The rotation a return trip doubles

Send a beam of light through a plate of quartz cut across its optic axis and the plane of polarisation comes out turned — 21.7 degrees for every millimetre of quartz at the sodium yellow line. Send it through a rod of a heavy glass sitting in a magnetic field and the plane comes out turned too. Both effects are linear in the path, both leave the light otherwise unaltered, and to any measurement made on the emerging beam they are the same phenomenon.

Put a mirror at the far end and send the light back, and they cease to be the same phenomenon at all.

What a rotation of the plane actually is #

A plane-polarised wave is not a fundamental object. It is the sum, in equal parts, of a left-handed and a right-handed circular wave.

What “polarisation” names is the direction the electric field of a wave oscillates along, and the basis chosen to describe it is free. Two perpendicular planes describe the same set of states as two opposite circles, and every statement in this essay can be made in either. The circular basis is the one that makes the Faraday effect trivial, which is the whole reason for introducing it: in that basis the medium simply gives the two components different speeds.

If a medium has different refractive indices for the two circular components, they accumulate different phases over a path $L$, and their sum is a plane at an angle

$$\phi = \frac{\pi L}{\lambda}\,(n_L - n_R).$$

That is circular birefringence, and it is all that either mechanism is. Nothing in the beam is twisted, no energy is exchanged, and the emerging wave is as clean as the one that went in.

Two ways of getting the two indices apart #

A handed structure. Quartz is built from helices of silicon–oxygen tetrahedra, and a helix is not the same as its mirror image. A circular wave whose handedness matches the structure’s interacts with it differently from one that does not, so the two indices differ. Left-handed and right-handed quartz exist as separate crystals and rotate in opposite senses.

A magnetic field. Put any transparent material in a field along the direction of travel, and the electrons that respond to the light are also precessing about the field — a field that does no work on them and only turns them, which is exactly the effect wanted, since a rotation without absorption is what is being asked for. The two circular components drive that precession in opposite senses — one with the precession and one against — so they see different resonant denominators and hence different indices.

Three charges of different momentum in the same field go round circles of different size in the same time — one turn in 3.57 nanoseconds, whatever the speed — because the period depends on the field and on nothing else. That same frequency appears in the Faraday effect. The electron’s response to the light is shifted by it in one sense for one circular component and against it for the other, so the split between the two indices comes out proportional to the field.

The second mechanism explains the linearity in the field and the fact that it works in any material at all. The Verdet constant collects the material’s part of it, and the rotation is $\phi = V B L$.

The difference that does not show on the way through #

Now the point. Both rotations are described by the same expression and the same figure, and there is one question that separates them: what happens to the sense of the rotation when the beam is reversed.

For quartz, the handedness that matters is the handedness of the structure relative to the direction of travel. Reverse the beam and that relationship reverses, so the rotation reverses in the laboratory and the round trip returns exactly what went in — at every wavelength, every path length and every temperature. That is not an approximation; it follows from the symmetry.

For a Faraday rotator, the handedness that matters is set by the magnetic field, which points the same way whichever direction the light is going. So the second pass rotates the same way in the laboratory as the first did, and the round trip is twice the single pass.

The deep reason is that a magnetic field is odd under time reversal. That is a rare property in optics, and its rarity is what makes the essay’s conclusion an impossibility proof rather than a preference — most of physics is symmetric under reversal, and the exceptions that are not are usually statistical rather than mechanical. Run a film of any electrostatic or optical situation backwards and it remains a possible situation. Run a film of a current backwards and the current flows the other way, so the field it makes reverses — and the field itself is made of nothing but those currents, since it is a field with no ends and no sources of its own. A medium whose behaviour depends on a magnetic field is therefore a medium whose behaviour is not the same run backwards, and that is exactly the property reciprocity assumes and Faraday rotation lacks.

The field is what breaks the symmetry, and it is worth being exact about how. A magnetic field is electricity seen from a moving frame, so the thing producing it is moving charge — and moving charge reverses under time reversal while the geometry of the apparatus does not. That asymmetry between the field and the glass is the entire source of the non-reciprocity: reverse the light and the medium is unchanged, which is not true of any effect that depends on the direction of travel alone.

Why nothing else can do it #

Reciprocity is a strong theorem and it applies to nearly everything in an optics catalogue, including the reflection that picks out one plane at Brewster’s angle. Its content is that the transmission from a point A to a point B through a linear, passive, time-invariant system equals the transmission from B to A. Lenses obey it, mirrors obey it, waveplates obey it, gratings obey it, and — crucially — so do quartz, sugar solution and every naturally active substance.

The consequence is worth stating as an impossibility. No arrangement of lenses, mirrors, prisms, polarisers, waveplates and naturally active crystals, of any complexity whatever, can pass light in one direction and block it in the other. Not inefficiently; not at all. The proof is one line of reciprocity and it does not depend on the arrangement.

The three words in the theorem #

The impossibility above is worth stating with its conditions attached, because the conditions are the map of every way round it.

Reciprocity holds for a system that is linear, passive and time-invariant. Each of those is doing work, and each can be given up.

Time-invariant is what a magnetic field gives up — not because the field changes, but because the field is odd under reversing time, so a description of the system and a description of it run backwards are not the same description. That is the route this essay is about, and it is the only one available with ordinary components sitting still.

Time-invariant can also be given up more literally, by modulating a medium’s properties in a travelling-wave pattern. A refractive index made to vary as a wave running along a waveguide couples forward-going light to one set of modes and backward-going light to another, and the asymmetry is in the modulation’s direction rather than in any material. Such devices need no magnet and need a driving signal instead, and they are the main line of work on isolators for integrated optics, where a permanent magnet is an unwelcome object to put on a chip.

Linear can be given up. A medium whose transmission depends on the intensity in it treats a strong forward beam differently from a weak backward one, which looks like isolation. It is a weaker thing than it appears: such a device passes a strong beam either way, and fails precisely in the case an isolator is usually bought for, where a strong reflection is coming back.

Passive can be given up, by including a source or an amplifier — which is what an optical circulator built round an amplifier does, at the cost of noise and power.

What the list makes clear is that the theorem is not a statement about optics but about symmetry, and that every device evading it is identifiable by which word it broke. A component with a magnet broke time-reversal; one with a modulator broke time-invariance; one with a threshold broke linearity. There is no fifth option, and a proposal that names none of them is a proposal with an error in it.

The device the asymmetry buys #

Which leaves exactly one construction. A polariser, a Faraday rotator set to 45 degrees, and a second polariser at 45 degrees to the first.

Forward: light passes the first polariser, is turned 45 degrees, and meets the second polariser aligned with it. Everything gets through.

Backward: light passes the second polariser at 45 degrees, is turned a further 45 degrees in the same laboratory sense, and arrives at the first polariser at 90 degrees to it. Nothing gets through.

The tolerance is why the device is practical. Both the loss forward and the leak backward are second order in the misadjustment, so a rotator good to a degree — easily achieved, since the rotation depends on a magnet and a temperature — gives 35 decibels of isolation. Two in series give seventy.

Every laser bench has one, sitting immediately after the source, because a laser is an amplifier and an amplifier that receives its own output back is an oscillator. Optical fibre links have them for the same reason. And the component is always identifiable by weight: it contains a permanent magnet, and there is no way to make it lighter.

The size of the effect, and what has to be spent to get it #

The numbers are worth putting beside each other, because they explain why the two mechanisms are used for completely different things.

Quartz gives 21.7 degrees per millimetre at 589 nanometres, from nothing but a crystal. A sugar solution at 100 grams per litre gives about 6.6 degrees in a decimetre. Both are free in the sense that no field, no current and no power supply is involved.

Terbium gallium garnet — the best ordinary Faraday material in the visible — has a Verdet constant of 134 radians per tesla-metre at 633 nanometres, which is 7.7 degrees per millimetre at one tesla. Getting 45 degrees therefore takes six millimetres of it and a field of a tesla, and a permanent magnet arrangement producing a tesla over six millimetres of bore is a substantial object weighing a few hundred grams. Water, by comparison, has a Verdet constant of 3.7, so a metre of water at one tesla turns the plane by twelve degrees.

So the naturally active material is stronger, cheaper and lighter, and it is useless for the one job that matters, because it is reciprocal. The whole value of the magnetic effect is in a property that costs a magnet and appears on no specification of the rotation itself.

What the effect is used to measure #

Two of the effect’s uses are measurements rather than devices, and both exploit the same linearity.

Sugar. The rotation of a solution is proportional to the concentration of the dissolved sugar, and a polarimeter reading it is the standard method for a sugar refinery. The name dextrose records which way its solution rotates.

Magnetic fields at a distance. The rotation of light passing through a magnetised medium measures the field integrated along the path — a line integral of a field, which is the same kind of quantity Ampère’s law equates to an enclosed current and which is read here without any wire to put a loop around — which is how the field in a plasma is measured without a probe in it, and how the interstellar magnetic field is measured from the rotation of light from distant sources. The measurement is a length times a field times a density, so it takes some untangling — but no other technique reads a field along a line of sight at all.

Faraday’s glass #

The effect is worth its history, because it was not a curiosity when it was found — it was the first evidence that light and electromagnetism have anything to do with each other.

Faraday spent years looking for it. He was convinced that the forces of nature were connected and that a magnetic field ought to do something to light, and he tried repeatedly and failed. The successful experiment in 1845 used a piece of dense lead borate glass he had made himself years earlier while working on optical glass for the Royal Society — a heavy material with a large Verdet constant, though nobody yet had the concept — placed between the poles of an electromagnet with polarised light passing along the field.

The plane turned. Faraday’s notebook entry records that he had “magnetised a ray of light”, which is not what happened and is a fair description of the significance: a magnetic field had altered the behaviour of light, and until that afternoon there was no experimental reason to think the two subjects were related at all.

Twenty years later Maxwell’s equations made the connection structural rather than suggestive, and the Faraday effect became a consequence rather than a clue. But the order matters. The unification was proposed by somebody who had been shown, by an experiment nobody else thought worth doing, that the two things were not independent.

There is a second thing worth taking from it. The experiment worked because Faraday happened to have made a material with an unusually strong response, for an entirely unrelated reason, a decade earlier. Most of the transparent substances he had tried first give a rotation too small to see with the fields he could produce, and had he not had the lead glass on a shelf the effect would have waited for somebody else.

Reading a field across a galaxy #

The astronomical use deserves its own arithmetic, because the wavelength dependence is what makes an awkward measurement into a clean one.

Light from a distant source crossing magnetised plasma has its plane of polarisation rotated, and the rotation is proportional to the wavelength squared — because the Verdet constant of a plasma goes as $\lambda^2$, from the electrons’ response far below their own resonances. So the observed angle is

$$\chi(\lambda) = \chi_0 + \mathrm{RM}\,\lambda^2,$$

with $\chi_0$ the angle the light left with and RM — the rotation measure — an integral along the line of sight of the electron density times the component of the field along the path.

That form is the gift. Observe the same source at several wavelengths, plot the measured angle against $\lambda^2$, and the result is a straight line whose slope is the rotation measure and whose intercept is the intrinsic angle. Two quantities that no single observation could separate are separated by the fact that one of them depends on wavelength in a known way and the other does not.

The quantity recovered is a product of a density and a field, integrated along a path, so it is not a field measurement on its own. Combined with the dispersion measure — the same line of sight’s electron content, obtained from the frequency-dependent arrival time of a pulsed source — the density divides out and what is left is a field averaged along the path, weighted by density.

That is how the magnetic field of the Milky Way has been mapped, and of the medium between galaxies. It is the only technique that reads a field along a line of sight rather than inferring one from what it does to something, and it exists because a magnetic field is odd under time reversal.

Where the model stops #

The Verdet constant is not a constant. It varies as roughly the inverse square of the wavelength away from resonances, and steeply near them, so a rotator set to 45 degrees at one wavelength is not at 45 at another. An isolator is specified for a wavelength and degrades either side of it.

Nor is it temperature-independent. Terbium gallium garnet’s Verdet constant falls by about a part in three hundred per kelvin, and a high-power isolator heats up, so the rotation drifts during operation and the isolation degrades. The high-power ones are temperature-compensated with a second element.

The linear regime is assumed throughout. At the intensities inside a high-power amplifier the medium’s index depends on the light in it, the beam self-focuses, and the rotation stops being a property of the path alone.

And the medium is assumed non-absorbing. In an absorbing medium the two circular components are absorbed differently as well as delayed differently — circular dichroism — and the emerging beam is elliptical rather than plane. That is a nuisance for an isolator and a diagnostic for a chemist, since it is the standard method for determining the handedness of a molecule in solution.

What the pictures cannot show #

The reciprocity figure draws two horizontal lines at 0 and 2, and the drawing conceals the fact that they are exact rather than approximate. Both are consequences of symmetries — one of the beam’s reversal, one of the field’s oddness under time reversal — so neither is a fitted number, and no measurement will find 0.02 or 1.98 except through an imperfection in the apparatus.

Nor can any figure here show a photon doing anything. Everything above is classical: two indices, two circular components, a phase difference. The quantum account attaches the handedness to the photon’s spin and produces the same numbers, and the classical drawing is the one that makes the mechanism visible.

Where this ladder goes next #

The three rungs below this one establish what polarisation is, what a reflection does to it, and what a birefringent crystal does to it. This rung is about something none of those needed: a symmetry of the whole situation rather than a property of the medium. Whether an optical component looks the same run backwards is not a question about glass, and it decides what can be built.

The habit worth carrying away is the test. Before believing a device can be built, ask whether the physics it uses is the same run backwards. Reciprocity is the cheapest impossibility proof in optics: it rules out one-way windows, passive optical diodes, and the perpetual-motion schemes built from them, in a line. And when a device of that kind is nevertheless wanted, the question becomes a shopping list — what is available that is odd under time reversal? — and the answer, in ordinary optics, is a magnetic field and very little else.

What is left on this ladder is the effect’s inverse: a material whose magnetisation is changed by the light passing through it, which turns the same coupling into a way of writing a magnetic bit with a laser pulse.