Linear polarization

Mathematical description

The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is (cgs units)

: \mathbf{E} ( \mathbf{r} , t ) = |\mathbf{E}| \mathrm{Re} \left { |\psi\rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right }

: \mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t )/c

for the magnetic field, where k is the wavenumber,

: \omega_{ }^{ } = c k

is the angular frequency of the wave, and c is the speed of light.

Here \mid\mathbf{E}\mid is the amplitude of the field and

: |\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \ \psi_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix}

is the Jones vector in the x-y plane.

The wave is linearly polarized when the phase angles \alpha_x^{ } , \alpha_y are equal,

: \alpha_x = \alpha_y \ \stackrel{\mathrm{def}}{=}\ \alpha .

This represents a wave polarized at an angle \theta with respect to the x axis. In that case, the Jones vector can be written

: |\psi\rangle = \begin{pmatrix} \cos\theta \ \sin\theta \end{pmatrix} \exp \left ( i \alpha \right ) .

The state vectors for linear polarization in x or y are special cases of this state vector.

If unit vectors are defined such that

: |x\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} 1 \ 0 \end{pmatrix}

and

: |y\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} 0 \ 1 \end{pmatrix}

then the polarization state can be written in the "x-y basis" as

: |\psi\rangle = \cos\theta \exp \left ( i \alpha \right ) |x\rangle + \sin\theta \exp \left ( i \alpha \right ) |y\rangle = \psi_x |x\rangle + \psi_y |y\rangle .

References

References

1. A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe", read 9 December 1822; printed in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), ''Oeuvres complètes d'Augustin Fresnel'', vol. 1 (1866), pp.{{nnbsp731–51; translated as "Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis", {{Zenodo. 4745976, 2021 (open access); §9.