On Physical Lines of Force/Modern derivations/Faraday rotation today

On Physical Lines of Force → Modern derivations → Faraday rotation

What Maxwell had

Part IV is the one place in the paper where no 1861 formula can be printed here at all: the mathematical investigation of the rotation is not recoverable for this edition, and the transcription breaks off mid-sentence. What survives is Maxwell's summary of the answer — that the rotation of the plane of polarisation is proportional to six things at once:

	the rotation is proportional to	modern counterpart
(A)	the thickness of the substance	the path length $d$
(B)	the resolved part of the magnetic force along the ray	$H_{\parallel }$
(C)	the index of refraction of the ray	dispersion, inside the modern constant
(D)	inversely, the square of the wavelength in air	$1/{\lambda }^2$, inside the modern constant
(E)	the mean radius of the vortices	no counterpart survives
(F)	the capacity for magnetic induction	$\mu$, and see (B)

Put the six together and Maxwell's claim is that the rotation $\theta$ goes as $d{H}_{\parallel }\mu ir/{\lambda }^2$, with $i$ the index of refraction and $r$ the vortex radius.

Provenance: Part IV's mathematics is not recoverable for this edition and no equation from it is printed here. The list above paraphrases Maxwell's six stated results, which are quoted in full on The six results; the combined proportionality is an editorial assembly of them, not a printed formula. See Sources.

Notice (B) and (F) sitting in the same product. $\mu H_{\parallel }$ is the magnetic induction $B_{\parallel }$. Maxwell listed the force and the permeability as two separate proportionalities; a modern account writes the one quantity they multiply to.

What it says today

The modern statement is as compact as physics gets:

$$\theta =VBd$$

The rotation is the product of the path length $d$, the component of the magnetic induction along the ray, and a number $V$ characteristic of the material — the Verdet constant, in radians per tesla per metre, named for the same Verdet whose measurements had confirmed Maxwell's (A) and (B) by 1862. For a field varying along the path, $\theta =V\int 𝐁\cdot d\ell$.

Compare with the list above. (A) is the factor $d$; (B) with (F) is the factor $B$. Two of Maxwell's six proportionalities are reproduced exactly, and they are the two still stated as a law rather than as a property of a substance. (C) and (D) are not discarded but absorbed: they describe how $V$ depends on colour, and the classical relation, usually credited to Becquerel, gives

$$V\propto \lambda \frac{dn}{d\lambda }$$

Far from any absorption band the index falls slowly with wavelength, roughly as $n=a+b/{\lambda }^2$, and this makes $V$ go as $1/{\lambda }^2$ — Maxwell's (D), recovered exactly. Nearer a band the dispersion steepens, and $V$ then rises faster than $1/{\lambda }^2$ toward the blue. That excess is precisely what Maxwell's model predicted and the received rule did not, and it is why he insisted on carrying (C) alongside (D). One warning: the sign convention for $V$ is not standardised. Sources disagree about which sense of rotation counts as positive, so check what a table means before trusting its minus signs.

Two circles make a line

The mechanism has nothing to do with vortices and everything to do with a piece of geometry that is worth doing in full, because it is short and the result is exact.

Plane-polarised light is the sum of two circularly polarised waves of equal amplitude turning in opposite senses. Let the light run along $z$ and be polarised along $x$ where it enters the medium:

$$𝐄=\frac{E_0}{2}[\hat{x}\cos {\varphi }_{+}-\hat{y}\sin {\varphi }_{+}]+\frac{E_0}{2}[\hat{x}\cos {\varphi }_{-}+\hat{y}\sin {\varphi }_{-}],{\varphi }_{\pm }=\omega t-k_{\pm }z$$

At $z=0$ the two phases are equal, the $\hat{y}$ parts cancel and the $\hat{x}$ parts add: $𝐄=E_0\hat{x}\cos \omega t$, a plane wave polarised along $x$, as required.

In a magnetised medium the two senses travel at different speeds, so $k_{+}\ne k_{-}$. Follow them to depth $d$. Write $\overline{\varphi }=\omega t-\overline{k}d$ for the average phase, with $\overline{k}=\frac{1}{2}(k_{+}+k_{-})$, and let

$$\theta =\frac{1}{2}(k_{-}-k_{+})d$$

so that ${\varphi }_{\pm }=\overline{\varphi }\pm \theta$. Adding the two waves and using $\cos (\overline{\varphi }+\theta )+\cos (\overline{\varphi }-\theta )=2\cos \overline{\varphi }\cos \theta$ together with $\sin (\overline{\varphi }-\theta )-\sin (\overline{\varphi }+\theta )=-2\cos \overline{\varphi }\sin \theta$:

$$𝐄=E_0\cos \overline{\varphi }[\hat{x}\cos \theta -\hat{y}\sin \theta ]$$

Read that carefully. The two components are still exactly in phase — they share the single factor $\cos \overline{\varphi }$ — so the light is still plane polarised, not elliptical. But the fixed direction it oscillates along is no longer $\hat{x}$. It has been turned through the angle $\theta$, which is half the phase difference the two circular components accumulated:

$$\theta =\frac{\omega d}{2c}(n_{-}-n_{+})=\frac{\pi d}{{\lambda }_0}(n_{-}-n_{+})$$

A difference in refractive index between the two circular senses is called circular birefringence, and this calculation is the whole of the connection between it and a rotated plane.

Why magnetism turns the plane of polarization

Why the two senses differ

What makes $n_{+}\ne n_{-}$ is the response of the medium's bound electrons. Light drives an electron in a circle; a magnetic field along the ray exerts on that circulating electron a force $-e𝐯\times 𝐁$ directed along the radius — see the Lorentz force — and that force adds to the binding for one sense of circulation and subtracts from it for the other. The effective resonance frequency is shifted by roughly the Larmor frequency ${\omega }_L=eB/2m_e$, one way for each sense. Different resonance, different polarisability, different refractive index.

So the modern account locates the handedness in the orbits of electrons rather than in the spin of cells of ether. But note what has not changed: the effect still requires something in the medium to be circulating, and the field still has to tell it which way.

Non-reciprocity, and why it matters

Here is the property that makes Faraday rotation different from every other kind of optical rotation, and Maxwell states it as sharply as anyone since:

In both cases the absolute direction of the rotation is the same, whether the light passes from north to south or from south to north,— a fact which distinguishes this phenomenon from the rotation produced by quartz, turpentine, &c., in which the absolute direction of rotation is reversed when that of the light is reversed.

	optical activity (quartz, sugar)	Faraday rotation
what fixes the sense	the handedness of the molecule	the direction of $𝐁$
send the light back	rotation reverses	rotation does not reverse
after a return pass	the two cancel: net zero	the two add: net $2\theta$
the word for it	reciprocal	non-reciprocal

The reason is in the formula. In sugar solution the sense is tied to the direction of travel, so reversing the travel reverses the sense and the mirror undoes the outward rotation. In a magnetised medium the sense is tied to $𝐁$, which the mirror does not touch; the return pass turns the plane the same absolute way again.

The optical isolator

Non-reciprocity is a rare and valuable thing, and it has one famous application. An optical isolator — a one-way street for light — is three components in a line:

1. a polariser, transmitting at $0^{\circ }$;
2. a Faraday rotator, a short block of high-Verdet material in a permanent magnet, cut to give exactly $45^{\circ }$;
3. a second polariser, set at $45^{\circ }$.

Light going forward passes the first polariser at $0^{\circ }$, is turned to $45^{\circ }$, and passes the second. Light coming back passes the second polariser at $45^{\circ }$, goes through the rotator, and is turned a further $45^{\circ }$ in the same absolute sense — to $90^{\circ }$ — where it meets the first polariser at $0^{\circ }$ and is extinguished. Were the rotation reciprocal, the return beam would arrive back at $0^{\circ }$ and sail through, and no such device could exist.

Every laser that would be destabilised by its own back-reflection sits behind one of these, built from terbium-doped glasses and garnets: a direct working descendant of the effect Faraday found in 1845 with a lump of heavy glass and an electromagnet.

What survived, and what did not

Maxwell's mechanism did not survive. It placed the rotation in the angular momentum of ether vortices, and its numbers depended on a vortex radius that has no referent. See What became of the vortices.

The character of the inference did, and on both counts it was right: Faraday rotation does show that something in a magnetised medium is genuinely turning, and magnetism is indeed associated with angular momentum — of electrons, orbital and intrinsic, rather than of cells of a fluid. That much was correctly read off Faraday's experiment, in 1862.

Result (E) closes a loop. Prop. I of Part I had shown that the magnetic behaviour of the medium does not depend on the size of the vortices — which is what let Maxwell say "molecular" without saying how small, and the parameter then dropped out of Parts I, II and III entirely. In Part IV it reappears, in one phenomenon and one only, so that the model which had declined to specify the quantity now offered the sole means of measuring it. The quantity turned out not to exist. The discipline of the move is exemplary all the same.

Where the conventions differ

	Maxwell 1861	modern
the statement	six separate proportionalities	one constant: $\theta =VBd$
the field in it	$H_{\parallel }$ and $\mu$, listed apart	$B_{\parallel }$, the product of the two
colour dependence	(C) and (D), as explicit factors	folded into $V(\lambda )$
vortex radius (E)	a real parameter of the medium	no counterpart
the mechanism	angular momentum of ether vortices	magnetically split resonances of bound electrons
sign of the rotation	fixed by the vortex sense chosen in Part II	fixed by the sign convention chosen for $V$, which varies between sources
units	thickness and magnetic force in his own system	$V$ in rad T⁻¹ m⁻¹

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In the paper: Why magnetism must be rotatory · The six results · What became of the vortices