Linear polarization


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{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 alpha_x ight ) \ sin heta exp left ( i alpha_y ight ) 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 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 alpha 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 alpha ight ) |x angle + sin heta exp left ( i alpha ight ) |y angle = psi_x |x angle + psi_y |y angle .

References

*cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 0-471-30932-X

See also

*Polarization of classical electromagnetic waves
* Circular polarization
* Elliptical polarization


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