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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.

Contents

Mathematical description of linear polarization

References

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)
:$mathbf\{E\}\; (\; mathbf\{r\}\; ,\; t\; )\; =\; mid\; mathbf\{E\}\; mid\; mathrm\{Re\}\; left\; \{\; |psi$
angle exp left [ i left ( kz-omega t
ight )
ight ]
ight }
:$mathbf\{B\}\; (\; mathbf\{r\}\; ,\; t\; )\; =\; hat\; \{\; mathbf\{z\}\; \}\; imes\; mathbf\{E\}\; (\; mathbf\{r\}\; ,\; t\; )$
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
:$|psi$
angle stackrel{mathrm{def}}{=} egin{pmatrix} psi_x \ psi_y end{pmatrix} = egin{pmatrix} cos heta exp left ( i lpha_x
ight ) \ sin heta exp left ( i lpha_y
ight ) end{pmatrix}
is the Jones vector in the x-y plane.
The wave is linearly polarized when the phase angles are equal,
:.
This represents a wave polarized at an angle $heta$ with respect to the x axis. In that case the Jones vector can be written
:$|psi$
angle = egin{pmatrix} cos heta \ sin heta end{pmatrix} exp left ( i lpha
ight ) .
The state vectors for linear polarization in x or y are special cases of this state vector.
If unit vectors are defined such that
:$|x$
angle stackrel{mathrm{def}}{=} egin{pmatrix} 1 \ 0 end{pmatrix}
and
:$|y$
angle stackrel{mathrm{def}}{=} egin{pmatrix} 0 \ 1 end{pmatrix}
then the polarization state can written in the "x-y basis" as
:$|psi$
angle = cos heta exp left ( i lpha
ight ) |x
angle + sin heta exp left ( i lpha
ight ) |y
angle = psi_x |x
angle + psi_y |y
angle .

References

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

See also

★ Polarization of classical electromagnetic waves

★ Circular polarization

★ Elliptical polarization