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Unpolarized light

Unpolarized light, also referred to as natural light, is a form of electromagnetic radiation in which the electric field vectors oscillate randomly in all directions perpendicular to the propagation direction, with no preferred orientation or plane of polarization.^[1] This random distribution arises from the incoherent superposition of waves emitted by numerous atomic or molecular sources, making it the predominant type of light produced by common sources such as the Sun, stars, and incandescent bulbs.^[2] As a result, unpolarized light exhibits isotropic properties, meaning its polarization state averages out over time scales faster than human detection, leading to no observable net polarization effects.^[2] In contrast to polarized light, where the electric field is confined to a specific plane (linear polarization) or follows a helical path (circular or elliptical polarization), unpolarized light can be conceptualized as an equal mixture of all possible polarization states.^[3] When unpolarized light encounters a linear polarizer, it becomes linearly polarized along the polarizer's transmission axis, with the transmitted intensity reduced to exactly half of the incident intensity due to the equal projection of components onto the allowed and blocked directions.^[4] This behavior is fundamental to devices like polarizing filters and is described by the principles underlying Malus's law applied to unpolarized incident light.^[5] Unpolarized light plays a central role in various optical phenomena and applications, including the polarization of light through scattering (as in the blue sky) or selective reflection at Brewster's angle, which can convert portions of it into polarized light.^[6] In everyday technologies, such as liquid crystal displays (LCDs) and polarized sunglasses, unpolarized ambient light is manipulated to control visibility and reduce glare by exploiting these polarization differences. Understanding unpolarized light is essential in fields like optics, photography, and materials science, where distinguishing it from polarized states enables precise control over light-matter interactions.^[7]

Fundamentals of Light Polarization

Polarization Basics

Light consists of electromagnetic waves, in which the electric and magnetic fields oscillate perpendicular to the direction of propagation, making light a transverse wave.^[8] The electric field vector, which is stronger and more commonly used to describe polarization, vibrates in a plane perpendicular to the wave's travel direction, while the magnetic field vibrates in a mutually perpendicular plane.^[9] Polarization describes the specific orientation and pattern of this electric field oscillation as the wave propagates.^[10] In linearly polarized light, the electric field oscillates back and forth along a fixed line within the transverse plane, such as horizontally or vertically relative to the propagation direction. Figure 1 (conceptual illustration): The electric field vector (arrow) points consistently in one direction, like a rope shaken up and down. Circular polarization occurs when the electric field rotates at a constant angular velocity, forming a helix along the propagation path; this arises from two equal-amplitude linear components, offset by a 90-degree phase difference, resulting in right-handed or left-handed rotation depending on the sense. Figure 2 (conceptual illustration): The vector tip traces a circle in the transverse plane. Elliptical polarization is the general case, where the electric field traces an ellipse due to unequal amplitudes of the linear components or a phase difference not exactly 90 degrees. Figure 3 (conceptual illustration): The vector path forms an elongated or flattened ellipse, encompassing both linear and circular as special limits. The foundational understanding of polarization emerged from experiments on double refraction in the early 19th century. In 1811, French physicist François Arago observed that a quartz crystal rotates the plane of polarization of light passing through it, a phenomenon linked to birefringence in the crystal.^[11] Collaborating with Augustin-Jean Fresnel, Arago conducted further studies around 1812–1815 on the interference of light rays from birefringent materials, discovering that two beams polarized in perpendicular planes do not interfere, providing key evidence for the transverse wave nature of light.^[11] These findings by Arago and Fresnel established polarization as an intrinsic property of light waves.^[1] These polarization states—linear, circular, and elliptical—represent the possible orientations and behaviors of the electric field vector in any light wave, forming the basis for analyzing how light interacts with matter.^[1]

Polarized versus Unpolarized Light

Polarized light is characterized by the electric field vectors oscillating in a coherent manner along a fixed orientation perpendicular to the direction of propagation, such as vertical or horizontal linear polarization.^[2]^[1] In contrast, unpolarized light consists of electric field vectors that vibrate with equal probability in all orientations perpendicular to the propagation direction, resulting in no preferred net polarization axis.^[12]^[2] The fundamental distinction lies in the behavior of the electric field: in polarized light, the vector traces a definite path—linear, circular, or elliptical—due to its coherent nature, whereas unpolarized light can be viewed as a superposition of two orthogonal polarizations (e.g., horizontal and vertical components) with random, uncorrelated phases that change rapidly over time.^[2]^[1] This randomness arises because typical light sources emit waves from many atoms or molecules out of phase, averaging out any specific orientation within the detection time scale.^[2] Experimentally, the difference is evident when light passes through a linear polarizer: unpolarized light transmits half its incident intensity regardless of the polarizer's orientation, as the device selects one random component equally from all directions, while polarized light's transmission varies with the angle between its polarization direction and the polarizer's axis, following Malus's law qualitatively by being maximally transmitted when aligned and fully blocked when perpendicular.^[5]^[4] Between these extremes lies partial polarization, where light exhibits some degree of preferred orientation but not fully coherent, forming a continuum from fully unpolarized to fully polarized states.^[12]^[2]

Properties of Unpolarized Light

Orientation Randomness

Unpolarized light is characterized by the random orientation of its electric field vector perpendicular to the propagation direction, with the direction decorrelating over the coherence time, which is much shorter than those perceptible to human observers but longer than a single optical cycle. For visible light, the electric field oscillates at frequencies around 400 to 750 THz, corresponding to periods of approximately 1.3 to 2.5 femtoseconds per cycle, while the polarization state randomizes due to incoherent superposition on timescales of femtoseconds to picoseconds depending on spectral bandwidth.^[13] This ensures that over any practical observation time, there is no statistically preferred direction for the electric field, resulting in an average absence of polarization.^[13]^[10] Spatially, unpolarized light exhibits randomness across the beam profile, where individual photons or wave packets possess independent and randomly oriented electric fields perpendicular to the propagation direction.^[14] This spatial variation means that while small regions might appear locally polarized, the overall beam shows no consistent orientation, reinforcing the unpolarized nature.^[13] From a statistical perspective, unpolarized light represents an incoherent mixture of all possible polarization states, with equal weighting across linear, circular, and elliptical polarizations.^[14] The orthogonal components of the electric field—typically resolved along two perpendicular axes—lack a fixed phase relationship, leading to no stable interference between them and thus no polarization-specific optical effects.^[10]^[13] The degree of polarization (DOP) provides a quantitative measure of this randomness, defined as zero for fully unpolarized light to signify complete isotropy in orientation.^[15] Qualitatively, a DOP of zero indicates that the light's properties average out over all directions, with no dominant polarized fraction; this can be assessed using Stokes parameters for precise evaluation.^[16]

Intensity and Statistical Nature

The intensity of unpolarized light is defined as the total energy flux through a surface, representing the power per unit area independent of any specific polarization direction. This total intensity, often denoted as I, encompasses the contributions from all possible polarization components without preferential alignment, making it equivalent to the flux measured for light of the same total energy but in a definite polarization state.^[17] For instance, in radio astronomy contexts, the specific intensity I_\nu is the power per unit area per unit solid angle per unit frequency, and for unpolarized radiation, it remains unaffected by polarization parameters such as the Stokes parameters Q, U, and V, which are zero.^[18] Unpolarized light exhibits a fundamentally statistical nature, arising from a random superposition of polarization states described by probability distributions over the electric field components. In classical optics, these components are modeled as zero-mean, uncorrelated, stationary Gaussian random processes, leading to an equal likelihood of all linear and circular polarization orientations.^[19] For thermal sources, such as blackbody radiation, the underlying photon statistics follow a Bose-Einstein distribution in the quantum description, with the classical low-frequency limit yielding the Rayleigh-Jeans distribution for the energy per mode, where polarization states are equally populated due to the random emission from atomic dipoles.^[20] This results in phasor fluctuations that obey a Rayleigh distribution, with phase angles uniformly distributed, reflecting the incoherent mixing of wave trains over short coherence times on the order of 1-10 femtoseconds for broadband visible thermal sources.^[21] Due to the extremely rapid decorrelation of polarization states on femtosecond timescales (rates on the order of 10^{14} Hz for broadband sources), unpolarized light is ergodic, meaning time averages of observables, such as intensity or field correlations, equal ensemble averages over many realizations.^[13] This property allows single-beam measurements to represent statistical ensembles effectively, as the system's rapid evolution samples the full distribution of states. In terms of spatial coherence for unpolarized beams from extended incoherent sources, the van Cittert-Zernike theorem qualitatively describes how the degree of coherence decreases with separation, determined by the Fourier transform of the source's angular intensity distribution, leading to a coherence length on the order of \lambda / \delta \alpha, where \delta \alpha is the source's angular size.^[22]^[23] Measuring the intensity of unpolarized light poses challenges because standard intensity meters detect only the total energy flux and cannot distinguish it from fully polarized light of equivalent intensity without additional polarization analysis.^[24] To reveal the unpolarized nature, polarizers must be introduced; for example, passing unpolarized light through a linear polarizer halves the transmitted intensity on average, as it projects onto a single orientation, whereas polarized light's transmission varies with alignment per Malus's law.^[4] This indistinguishability in direct intensity measurements underscores the need for polarimetric techniques to probe the underlying statistical mixture.

Mathematical Descriptions

Coherency Matrix

The coherency matrix serves as the primary mathematical tool for describing the polarization state of quasi-monochromatic light fields through their correlation functions. It is a 2×2 Hermitian matrix whose elements represent the time-averaged correlations between the orthogonal components of the electric field. For a quasi-monochromatic field, the coherency matrix \rho is defined as \rho = \langle \mathbf{E} \mathbf{E}^\dagger \rangle, where \mathbf{E} = (E_x, E_y)^T is the Jones vector representing the transverse electric field components, \dagger denotes the Hermitian transpose, and \langle \cdot \rangle indicates an ensemble average or time average over a period much longer than the optical cycle but shorter than variations in the field's statistics. The matrix elements are explicitly given by:

\rho = \begin{pmatrix} \langle E_x E_x^* \rangle & \langle E_x E_y^* \rangle \\ \langle E_y E_x^* \rangle & \langle E_y E_y^* \rangle \end{pmatrix},

where the off-diagonal terms capture phase correlations between the x- and y-components, and the diagonal terms correspond to the intensities along each axis. This formulation arises from the analytic signals of the real electric field components E_x(t) and E_y(t), typically represented as complex amplitudes with random phases and amplitudes for partially coherent light. In the unpolarized case, the fields E_x and E_y are uncorrelated, leading to vanishing cross-terms \langle E_x E_y^* \rangle = 0, while the intensities satisfy \langle |E_x|^2 \rangle = \langle |E_y|^2 \rangle = I/2, where I is the total intensity. For unpolarized light, the coherency matrix simplifies to a diagonal form proportional to the identity matrix:

\rho = \frac{I}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.

This structure reflects the equal and uncorrelated contributions from both polarization components, embodying the random orientation characteristic of unpolarized light. The trace of \rho equals the total intensity I, and the determinant is (I/2)^2, indicating maximum depolarization. Unlike the Jones vector formalism, which applies only to fully coherent and polarized states, the coherency matrix naturally accommodates partial coherence and unpolarized conditions by incorporating statistical averages, making it essential for realistic light sources exhibiting fluctuations. This matrix representation is equivalent to the Stokes parameters in vector form but provides a direct bilinear form for field correlations, facilitating computations in quantum and statistical optics. Under the action of a polarizer or any linear optical element, the coherency matrix transforms via similarity: \rho' = T \rho T^\dagger, where T is the 2×2 Jones matrix of the element. For unpolarized incident light, this yields a partially polarized output whose coherency matrix retains the diagonal structure but with reduced intensity, typically halving the power through an ideal linear polarizer.

Stokes Parameters

The Stokes parameters form a four-component vector that quantifies the polarization state of light in terms of measurable intensities, originally introduced by George Gabriel Stokes in 1852.^[25] The vector is denoted as \mathbf{S} = (S_0, S_1, S_2, S_3), where S_0 represents the total intensity of the light beam, equivalent to the sum of the intensities of orthogonally polarized components.^[26] The parameter S_1 measures the excess of horizontally polarized light over vertically polarized light, S_2 captures the difference between linear polarization at +45° and -45° orientations, and S_3 indicates the imbalance between right-handed and left-handed circular polarization.^[25] For unpolarized light, which lacks any preferred orientation of the electric field vector, the Stokes vector simplifies to \mathbf{S} = (I, 0, 0, 0), where I is the total intensity; the zero values for S_1, S_2, and S_3 reflect the absence of net linear or circular polarization components.^[26] This representation highlights the random, isotropic nature of unpolarized light's polarization, where averaging over many cycles yields no preferential direction.^[26] The Stokes parameters are mathematically linked to the coherency matrix \boldsymbol{\rho}, a Hermitian matrix describing the second-order statistical properties of the electric field, through the relations:

\begin{align} S_0 &= \rho_{xx} + \rho_{yy}, \\ S_1 &= \rho_{xx} - \rho_{yy}, \\ S_2 &= 2 \Re(\rho_{xy}), \\ S_3 &= 2 \Im(\rho_{xy}), \end{align}

where \rho_{xx} and \rho_{yy} are the auto-correlation intensities along the x and y axes, and \rho_{xy} is the complex cross-correlation.^[27] These connections allow the Stokes vector to serve as a simpler, intensity-based alternative to the full coherency matrix for experimental analysis, particularly when dealing with partially coherent or unpolarized beams.^[27] The degree of polarization P is defined as

P = \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0},

which equals zero for fully unpolarized light, confirming the lack of coherent polarization structure.^[26] In practice, the Stokes parameters for unpolarized light are determined experimentally by passing the beam through a sequence of linear polarizers and quarter-wave plates, followed by intensity measurements at specific orientations (e.g., 0°, 45°, 90° without the plate, and 45° with the plate fast axis at 45° to induce circular components).^[26] For instance, S_0 = I(0^\circ) + I(90^\circ), S_1 = I(0^\circ) - I(90^\circ), S_2 = 2I(45^\circ) - S_0, and S_3 = 2I(45^\circ, \text{quarter-wave}) - S_0, where I(\theta) denotes the measured intensity at polarizer angle \theta.^[26] This method directly verifies the (I, 0, 0, 0) form for sources like sunlight or incandescent lamps.^[26]

Poincaré Sphere

The Poincaré sphere provides a powerful geometric visualization for the states of polarization in light, originally proposed by Henri Poincaré in 1892 as a method to represent the full range of polarization phenomena. It is constructed as a sphere of unit radius in the three-dimensional Cartesian space spanned by the Stokes parameters S_1, S_2, and S_3, where the sphere's radius is scaled by the total intensity S_0.^[28]^[29]^[30] Fully polarized pure states are depicted as points on the sphere's surface, while partially polarized states occupy points in the interior, with the distance from the origin indicating the degree of polarization.^[29]^[31] Unpolarized light is uniquely represented at the origin of the Poincaré sphere, corresponding to the coordinates (0, 0, 0), where it manifests as an equal mixture of all possible polarization states, yielding no net polarization preference.^[29]^[16]^[31] This central position reflects the complete randomness in orientation, distinguishing it from any polarized or partially polarized light.^[32] Key visualizations on the sphere include the north pole, which denotes right-circular polarization, and the south pole for left-circular polarization, while the equatorial plane encompasses all linear polarization orientations.^[29]^[30]^[16] Optical elements act on these representations through specific geometric operations: retarders induce rotations of the state point around the sphere, preserving the degree of polarization, whereas polarizers cause projections of the point toward the sphere's surface along lines connecting to their transmission axes.^[29]^[31] The Poincaré sphere's representation also connects to the underlying coherency matrix \rho, where the eigenvalues of \rho—which sum to S_0—determine the radial distance from the origin, with equal eigenvalues signaling the unpolarized state at the center.^[29] This geometric framework thus offers an intuitive means to analyze polarization mixtures, highlighting how unpolarized light serves as the invariant core amid transformations affecting polarized components.^[31]

Sources and Applications

Natural and Artificial Sources

Natural sources of unpolarized light primarily arise from thermal emission processes in hot plasmas, where atomic emitters are oriented randomly, leading to no preferred direction for the electric field oscillations. Sunlight exemplifies this, as the Sun's photosphere—a hot plasma at approximately 5800 K—emits blackbody radiation through incoherent processes involving countless atoms vibrating and radiating independently over very short timescales, on the order of 10^{-8} seconds per emission event.^[2] This rapid succession of emissions with varying polarization directions results in an average that appears unpolarized to detectors, as the changes occur faster than can be resolved.^[2] Starlight follows a similar mechanism, with stellar atmospheres producing thermal radiation from randomly oriented atomic and ionic emitters in their plasmas.^[17] Artificial sources also commonly generate unpolarized light through mechanisms mimicking thermal emission or incoherent recombination. Incandescent bulbs operate by heating a tungsten filament to around 2500–3000 K, approximating blackbody radiation where the filament's atoms emit light isotropically due to random thermal motions and orientations, yielding unpolarized output.^[17] Similarly, standard light-emitting diodes (LEDs) without integrated polarizers produce unpolarized light via spontaneous emission in semiconductors, where electron-hole recombination generates photons with random polarization directions, as the emission lacks a coherent phase or preferred orientation.^[33] Unpolarized light can also emerge from depolarization processes that randomize the polarization of initially polarized beams. Rayleigh scattering partially depolarizes incident polarized light, while for unpolarized light like sunlight, it induces polarization. Multiple reflections and scatterings further enhance depolarization; for instance, light undergoing successive diffuse reflections off rough surfaces or multiple scattering in particulate media, such as clouds in the atmosphere via repeated Rayleigh interactions with molecules like nitrogen and oxygen, experiences progressive randomization of its electric field direction, leading to effective depolarization.^[34]^[35] In contrast, lasers typically emit coherent, polarized light due to stimulated emission aligning photon polarizations, but this can be depolarized through methods like propagation through turbid media with strong multiple scattering, which mixes polarization states.^[10]^[36] From a quantum perspective, unpolarized light consists of photons in random polarization states, equivalent to a statistical mixture where half the photons are in horizontal linear polarization and half in vertical, with no fixed phase relation between them. This randomness arises because each photon's spin projection (helicity) projects onto these bases incoherently, reflecting the independent emission events in the source.^[37]

Practical Implications

Sunlight, as a primary source of unpolarized light, ensures equitable visibility for observers regardless of the orientation of polarized sunglasses or viewing angles, since its random polarization components pass through linear polarizers at approximately half intensity without directional bias.^[38] This property contrasts with glare from reflective surfaces, which becomes partially polarized and can be selectively reduced by such filters, allowing unfiltered photography of direct sunlight or diffuse scenes to capture natural color and detail without artificial contrast enhancement.^[10] In astronomy, the unpolarized nature of starlight facilitates unbiased spectroscopic analysis, as the random orientations do not introduce polarization-dependent distortions in spectral lines, enabling accurate measurement of stellar compositions and velocities.^[39] Similarly, in biology, multiple scattering events within tissues rapidly depolarize incident light, producing predominantly unpolarized output that mimics natural illumination and aids in non-invasive imaging techniques like optical coherence tomography, where polarization contrast reveals structural details without interference from initial beam orientation.^[40] Liquid crystal displays (LCDs) rely on unpolarized backlights from LEDs or fluorescents, which are diffused and depolarized via light guides to achieve uniform illumination across the panel, preventing visible polarization-induced patterns or hotspots that could degrade image quality.^[41] In 3D cinema systems, projectors start with unpolarized light sources and encode left- and right-eye images using orthogonal circular polarizations to minimize crosstalk and viewing conflicts, such as those arising from head tilts that would disrupt linear polarization schemes.^[42] Partial depolarization in fiber optic communications poses challenges by scrambling polarization states over long distances due to birefringence and nonlinear effects, leading to signal impairments in polarization-division-multiplexed systems and necessitating compensators to maintain data integrity.^[43] This underscores the motivation for polarizers in displays, where they convert the incoming unpolarized backlight into a controllable linear state, enabling the liquid crystals to modulate intensity and color effectively for high-contrast visuals.^[41] Early 19th-century optics experiments, such as those by Étienne-Louis Malus in 1808 observing polarization upon reflection and subsequent work by Augustin-Jean Fresnel and François Arago in 1819 demonstrating interference of polarized rays, revealed the unpolarized character of natural light through its consistent half-transmission via analyzers regardless of rotation, motivating the development of wave theories and polarization tools.^[44]

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