Linear polarization

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Diagram of the electric field of a light wave (blue), linear-polarized along a plane (purple line), and consisting of two orthogonal, in-phase components (red and green waves)

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[edit]

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.


  \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}


   |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 .


  • Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X. 
  1. ^ Shapira, Joseph; Shmuel Y. Miller (2007). CDMA radio with repeaters. Springer. p. 73. ISBN 0-387-26329-2. 

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See also[edit]

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