Circular Polarization - Mathematical Description

Mathematical Description

The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is

where k is the wavenumber,

is the angular frequency of the wave, is an orthogonal matrix whose columns span the transverse x-y plane and is the speed of light.

Here

is the amplitude of the field and

is the Jones vector in the x-y plane.

If is rotated by radians with respect to and the x amplitude equals the y amplitude the wave is circularly polarized. The Jones vector is

where the plus sign indicates left circular polarization and the minus sign indicates right circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.

If basis vectors are defined such that

and

then the polarization state can be written in the "R-L basis" as

where

and