Polarization

5.1 States of Polarization

For a plane wave propagating in the $z$-direction:

$\mathbf{E} = E_{0x}\cos(kz - \omega t)\,\hat{\mathbf{x}} + E_{0y}\cos(kz - \omega t + \delta)\,\hat{\mathbf{y}}$

• Linear polarization: $\delta = 0$ or $\delta = \pi$. The E-field oscillates along a fixed line.
• Circular polarization: $E_{0x} = E_{0y}$ and $\delta = \pm\pi/2$. Right-handed ($\delta = -\pi/2$) or left-handed ($\delta = +\pi/2$).
• Elliptical polarization: General case. The tip of $\mathbf{E}$ traces an ellipse.

5.2 Malus”s Law

When linearly polarised light of intensity $I_0$ passes through a polariser at angle $\theta$ to the Polarisation direction:

$I = I_0 \cos^2\theta$

Proof. The component of $\mathbf{E}$ along the polariser axis is $E\cos\theta$. Since $I \propto E^2$: $I = I_0 \cos^2\theta$. $\blacksquare$

5.3 Birefringence and Wave Plates

Birefringent crystals (e.g., calcite) have two refractive indices: $n_o$ (ordinary ray) and $n_e$ (extraordinary ray). The two rays have orthogonal polarisations and different phase velocities.

A wave plate of thickness $t$ introduces a relative phase shift between the two polarisation Components:

$\Delta\phi = \frac{2\pi}{\lambda}(n_o - n_e)\,t$

Quarter-wave plate (QWP): $\Delta\phi = \pi/2$So $t_{\mathrm{QWP} = \lambda/(4|n_o - n_e|)}$. Converts linear polarisation at $45°$ to the fast/slow axes into circular polarisation, and vice Versa.

Half-wave plate (HWP): $\Delta\phi = \pi$So $t_{\mathrm{HWP} = \lambda/(2|n_o - n_e|)}$. Rotates the plane of linear polarisation by $2\theta$Where $\theta$ is the angle between the Input polarisation and the fast axis.

:::caution Common Pitfall A quarter-wave plate only produces circular polarisation when the input linear polarisation is at Exactly $45°$ to the fast and slow axes. For other input angles, the output is elliptically Polarised. A half-wave plate rotates linear polarisation by $2\theta$Not $\theta$.

5.4 Optical Activity

Certain materials (sugars, quartz) rotate the plane of linearly polarised light. The specific Rotation is:

$[\alpha] = \frac{\theta}{c \cdot l}$

Where $\theta$ is the rotation angle, $c$ is the concentration, and $l$ is the path length.

Optical activity arises from the helical structure of molecules, which gives different refractive Indices for left- and right-circularly polarised light (circular birefringence). If $n_L$ and $n_R$ Are the refractive indices for left and right circular polarisation:

$\theta = \frac{\pi l}{\lambda}(n_L - n_R)$

Optical activity is reciprocal: if the beam is reflected back through the medium, the rotation Is cancelled.

5.5 Brewster’s Angle

At the Brewster angle $\theta_B$The reflected beam for p-polarised light vanishes ($r_p = 0$):

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

Proof. Setting $r_p = 0$ requires $n_2\cos\theta_i = n_1\cos\theta_t$. Using Snell’s law $n_1\sin\theta_i = n_2\sin\theta_t$:

$\frac{\cos\theta_i}{\sin\theta_i} = \frac{\cos\theta_t}{\sin\theta_t} \implies \cot\theta_i = \cot\theta_t \implies \theta_i = \theta_t'$

Where $\theta_t' = 90° - \theta_t$. This gives $\theta_i + \theta_t = 90°$So:

$\tan\theta_i = \frac{n_2}{n_1} \quad \blacksquare$

At Brewster’s angle, the reflected beam is purely s-polarised, and the reflected and refracted beams Are perpendicular ($\theta_B + \theta_t = 90°$). This principle is used in Brewster windows and Polarisation by reflection.

For an air-glass interface ($n_1 = 1$, $n_2 = 1.5$): $\theta_B = \arctan(1.5) = 56.3°$.

5.6 Faraday Rotation

In a magneto-optical material with a magnetic field $\mathbf{B}$ applied along the propagation Direction, the plane of polarisation rotates by:

$\theta_F = V B l$

Where $V$ is the Verdet constant (rad/(T$\cdot$M)), $B$ is the magnetic field strength, and $l$ is the path length through the material.

Mechanism. The magnetic field induces circular birefringence: left and right circular polarisations Experience different refractive indices ($n_L \neq n_R$). Unlike natural optical activity, Faraday Rotation is non-reciprocal: if the beam is reflected back through the medium, the rotation Doubles rather than cancelling.

Applications. Optical isolators (one-way light valves), optical circulators, and magneto-optical Sensors. An optical isolator consists of a polariser, a Faraday rotator set to rotate by $45°$And An analyser at $45°$ to the polariser. Forward-propagating light is transmitted; backward light Is rotated by another $45°$ (total $90°$) and blocked by the analyser.

5.7 Polarization by Scattering

When light is scattered by particles much smaller than the wavelength (Rayleigh scattering), the Scattered light is partially polarised. Light scattered at $90°$ to the incident direction is completely linearly polarised in the plane perpendicular to the scattering plane.

Proof. Consider an incident unpolarised beam propagating along $\hat{\mathbf{z}}$. The $\mathbf{E}$-field oscillates in the $xy$-plane. An observer along $\hat{\mathbf{x}}$ (scattering Angle $90°$) receives radiation from the accelerating electrons. The dipole radiation pattern of an Oscillator along $\hat{\mathbf{y}}$ has zero intensity along $\hat{\mathbf{y}}$ but maximum along $\hat{\mathbf{x}}$. The oscillator along $\hat{\mathbf{x}}$ radiates zero along its own axis. Thus The observer along $\hat{\mathbf{x}}$ sees only the $y$-component: the scattered light is Polarised along $\hat{\mathbf{y}}$. $\blacksquare$

This explains why the sky is polarised at $90°$ from the sun and why polarising sunglasses reduce Glare from horizontal surfaces (Brewster’s angle reflection from road/water).

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