Faraday and Magneto-Optical Effects

Put a block of dense glass inside a strong magnet, aim linearly polarized light straight down the field lines, and the plane of polarization arrives at the far face twisted — rotated by an angle that grows with the field and the path length. Michael Faraday found this in September 1845, in a bar of dense lead-borosilicate (“heavy”) glass, after years of hunting for a magnetic action on light: the first hard evidence that light and magnetism are the same subject, and the seed of Maxwell's electromagnetic theory of light twenty years later. The rotation obeys one strikingly simple law, $\beta = V\,B\,d$ : axial field times length times a material constant, the Verdet constant. But the effect matters far beyond its size because it is nonreciprocal. Natural optical activity, the twist a sugar solution gives, is set by the handedness of molecules, so a beam sent back through unwinds what it wound and a round trip cancels. Faraday rotation is set by an external field that ignores the beam direction, so a reflected beam rotates the same way again and returns twisted twice as far, not zero. No wave plate, prism, or mirror can imitate that broken symmetry, and it is the basis of the optical isolator — the one-way valve that guards every high-power laser and coherent fibre link from its own back-reflection.

Key takeaways

A magnetic field along the direction of travel rotates the plane of linear polarization by $\beta = V B d$ — axial field times length times the Verdet constant $V$ .
The mechanism is magnetic circular birefringence: the field makes the indices for left- and right-circular light unequal ( $n_- \neq n_+$ ), and a linear state is the sum of those two counter-rotating circles.
The effect is nonreciprocal — its sense is fixed by $\mathbf{B}$ , not $\mathbf{k}$ — so a round trip doubles the rotation ( $45° \to 90°$ ) instead of cancelling it, unlike reciprocal natural optical activity (recalled by title from Optical Activity).
That nonreciprocity builds the optical isolator: a 45° Faraday rotator between polarizers passes light forward and blocks its return, protecting lasers from back-reflection.
Family cousins: the magneto-optic Kerr effect (MOKE, rotation on reflection) and the Voigt / Cotton–Mouton effect (transverse field, birefringence $\propto B^2$ ).
Key idea: an axial field lifts the degeneracy of the two circular polarizations, and because the field — not the light's direction — sets the handedness, the rotation is nonreciprocal, doubling on a round trip and making one-way optical devices possible.

The core insight

Linear polarization is an equal sum of left- and right-circular light. A field along the propagation direction makes the two circular components travel at slightly different speeds ( $n_- \neq n_+$ ), so their recombined axis emerges rotated by $\beta = V B d$ . The handedness is set by the external field, an axial vector unchanged by reversing the beam, so the rotation is nonreciprocal: a round trip adds rather than cancels — the physics of the optical isolator.

Where this lives today

The Faraday rotator's headline job is laser protection. Back-reflection into a laser cavity or fibre amplifier causes mode-hopping, noise, and — at high power — damage, so nearly every serious laser carries a Faraday isolator: a terbium-garnet rod in a permanent-magnet housing set to 45°, between crossed-at-45° polarizers, rejecting the return by 30–40 dB. Swap the polarizers for polarizing beam splitters and the same nonreciprocity becomes an optical circulator, routing light one way only around three or four ports — the component that lets a single fibre carry traffic both directions. Push the Verdet constant up near an atomic resonance and the effect becomes an instrument: optically pumped atomic magnetometers detect femtotesla fields — rivalling SQUIDs without cryogenics — by reading the Faraday rotation of a probe beam through spin-polarized alkali vapour. And the reflection cousin, MOKE, images magnetic domains and traces thin-film hysteresis loops — the physics that once read magneto-optical discs.

The physical picture

Think of a linear polarization as two vectors of equal length spinning in opposite senses — one left-circular, one right-circular — whose sum is always a line. Send them down a magnetic field and the orbiting electrons in the glass feel a Lorentz force that makes one sense of circulation slightly "stiffer" and the other slightly "softer". The two circular components now travel at different phase speeds; downstream one has slipped ahead, and re-adding the spinning vectors gives a resultant line in a new direction — rotated by half the accumulated phase difference. The light stays linear; only its axis has turned. The crucial twist is on the way back. The sense of electron gyration is dictated by $\mathbf{B}$ , an axial vector: reverse the propagation direction and the field is unchanged, so a beam retracing its path rotates the same way again — the two passes add. A sugar solution, whose handedness is baked into its molecules relative to the beam, does the opposite: reverse the beam and the twist reverses, so a round trip unwinds to zero. That is the whole difference between a nonreciprocal and a reciprocal rotator, and the figure below makes it the punchline.

Figure

A round trip through a Faraday rotator doubles the rotation to 90° and is blocked; through natural optical activity it cancels to 0° and passes

Because the rotation is simply proportional to the axial field, the whole magneto-optic response collapses into one number — the Verdet constant $V$ — and rotation versus field is a straight line through the origin of slope $Vd$ . That linearity makes the effect a clean tool: measure the slope and you have calibrated the material; measure the rotation of a probe beam and you have measured the field. The next figure shows that line for the workhorse isolator crystal, terbium gallium garnet.

Figure

Faraday rotation is linear in field: β = V·B·d for a TGG rotator, reaching 45° at B = 1 T for a 19.6 mm rod

The core equations

The empirical heart of the subject is one proportionality: rotation equals field times length times a material constant. Faraday measured the proportionality; Verdet, in 1854, pinned down the constant that now carries his name and fixes the sign and size of the response for each material and wavelength.

\beta = V\,B\,d \qquad (1)

where: $\beta$ — angle through which the plane of polarization rotates (rad); $V$ — Verdet constant of the medium (rad·T⁻¹·m⁻¹), wavelength- and temperature-dependent, carrying the sign of the effect; $B$ — magnetic flux density along the propagation direction (T); $d$ — path length through the magnetized medium (m). Only the component of $\mathbf{B}$ parallel to $\mathbf{k}$ enters; a purely transverse field produces no rotation at all. That empirical law comes from something microscopic: circular birefringence. Decompose the linear state into its two circular halves and let each accumulate its own phase over the length $d$ ; the recombined axis bisects the two phases, so the rotation is exactly half the phase difference between the circular components.

\beta = \frac{\omega d}{2c}\,\bigl(n_- - n_+\bigr) \qquad (2)

where: $\omega$ — optical angular frequency; $c$ — vacuum speed of light; $n_\pm$ — indices seen by the right/left circular components. The field hides inside $n_- - n_+$ : with no field the indices are equal and $\beta = 0$ ; an axial $B$ splits them in proportion to itself, so $n_- - n_+ \propto B$ and (2) reproduces $\beta = VBd$ . Everything that makes the effect useful is packed into the return pass, and Jones calculus states it in one line. A rotator is the matrix $R(\beta)$ ; the only question is the sign of $\beta$ on the return. The field fixes the Faraday sense, so both passes add; natural optical activity flips sign with the beam, so its return cancels the forward pass.

R(\beta)\,R(\beta) = R(2\beta) \quad\text{(Faraday)}, \qquad R(\beta)\,R(-\beta) = \mathbb{1} \quad\text{(optical activity)} \qquad (3)

where: $R(\beta)$ — the $2\times2$ SO(2) rotation matrix with entries $\cos\beta$ on the diagonal and $\mp\sin\beta$ off it (determinant $+1$ , real orthogonal); $\mathbb{1}$ — the identity. The single sign difference between the two right-hand sides is the entire distinction between a device that can be one-way and one that cannot. Feed that round-trip doubling into a pair of polarizers and you get the isolator's transmission. Set the rotator to 45° and orient the output polarizer to the forward-rotated light: the forward beam passes in full, while any return is rotated a further 45° to reach the input polarizer crossed at 90°, where Malus's law extinguishes it.

T_{\text{fwd}} = \cos^2(0°) = 1, \qquad T_{\text{bwd}} = \cos^2(90°) = 0 \qquad (4)

where: $T_{\text{fwd}}, T_{\text{bwd}}$ — power transmission forward and backward. Real isolators reach 30–40 dB of backward rejection rather than a perfect zero, limited by polarizer extinction and how exactly the rotation holds at 45° across wavelength and temperature.

Derivation AFaraday rotation from magnetic circular birefringence

Our goal is to turn the empirical law $\beta = VBd$ into physics: to show that a magnetic field makes the two circular polarizations travel at different speeds, that recombining them gives a rotated linear state, and that the rotation is linear in the field because a magnetized electron's resonance shifts in proportion to $B$ . We start from the geometry — a linear state is a sum of circles — then feed in the electron's equation of motion.

Step 1. Write the linear input as an equal-weight sum of circular basis states, $\hat{x} = \tfrac{1}{2}(\hat{e}_+ + \hat{e}_-)$ with $\hat{e}_\pm = \hat{x} \pm i\hat{y}$ . Why: the circular states are the propagation eigenmodes when $\mathbf{B}\parallel\mathbf{k}$ , so this is the basis in which the problem is diagonal.

Step 2. Propagate each circle across the length $d$ with its own index, $E_\pm(d) = E_\pm(0)\,e^{i n_\pm \omega d/c}$ ; recombining, the resultant linear axis has rotated by half the phase difference, $\beta = \tfrac{1}{2}(\phi_- - \phi_+) = \tfrac{\omega d}{2c}(n_- - n_+)$ , which is equation (2). Why: two counter-rotating vectors of equal length always sum to a line, and advancing one relative to the other swings that line by half their relative phase — exactly, not approximately.

Step 3. Find $n_- - n_+$ from the electron's motion. Model each bound electron as a damped oscillator of resonance $\omega_0$ driven by the optical field, with the Lorentz force from $\mathbf{B} = B\hat{z}$ added: $m_e\ddot{\mathbf{r}} + m_e\gamma\dot{\mathbf{r}} + m_e\omega_0^2\mathbf{r} = -e\mathbf{E} - e\,\dot{\mathbf{r}}\times\mathbf{B}$ . In the circular basis $r_\pm = x \pm iy$ the magnetic term is diagonal. Why: the $\dot{\mathbf{r}}\times\mathbf{B}$ coupling mixes $x$ and $y$ motion, and the circular basis — the one the light already splits into — is what undoes the mixing.

Step 4. Solve in steady state; the cyclotron shift $\omega_c = eB/m_e$ moves the effective resonance oppositely for the two handednesses, giving $\varepsilon_\pm = 1 + \dfrac{Ne^2/m_e\varepsilon_0}{\omega_0^2 - \omega^2 \mp \omega\omega_c}$ and $n_\pm = \sqrt{\varepsilon_\pm}$ . Why: one circular sense co-rotates with the electron's magnetic precession and one counter-rotates, so the field pushes their resonances apart — a splitting odd in $B$ .

Step 5. Expand to first order in the small field: the indices behave like the field-free index at Larmor-shifted frequencies, $n_\pm(\omega) \approx n(\omega \mp \omega_L)$ with $\omega_L = eB/2m_e$ , so $n_- - n_+ \approx 2\omega_L\,\dfrac{\mathrm{d}n}{\mathrm{d}\omega} \propto B$ . Equation (2) then gives $\beta = VBd$ and the Becquerel relation $V = -\dfrac{e}{2m_e c}\,\lambda\,\dfrac{\mathrm{d}n}{\mathrm{d}\lambda}$ . Why: the rotation inherits its sign and wavelength dependence from the ordinary dispersion — large near a resonance, small and $\propto 1/\lambda^2$ far from one. *The three key results — the geometric relation $\beta = \tfrac{\omega d}{2c}(n_- - n_+)$ , the split permittivities $\varepsilon_\pm$ with cyclotron shift $\omega_c = eB/m_e$ , and the first-order proportionality $n_- - n_+ \propto B \Rightarrow \beta = VBd$ — are all consistent with one another symbolically.*

Go deeper · Derivation B · 1 minNonreciprocity: why a round trip doubles the rotation

Derivation BNonreciprocity: why a round trip doubles the rotation

The single most important fact about Faraday rotation is that it does not undo itself on the way back, and this derivation isolates why: pure Jones-matrix bookkeeping for a forward pass, a mirror, and a return pass, compared side by side for the field-driven and the chirality-driven rotator. The whole result turns on one sign — whether the return carries $+\beta$ or $-\beta$ — set by the parity of the handedness under $\mathbf{k}\to-\mathbf{k}$ .

Step 1. For the Faraday effect the rotation sense is tied to the axial vector $\mathbf{B}$ , which is unchanged when the propagation direction reverses, so both passes apply $R(+\beta)$ in the fixed laboratory frame. Why: an axial vector is even under the inversion that reversing $\mathbf{k}$ implies, so the medium looks identically handed to the returning beam — nothing about the twist flips.

Step 2. Compose the two passes: $R(+\beta)\,R(+\beta) = R(2\beta)$ . A beam reflected back through a 45° rotator returns at 90°, not 0°. Why: rotations about the same axis add their angles (the SO(2) group law), so identical-sign passes accumulate.

Step 3. For natural optical activity the handedness is fixed by the molecular structure relative to the beam, so it reverses when the beam does: the return applies $R(-\beta)$ , and $R(+\beta)\,R(-\beta) = \mathbb{1}$ brings the polarization home unchanged. Why: the twist is chiral and rides with the propagation direction, so retracing the path retraces the twist — the passes are inverses.

Step 4. Read off the consequence: only the Faraday effect breaks reciprocity, because $\mathbf{B}$ is odd under time reversal. It encodes broken time-reversal symmetry — exactly what a one-way device needs and what no lossless reciprocal element (wave plate, crystal, mirror, chiral medium) can supply. Why: reciprocity theorems forbid a lossless reciprocal network from transmitting differently in the two directions, so an isolator demands a broken symmetry, and the field is the cheapest source of one. *The matrix identities $R(\beta)R(\beta) = R(2\beta)$ and $R(\beta)R(-\beta) = \mathbb{1}$ follow directly from the SO(2) group law, and the 45° round-trip doubling to 90° is their immediate numerical consequence.*

Limiting cases

Case	Result	Meaning
Longitudinal field, $\mathbf{B}\parallel\mathbf{k}$ (Faraday)	$\beta = VBd \propto B$	pure circular birefringence; rotation linear in $B$ and nonreciprocal — the device-relevant geometry
Transverse field, $\mathbf{B}\perp\mathbf{k}$ (Voigt / Cotton–Mouton)	$\Delta n \propto B^2$	magnetically induced linear birefringence, quadratic in $B$ and reciprocal; converts linear to elliptical rather than rotating it
Far from any resonance (transparent glass)	$V$ small, $V \propto 1/\lambda^2$	diamagnetic response tracking normal dispersion; silica $V \approx 4$ rad·T⁻¹·m⁻¹ at 589 nm
Near an atomic resonance	$n_- - n_+$ large and steeply dispersive	resonantly enhanced Verdet constant — the basis of alkali-vapour magnetometers and Faraday filters
Paramagnetic (Curie) medium, e.g. TGG	$V \propto 1/T$ and large	aligned magnetic moments give a big, temperature-dependent $V$ — the ideal isolator crystal
Reflection from a magnetized surface	MOKE: $\theta_K \propto \mathrm{Im}\{r_+ - r_-\}$	rotation and ellipticity imprinted on reflection; images magnetic domains

Common trap“Faraday rotation is just optical activity with a magnet — a round trip should undo it.”

It looks identical going forward: both rotate a linear plane by circular birefringence, and a single pass through either can give the same 45°. The difference shows up only on the return, and it is total. Natural optical activity (recalled by title from Optical Activity) is reciprocal — its handedness rides with the beam, so retracing the path unwinds the twist to 0°. Faraday rotation is nonreciprocal — its handedness is nailed to an external field, so the return adds another +45° and the round trip lands at 90°. Assume the sugar-solution intuition and you will "prove" no isolator can exist. The field breaks the time-reversal symmetry reciprocity assumes; that broken symmetry, not the rotation itself, is what a one-way device is built from. And watch the geometry — only the field component along $\mathbf{k}$ rotates; a transverse field gives the Voigt effect ( $\propto B^2$ ) instead.

Worked examples

1 · Sizing a TGG optical isolator (confidence builder)

A terbium-gallium-garnet (TGG) rod has $V = 40$ rad·T⁻¹·m⁻¹ at 1064 nm and sits in a $B = 1$ T field along its axis. How long must it be to rotate the polarization by the 45° an isolator needs, and what happens to a beam that reflects back through it?

Step 1. Convert the target to radians: $45° = \pi/4 = 0.785$ rad — the rotation the forward beam must acquire so the analyzer can pass it. The isolator's design angle is always 45°, because the return pass doubles it into a blocking 90°.

Step 2. Invert $\beta = VBd$ for the length: $d = \dfrac{\beta}{VB} = \dfrac{0.785}{(40)(1)} = 0.0196\ \mathrm{m} = \mathbf{19.6\ mm}$ . A two-centimetre crystal in a one-tesla field is exactly the scale of a real tabletop isolator.

Step 3. Check the round trip: a beam reflected straight back travels the same 19.6 mm in the same field and picks up another 45°, returning at $45° + 45° = \mathbf{90°}$ — crossed with the input polarizer. Forward it transmits ( $\cos^2 0° = 1$ ); backward it is extinguished ( $\cos^2 90° = 0$ ).

Step 4. The trade-off: fix the length at a stock 25 mm instead and the field drops to $B = 0.785/(40 \times 0.025) = 0.785$ T — still a hefty magnet. Since $V$ is a material given there is no cheap way out, which is why isolators are chunky magnetic cylinders. A 19.6 mm TGG rod at 1 T rotates 45° forward and blocks the doubled 90° return.

2 · Round-trip doubling versus cancellation, in numbers (exam level)

Compare what a mirror-terminated round trip does to a beam through (a) a 45° Faraday rotator and (b) a sugar solution that also rotates 45° per forward pass. Score each with Malus's law at an input polarizer set to 0°.

Step 1. Faraday, forward: input at 0°, rotate +45°, so the beam is at 45° at the mirror — nothing lost yet. The forward pass is identical for both media; the difference is entirely in the return.

Step 2. Faraday, return: the reflected beam re-enters at 45° and, the sense being field-fixed, gains another +45°, exiting at $45° + 45° = 90°$ . At the 0° input polarizer, $T = \cos^2 90° = \mathbf{0}$ — total extinction. Round-trip rotation $2\beta = 90°$ .

Step 3. Sugar solution, return: the beam re-enters at 45°, but a chiral medium's sense reverses with the beam, so the return applies $-45°$ , back to $45° - 45° = 0°$ . At the 0° polarizer, $T = \cos^2 0° = \mathbf{1}$ — full transmission. Round-trip rotation $\beta - \beta = 0°$ .

Step 4. The moral: the same 45° single-pass rotation gives opposite round-trip results — 90° and total block for Faraday, 0° and free return for the sugar — solely because of the sign the return carries. An isolator must be a magneto-optic rotator; no reciprocal stack can fake it. Faraday round trip: 90°, blocked; optical activity: 0°, passed.

Physics of the portrait

The portrait here is a phenomenon, not a person — faraday rotation — because the twist itself became a tool that reshaped optics. Faraday's 1845 measurement was a philosophical bombshell (light responds to magnetism!) long before it was a technology; the technology waited on Verdet's constant and, finally, on the laser, which needed a one-way valve and found one ready-made in a magnetized crystal.

Origins · 1 minfaraday rotation and how this was discovered

faraday rotation

discovered 1845 · quantified by Verdet 1854 · magneto-optics and nonreciprocal devices

Faraday rotation is the magnetically induced twist of a light beam's polarization plane, and its history is a three-act play. On 13 September 1845, Michael Faraday — bookbinder's apprentice turned the greatest experimentalist of his century — placed a bar of dense lead-borosilicate (“heavy”) glass between the poles of an electromagnet, sent polarized light along the field, and watched the plane rotate. He grasped its weight at once, writing that he had "magnetised and electrified a ray of light": the first direct link between electromagnetism and optics, feeding straight into Maxwell's unification. In the second act Émile Verdet, in the 1850s, made the effect quantitative — establishing $\beta = VBd$ and measuring the constant that bears his name across materials and wavelengths. The third act neither man could foresee: the effect's nonreciprocity would become indispensable a century later, guarding every high-power laser and coherent link with a terbium-garnet valve that lets light out but never back in.

History

1845

— Faraday sends polarized light through dense lead-borosilicate (“heavy”) glass in a magnetic field and sees the plane rotate — the "magnetisation of light" and the first bridge between optics and electromagnetism

1854

— Verdet makes the effect quantitative, establishing β = V·B·d and measuring the Verdet constant across materials and wavelengths

1877

— John Kerr discovers the magneto-optic Kerr effect (MOKE): polarization rotates on reflection from a magnetized surface

1898

— Woldemar Voigt describes magneto-optic birefringence for a transverse field — the Voigt effect, quadratic in B

1992

— magneto-optical recording reaches consumers in the MiniDisc, reading stored bits by the Kerr rotation of reflected light

Going further

Texts. Hecht, Optics, 5th ed., §8.11 (optical activity and the Faraday effect) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §6.4 (magneto-optics and isolators) · Jackson, Classical Electrodynamics, 3rd ed., §7.6 (waves in a magnetized medium) · Foot, Atomic Physics, Ch. 7 (Zeeman effect and magneto-optical phenomena) · Loudon, The Quantum Theory of Light, 3rd ed. (dispersion and circular birefringence) · Demtröder, Laser Spectroscopy, Vol. 1 (Faraday/Voigt spectroscopy and Faraday filters). Key anchor values: silica $V \approx 4$ rad·T⁻¹·m⁻¹ at 589 nm and TGG $V \approx 40$ rad·T⁻¹·m⁻¹ at 1064 nm. The magnetized Lorentz-oscillator derivation gives circular indices with cyclotron shift $\omega_c = eB/m_e$ , and the Becquerel formula $V = -(e/2m_e c)\,\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ is the standard diamagnetic result. The Jones-matrix nonreciprocity identity $R(\beta_1)R(\beta_2) = R(\beta_1+\beta_2)$ underlies the 45°-isolator extinction analysis.

Prerequisites. Circular Polarization · Linear Polarization · Birefringence · Jones Calculus · Polarizers and Extinction Ratio. Leads to. Mueller Calculus · Stokes Parameters and the Poincaré Sphere · Atomic Magnetometers · Optical Isolators and Circulators.

Is Faraday rotation the same thing as natural optical activity?

Only in the forward direction. Both rotate a linear plane by circular birefringence, and a single pass through either can rotate by 45°. The difference is reciprocity. Natural optical activity's handedness is fixed by the molecules relative to the beam, so it reverses when the beam reverses and a round trip cancels to 0°. Faraday rotation's handedness is fixed by an external field that ignores the beam direction, so a round trip adds, reaching 90°. That is why only the Faraday effect can build a one-way device — the two are identical on a single pass and physically worlds apart on the return.

Why must the magnetic field be along the direction the light travels?

Because rotation comes from circular birefringence, and the circular polarizations are propagation eigenmodes only when the field is parallel to $\mathbf{k}$ . There the field splits $n_+$ from $n_-$ in proportion to itself, giving $\beta = VBd$ ; tilt the field and only its component along $\mathbf{k}$ counts. A purely transverse field gives no rotation at all — it induces linear birefringence instead, the Voigt (Cotton–Mouton) effect, quadratic in $B$ , which turns linear light elliptical.

Why are Faraday isolators always so bulky and magnetic?

The numbers force it. The rotation needed is a fixed 45°, and $\beta = VBd$ leaves only two knobs — the length $d$ and the field $B$ — since $V$ is a material constant. Even for terbium gallium garnet, one of the strongest transparent Verdet materials ( $V \approx 40$ rad·T⁻¹·m⁻¹), 45° takes about a two-centimetre rod in a one-tesla field. Trade a longer crystal for a weaker magnet if you like, but you cannot escape the product $VBd = 45°$ — a compact isolator is inescapably a dense cylinder of glass and magnet.

What sets the sign and size of the Verdet constant?

The material's ordinary dispersion. The Becquerel relation, $V = -(e/2m_e c)\,\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ , ties $V$ to how steeply the index changes with wavelength: near a resonance the index varies fast and $V$ is large; in a clear glass far from resonance it is small and smooth, roughly $\propto 1/\lambda^2$ . The sign follows the dispersion — why some materials rotate one way and some the other — and paramagnetic terbium compounds, their aligned moments boosting the response, give the large Verdet constants isolators exploit.

Does the magneto-optic Kerr effect (MOKE) rotate light the same way?

It is the reflection relative of the Faraday effect. Where Faraday rotation accumulates as light transmits through a magnetized medium, MOKE appears when light reflects from a magnetized surface: the two circular components acquire different complex reflection coefficients, $r_+ \neq r_-$ , whose difference imprints a small rotation and ellipticity on the reflected beam. Because that rotation tracks the surface magnetization, MOKE images magnetic domains, measures thin-film hysteresis loops, and once read magneto-optical discs; its polar, longitudinal, and transverse geometries probe different magnetization components.

What is the factor of two between polarization rotation and the Poincaré sphere?

On the Poincaré sphere orthogonal polarizations sit at antipodes, so the sphere maps polarization ellipses through double angles: a physical rotation by $\beta$ is a $2\beta$ turn of the Stokes vector. Faraday birefringence rotates the Stokes vector about the polar $S_3$ axis, spinning linear states around the equator; linear birefringence — the Voigt effect — rotates about an equatorial axis instead, converting linear to elliptical. Keeping the factor of two straight is the usual snag when moving between the two pictures.