Polarization

The polarization of an electromagnetic wave is the direction of the wave’s oscillating electric field.

Light is considered polarized if the component electromagnetic waves all share the same polarization, i.e., if the electric fields all oscillate in the same direction.

A polarizer is a device that constrains the electromagnetic waves passing through it to a certain polarization. Physically, a polarizer may be implemented as a series of thin conductive strips. An incident electric field in the direction of the strips will cause electrons in the conductor to move and create an opposite electric field, effectively canceling it. There is nowhere for the electrons to move perpendicular to the strips (in the so-called “easy axis”) so an incident electric field in this direction passes through unaffected.

The figure below shows a polarizer in the $x,y$ plane, an incident wave coming from the left, and the polarized wave after it passes through the polarizer. The arrows $\mathbf E_i$ and $\mathbf B_i$ show the direction of the incident electric and magnetic fields, while $\mathbf E_p$ and $\mathbf B_p$ show the direction of the polarized electric and magnetic fields.

Consider a polarizer lying in the $x,y$ plane and an incident light beam in the $+z$ direction towards us, as pictured below. The easy axis of the polarizer is at an angle $\theta$ from the horizontal.

The electric field of the incident wave has some component in the $x$ and $y$ direction

\begin{align*} \mathbf E_x &= E_{0,x} \cos(\mathcal Kz-\omega t) \mathbf{\hat x} \\ \mathbf E_y &= E_{0,y} \cos(\mathcal Kz-\omega t) \mathbf{\hat y} \\ \mathbf E &= \mathbf E_x + \mathbf E_y = \mathbf E_0 \cos(\mathcal Kz-\omega t). \end{align*}

We wish to find the resulting electric field once the wave passes through the polarizer. The magnitude of the electric field that is allowed through the polarizer is the component of the incident field in the direction of the easy axis. The direction of the resulting field is in the direction of the easy axis.

Effectively this is a projection of the incident electric field vector onto the easy axis vector. Let $\mathbf A = \cos \theta\, \mathbf{\hat x} + \sin \theta\, \mathbf{\hat y}$ be the unit vector in the direction of the easy axis. Then our polarized electric field is

\mathbf E_{\text{polarized}} = (\mathbf E \cdot \mathbf A) \mathbf A.

Let’s expand this expression.

\begin{align*} \mathbf E_\text{polarized} &= (E_{0,x} \cos \theta + E_{0,y} \sin \theta) \begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix} \cos(\mathcal Kz -\omega t) \\ &= \begin{pmatrix} E_{0,x} \cos^2 \theta + E_{0,y} \cos\theta\sin\theta \\ E_{0,x} \cos\theta\sin\theta + E_{0,y} \sin^2\theta \end{pmatrix} \cos(\mathcal Kz -\omega t). \end{align*}

It would be more convenient if we could express $\mathbf E_\text{polarized}$ in terms of a single operation on $\mathbf E$ , as opposed to two products. Considering the above vector, we see that it can simply be expressed as a matrix-vector product

\mathbf E_\text{polarized} = P_\theta \cdot \mathbf E = \begin{pmatrix} \cos^2 \theta & \cos\theta\sin\theta \\ \cos\theta\sin\theta & \sin^2\theta \end{pmatrix} \mathbf E

where $P_\theta$ is defined as the polarization matrix for a linear polarizer at angle $\theta$ from the $x$ -axis.

Wave plate

A wave plate allows light polarized along its “fast axis” to travel through it more quickly than light polarized along the “slow axis”. The phase difference caused by a wave plate generally depends on the wavelength of the incident light.

Most commonly we will look at a quarter wave plate, which delays the electric field along the slow axis by ¼ period relative to the fast axis. Consider for example a quarter wave plate with fast axis $\mathbf{\hat x}$ and slow axis $\mathbf{\hat y}$ . The electric field after passing through the wave plate is

\begin{align*} \mathbf E_\text{delayed} &= \mathbf E_0 (e^{i(kz-\omega t)} \mathbf{\hat x} + e^{i(kz-\omega t+\pi/2)} \mathbf{\hat y}) \\ &= \mathbf E_0 \begin{pmatrix} 1 \\ i \end{pmatrix} e^{i(kz-\omega t)}. \end{align*}

Generally, we can think of a quarter wave plate as the sum of two orthogonal polarizers, one along the fast axis and one along the slow axis. The slow axis polarizer adds a quarter period phase shift to the light that passes through, giving us $Q_\theta = P_\theta + iP_{\theta + \pi/2}$

If the fast axis is at angle $\theta$ with respect to the $x$ axis, we can write a matrix to describe the effect of the wave plate on an incident wave

\begin{align*} Q_\theta &= \begin{pmatrix} \cos^2\theta + i \sin^2\theta & \cos\theta\sin\theta-i\sin\theta\cos\theta \\ \cos\theta\sin\theta-i\sin\theta\cos\theta & \sin^2\theta+i\cos^2\theta \end{pmatrix} \end{align*}

\mathbf E_\text{delayed} = Q_\theta \cdot \mathbf E.

A quarter wave plate turns unpolarized light into circularly polarized light. The electric field of circularly polarized light rotates in time, sweeping a circle. The direction of the rotation is from the positive fast axis to the positive slow axis.