Polarization and Poincare circle.

[yungman]

What is the theory behind mapping of the latitude and longitude of the sphere in the Poincare Circle to the polarization of the TEM wave?

That is, why:

1) Linear polarization when ε=0 deg?
2) Circular polarization when ε=+/- 45 deg?
3) Elliptical when ε is not 0 or +/- 45 deg?
4) RH rotation if ε=-ve. and LH rotation if ε=+ve.?

Where 2ε= latitude.

Thanks

Alan

 

[Andy Resnick]

The first step is to represent the polarization state by a complex number: the general elliptic state with azimuth θ and ellipticity ε is combined into χ= tan(ε +π/4)exp(-i*2θ.) This maps polarization states to the "cartesian complex plane", and recall that -π/2 ≤ θ < π/2 and -π/4 ≤ ε ≤ π/4. χ= 0 refers to left-circularly polarized light, χ=∞ is right-circularly polarized light.

To construct the Poincare sphere, perform a stereographic mapping of the plane to a unit sphere: latitudes on the sphere then correspond to circles of constant ε on the complex plane and longitude corresponds to lines of constant azimuth on the plane.

Azzam and Bashara's "Ellipsometry and Polarized Light" is an excellent resource for this material.

 

[yungman]

Thanks for your reply.

The book is way to expensive as This is only a small part of my study in antenna theory.

Can you show me how the polarity of the ellipticity angle relate to the direction of rotation? That is, why +ve ε gives rise to Left hand rotation and -ve ε gives rise to Right hand rotation.

Thanks

 

[Andy Resnick]

If I understand your question, it's a sign convention.

 

[yungman]

If I understand your question, it's a sign convention.

No so much about convention, but rather why ε affect the direction of rotation.

$$\epsilon=\frac {E_{max}}{E_{min}}$$

How do you justify +ve or -ve of $E_{max},E_{min}$?

Thanks