Introduction

This article describes the deviation of the formulas used in GLWaves for calculations concerning the polarisation ellipse.

We consider a wave superposed of two linearely polarized waves, one in X direction, the other one in Y direction. They have 90 in between. These are defined by

 (1)

Both waves have their own amplitude and phase offset. The instantaneous angle is written here for simplicity. A real wave has . If we superpose both waves we will get an elliptically polarized wave. The E vector describes an ellipse. Silently we assume the x and y coordinates being proportional to the x and y component of the field strength vector, respectively.

This ellipse is centered at the origin. Its major axis has the length , its minor axis the lenth . The major axis' inclination to the axis is denoted by . The rectangle described by the amplitudes of the E field components has a hypotenuse angle called , the rectangle streched by the ellipse axes has an angle known as .

Ellipse parameters

Handedness

The angle is obviously given by

 (2)

Ellipse axes

Given are the amplitudes , and the phase offsets and . We search the inclination angle and the ellipse axes lengths and . Therefore we define the length of the compound E vector

 (3)

The maxium of the length appears in the the major axis' direction. The derivation of the length1 by the angle has to be zero.

 (4)

This supplies the angle when the superposed waves point to the major axis

 (5)

where , the phase difference.

Equations with 2 give the x and y coordinate of the major axis

 (6)

whereas results in the coordinates of the minor axis

 (7)

The length of the axes is then given by

 (8)

Inclination

The angle specifying the inclination of the ellipse now is easily determined by

 (9)

Ellipticity

The ellipticity results to

 (10)

Wave parameters

This section is a treatise on the derivation of the wave parameters , , and when the ellipse parameters , or are given.

Given handedness

is determined by the component amplitudes and (see eq. ()) so we have to reversely calculate both amplitudes from the given . Therefore we need a second equation. We can take one of these
1.    const.
2.    const.
3.    const.
The most meaningful is variant . First the diagonal of the current rectangle is determined as

 (11)

what we directly use in the equations for the new amplitudes

 (12)

(remember the unit circle and the trigonometric functions). The phase offsets don't depend on .

Finally we recalculate and since they change with . The calculation is given in subsections and , respectively.

Given inclination

Coordinate system rotation

The polarisation ellipse can be described as a horizontal ellipse in a rotated coordinate system (rotated by the inclination angle ). The coordinate transformation will be derived now.

The coordinates in the base coordinate system and are rendered as

 (13)

In the rotated coordinate system the coordinates are given by

 (14)

which can be simplified to the coordinate transformation

 (15)

The backward transformation

 (16)

Component maxima

In the rotated coordinate system the ellipse is given by

 (17)

In the base coordinate system this equals to

 (18)

where the angle refers to the rotated coordinate system.

The components' amplitudes appear when these coordinates have their maximum. The angle relative to the major axis points to the amplitude of , denotes the amplitude of .

 (19)

Using that result the field strength components are given by

 (20)

Phase difference

When the x component of the elliptically polarized electromagnetic wave takes its maximum (this is at ), the y component's value must be (compare with eq. ())

 (21)

with being the phase difference between and . Solving this equation for finishes the problem.

 (22)

Attention: be careful with this . Little inaccuracies and the granularity of floating point calculations can lead to an argument raising a floating point exception.

Since changing the field strength components also affects the handedness , it has to be calculated as described in section . doesn't change because the ellipse is rotated as is.

Given ellipticity

The ellipticity is determined by the ellipse axes and so we have to reversely calculate both from the given . Therefore we again need a second equation.
1.    const.
2.    const.
3.    const.
Item number gives the best result. The diagonal of the ellipticity rectangle derives to

 (23)

The new axes lengths thus become

 (24)

(recall unit circle and trigonometric functions). The values , , , and ( is held constant when is changed) are given by the equations derived in section .

Footnotes

... length1
Here we differentiate the square of the length . This is allowed since the length is surely greater than zero and the square is a monotone transformation on the positive axis. It simplifies the calculation enormously.
...#tex2html_wrap_inline502#2
we simply use the absolute values instead of the vectors here
Johann Glaser Johann.Glaser@gmx.at