Understanding Direction of Wave Propagation with Poincaire Sphere

[ask_LXXXVI]

My doubt arises because I couldn't understand how a Poincaire Sphere gives us info about direction of propagation of wave. Now as we know Poincaire sphere is a neat way of showing the polarization state of an TEM wave. Now ,let me give an example to illustrate my doubt : -

Let us take a uniform plane TEM wave traveling in +z direction which is composed of two linearly polarised TEM waves , one whose electric field lies in X direction , the other whose electric field lies in Y direction . Let us take the case of circular polarisation
so we take ,

E_x = E₀ cos(\omega t - \beta z) a_x
E_y = E₀ cos(\omega t - \beta z + \pi\2) a_y

Now the resultant TEM wave has the Electric field vector left handed circularly polarised .
On the Poincaire Sphere this will be given by the north pole point.
Suppose we had same wave but traveling in -ve z direction.
E_x = E₀ cos(\omega t + \beta z) a_x
E_y = E₀ cos(\omega t + \beta z + \pi\2) a_y
So the wave should be right handed circularly polarised(I hope I am right here ) . Now as far as I know , while plotting the polarisation state on the Poincaire sphere we simply take the Phase difference between the two waves and the ratio of their magnitudes into consideration .So according to me , this wave will also be plotted on the north pole point.

Isn't this a discrepancy (maybe of my understanding) that a right handed circular polarised wave is getting plotted to north pole. So do we invert the Poincaire sphere if we reverse direction of propagation of wave?

My doubt basically boils down to this - while plotting Polarisation state of a TEM wave on a Poincaire sphere , how is the direction of propagation of the TEM wave taken into account ?

 

[ask_LXXXVI]

please help in solving my doubt! I hope I was clear in stating the doubt!