Editing Linear polarization (section)

==Mathematical description==
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 ) = |\mathbf{E}|  \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 )/c   </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  \ \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}   </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 \ \stackrel{\mathrm{def}}{=}\   \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  \ \stackrel{\mathrm{def}}{=}\    \begin{pmatrix} 1    \\ 0  \end{pmatrix}    </math>

and

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

then the polarization state can be 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>.