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 leftcircularly polarized light, χ=∞ is rightcircularly 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.
