the wrong kind gives a relation with a factor in it rather than an equality. In acoustics that is the difference between a volume source and a force source; in electromagnetism it is the difference between an electric and a magnetic dipole. The bookkeeping is dull and it is exactly where a careless application of the theorem produces a wrong factor of the impedance.

And absorption is not a loophole, however tempting it looks. A wedge of absorber thick at one end and thin at the other transmits equally both ways; the argument that it “must” attenuate more in one direction confuses the geometry of the wedge with the path length through it, which is the same in both directions for the same ray.

The elementary case is a pulse meeting a change of medium. The transmitted fraction is the same whichever side the pulse arrives from — that is reciprocity at its smallest — while the reflected one differs in sign, which is not a violation of anything: reciprocity is a statement about transmission from one place to another, and the two reflections happen at different places.

The check that would have caught a wrong version #

It is easy to write down a transfer-matrix calculation that agrees with itself for the wrong reason, so it is worth saying what was tested and what was not.

The two numbers compared here come from two matrix products taken in opposite orders over a stack drawn once at random. If the code had accidentally computed the same product twice, the agreement would be an identity of the arithmetic rather than a fact about waves — so the calculation also reports the reflected phases from the two sides, which come from the same two matrices and differ by 1.6 radians. Two quantities extracted from one pair of matrices, one identical and one not, is the pattern that distinguishes a theorem from a tautology.

The stack is also checked for having no accidental symmetry: nine thicknesses and nine wavenumbers drawn independently, with no pair matching. A palindromic stack would be reciprocal for a trivial reason, and a figure whose subject is a non-trivial symmetry has to avoid supplying a trivial one.

The history, which is older than the wave equation people prove it from #

Helmholtz stated the acoustic version in 1860 and Rayleigh generalised it in 1873, in the Theory of Sound, with the observation that it holds for any linear system with a symmetric matrix of coefficients — which for a mechanical system means any system whose energy is a quadratic form, and that is all of them within the linear approximation.

Lorentz gave the electromagnetic version in 1896, and it is his name that stays attached in optics. What Lorentz added was the identification of the exception: media in which the permittivity or permeability is not a symmetric tensor, which is exactly what a magnetic field produces and what Faraday had discovered experimentally in 1845 without anybody realising what it implied about one-way devices.

The chronology is worth the paragraph. The exception was found first, by fifteen years, and sat unexplained; the rule came second; and it took until the middle of the twentieth century, with microwave ferrites, before anybody built the device the two facts together make possible.

Huygens’ construction carries the same symmetry without any algebra in it. Every point of a front acts as a source of the next, and the construction is unchanged under exchanging where the wave came from with where it is going — the wavelets are the same wavelets and the envelope is the same envelope. That is the theorem in the form it was known long before there was a wave equation to prove it from.

What the pictures cannot show #

The computation here is one-dimensional and monochromatic, which is a considerable simplification of the general claim. It is enough to be a genuine test — nine random slabs with no symmetry is not a case anybody would expect to be special — and it is not a proof.

What a one-dimensional stack cannot exhibit is the more interesting form of the theorem, which involves polarisation and direction. In three dimensions the reciprocal statement relates a source of one polarisation at one place to a detector of another polarisation somewhere else, and the exchange has to swap those too. Getting that bookkeeping right is where the practical subtleties live, and it is where an antenna engineer’s version of the theorem differs in appearance from an acoustician’s.

What lets several sources be handled at once is linearity. Reciprocity applies to each source–detector pair separately, and linearity is what permits the pairs to be added, so the theorem survives interference intact. It would not survive a medium whose response depended on the total amplitude — which is why every nonlinear optical device is a candidate for breaking it, and why the ones that do are the useful ones.

The ladder from here #

Later rungs on this anchor: the reciprocity theorem for elastic waves, where the polarisation bookkeeping is genuinely involved and the exchange includes swapping force directions; time-reversal mirrors, which use the theorem constructively by re-emitting a recorded field reversed and focusing it back on its source through an unknown medium; the relation between reciprocity and the fluctuation–dissipation theorem, where the symmetry of the response function is the same symmetry; and non-reciprocal metamaterials, where the modulation loophole is exploited deliberately.

The neighbouring ladders are the equation that lets a shape travel, whose second time derivative is what the whole argument rests on; what happens where the medium changes, where the asymmetry of the reflection first appears; and the answer that cannot come first, which is the other great constraint that follows from the structure of the equation rather than from any material in it.