Deep Dive: Polarization

The Polarization Vector

The polarization of an EM wave is defined by the direction of the electric field vector $\vec{E}$. We characterize this using a complex displacement vector $\vec{n}$ (the polarization vector). For a plane wave:

$$ \tilde{\mathbf{E}}(z, t) = \tilde{E}_0 e^{i(kz-\omega t)} \hat{n} $$

If $\hat{n}$ is a real unit vector, the wave is linearly polarized. If $\hat{n}$ is complex, the tip of $\vec{E}$ traces out an ellipse.

Types of Polarization

Linear The E-field oscillates in a single fixed direction. The polarization vector $\hat{n}$ is real.

Circular The E-field tip traces a circle. Two orthogonal waves with equal amplitude and $90^\circ$ phase shift.

Elliptical The most general case where amplitudes are unequal or the phase shift is not $90^\circ$.

Handedness Convention

Looking towards the source, if the vector rotates clockwise, it is Right-Handed (RHCP). Using the $(\hat{x} + i\hat{y})$ convention corresponds to a specific rotation direction that depends on whether you choose $kz - \omega t$ or $\omega t - kz$ in the exponent.

Mathematical Descriptions

Linear

A wave polarized at an angle $\theta$ to the $x$-axis: $$ \vec{E}(z,t) = E_0 \cos(kz - \omega t) (\cos\theta \hat{x} + \sin\theta \hat{y}) $$

Circular

Two orthogonal waves with a $90^\circ$ phase shift:

$$ \tilde{\mathbf{E}}_{R/L} = E_0 e^{i(kz-\omega t)} (\hat{x} \pm i\hat{y}) $$

In real terms: $$ \vec{E}(z,t) = E_0 [\cos(kz-\omega t)\hat{x} \mp \sin(kz-\omega t)\hat{y}] $$