What Are Stokes Parameters?

Stokes parameters are a set of four real-valued quantities that together fully describe the polarization state and intensity of an electromagnetic wave, including completely polarized, partially polarized, and unpolarized light. Introduced by George Gabriel Stokes in 1852, they represent the polarization of a light beam through observable time-averaged irradiance measurements rather than through the instantaneous electric field amplitudes and phases that cannot be directly measured at optical frequencies. The four parameters are conventionally labeled I, Q, U, and V, and together form the Stokes vector, a column matrix that serves as the fundamental descriptor of polarization in optics, astronomy, radar polarimetry, and remote sensing. The ability to characterize partially polarized light is the key advantage of the Stokes representation over the polarization ellipse, which applies only to fully coherent, completely polarized beams.

The parameters are defined through four distinct intensity measurements made with combinations of linear and circular polarization filters. Because each parameter is an intensity difference, they are directly accessible to detectors without requiring phase-sensitive interferometry, making the Stokes formalism practical for optical instrumentation and calibration.

The Four Parameters and Their Physical Meaning

The first Stokes parameter, I, represents the total intensity of the beam and is always non-negative. The second parameter, Q, describes the preponderance of linear polarization along the horizontal axis over linear polarization along the vertical axis: Q greater than zero indicates a horizontally polarized component, and Q less than zero indicates vertical polarization. The third parameter, U, describes the preponderance of linear polarization at positive 45 degrees over linear polarization at negative 45 degrees, capturing diagonal polarization orientations not covered by Q. The fourth parameter, V, describes the preponderance of right circular polarization over left circular polarization. For completely polarized light, the relationship I squared equals Q squared plus U squared plus V squared holds exactly; for partially polarized or unpolarized light, I squared is strictly greater than the sum of the squared polarization parameters. The physical basis and measurement protocol for each parameter are described in the SPIE Field Guide to Stokes polarization parameters and in the Ocean Optics Web Book on Stokes vectors.

The Poincaré Sphere

The Poincaré sphere is a geometric representation of all polarization states in three-dimensional space using Q, U, and V as Cartesian coordinates normalized by I. Completely polarized states map to points on the surface of the sphere: the equator represents all linear polarization states, the north pole represents right circular polarization, the south pole represents left circular polarization, and elliptical polarization states occupy intermediate latitudes. Partially polarized light maps to points inside the sphere, with unpolarized light at the center. The great circle routes on the Poincaré sphere correspond to the action of wave retarders and polarizers, giving optical designers a geometric tool for understanding how optical elements transform polarization states. Antipodal points on the sphere represent orthogonal polarization states.

Measurement and Instrumentation

A polarimeter measures the Stokes vector by recording a minimum of four intensity values with different polarizer and wave plate configurations. Division-of-amplitude and division-of-aperture polarimeters sample multiple polarization states simultaneously, enabling snapshot measurement of dynamic polarization changes. In radar and microwave remote sensing, the Stokes parameters describe the polarization of scattered or emitted radiation from surfaces and atmospheric constituents, providing information about surface roughness, moisture content, vegetation structure, and cloud properties. The Mueller matrix, a 4x4 real matrix, describes how an optical element transforms an incident Stokes vector into an output Stokes vector, forming the basis for polarization ray tracing in optical system design, as implemented in tools such as Ansys Optics for Stokes parameter calculation.

Applications

Stokes parameters have applications across a wide range of optical and remote sensing domains, including:

• Astronomical polarimetry for measuring magnetic field orientations in stellar and galactic objects
• Radar polarimetry for land cover classification, sea state estimation, and target recognition
• Biomedical tissue characterization using polarization-sensitive optical coherence tomography
• Optical fiber sensing of stress and birefringence along transmission lines
• Atmospheric remote sensing of aerosol composition using polarized lidar
• Ellipsometry for thin film thickness and optical constant measurement in semiconductor fabrication