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Circular polarization may be referred to as ''right'' or ''left'', depending on the direction in which the electric field vector rotates. Unfortunately, two opposing, historical conventions exist. In [[physics]] and [[astronomy]], polarization is defined as seen from the ''receiver'', such as a [[telescope]] or [[radio telescope]]. By this definition, if you could stop time and look at the electric field along the beam, it would trace a helix which is the same shape as the same-handed screw. For example, right circular polarization produces a right threaded (or forward threaded) [[screw]]. In the U.S., [[Federal Standard 1037C]] also defines the handedness of circular polarization in this manner. In [[electrical engineering]], however, it is more common to define polarization as seen from the ''source'', such as from a transmitting antenna. To avoid confusion, it is good practice to specify "as seen from the receiver" (or transmitter) when polarization matters.
==Mathematical description of circular polarization==
The [[Classical physics | classical]] [[sinusoidal]] plane wave solution of the [[electromagnetic wave equation]] for the [[Electric field | electric]] and [[Magnetic field | magnetic]] fields is (cgs units)
:<math> \mathbf{E} ( \mathbf{r} , t ) = \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\psi\rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \} </math>
:<math> \mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t ) </math>
for the magnetic field, where k is the [[wavenumber]],
:<math> \omega_{ }^{ } = c k</math>
is the [[angular frequency]] of the wave, and <math> c </math> is the [[speed of light]].
Here
:<math> \mid \mathbf{E} \mid </math>
is the [[amplitude]] of the field and
:<math> |\psi\rangle \equiv \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} </math>
is the [[Jones vector]] in the x-y plane.
If <math> \alpha_y </math> is rotated by <math> \pi / 2 </math> radians with respect to <math> \alpha_x </math> and the x amplitude equals the y amplitude the wave is [[Circular polarization | circularly polarized]]. The Jones vector is
:<math> |\psi\rangle = {1\over \sqrt{2}} \begin{pmatrix} 1 \\ \pm i \end{pmatrix} \exp \left ( i \alpha_x \right ) </math>
where the plus sign indicates right circular polarization and the minus sign indicates left circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.
If unit vectors are defined such that
:<math> |R\rangle \equiv \begin{pmatrix} 1 \\ i \end{pmatrix} </math>
and
:<math> |L\rangle \equiv \begin{pmatrix} 0 \\ -i \end{pmatrix} </math>
then the polarization state can written in the "R-L basis" as
:<math> |\psi\rangle = \left ( {\cos\theta -i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) |R\rangle + \left ( {\cos\theta + i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) |L\rangle = \psi_R |R\rangle + \psi_L |L\rangle </math>
where
:<math> \psi_R \equiv \left ( {\cos\theta -i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) </math>
and
:<math> \psi_L \equiv \left ( {\cos\theta +i\sin\theta \over \sqrt{2} } \right ) \exp \left ( i \alpha_x \right ) </math>.
== FM radio ==
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