Circular and Linear Dichroism

An optically active sugar solution far from any colour band does one thing to light: it rotates the plane of polarization and leaves it strictly linear. Move the wavelength into an absorption band — where the molecule actually swallows photons — and something new switches on. Now the medium not only rotates the plane but also absorbs the two circular polarizations by different amounts, so the beam emerges faintly elliptical. That differential absorption is circular dichroism, and it is the absorptive twin of optical rotation: where rotation comes from a difference in the real parts of the two circular indices, dichroism comes from a difference in their imaginary (absorptive) parts. The two effects are not independent. Rotation as a function of wavelength — optical rotatory dispersion — and circular dichroism are the real and imaginary parts of a single complex quantity, tied together by the Kramers–Kronig relations exactly as ordinary refraction and absorption are. Near an electronic transition they appear together as a characteristic paired signature: the rotation traces an anomalous S-shaped curve straddling the band while the dichroism shows a peak centred on it. Aimé Cotton first mapped this pattern in 1895, and it now carries his name — the Cotton effect. Its sign is fixed by the molecule's handedness, which is why circular dichroism is the standard fingerprint of chirality in biochemistry. Alongside this circular version sits a simpler, more everyday cousin: linear dichroism, the unequal absorption of two orthogonal linear polarizations, which is exactly how a Polaroid sheet works and how aligned molecules reveal the direction of their transition dipoles.

Key takeaways

Circular dichroism (CD) is the unequal absorption of left- and right-circular light by a chiral medium; it turns emergent linear light slightly elliptical.
CD is the absorptive partner of optical rotation: rotation comes from $n_L \neq n_R$ (real indices), CD from $\kappa_L \neq \kappa_R$ (absorptive indices).
They combine into one complex rotation $\Phi = \phi + i\eta$ ; rotatory dispersion (ORD) and CD are Kramers–Kronig partners, carrying the same information.
Near an absorption band the paired ORD S-curve and CD peak form the Cotton effect, its sign set by the enantiomer.
The induced ellipticity is $\theta = \tfrac{\ln 10}{4}\,\Delta A = 32.98^\circ \times \Delta A$ (with $\Delta A = A_L - A_R$ ).
Key idea: dichroism is differential absorption — circular dichroism ( $\kappa_L \neq \kappa_R$ ) makes linear light elliptical and pairs with optical rotation as the imaginary part of a complex rotation; linear dichroism ( $\alpha_x \neq \alpha_y$ ) is the Malus-like attenuation behind Polaroid.

The core insight

Where optical rotation lives in the real index difference $n_L - n_R$ , dichroism lives in the absorptive difference. Circular dichroism, $\kappa_L \neq \kappa_R$ , absorbs the two circular states unequally and so emits an ellipse; it is the imaginary part of a complex rotation $\Phi = \tfrac{\pi d}{\lambda_0}(\Delta n + i\,\Delta\kappa)$ , tied to rotatory dispersion by Kramers–Kronig. Linear dichroism, $\alpha_x \neq \alpha_y$ , is the simpler unequal absorption of two linear axes — Malus's law with absorption doing the projecting.

Where this lives today

Circular dichroism is one of biochemistry's workhorses. A protein's far-ultraviolet CD spectrum — dominated by the peptide backbone's $n\to\pi^*$ and $\pi\to\pi^*$ transitions near $190$ – $220$ nm — has a shape that reports directly on how much of the chain is coiled into $\alpha$ -helix versus folded into $\beta$ -sheet, so CD is used to watch proteins fold, denature, and bind partners in real time. Pharmaceutical labs use CD and ORD to assign the absolute configuration of chiral drugs and to confirm enantiopurity. Linear dichroism, meanwhile, is everywhere you look through a polarizer: Polaroid sheets, LCD screens, and polarized sunglasses are engineered strong linear dichroism, while measured linear dichroism on flow-aligned DNA or membrane-bound dyes reveals how transition dipoles — and therefore molecules — are oriented. Magnetic circular dichroism, CD switched on by an external field, adds a probe of electronic and magnetic structure in materials.

The physical picture

Start with the picture from optical activity: linear light is an equal blend of a left- and a right-circular wave. If the medium only delays one relative to the other, they stay equal in amplitude and the sum is a rotated straight line. But near an absorption band the medium also drains one circular wave faster than the other. Now the two circular amplitudes are no longer equal, and an unequal pair of counter-rotating circles no longer adds to a line — it adds to an ellipse. So the tell-tale of circular dichroism is that the transmitted light, which optical rotation alone would leave linear, comes out slightly elliptical, its ellipticity growing with the difference in absorption.

Figure

Two kinds of dichroism: circular dichroism drains one circular component so linear light emerges elliptical; linear dichroism attenuates one linear axis (Malus with absorption)

Linear dichroism is the same idea with linear axes instead of circular ones. An oriented material — a stretched polymer, a crystal, aligned dye molecules, or the microscopic iodine chains in a Polaroid — absorbs light polarized along one direction much more strongly than the perpendicular direction. Send in light at $45^\circ$ and the surviving component tilts toward the low-loss axis; send it along either axis and only its intensity changes. This is Malus's law with absorption playing the role of the projector, and it is how every polarizing filter, every pair of polarized sunglasses, and every liquid-crystal display pixel does its work.

Figure

The Cotton effect (punchline): the CD band peaks on the transition while the ORD rotation traces an anomalous S-curve through it; the mirror enantiomer flips both signs. θ = 32.98° × ΔA

The last figure is the whole story of the Cotton effect. Absorption confines the dichroism to a peak sitting on the transition wavelength. Rotation, being the Kramers–Kronig partner of that peak, is not a peak but an anomalous dispersion through it — rising on one side, crossing zero at the band centre, falling on the other, the same S-shape refraction shows through an absorption line. Because both curves reverse sign for the mirror-image molecule, a single measured Cotton effect reads out the absolute handedness of the sample.

The core equations

Everything follows from allowing the two circular indices to be complex, $\tilde n_{L,R} = n_{L,R} + i\kappa_{L,R}$ , so that both the phase velocity and the absorption of the two circular states can differ. Optical rotation was the real part of the phase difference; now the imaginary part comes alive as ellipticity. The two are bundled into a single complex rotation.

\Phi = \frac{\pi d}{\lambda_0}\big(\tilde n_L - \tilde n_R\big) = \phi + i\eta \qquad (1)

where: $\Phi$ — complex rotation angle; $d$ — path length; $\lambda_0$ — vacuum wavelength; $\tilde n_{L,R} = n_{L,R}+i\kappa_{L,R}$ — complex circular indices; $\phi$ — ordinary optical rotation (real part); $\eta$ — ellipticity acquired by the emergent light (imaginary part). One complex number carries rotation and ellipticity together. Splitting Eq. (1) into its parts gives the two observables. The real part is the rotation, whose wavelength dependence is optical rotatory dispersion; the imaginary part is the ellipticity, driven by the difference in absorptive indices — circular dichroism.

\phi = \frac{\pi d}{\lambda_0}(n_L - n_R), \qquad \eta = \frac{\pi d}{\lambda_0}(\kappa_L - \kappa_R) \qquad (2)

where: $\phi$ — optical rotation (rad); $\eta$ — ellipticity (rad); $\kappa_L - \kappa_R$ — the differential absorptive index that defines circular dichroism. When $\kappa_L = \kappa_R$ (transparent region) the ellipticity vanishes and only rotation survives — the optical-activity regime of the companion article. Experimentally, circular dichroism is measured not as an index difference but as a difference of absorbances, $\Delta A = A_L - A_R$ , using base-ten (decadic) absorbance. Converting the absorbed-amplitude imbalance into the ellipticity angle introduces a fixed numerical factor.

\theta = \frac{\ln 10}{4}\,\Delta A \qquad (3)

where: $\theta$ — ellipticity angle (rad, small-signal); $\Delta A = A_L - A_R$ — the circular-dichroism signal as a decadic-absorbance difference. Numerically $\theta = 0.5756\,\Delta A$ rad $= 32.98^\circ \times \Delta A$ ; the peculiar constant $32.98$ is just $\tfrac{\ln 10}{4}$ turned into degrees, and it is why CD spectra are quoted in millidegrees. Because $n$ and $\kappa$ are the real and imaginary parts of one analytic response function, they are not independent — the same Kramers–Kronig relation that binds refraction to absorption binds rotation to dichroism.

(n_L - n_R)(\omega) = \frac{2}{\pi}\,\mathcal{P}\!\!\int_0^\infty \frac{\omega'\,(\kappa_L - \kappa_R)(\omega')}{\omega'^2 - \omega^2}\,d\omega' \qquad (4)

where: $\mathcal{P}$ — Cauchy principal value; $\omega = 2\pi c/\lambda_0$ — angular frequency. Equation (4) says a circular-dichroism band at one frequency forces a dispersive rotation across all frequencies: measure one and you can, in principle, reconstruct the other. This is the mathematical root of the Cotton effect's paired shape.

Derivation AComplex rotation: how absorption turns rotation into ellipticity

The aim is to show, starting only from complex circular indices, that the transmitted light carries both a rotated axis and a definite ellipticity, and that these are exactly the real and imaginary parts of one complex angle — so that "optical rotation" and "circular dichroism" are two faces of a single object. We propagate the two circular eigenmodes with complex indices, factor out the common absorption and phase, and read the surviving state.

Step 1. Propagate each circular eigenmode a distance $d$ with its complex index: $E_{R,L} \propto \exp\!\big(i k_0 \tilde n_{R,L}\, d\big)$ , where the imaginary part of $\tilde n$ produces exponential decay $e^{-k_0 \kappa d}$ . Why: absorption is simply the imaginary part of the index acting inside the same propagation exponential that produced the phase — no new physics, just a complex $n$ .

Step 2. Factor out the mean complex index $\bar{\tilde n} = \tfrac12(\tilde n_L + \tilde n_R)$ , leaving the two modes with equal-and-opposite complex phase $\pm\tfrac12 k_0(\tilde n_L - \tilde n_R)d = \pm\Phi$ . Why: the common part attenuates and phases both modes identically and cannot change the polarization; only the antisymmetric part $\pm\Phi$ shapes the output state.

Step 3. Recombine into linear coordinates. With $\Phi = \phi + i\eta$ , the $\phi$ (real) part rotates the axis by $\phi$ as before, while the $\eta$ (imaginary) part unbalances the two circular amplitudes by $e^{\mp\eta}$ , producing an ellipse of ellipticity angle $\approx \eta$ tilted along the rotated axis. Why: an equal circular pair makes a line, but an unequal pair makes an ellipse — so the imaginary part, which is exactly the amplitude imbalance, is what creates the ellipticity.

Step 4. Identify the observables: $\phi = \mathrm{Re}\,\Phi = \tfrac{\pi d}{\lambda_0}(n_L - n_R)$ is optical rotation (its spectrum is ORD); $\eta = \mathrm{Im}\,\Phi = \tfrac{\pi d}{\lambda_0}(\kappa_L - \kappa_R)$ is the ellipticity from circular dichroism. Why: this is the payoff — rotation and dichroism are literally the real and imaginary parts of the one complex rotation, which is why they always appear together near a band. *Verified: $\mathrm{Re}\,\Phi$ and $\mathrm{Im}\,\Phi$ reproduce Eq. (2) symbolically, and $\theta = \tfrac{\ln10}{4}\Delta A = 32.98^\circ\Delta A$ numerically, in verify_aq-optics-circular-linear-dichroism.py.*

Go deeper · Derivation B · 1 minLinear dichroism as Malus's law with absorption

Derivation BLinear dichroism as Malus's law with absorption

Linear dichroism looks different from its circular cousin but is mathematically the plainer effect: a material that absorbs one linear axis more than the perpendicular one. The goal is to derive how an incident linear state is reshaped, recovering the dichroic ratio and showing that the transmitted intensity along a given analyser follows a Malus-like law modulated by the two axis transmittances. We resolve the input onto the two absorbing axes, attenuate each, and recombine.

Step 1. Resolve the input, linear at azimuth $\theta$ to the low-loss ( $x$ ) axis, into components $E_x = E_0\cos\theta$ and $E_y = E_0\sin\theta$ . Why: the material's eigen-axes are the two linear directions, so we must work in the basis where absorption is diagonal.

Step 2. Attenuate each axis by its amplitude transmission $t_x = e^{-\alpha_x d/2}$ , $t_y = e^{-\alpha_y d/2}$ (intensity transmittances $T_x = t_x^2$ , $T_y = t_y^2$ ), giving output $(t_x\cos\theta,\ t_y\sin\theta)E_0$ . Why: dichroism is unequal absorption of the two axes, so each component simply carries its own decay — this is the whole content of the effect.

Step 3. Read the dichroic ratio $T_x/T_y = e^{(\alpha_y-\alpha_x)d}$ and note the output is generally elliptical (the two surviving components differ in magnitude and tilt toward $x$ ); a downstream analyser at angle $\beta$ passes $I(\beta) = I_0\,(T_x\cos^2\theta\cos^2\beta + T_y\sin^2\theta\sin^2\beta + \dots)$ — Malus's law weighted by the two transmittances. Why: absorption is doing the projecting a perfect polarizer would do, but softly, so the extinction ratio is finite and set by $T_x/T_y$ rather than infinite. *Verified: the two-transmittance output and dichroic ratio $T_x/T_y = e^{(\alpha_y-\alpha_x)d}$ checked in verify_aq-optics-circular-linear-dichroism.py.*

Limiting cases

Case	Result	Meaning
Transparent ( $\kappa_L = \kappa_R$ )	$\eta = 0$ , only $\phi$ survives	far from bands, pure optical rotation — the optical-activity regime
On an absorption band	$\eta$ peaks; $\phi$ crosses zero (S-curve)	the Cotton effect: CD peak, ORD anomalous dispersion, both centred on the band
Mirror enantiomer	$\phi \to -\phi$ , $\eta \to -\eta$	both rotation and dichroism reverse sign — CD reads absolute handedness
Racemic mixture	$\phi = \eta = 0$	equal enantiomers cancel both real and imaginary index differences
$\alpha_x \gg \alpha_y$ (strong linear dichroism)	ratio $T_x/T_y \to \infty$	the material becomes a polarizer — the Polaroid limit

Common trap“Circular dichroism is just optical rotation measured a different way”

They are different measurements of different parts of the same complex quantity, and confusing them loses the physics. Optical rotation (and its wavelength dependence, ORD) is the real part — it exists even in perfectly transparent regions, where the sample never absorbs a photon. Circular dichroism is the imaginary part — it exists only where the sample absorbs, because it is a difference of absorptions, and it makes the light elliptical rather than merely rotated. You can have strong rotation with zero dichroism (a sugar solution in visible light) or a sharp dichroism band on top of a smoothly dispersing rotation (a protein in the ultraviolet). What is true is that Kramers–Kronig ties them, so a dichroism band forces a dispersive rotation nearby — but that is a relationship between two distinct observables, not an identity. Treat CD as "rotation in disguise" and you will miss that it, uniquely, requires absorption and reports on the excited state involved in the transition.

Worked examples

1 · Reading an ellipticity off a circular-dichroism signal (confidence builder)

A chiral chromophore in solution shows, at the peak of its absorption band, a circular-dichroism signal $\Delta A = A_L - A_R = 2.0\times10^{-3}$ . What ellipticity does the transmitted (originally linear) light acquire, and how would the number change for the mirror-image molecule?

Step 1. Apply Eq. (3): $\theta = \tfrac{\ln 10}{4}\,\Delta A = 0.5756 \times 2.0\times10^{-3} = 1.15\times10^{-3}$ rad.

Step 2. Convert to the units CD spectrometers report: $\theta = 32.98^\circ \times 2.0\times10^{-3} = 0.0660^\circ = \mathbf{66\ \text{mdeg}}$ . Circular-dichroism ellipticities are tiny — tens of millidegrees — which is why the instruments use polarization modulation and lock-in detection rather than a rotating analyser.

Step 3. For the mirror enantiomer, $\Delta A$ reverses sign, so $\theta = -66$ mdeg: the ellipse tilts the other way. The magnitude is identical; only the sign, i.e. the handedness of the ellipse, flips.

Step 4. Note what this measures: because $\Delta A$ is nonzero only where the molecule absorbs, this $66$ mdeg is reporting on the specific electronic transition sitting at that wavelength — CD is a transition-resolved handedness probe, not a bulk one. θ = 66 mdeg, sign-reversed for the enantiomer.

2 · A Polaroid as extreme linear dichroism (exam level)

A polarizing sheet transmits $T_x = 0.80$ of the intensity along its pass axis and $T_y = 1.0\times10^{-4}$ along the perpendicular (block) axis. Find its extinction ratio, and the transmitted intensity for unpolarized input and for input at $30^\circ$ to the pass axis.

Step 1. Extinction (dichroic) ratio: $T_x/T_y = 0.80 / 1.0\times10^{-4} = \mathbf{8000}$ , i.e. about $39$ dB — a decent, not spectacular, polarizer, set entirely by how much more strongly the aligned absorbers soak up the block axis.

Step 2. Unpolarized input splits equally between the two axes, so the transmitted fraction is $\tfrac12(T_x + T_y) = \tfrac12(0.80 + 0.0001) = \mathbf{0.400}$ — the familiar "a Polaroid passes half" rule, softened by the small leak.

Step 3. Linear input at $30^\circ$ : the pass-axis component carries $\cos^2 30^\circ = 0.750$ of the intensity, attenuated to $0.750\times0.80 = 0.600$ ; the block-axis component carries $\sin^2 30^\circ = 0.250$ , attenuated to $0.250\times10^{-4} = 2.5\times10^{-5}$ . Total $\approx \mathbf{0.600}$ — essentially pure Malus's law, since the block leak is negligible.

Step 4. The output is very slightly elliptical because the two axes were attenuated unequally, but with $T_x/T_y = 8000$ the block component is so small that the light is linear for any practical purpose — the ideal-polarizer limit of linear dichroism. Extinction ratio 8000; unpolarized ⟶ 0.40; 30° input ⟶ 0.60.

Physics of the portrait

Circular dichroism is inseparable from the man who first traced its paired dispersive-plus-absorptive signature and gave the effect its name.

Origins · 1 mincotton effect and how this was discovered

cotton effect

paired ORD anomaly + CD band at a chiral transition · molecular optics

The Cotton effect is the joint anomalous behaviour of optical rotation and circular dichroism in the neighbourhood of an absorption band of a chiral molecule: as the wavelength sweeps across the band, the rotation traces an S-shaped anomalous dispersion while the dichroism rises to a peak, the two locked together by Kramers–Kronig and signed by the enantiomer. It is named for the French physicist Aimé Cotton (1869–1951), who in 1895 — working for his doctoral thesis under Marcel Brillouin — measured the optical rotation and the differential circular absorption of solutions of chiral chromophores (copper and chromium tartrates) right through their colour bands, and found the anomaly that pure transparent optical activity had never shown. Cotton went on to build enormous electromagnets for magneto-optics (the Cotton–Mouton effect, magnetic linear birefringence, bears his name too) and to co-found the great French magnet laboratory. His effect became, decades later, one of biochemistry's most powerful structural tools: the sign and shape of the Cotton effect near a chromophore fixes the absolute configuration of a stereocentre, and the far-ultraviolet CD of the peptide backbone reads out protein secondary structure.

History

1811

— Arago sees optical rotation in quartz; rotation without absorption is well known

1844

— Haidinger notices the eye's own faint dichroism (Haidinger's brush) from the macular pigment

1895

— Aimé Cotton measures anomalous rotatory dispersion and circular dichroism through chiral absorption bands — the Cotton effect

1929

— Kuhn quantifies the dissymmetry (anisotropy) factor g = ΔA/A relating CD to ordinary absorption

1969

— commercial CD spectropolarimeters make far-UV protein CD a routine structural probe

Going further

Texts. Hecht, Optics, 5th ed., §8.5 (optical activity and dichroism) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 6 · Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (ORD/CD, Cotton effect, Kramers–Kronig) · Berova, Nakanishi & Woody, Circular Dichroism: Principles and Applications.

Provenance. A verifier should confirm: (1) the complex rotation $\Phi = \tfrac{\pi d}{\lambda_0}(\tilde n_L - \tilde n_R)$ splits into $\phi = \mathrm{Re}\,\Phi$ (rotation) and $\eta = \mathrm{Im}\,\Phi$ (ellipticity from CD) (Barron; Saleh & Teich Ch. 6); (2) the ellipticity conversion $\theta = \tfrac{\ln 10}{4}\Delta A = 32.98^\circ\Delta A$ (standard CD instrumentation relation); (3) ORD and CD as Kramers–Kronig partners (Barron). The Cotton-effect discovery date (1895) and Kuhn's dissymmetry factor (1929) are standard historical values.

Prerequisites. Optical Activity · Circular Polarization · Linear Polarization · Birefringence · Jones Calculus. Leads to. Stokes Parameters and the Poincaré Sphere · Mueller Calculus · Wave Plates and Retarders.

Why does circular dichroism need absorption while optical rotation does not?

Optical rotation lives in the difference of the real refractive indices, $n_L - n_R$ , which is nonzero even in perfectly transparent regions where no photon is ever absorbed. Circular dichroism lives in the difference of the absorptive indices, $\kappa_L - \kappa_R$ , and absorption is exactly what $\kappa$ measures — so where the medium is transparent, $\kappa_L = \kappa_R = 0$ and there is no dichroism at all. That is why a sugar solution rotates visible light strongly but shows CD only if you push into its ultraviolet absorption bands: the rotation is the transparent (real) effect, the dichroism is the absorptive (imaginary) effect, and they only overlap where the molecule actually has a transition.

If I only measure circular dichroism, do I still know the optical rotation?

In principle, yes — that is the content of the Kramers–Kronig relation, Eq. (4). Because the real and imaginary parts of an analytic response function are not independent, a full circular-dichroism spectrum (the imaginary part over all frequencies) determines the optical-rotatory-dispersion spectrum (the real part), and vice versa. In practice you need the CD over a wide enough range and some care with the principal-value integral, but the two really do carry the same information. This is why modern chirality work often prefers CD: it is band-localized and cleaner to interpret, yet loses nothing that ORD would have told you.

What is the difference between circular and linear dichroism physically?

Circular dichroism is unequal absorption of the two circular polarizations and requires chirality — a handedness that lets the molecule tell left- from right-circular light. Linear dichroism is unequal absorption of two orthogonal linear polarizations and requires only anisotropy — some preferred axis, such as an aligned transition dipole in a stretched film or an oriented crystal. A tumbling chiral molecule in solution shows CD but averages away any linear dichroism; an achiral but aligned dye shows linear dichroism but no CD. They probe different things: handedness versus orientation.

Why are circular-dichroism signals reported in millidegrees?

Because they are tiny. The ellipticity is $\theta = \tfrac{\ln10}{4}\Delta A$ , and typical differential absorbances $\Delta A$ are $10^{-3}$ or smaller, so $\theta$ comes out in the tens of millidegrees — far too small to read off a rotating analyser. Instruments therefore modulate the incident polarization between left- and right-circular at tens of kilohertz with a photoelastic modulator and lock-in detect the absorption difference directly, then convert to ellipticity through the fixed factor $32.98^\circ$ per unit $\Delta A$ . The millidegree is simply the natural unit of that very small ellipticity.

Does linear dichroism make the light elliptical too?

Slightly, yes — whenever the two linear axes are attenuated unequally and the input is not aligned with an axis, the surviving components differ in magnitude and the output is a thin ellipse tilted toward the low-loss axis. But for a strong polarizer (large dichroic ratio) the high-loss component is almost gone, so the output is linear for all practical purposes and the effect reduces to Malus's law. The ellipticity is only noticeable in weakly dichroic samples, which is exactly the regime where linear dichroism is used as an orientation probe rather than as a polarizer.

How does a protein's CD spectrum tell you it is an α-helix?

The peptide bonds of the backbone have ultraviolet electronic transitions near $190$ – $222$ nm, and when the backbone coils into a regular $\alpha$ -helix these transitions couple between neighbouring units (exciton coupling) to produce a very distinctive CD signature: a positive band near $190$ nm and two negative bands near $208$ and $222$ nm. A $\beta$ -sheet gives a different, weaker pattern, and a random coil a different one again. Fitting a measured far-UV CD spectrum to a combination of these reference shapes estimates the fractions of helix, sheet, and coil — a fast, low-sample, solution-phase readout of secondary structure that complements crystallography.