Linear polarization: Difference between revisions

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Historically, the orientation of a polarized electromagnetic wave has been defined in the optical regime by the orientation of the electric vector, and in the [[radio]] regime, by the orientation of the magnetic vector.

==Mathematical description of linear polarization==

The [[Classical physics | classical]] [[sinusoidal]] plane wave solution of the [[electromagnetic wave equation]] for the [[Electric field | electric]] and [[Magnetic field | magnetic]] fields is (cgs units)

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

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

for the magnetic field, where k is the [[wavenumber]],

:<math> \omega_{ }^{ } = c k</math>

is the [[angular frequency]] of the wave, and <math> c </math> is the [[speed of light]].

Here

:<math> \mid \mathbf{E} \mid </math>

is the [[amplitude]] of the field and

:<math> |\psi\rangle \equiv \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} </math>

is the [[Jones vector]] in the x-y plane.

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

:<math> \alpha_x = \alpha_y \equiv \alpha </math>.

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

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

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

If unit vectors are defined such that

:<math> |x\rangle \equiv \begin{pmatrix} 1 \\ 0 \end{pmatrix} </math>

and

:<math> |y\rangle \equiv \begin{pmatrix} 0 \\ 1 \end{pmatrix} </math>

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

:<math> |\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 </math>.

== See also ==