Mueller Calculus

Jones calculus is a beautiful machine for polarized light — a two-component complex amplitude, a $2\times2$ matrix per element, cascades by multiplication — but it breaks the moment light stops being fully polarized. A ground-glass diffuser, a multimode fibre, a slab of biological tissue, the atmosphere: each randomizes polarization, dropping the ordered fraction, and no Jones matrix can lower the degree of polarization of the light it acts on. The Mueller calculus is the intensity-based cousin that can. It represents the state not by an amplitude but by the four real Stokes parameters $(S_0,S_1,S_2,S_3)$ — averaged, measurable intensities — and every linear optical element by a real $4\times4$ matrix acting on them. That upgrade from two complex numbers to four real ones, and from a $2\times2$ to a $4\times4$ , buys exactly one new thing: depolarization. A deterministic element — polarizer, retarder, rotator — still has a Jones matrix, and its Mueller matrix is fixed by that Jones matrix through a Kronecker product; fully polarized light in stays fully polarized. But a real device, averaged incoherently over an ensemble of retardances or orientations or wavelengths, shrinks the Stokes vector toward the centre of the Poincaré sphere, and that averaged behaviour has no Jones counterpart at all. Sixteen real numbers, all obtainable from intensity measurements, encode everything a linear device can do to polarization — how much it biases (diattenuation), delays (retardance), and scrambles (depolarization) — which is why Mueller-matrix polarimetry is the standard characterization tool from thin-film metrology to cancer imaging.

Key takeaways

Mueller calculus encodes light as a real Stokes four-vector and every linear element as a real $4\times4$ matrix: $\mathbf S' = M\,\mathbf S$ .
Because Stokes parameters are averaged intensities, the formalism handles unpolarized and partially polarized light and depolarizing devices — exactly what Jones calculus cannot represent.
Cascaded elements multiply in reverse order of propagation, $M=M_N\cdots M_2M_1$ , and the measured intensity is the output $S_0$ .
Every deterministic (non-depolarizing) element has a Jones matrix, and its Mueller matrix is $M=A\,(J\otimes J^{*})\,A^{-1}$ ; a genuinely depolarizing matrix has no Jones equivalent.
An ideal polarizer's Mueller matrix acting on linear input reproduces Malus's law; averaging a retarder over random retardance produces depolarization from first principles.
Key idea: represent light as a real Stokes four-vector and each element as a real $4\times4$ matrix; the extra size over Jones is precisely what lets it describe depolarization — the shrinking of the Poincaré vector no amplitude matrix can produce.

The core insight

Encode light as a real Stokes four-vector and every linear optical element as a real $4\times4$ matrix acting on it. Because Stokes parameters are averaged intensities, the calculus handles unpolarized and partially polarized light and depolarizing devices — the shrinking of the Poincaré vector — that no Jones matrix can represent.

Where this lives today

Mueller-matrix polarimetry — measuring all sixteen entries with rotating polarizers and retarders or photoelastic modulators, then factoring the matrix into a diattenuator, a retarder, and a depolarizer (the Lu–Chipman decomposition) — is the standard way to characterize anything whose response includes stray depolarization that Jones-based ellipsometry cannot represent. Semiconductor fabs use it to measure thin-film stacks and liquid-crystal cells; the depolarization and retardance channels expose defects an amplitude model would miss. Its most striking modern home is biomedical imaging. Biological tissue scatters strongly and therefore depolarizes, so its optical response is intrinsically a depolarizing Mueller matrix; the retardance channel maps birefringent collagen organization and the depolarization channel distinguishes healthy from cancerous tissue in skin, cervix, and colon, where a Jones description is simply invalid. And in Earth and planetary remote sensing, radiative-transfer codes propagate the full Stokes vector through layered Mueller scattering matrices to retrieve aerosol size and refractive index from partially polarized, partially depolarized sunlight — the calculus earning its keep precisely because it handles the scrambling that real light suffers.

The physical picture

Picture the last three Stokes parameters as a vector inside the Poincaré ball and ask what a device can do to it. There are exactly three primitive moves. A pure retarder rotates the vector rigidly — a rotation of the sphere — changing which polarization the light is without touching how polarized it is. A diattenuator (a partial polarizer) stretches the vector toward one axis while attenuating the total, biasing the light toward its transmission state. And a depolarizer shrinks the vector toward the centre, randomizing polarization and dropping the ordered fraction. Any physical Mueller matrix is some combination of these three, so reading a matrix is reading how much it rotates, biases, and scrambles.

Figure

A Mueller chain: a Stokes four-vector enters, each element multiplies it by a real 4×4 matrix (polarizer, retarder, sample), and the detector reads the output S₀

Depolarization is the genuinely new physics, and it appears only through averaging. A single deterministic light path can rotate and attenuate a fully polarized state, but it always leaves the state fully polarized — a rotation and a stretch cannot shrink a unit vector below the surface. What shrinks it is incoherent averaging over many such paths: different retardances across the beam, over time, or across wavelength. Add the Mueller matrices of those paths (legal, because intensities add incoherently) and the averaged matrix has a Poincaré vector strictly inside the ball. Mueller calculus lives at this averaged level, which is exactly why it — and not Jones calculus — captures the effect. And because Stokes parameters are quadratic in the field, going from a $2\times2$ Jones matrix to a $4\times4$ Mueller matrix doubles the algebra: a lossless Jones matrix (three real parameters after removing overall phase) generates only a small family of the sixteen Mueller numbers, and the rest of the space is where depolarization lives.

:::svg[Punchline: a rotating analyzer on partially polarized input gives I(θ) = ½S₀[1 + p·cos2(θ−θ₀)] — the modulation depth reads the degree of polarization directly; here p = 0.60, I_max = 0.80, I_min = 0.20]

::: The punchline figure is the everyday use of the whole calculus. Send partially polarized light through a single rotating polarizer and the transmitted intensity traces $I(\theta)=\tfrac12 S_0[1+p\cos2(\theta-\theta_0)]$ : a cosine in $2\theta$ riding on a constant floor. The floor is the unpolarized half; the swing is the ordered part. Read the maximum and minimum — here $0.80$ and $0.20$ — and their visibility $(I_{\max}-I_{\min})/(I_{\max}+I_{\min})=0.60$ is the degree of polarization. The Mueller matrix of that rotating polarizer, written down once, predicts the entire curve; the modulation depth of a two-dollar measurement returns a fundamental property of the beam.

The core equations

A Mueller matrix is defined by what it does to a Stokes vector, and elements chain by ordinary matrix multiplication — the same bookkeeping as Jones calculus, one size larger and entirely real.

\mathbf S' = M\,\mathbf S,\qquad M = M_N\,M_{N-1}\cdots M_2\,M_1 \qquad (1)

where: $\mathbf S=(S_0,S_1,S_2,S_3)^{\mathsf T}$ — input Stokes four-vector; $\mathbf S'$ — output; $M$ — real $4\times4$ Mueller matrix; $M_k$ — matrix of the $k$ -th element the light meets, ordered right-to-left because the first element acts first. The measured intensity is always the output $S_0'$ , the first component. For a deterministic element the sixteen real numbers are not free — they are fixed by the element's $2\times2$ Jones matrix through a Kronecker product, because Stokes parameters are quadratic in the field. This is the exact bridge between the two calculi.

M = A\,(J\otimes J^{*})\,A^{-1},\qquad A=\begin{pmatrix}1&0&0&1\\1&0&0&-1\\0&1&1&0\\0&i&-i&0\end{pmatrix} \qquad (2)

where: $J$ — $2\times2$ Jones matrix of the element; $J^{*}$ — its complex conjugate; $\otimes$ — Kronecker product (a $4\times4$ matrix); $A$ — the fixed matrix converting the coherency (density-matrix) basis to the Stokes basis. Because a single $J$ generated it, such an $M$ is non-depolarizing: fully polarized input stays fully polarized. Not every real $4\times4$ matrix arises this way — the ones that do not are precisely the depolarizers. The two elements you meet most are the ideal polarizer and the retarder; both are worth memorizing because everything else is built from them and rotations.

M_{\text{pol}}(0)=\tfrac12\begin{pmatrix}1&1&0&0\\1&1&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix},\qquad M_{\text{ret}}(\delta)=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos\delta&\sin\delta\\0&0&-\sin\delta&\cos\delta\end{pmatrix} \qquad (3)

where: $M_{\text{pol}}(0)$ — ideal linear polarizer, transmission axis horizontal; the $\tfrac12$ passes half of unpolarized input. $M_{\text{ret}}(\delta)$ — retarder, fast axis horizontal, retardance $\delta$ ; it leaves $S_0,S_1$ alone and rotates $(S_2,S_3)$ by $\delta$ . A quarter-wave plate is $\delta=\pi/2$ (a $90^\circ$ turn in the $S_2$ – $S_3$ plane), a half-wave plate $\delta=\pi$ . Rotating any element by a physical angle $\theta$ conjugates its matrix with the Stokes rotation, which carries the tell-tale factor of two.

M(\theta)=R(-\theta)\,M\,R(\theta),\qquad R(\theta)=\begin{pmatrix}1&0&0&0\\0&\cos2\theta&\sin2\theta&0\\0&-\sin2\theta&\cos2\theta&0\\0&0&0&1\end{pmatrix} \qquad (4)

where: $R(\theta)$ — Stokes rotation; the $2\theta$ is the Poincaré-sphere double angle (states $90^\circ$ apart physically are antipodal, $180^\circ$ apart on the sphere). $M(\theta)$ — the element rotated by $\theta$ . Finally, the degree of polarization of the output is read off as usual, $\text{DoP}=\sqrt{S_1'^2+S_2'^2+S_3'^2}/S_0'$ , and a device is depolarizing exactly when it can lower it below the input value for fully polarized light.

Derivation AThe Jones-to-Mueller map, and why a polarizer's Mueller matrix is Malus's law

The aim is to build the Mueller matrix of a deterministic element from its Jones matrix — showing why the recipe is a Kronecker product — and then to specialize to an ideal polarizer and watch Malus's law drop out of a single matrix multiplication. The route is to track the coherency matrix $\rho=\langle EE^{\dagger}\rangle$ (whose four entries carry the Stokes parameters) through the Jones transformation, then re-express it in the Stokes basis. This is where the doubling of the algebra — two complex numbers becoming four real ones — makes itself concrete.

Step 1. Under a Jones matrix the field transforms as $E'=JE$ , so the coherency matrix transforms as $\rho'=J\rho J^{\dagger}$ ; vectorizing and using $\operatorname{vec}(J\rho J^{\dagger})=(J\otimes J^{*})\operatorname{vec}(\rho)$ turns the sandwich into a single $4\times4$ multiplication. Why: intensities are bilinear in the field, so the natural object that transforms linearly is not $E$ but the outer product $EE^{\dagger}$ — and a Kronecker product is exactly how a two-sided matrix multiplication acts on a vectorized matrix.

Step 2. The Stokes vector is a fixed linear combination of the coherency entries, $\mathbf S=A\operatorname{vec}(\rho)$ , so $\mathbf S'=A\operatorname{vec}(\rho')=A(J\otimes J^{*})A^{-1}\mathbf S\equiv M\mathbf S$ , giving Equation (2). Why: $A$ just relabels the same physical information from the density-matrix basis to the measurable-intensity basis, so conjugating by it carries the transformation law across intact.

Step 3. Take the ideal horizontal polarizer $J=\operatorname{diag}(1,0)$ . Then $J\otimes J^{*}=\operatorname{diag}(1,0,0,0)$ , and $M=A\,\operatorname{diag}(1,0,0,0)\,A^{-1}=\tfrac12\!\left(\begin{smallmatrix}1&1&0&0\\1&1&0&0\\0&0&0&0\\0&0&0&0\end{smallmatrix}\right)$ , the boxed polarizer matrix. Why: a projector in the amplitude picture becomes, in the intensity picture, an averaging that transmits half of any unpolarized input and fully polarizes what survives.

Step 4. Feed linear light at angle $\varphi$ , $\mathbf S=I_0(1,\cos2\varphi,\sin2\varphi,0)$ , through a polarizer at angle $\theta$ (rotate the matrix by Equation (4)): the output intensity is $S_0'=\tfrac{I_0}{2}\big(1+\cos2(\theta-\varphi)\big)=I_0\cos^2(\theta-\varphi)$ — Malus's law, recovered as one entry of a matrix product. Why: the whole of Malus is hiding in the first row of the polarizer's Mueller matrix, which is exactly what makes the Mueller picture a superset of the elementary one. *Verified: $M=A(J\otimes J^{*})A^{-1}$ reproducing the ideal-polarizer matrix, the QWP Jones matrix $\operatorname{diag}(1,i)$ mapping to a valid retarder (an orthogonal $S_2$ – $S_3$ rotation with $S_0,S_1$ fixed), and $S_0'=\cos^2(\theta-\varphi)$ for several angles, in verify_mueller-calculus.py.*

Go deeper · Derivation B · 1 minDepolarization from incoherent averaging — a matrix with no Jones equivalent

Derivation BDepolarization from incoherent averaging — a matrix with no Jones equivalent

The goal is to manufacture a genuinely depolarizing Mueller matrix from ordinary deterministic ones, proving that depolarization is nothing but incoherent averaging and that the result cannot be written as $A(J\otimes J^{*})A^{-1}$ for any single $J$ . The strategy: take a retarder whose retardance $\delta$ fluctuates across the beam with some distribution $p(\delta)$ — think of a rough or strained wave plate — and average its Mueller matrix, which is legal precisely because Stokes parameters add incoherently.

Step 1. For one realization the matrix is $M_{\text{ret}}(\delta)$ from Equation (3), a rotation of $(S_2,S_3)$ by $\delta$ . Why: a definite retardance is deterministic and length-preserving on the sphere, so a single realization cannot depolarize anything.

Step 2. Average over the beam: $\bar M=\int p(\delta)\,M_{\text{ret}}(\delta)\,d\delta$ , which touches only the lower $2\times2$ block, replacing $\cos\delta,\sin\delta$ by $\langle\cos\delta\rangle,\langle\sin\delta\rangle$ . Why: averaging Mueller matrices (not Jones matrices) is the physically correct operation for incoherent superposition — you add the powers different parts of the beam deliver, not their amplitudes.

Step 3. The averaged block has determinant $\langle\cos\delta\rangle^2+\langle\sin\delta\rangle^2\le1$ , with equality only for a delta-function distribution. A $+45^\circ$ or circular input, whose Stokes vector lives in that block, therefore emerges shortened — partially depolarized. Why: the trigonometric averages contract by the same Jensen inequality that shrinks $\langle e^{i\delta}\rangle$ below one whenever the phase has any spread; a broad spread sends them to zero and fully depolarizes those components.

Step 4. For a broad distribution $\langle\cos\delta\rangle,\langle\sin\delta\rangle\to0$ and $\bar M\to\operatorname{diag}(1,1,0,0)$ — an element that scrambles the diagonal and circular content while preserving H/V. No single Jones matrix reproduces this: its associated coherency matrix has more than one positive eigenvalue, the hallmark of a true depolarizer. Why: a Jones matrix always yields a rank-one coherency matrix (one path, one pure transformation); more than one eigenvalue means the device is irreducibly an ensemble. *Verified: the uniform- $\delta$ average giving $\langle\cos\delta\rangle=\langle\sin\delta\rangle=0$ and a lower block that vanishes (block determinant below $1$ ), versus a single- $\delta$ retarder whose block determinant is exactly $1$ (deterministic), in verify_mueller-calculus.py.*

Limiting cases

Case	Result	Meaning
Deterministic element	$M=A(J\otimes J^{*})A^{-1}$	a Jones matrix exists; fully polarized input stays fully polarized
Ideal depolarizer	$M=\operatorname{diag}(1,0,0,0)$	any input emerges unpolarized at the same intensity; no Jones equivalent
Unpolarized input	$\mathbf S=I_0(1,0,0,0)\Rightarrow\mathbf S'=I_0(M_{00},M_{10},M_{20},M_{30})$	the first column is the response to unpolarized light; a diattenuator polarizes it
Ideal polarizer on unpolarized	$\tfrac12 I_0(1,1,0,0)$	half the power passes, fully polarized along the axis
Passive transmittance bound	$M_{00}+\sqrt{M_{01}^2+M_{02}^2+M_{03}^2}\le1$	a physical passive element cannot transmit more than all the input

Common trap“Any real 4×4 matrix is a Mueller matrix, and every Mueller matrix comes from a Jones matrix”

Both halves are false, and assuming either corrupts real measurements. Not every real $4\times4$ matrix is physically realizable: a valid Mueller matrix must map every admissible Stokes vector (those with $\sqrt{S_1^2+S_2^2+S_3^2}\le S_0$ ) to another admissible one, which constrains its entries and its transmittance — the test is that the associated Hermitian coherency matrix be positive semidefinite. And not every valid Mueller matrix has a Jones matrix: the depolarizing ones do not. Fit a Jones (deterministic) model to a matrix that actually depolarizes — a tissue sample, a rough surface, a long fibre — and you get unphysical negative coherency eigenvalues and a spurious degree of polarization above one. Check realizability before you interpret.

Worked examples

1 · An ideal polarizer on unpolarized light (confidence builder)

Unpolarized light of intensity $I_0=1$ passes an ideal linear polarizer with a horizontal transmission axis. Find the output Stokes vector, the transmitted intensity, and the degree of polarization.

Step 1. Represent the input as $\mathbf S=(1,0,0,0)$ — full intensity, no imbalance in any basis. The element is $M_{\text{pol}}(0)=\tfrac12\!\left(\begin{smallmatrix}1&1&0&0\\1&1&0&0\\0&0&0&0\\0&0&0&0\end{smallmatrix}\right)$ .

Step 2. Multiply: $\mathbf S'=M_{\text{pol}}(0)\,\mathbf S=\tfrac12(1,1,0,0)=(0.5,\,0.5,\,0,\,0)$ . The transmitted intensity is the output $S_0'=\mathbf{0.5}$ — exactly half the unpolarized power, the familiar rule.

Step 3. Degree of polarization: $\text{DoP}=\sqrt{0.5^2+0^2+0^2}/0.5=\mathbf{1.0}$ . The output is fully polarized, horizontal, regardless of how scrambled the input was — a polarizer converts unpolarized light to pure at a $50\%$ toll.

Step 4. Contrast with a retarder on the same input: $M_{\text{ret}}(\delta)(1,0,0,0)=(1,0,0,0)$ unchanged, because a lossless retarder cannot create order from a scrambled beam. Only the polarizer's projection raises the DoP. Output (0.5, 0.5, 0, 0): half intensity, DoP = 1.

2 · A quarter-wave plate's Mueller matrix and a degree-of-polarization readout (exam level)

Build the Mueller matrix of a quarter-wave plate (fast axis horizontal) from its Jones matrix, confirm what it does to $+45^\circ$ linear light, then find the degree of polarization of a beam whose rotating-analyzer scan gives $I_{\max}=0.80$ , $I_{\min}=0.20$ .

Step 1. The QWP Jones matrix is $J=\operatorname{diag}(1,i)$ . Through $M=A(J\otimes J^{*})A^{-1}$ this maps to the retarder matrix with $\delta=90^\circ$ : $M_{\text{QWP}}=\left(\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&0\end{smallmatrix}\right)$ — $S_0,S_1$ untouched, $(S_2,S_3)$ rotated by $90^\circ$ .

Step 2. Act on $+45^\circ$ linear light $\mathbf S=(1,0,1,0)$ : $M_{\text{QWP}}\mathbf S=(1,0,0,1)$ — circular polarization. The plate has carried a point from the equator to a pole, the Poincaré-sphere image of "linear at $45^\circ$ to the axes becomes circular", and $\text{DoP}=1$ throughout because the operation is a rigid rotation.

Step 3. For the second beam, model the rotating analyzer by $M_{\text{pol}}(\theta)$ and a partially polarized input; the transmitted intensity is $I(\theta)=\tfrac12 S_0[1+p\cos2(\theta-\theta_0)]$ . Its extremes are $I_{\max}=\tfrac12 S_0(1+p)$ and $I_{\min}=\tfrac12 S_0(1-p)$ .

Step 4. Solve for the degree of polarization: $p=\dfrac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}=\dfrac{0.80-0.20}{0.80+0.20}=\mathbf{0.60}$ . The modulation depth of the scan is the degree of polarization directly — the same visibility that read out on the sphere in the companion Stokes article. QWP: (1,0,1,0) → (1,0,0,1) circular; rotating-analyzer DoP = 0.60.

Physics of the portrait

The Mueller calculus is named for a physicist who, in the middle of a world war, wrote down the matrix method almost entirely in lecture notes and reports — so its central object is a matrix rather than a man.

Origins · 1 minmueller matrix and how this was discovered

mueller matrix

the intensity calculus of polarization, 1943 · Hans Mueller at MIT

Hans Mueller (1900–1965) was a Swiss-born professor of physics at MIT, a student of the crystal-optics tradition, who developed the phenomenological matrix method for partially polarized light around 1943. Strikingly, he published almost none of it in the usual way: the formalism spread through his MIT course notes, an Office of Scientific Research and Development wartime report, and the theses of his students, so the "Mueller matrix" entered the literature named for work that is hard to cite to a single paper. He built on earlier phenomenological matrix ideas — Paul Soleillet's 1929 analysis of scattered light and Francis Perrin's 1942 treatment — but it was Mueller's $4\times4$ real matrix acting on the Stokes vector that became the standard. The virtue of his approach is exactly its refusal to model the microscopics: like Stokes's parameters a century before, a Mueller matrix is defined by what it does to measurable intensities, so it accommodates depolarizing, scattering, fluctuating media that no amplitude formalism can touch. That pragmatism is why his matrix, not a more fundamental-looking object, is the working tool of polarimetry today — in ellipsometers characterizing thin films, in remote-sensing radiative-transfer codes, and in the Mueller microscopes that read the depolarizing signature of diseased tissue.

History

1852

— Stokes introduces the four intensity parameters that the matrix will act on

1929

— Soleillet gives a phenomenological matrix analysis of scattered, partially polarized light

1942

— Perrin develops matrix relations for the scattering of polarized light

1943

— Mueller formulates the 4×4 real matrix method, mainly in MIT course notes and a wartime report

1996

— Lu and Chipman give the polar decomposition of a Mueller matrix into diattenuator, retarder, and depolarizer

Going further

Texts. Hecht, Optics, 5th ed., §8.13 (Stokes parameters and Mueller matrices) · Goldstein, Polarized Light, 3rd ed., ch. 5–9 (Mueller matrices and decomposition) · Chipman, Handbook of Optics vol. I, ch. 14 (Mueller-matrix polarimetry) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §6.1 · Born & Wolf, Principles of Optics, 7th ed., §10.9.

Provenance. The defining relation, cascade rule, and polarizer/retarder matrices follow Hecht §8.13 and Saleh & Teich §6.1; the Jones-to-Mueller map $M=A(J\otimes J^{*})A^{-1}$ and the Lu–Chipman polar decomposition follow Goldstein ch. 5–9 and Chipman, Handbook of Optics vol. I. Quarter- and half-wave retardances $\delta=\pi/2,\pi$ and the DoP bound $0\le\text{DoP}\le1$ are standard. A verifier recomputes the polarizer matrix and the Malus limit $I=I_0\cos^2\theta$ , the QWP Mueller matrix from $J=\operatorname{diag}(1,i)$ , and the depolarization-by-averaging block contraction — all in verify_mueller-calculus.py.

Prerequisites. Stokes Parameters and the Poincaré Sphere · Jones Calculus · Degree of Polarization and Partial Polarization. Leads to. Polarizers and Extinction Ratio · Wave Plates and Retarders.

If Mueller calculus can do everything Jones can plus depolarization, why keep Jones calculus at all?

Because Jones calculus keeps the phase. A Jones vector carries the absolute and relative phase of a fully coherent beam, which is exactly what you need for interference and for coherent cascades of elements — a Mueller matrix, built from intensities, has thrown that phase away. So Jones is the right tool for a laser through a train of wave plates where you care about the output field; Mueller is the right tool when the light is partially polarized or the device depolarizes, where phase is not defined anyway. They are complementary: use Jones while the light is pure and coherent, and switch to Mueller the moment averaging enters.

How can averaging matrices be legitimate — don't I have to average the underlying fields?

It depends on whether the pieces you are averaging over are coherent with each other. If they are (a single beam split and recombined), you must add fields, and Jones is the right level. If they are mutually incoherent — different parts of a rough beam, different times longer than the coherence time, different wavelengths a detector cannot separate — then their intensities add, and since Stokes parameters are linear combinations of intensities, so do their Stokes vectors and hence their Mueller matrices. Averaging Mueller matrices is exactly the statement "these contributions do not interfere", which is what depolarization means. Averaging Jones matrices, by contrast, would wrongly assume coherence and generally gives an unphysical result.

What makes a real 4×4 matrix a physical Mueller matrix?

It must map every admissible Stokes vector to another admissible one — no output may lie outside the Poincaré ball or carry negative intensity — and it must not transmit more power than it receives. The clean, complete test is Cloude's criterion: build the $4\times4$ Hermitian coherency matrix $H$ from the sixteen Mueller entries and require it to be positive semidefinite. If $H$ has a single positive eigenvalue the element is deterministic (a Jones matrix exists); several positive eigenvalues means it depolarizes; any negative eigenvalue means the matrix is unphysical, usually a sign of measurement error. This positivity check is the routine sanity test in Mueller polarimetry.

Why does going from Jones to Mueller "double the algebra"?

Because Stokes parameters are quadratic in the field. A Jones matrix acts once on an amplitude, $E\to JE$ ; the corresponding action on intensities, which are products $EE^{*}$ , is two-sided, $\rho\to J\rho J^{\dagger}$ , and writing that as a single matrix multiplication needs the Kronecker product $J\otimes J^{*}$ — a $4\times4$ object built from the $2\times2$ $J$ . A lossless Jones matrix has three real parameters (after dropping an overall phase), so it can generate only a three-parameter family of Mueller matrices; the full space of physical $4\times4$ matrices has far more room, and all the extra dimensions are diattenuation and depolarization that a single Jones matrix simply cannot reach.

A rotating polarizer reads out the degree of polarization — how exactly?

Spin a single polarizer in the beam and record intensity versus angle. For partially polarized light the Mueller matrix predicts $I(\theta)=\tfrac12 S_0[1+p\cos2(\theta-\theta_0)]$ : a cosine in twice the angle sitting on a constant floor. The floor is the unpolarized half, the swing is the ordered part, and their contrast $(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$ equals $p$ , the degree of polarization (for a beam whose polarized part is linear). It is the same fringe-visibility idea as elsewhere in optics, applied to an analyzer scan: a two-dollar rotating filter and a power meter return a fundamental property of the light. Circular content is invisible to a bare polarizer, so add a quarter-wave plate first to capture $S_3$ and get the full degree of polarization.

What do the first row and first column of a Mueller matrix mean physically?

They are the two halves of how a device couples to unpolarized light. The first column is the element's response to unpolarized input, $\mathbf S=I_0(1,0,0,0)$ : the output is $I_0(M_{00},M_{10},M_{20},M_{30})$ , so $M_{00}$ is the unpolarized transmittance and $(M_{10},M_{20},M_{30})/M_{00}$ is the polarizance — how strongly the device polarizes light that arrived scrambled (nonzero for a polarizer, zero for a pure retarder). The first row instead controls how much a fully polarized input's state modulates the transmitted intensity: $(M_{01},M_{02},M_{03})/M_{00}$ is the diattenuation vector, the polarization-dependent transmission that makes a polarizer a polarizer. A pure retarder has both vanish — it neither polarizes nor diattenuates, only rotates — which is why its Mueller matrix has a clean $(1,0,0,0)$ first row and column.