Definitions

# 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.

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)
$mathbf\left\{E\right\} \left(mathbf\left\{r\right\} , t \right) = mid mathbf\left\{E\right\} mid mathrm\left\{Re\right\} left \left\{ |psirangle exp left \left[i left \left(kz-omega t right \right) right \right] right \right\}$

$mathbf\left\{B\right\} \left(mathbf\left\{r\right\} , t \right) = hat \left\{ mathbf\left\{z\right\} \right\} times mathbf\left\{E\right\} \left(mathbf\left\{r\right\} , t \right)$

for the magnetic field, where k is the wavenumber,

$omega_\left\{ \right\}^\left\{ \right\} = c k$

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

Here

$mid mathbf\left\{E\right\} mid$

is the amplitude of the field and

$|psirangle stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} begin\left\{pmatrix\right\} psi_x psi_y end\left\{pmatrix\right\} = begin\left\{pmatrix\right\} costheta exp left \left(i alpha_x right \right) sintheta exp left \left(i alpha_y right \right) end\left\{pmatrix\right\}$

is the Jones vector in the x-y plane.

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

$alpha_x = alpha_y stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} alpha$.

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

$|psirangle = begin\left\{pmatrix\right\} costheta sintheta end\left\{pmatrix\right\} exp left \left(i alpha right \right)$.

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

If unit vectors are defined such that

$|xrangle stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} begin\left\{pmatrix\right\} 1 0 end\left\{pmatrix\right\}$

and

$|yrangle stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} begin\left\{pmatrix\right\} 0 1 end\left\{pmatrix\right\}$

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

$|psirangle = costheta exp left \left(i alpha right \right) |xrangle + sintheta exp left \left(i alpha right \right) |yrangle = psi_x |xrangle + psi_y |yrangle$.

## References

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