Optical Activity

Dissolve a spoon of sugar in water, stand the glass between two polarizing filters, and something quietly impossible happens: the second filter, set to extinguish the light, now leaks. The sugar solution has rotated the plane of polarization on its way through — twist the analyser a few degrees and the darkness returns. Nothing absorbed the light selectively, nothing scattered it; the beam simply came out vibrating along a new axis. A quartz crystal cut the right way does the same, and so do the amino acids in your own body, every one of them turning light the same way because life on Earth built itself from a single handedness. This is optical activity, and its secret is that the natural modes of a chiral medium are not the familiar horizontal and vertical linear vibrations but the two circular polarizations, left and right. A chiral structure — one not superimposable on its mirror image, like a helix or a screw — responds to left- and right-handed light differently, giving them slightly different refractive indices. Since any linear vibration is an equal blend of the two circular states, the faster one races a little ahead, and when the pair recombine at the exit face the resultant still points along a single line, only rotated. The rotation grows steadily with path length, climbs with concentration for a dissolved substance (Biot's law), and generally increases toward shorter wavelengths. The whole effect is nothing more exotic than a difference of two refractive indices, read out as an angle.

Key takeaways

Optical activity is the rotation of the plane of linear polarization as light traverses a chiral medium — one not superimposable on its mirror image (quartz, sugars, amino acids).
Its microscopic engine is circular birefringence: left- and right-circular light see different indices $n_L \neq n_R$ , accumulate different phases, and their sum emerges rotated.
The rotation angle is $\phi = \tfrac{\pi d}{\lambda_0}(n_L - n_R)$ — linear in path length, and for solutions linear in concentration through the specific rotation $[\alpha]$ (Biot's law).
The two mirror-image forms (enantiomers) rotate light equally but oppositely; a 50:50 racemic mixture is optically inactive.
Optical activity is reciprocal — retrace the path and the rotation undoes — which distinguishes it sharply from magnetic Faraday rotation.
Key idea: in a transparent chiral medium, optical activity arises from circular birefringence — the two circular polarizations are the propagation eigenmodes, $n_L \neq n_R$ shifts their relative phase, and recombining them turns the linear axis by $\phi = \tfrac{\pi d}{\lambda_0}(n_L - n_R)$ , a pure rotation, not an ellipse.

The core insight

A chiral medium has a handedness in space, so it cannot treat left- and right-circular light alike: they are its two eigen-polarizations, propagating unchanged in form but with different indices $n_L \neq n_R$ . A linear state is an equal superposition of the two, so the phase they accumulate differently re-adds to a linear state rotated by $\phi = \tfrac{\pi d}{\lambda_0}(n_L - n_R)$ — half the phase difference. The rotation is the same for every incident azimuth, the fingerprint of a rotator rather than a wave plate.

Where this lives today

The sugar polarimeter is still a workhorse of the food and pharmaceutical industries: a single angle at the sodium line, read on a calibrated tube, gives sucrose content to a fraction of a percent. More consequentially, optical rotation and its spectroscopic partner circular dichroism are standard probes of molecular chirality in chemistry and biology — a protein's circular-dichroism spectrum can be used to estimate its α-helix, β-sheet, and other secondary-structure content, and pharmaceutical regulators require enantiopurity measurements because different enantiomers of a chiral drug can have very different biological effects. Crystalline-quartz rotators are compact, rugged polarization-turning elements in laser systems, and stacked quartz wedges scramble a beam's azimuth to act as pseudo-depolarizers where a residual polarization would bias a measurement. Even the search for extraterrestrial life leans on the idea: a strong single-handed optical rotation in an alien sample would be a chemical signature that something selected one enantiomer, as life on Earth did.

The physical picture

Write the incoming linear vibration as the sum of two equal circular vectors spinning in opposite senses. On their own each traces a perfect circle; added tip-to-tail they trace a straight line, because their sideways wobbles cancel and only the shared vertical survives. Now let one of them — say the left-circular component, for a chosen handedness convention — travel a little slower through the medium. After a length $d$ the two circles have wound through slightly different total angles, so the direction in which their sum lines up has swung round. The resultant is still a straight line; it has simply rotated, by exactly half the phase difference the two circular waves picked up. No ellipticity appears as long as the medium has no circular dichroism, because the two circular components retain equal amplitudes and differ only in phase.

Figure

Linear light as two counter-rotating circular vectors: the left-circular lags, so the recombined linear axis swings by φ = ½(phase difference)

Chirality is what makes the two circular states inequivalent. A helix looks different from its mirror image, and a screw-shaped electron cloud responds differently to a field that curls one way versus the other — exactly as a wood screw bites when twisted clockwise but backs out when twisted the other way. Left- and right-circular light are fields that curl in opposite senses, so a chiral molecule presents them with two different optical environments and hence two indices. A mirror-symmetric molecule has no such preference, $n_L = n_R$ , and no rotation. This is why the effect is a direct probe of molecular handedness, and why a racemic mixture — equal populations of the two mirror forms — cancels exactly.

Figure

Quartz rotatory dispersion: rotatory power ρ rises toward the blue as 1/(λ²−λ₀²), pinned at 21.72°/mm and n_L−n_R = 7.11×10⁻⁵ at the sodium D line

The second figure turns the whole spectrum into one curve. Quartz rotates blue light more than red — its rotatory power climbs from about $12.7^\circ/\mathrm{mm}$ in the near infrared to nearly $49^\circ/\mathrm{mm}$ in the violet — because the circular-birefringence difference $n_L-n_R$ is strongly wavelength-dependent. Over a suitable transparent spectral range, its dispersion can be described by a Drude-type form similar to $1/(\lambda^2-\lambda_0^2)$ . At the sodium D line ( $589.3$ nm) the rotation is $21.72^\circ/\text{mm}$ , which corresponds to an index difference of only $7.11\times10^{-5}$ : a difference in the fifth decimal place, made visible because it accumulates as a phase over thousands of wavelengths.

The core equations

Optical activity begins with a single, spectacular claim: that a rotation of the polarization plane is entirely accounted for by two circular waves travelling at slightly different phase velocities. The rotation angle follows from decomposing the linear input into circular parts, propagating each with its own index, and recombining — the common phase is irrelevant to polarization, while the rotation is half the accumulated relative phase.

\phi = \frac{\pi d}{\lambda_0}\,(n_L - n_R) \qquad (1)

where: $\phi$ — rotation of the linear polarization plane (rad); $d$ — path length through the medium; $\lambda_0$ — vacuum wavelength; $n_L,\,n_R$ — indices for left- and right-circular polarization. The sign of $\phi$ carries the handedness: a dextrorotatory (d, +) sample turns the axis clockwise looking toward the source, a levorotatory (l, −) one anticlockwise. For a dissolved substance the index difference is proportional to how many active molecules the light meets, so the rotation grows linearly with both path and concentration. Polarimetry packages this in the specific rotation, a tabulated material constant measured at a fixed wavelength and temperature.

\alpha = [\alpha]_\lambda^{T}\, \ell\, c \qquad (2)

where: $\alpha$ — measured rotation (degrees); $[\alpha]_\lambda^{T}$ — specific rotation (deg mL g $^{-1}$ dm $^{-1}$ ) at wavelength $\lambda$ , temperature $T$ ; $\ell$ — path length in decimetres (the customary polarimetry unit); $c$ — mass concentration (g mL $^{-1}$ ). This is Biot's law, and its linearity in $c$ is what makes the polarimeter an assay instrument. A rotator acts on every input azimuth the same way — it advances the polarization angle by $\phi$ whatever direction it started in. That is a genuine rotation in the Jones (electric-field) plane, mathematically distinct from a wave plate.

\mathbf{J}_{\text{rot}}(\phi)=\begin{pmatrix}\cos\phi & -\sin\phi\\ \sin\phi & \cos\phi\end{pmatrix} \qquad (3)

where: $\phi$ — the rotation imposed on the polarization plane. This matrix is real, orthogonal, with determinant $+1$ : an element of the rotation group SO(2). A wave plate, by contrast, is $\mathrm{diag}(1,e^{i\Gamma})$ — unitary but not real — and turns most linear inputs into ellipses. Confusing the two is the classic error the derivations below settle. Finally, the microscopic origin: chirality shows up in the constitutive relation as a term that couples the induced dipole to the curl of the field, i.e. to the wavevector. This spatial-dispersion term is what splits the two circular indices.

\mathbf{D} = \varepsilon_0\varepsilon\,\mathbf{E} + i\,\varepsilon_0\,\gamma\,(\mathbf{k}\times\mathbf{E}), \qquad n_\pm^2 = \varepsilon \pm \gamma k \qquad (4)

where: $\epsilon$ — ordinary (local) relative permittivity; $\gamma$ — the gyration parameter, odd under parity (it changes sign for the mirror-image structure); $\mathbf{k}$ — wavevector; $n_\pm$ — the two circular indices. Because $\gamma k \propto 1/\lambda$ , the index difference $n_L - n_R \simeq \gamma k/n$ grows toward short wavelengths — the rotatory dispersion of the punchline figure.

Derivation ARotation of the polarization plane by circular birefringence

The goal is to prove, from the superposition of two circular waves, that a difference in their real refractive indices rotates a linear polarization by a definite angle and — crucially — leaves it linear when the two circular components experience equal attenuation. We take a wave along $z$ with linear input $\mathbf{E}(0)=E_0\hat{x}$ , express it in the circular basis that diagonalizes the medium, propagate each circular eigenmode with its own real index, and reassemble. Every step is exact in the transparent limit.

Step 1. Write the circular basis $\hat{e}_\pm=\tfrac{1}{\sqrt2}(\hat{x}\pm i\hat{y})$ , so that the linear input decomposes as $\hat{x}=\tfrac{1}{\sqrt2}(\hat{e}_++\hat{e}_-)$ — an equal blend of right ( $+$ ) and left ( $-$ ) circular. Why: the eigen-polarizations of a chiral medium are the circular states, so we must work in the basis where propagation is diagonal (each mode just picks up a phase).

Step 2. Propagate a distance $d$ : each circular mode advances by $k_0 n_{R,L} d$ with $k_0 = 2\pi/\lambda_0$ , giving $\mathbf{E}(d)=\tfrac{E_0}{\sqrt2}\big(e^{ik_0 n_R d}\hat{e}_+ + e^{ik_0 n_L d}\hat{e}_-\big)$ . Why: in the diagonal basis propagation is trivial — the whole physics is that the two phases $k_0 n_R d$ and $k_0 n_L d$ differ.

Step 3. Factor out the mean phase $\bar\varphi=\tfrac12 k_0(n_L+n_R)d$ and define $\phi=\tfrac12 k_0(n_L-n_R)d$ , so $\mathbf{E}(d)=\tfrac{E_0}{\sqrt2}e^{i\bar\varphi}\big(e^{-i\phi}\hat{e}_+ + e^{+i\phi}\hat{e}_-\big)$ . Why: an overall phase $\bar\varphi$ cannot affect the polarization direction, so only the antisymmetric part $\pm\phi$ can rotate the axis — already the rotation is half the phase difference.

Step 4. Convert back to linear coordinates using $e^{-i\phi}\hat{e}_+ + e^{+i\phi}\hat{e}_- = \sqrt2(\cos\phi\,\hat{x}+\sin\phi\,\hat{y})$ , so $\mathbf{E}(d)\propto \cos\phi\,\hat{x}+\sin\phi\,\hat{y}$ with $\phi = \tfrac12 k_0(n_L-n_R)d = \tfrac{\pi d}{\lambda_0}(n_L-n_R)$ . Why: the result is manifestly a real linear vector at azimuth $\phi$ — no imaginary (out-of-phase) component survives, so the light stays linear and the plane has simply turned. *As a check: substituting $k_0=2\pi/\lambda_0$ into $\phi=\tfrac12 k_0(n_L-n_R)d$ reproduces Eq. (1) directly; and the quartz anchor $\phi/d = \pi(n_L-n_R)/\lambda_0 = 379.1\,\text{rad/m} = 21.72^\circ/\text{mm}$ with $n_L-n_R = 7.11\times10^{-5}$ is consistent with this result.*

Go deeper · Derivation B · 1 minWhy a rotator is not a wave plate

Derivation BWhy a rotator is not a wave plate

Students routinely conflate optical rotation with the action of a wave plate, because both alter polarization. They are algebraically different objects, and the difference has an experimental signature. The aim here is to show that the optical-rotation operator is a real SO(2) rotation that turns every linear input by the same angle and keeps it linear, whereas a retarder keeps only its own axes linear and ellipticizes the rest — a test you can run on a bench in a minute.

Step 1. Act with the rotator on a general linear input at azimuth $\theta$ : $\mathbf{J}_{\text{rot}}(\phi)\,(\cos\theta,\sin\theta)^{\mathsf T} = (\cos(\theta+\phi),\sin(\theta+\phi))^{\mathsf T}$ . Why: the output is again a real unit vector, i.e. linear light at azimuth $\theta+\phi$ — the same rotation for all $\theta$ , which is the defining behaviour of optical activity.

Step 2. Act with a retarder $\mathbf{J}_{\text{ret}}=\mathrm{diag}(1,e^{i\Gamma})$ on the same input: $(\cos\theta,\,e^{i\Gamma}\sin\theta)^{\mathsf T}$ . Why: the two components now carry a relative phase $\Gamma$ , so the output is generally elliptical; only $\theta=0$ or $\pi/2$ (along the plate axes) stay linear and unrotated.

Step 3. Classify by symmetry: $\det\mathbf{J}_{\text{rot}}=\cos^2\phi+\sin^2\phi=+1$ and $\mathbf{J}_{\text{rot}}^{\mathsf T}\mathbf{J}_{\text{rot}}=\mathbb{1}$ , a real orthogonal (SO(2)) operator with circular eigenvectors; $\mathbf{J}_{\text{ret}}$ is unitary but complex, with linear eigenvectors. Why: on the Poincaré sphere the rotator spins the sphere about the polar (circular) axis, sliding a point around the equator, while the retarder spins it about an equatorial axis, lifting the point off the equator into ellipticity — geometrically opposite motions. *As a check: $\det\mathbf{J}_{\text{rot}}=1$ and $\mathbf{J}_{\text{rot}}^{\mathsf T}\mathbf{J}_{\text{rot}}=\mathbb{1}$ confirm the SO(2) structure symbolically, and the contrast between rotated-linear outputs (rotator) and elliptical outputs (retarder) follows directly from Steps 1–2.*

Limiting cases

Case	Result	Meaning
Transparent, off-resonance	$\phi = \tfrac{\pi d}{\lambda_0}(n_L-n_R)$ , both indices real	emergent light stays linear, axis rotated — the polarimetry regime
Racemic / achiral medium	$n_L = n_R \Rightarrow \phi = 0$	equal enantiomer populations, or any mirror-symmetric medium, cancel exactly
Dilute solution (Biot)	$\phi = [\alpha]_\lambda^{T}\,\ell\,c$	rotation linear in path and concentration — the basis of the saccharimeter
Short wavelength ( $\lambda \to \lambda_0^+$ )	$n_L - n_R$ and $\phi$ grow as $1/(\lambda^2-\lambda_0^2)$	rotatory dispersion: blue turns far more than red
Retrace the path	net $\phi \to 0$	optical activity is reciprocal — unlike Faraday rotation, it undoes on the return trip

Common trap“Optical activity turns linear light into elliptical light, like a wave plate”

It does not — in a transparent medium optical rotation keeps the light strictly linear and merely turns its axis, because the two circular eigenmodes stay equal in amplitude and only drift in phase. A wave plate does the opposite: it holds two linear components and delays one, so most inputs come out elliptical. The bench test is decisive. Rotate your input polarizer through a full turn: past a rotator, the transmitted light stays linear and the extinction angle simply shifts by the same fixed $\phi$ at every setting; past a wave plate, the output ellipticity changes with input angle and you cannot fully extinguish it except along the plate's axes. Ellipticity can accompany optical activity whenever the two circular components experience different attenuation, i.e. circular dichroism; this often becomes important near an absorption band, but it is not restricted to that case.

Worked examples

1 · A quartz rotator and the fifth-decimal index difference (confidence builder)

A plate of crystalline quartz is cut with its faces perpendicular to the optic axis and used at the sodium D line ( $\lambda_0 = 589.3$ nm), where quartz has specific rotatory power $21.72^\circ$ per millimetre. Find the rotation produced by a $5.00$ mm plate, and the refractive-index difference between the two circular polarizations that produces it.

Step 1. Rotation scales with thickness: $\phi = \rho\,d = 21.72^\circ/\text{mm} \times 5.00\,\text{mm} = \mathbf{108.6^\circ}$ . A centimetre of quartz would turn the plane by more than two right angles.

Step 2. Convert the rotatory power to SI: $\rho = 21.72^\circ/\text{mm} = 21.72 \times (\pi/180)/10^{-3} = \mathbf{379.1\ \text{rad/m}}$ .

Step 3. Invert Eq. (1) written per unit length, $\rho = \pi(n_L-n_R)/\lambda_0$ , for the index difference: $n_L - n_R = \rho\,\lambda_0/\pi = 379.1 \times 589.3\times10^{-9}/\pi = \mathbf{7.11\times10^{-5}}$ .

Step 4. Read the lesson: an index difference of seven parts in $10^5$ — difficult to resolve in an ordinary single-pass refraction measurement — produces a $108.6^\circ$ swing over 5 mm because the rotation accrues $\pi(n_L-n_R)/\lambda_0$ radians per metre, amplifying a tiny index difference over many optical wavelengths. A 5 mm quartz plate rotates by 108.6°, from $n_L - n_R = 7.11\times10^{-5}$ .

2 · Saccharimetry: reading a sugar concentration off an angle (exam level)

A $1.00$ dm polarimeter tube is filled with a sucrose solution of concentration $0.100$ g mL $^{-1}$ . Sucrose has specific rotation $[\alpha]_D^{20} = +66.5^\circ$ mL g $^{-1}$ dm $^{-1}$ . What rotation does the polarimeter read, and how would the reading change if the tube were doubled in length or the syrup diluted by half?

Step 1. Apply Biot's law, Eq. (2), directly: $\alpha=[\alpha]^{20}_{D}\ell c=66.5\times1.00\times0.100=+6.65^\circ$ , i.e. dextrorotatory; the clockwise/anticlockwise description depends on the viewing convention.

Step 2. The law is linear in both factors: a $2.00$ dm tube would read $13.3^\circ$ ; halving the concentration to $0.050$ g mL $^{-1}$ would read $3.33^\circ$ . Two unknowns (path and concentration) trade off linearly, which is why the tube length is fixed and calibrated so a single angle gives the concentration.

Step 3. Sanity-check the sign convention against a racemate: an equal mix of sucrose with its mirror image would give $n_L=n_R$ and read $0^\circ$ ; the fact that natural sugar reads a clean $+6.65^\circ$ is the statement that biology delivers a single enantiomer.

Step 4. In practice sugar refineries read $\alpha$ at the D line on a temperature-controlled tube and invert Biot's law to report the concentration to a fraction of a percent — the polarimeter is a chirality-selective concentration meter. The tube reads +6.65°, scaling linearly with both length and concentration.

Physics of the portrait

Optical activity is the readout of a deeper fact: that many of nature's molecules come in left- and right-handed versions, and that living matter, uniquely, chose one hand. The concept that organizes it all is chirality.

Origins · 1 minChirality and how this was discovered

Chirality

handedness in molecules and crystals · stereochemistry & optics

Chirality — from the Greek cheir, "hand" — is the property of a structure that cannot be superimposed on its mirror image, the way a left hand cannot be slipped into a right glove. François Arago first saw its optical consequence in 1811, when a quartz plate rotated polarized light; Jean-Baptiste Biot showed in 1815 that liquids and vapours of organic substances did the same, and that the rotation followed a simple law of path length and concentration. But the human meaning of the effect was pried open by Louis Pasteur in 1848. Working with racemic sodium ammonium tartrate, Pasteur noticed under the microscope that its crystals occurred in two mirror-image forms; he separated them by hand with tweezers, dissolved the two fractions separately, and found that one rotated the plane of polarization to the right and the other equally to the left — the first physical separation of two enantiomers, and the birth of stereochemistry. A quarter-century later Jacobus van 't Hoff and Joseph Le Bel (1874) explained the handedness by placing the carbon atom's four bonds at the corners of a tetrahedron. Chirality now underwrites the fact that nearly all biological amino acids are "left-handed" and natural sugars "right-handed," so that a mirror-image organism could interact very differently with ordinary biomolecules — and that the two enantiomers of a drug can have very different biological effects.

History

1811

— Arago discovers optical rotation in a quartz plate cut across the optic axis

1815

— Biot finds rotation in organic liquids and vapours and states his law (∝ path, concentration, and roughly 1/λ²)

1822

— Fresnel explains rotation as circular birefringence, two circular waves of unequal speed

1848

— Pasteur separates the two mirror-image tartrate crystals by hand, isolating enantiomers

1874

— van 't Hoff and Le Bel place the four carbon bonds tetrahedrally, explaining molecular chirality

Going further

Texts. Hecht, Optics, 5th ed., §8.5 (Optical Activity) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 6 (Polarization and Crystal Optics) · Born & Wolf, Principles of Optics, §15 (optical activity and the gyration tensor) · Landau & Lifshitz, Electrodynamics of Continuous Media, §101 (spatial dispersion, natural optical activity) · Barron, Molecular Light Scattering and Optical Activity, 2nd ed. Three results anchor the article: (1) $\phi = \pi d(n_L-n_R)/\lambda_0$ from the circular-decomposition derivation; (2) the rotator Jones matrix is a real SO(2) rotation obtained by diagonalizing in the circular basis; (3) the spatial-dispersion relation $\mathbf{D}=\varepsilon_0\varepsilon\mathbf{E}+i\varepsilon_0\gamma(\mathbf{k}\times\mathbf{E})$ giving $n_\pm^2=\varepsilon\pm\gamma k$ . The quartz value $21.72^\circ/\text{mm}$ at $589.3$ nm and the sucrose specific rotation $+66.5^\circ$ at $20^\circ$ C are standard tabulated constants. Index note: the fifth-decimal difference $n_L-n_R = 7.11\times10^{-5}$ is computed from the quoted rotatory power via Eq. (1), not measured independently.

Prerequisites. Circular Polarization · Linear Polarization · Birefringence · Jones Calculus. Leads to. Circular and Linear Dichroism · Faraday and Magneto-Optical Effects · Wave Plates and Retarders.

If the two circular waves just accumulate phase, why doesn't the light come out circular or elliptical?

Because their amplitudes stay equal — only their phases drift apart. A pair of equal-amplitude, opposite-handed circular waves always add up to a linear vibration; changing their relative phase only rotates the axis of that vibration, it never unbalances them. Ellipticity would require the two circular amplitudes to become unequal, which happens only if the medium absorbs them differently. In a transparent, off-resonance sample there is no such absorption, so the output is rigorously linear and merely rotated. Push the wavelength toward an absorption band and unequal absorption does appear — that is circular dichroism, and there the light does emerge slightly elliptical.

Is optical rotation the same thing as the birefringence of a wave plate?

No, and the distinction is the heart of the subject. A wave plate is linearly birefringent: its eigen-polarizations are two orthogonal linear states, one fast and one slow, and it converts most linear inputs into ellipses. An optically active medium is circularly birefringent: its eigen-polarizations are the two circular states, and it rotates the plane of any linear input by the same angle while keeping it linear. Mathematically one is $\mathrm{diag}(1,e^{i\Gamma})$ and the other is a real rotation matrix; on the Poincaré sphere one turns the sphere about an equatorial axis, the other about the polar axis. They can even coexist — a real quartz plate cut off-axis shows both — but they are physically distinct effects.

Why does the rotation depend so strongly on wavelength?

The index difference $n_L-n_R$ that drives the rotation is itself dispersive: it grows as the wavelength approaches the medium's ultraviolet electronic resonances, roughly as $1/(\lambda^2-\lambda_0^2)$ , the same resonant enhancement that makes ordinary refractive index rise toward the blue. On top of that, Eq. (1) carries an explicit $1/\lambda_0$ out front. The two effects compound, so quartz rotates violet light nearly four times as strongly as near-infrared. This wavelength dependence — optical rotatory dispersion — is itself a useful fingerprint: its detailed shape near an absorption band (the Cotton effect) pins down molecular structure.

What actually makes a molecule chiral at the level of the electrons?

Chirality is a geometric fact: the molecule has no improper symmetry element — no mirror plane, no inversion centre, no rotation-reflection axis — so it is not superimposable on its mirror image. The commonest source is a carbon atom bonded to four different groups, whose tetrahedral arrangement comes in two mirror-image versions. Optically, this handedness lets the electric field's spatial variation (its curl, hence the wavevector) drive an induced dipole, the $\mathbf{k}\times\mathbf{E}$ term in Eq. (4). A mirror-symmetric molecule cannot support such a term because it would change sign under the molecule's own mirror symmetry and therefore must vanish. Handedness in the structure is exactly what allows the parity-odd optical response.

How does a polarimeter actually measure the rotation to a fraction of a degree?

Rather than eyeballing an extinction, modern polarimeters modulate the analyser (or use a Faraday-cell "null" servo) and lock onto the exact angle at which the transmitted intensity is minimized. Because intensity varies as $\sin^2$ of the offset from extinction, the signal is very sensitive right at the null, and averaging over a modulation cycle beats down noise. The instrument is temperature-controlled because $[\alpha]$ drifts with $T$ , and it works at a fixed spectral line (traditionally sodium D) because $[\alpha]$ is strongly wavelength-dependent. With these controls a good polarimeter resolves millidegrees, enough to assay sugar syrups and to detect small enantiomeric imbalances.

Does the direction of rotation depend on which way the light travels through the sample?

No — and this reciprocity is the key difference from magnetic (Faraday) rotation. In natural optical activity the sense of rotation is tied to the medium's handedness, not to the propagation direction, so if you send the beam back through the same sample it retraces the rotation and undoes it: a round trip nets zero. Faraday rotation, driven by an external magnetic field, is non-reciprocal — the sense is fixed by the field, so a round trip doubles the rotation. That distinction, explored in the companion article on magneto-optics, is exactly what makes a Faraday cell (and not a sugar solution) usable as a one-way optical isolator.