### 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.

The orientation of a linearly polarized electromagnetic wave is defined by the direction of the electric field vector.^{[1]} For example, if the electric field vector is vertical (alternately up and down as the wave travels) the radiation is said to be vertically polarized.

## 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

- |\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

## External links

- Animation of Linear Polarization (on YouTube)
- Comparison of Linear Polarization with Circular and Elliptical Polarizations (YouTube Animation)

## See also

- Sinusoidal plane-wave solutions of the electromagnetic wave equation
- Polarization
- Photon polarization

This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".