A field lets every class rotate light

Fifteen may rotate light counted the crystal classes whose symmetry lets a crystal turn the plane of polarisation of light passing through it. The answer turned on a single determinant. The gyration tensor that carries the effect is axial, so a mirror or an inversion multiplies it by minus one on top of whatever the rotation part of the operation does. An inversion centre then forces the tensor to equal its own negative, and seventeen of the thirty-two classes have none. Quartz rotates light and calcite does not, and calcite’s inversion centre is the whole reason.

In 1845 Michael Faraday put a block of heavy lead-borate glass between the poles of an electromagnet, sent polarised light through it along the field, and watched the plane of polarisation turn when the current came on. Glass is isotropic and has a centre of symmetry at every point that matters optically, and it rotated light anyway. The field had supplied something the glass lacked, and what a crystal keeps in a field says in general what that is: the crystal and the field together have only the symmetry the two share. This essay asks which of the thirty-two classes permit the Faraday effect, and the answer is all of them, with an inversion centre or without. It then asks what distinguishes a rotation the field supplies from one the structure supplies, and why most crystals cannot use a rotation they are permitted to have.

What the field supplies #

Both rotations live in the same place. A transparent crystal’s response to light is its dielectric tensor, and optical rotation is carried by the tensor’s antisymmetric imaginary part. That part has three independent entries and can be written as a vector $G$, with $\varepsilon^a_{ij} = i\,e_{ijk}G_k$. Light travelling along a unit direction $\hat k$ has its plane of polarisation turned at a rate proportional to $G\cdot\hat k$. The vector $G$ is axial, because it is built from the antisymmetric part of a tensor, and it changes sign under time reversal, because the Onsager relations make the dielectric tensor’s antisymmetric part odd when every motion is reversed, the operation the operation that reverses time adds to a point group.

Anything that produces $G$ has to supply an axial, time-odd vector. Natural activity supplies it from the wavevector. The light’s own $k$ is a polar vector, odd under time reversal, and a linear relation $G_i = g_{ij}k_j$ between a polar vector and an axial one needs a tensor $g$ that is axial. That tensor is even under time reversal, because the time-oddness of $k$ and of $G$ cancel. Its law is $g' = \det(M)\,MgM^{\mathsf T}$, and the determinant is the one the census of fifteen turned on.

A magnetic field supplies the vector directly. $B$ is axial and odd under time reversal, exactly like $G$, so the relation $G_i = f_{ij}B_j$ needs a tensor $f$ that is polar and even. No determinant enters its law, $f' = MfM^{\mathsf T}$, which is the law of the dielectric tensor itself. The one difference is that $f$ need not be symmetric, since nothing ties $G_1$’s response to $B_2$ to $G_2$’s response to $B_1$. So $f$ is a general second-rank tensor with nine entries, and the number of them a class leaves free is the average of its character, $\operatorname{tr}(M)^2$, over the class. That is Neumann’s principle as one sum, applied to a tensor nobody had reason to count until a field was switched on.

A count that cannot reach zero #

That average has a lower bound that holds before any class is examined. The identity contributes $\operatorname{tr}(I)^2 = 9$, every other operation contributes a square, and the mean of the character over a group is the dimension of the subspace it fixes, an integer. The subspace is never empty, because the unit tensor $\delta_{ij}$ is carried to itself by every orthogonal matrix. An isotropic material has exactly that one component, the Verdet constant. So no class, and no material, forbids the Faraday effect.

The picture at the head of this essay puts the two counts side by side for all thirty-two classes, and they share almost nothing but the identity’s contribution. The Faraday count depends only on the Laue class, the class with an inversion added, because an inversion leaves a polar second-rank tensor unchanged. Both triclinic classes have nine components and all three monoclinic classes five. The orthorhombic classes have three, the tetragonal, trigonal and hexagonal classes two or three, and the five cubic classes one. The natural count depends on the handedness of the operations and follows none of those groupings. Centrosymmetric $\bar{1}$ has nine Faraday components and no gyration component at all, while $\bar{4}3m$, which has no centre, has one Faraday component and no gyration component either.

The two counts are independent measurements of the same classes and they agree with independent checks. The Faraday count by characters was set against a projector, the average of $MEM^{\mathsf T}$ over each class for each of the nine unit tensors $E$, done in Cartesian coordinates so that the hexagonal classes are not averaged in a skewed basis. The ranks agree in all thirty-two classes. The gyration count computed alongside reproduces the census of fifteen, row for row.

Two parts, and a turn across the ray #

A general second-rank tensor is a symmetric tensor plus an antisymmetric one, and the class decides the two separately, as it decides every split of the parts a property splits into. The symmetric part has the dielectric tensor’s six entries and its shape, which three optical characters sorted into a sphere, a spheroid and a general ellipsoid: one, two or three independent values in the principal frame, and more when the frame is not fixed. The antisymmetric part has three entries, and an antisymmetric tensor in three dimensions is a disguised axial vector $a$, with $f^a_{ij} = e_{ijk}a_k$. Its contribution to the rotation is $G = a \times B$.

The character of the antisymmetric part is $\tfrac12\bigl(\operatorname{tr}(M)^2 - \operatorname{tr}(M^2)\bigr)$, and for an orthogonal matrix that equals $\det(M)\operatorname{tr}(M)$, the character of an axial vector. The identity was checked class by class rather than trusted. So the transverse term exists exactly in the thirteen classes that leave some axial vector alone: $1$, $\bar{1}$, $2$, $m$, $2/m$, $3$, $\bar{3}$, $4$, $\bar{4}$, $4/m$, $6$, $\bar{6}$ and $6/m$. These are the classes of which magnetism a class permits that allow a ferromagnetic moment, and the coincidence is no accident. An axial vector left alone by the class is a direction the class has already picked out, and the field’s turn about it needs nothing else.

The transverse term changes what an experiment sees. With $B$ along the ray only the symmetric part matters, since $a\times B$ is then perpendicular to the ray and $G\cdot\hat k$ does not see it. With $B$ across the ray, the geometry an ordinary Faraday measurement avoids, the symmetric part contributes only through its entry linking the two directions, and the antisymmetric part contributes directly. In the centrosymmetric class $4/m$ a field perpendicular to the four-fold axis, with the ray perpendicular to both, gives a rotation proportional to the one antisymmetric component. In $4/mmm$ the same geometry gives nothing. The pictured pattern is the whole content of that sentence.

A mirror that sends the light back #

Permission says the rotation may be there. It does not say which of the two causes produced a rotation that is measured. The two differ in one respect, and that one difference can be measured. Reversing the direction of the light reverses $k$ and leaves $B$ where it was. Natural rotation, expressed as a turn about the ray, has the same sense whichever way the ray runs, so in the laboratory it turns one way going out and the other way coming back. Faraday rotation is tied to the field, turns the same way about the field whichever way the light goes, and so accumulates.

The round trip is computed here rather than assumed, by Jones propagation on the Poincaré sphere, with quartz’s rates at the sodium line as the stated inputs. Its natural rotation is 21.7 degrees per millimetre along the optic axis, and its Verdet constant is 4.83 radians per tesla-metre. Two millimetres of quartz in a field of one tesla, with the ray and the field along the axis, turn the plane by 43.95 degrees going out: 43.40 from the structure and 0.55 from the field. After a mirror and the return pass the natural part is gone, to rounding error, and the Faraday part has doubled to 1.11 degrees. The two causes that were added going out are separated by coming back. The same separation follows from reversing the field instead, which flips the Faraday rotation and leaves the natural one, but the round trip needs no magnet that can be reversed and so it is the cleaner experiment.

This is the property that makes the effect worth building devices around. A Faraday rotator set to 45 degrees, between polarisers at 0 and 45 degrees, passes light going one way. Light coming back is turned a further 45 degrees rather than back by 45, arrives at the first polariser crossed, and is stopped. That is an optical isolator, and it is possible only because the rotation is odd under time reversal. A naturally active crystal of any thickness between the same polarisers passes light equally in both directions. The Onsager relations say why: $\varepsilon_{ij}(k, B) = \varepsilon_{ji}(-k, -B)$, so a material is reciprocal unless something in it, a field or a magnetic order, is odd under time reversal.

Why an isolator is cut from a cubic crystal #

Every class permits the Faraday effect, but most crystals cannot deliver it along most directions, for a reason that has nothing to do with the Faraday tensor. A crystal that is not cubic has linear birefringence. Its two polarisation eigenmodes travel at different speeds, and a linear birefringence even a thousand times larger than the rotation per unit length does not add to it. It overwhelms it. On the Poincaré sphere a rotation of the plane of polarisation is a turn about the circular axis, and linear birefringence is a turn about a linear axis. The two combine into a turn about their resultant. When the linear part dominates, the resultant sits almost on the linear axis, and a beam that starts linearly polarised along an eigenmode circles that axis on a small cone. Its azimuth never leaves a narrow band.

Along the optic axis of a uniaxial crystal there is no linear birefringence, and the whole rotation arrives. The calculation confirms it exactly. Off the axis the linear retardance grows as the square of the angle, and the rotation is quenched. The figure measures how fast, for quartz’s indices, with $n_o = 1.5443$ and a birefringence of 0.0091.

The half-way points come in two regimes, and the regime is not set by the size of the rotation alone. For quartz’s natural rotation, 379 radians per metre, a one-centimetre path would turn the plane far past a quarter turn. The azimuth is then confined to the cone, and half of it is lost once the linear retardance rate passes twice the rotation rate. That predicts 5.07 degrees, and the propagation gives 5.31. The Faraday rotation at one tesla is eighty times weaker, and over one centimetre it would turn the plane less than three degrees even unobstructed. The cone is then not the limit, because the beam never gets far enough round it. What limits it is that the azimuth grows as $\rho\,\sin(\delta z)/\delta$ and falls behind the unobstructed $\rho z$ once $\delta z$ is no longer small. Half is lost where $\delta L = 1.895$, the root of $\sin x = x/2$. That predicts 2.53 degrees for one centimetre, against 2.62 computed.

So the weaker rotation needs the better alignment, and the crystal’s length sets how much better. A one-millimetre plate forgives 8.3 degrees of misalignment, a five-centimetre rod 1.2. A longer crystal gives more rotation per tesla and less tolerance per degree, and the two trade exactly. That is the reason Faraday rotators are not cut from quartz, although quartz is permitted a rotation along its axis.

The same criterion turns into a solid angle once it is integrated over directions around each optic axis. For Faraday rotation at one tesla over one centimetre, a cubic crystal rotates in every direction. A uniaxial crystal rotates inside a cone 2.5 degrees about its axis, one direction in a thousand. A biaxial crystal fares far worse than twice that, although it has two optic axes, and the reason is geometric. Near a biaxial optic axis the birefringence grows in proportion to the angle, where near a uniaxial one it grows as the square. Its two cones are 0.11 degrees wide and cover 3.8 millionths of the sphere. The biaxial indices here are a stated model, quartz’s range split evenly, and not a real crystal. The contrast between linear and quadratic growth is general, though, and it is the conical-refraction geometry of a biaxial optic axis showing up in a count.

That is why the working Faraday materials are cubic. Terbium gallium garnet and yttrium iron garnet are both garnets, of class $m\bar{3}m$, whose indicatrix is a sphere. Of the thirty-two classes that permit the effect, the five cubic ones are the only classes whose crystals deliver it along any direction the designer chooses. They are also the classes with the fewest components, one each. The same fact that makes them useful, that nothing picks a direction, leaves them a single number to measure.

Rotation with no field applied #

The field is one source of an axial, time-odd vector. A crystal with a spontaneous magnetisation carries its own, and $G$ can then be linear in the magnetisation with no field applied. The two vectors obey one law, both axial and both odd under time reversal. A magnetic point group therefore permits spontaneous rotation exactly when it permits a magnetisation. The count is not a second calculation that happens to agree with the first. It is the same character.

Of the 122 magnetic point groups that which magnetism a class permits derives, 31 permit it: thirteen ordinary groups, eighteen black-and-white ones, and no grey group. The grey groups contain time reversal itself as an operation, and a time-odd vector left alone by time reversal is zero. A paramagnet in zero field cannot rotate light by this mechanism, and a ferromagnet can. The magneto-optical Kerr effect, the same rotation seen in reflection, is how the magnetisation of thin films is imaged, and it was how magneto-optical discs were read. It is permitted by the same thirty-one groups for the same reason.

There is also a term that needs both the ray and the field at once, a change in the refractive index linear in each, $\Delta n \propto \chi_{kl}k_kB_l$. The tensor $\chi$ is axial and even under time reversal, with the character $\det(M)\operatorname{tr}(M)^2$. It is permitted in eighteen classes. Its symmetric part is the gyration tensor’s twin and lives in the fifteen that rotate light naturally. Its antisymmetric part is a polar vector, giving a term in $P\cdot(k\times B)$, and it adds $4mm$, $3m$ and $6mm$, three of the ten polar classes with no natural activity. The first is magneto-chiral anisotropy, predicted by Baranova and Zel’dovich in 1979 and observed by Rikken and Raupach in 1997. Both are differences between light going one way and light going the other, in a field, with no magnetic order at all. Whether a textured aggregate can carry the first where no single grain does, as a fibre of achiral crystals found for natural rotation, is the same averaging question, and it is not computed here.

The conventions the numbers rest on #

The rotation is measured in the laboratory frame, about the field. A natural rotation is conventionally quoted about the ray, looking towards the source, and with that convention it is the same in both directions. The figures here quote every angle about a fixed laboratory axis, because only in that frame does a round trip add or cancel. Changing convention changes the sign of the natural part on the return leg and nothing else.

The sign of the Verdet constant is a convention layered on a physical sign. Diamagnetic glasses and quartz rotate one way and paramagnetic terbium compounds the other. The symmetry argument says the isotropic component exists. It says nothing about its sign.

The quench is computed for a non-absorbing crystal with its rotation and birefringence taken as independent. Near an absorption band both are dispersive, circular dichroism accompanies the rotation, and a real Faraday measurement also reports an ellipticity. The Jones model keeps the geometry and drops those.

What the census cannot say #

The counts are exact integers, computed from the site’s own enumeration of the thirty-two classes and of the 122 magnetic point groups, with the characters averaged over the matrices and checked against projectors. They are permissions and never predictions, the distinction what the count does not say draws for every tensor. A class with nine components is permitted nine independent numbers and may have any of them vanishingly small, and the Verdet constant of a real crystal is a property of its electrons that no symmetry argument sizes. The quartz numbers are literature values used as inputs. The quench and round-trip angles are what those inputs imply under the model, not measurements made here.

The figures cannot show the time-reversal argument itself. They show its consequences, a count that is never zero and a curve that doubles instead of cancelling. The reason a field can stand in for a handed structure is the pairing of an axial, time-odd cause with an axial, time-odd effect, and that is algebra about transformation laws. It is established by the character identity the calculation checks, not by any drawing.

Who found it, and when #

Faraday found the effect in 1845, in a heavy glass he had made years earlier for other purposes, as the first evidence that light and magnetism interact. Émile Verdet measured its proportionality to the field and to the path through the 1850s, and the constant carries his name. Lars Onsager’s reciprocal relations of 1931, extended to magnetic fields by Hendrik Casimir in 1945, supplied the general rule the round trip illustrates: a response is symmetric under exchange of cause and effect with every time-odd quantity reversed. Isolators built on garnets became standard components of fibre-optic systems in the 1980s.

What the calculation has to refuse #

The first refused claim is the analogy that makes the effect surprising. Natural rotation and Faraday rotation both turn the plane, and an inversion centre kills the first. In $\bar{1}$ the gyration tensor has no component and the Faraday tensor has nine, so the analogy fails in the most symmetric-looking way available. The second refusal is the check that the difference is the determinant and nothing else. Restore it, and the law becomes that of an axial tensor, which fourteen classes would forbid. That is the wrong answer glass alone refutes.

Still open: the monoclinic crystals that can show the transverse term #

The transverse term has a difficulty the census does not show, and the quench is what exposes it. In the uniaxial classes that have it, $4/m$ among them, the axial vector $a$ lies along the unique axis, which is also the optic axis. The rotation it contributes, $G = a\times B$, is perpendicular to that axis. A ray sees it only in proportion to the ray’s component across the axis, zero on the axis itself. So the transverse term can be seen only where the birefringence is quenching it, and at the 2.6 degrees a one-centimetre crystal tolerates it is reduced by the sine of that angle as well. In a uniaxial crystal it is permitted and all but unobservable as a rotation.

The monoclinic classes are different, because there symmetry fixes one principal direction of the indicatrix and leaves the other two free, as the axes a class pins down found. The vector $a$ lies along the two-fold axis. If that axis happens to be the direction of the intermediate refractive index, both optic axes lie in the plane perpendicular to it. A ray along an optic axis, with the field across the ray in that plane, then sees the whole transverse term and no birefringence at all. Whether the two-fold axis is the intermediate one is a fact about each material, with no symmetry argument to decide it, and it can change with wavelength. How many monoclinic crystals have that optical orientation, and so could show a Faraday rotation with the field across the light, is a question for tables of measured indices, and it has not been asked here.