Polarization states

The polarization state of light describes how the electric-field vector orients and evolves over time at a fixed point in space. Light has two transverse degrees of freedom (perpendicular to the propagation direction); the polarization state describes the relationship between them.

Categories of polarization.

Linear polarization. The electric field oscillates along a single direction in the transverse plane:

$$\mathbf{E}(t) \;=\; E_0 \, \hat{\mathbf{e}}_\theta \cos(\omega t),$$

where $\hat{\mathbf{e}}_\theta$ is the polarization axis. The two extreme cases — horizontal and vertical — are the standard basis for linear polarization. Diagonal polarization at 45° is a superposition of equal-amplitude horizontal and vertical components in phase.

Circular polarization. The electric field rotates at the optical frequency, maintaining constant magnitude:

$$\mathbf{E}(t) \;=\; E_0 [\hat{\mathbf{x}} \cos(\omega t) \pm \hat{\mathbf{y}} \sin(\omega t)].$$

The two signs correspond to right- and left-circular polarization (sense of rotation depends on convention). Circular polarization is a superposition of equal-amplitude horizontal and vertical components 90° out of phase.

Elliptical polarization. The general case: the electric-field vector traces out an ellipse in the transverse plane. Linear and circular polarization are limiting cases of elliptical (degenerate ellipse → line; circular = special equal-amplitude case).

Unpolarized light. The electric-field orientation is random and changes faster than any practical measurement timescale. Sunlight and thermal incandescent light are unpolarized. Mathematically: equal probability density on every linear polarization direction (and zero net circular component).

Partially polarized light. Most practical sources have polarization that is partially defined: some fraction of the power has a specific polarization (called the polarized component), and the rest is unpolarized. The Stokes parameters provide the natural description of partially-polarized light.

Decomposition and synthesis.

Polarization	Decomposition (horiz/vert basis)
H linear	100% H
V linear	100% V
+45° linear	50% H + 50% V, in phase
$-45°$ linear	50% H + 50% V, 180° out of phase
Right-circular	50% H + 50% V, V leads H by 90°
Left-circular	50% H + 50% V, H leads V by 90°
1:2 elliptical at 30°	Mix of H and V with specific phase and amplitude

Polarization manipulation elements.

Element	Function
Linear polarizer	Transmits only one linear polarization; absorbs/reflects the other
Circular polarizer	Transmits only one circular polarization
Quarter-wave plate	Converts linear ↔ circular polarization
Half-wave plate	Rotates linear polarization by twice the plate angle
Faraday rotator	Magnetic-field-induced rotation; non-reciprocal
Optical activity (sugar solutions, quartz)	Reciprocal rotation; reverses with direction
Polarization scrambler	Time-varying polarization conversion; randomizes
Polarization beam splitter (PBS)	Transmits one linear polarization, reflects orthogonal

Generation of standard polarization states.

• Linear polarization: from a linear polarizer (Glan-Thompson, sheet polarizer, wire-grid) or from sources with intrinsic polarization (most lasers)
• Circular polarization: linear polarizer + quarter-wave plate oriented at 45° to its axis
• Elliptical polarization: linear polarizer + arbitrary-angle quarter-wave plate
• Unpolarized: incandescent lamp, scrambled laser, sunlight after reflection from many surfaces

Polarization in light-matter interactions.

• Optical absorption: many molecules have polarization-dependent absorption (selection rules); the orientation of polarized light determines absorbed power
• Optical rotation in chiral media: glucose solutions and quartz crystals rotate linear polarization
• Birefringence: anisotropic media have refractive index that depends on polarization
• Polarization in scattering: Rayleigh scattering polarizes scattered light at 90° to incident direction; useful for atmospheric studies

Polarization in telecom and photonics.

• Polarization-division multiplexing (PDM): in coherent optical transmission, two independent data streams travel on the two orthogonal polarization states of a single wavelength. Demultiplexed in the coherent receiver via DSP.
• Polarization-mode dispersion (PMD): in single-mode fiber, the two polarization states travel slightly differently — a noise source in long-distance transmission
• Polarization-maintaining fiber: specially-designed birefringent fiber maintains a defined linear polarization through propagation
• Polarization-dependent loss (PDL): real components attenuate the two polarizations differently; an impairment in transmission

Standard measurement.

A polarization-state analyzer (PSA) or Stokes polarimeter measures the polarization state. Modern instruments (Thorlabs PAX series, Schäfter+Kirchhoff PolaScope) provide:

• Stokes parameters (4 measurements)
• Degree of polarization (DOP)
• Ellipticity and azimuth angles
• Real-time Poincaré sphere visualization

Typical accuracy: $< 1°$ azimuth, $< 0.5°$ ellipticity, $< 1\%$ DOP.

Polarization at boundaries. Reflection and transmission at interfaces are polarization-dependent:

• s-polarization (TE): electric field perpendicular to plane of incidence
• p-polarization (TM): electric field in plane of incidence
• Brewster's angle: p-polarization fully transmits, s-polarization partially reflects
• Fresnel coefficients: $r_s, r_p, t_s, t_p$ describe these polarization-dependent quantities

This polarization-dependent reflection underlies polarization-sensitive optics: PBS cubes, polarizing thin-film coatings, Brewster-angle windows.

Quantum polarization. The polarization state of a single photon is the quantum-mechanical analog of classical polarization. The state space is the 2D complex Hilbert space (the "qubit"). Bell-state pairs of polarization-entangled photons are central to quantum information experiments.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 6 (polarization optics); Born & Wolf, Principles of Optics (7th ed., 1999), Ch. 1 for the canonical treatment; Goldstein, Polarized Light (3rd ed., 2010) for comprehensive engineering reference.