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Derivation

Polarization States and the Jones Calculus

D-203 Home PU-206 Threads light · symmetry · fields Depends on plane-wave-solutions, matrix-multiplication-and-determinant
Statement

For a monochromatic plane wave propagating along \(+\hat{\mathbf z}\), the transverse electric field is fully specified by the complex amplitudes of its two Cartesian components. Collecting these into a two-component complex column vector \(\mathbf J = \begin{pmatrix} E_x \\ E_y \end{pmatrix}\) — the Jones vector — the action of any lossless or lossy non-depolarizing optical element is a linear map \(\mathbf J' = M\,\mathbf J\) with \(M\) a \(2\times 2\) complex matrix. We derive the Jones matrices of the ideal linear polarizer and of the general retarder (wave plate) from first principles, and show that the normalized Jones vector, equivalently the relative amplitude ratio and phase \(\delta = \arg E_y - \arg E_x\), classifies every state of fully polarized light as linear, circular, or elliptical.

Why it matters

Polarization is a two-state quantum-like degree of freedom carried by classical light. The Jones calculus reduces the propagation of fully polarized light through arbitrarily many polarizers, retarders, and rotators to a single matrix product, so that a whole optical train collapses to one \(2\times 2\) matrix. This is the workhorse of optical metrology, liquid-crystal displays, ellipsometry, and polarization-based quantum-optics experiments.

The formalism is also the historical and conceptual bridge to the qubit: the two-dimensional complex Hilbert space of Jones vectors is mathematically identical to a spin-\(\tfrac12\) system, with the Poincaré sphere playing the role of the Bloch sphere. Understanding Jones calculus is understanding the geometry of a single two-level system.

Assumptions
Monochromatic plane wave.A single well-defined frequency \(\omega\) and wavevector \(\mathbf k = k\hat{\mathbf z}\); if dropped, the phases of the two components drift independently and the light is partially polarized, requiring the \(4\times 4\) Mueller / Stokes description instead.
Fully (coherent) polarization.The \(x\) and \(y\) components maintain a fixed relative phase over the observation time; if dropped, a single Jones vector cannot represent the state and one needs the \(2\times2\) coherency (density) matrix.
Transverse field, paraxial propagation along \(\hat{\mathbf z}\).Only \(E_x,E_y\) are dynamical; if dropped (tightly focused or off-axis beams) a longitudinal \(E_z\) appears and the two-component reduction fails.
Non-depolarizing, linear, deterministic elements.Each element maps one pure input state to one pure output state; if dropped (scattering, multiple incoherent reflections) the map is not a single Jones matrix but a convex mixture, again demanding the Mueller formalism.
Common time convention \(e^{-i\omega t}\).Fixes the sign of retarder phases; if the opposite convention \(e^{+i\omega t}\) is silently mixed in, the handedness of circular states and the sign of every retardance flips.
Derivation
1
\[ \mathbf E(\mathbf r,t) = \operatorname{Re}\!\left[\big(E_x\,\hat{\mathbf x} + E_y\,\hat{\mathbf y}\big)\,e^{\,i(kz-\omega t)}\right],\qquad E_x,E_y\in\mathbb C \]
Prior result (plane-wave solutions): a transverse plane wave along \(\hat{\mathbf z}\) has no \(z\)-component; both surviving components share the same propagator \(e^{i(kz-\omega t)}\), so all polarization information sits in the two complex constants. A
2
\[ \mathbf J \equiv \begin{pmatrix} E_x \\ E_y \end{pmatrix} = \begin{pmatrix} a_x\,e^{i\varphi_x} \\ a_y\,e^{i\varphi_y} \end{pmatrix},\qquad a_x,a_y\ge 0 \]
Definition: strip the common propagator and collect the two amplitudes into a column vector. Writing each in polar form isolates the two real amplitudes and two phases. A
3
\[ \mathbf J = e^{i\varphi_x}\begin{pmatrix} a_x \\ a_y\,e^{i\delta} \end{pmatrix},\qquad \delta \equiv \varphi_y - \varphi_x \]
Factor the global phase \(e^{i\varphi_x}\). Physical polarization is invariant under a common phase, so only the relative phase \(\delta\) and the amplitude ratio \(a_y/a_x\) are observable — three real numbers reduce to two. B
4
\[ \hat{\mathbf J} = \frac{1}{\sqrt{a_x^2 + a_y^2}}\begin{pmatrix} a_x \\ a_y\,e^{i\delta} \end{pmatrix},\qquad \hat{\mathbf J}^\dagger \hat{\mathbf J} = 1 \]
Normalize by the (time-averaged) intensity \(I \propto |E_x|^2+|E_y|^2 = \mathbf J^\dagger \mathbf J\). The normalized Jones vector lives on the unit sphere in \(\mathbb C^2\) modulo global phase — exactly the state space of one qubit. B
5
\[ \mathbf J' = M\,\mathbf J,\qquad M\in\mathbb C^{2\times 2} \]
Linearity of Maxwell's equations in the field, plus the assumption of a deterministic non-depolarizing element, forces the input–output relation to be linear and local in the two-component space; the most general such map is a \(2\times 2\) complex matrix. Two cascaded elements act as \(M_2 M_1\) (prior result: matrix multiplication is associative, order = last element leftmost). B
6
\[ P_x = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \]
Ideal linear polarizer with transmission axis along \(\hat{\mathbf x}\): by definition it passes \(E_x\) untouched and fully extinguishes \(E_y\). This is a projector — \(P_x^2 = P_x\), \(\det P_x = 0\) (prior result: a singular matrix, information is irreversibly lost). A
7
\[ R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \phantom{-}\cos\theta \end{pmatrix},\qquad P(\theta) = R(\theta)\,P_x\,R(-\theta) \]
To orient any element at angle \(\theta\), rotate the field into the element's eigenbasis, apply the element, rotate back. \(R(\theta)\) is the standard passive-to-active \(2\times2\) rotation; \(R(-\theta)=R(\theta)^{-1}=R(\theta)^{\mathsf T}\). B
8
\[ P(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\!\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\!\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} = \begin{pmatrix} \cos^2\theta & \sin\theta\cos\theta \\ \sin\theta\cos\theta & \sin^2\theta \end{pmatrix} \]
Carry out the two matrix products (prior result: matrix multiplication). The result is the projector \(|\hat{\mathbf e}_\theta\rangle\langle\hat{\mathbf e}_\theta|\) onto the unit vector \((\cos\theta,\sin\theta)^{\mathsf T}\); still idempotent, still \(\det = 0\). C
9
\[ W = \begin{pmatrix} e^{-i\Gamma/2} & 0 \\ 0 & e^{+i\Gamma/2} \end{pmatrix},\qquad \Gamma = \frac{2\pi}{\lambda}\,(n_e - n_o)\,d \]
A retarder (wave plate) is a birefringent slab: fast axis (index \(n_o\)) along \(\hat{\mathbf x}\), slow axis (index \(n_e\)) along \(\hat{\mathbf y}\). Each eigen-component accumulates its own phase \(k_i d = 2\pi n_i d/\lambda\) over thickness \(d\); factoring the mean phase leaves the symmetric retardance \(\Gamma\) split as \(\mp\Gamma/2\). It is unitary, \(W^\dagger W = \mathbf 1\), \(|\det W| = 1\): lossless. B
10
\[ \text{QWP: } \Gamma=\tfrac{\pi}{2},\quad W_{\lambda/4}=e^{-i\pi/4}\!\begin{pmatrix}1&0\\0&i\end{pmatrix};\qquad \text{HWP: } \Gamma=\pi,\quad W_{\lambda/2}=e^{-i\pi/2}\!\begin{pmatrix}1&0\\0&-1\end{pmatrix} \]
Specialize \(\Gamma\) to the quarter- and half-wave cases (drop the physically irrelevant global phase). The QWP converts linear at \(45^\circ\) to circular; the HWP mirrors linear polarization about its fast axis. B
11
\[ \hat{\mathbf J} = \begin{pmatrix} \cos\psi \\ \sin\psi\,e^{i\delta} \end{pmatrix} \;\Longrightarrow\; \begin{cases} \delta = 0,\pi: & \text{linear} \\ \psi=\tfrac\pi4,\ \delta=\pm\tfrac\pi2: & \text{circular} \\ \text{otherwise}: & \text{elliptical} \end{cases} \]
Write the general normalized state with \(a_x=\cos\psi,\ a_y=\sin\psi\). Tracing \(\big(\operatorname{Re}E_x,\operatorname{Re}E_y\big)\) over one period gives an ellipse; the three special values of \((\psi,\delta)\) degenerate it to a line or a circle. This is the complete classification of fully polarized light. C
Result
\[ \mathbf J' = M\,\mathbf J,\quad P(\theta)=\begin{pmatrix}\cos^2\theta&\sin\theta\cos\theta\\\sin\theta\cos\theta&\sin^2\theta\end{pmatrix},\quad W(\Gamma)=\begin{pmatrix}e^{-i\Gamma/2}&0\\0&e^{+i\Gamma/2}\end{pmatrix} \]

Reading. Every state of fully polarized monochromatic light is a two-component complex Jones vector, fixed by an amplitude ratio and a single relative phase \(\delta\). A polarizer is a rank-one projector (\(\det=0\), lossy, idempotent); a retarder is a unitary diagonal matrix in its eigenbasis (\(|\det|=1\), lossless). An optical train is the ordered product of these matrices, last element on the left. The relative phase \(\delta\) and amplitude ratio \(\tan\psi\) sort the state into linear, circular, or elliptical.

Units check. Jones vectors carry units of field, \(\mathrm{V\,m^{-1}}\); the matrices \(M\) are dimensionless, so \(\mathbf J'=M\mathbf J\) is dimensionally consistent. The retardance \(\Gamma=2\pi(n_e-n_o)d/\lambda\) is \([\text{dimensionless}]\times[\text{length}]/[\text{length}] = \text{radians}\), a pure number, as a phase must be. Intensity \(I=\mathbf J^\dagger\mathbf J\) has units \((\mathrm{V\,m^{-1}})^2\), proportional to irradiance.

Limiting cases
  • \(\Gamma\to 0\): \(W\to\mathbf 1\), the plate is optically absent — a "zero-wave" plate.
  • Two crossed polarizers \(P(0)P(\tfrac\pi2)=0\): total extinction (Malus' law at \(90^\circ\)).
  • Parallel polarizers \(P(\theta)P(\theta)=P(\theta)\): idempotence, a second identical polarizer does nothing.
  • Malus' law: input \(\hat{\mathbf x}\) through \(P(\theta)\) gives transmitted intensity \(I=I_0\cos^2\theta\).
  • QWP then QWP with aligned axes \(=\) HWP: \(\big(\operatorname{diag}(1,i)\big)^2=\operatorname{diag}(1,-1)\).
  • Circular states are eigenvectors of any rotator \(R(\theta)\) with eigenvalue \(e^{\mp i\theta}\): rotating the frame only adds a phase.
Breaks when
  • Partially polarized or unpolarized light. A single \(\mathbf J\) assumes a definite relative phase. Natural light has fluctuating \(\delta\); the state is then a statistical mixture and only the \(2\times2\) coherency matrix (or the four Stokes parameters with the \(4\times4\) Mueller matrix) describes it. Jones calculus silently gives wrong intensities here.
  • Depolarizing or scattering media. Multiple incoherent paths, rough surfaces, or diffuse scatterers map a pure state to a mixed one. No single \(2\times2\) matrix can reduce the purity of a Jones vector, so the description fails.
  • Non-transverse / strongly focused fields. Tight focusing or near-field optics generate a longitudinal \(E_z\); the field is no longer two-dimensional and the reduction to \(\mathbb C^2\) collapses.
  • Nonlinear optics. At high intensity the response depends on the field amplitude (e.g. self-phase modulation, four-wave mixing); \(M\) ceases to be field-independent and the superposition that underpins step 5 is lost.
Failure modes
  • Multiplying matrices in the wrong order. Light meets element 1 first, but its matrix goes on the right: the train is \(M_n\cdots M_2 M_1\). Writing \(M_1 M_2\) reverses the physical sequence.
  • Dropping the relative phase. Treating \(\mathbf J\) as a real vector of amplitudes loses \(\delta\) entirely, so circular and \(45^\circ\)-linear light become indistinguishable — a fatal error.
  • Sign of the retardance / time convention. Mixing \(e^{-i\omega t}\) and \(e^{+i\omega t}\) flips the sign of \(\Gamma/2\) and swaps left- for right-circular light.
  • Forgetting to normalize before computing intensity ratios, or conversely normalizing away a global phase that was needed to interfere two beams coherently downstream.
  • Using Jones matrices for a depolarizer (e.g. modeling a diffuser or a partial reflection as a \(2\times2\) matrix) — a category error that only Mueller/Stokes can handle.
  • Confusing the polarizer angle convention (axis vs. blocked direction), giving \(\cos^2\theta\) where \(\sin^2\theta\) is meant.
Discussion

The deepest content of the Jones calculus is that fully polarized light is a genuine two-level quantum-like system. The normalized Jones vector modulo global phase is a point on the Poincaré sphere, which is nothing other than the Bloch sphere of a qubit: linear states sit on the equator, right- and left-circular states at the two poles, and elliptical states everywhere in between. Retarders are unitary \(SU(2)\) rotations of this sphere, while polarizers are non-unitary projective measurements. Every polarization experiment is thus a hands-on single-qubit laboratory.

The split between lossless and lossy elements is exactly the split between unitary and non-unitary matrices. A retarder has \(W^\dagger W=\mathbf 1\), so it conserves intensity, \(\mathbf J'^\dagger\mathbf J' = \mathbf J^\dagger\mathbf J\); its determinant is a pure phase. A polarizer has \(\det P=0\); it is a rank-one projector that irreversibly discards one dimension of information. The determinant, introduced abstractly as the area-scaling factor of a linear map, here acquires a physical reading: \(|\det M|^2\) is the intensity-transmission factor of the element for unpolarized input.

The rotator \(R(\theta)\) illuminates the symmetry thread. Because circular states are its eigenvectors with eigenvalue \(e^{\mp i\theta}\), the two circular polarizations are the natural basis adapted to rotational symmetry about the propagation axis — they carry definite spin angular momentum \(\pm\hbar\) per photon. Choosing the circular basis \(\{|L\rangle,|R\rangle\}\) instead of \(\{|H\rangle,|V\rangle\}\) diagonalizes rotations, precisely as choosing angular-momentum eigenstates diagonalizes rotations in quantum mechanics.

At the most rigorous level, the set of realizable Jones matrices for lossless elements is \(U(2)=U(1)\times SU(2)\): the \(U(1)\) factor is the physically irrelevant global phase, and the \(SU(2)\) factor is the double cover of the rotation group \(SO(3)\) acting on the Poincaré sphere. This is why a half-wave plate rotated by \(\theta\) rotates the linear polarization by \(2\theta\): the two-to-one homomorphism \(SU(2)\to SO(3)\) doubles angles. The same factor of two is the geometric (Pancharatnam–Berry) phase acquired when a state is carried around a closed loop on the Poincaré sphere — equal to half the enclosed solid angle — the optical analogue of Berry's phase for a spin in a slowly varying field.

Common misconceptions. (i) A polarizer does not "rotate" light into its axis — it projects, discarding the orthogonal component and its energy; only a rotator or retarder redistributes without loss. (ii) Circular light is not "two linear beams" in any observable sense until you choose a basis; the state is basis-independent. (iii) The Jones matrix of an element depends on the chosen coordinate axes and time convention, so published matrices differ by signs and global phases that are physically harmless but must be applied consistently across a whole calculation.

Worked examples
1
Producing circular light with a quarter-wave plate. Horizontally polarized light of intensity \(I_0=1.0\ \mathrm{W\,m^{-2}}\) is sent through a QWP whose fast axis is at \(45^\circ\). Find the output state and confirm it is circular. A
\[ \hat{\mathbf J}_{\text{in}} = \begin{pmatrix} 1 \\ 0 \end{pmatrix},\qquad W_{45^\circ} = R(45^\circ)\begin{pmatrix}1&0\\0&i\end{pmatrix}R(-45^\circ) \]
2
\[ W_{45^\circ} = \tfrac12\begin{pmatrix}1&-1\\1&1\end{pmatrix}\begin{pmatrix}1&0\\0&i\end{pmatrix}\begin{pmatrix}1&1\\-1&1\end{pmatrix} = \tfrac12\begin{pmatrix}1+i & 1-i \\ 1-i & 1+i\end{pmatrix} \]
Insert \(\cos45^\circ=\sin45^\circ=1/\sqrt2\) and multiply the three matrices (global phase \(e^{-i\pi/4}\) dropped). B
3
\[ \hat{\mathbf J}_{\text{out}} = \tfrac12\begin{pmatrix}1+i & 1-i \\ 1-i & 1+i\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \tfrac12\begin{pmatrix}1+i\\1-i\end{pmatrix} = \tfrac{1}{\sqrt2}\,e^{i\pi/4}\begin{pmatrix}1\\-i\end{pmatrix} \]
Apply the matrix; factor the common phase. The residual \(\begin{pmatrix}1\\-i\end{pmatrix}\) has \(\psi=45^\circ,\ \delta=-\pi/2\): right-circular. B
\[ \hat{\mathbf J}_{\text{out}} = \tfrac{1}{\sqrt2}\begin{pmatrix}1\\-i\end{pmatrix},\qquad I_{\text{out}} = I_0 = 1.0\ \mathrm{W\,m^{-2}} \]

Reading. A QWP at \(45^\circ\) turns horizontal linear light into right-circular light; the plate is unitary so no energy is lost.

Units check. The matrix is dimensionless and unitary, so \(I_{\text{out}}=\mathbf J^\dagger\mathbf J = I_0\), preserving \(\mathrm{W\,m^{-2}}\).

1
Malus' law through a rotated polarizer. Vertically polarized light of intensity \(I_0=2.0\ \mathrm{W\,m^{-2}}\) passes through a linear polarizer whose transmission axis is at \(\theta=30^\circ\) from vertical. Find the transmitted intensity and state. A
\[ \hat{\mathbf J}_{\text{in}} = \begin{pmatrix}0\\1\end{pmatrix},\qquad P(\theta)=\begin{pmatrix}\cos^2\theta & \sin\theta\cos\theta \\ \sin\theta\cos\theta & \sin^2\theta\end{pmatrix} \]
2
\[ \mathbf J_{\text{out}} = P(\theta)\begin{pmatrix}0\\1\end{pmatrix} = \begin{pmatrix}\sin\theta\cos\theta \\ \sin^2\theta\end{pmatrix} = \sin\theta\begin{pmatrix}\cos\theta\\\sin\theta\end{pmatrix} \]
The output points along the transmission axis (as a projector demands), scaled by \(\sin\theta\) — the projection of the vertical input onto the axis. B
3
\[ \frac{I_{\text{out}}}{I_0} = |\sin\theta|^2 = \sin^2 30^\circ = \left(\tfrac12\right)^2 = \tfrac14,\qquad I_{\text{out}} = 0.25\times 2.0 = 0.50\ \mathrm{W\,m^{-2}} \]
Intensity is \(\mathbf J_{\text{out}}^\dagger\mathbf J_{\text{out}} = \sin^2\theta\); here the input is at \(90^\circ-\theta=60^\circ\) to the axis, so \(\cos^2 60^\circ=\sin^2 30^\circ=\tfrac14\), consistent with Malus. A
\[ I_{\text{out}} = 0.50\ \mathrm{W\,m^{-2}},\qquad \hat{\mathbf J}_{\text{out}} = \begin{pmatrix}\cos 30^\circ\\\sin 30^\circ\end{pmatrix} \]

Reading. The polarizer transmits one quarter of the incident power and re-orients the surviving light along its own axis; the discarded three quarters are absorbed.

Units check. \(I_{\text{out}}=I_0\sin^2\theta\): dimensionless factor times \(\mathrm{W\,m^{-2}}\) gives \(\mathrm{W\,m^{-2}}\).

Problems
  1. Show that the ideal linear polarizer \(P(\theta)\) is idempotent, \(P(\theta)^2 = P(\theta)\), and compute \(\det P(\theta)\).
    Solution Using \(P(\theta)=\begin{pmatrix}\cos^2\theta&\sin\theta\cos\theta\\\sin\theta\cos\theta&\sin^2\theta\end{pmatrix}\), the \((1,1)\) entry of \(P^2\) is \(\cos^4\theta+\sin^2\theta\cos^2\theta=\cos^2\theta(\cos^2\theta+\sin^2\theta)=\cos^2\theta\). The \((1,2)\) entry is \(\cos^2\theta\sin\theta\cos\theta+\sin\theta\cos\theta\sin^2\theta=\sin\theta\cos\theta\). By symmetry all entries reproduce \(P(\theta)\), so \(P^2=P\). The determinant is \(\cos^2\theta\sin^2\theta-(\sin\theta\cos\theta)^2=0\). A projector is singular and non-invertible: transmitted light cannot be "un-polarized."
  2. Two ideal polarizers are crossed (\(0^\circ\) and \(90^\circ\)). A third is inserted between them at \(45^\circ\). Unpolarized light of intensity \(I_0\) enters. Find the final intensity. (An ideal polarizer transmits half of unpolarized light.)
    Solution After the first polarizer: \(I_1=\tfrac12 I_0\), polarized at \(0^\circ\). Through the \(45^\circ\) polarizer, Malus gives \(I_2=I_1\cos^2 45^\circ=\tfrac12 I_1=\tfrac14 I_0\), now polarized at \(45^\circ\). Through the \(90^\circ\) polarizer, the angle from \(45^\circ\) to \(90^\circ\) is \(45^\circ\): \(I_3=I_2\cos^2 45^\circ=\tfrac12 I_2=\tfrac18 I_0\). Result: \(I_3=\tfrac18 I_0\). With the middle polarizer removed the crossed pair gives zero, so inserting it lets light through — a classic demonstration that projection is not "filtering."
  3. A half-wave plate has its fast axis at angle \(\alpha\). Show that it maps linear polarization at angle \(\phi\) to linear polarization at angle \(2\alpha-\phi\).
    Solution The HWP in its own axes is \(\operatorname{diag}(1,-1)\) (global phase dropped). Rotated to angle \(\alpha\): \(W=R(\alpha)\operatorname{diag}(1,-1)R(-\alpha)=\begin{pmatrix}\cos2\alpha&\sin2\alpha\\\sin2\alpha&-\cos2\alpha\end{pmatrix}\). Acting on the input \((\cos\phi,\sin\phi)^{\mathsf T}\): first component \(\cos2\alpha\cos\phi+\sin2\alpha\sin\phi=\cos(2\alpha-\phi)\); second \(\sin2\alpha\cos\phi-\cos2\alpha\sin\phi=\sin(2\alpha-\phi)\). So the output is linear at \(2\alpha-\phi\), a mirror reflection of the input about the fast axis. Note the doubling of \(\alpha\), the \(SU(2)\to SO(3)\) signature.
  4. Light in the state \(\hat{\mathbf J}=\tfrac{1}{\sqrt2}\begin{pmatrix}1\\i\end{pmatrix}\) (left-circular) passes through a quarter-wave plate with fast axis along \(\hat{\mathbf x}\). Identify the output state.
    Solution \(W_{\lambda/4}=\operatorname{diag}(1,i)\) (dropping \(e^{-i\pi/4}\)). Then \(\mathbf J_{\text{out}}=\tfrac{1}{\sqrt2}\begin{pmatrix}1&0\\0&i\end{pmatrix}\begin{pmatrix}1\\i\end{pmatrix}=\tfrac{1}{\sqrt2}\begin{pmatrix}1\\i^2\end{pmatrix}=\tfrac{1}{\sqrt2}\begin{pmatrix}1\\-1\end{pmatrix}\). This is linear polarization at \(-45^\circ\) (i.e. \(135^\circ\)). A QWP converts circular light back to linear, oriented at \(\mp45^\circ\) depending on handedness — the inverse of Worked Example 1.
  5. A retarder of retardance \(\Gamma\) has fast axis along \(\hat{\mathbf x}\). Horizontal light \((1,0)^{\mathsf T}\) enters. For what \(\Gamma\) is the output still exactly horizontal, and what is the smallest nonzero such \(\Gamma\)? Then find the output for \(\Gamma=\pi/3\).
    Solution \(W=\operatorname{diag}(e^{-i\Gamma/2},e^{+i\Gamma/2})\) acting on \((1,0)^{\mathsf T}\) gives \((e^{-i\Gamma/2},0)^{\mathsf T}\) — always along \(\hat{\mathbf x}\) for every \(\Gamma\), since horizontal light lies on the fast eigen-axis and merely picks up a phase. So the polarization is unchanged for all \(\Gamma\); the smallest nonzero \(\Gamma\) returning the same absolute phase state (mod global phase) is any value, but for the field itself to repeat, \(\Gamma=2\pi\) (a full-wave plate). For \(\Gamma=\pi/3\): output \(=e^{-i\pi/6}(1,0)^{\mathsf T}\), still horizontal, intensity unchanged. Lesson: an eigen-polarization of a retarder is invariant up to phase — retarders only act on superpositions of both axes.