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[[File:Polarisation ellipse2.svg|250px|right]]
At a fixed point in space (or for fixed z), the electric vector <math> \mathbf{E} </math> traces out an ellipse in the x-y plane. The semi-major and semi-minor axes of the ellipse have lengths aA and bB, respectively, that are given by
:<math> aA=|\mathbf{E}|\sqrt{\frac{1+\sqrt{1-\sin^2(2\theta)\sin^2\beta}}{2}}</math>
and
:<math> bB=|\mathbf{E}|\sqrt{\frac{1-\sqrt{1-\sin^2(2\theta)\sin^2\beta}}{2}}</math>,
where <math>\beta =\alpha_y-\alpha_x</math>.
The orientation of the ellipse is given by the angle <math>\phi </math> the semi-major axis makes with the x-axis. This angle can be calculated from
:<math> \tan2\phi=\tan2\theta\cos\beta</math>.
If <math>\beta= 0</math>, the wave is [[linear polarization|linearly polarized]]. The ellipse collapses to a straight line <math>(aA=|\mathbf{E}|, bB=0</math>) oriented at an angle <math>\phi=\theta</math>. This is the case of superposition of two simple harmonic motions (in phase), one in the x direction with an amplitude <math>|\mathbf{E}| \cos\theta</math>, and the other in the y direction with an amplitude <math>|\mathbf{E}| \sin\theta </math>. When <math>\beta</math> increases from zero, i.e., assumes positive values, the line evolves into an ellipse that is being traced out in the counterclockwise direction (looking into the propagating wave); this then corresponds to Left-Handed Elliptical Polarization; the semi-major axis is now oriented at an angle <math>\phi\neq\theta </math>. Similarly, if <math>\beta</math> becomes negative from zero, the line evolves into an ellipse that is being traced out in the clockwise direction; this corresponds to Right-Handed Elliptical Polarization.
If <math>\beta=\pm\pi/2</math> and <math>\theta=\pi/4</math>, <math> aA=bB=|\mathbf{E}|/\sqrt{2}</math>, i.e.,the wave is [[circular polarization|circularly polarized]]. When <math>\beta=\pi/2</math>, the wave is left-circularly polarized, and when <math>\beta=-\pi/2</math>, the wave is right-circularly polarized.
==See also==
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