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.

1 Mathematical description of linear polarization
2 Applications of linear polarization
3 References
4 External links
5 See also

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)

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

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

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 written in the "x-y basis" as

Applications of linear polarization

A research group at the MIT Media Lab has reported the use of polarization field synthesis to create a dynamic light field display. The prototype display is composed of multiple LCD layers, each acting as polarization rotators, enclosed by a pair of crossed linear polarizers. A Four-dimensional light-field can be emitted by algorithmically determining the optimal rotations to be applied at each layer of the display. ^[1]

References

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

^ Lanman, Douglas; Gordon Wetzstein, Matthew Hirsch, Wolfgang Heidrich, Ramesh Raskar (6). "Polarization fields: dynamic light field display using multi-layer LCDs". ACM Transactions on Graphics (TOG) 30 (6). doi:10.1145/2070781.2024220.

Linear polarization

Contents

Mathematical description of linear polarization

Applications of linear polarization

References

External links

See also