2.3 Polarization

Types of polarization

Polarization describes the orientation of the electric field vector in an electromagnetic wave. Because EM waves are transverse, the electric field oscillates perpendicular to the propagation direction, and how that field vector moves over time defines the polarization state.

The three main types are linear, circular, and elliptical. Each corresponds to a distinct pattern traced by the tip of the electric field vector as seen looking along the propagation direction.

Linear polarization

In linear polarization, the electric field oscillates back and forth along a single fixed plane perpendicular to the propagation direction. That plane can be horizontal, vertical, or at any angle.

You encounter linearly polarized light when:

• Light passes through a polarizing filter (Polaroid), which transmits only the component along its transmission axis
• Light reflects from a flat surface at Brewster's angle, where the reflected beam is polarized perpendicular to the plane of incidence

Circular polarization

Here the electric field vector rotates in a circle (constant magnitude, steadily changing direction) in the plane perpendicular to propagation. Looking toward the source:

• Right-handed circular polarization (RCP): the field rotates clockwise
• Left-handed circular polarization (LCP): the field rotates counterclockwise

To produce circularly polarized light, pass linearly polarized light through a quarter-wave plate with its fast axis at 45° to the incoming polarization direction. The plate introduces a 90° phase shift between the two orthogonal field components, converting the linear oscillation into circular rotation.

Elliptical polarization

Elliptical polarization is the most general case. The tip of the electric field vector traces out an ellipse. The shape of that ellipse depends on:

• The relative amplitudes of the two orthogonal field components
• The phase difference between them

Both linear and circular polarization are special cases of elliptical polarization. Linear corresponds to a phase difference of 0° or 180° (the ellipse collapses to a line), and circular corresponds to equal amplitudes with a 90° phase difference (the ellipse becomes a circle).

Mathematical representation

Polarization states can be described mathematically using Jones vectors (for fully polarized light) or Stokes parameters (which also handle partial polarization and unpolarized light).

Jones vectors

A Jones vector is a complex two-component column vector representing the amplitude and phase of the two orthogonal electric field components of a polarized wave:

$\mathbf{J} = \begin{pmatrix} E_x \\ E_y \end{pmatrix}$

Each element is a complex number whose magnitude gives the amplitude and whose argument gives the phase. For example, horizontal linear polarization is $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and right circular polarization is $\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix}$.

Optical elements (polarizers, wave plates) are represented by 2×2 Jones matrices. To find the output polarization, you multiply the Jones matrix of the element by the input Jones vector.

Limitation: Jones vectors only work for fully polarized, coherent light.

Stokes parameters

Stokes parameters are four real numbers $(S_0, S_1, S_2, S_3)$ that fully describe any polarization state, including partially polarized and unpolarized light.

• $S_0$: total intensity
• $S_1$: preference for horizontal ($+$) vs. vertical ($-$) linear polarization
• $S_2$: preference for +45° ($+$) vs. -45° ($-$) linear polarization
• $S_3$: preference for right-circular ($+$) vs. left-circular ($-$) polarization

For fully polarized light, $S_0^2 = S_1^2 + S_2^2 + S_3^2$. For partially polarized light, $S_0^2 > S_1^2 + S_2^2 + S_3^2$. This makes Stokes parameters far more versatile than Jones vectors when dealing with real-world, partially polarized beams.

Polarization of electromagnetic waves

Transverse nature of EM waves

Electromagnetic waves are transverse: both the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ oscillate in planes perpendicular to the propagation direction $\mathbf{k}$, and they are perpendicular to each other. The relationship is $\mathbf{B} = \frac{1}{c}\hat{k} \times \mathbf{E}$.

Because $\mathbf{B}$ is always determined by $\mathbf{E}$ and $\hat{k}$, polarization is defined entirely by the behavior of the electric field.

Electric field orientation

• In a linearly polarized wave, $\mathbf{E}$ oscillates along a fixed direction in the transverse plane.
• In a circularly polarized wave, $\mathbf{E}$ rotates at constant magnitude, tracing a circle.
• In the general elliptical case, $\mathbf{E}$ traces an ellipse.

Any polarization state can be decomposed into two orthogonal linear components with specific amplitudes and a relative phase.

Polarization by reflection

When light reflects from a surface, the reflected beam's polarization depends on the angle of incidence and the refractive indices of the two media.

Brewster's angle

Brewster's angle $\theta_B$ is the incidence angle at which the reflected light is completely s-polarized (perpendicular to the plane of incidence). It satisfies:

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

where $n_1$ is the refractive index of the incident medium and $n_2$ is that of the reflecting medium. For an air-glass interface ($n_2 \approx 1.5$), $\theta_B \approx 56.3°$.

At Brewster's angle, the reflected and refracted rays are perpendicular to each other (they form a 90° angle). The p-polarized component has zero reflectance, so only the s-component reflects.

Fresnel equations

The Fresnel equations give the reflection and transmission amplitude coefficients for the two polarization components at a dielectric interface:

• s-polarization ($\mathbf{E}$ perpendicular to the plane of incidence):

$r_s = \frac{n_1 \cos\theta_i - n_2 \cos\theta_t}{n_1 \cos\theta_i + n_2 \cos\theta_t}$

• p-polarization ($\mathbf{E}$ parallel to the plane of incidence):

$r_p = \frac{n_2 \cos\theta_i - n_1 \cos\theta_t}{n_2 \cos\theta_i + n_1 \cos\theta_t}$

where $\theta_i$ is the angle of incidence and $\theta_t$ is the refraction angle (from Snell's law). These equations let you calculate the polarization state and intensity of reflected and transmitted beams at any angle.

Polarization by refraction

Birefringent materials

Birefringent materials (such as calcite and quartz) have anisotropic crystal structures, meaning the refractive index depends on the polarization direction and propagation direction of the light. When unpolarized light enters a birefringent crystal, it splits into two orthogonally polarized beams:

• The ordinary ray (o-ray), which sees a constant refractive index $n_o$ regardless of direction and obeys Snell's law normally
• The extraordinary ray (e-ray), which sees a direction-dependent refractive index $n_e(\theta)$ and does not generally obey Snell's law

Ordinary vs extraordinary rays

The two rays travel at different speeds inside the crystal and refract at different angles. This separation is called double refraction (or birefringence). The magnitude of the birefringence is characterized by $\Delta n = n_e - n_o$.

If $n_e > n_o$, the crystal is called positive uniaxial; if $n_e < n_o$, it's negative uniaxial. Calcite is negative uniaxial ($n_o = 1.658$, $n_e = 1.486$), which is why it produces such a dramatic double image when you look through it.

Polarization by scattering

Rayleigh scattering

Rayleigh scattering occurs when light interacts with particles much smaller than its wavelength (e.g., $\text{N}_2$ and $\text{O}_2$ molecules in the atmosphere). The scattered light is partially polarized, with maximum polarization perpendicular to the scattering plane.

The scattering intensity scales as $\lambda^{-4}$, so shorter wavelengths scatter much more strongly. This is why the sky appears blue: blue light ($\sim$450 nm) scatters roughly 5.5 times more than red light ($\sim$700 nm). If you look at the sky 90° from the sun through a polarizing filter, you'll see strong polarization.

Mie scattering

Mie scattering applies when particle sizes are comparable to the wavelength (e.g., water droplets, aerosols, dust). The polarization of Mie-scattered light depends on particle size, shape, refractive index, and scattering angle. Unlike Rayleigh scattering, Mie scattering doesn't have a simple analytical formula and generally requires numerical solutions to Maxwell's equations. It tends to scatter all wavelengths more equally, which is why clouds (large water droplets) appear white rather than blue.

Polarizing devices

Polarizing filters

A linear polarizing filter has a transmission axis. It transmits the component of the electric field parallel to that axis and absorbs the perpendicular component. For linearly polarized light incident at angle $\theta$ to the transmission axis, the transmitted intensity follows Malus's law:

$I = I_0 \cos^2\theta$

Polaroid sheets work by dichroic absorption: long-chain polymer molecules aligned in one direction preferentially absorb one polarization component. Applications include sunglasses, photography (reducing reflections), and LCDs.

Wave plates

Wave plates (retarders) introduce a phase difference between the fast and slow axis components of the transmitted light, changing the polarization state without absorbing energy.

• Quarter-wave plate ($\lambda/4$): introduces a 90° phase shift. Converts linear polarization to circular (and vice versa) when the input is at 45° to the fast axis.
• Half-wave plate ($\lambda/2$): introduces a 180° phase shift. Rotates the plane of linear polarization by twice the angle between the input polarization and the fast axis.

Wave plates are wavelength-specific because the phase retardation depends on $\Delta n \cdot d / \lambda$, where $d$ is the plate thickness.

Polarizing beamsplitters

Polarizing beamsplitters separate an incident beam into two orthogonally polarized output beams. One polarization is transmitted and the other is reflected.

Common types:

• Cube beamsplitters: two prisms cemented together with a dielectric multilayer coating at the interface that reflects s-polarization and transmits p-polarization
• Wollaston prisms: two birefringent wedges cemented together; the o-ray and e-ray diverge symmetrically as they exit

These are used in polarization imaging, interferometry, and quantum optics experiments.

Applications of polarization

Polarizing sunglasses

Glare from horizontal surfaces (water, roads, snow) is predominantly horizontally polarized because of reflection near Brewster's angle. Polarizing sunglasses use vertically oriented transmission axes to block this horizontally polarized glare while transmitting other light, significantly improving contrast and visual comfort.

Liquid crystal displays (LCDs)

An LCD pixel works by controlling the polarization rotation of light:

1. A backlight produces unpolarized light.
2. A rear polarizer linearly polarizes the light.
3. The liquid crystal layer rotates the polarization by an amount that depends on the applied voltage.
4. A front polarizer (crossed relative to the rear) either blocks or transmits the light depending on how much the liquid crystal rotated it.

By varying the voltage on each sub-pixel, the display controls brightness. Color filters over the sub-pixels produce the full color image.

Polarimetry in astronomy

Many astrophysical processes produce polarized light: scattering in stellar atmospheres, synchrotron radiation from relativistic electrons spiraling in magnetic fields, and reflection from planetary surfaces. Measuring the polarization (polarimetry) reveals information about magnetic field geometry, dust grain alignment, and the physical environment of the source that intensity measurements alone cannot provide.

Interaction with matter

Dichroism

Dichroism is the selective absorption of one polarization component over another. A dichroic material has an absorption coefficient that depends on the polarization direction relative to the material's optical axis. Polaroid sheets are a practical example: they strongly absorb light polarized along one direction while transmitting the orthogonal polarization. Natural dichroic crystals include tourmaline and cordierite.

Optical activity

Optically active materials rotate the plane of polarization of linearly polarized light as it propagates through them. This happens because the material has different refractive indices for left and right circularly polarized light (circular birefringence).

The rotation angle $\phi$ is given by:

$\phi = [\alpha] \cdot l \cdot c$

where $[\alpha]$ is the specific rotation (a material property, dependent on wavelength and temperature), $l$ is the path length, and $c$ is the concentration (for solutions). Quartz and sugar solutions are classic examples. This property is widely used in chemistry and pharmacology to identify and quantify chiral molecules.

Faraday effect

The Faraday effect is the rotation of the plane of polarization of linearly polarized light by a magnetic field applied parallel to the propagation direction. The rotation angle is:

$\phi = V B l$

where $V$ is the Verdet constant (material-dependent), $B$ is the magnetic field strength, and $l$ is the path length.

A key distinction from optical activity: the Faraday rotation is non-reciprocal. If light passes through the material and then reflects back, the rotation doubles rather than canceling. This property makes the Faraday effect essential for building optical isolators, which allow light to pass in one direction only, protecting lasers from destabilizing back-reflections.

Polarization in antennas

Polarization diversity

In wireless communications, the signal's polarization can change unpredictably due to multipath propagation (reflections, scattering). Polarization diversity combats this by using two orthogonally polarized antennas (e.g., one horizontal, one vertical) at the receiver. Since fading on one polarization channel is largely independent of the other, the system can select or combine the stronger signal, improving reliability.

Polarization matching

For maximum power transfer, the receiving antenna's polarization must match the incoming wave's polarization. A polarization mismatch between transmitter and receiver causes signal loss described by the polarization loss factor:

$\text{PLF} = |\hat{\rho}_r \cdot \hat{\rho}_w|^2$

where $\hat{\rho}_r$ and $\hat{\rho}_w$ are the polarization unit vectors of the receiving antenna and the incoming wave. Complete mismatch (e.g., a vertically polarized antenna receiving a horizontally polarized wave) gives PLF = 0, meaning total signal loss. Circularly polarized antennas are often used in satellite communications because they avoid the orientation-dependent mismatch problems of linear polarization.

Quantum aspects of polarization

Photon polarization states

In quantum mechanics, a single photon's polarization is described by a state vector in a two-dimensional Hilbert space. Using Dirac notation, horizontal and vertical polarization form a basis:

$|\psi\rangle = \alpha|H\rangle + \beta|V\rangle$

where $|\alpha|^2 + |\beta|^2 = 1$. This is mathematically identical to a qubit, which is why photon polarization is one of the most common physical implementations of quantum bits.

Optical elements act as operators on this state: a polarizer projects onto a particular state, a wave plate applies a unitary rotation. Measuring the polarization collapses the superposition into one of the basis states with probabilities $|\alpha|^2$ and $|\beta|^2$.

Entanglement of polarized photons

Polarization-entangled photon pairs can be generated through spontaneous parametric down-conversion (SPDC): a pump photon entering a nonlinear crystal splits into two lower-energy photons whose polarizations are quantum-mechanically correlated. A typical entangled state is the Bell state:

$|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|H\rangle|H\rangle + |V\rangle|V\rangle)$

Measuring one photon's polarization instantly determines the other's, regardless of separation distance. This isn't faster-than-light communication (no information is transmitted), but it does produce correlations that violate Bell inequalities, confirming that the entanglement is genuinely quantum mechanical.

Polarization entanglement is a core resource in quantum key distribution (QKD), quantum teleportation, and photonic quantum computing.