Inconsistency between Jones vectors and Stokes parameters

Question

Plane wave equation : \begin{equation} \vec{E}(z,t)=\left| \begin{array}{c} E_{0}^{x}e^{i\delta_{x}} \\ E_{0}^{y}e^{i\delta_{y}} \\ 0 \\ \end{array}\right|e^{i(\omega t -kz)}\quad,\quad E_0^{x,y} \in\mathbb{R} \end{equation} which can be rewritten at $z=0$ and changing the time origin : \begin{equation} \vec{E}(0,t)=\left| \begin{array}{c} E_{0}^{x} \\ E_{0}^{y}e^{i\delta} \\ 0 \\ \end{array}\right|e^{i(\omega t)}\quad,\quad E_0^{x,y} \in\mathbb{R} \end{equation} with $\delta = \delta_y - \delta_x \in [0,2\pi [$.

The $z$ axis is in the direction of the Poynting vector of the plane wave (same direction as $\vec{k}$), as on following picture.

The Jones vectors are a representation of degenerate light polarization states :

• horizontal : the $x$ axis, in the sequel $E_H = E_x$

  \begin{equation} | H \rangle = (1,0)^T \end{equation}

• vertical : the $y$ axis, in the sequel $E_V = E_y$

  \begin{equation} | V \rangle = (0,1)^T \end{equation}

• left circular (clockwise) \begin{equation} | L \rangle = \frac{1}{\sqrt{2}} (1,i)^T \qquad (1) \end{equation}

• right circular (counter clockwise) \begin{equation} | R \rangle = \frac{1}{\sqrt{2}} (1,-i)^T \qquad (2) \end{equation}

For left and right circular polarizations, I follow the convention of increasing phase ($\omega t - kz$) with time.

That is to note horizontal and vertical polarizations can be expressed with left and right circular polarizations : \begin{equation} |H\rangle = \frac{1}{\sqrt{2}} (|R\rangle + |L\rangle) \end{equation} and \begin{equation} |V\rangle = \frac{i}{\sqrt{2}} (|R\rangle - |L\rangle) \end{equation} In latter equation, the phase factor $i$ could be omitted, but I prefer to keep it, otherwise there is no more consistency between Jones vectors.

With $E_H$ and $E_V$, one can derive Stokes parameters : \begin{equation} \begin{split} S_{0} &= (E_{0}^{H})^{2} + (E_{0}^{V})^{2} \newline S_{1} &= (E_{0}^{H})^{2} - (E_{0}^{V})^{2} \newline S_{2} &= 2E_{0}^{H}E_{0}^{V}\cos(\delta) \newline S_{3} &= 2E_{0}^{H}E_{0}^{V}\sin(\delta) \end{split} \end{equation}

and $\delta = \delta_y - \delta_x = \delta_V - \delta_H $

For circular polarizations, I consider $E_0^x = E_0^y = \frac{E_0}{\sqrt{2}}$, where $(E_0)^2$ can be interpreted as the total light intensity.

Left circular polarization is characterized by $\delta = \frac{\pi}{2}$ and right one by $\delta = - \frac{\pi}{2}$. Then, for left circular polarization, $S_3 = (E_0)^2$ and for right circular polarization, $S_3 = - (E_0)^2$. This explains the position of left and right circular polarizations one the Poincaré sphere.

But $S_3$ can also be expressed as : \begin{equation} S_3 = i(E_{H}E_{V}^{\star}- E_{H}^{\star}E_{V}) \end{equation}

If I replace $E_H$ and $E_V$ by their decomposition (Eqs. 1 and 2) in the $(E_L, E_R)$ basis (I implicitely identify the complex electric field $E$ with Jones vectors), one get : \begin{equation} \begin{split} S_3 &=i(E_{H}E_{V}^{\star}- E_{H}^{\star}E_{V})\newline &= \frac{i}{2}[(E_R + E_L)(-i)(E_R^\star - E_L^\star)] - (E_R^\star + E_L^\star)(i)(E_R - E_L)]\newline &=\frac{1}{2}[(E_R + E_L)(E_R^\star - E_L^\star)] + (E_R^\star + E_L^\star)(E_R - E_L)]\newline &=|E_R|^2 - |E_L|^2 \end{split} \end{equation}

However, this would mean that $S_3= (E_0)^2$ for right circular polarization and $S_3 = - (E_0)^2$ for left one, which contradicts the previous results and the mapping on the Poincaré sphere !

I must be missing something, where is my error ?

Answer 1

As you explain, your definitions for $|H\rangle, |V\rangle, |R\rangle$ and $|L\rangle$ are related by: $$ \begin{align} |H\rangle &= \frac{1}{\sqrt{2}} (|R\rangle + |L\rangle) \\ |V\rangle &= \frac{i}{\sqrt{2}} (|R\rangle - |L\rangle) \tag{1} \end{align} $$ But that means that for the coefficients used with these basis vectors you have: $$ \begin{align} E_H &= \frac{1}{\sqrt{2}} (E_R + E_L) \\ E_V &= \frac{1}{i\sqrt{2}} (E_R - E_L) \quad \color{red}{\bf \Leftarrow\text{!!}} \tag{2} \end{align} $$ where the second line has $1/i$ instead of the factor $i$ in $(1)$. (If the basis has covariant vectors then the coefficients are contravariant! Or you could call it the difference between an active and a passive transformation.) You forgot to use that in evaluating your second definition of $S_3$ (below the 2nd image) where you erroneously use $E_V=\tfrac{i}{\sqrt{2}}(E_R - E_L)$.

To see that $(1)$ really leads to $(2)$ with the surprising $1/i$ factor, just solve for: $$ E = E_H \binom{1}{0}+E_V \binom{0}{1} = E_L\ \frac1{\sqrt{2}} \binom{1}{i}+E_R\ \frac1{\sqrt{2}} \binom{1}{-i} $$ which will unambiguously give you $E_H$ and $E_V$ expressed in $E_R$ and $E_L$