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# Linear polarization

In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. See polarization for more information.

### Additional recommended knowledge

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 sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is (cgs units) $\mathbf{E} ( \mathbf{r} , t ) = \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\psi\rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \}$ $\mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t )$

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 θ 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 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

• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.