The handedness a mirror cannot keep

The term that made light completed Maxwell’s equations, the potentials that are not unique wrote them in terms of quantities that carry a redundancy, and the two equations that are not laws of motion separated the two that evolve the fields from the two that constrain them. The symmetry one missing charge would complete then noticed that in empty space the equations are unchanged if the electric field is rotated into the magnetic field, and that only the absence of magnetic charge stops the symmetry holding in the presence of sources.

That essay left a question it named but did not answer. The conservation law a symmetry hands over established that every continuous symmetry of a system comes with a conserved quantity, and the rotation of electric into magnetic field is continuous — any angle will do. So something must be conserved by light in empty space because of it. The something is not energy or momentum, which come from symmetries of time and space. It is the handedness of the light: the number of photons spinning one way along their direction of travel, less the number spinning the other way. And the conservation is broken, visibly and measurably, at almost every surface light meets, which is where it becomes interesting.

A rotation of one field into the other #

In empty space, with no charges or currents, Maxwell’s equations say that a changing magnetic field curls the electric field and a changing electric field curls the magnetic field, and that neither has sources. Written with the magnetic field multiplied by $c$ so that the two have the same units, the pair of equations is symmetric: swap $\mathbf{E}$ for $c\mathbf{B}$ and $c\mathbf{B}$ for $-\mathbf{E}$ and the equations come back unchanged. That swap is a rotation by 90°, and any intermediate angle works too:

$$\mathbf{E} \to \mathbf{E}\cos\theta + c\mathbf{B}\sin\theta, \qquad c\mathbf{B} \to c\mathbf{B}\cos\theta - \mathbf{E}\sin\theta .$$

Every solution of the equations is carried into another solution. The energy density, $\tfrac12\varepsilon_0(E^2 + c^2B^2)$, is unchanged, and so is the momentum density, since the cross product of the two rotated fields is the cross product of the two originals.

The figure shows what the rotation does to a plane wave. In a light wave the electric and magnetic fields are perpendicular to each other and to the direction of travel, and of equal size once the magnetic one is multiplied by $c$. Rotating each into the other by an angle $\theta$ therefore turns the pair together about the direction of travel by the same angle. A linearly polarised wave comes out linearly polarised at a new angle. A circularly polarised wave, whose fields are already turning steadily about the direction of travel, comes out looking exactly like itself a little earlier or a little later — the rotation is equivalent to a shift of phase, forward for one handedness and backward for the other.

That is the key to what the symmetry conserves. The states that a symmetry leaves unchanged apart from a phase are the ones it distinguishes, and here they are the two circular polarisations. The conserved quantity is the difference between them, the helicity: for a beam, the number of photons whose spin points along their direction of travel less the number whose spin points against it. Two states where the counting says three found that a photon has exactly two such states and no third, and duality symmetry is the reason their difference is a conserved count rather than merely a label.

What a mirror does to handedness #

In empty space helicity is conserved trivially, because a free photon never changes. The content of the law appears when light meets matter, and the first surprise is a mirror.

Shine right-handed circularly polarised light straight at a metal mirror and it comes back left-handed. That is not because the mirror does anything to the rotation of the field. The field at the mirror keeps turning the same way in space; what reverses is the direction of travel, and handedness is measured relative to the direction of travel. Photon by photon, the spin angular momentum about the beam’s axis is unchanged while the momentum has been reversed, so the helicity — spin projected on momentum — has changed sign. A perfect conductor does this at every angle of incidence, as the flat line at zero in the figure shows.

Glass does it at normal incidence too, and then less and less as the angle grows. At Brewster’s angle only the component polarised perpendicular to the plane of incidence is reflected, so the reflected light is linear and splits evenly between the two handednesses; beyond it glass keeps a growing share, and at grazing incidence it reflects almost everything with the handedness kept. In duality terms, glass breaks the symmetry at every angle but the last. The breaking is not subtle. Ordinary matter responds strongly to the electric field of light — glass has a relative permittivity of 2.25 at visible frequencies — and hardly at all to the magnetic field, with a permeability indistinguishable from one. A material that treats the two fields so differently cannot respect a symmetry that rotates one into the other.

A material that keeps it #

What would a material have to be like to keep the symmetry? The rotation mixes $\mathbf{E}$ and $c\mathbf{B}$; inside matter the corresponding fields are $\mathbf{D} = \varepsilon\varepsilon_0\mathbf{E}$ and $\mathbf{B} = \mu\mu_0\mathbf{H}$, and the equations are unchanged by the rotation only if the relative permittivity $\varepsilon$ equals the relative permeability $\mu$. The refractive index of such a medium is $\sqrt{\varepsilon\mu} = \varepsilon$, which can be anything, but its wave impedance, $\sqrt{\mu/\varepsilon}$ times that of empty space, is exactly one.

The figure’s top line is such a medium with the same index as the glass, 1.5. It keeps the handedness of reflected light at every angle, and the reason is visible in the Fresnel coefficients: with equal impedances the reflection coefficients for the two linear polarisations are equal at every angle, and a circular wave, which is an equal mixture of the two with a quarter-cycle between them, is reflected as the same mixture. The dashed curve, a medium with $\mu = 1.2$ and $\varepsilon = 1.88$, sits between glass and the dual medium, because it breaks the symmetry less.

Transmission disturbs the handedness far less than reflection does, because the transmission coefficients for the two polarisations never differ by much. Even so, glass converts a small part of a circular beam into the other handedness at steep angles, 1.5 per cent of the transmitted power at 80°, and the figure shows it drooping towards grazing incidence. The dual medium converts none. Light entering it keeps its handedness whatever the angle, and so, by the symmetry, does light leaving it.

The surface that sends nothing straight back #

There is a consequence of keeping handedness that sounds like a separate fact and is the same one.

The figure plots how much light each surface reflects. Glass reflects four per cent at normal incidence, the familiar reflection from a window, and its two polarisations separate at larger angles. The dual medium reflects nothing at all at normal incidence, because its impedance is that of empty space, and then reflects s and p light equally at oblique angles, rising to total reflection at grazing incidence. A surface can therefore bend light by exactly as much as glass does and still send nothing straight back.

The two properties are linked by angular momentum. A wave reflected straight back with its handedness unchanged has had its spin about the beam’s axis reversed: the direction of travel flipped, the handedness kept, so the spin must have flipped. A flat surface, or any object with complete symmetry about the beam’s axis, cannot take up that angular momentum, so such a reflection is forbidden, and a surface that keeps handedness can only keep it by not reflecting straight back at all. What happens where the medium changes found that a wave sees only the impedance at a join; the duality argument says the same thing more strongly, that a join between media of equal impedance cannot return a wave along its path for any reason at all.

A particle with no back #

The same argument applies to a single small particle, and there it produces a result with a name and a history.

A particle much smaller than the wavelength scatters light as a small oscillating electric dipole, and, if it responds to the magnetic field, as a small magnetic dipole too. An ordinary particle of glass or water is almost purely electric, and it scatters as much light backward as forward, the pattern that makes the sky equally bright in the directions towards and away from the Sun. For circularly polarised light the backward half of that pattern has its handedness reversed, as a mirror’s has.

In 1983 Milton Kerker and his colleagues asked what happens to a particle whose magnetic response equals its electric one. The electric and magnetic dipoles then radiate in phase in the forward direction and cancel exactly in the backward direction, and the particle scatters nothing back towards its source. The figure draws the pattern turning from a symmetric peanut into a forward-pointing heart as the magnetic response grows. In the language of duality, the particle with equal responses respects the symmetry, keeps the handedness of every photon it scatters, and so cannot send any straight back. Kerker’s condition was a curiosity for decades, because no natural particle has a significant magnetic response at optical frequencies. It stopped being one when particles of silicon a few hundred nanometres across, whose internal resonances give them a strong magnetic dipole response, were shown to meet it at particular wavelengths, and it now guides the design of surfaces made of such particles that transmit without reflecting.

Why nothing natural answers the magnetic field of light #

The obstacle to building a dual medium is that nature supplies almost no magnetic response at the frequencies of light, and the reason is a ratio of speeds. An atom responds to the electric field of a light wave by shifting its electrons, and the force on an electron is $e\mathbf{E}$. It responds to the magnetic field through the force $e\mathbf{v}\times\mathbf{B}$, and since $cB = E$ in the wave, that force is smaller by the electron’s speed over $c$ — about the fine-structure constant, $1/137$, for an electron in an atom. The magnetic dipole it induces is smaller by the same factor again, so the magnetic response is some $10^{-4}$ or $10^{-5}$ of the electric one. Landau and Lifshitz remarked on this in their treatment of continuous media: above the frequencies of microwaves, the permeability of every natural material is one, and it is not worth writing down.

At radio and microwave frequencies magnetic materials exist, because the spins in ferrites can follow the field, and duality-respecting surfaces were demonstrated there first. At optical frequencies the only route is to build a magnetic response out of electric ones. A current flowing round a loop is a magnetic dipole — the loop that behaves like a needle showed that at a distance it is indistinguishable from a bar magnet — and a displacement current will do as well as a conduction current. A sphere of a material with a high refractive index, lit by light whose wavelength inside the sphere is comparable with its diameter, carries an electric field that curls round inside it. The curl is a loop of displacement current, and the sphere acquires a magnetic dipole as large as its electric one.

For silicon, with an index near 3.5 in the near infrared, the first such magnetic resonance of a sphere 200 nanometres across falls at roughly 700 to 800 nanometres, where the wavelength inside the silicon is about the sphere’s diameter. Just to the long-wavelength side of it the magnetic and electric dipoles are equal in size and phase, Kerker’s condition is met, and the sphere scatters forward only. The magnetic response was manufactured from the electric one by shape, which is the whole method of metamaterials: a medium whose permeability differs from one at optical frequencies is always a structure of electric responses arranged so that their currents circulate.

Two symmetries, and which one matter breaks #

It is worth being careful about what has been shown. Duality symmetry holds exactly for the free field. It fails in the presence of electric charges without magnetic ones, which is the failure the previous argument about monopoles was about. It also fails in the presence of ordinary matter, which is made of those charges and so responds to the electric field of light far more strongly than to the magnetic one. The second failure is the one visible in the laboratory, and it has a clean signature: wherever light changes its handedness, matter has broken the duality symmetry there.

That gives the conserved quantity a practical use. In an experiment where helicity is conserved, the only way to change a beam’s handedness is to break the symmetry somewhere, so a change of handedness locates a place where electric and magnetic responses differ. A structure built to respect the symmetry — an array of silicon particles tuned to Kerker’s condition, a lens made of such a medium — passes circularly polarised light without scrambling it, which matters in measurements that distinguish mirror-image molecules by how they absorb light of the two handednesses. The effect such molecules have on circular light is weak, a part in a thousand or less, and a surface that converted a part in a hundred of the light into the other handedness would swamp it.

The helicity of light is also different from two quantities with similar names that appear in other parts of physics. The knot the field cannot untie described the magnetic helicity of a plasma, which measures how its field lines are linked and is conserved by a perfect conductor rather than by empty space. And the angular momentum that the angular momentum that is in nothing at all found stored in a static field is total angular momentum, which includes the orbital part; helicity is only the spin part projected on the direction of travel.

What the curves cannot show #

The figures treat each medium as a single flat interface with a real, frequency-independent permittivity and permeability. Real materials absorb, and their responses vary with frequency, so a medium with $\varepsilon = \mu$ at one wavelength will not have it at another; the silicon particles meet Kerker’s condition in a band a few tens of nanometres wide. The small-particle patterns keep only the two dipole terms, which is correct for a particle much smaller than the wavelength and fails as it grows, when higher multipoles appear and the condition becomes a set of conditions, one for each pair of multipoles.

Nor do the figures show what happens in the near field of a surface, where evanescent waves carry helicity that does not propagate, or in structured media whose response depends on direction. And they do not show a medium with $\varepsilon = \mu = -1$, which is dual too and which the ray on the wrong side of the normal described as the ideal of negative refraction: its perfect lens is, among other things, a device that conserves helicity.

Still open: what the conserved quantity is, exactly #

The helicity of a light beam is conserved, but writing it as a local density — an amount per unit volume that flows from place to place — is surprisingly delicate. The natural expression involves the vector potential, which is not unique, and a density built from a non-unique quantity is suspect. Expressions that avoid the problem exist, using both an electric and a magnetic vector potential, and they give the right total. Whether there is a single correct local density, how it behaves in dispersive matter, and how it relates to a family of other conserved quantities of the free field found in 1964 and called the zilch, whose physical meaning was obscure for half a century, are questions still being argued in the literature on the angular momentum of light.

The habit worth carrying away is to ask what a symmetry hands over and then where it is broken. Maxwell’s equations in empty space cannot tell an electric field from a magnetic one, so light conserves its handedness; every mirror reverses it because matter answers the electric field and not the magnetic, and a material that answered both alike would keep every photon’s handedness — and could therefore send nothing straight back. The reflection in a window is, among other things, a measurement of how unequally glass treats the two fields.