# How do we get from LHP to RCP on the poincare sphere?

A paper that I was reading wanted to transform the jones matrix of linearly horizontal polarization(LHP) to right circular polarization(RCP).

The paper states:

Consider... $$J_{LHP}\to J_{RCP}$$... In order to get to RCP from an initial LHP we must first rotate down to the equator to L+45. This transformation takes its path all the way around the sphere in a helical, descending manner and is physically accomplished by rotating a Waveplates in the plane of polarization.

The author doesn't provide information on how we finally reach RCP. From my understanding we reached a linear state($$L+45$$) through a path that traces elliptical states. How do we arrive to RCP or am I missing something?

The left-handed and right handed circular light are described by the Jones vector \begin{align} \vert LHP\rangle&=\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1\\ i\end{array}\right)\, ,\\ \vert RHP\rangle&=\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1\\ -i\end{array}\right) \end{align} so a rotation about the $$\hat z$$ axis $$R_z(\theta)=e^{-i\theta \sigma_z/2}$$ by and angle $$\theta=\pi$$ \begin{align} R_z(\pi)=\left(\begin{array}{cc} e^{i\pi/2} & 0 \\ 0&e^{-i\pi/2} \end{array}\right) =\left(\begin{array}{cc} i& 0 \\ 0&-i \end{array}\right)=i \left(\begin{array}{cc} 1 & 0 \\ 0&-1 \end{array}\right) \end{align} acting on $$\vert LHP\rangle$$ will yield $$\vert RHP\rangle$$ up to global inessential phase.
Basically the Jones vectors for the left- and right-handed polarization are eigenstates of $$\sigma_y$$ so lie opposite on the equator of the Poincaré sphere; a rotation about $$R_z$$ tracing a path along the equator should transform one into the other.