Polarization

Polarization States

Polarization specifies how the transverse electric field varies at a fixed point in space. A wave traveling in the positive $z$ direction has real field

The electric field has no $z$ component in the ideal plane-wave model. The magnetic field is perpendicular to both $\vec{E}$ and the propagation direction. Polarization is therefore described by the pair of transverse components, their amplitudes, and their relative phase. The intensity and wavelength alone do not determine the polarization.

A single-frequency wave has a common angular frequency, so write

Only the phase difference matters:

A common phase shift changes the choice of time origin but leaves the polarization state unchanged. The amplitudes $E_{0x}$ and $E_{0y}$ may be unequal. A state with one component equal to zero is linearly polarized along the remaining component direction.

At fixed $z$, eliminate time between the two component expressions. The endpoint of the electric-field vector obeys the polarization-ellipse equation

The equation gives a line when $\delta =0$ or $\pi$. It gives an ellipse for a general phase difference. Circular polarization is the special ellipse with equal component amplitudes and a quarter-cycle phase difference. An optical detector that averages over many cycles responds to intensity; it does not directly display the instantaneous ellipse without a phase-sensitive measurement or a controlled analyzer sequence.

Linear, circular, and elliptical states

For linear polarization, the component phase difference is zero or $\pi$. The two components increase and decrease together or in opposite directions, so their ratio is constant except at the zero crossings. The polarization azimuth $\psi$ follows

for in-phase components. The azimuth identifies a line, not an oriented arrow. Rotating the electric field by $180^{\circ }$ describes the same linear polarization line.

For circular polarization,

At a fixed point, the field magnitude is constant:

The electric-field direction rotates once during every wave period. The sign of the phase difference determines the rotation sense. Names such as right-circular and left-circular depend on the viewing convention used by a field, optics, or radio community. State the propagation direction, the observer's viewing direction, and the component phase convention whenever the handedness name matters.

Elliptical polarization covers unequal amplitudes with a quarter-cycle phase difference and equal amplitudes with a phase difference other than a quarter cycle. The major-axis orientation, axial ratio, and rotation sense specify an ideal fully polarized ellipse. The axial ratio is the minor-axis amplitude divided by the major-axis amplitude and lies between zero and one. Zero gives linear polarization; one gives circular polarization.

Partially polarized light has a stable polarized component mixed with an uncorrelated component. It cannot be represented by one deterministic electric-field ellipse over a long averaging interval. A single short time record may look elliptical, while averaged analyzer data show a reduced degree of polarization. Stokes-style intensity measurements separate the total intensity from the degree and type of polarization later in this lesson.

Phasor representation

Complex amplitudes compress component amplitude and phase into one expression:

The complex ratio ${\tilde{E}}_y/{\tilde{E}}_x$ contains the amplitude ratio and the phase difference. Under this time convention, the component phase factors associated with the earlier cosine expressions are $e^{-i{\delta }_x}$ and $e^{-i{\delta }_y}$. A real ratio gives linear polarization. A ratio of magnitude one and phase $\pm \pi /2$ gives circular polarization. The notation is compact, but it does not remove the need to state the $x$ and $y$ axes and the propagation direction.

Polarizers and Measurements

An ideal linear polarizer transmits the electric-field component along one transmission axis and removes the perpendicular component. A wire-grid polarizer illustrates the mechanism at microwave wavelengths. Electric field parallel to conducting wires drives current and is absorbed or reflected. Electric field perpendicular to the wires drives much less current and can pass. In an absorbing optical sheet, aligned conducting molecular chains play the corresponding role; the transmission axis is perpendicular to the strongly absorbing chain direction.

For linearly polarized input of intensity $I_0$, an analyzer with transmission axis at angle $\theta$ to the input polarization transmits

This is Malus's law. The field amplitude transmitted by the analyzer is $E_0\cos \theta$; intensity is proportional to squared field amplitude. The intensity is unchanged by reversing the analyzer axis through $180^{\circ }$, which is consistent with a linear-polarization axis rather than an oriented vector.

Unpolarized input has no preferred transverse axis over the averaging interval. The mean projection of its intensity onto an ideal polarizer is one half:

The light leaving the first polarizer is linearly polarized along its transmission axis. If a second ideal analyzer follows at relative angle $\theta$,

The factor one half belongs to unpolarized input entering the first ideal polarizer. It must not be inserted again when the input to an analyzer is already known to be linearly polarized.

The analyzer scan is a direct polarization measurement. Rotate the analyzer through at least $180^{\circ }$, record the detector signal at known angles, and fit the sinusoidal form. A fully linearly polarized input gives a high modulation contrast. An unpolarized input gives no ideal angle dependence after a single analyzer because every axis receives the same average intensity. A partly polarized input produces a nonzero offset and a reduced sinusoidal modulation.

Crossed ideal polarizers have perpendicular transmission axes and zero transmitted intensity for a linearly polarized input aligned with the first axis. A third polarizer placed between them can transmit light because it changes the projection sequence. For three ideal sheets at axes $0^{\circ }$, $45^{\circ }$, and $90^{\circ }$, unpolarized incident intensity becomes

The middle sheet does not add energy. It prepares a component along the final analyzer axis that the crossed two-sheet system lacks.

Real polarizers and detector records

Real polarizers transmit a finite perpendicular component and lose some parallel component. A practical analyzer model for a linearly polarized input is

$I_{\parallel }$ and $I_{\perp }$ include polarizer transmission and detector response. $I_{\mathrm{dark}}$ includes detector offset and background measured with the source blocked. The extinction ratio is commonly reported as $I_{\min }/I_{\max }$ after background subtraction. A finite minimum does not establish partial source polarization until the analyzer's own extinction and detector offset have been measured.

Angular zero is a measurement parameter. Find the maximum angle from a coarse scan, then repeat a fine scan near the maximum and record the mechanical reference. A constant zero-angle error shifts the fitted polarization azimuth. A slowly drifting source level can change the apparent contrast if angles are scanned only in one direction. Interleave reference angles or perform forward and reverse scans to expose drift.

Stokes Analysis

Intensity measurements can describe a fully polarized ellipse or a partially polarized beam without sampling the optical carrier phase directly. With complex transverse amplitudes ${\tilde{E}}_x$ and ${\tilde{E}}_y$, define one common Stokes convention:

$S_0$ is proportional to total intensity. $S_1$ compares horizontal and vertical linear components. $S_2$ compares the two diagonal linear bases. $S_3$ distinguishes the two circular bases. Some optics texts use the opposite sign for $S_3$ because they choose a different viewing or time-dependence convention. The chosen sign must accompany any reported circular-polarization label.

The degree of polarization is

An ideal deterministic polarization ellipse has $\mathcal{P}=1$. A completely unpolarized beam has $S_1=S_2=S_3=0$ and $\mathcal{P}=0$. Intermediate values describe a beam whose time-averaged intensity contains both a polarized component and an unpolarized component. A measured value above one signals inconsistent calibration, background subtraction, or uncertainty treatment.

An intensity instrument can recover the Stokes values from calibrated analyzer settings. Let $I_H$, $I_V$, $I_D$, and $I_A$ denote background-corrected intensities through horizontal, vertical, diagonal, and antidiagonal linear analyzers. Let $I_R$ and $I_L$ denote calibrated right- and left-circular analyzer intensities under the stated $S_3$ convention. Then

The pairs must use the same detector gain and source normalization. The redundant sums $I_D+I_A$ and $I_R+I_L$ should agree with $S_0$ within uncertainty. A disagreement indicates analyzer loss differences, an uncorrected background, source drift, or a detector response change between measurements.

Linear analyzers alone cannot determine $S_3$. A quarter-wave plate followed by a linear analyzer converts circular-basis content into a linear intensity difference. The plate axis and analyzer axis must be calibrated together. A quarter-wave plate with retardance different from $\pi /2$ or an axis error mixes $S_1$, $S_2$, and $S_3$, so the measured circular component requires an instrument correction or a stated approximation.

The polarization-ellipse orientation and ellipticity can be derived from normalized Stokes values when $\mathcal{P}$ is close to one. Under the convention used here,

$\psi$ is the ellipse orientation. $\chi$ is the ellipticity angle, with zero for linear polarization and magnitude $45^{\circ }$ for circular polarization. The two-argument angle function retains the correct quadrant of the orientation. For partially polarized data, report the degree of polarization with these derived angles or fit the fully polarized component separately; an arbitrary partially polarized beam does not have one complete deterministic ellipse.

Data uncertainty

Subtract dark and background signals before forming differences. If independent intensity uncertainties are $u(I_H)$ and $u(I_V)$, the difference uncertainty is

The same form applies to $S_2$ and $S_3$ pairs. Normalizing by a measured $S_0$ creates shared uncertainty among all normalized components. A small signal difference obtained by subtracting two large intensities has a larger relative uncertainty than either raw intensity alone. Report raw intensities, background values, analyzer angles, and the normalization method with the Stokes result.

Reflection and Birefringence

Reflection from a transparent boundary separates polarization components differently. At a particular incidence angle, called the Brewster or polarizing angle, the reflected wave is linearly polarized perpendicular to the plane of incidence. The reflected and refracted rays are perpendicular at that angle. Combining this geometry with Snell's law gives

where the incident wave travels from index $n_1$ into index $n_2$. This relation is a special reflection condition; the general laws of reflection and refraction belong to the optics lesson on interfaces. The transmitted wave at the Brewster angle is generally only partially polarized because it contains most of the incident power.

The physical origin of the Brewster null for one incident polarization can be stated in electric-dipole terms. The refracted electric field drives bound charges near the boundary. An oscillating charge has no dipole-radiation intensity along its oscillation axis. At the Brewster geometry, the direction of the reflected ray lies along the driven charge motion for the component parallel to the incidence plane, so that component has no reflected contribution. The perpendicular component remains and defines the polarization of the reflected light.

For air with $n_1\approx 1.00$ and glass with $n_2=1.50$,

The refracted angle is $33.7^{\circ }$, giving the required $90.0^{\circ }$ sum. Reflected glare from a horizontal water, road, or snow surface has a substantial horizontal electric-field component near its Brewster geometry. Sunglasses with a vertical transmission axis reduce that component. Surface roughness, wavelength, and a spread of incidence angles prevent a real scene from being perfectly linearly polarized.

Scattering can also produce polarization. An incident wave drives charge oscillations in a small particle or molecule. The scattered radiation follows the electric-dipole pattern of those driven oscillations. With an incident beam along $z$ and electric components in the $x$ and $y$ directions, light observed along $x$ cannot receive radiation from the $x$ oscillation, because that is its dipole axis. The remaining $y$ oscillation produces scattered light polarized along $y$.

At a right-angle scattering view, the scattered electric field is perpendicular to both the incident propagation direction and the scattered propagation direction. The degree of polarization decreases away from this geometry because both driven dipole components can contribute. Molecular anisotropy, multiple scattering, and surface reflections also reduce the ideal single-scattering result. The connection to the source is the same electric-dipole radiation pattern developed in Dipole Radiation.

Birefringence and wave plates

An isotropic transparent material has one refractive index for a chosen propagation direction. A birefringent material has different phase velocities for two perpendicular polarization eigenaxes. In a bulk crystal, those components may emerge as spatially separated ordinary and extraordinary rays. In a plate cut for propagation along a chosen direction, the components can remain collinear while accumulating different phase.

A plane-parallel wave plate requires resolution of the incident field along fast and slow axes. If $n_s-n_f=\Delta n$ is positive, the slow-axis component accumulates an extra phase delay

where $d$ is plate thickness and ${\lambda }_0$ is vacuum wavelength. The material dispersion means that $\Delta n$ and therefore retardance generally vary with wavelength. One plate cannot be a mathematically exact quarter-wave plate for every color.

A quarter-wave plate has

and a half-wave plate has

The lowest-order thicknesses are

Integer multiples of full-wave retardance can be added when manufacturing, absorption, or mechanical thickness requires them. The phase difference, not the plate name alone, determines the output polarization.

With a linearly polarized input at $45^{\circ }$ to the plate axes, the fast and slow components have equal amplitude. A quarter-wave plate produces a circular state at its design wavelength. If the input angle differs from $45^{\circ }$, the components have unequal amplitude and the output is elliptical. Reversing the plate or rotating it by $90^{\circ }$ changes the relative-delay convention and reverses the circular handedness under a fixed viewing convention.

A half-wave plate with linearly polarized input azimuth $\alpha$ and fast-axis azimuth $\beta$ leaves linear polarization with azimuth

The plate reflects the input polarization line about its fast axis. An input at $45^{\circ }$ to a half-wave plate axis rotates by $90^{\circ }$. This rule applies to an ideal plate at its design wavelength and with axes known relative to the input reference.

Crossed-polarizer phase analysis

Place a birefringent plate between an ideal input polarizer and a crossed ideal analyzer. Set the input polarization at $45^{\circ }$ to the fast and slow axes. The plate receives equal component amplitudes. If $I_P$ is the intensity after the input polarizer, the two components at the plate output have the common amplitude factor $\sqrt{I_P/2}$ and a relative phase $\Delta \varphi$. Projection onto the crossed analyzer gives

With the output analyzer parallel to the input polarizer, the corresponding result is

The two intensities sum to $I_P$ for ideal lossless elements. A quarter-wave delay gives equal intensities in the parallel and crossed analyzer channels. A half-wave delay gives maximum intensity through crossed polarizers and a null through parallel polarizers. The same material can give different results at different wavelengths because its retardance depends on ${\lambda }_0$.

Optical retardance is often specified in waves or radians. The phase model uses the vacuum wavelength because the refractive-index difference already contains the material phase velocity. A measured plate with nominal retardance $\Delta {\varphi }_0$ at design wavelength ${\lambda }_d$ has a first approximation

The second factor represents material dispersion. It cannot be omitted across a broad spectrum. White light between crossed polarizers and a birefringent plate can therefore show colors: some wavelength bands have near-half-wave retardance and transmit, while others remain near-full-wave retardance and are suppressed.

Stress can produce birefringence in nominally isotropic glass and plastic. Spatially varying stress changes the local product $\Delta nd$, so a sample between crossed polarizers transmits different wavelengths and intensities at different positions. The image records stress structure through polarization retardance. It is not a direct map of mechanical stress until the photoelastic coefficient, thickness, and optical geometry are calibrated.

Worked Reductions

Measurement Protocol and Checks

A polarization measurement begins by fixing a coordinate system. Mark a horizontal reference axis in the laboratory frame, define positive analyzer rotation, and identify the wave propagation direction. Mount the source, polarizer, wave plate, analyzer, and detector so that their optical axes share a common beam line. A tilt of a plate or polarizer changes the projected axis and can introduce reflection loss or unintended retardance. Record the reference plane where intensity is measured, especially when an optical fiber, aperture, or imaging lens lies between the analyzer and detector.

Measure detector background with the source blocked and with the same integration time used for the polarization scan. Subtract this background only when it is stable and when subtraction does not drive low signals below the detector's reliable range. A dark signal can include electronic offset, room light, fluorescence, and stray reflection from the apparatus. If it drifts, interleave dark measurements throughout the scan instead of using one value measured at the beginning.

A beam described by Stokes values in the laboratory horizontal-vertical basis has ideal linear-analyzer transmission at azimuth $\alpha$

The circular component $S_3$ does not appear. A linear-analyzer sweep therefore determines total intensity, linear polarization magnitude, and linear azimuth, but it cannot determine circular polarization. The extrema of this scan are

The contrast of the sweep measures the linear polarized portion. A circularly polarized beam produces a constant ideal linear-analyzer reading because every linear axis receives the same average intensity.

Use a coarse angular sweep to locate maxima and minima, then a finer sweep over at least one complete 180-degree period. Fit the model to all data points rather than using only one maximum and one minimum. A fit exposes a zero-angle offset, unequal source levels, and detector noise. If the source is not stable, take a reference reading at a fixed analyzer angle after each several scan points and divide each raw point by an interpolated source reference before fitting.

Detector response must remain linear over the scan. A saturated detector clips the maximum and falsely lowers the inferred degree of linear polarization. At the opposite end, quantization and dark noise can raise the apparent minimum. Select source power, neutral attenuation, integration time, and detector gain so that both maximum and minimum readings lie in the calibrated linear range. Keep detector bandwidth fixed when comparing wavelength-dependent data.

Rotating a wave plate before the analyzer gives phase sensitivity. With a nominal quarter-wave plate, take a linear-analyzer sweep at several plate angles. A circular input becomes strongly modulated when the plate axes are near the appropriate 45-degree orientation, while a purely linear input has a different modulation dependence. A full instrument calibration uses known linear and known circular standards, or an optical model with fitted plate retardance, diattenuation, analyzer extinction, and angular zero.

Calibration entries

Each measurement record should include the following entries:

• Axis references. State the laboratory horizontal direction, analyzer zero, plate fast-axis mark, source propagation direction, and detector location.
• Spectral condition. Record source center wavelength or frequency, bandwidth, plate design wavelength, and any spectral selection element.
• Intensity corrections. Record dark signal, background procedure, detector gain, integration time, linearity range, and reference-source normalization.
• Component order. Reversing a wave plate changes the sign of its retardance under a stated convention. Record the beam-side order of plate and analyzer.
• Fit and residuals. Save raw angles and corrected intensities, fitted model parameters, residuals, and uncertainties rather than retaining only a plotted curve.

These entries separate a physical change in polarization from an analyzer-axis shift, spectral mismatch, or detector artifact. They are also required for comparison between measurements made on different days or with a different plate.

Suppose each of the six background-corrected readings has independent standard uncertainty $0.010\mathrm{mW}$. The uncertainty of a difference such as $S_1$ is

The same uncertainty applies to $S_2$ and $S_3$ for equal independent readings. The sum $S_0=I_H+I_V$ also has $0.014$ mW uncertainty before any common-mode source normalization error. A source-level drift shared by both members of a pair is correlated, so treating it as independent random noise would underestimate uncertainty. Reference normalization and repeated pair measurements are needed to characterize that drift.

The physical consistency condition is

The worked values satisfy it because $0.29\mathrm{m}{\mathrm{W}}^2<1.00\mathrm{m}{\mathrm{W}}^2$. An experimental value slightly outside the boundary can arise from noise. Report the raw measurements and uncertainty, then use a constrained fit if a physical Stokes estimate is required. Clipping one component by hand leaves the calibration problem unresolved and changes the inferred state without a documented model.

This record alone does not distinguish whether the residual minimum comes from the source, the analyzer, or other optics. Measure the same analyzer with a known high-extinction linear source, rotate the detector or cable path to check background pickup, and repeat the scan after rotating the source polarization. A source whose residual minimum follows the analyzer axes points to finite analyzer extinction. A residual that follows the source or sample orientation can indicate partial polarization or birefringent leakage.

Retardance measurement

The crossed-polarizer plate formula gives a direct retardance estimate when the input polarizer is at $45^{\circ }$ to the plate eigenaxes and element losses have been measured. Define the normalized crossed transmission

A measured value $R=0.500$ is consistent with quarter-wave retardance $\Delta \varphi =\pi /2$, but it is also consistent with $3\pi /2$ and higher odd-quarter-wave orders. Plate thickness, approximate birefringence, or a wavelength scan resolves the order. A value near zero can represent zero, full-wave, or multiple-full-wave retardance; it does not establish that the plate has no birefringence.

The local sensitivity is

Near zero or full-wave retardance, this derivative is small. A small intensity error then maps to a large retardance uncertainty. Quarter-wave regions have greater local intensity sensitivity. A phase measurement should therefore avoid operating only at a transmission extremum when the aim is to estimate a small retardance change.

Retardance error has three common sources. Thickness error changes $d$ directly. Wavelength or spectral-bandwidth error changes the phase factor and averages several retardances together. Axis error changes the component amplitudes, so a plate that has the correct phase delay can still fail to generate circular output. A narrowband source, a calibrated axis mark, and a plate-angle scan separate these effects more effectively than a single crossed-polarizer reading.

Use the following checks for a nominal quarter-wave plate:

• Place a known $45^{\circ }$ linear input before the plate. An output analyzer should show nearly constant intensity if the output is close to circular.
• Rotate the plate by $90^{\circ }$. The analyzer modulation should remain similar while the circular Stokes sign reverses under a fixed coordinate convention.
• Replace the plate with a known half-wave plate. The output should remain linear and its analyzer maximum should rotate at twice the plate-angle change.
• Repeat at a nearby wavelength. A measurable change in modulation or circular-channel balance is expected from retardance dispersion.

Handedness is another common reporting failure. The same physical rotating field can be called right- or left-circular under conventions that differ by observer direction or time dependence. A complete record avoids ambiguity by listing $E_x$ and $E_y$ with their phase difference, the sign used for $S_3$, and the view direction. For example, the formula $E_x=E_0\cos \omega t$, $E_y=E_0\sin \omega t$ at fixed $z$ specifies the field rotation without a handedness name.

Data-reduction audit

Before assigning a polarization state, check the following relationships against the actual record:

• Transverse basis. The reported $x$ and $y$ axes are perpendicular to propagation. A coordinate change rotates $S_1$ and $S_2$; it does not create or remove polarization.
• Intensity basis sums. Check $I_H+I_V$, $I_D+I_A$, and $I_R+I_L$ against the same calibrated total intensity. Unequal sums require an instrument correction or a source normalization before taking differences.
• Physical Stokes bound. Check $S_1^2+S_2^2+S_3^2\le S_0^2$ with uncertainty. The degree of polarization must remain in the interval from zero to one.
• Analyzer model. Use Malus's law only after the first polarizer has created a known linear input. Include finite extinction and background when a measured null is used.
• Plate model. A wave plate changes relative phase. Circular output additionally requires equal component amplitudes; a plate at the wrong input angle gives an ellipse.
• Spectral model. Report wavelength and bandwidth whenever retardance, birefringence, or Brewster-angle data are compared.

The audit distinguishes a physical polarization result from a set of detector readings. It also identifies the model used at each stage: ideal components for a derivation, finite-extinction components for a bench measurement, or a calibrated Stokes instrument for a reported state.

An ideal linear analyzer at $40.0^{\circ }$ has predicted intensity

in the same proportional units. The circular Stokes component does not enter this linear-analyzer prediction. A quarter-wave plate before the analyzer changes the basis and makes the circular component observable as an intensity change. This separation explains why a scan with one rotating linear analyzer cannot distinguish circular from unpolarized light when their total and linear Stokes components match.

Optical element order matters because a polarizer selects amplitude along an axis while a wave plate changes relative phase between its eigenaxes. A polarizer followed by a quarter-wave plate can produce circular polarization when the selected linear axis is at $45^{\circ }$ to the plate axes. Reversing the order generally gives a different result: the plate first changes the input state, then the polarizer removes one component and returns the beam to a linear state along its own axis. The components are linear operators on field amplitudes, but their matrices need not commute.

Instruments with multiple polarizing elements should be modeled in their physical order. A reflection before a wave plate can change the propagation reference and circular handedness convention. A fiber or stressed window before an analyzer can add unknown birefringence. An analyzer after an imaging system can sample a spatially varying polarization field rather than one uniform state. State whether the reported Stokes values describe one detector pixel, an aperture average, or an image-region average.

Final measurement checklist

Use this concise sequence before releasing a polarization result:

1. Stabilize the source and measure dark/background at the selected integration time.
2. Verify detector linearity with a known attenuation change.
3. Calibrate analyzer zero and plate fast-axis marks with a known linear reference.
4. Acquire horizontal-vertical, diagonal-antidiagonal, and circular-basis intensity pairs with source-reference checks.
5. Apply background, gain, and source-drift corrections before forming Stokes differences.
6. Check pair sums, the physical Stokes bound, analyzer residuals, and the stated polarization convention.
7. Report wavelength, bandwidth, coordinate axes, component order, raw data, fitted state, and uncertainty.

The sequence preserves the distinction between a clean textbook state and a measured optical signal. It also localizes whether a discrepancy belongs to source polarization, element retardance, mechanical alignment, or detector response.

Spatial and spectral averaging

A detector reports an aperture- and bandwidth-weighted polarization state. For sampled spatial or spectral components, the measured Stokes vector has the form

with weights set by collection geometry, spectral response, and integration time. Components can have fully polarized local states while their vector sum has a reduced degree of polarization.

• Spatial selection. Reduce the aperture or image the sample onto the detector when a local state is required. A detector collecting several image points adds their Stokes vectors; neighboring points with opposite circular components or different linear axes can lower the measured degree of polarization.
• Spectral and coherence limits. Pair a quarter-wave plate with its design wavelength and stated source bandwidth. A broadband source through a dispersive plate can produce well-defined ellipses in narrow bands while the unresolved detector reports their average. Malus-law intensity through two polarizers does not require a stable optical carrier phase. Wave-plate conversion of one monochromatic component does require a defined relative phase between its fast and slow components.
• Path-dependent changes. Nonnormal windows, metallic mirrors, fiber bends, and stressed mounts can add diattenuation or birefringence outside the intended sample model. Measure a reference path with the same windows, lenses, and detector geometry before assigning all observed Stokes change to the sample.
• Quantity and units. Field amplitude, optical power, irradiance, detector current, and normalized Stokes component are different observables. A calibration coefficient connects them only over its stated wavelength range, dynamic range, and detector configuration. An ideal wave plate conserves intensity while changing relative phase; an ideal polarizer changes both transmitted intensity and polarization; reflection and scattering can redistribute energy among directions.
• Report. State the spatial region, wavelength range, time average, coordinate axes, propagation direction, normalized Stokes vector, degree of polarization, analyzer and wave-plate order, circular-basis convention, raw basis-pair intensities, and uncertainty. “Elliptically polarized” is incomplete without those conditions.

Consider two equal-intensity, mutually incoherent image regions: one horizontally linear and one vertically linear. Their summed intensity is nonzero, while their opposite linear Stokes contributions cancel. The aperture can therefore report a reduced or zero degree of polarization even though each resolved region is fully polarized. A coherent superposition is different because its relative phase carries additional polarization information. State whether the detector averages incoherent regions, resolved coherent components, or both before interpreting a reduced Stokes vector as depolarization by the sample.

Detector linearity matters in the same comparison. Acquire the basis-pair intensities at two source levels related by a calibrated attenuation and verify that the corrected Stokes differences scale with the total intensity. A detector offset can mimic a small linear or circular component when one basis intensity is weak. Retain dark records and analyzer-angle repeats with the same aperture and bandwidth as the final state measurement.

The reported state is therefore a property of the stated collection procedure as well as of the optical field.