---
title: Polarization
module: Maxwell’s Equations and Electromagnetic Waves
moduleNumber: 10
lessonNumber: 5
order: 1005
summary: >
  A plane wave still leaves one thing free: which way its electric field points as it
  oscillates. That freedom is polarization, set entirely by the relative amplitude and phase
  of the two transverse field components — in phase gives a line, equal amplitudes a quarter
  cycle apart give a circle, everything else an ellipse. We work out how a linear analyzer
  reads a state through Malus's law $I=I_0\cos^2\theta$, why that scan alone cannot tell
  circular light from unpolarized, and how a quarter-wave plate plus a few analyzer settings
  recover the full Stokes vector and the degree of polarization.
topics: [Maxwell’s Equations and Electromagnetic Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 30 — Maxwell’s Equations and Electromagnetic Waves; §30-4 Electromagnetic Radiation"
  - book: Tipler & Mosca
    ref: "Ch. 31 — Properties of Light; §31-4 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

$$
\vec E(z,t)=E_x(z,t)\,\hat x+E_y(z,t)\,\hat y.
$$

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

$$
E_x=E_{0x}\cos(kz-\omega t+\delta_x),
\qquad
E_y=E_{0y}\cos(kz-\omega t+\delta_y).
$$

Only the phase difference matters:

$$
\delta=\delta_y-\delta_x.
$$

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

$$
\left(\frac{E_x}{E_{0x}}\right)^2
+\left(\frac{E_y}{E_{0y}}\right)^2
-2\left(\frac{E_x}{E_{0x}}\right)
\left(\frac{E_y}{E_{0y}}\right)\cos\delta
=\sin^2\delta.
$$

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.

$$
% caption: The polarization-ellipse equation gives a straight line when the phase difference is $0$ or $\pi$, a circle for equal amplitudes a quarter cycle apart, and an ellipse in the general case. The arrow marks the field endpoint that traces each curve once per period.
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$$

### 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

$$
\tan\psi=\frac{E_{0y}}{E_{0x}}
$$

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,

$$
E_{0x}=E_{0y}=E_0,
\qquad
\delta=\pm\frac{\pi}{2}.
$$

At a fixed point, the field magnitude is constant:

$$
E_x^2+E_y^2=E_0^2.
$$

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.

$$
% caption: Unpolarized light has random transverse field directions that average to no preferred axis; a polarized beam has a stable axis an analyzer can track. Partial polarization mixes the two.
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$$

### Phasor representation

Complex amplitudes compress component amplitude and phase into one expression:

$$
\widetilde{\vec E}
=\widetilde E_x\hat x+\widetilde E_y\hat y,
\qquad
\vec E(z,t)=\Re\!\left\{\widetilde{\vec E}e^{i(\omega t-kz)}\right\}.
$$

The complex ratio $\widetilde E_y/\widetilde E_x$ 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.

$$
% caption: A wire-grid polarizer. Field parallel to the wires drives current and is absorbed (low transmission); field perpendicular to the wires passes. The transmission axis is perpendicular to the wires.
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\node[acc] at (4.45,1.30) {high};
\node[above] at (2.90,2.95) {$E$ across wires};
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\end{tikzpicture}
$$

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

$$
I(\theta)=I_0\cos^2\theta.
$$

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.

$$
% caption: Malus's-law geometry. The incoming linear field $E$ is resolved onto the analyzer transmission axis; the transmitted amplitude is $E\cos\theta$ and the transmitted intensity is $I_0\cos^2\theta$.
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$$

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

$$
I_1=\frac{I_{\mathrm{unpol}}}{2}.
$$

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

$$
I_2=\frac{I_{\mathrm{unpol}}}{2}\cos^2\theta.
$$

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.

$$
% caption: Malus's law $I=I_0\cos^2\theta$. Transmission is maximal when the analyzer axis aligns with the input polarization, zero when the axes are crossed, and repeats every $180^\circ$.
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$$

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

$$
I_3=\frac{I_0}{2}\cos^2(45^\circ)\cos^2(45^\circ)=\frac{I_0}{8}.
$$

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

$$
% caption: Three ideal polarizers at $0^\circ$, $45^\circ$, and $90^\circ$. The middle sheet prepares a component along the final axis, so the system transmits $I_0/8$ even though the first and last axes are crossed.
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$$

### 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

$$
I_{\mathrm{meas}}(\theta)
=I_{\parallel}\cos^2(\theta-\psi)
+I_{\perp}\sin^2(\theta-\psi)
+I_{\mathrm{dark}}.
$$

$I_{\parallel}$ 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 $\widetilde E_x$ and $\widetilde E_y$, define one common Stokes convention:

$$
\begin{aligned}
S_0&=|\widetilde E_x|^2+|\widetilde E_y|^2,\\
S_1&=|\widetilde E_x|^2-|\widetilde E_y|^2,\\
S_2&=2\Re\!\left(\widetilde E_x\widetilde E_y^\ast\right),\\
S_3&=2\Im\!\left(\widetilde E_x\widetilde E_y^\ast\right).
\end{aligned}
$$

$S_0$ 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

$$
\mathcal P=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0},
\qquad 0\leq\mathcal P\leq1.
$$

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.

$$
% caption: Poincaré representation (the $S_1$–$S_3$ plane). The center is unpolarized light, the outer circle holds fully polarized states with $\mathcal P=1$, and interior points are partially polarized; the $S_1$ axis carries linear bases and the $S_3$ axis carries circular bases.
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$$

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

$$
\begin{aligned}
S_0&=I_H+I_V,\\
S_1&=I_H-I_V,\\
S_2&=I_D-I_A,\\
S_3&=I_R-I_L.
\end{aligned}
$$

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,

$$
\psi=\frac12\atanTwo(S_2,S_1),
\qquad
\chi=\frac12\arcsin\!\left(\frac{S_3}{S_0}\right).
$$

$\psi$ 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

$$
u(S_1)=\sqrt{u(I_H)^2+u(I_V)^2}.
$$

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

$$
\tan\theta_B=\frac{n_2}{n_1},
$$

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.

$$
% caption: Brewster geometry. At the polarizing angle the reflected and refracted rays are perpendicular, and the reflected wave is linearly polarized perpendicular to the plane of incidence, with $\tan\theta_B=n_2/n_1$.
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$$

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 $n_1\approx1.00$ and glass with $n_2=1.50$,

$$
\theta_B=\arctan(1.50)=56.3^\circ.
$$

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 $y$.

$$
% caption: Polarization by scattering. An unpolarized beam along $z$ drives transverse charge motion; light scattered along $x$ is linearly polarized along $y$, because the $x$-directed dipole radiates nothing along its own axis.
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\draw[->,acc,very thick] (4.40,1.60)--(3.95,1.15) node[below] {$E$};
\end{tikzpicture}
$$

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](/electricity-and-magnetism/maxwell-electromagnetic-waves/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.

$$
% caption: Double refraction in a birefringent crystal. The two polarization eigenmodes travel at different phase velocities and can emerge as spatially separated ordinary and extraordinary rays with perpendicular polarizations.
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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

$$
\Delta\phi=\frac{2\pi}{\lambda_0}\Delta n\,d,
$$

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

$$
\Delta\phi=\frac{\pi}{2}\pmod{2\pi},
$$

and a half-wave plate has

$$
\Delta\phi=\pi\pmod{2\pi}.
$$

The lowest-order thicknesses are

$$
d_{\lambda/4}=\frac{\lambda_0}{4\Delta n},
\qquad
d_{\lambda/2}=\frac{\lambda_0}{2\Delta n}.
$$

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.

$$
% caption: A quarter-wave plate. Linear input at $45^\circ$ to the fast and slow axes splits into equal components that emerge a quarter cycle apart, producing circular polarization at the design wavelength.
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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

$$
\alpha_{\mathrm{out}}=2\beta-\alpha.
$$

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 $\Delta\phi$. Projection onto the crossed analyzer gives

$$
I_{\perp}=I_P\sin^2\!\left(\frac{\Delta\phi}{2}\right).
$$

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

$$
I_{\parallel}=I_P\cos^2\!\left(\frac{\Delta\phi}{2}\right).
$$

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 $\Delta\phi_0$ at design
wavelength $\lambda_d$ has a first approximation

$$
\Delta\phi(\lambda_0)
\approx\Delta\phi_0\frac{\lambda_d}{\lambda_0}
\frac{\Delta n(\lambda_0)}{\Delta n(\lambda_d)}.
$$

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 $\Delta n\,d$, 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

> **Worked example (two-polarizer intensity).** Unpolarized light of intensity
> $I_0=3.00\ \mathrm{W\,m^{-2}}$ passes through an ideal polarizer and then an ideal analyzer
> at $60.0^\circ$ to the first axis. The first sheet transmits
>
> $$
> I_1=\frac{I_0}{2}=1.50\ \mathrm{W\,m^{-2}},
> $$
>
> and the analyzer applies Malus's law,
>
> $$
> I_2=I_1\cos^2(60.0^\circ)
> =(1.50)(0.500)^2
> =0.375\ \mathrm{W\,m^{-2}}.
> $$
>
> The result carries two separate reductions: the one-half average over unpolarized input at
> the first sheet, and the $\cos^2$ projection at the second. Starting from
> $I_0\cos^2(60.0^\circ)$ would drop the first factor and double the answer; reusing $I_0/2$
> at the analyzer would undercount a beam that is already linearly polarized.

> **Worked example (Brewster angle for an air–glass interface).** A wave incident from air
> into glass with $n_1=1.00$ and $n_2=1.50$ has polarizing angle
>
> $$
> \theta_B=\arctan\!\left(\frac{1.50}{1.00}\right)=56.3^\circ.
> $$
>
> The refracted ray is perpendicular to the reflected ray, so
>
> $$
> \theta_2=90.0^\circ-56.3^\circ=33.7^\circ,
> $$
>
> which Snell's law confirms: $(1.00)\sin(56.3^\circ)\approx(1.50)\sin(33.7^\circ)$, both
> sides near $0.832$. This locates the angle at which the reflected wave is fully linearly
> polarized in the ideal transparent-interface model. It says nothing about the reflected
> power without the Fresnel coefficients, and it excludes surface roughness and absorption.

> **Worked example (quarter-wave plate thickness).** A crystal has birefringence
> $\Delta n=0.0090$ at vacuum wavelength $\lambda_0=550\ \mathrm{nm}$. The lowest-order
> quarter-wave thickness is
>
> $$
> d_{\lambda/4}
> =\frac{550\times10^{-9}\ \mathrm{m}}{4(0.0090)}
> =1.53\times10^{-5}\ \mathrm{m}
> =15.3\ \mathrm{\mu m}.
> $$
>
> Circular output needs both this quarter-wave delay and equal input components along the
> material axes. An input at any other angle gives unequal components and an elliptical
> output even at the correct retardance, and a wavelength shift moves the retardance off
> $\pi/2$, turning the intended circle into an ellipse.

## 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 $\alpha$

$$
I(\alpha)
=\frac12\left(S_0+S_1\cos2\alpha+S_2\sin2\alpha\right).
$$

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

$$
I_{\max,\min}
=\frac12\left(S_0\pm\sqrt{S_1^2+S_2^2}\right).
$$

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.

> **Worked example (Stokes reduction from six intensities).** Six background-corrected
> analyzer readings, in milliwatts, are taken for one spectrally narrow beam:
>
> $$
> I_H=0.65,\quad I_V=0.35,\quad
> I_D=0.70,\quad I_A=0.30,\quad
> I_R=0.60,\quad I_L=0.40.
> $$
>
> All three pair sums equal 1.00 mW, so under the stated circular-basis convention
>
> $$
> S_0=1.00\ \mathrm{mW},
> \qquad
> S_1=0.30\ \mathrm{mW},
> \qquad
> S_2=0.40\ \mathrm{mW},
> \qquad
> S_3=0.20\ \mathrm{mW},
> $$
>
> and the degree of polarization is
>
> $$
> \mathcal P
> =\frac{\sqrt{(0.30)^2+(0.40)^2+(0.20)^2}}{1.00}
> =0.539.
> $$
>
> The beam is partially polarized: an excess of horizontal over vertical, an excess in one
> diagonal basis, and a nonzero circular-basis difference, but $\mathcal P$ well below one, so
> the readings do not describe a single fully polarized ellipse. The linear-orientation angle
> of the polarized part is
>
> $$
> \psi=\frac12\atanTwo(0.40,0.30)=26.6^\circ.
> $$
>
> The sign of $S_3$ becomes a right- or left-handed name only once the circular-basis
> convention is stated in the record.

$$
% caption: The six analyzer intensities of the worked reduction ($I_H=0.65$, $I_V=0.35$, $I_D=0.70$, $I_A=0.30$, $I_R=0.60$, $I_L=0.40$ mW). Each basis pair sums to $1.00$ mW; the paired differences give $S_1$, $S_2$, and $S_3$.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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$$

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

$$
u(S_1)=\sqrt{(0.010)^2+(0.010)^2}\ \mathrm{mW}
=0.014\ \mathrm{mW}.
$$

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

$$
S_1^2+S_2^2+S_3^2\leq S_0^2.
$$

The worked values satisfy it because $0.29\ \mathrm{mW^2}<1.00\ \mathrm{mW^2}$.
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.

> **Worked example (extinction ratio and contrast).** An analyzer scan has a maximum raw
> reading of $0.825\ \mathrm{mW}$, a minimum of $0.025\ \mathrm{mW}$, and a blocked-source
> background of $0.005\ \mathrm{mW}$. Background correction gives $I_{\max}=0.820\ \mathrm{mW}$
> and $I_{\min}=0.020\ \mathrm{mW}$, so the extinction ratio is
>
> $$
> \rho=\frac{I_{\min}}{I_{\max}}
> =\frac{0.020}{0.820}
> =2.44\times10^{-2}
> $$
>
> and the modulation contrast is
>
> $$
> C=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}
> =\frac{0.820-0.020}{0.820+0.020}
> =0.952.
> $$

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

$$
R=\frac{I_{\perp}}{I_P}=\sin^2\!\left(\frac{\Delta\phi}{2}\right).
$$

A measured value $R=0.500$ is consistent with quarter-wave retardance
$\Delta\phi=\pi/2$, 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

$$
\frac{dR}{\d(\Delta\phi)}=\frac12\sin\Delta\phi.
$$

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 $S_1^2+S_2^2+S_3^2\leq S_0^2$ 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.

> **Worked example (Stokes vector of an elliptical state).** A fully polarized monochromatic
> state has component amplitudes
>
> $$
> E_{0x}=2.00\ \mathrm{V\,m^{-1}},
> \qquad
> E_{0y}=1.00\ \mathrm{V\,m^{-1}},
> \qquad
> \delta=60.0^\circ.
> $$
>
> The amplitudes are unequal and the phase difference is neither zero nor a quarter cycle, so
> the state is elliptical. Its Stokes values in intensity-proportional units are
>
> $$
> \begin{aligned}
> S_0&=2.00^2+1.00^2=5.00,\\
> S_1&=2.00^2-1.00^2=3.00,\\
> S_2&=2(2.00)(1.00)\cos60.0^\circ=2.00,\\
> S_3&=2(2.00)(1.00)\sin60.0^\circ=3.46.
> \end{aligned}
> $$
>
> The full-polarization check holds exactly,
>
> $$
> S_1^2+S_2^2+S_3^2
> =9.00+4.00+12.0
> =25.0=S_0^2,
> $$
>
> so $\mathcal P=1$. The orientation and ellipticity angles are
>
> $$
> \psi=\frac12\atanTwo(2.00,3.00)=16.8^\circ,
> \qquad
> \chi=\frac12\arcsin\!\left(\frac{3.46}{5.00}\right)=21.9^\circ.
> $$
>
> The major-axis orientation depends on both the amplitude ratio and the component phase; the
> Stokes reduction folds that phase in through $S_2$ and $S_3$, and the sign of $S_3$ fixes the
> rotation sense once the handedness convention is stated.

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

$$
I(40.0^\circ)
=\frac12\left[5.00+3.00\cos80.0^\circ+2.00\sin80.0^\circ\right]
=3.75
$$

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

$$
\vec S_{\mathrm{meas}}=\sum_j w_j\vec S_j,
\qquad
w_j\ge0,
$$

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.

| averaging mechanism | physical cause | measurement consequence |
|---|---|---|
| spatial aperture | different image points have different axes or handedness | Stokes differences cancel while total intensity adds |
| spectral bandwidth | retardance varies with wavelength | one ellipse does not represent the broadband average |
| temporal integration | state fluctuates during the exposure | detector reports the time-averaged Stokes vector |
| optical path | reflections, stress, or oblique transmission modify components | sample and reference paths require matched optics |

- **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.