Physics note

Polarization of light

Polarization of light, and related things

Introduction

Polarization describes the orientation and temporal evolution of the electric field vector of an electromagnetic wave. At the classical level, it is characterized by the Jones vector and Jones calculus. At the quantum level, photon polarization is a two-level quantum degree of freedom central to optical experiments, quantum information processing, and the study of light-matter interactions in atomic media. Polarization effects, combined with coherent laser fields, enable control of atomic transitions through optical pumping and polarization-dependent selection rules.

1. Classical Polarization: The Ellipse and the Jones Vector

1.1 The Electromagnetic Plane Wave and the Polarization Ellipse

Consider a monochromatic, uniform plane wave of angular frequency $\omega$ propagating in the $+z$ direction in free space. The electric field vector lies entirely in the $xy$-plane and can be written as \begin{equation} \mathbf{E}(z, t) = \operatorname{Re}\left[ \begin{pmatrix} E_{0x} e^{i\delta_x} \\ E_{0y} e^{i\delta_y} \end{pmatrix} e^{i(kz - \omega t)} \right], \label{eq:E_field} \end{equation} where $E_{0x}, E_{0y} \ge 0$ are the real amplitudes along the $x$ and $y$ axes, and $\delta_x, \delta_y$ are the corresponding phases. The physical electric field at a fixed plane $z = 0$ traces out an ellipse as a function of time. To see this, we define the relative phase \begin{equation} \delta = \delta_y - \delta_x, \end{equation} and write the components as \begin{equation} E_x(t) = E_{0x} \cos(\omega t - \delta_x), \qquad E_y(t) = E_{0y} \cos(\omega t - \delta_y). \end{equation} Eliminating the time dependence yields the equation of an ellipse: \begin{equation} \frac{E_x^2}{E_{0x}^2} + \frac{E_y^2}{E_{0y}^2} - 2 \frac{E_x E_y}{E_{0x} E_{0y}} \cos\delta = \sin^2\delta. \label{eq:ellipse} \end{equation} The ellipse is characterized by three independent parameters: the amplitudes $E_{0x}$, $E_{0y}$ (or their ratio), the relative phase $\delta$, and the absolute phase (which determines the orientation of the major axis in the $xy$-plane). The sense of rotation (handedness) is determined by the sign of $\sin\delta$: for $0 < \delta < \pi$, the ellipse is traced in a right-handed sense (clockwise when viewed by an observer looking into the oncoming beam); for $-\pi < \delta < 0$, it is left-handed.

The main special cases are easy to keep in mind. Linear polarization occurs when the two components are in phase or exactly out of phase, so the field traces a straight line. Circular polarization occurs when the two orthogonal components have equal amplitude and a quarter-cycle phase difference, so the field traces a circle. Elliptical polarization is the general case in between, where the field traces an ellipse with a definite handedness.

1.2 The Jones Vector: A Complex Two-Component Representation

For fully polarized light, the polarization state is completely specified by the complex amplitude of the electric field in a chosen basis. The Jones vector is defined as \begin{equation} \boxed{| J \rangle = \begin{pmatrix} E_x \\ E_y \end{pmatrix} = \begin{pmatrix} E_{0x} e^{i\delta_x} \\ E_{0y} e^{i\delta_y} \end{pmatrix}}. \label{eq:jones_vector} \end{equation} The notation $| J \rangle$ (a "ket") is deliberately chosen to emphasize the mathematical analogy with quantum state vectors, though here it is a classical vector. The Jones vector contains both amplitude and phase information. The overall phase factor is usually irrelevant for describing the polarization state; it is the relative amplitude and phase that matter. The intensity of the wave is proportional to the squared norm: \begin{equation} I = |E_x|^2 + |E_y|^2 = E_{0x}^2 + E_{0y}^2. \end{equation} It is customary to work with normalized Jones vectors, for which $I = 1$.

The most common normalized Jones vectors are:

1.3 Orthogonality and the Polarization Basis

Two Jones vectors $| J_1 \rangle$ and $| J_2 \rangle$ are said to represent orthogonal polarization states if their Hermitian inner product vanishes: \begin{equation} \langle J_1 | J_2 \rangle = E_{1x}^* E_{2x} + E_{1y}^* E_{2y} = 0. \end{equation} For example, H and V are orthogonal; D and A are orthogonal; R and L are orthogonal. Any pair of orthogonal polarization states forms a basis for the two-dimensional polarization space. A general elliptically polarized state can be decomposed onto any such basis, analogous to the decomposition of a spin-1/2 state onto the $\sigma_z$ or $\sigma_x$ eigenbases.

[Figure 1: (a) The polarization ellipse traced by the electric field vector in the $xy$-plane. The parameters $E_{0x}$, $E_{0y}$, and $\delta$ define the ellipse. (b) The Poincaré sphere representation: all fully polarized states lie on the surface of the unit sphere, with linear states on the equator and circular states at the poles. The Jones vectors for H, V, D, A, R, and L are indicated.]

2. Jones Calculus: Matrix Representation of Polarization Elements

2.1 The Jones Matrix

A non-depolarizing, linear optical element that transforms an input polarization state $| J_{\text{in}} \rangle$ into an output state $| J_{\text{out}} \rangle$ is represented by a $2 \times 2$ complex Jones matrix $\mathbf{J}$: \begin{equation} \boxed{| J_{\text{out}} \rangle = \mathbf{J} \, | J_{\text{in}} \rangle}. \label{eq:jones_matrix} \end{equation} The Jones matrix is the classical analog of the unitary evolution operator acting on a two-level quantum system. If the optical element is lossless, $\mathbf{J}$ is unitary: $\mathbf{J}^\dagger \mathbf{J} = \mathbb{1}$. If there is polarization-dependent loss (dichroism), $\mathbf{J}$ is non-unitary but remains linear.

A sequence of $N$ optical elements with Jones matrices $\mathbf{J}_1, \mathbf{J}_2, \dots, \mathbf{J}_N$ (encountered in order from first to last) produces an overall transformation \begin{equation} \mathbf{J}_{\text{total}} = \mathbf{J}_N \cdots \mathbf{J}_2 \mathbf{J}_1. \end{equation} Note the ordering: matrices are applied from right to left, as in quantum mechanics. This composition property is a major strength of Jones calculus.

2.2 The Ideal Linear Polarizer

An ideal linear polarizer with its transmission axis along the $x$-direction transmits only the $x$-component of the field and completely absorbs the $y$-component. Its Jones matrix is \begin{equation} \boxed{\mathbf{P}_H = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}}. \label{eq:polarizer_H} \end{equation} For a polarizer with transmission axis at an angle $\theta$ from the $x$-axis, the Jones matrix is obtained by rotating the coordinate system: \begin{equation} \mathbf{P}(\theta) = \mathbf{R}(-\theta) \, \mathbf{P}_H \, \mathbf{R}(\theta) = \begin{pmatrix} \cos^2\theta & \sin\theta\cos\theta \\ \sin\theta\cos\theta & \sin^2\theta \end{pmatrix}, \label{eq:polarizer_theta} \end{equation} where $\mathbf{R}(\theta)$ is the rotation matrix defined below. A polarizing beam splitter (PBS) can be modeled as a device that transmits H-polarized light and reflects V-polarized light into orthogonal output ports (see Section 2.6).

2.3 Waveplates (Retarders) and the Phase Shift Operator

A waveplate (or retarder) is an optical element made of a birefringent material—typically a uniaxial crystal such as quartz, calcite, or magnesium fluoride—in which the refractive index depends on the polarization direction of the light. The crystal has two orthogonal principal axes, called the fast axis and the slow axis. Light polarized along the fast axis experiences a refractive index $n_f$ and propagates with phase velocity $c/n_f$; light polarized along the slow axis experiences $n_s > n_f$ and travels more slowly. After traversing a crystal of thickness $d$, the two polarization components accumulate a relative phase difference (the retardance): \begin{equation} \boxed{\Gamma = \frac{2\pi}{\lambda} (n_s - n_f) d}, \label{eq:retardance} \end{equation} where $\lambda$ is the vacuum wavelength. The waveplate thus imposes a phase shift $+\Gamma/2$ on the slow-axis component and $-\Gamma/2$ on the fast-axis component (relative to the mean phase). The Jones matrix for a waveplate with its fast axis along the $x$-direction is the phase shift operator: \begin{equation} \boxed{\mathbf{W}_0(\Gamma) = \begin{pmatrix} e^{-i\Gamma/2} & 0 \\ 0 & e^{+i\Gamma/2} \end{pmatrix} = e^{-i\Gamma/2} \begin{pmatrix} 1 & 0 \\ 0 & e^{i\Gamma} \end{pmatrix}}. \label{eq:waveplate_zero} \end{equation} The overall phase factor $e^{-i\Gamma/2}$ is often dropped since it does not affect the polarization state (though it can be important in interferometric setups). For a waveplate with its fast axis at an angle $\phi$ from the $x$-axis, we apply the rotation transformation: \begin{equation} \boxed{\mathbf{W}_\phi(\Gamma) = \mathbf{R}(-\phi) \, \mathbf{W}_0(\Gamma) \, \mathbf{R}(\phi)}, \label{eq:waveplate_phi} \end{equation} where the rotation matrix is \begin{equation} \mathbf{R}(\theta) = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}. \label{eq:rotation} \end{equation}

Two especially useful waveplate cases are worth keeping in mind. A half-wave plate has $\Gamma = \pi$ and flips the polarization ellipse about its fast axis. It is often used to rotate the direction of linear polarization or to reverse the handedness of circular polarization. A quarter-wave plate has $\Gamma = \pi/2$ and is especially useful for converting linear polarization into circular polarization, and vice versa, depending on its orientation relative to the input field.

2.4 The Phase Shift Operator in the Language of Coherent States

The classical Jones calculus has a direct quantum analog. In the quantum description, the two orthogonal polarization modes (e.g., H and V) are independent quantized field modes with annihilation operators $\hat{a}_H$ and $\hat{a}_V$. A coherent state in a general polarization state is a two-mode coherent state: \begin{equation} | \alpha_H, \alpha_V \rangle = e^{-(|\alpha_H|^2 + |\alpha_V|^2)/2} \sum_{n_H, n_V = 0}^\infty \frac{\alpha_H^{n_H} \alpha_V^{n_V}}{\sqrt{n_H! n_V!}} | n_H, n_V \rangle, \end{equation} where $\alpha_H$ and $\alpha_V$ are the complex coherent amplitudes for the two polarization modes. The classical Jones vector components are proportional to these amplitudes: $E_x \propto \alpha_H$, $E_y \propto \alpha_V$.

A lossless polarization element is represented by a unitary operator $\hat{U}$ acting on the two-mode Hilbert space. For a waveplate with fast axis along $x$ and retardance $\Gamma$, the unitary operator is \begin{equation} \boxed{\hat{U}_{\text{WP}}(\Gamma) = e^{-i\Gamma (\hat{a}_H^\dagger \hat{a}_H - \hat{a}_V^\dagger \hat{a}_V)/2}}. \label{eq:U_waveplate} \end{equation} This operator is the exponential of the Stokes operator $\hat{S}_1 = \hat{a}_H^\dagger \hat{a}_H - \hat{a}_V^\dagger \hat{a}_V$. Under this unitary transformation, the field operators evolve as \begin{align} \hat{U}^\dagger \hat{a}_H \hat{U} &= e^{i\Gamma/2} \hat{a}_H, \label{eq:U_aH} \\ \hat{U}^\dagger \hat{a}_V \hat{U} &= e^{-i\Gamma/2} \hat{a}_V. \end{align} For a coherent state input, the unitary acts on the coherent amplitudes exactly as the Jones matrix acts on the Jones vector: \begin{equation} \hat{U}_{\text{WP}}(\Gamma) | \alpha_H, \alpha_V \rangle = | e^{-i\Gamma/2} \alpha_H, e^{+i\Gamma/2} \alpha_V \rangle. \label{eq:U_coherent} \end{equation} Thus, for coherent-state inputs, the quantum transformation of the amplitudes is identical to the classical Jones matrix transformation (up to an irrelevant global phase). This is a special case of the general correspondence between classical linear optics and unitary Gaussian operations on coherent states. The phase shift operator $\hat{U}_{\text{WP}}$ is the quantum mechanical implementation of the waveplate.

A waveplate with its fast axis rotated by angle $\phi$ is described by the unitary \begin{equation} \hat{U}_{\text{WP}}(\Gamma, \phi) = e^{-i\phi \hat{S}_2} \, \hat{U}_{\text{WP}}(\Gamma) \, e^{i\phi \hat{S}_2}, \end{equation} where $\hat{S}_2 = \hat{a}_H^\dagger \hat{a}_V + \hat{a}_V^\dagger \hat{a}_H$ is the Stokes operator generating rotations in the H/V basis. The rotation operator $e^{i\theta \hat{S}_2/2}$ corresponds precisely to the rotation matrix $\mathbf{R}(\theta)$ of the classical Jones calculus.

2.5 Polarization Rotators: Optical Activity

Some materials (e.g., sugar solutions, quartz along the optic axis) exhibit optical activity: they rotate the plane of linear polarization without introducing ellipticity. This effect arises from a difference in refractive indices for left- and right-circular polarizations (circular birefringence). The Jones matrix for a rotator that rotates the polarization by an angle $\theta$ is simply the rotation matrix: \begin{equation} \boxed{\mathbf{J}_{\text{rot}}(\theta) = \mathbf{R}(\theta) = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}}. \label{eq:rotator} \end{equation} In the quantum picture, the polarization rotator is implemented by the unitary \begin{equation} \hat{U}_{\text{rot}}(\theta) = e^{-i\theta \hat{S}_3}, \end{equation} where $\hat{S}_3 = i(\hat{a}_H^\dagger \hat{a}_V - \hat{a}_V^\dagger \hat{a}_H)$ is the third Stokes operator, proportional to the circular polarization degree. Optical rotation is the basis of polarimetry, a sensitive analytical technique for measuring concentrations of chiral molecules.

2.6 Polarizing Beam Splitters (PBS) and the Quantum Description

A polarizing beam splitter (PBS) is a crucial component that spatially separates orthogonal polarization components. An ideal PBS transmits horizontally polarized light (H) and reflects vertically polarized light (V). Unlike a waveplate or rotator, a PBS is a four-port device with two input modes and two output modes. The classical Jones calculus is inadequate to describe a PBS; one needs to consider spatial modes as well as polarization. In the quantum description, the PBS is a linear optical element that transforms the input field operators $(\hat{a}_{H,\text{in}}, \hat{a}_{V,\text{in}})$ into output operators $(\hat{a}_{H,\text{out}}, \hat{a}_{V,\text{out}})$ for the transmitted port, and $(\hat{b}_{H,\text{out}}, \hat{b}_{V,\text{out}})$ for the reflected port: \begin{align} \hat{a}_{H,\text{out}} &= \hat{a}_{H,\text{in}}, \qquad \hat{a}_{V,\text{out}} = \hat{b}_{V,\text{in}}, \\ \hat{b}_{H,\text{out}} &= \hat{b}_{H,\text{in}}, \qquad \hat{b}_{V,\text{out}} = \hat{a}_{V,\text{in}}. \end{align} The PBS swaps the V component of the transmitted input with the H component of the reflected input. This operation is unitary on the four-mode Hilbert space. PBSs are essential for implementing polarization measurements (by converting polarization information into spatial which-path information) and for generating polarization entanglement in quantum optics experiments.

[Figure 2: (a) The operation of a half-wave plate (HWP) and a quarter-wave plate (QWP) on linear and circular polarization states. (b) A polarizing beam splitter (PBS) separating H and V components into orthogonal output ports. (c) The quantum circuit representation of a PBS as a mode-swapping operation.]

3. Phase Shifts from Atomic Media: Birefringence and EIT

3.1 Linear Susceptibility and Refractive Index

When light propagates through an atomic vapor, the atoms respond to the electric field by developing an induced polarization, which in turn modifies the propagation of the field. For a weak probe field, the medium's response is characterized by the linear susceptibility $\chi(\omega)$, a complex quantity whose real and imaginary parts determine the refractive index and the absorption coefficient, respectively: \begin{equation} n(\omega) = 1 + \frac{1}{2} \operatorname{Re}[\chi(\omega)], \qquad \alpha(\omega) = \frac{\omega}{c} \operatorname{Im}[\chi(\omega)]. \end{equation} For a two-level atom, the susceptibility is \begin{equation} \chi(\omega) = -\frac{\mathcal{N} |d_{eg}|^2}{\varepsilon_0 \hbar} \frac{1}{\Delta + i\Gamma/2}, \end{equation} where $\mathcal{N}$ is the atomic density, $d_{eg}$ is the dipole matrix element, $\Delta = \omega - \omega_0$ is the detuning, and $\Gamma$ is the spontaneous emission rate. The real part $\operatorname{Re}[\chi]$ gives a dispersive line shape, while $\operatorname{Im}[\chi]$ gives an absorptive Lorentzian.

3.2 Polarization-Dependent Phase Shifts and Atomic Birefringence

If the atomic medium is prepared in an anisotropic state—for example, by optical pumping with circularly polarized light that orients the atomic dipoles—the susceptibility becomes polarization-dependent. The medium exhibits atomic birefringence: the refractive index for right-circular polarization ($n_R$) differs from that for left-circular polarization ($n_L$). The difference $\Delta n = n_R - n_L$ leads to a relative phase shift between the R and L components of a probe beam, rotating its linear polarization. This is the principle of polarization spectroscopy and nonlinear magneto-optical rotation (NMOR).

Quantitatively, for a probe beam propagating along the $z$-direction through an atomic vapor of length $L$, the Jones matrix for circular birefringence is \begin{equation} \mathbf{J}_{\text{atomic}} = \begin{pmatrix} \cos(\Delta\phi/2) & \sin(\Delta\phi/2) \\ -\sin(\Delta\phi/2) & \cos(\Delta\phi/2) \end{pmatrix}_{\text{lin}}, \end{equation} where $\Delta\phi = (2\pi L / \lambda)(n_R - n_L)$ is the circular phase difference. In the R/L basis, this is simply $\operatorname{diag}(e^{i\Delta\phi/2}, e^{-i\Delta\phi/2})$, analogous to the waveplate matrix (\ref{eq:waveplate_zero}) but in the circular basis. The rotation angle of a linearly polarized probe is $\Delta\phi/2$.

3.3 EIT-Enhanced Phase Shifts and Slow-Light Polarization Rotation

As we discussed in the notes on dark states and EIT, electromagnetically induced transparency dramatically modifies the susceptibility of a medium: the absorption vanishes on two-photon resonance, while the dispersion becomes very steep. For a $\Lambda$-system under EIT conditions, the susceptibility for the probe field is given by (see the EIT notes, Eq. (\ref{eq:chi_EIT})): \begin{equation} \chi_{\text{EIT}}(\Delta_p) = \frac{i \mathcal{N} |d_{31}|^2}{\varepsilon_0 \hbar} \, \frac{1}{\Gamma/2 + i\Delta_p + \dfrac{|\Omega_c|^2/4}{\gamma_{12} + i\delta}}. \end{equation} On two-photon resonance ($\delta = 0$) and at exact probe resonance ($\Delta_p = 0$), $\operatorname{Im}[\chi_{\text{EIT}}] = 0$ (perfect transparency), while the slope of $\operatorname{Re}[\chi_{\text{EIT}}]$ can be extremely large when $\gamma_{12}$ is small. This steep dispersion implies that a small detuning produces a large refractive index change.

If the EIT medium is prepared such that the control field couples only to one circular polarization component (or one Zeeman sublevel), the EIT window exists only for that polarization. The medium then acts as a giant circular birefringence element: one circular polarization experiences a large, steep dispersion, while the orthogonal polarization propagates as in a two-level medium with normal dispersion and absorption. This can produce enormous polarization rotations—by many radians—for a very small magnetic field or laser detuning. Such EIT-enhanced polarization rotations are the basis of ultrasensitive atomic magnetometers and are also being explored for quantum non-demolition measurements of atomic spin states.

In the quantum operator picture, the unitary transformation for an EIT-based polarization rotator is \begin{equation} \hat{U}_{\text{EIT}}(\phi) = e^{-i\phi \hat{S}_3}, \end{equation} exactly the same form as the rotator operator, but with a phase $\phi$ that can be controlled by the control field intensity, detuning, or an external magnetic field. This enables fast, coherent control of the polarization state of single photons and weak coherent pulses—an essential capability for photonic quantum logic.

[Figure 3: (a) Real and imaginary parts of the linear susceptibility for a two-level atom (dispersive and absorptive line shapes). (b) The EIT susceptibility: zero absorption at resonance and steep dispersion. (c) Polarization rotation of a weak probe beam passing through an EIT medium with control-field-induced circular birefringence.]

4. Applications of Jones Calculus: Worked Examples

4.1 Example 1: A QWP Followed by a Linear Polarizer

Consider a quarter-wave plate with its fast axis horizontal ($\phi = 0$), followed by a linear polarizer with transmission axis at $45^\circ$ to the horizontal. The input beam is linearly polarized at an arbitrary angle $\theta$: $| J_{\text{in}} \rangle = \begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix}$. The overall Jones matrix is $\mathbf{J} = \mathbf{P}(45^\circ) \mathbf{W}_{\text{QWP}}(0^\circ)$. Using (\ref{eq:QWP}) with $\phi = 0$ and (\ref{eq:polarizer_theta}) with $\theta = 45^\circ$, \begin{equation} \mathbf{W}_{\text{QWP}}(0^\circ) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1-i & 0 \\ 0 & 1+i \end{pmatrix}, \qquad \mathbf{P}(45^\circ) = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}. \end{equation} The output intensity as a function of input polarization angle $\theta$ is $I_{\text{out}}(\theta) = |\mathbf{J} | J_{\text{in}} \rangle|^2$. The result is a sinusoidal modulation with period $\pi$, enabling the precise measurement of the input polarization angle. This is the principle of a rotating-analyzer ellipsometer.

4.2 Example 2: Coherent-State Input Through a Birefringent Medium

We now treat the same problem in the quantum coherent-state picture. An input coherent state $| \alpha_H, \alpha_V \rangle$ with $\alpha_H = \alpha_0 \cos\theta$, $\alpha_V = \alpha_0 \sin\theta$ passes through a birefringent crystal of retardance $\Gamma$, with fast axis horizontal. The unitary operator is $\hat{U} = e^{-i\Gamma (\hat{n}_H - \hat{n}_V)/2}$. The output state is \begin{equation} | \psi_{\text{out}} \rangle = \hat{U} | \alpha_H, \alpha_V \rangle = | e^{-i\Gamma/2} \alpha_0 \cos\theta, e^{+i\Gamma/2} \alpha_0 \sin\theta \rangle. \end{equation} The mean photon number in the horizontal mode is $\langle \hat{n}_H \rangle = |\alpha_H|^2 = \alpha_0^2 \cos^2\theta$, unchanged by the waveplate. The expectation value of the Stokes operator $\hat{S}_1 = \hat{n}_H - \hat{n}_V$ is \begin{equation} \langle \hat{S}_1 \rangle = \alpha_0^2 (\cos^2\theta - \sin^2\theta) = \alpha_0^2 \cos(2\theta). \end{equation} If we measure the intensity after a polarizer at $45^\circ$ (corresponding to measuring $\hat{S}_2$), we find \begin{equation} \langle \hat{S}_2 \rangle = \alpha_0^2 \sin(2\theta) \cos\Gamma, \end{equation} which exhibits interference fringes as a function of $\Gamma$. This is precisely the quantum version of the classical Jones calculus result. The equivalence is exact because the input is a coherent state; for non-classical inputs (e.g., Fock states), the quantum treatment yields results that cannot be obtained from classical Jones calculus.

4.3 Example 3: A Polarization Mach–Zehnder Interferometer

A powerful configuration uses polarization as the "which-path" degree of freedom. Consider a Mach–Zehnder interferometer where a PBS at the input splits H and V into two spatial paths, each of which may contain a phase shifter or other element, and the paths are recombined at a second PBS. The overall transformation is a unitary mixing of the H and V modes. With a phase difference $\Delta\phi$ between the two arms, the Jones matrix for the interferometer (in the H/V basis) is \begin{equation} \mathbf{J}_{\text{MZ}} = \begin{pmatrix} e^{i\Delta\phi/2} & 0 \\ 0 & e^{-i\Delta\phi/2} \end{pmatrix}, \end{equation} which is exactly the matrix of a waveplate with retardance $\Delta\phi$. By measuring the output polarization, one can infer the phase difference $\Delta\phi$ with quantum-limited sensitivity. When squeezed light is injected into the unused input port, the phase sensitivity can surpass the SQL—this is the principle of polarization-squeezed interferometry.

[Figure 4: (a) A polarization Mach–Zehnder interferometer using two PBSs and a birefringent phase shifter. (b) The Poincaré sphere trajectory of the polarization state as the phase difference is varied. (c) The measured intensity at the output as a function of the phase difference, demonstrating interference fringes.]

5. The Stokes Parameters and the Poincaré Sphere

5.1 Classical Stokes Parameters

While the Jones vector fully describes fully polarized light, partially polarized light requires the Stokes parameters, defined as \begin{align} S_0 &= |E_x|^2 + |E_y|^2 = I_{\text{total}}, \\ S_1 &= |E_x|^2 - |E_y|^2 = I_H - I_V, \\ S_2 &= 2 \operatorname{Re}(E_x^* E_y) = I_D - I_A, \\ S_3 &= 2 \operatorname{Im}(E_x^* E_y) = I_R - I_L. \end{align} For fully polarized light, $S_0^2 = S_1^2 + S_2^2 + S_3^2$, and the normalized vector $(S_1/S_0, S_2/S_0, S_3/S_0)$ lies on the surface of the unit Poincaré sphere. Partially polarized light satisfies $S_0^2 > S_1^2 + S_2^2 + S_3^2$, and the degree of polarization is $\mathcal{P} = \sqrt{S_1^2 + S_2^2 + S_3^2} / S_0$.

5.2 Quantum Stokes Operators

In quantum optics, the Stokes parameters are promoted to operators: \begin{align} \hat{S}_0 &= \hat{a}_H^\dagger \hat{a}_H + \hat{a}_V^\dagger \hat{a}_V = \hat{n}_{\text{total}}, \\ \hat{S}_1 &= \hat{a}_H^\dagger \hat{a}_H - \hat{a}_V^\dagger \hat{a}_V, \\ \hat{S}_2 &= \hat{a}_H^\dagger \hat{a}_V + \hat{a}_V^\dagger \hat{a}_H, \\ \hat{S}_3 &= i(\hat{a}_H^\dagger \hat{a}_V - \hat{a}_V^\dagger \hat{a}_H). \end{align} These operators satisfy the commutation relations of angular momentum: \begin{equation} [\hat{S}_i, \hat{S}_j] = 2i \varepsilon_{ijk} \hat{S}_k, \end{equation} and the Casimir invariant is $\hat{S}_1^2 + \hat{S}_2^2 + \hat{S}_3^2 = \hat{S}_0 (\hat{S}_0 + 2)$. The Stokes operators form a Schwinger boson representation of the SU(2) algebra. Polarization transformations—waveplates, rotators, phase shifters—correspond to SU(2) rotations generated by the Stokes operators. The Poincaré sphere is thus the Bloch sphere for the polarization degree of freedom. This deep connection between polarization optics and quantum angular momentum is a recurring and fruitful theme in quantum optics.

References

  1. R. C. Jones, "A new calculus for the treatment of optical systems," Journal of the Optical Society of America 31, 488–493 (1941). The original papers introducing Jones calculus. A series of eight papers by Jones established the systematic matrix treatment of polarization optics.
  2. E. Hecht, Optics, 5th ed. (Pearson, 2017). Chapters 8 and 9 provide a clear, comprehensive introduction to polarization, Jones matrices, waveplates, and the Poincaré sphere at the advanced undergraduate level.
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999). Chapter 1 and Chapter 14 provide the authoritative treatment of polarization, the Jones and Müller–Stokes calculi, and crystal optics.
  4. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2005). Chapter 6 treats the quantum Stokes operators, polarization squeezing, and the SU(2) structure of polarization transformations.
  5. D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, Berlin, 2008). Chapter 8 provides an advanced treatment of the quantum theory of polarization, including the Stokes operators and polarization entanglement.
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995). Chapter 6 contains a rigorous treatment of the classical and quantum coherence properties of polarized light.
  7. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Reviews of Modern Physics 74, 1153–1201 (2002). A comprehensive review of atomic birefringence, NMOR, and polarization spectroscopy in atomic vapors.
  8. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Reviews of Modern Physics 77, 633–673 (2005). Section V discusses EIT-enhanced polarization rotations and cross-phase modulation for quantum information applications.
  9. J. L. O'Brien, A. Furusawa, and J. Vučković, "Photonic quantum technologies," Nature Photonics 3, 687–695 (2009). A review of integrated photonic circuits for quantum information, in which polarization and path encoding are implemented using on-chip waveplates and beam splitters.
  10. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007). Chapter 1 covers the Jones calculus and wave propagation in birefringent and optically active crystals.