# Coherency matrix of partially polarized light incomplete?

The electric field of quasi-monochromatic, partially polarized light can be expressed by the following random process (Goodman, Statistical optics)
$$\bar{E}(t,\bar{x})=u_{x}(t,\bar{x})\bar{e}_{x}+u_{y}(t,\bar{y})\bar{e}_{y}$$
$$u_{x}(t,\bar{x})=\Psi_{x} e^{i(\bar{k}\cdot\bar{x}-\omega t)}$$
$$u_{y}(t,\bar{x})=\Psi_{y} e^{i(\bar{k}\cdot\bar{x}-\omega t)}$$
where $\Psi_{x}$ and $\Psi_{y}$ are radom phasor sums which are circular complex Gaussian random variable. The joint statistics of $u_{x}=a+bi$ and $u_{y}=c+di$ describe the polarization state. Knowing that $E(u_{x})=E(u_{y})=0$, the covariance matrix of these two complex is given by
$$C=\begin{bmatrix} E(aa)&E(ac)&E(ab)&E(ad)\\ E(ca)&E(cc)&E(cb)&E(cd)\\ E(ba)&E(bc)&E(bb)&E(bd)\\ E(da)&E(dc)&E(db)&E(dd) \end{bmatrix}=\begin{bmatrix} E(aa)&E(ac)&0&E(ad)\\ E(ac)&E(cc)&E(bc)&0\\ 0&E(bc)&E(aa)&E(bd)\\ E(ad)&0&E(bd)&E(cc) \end{bmatrix}$$
This matrix has $6$ free parameters. However, one often states that the polarization is determined by the coherency matrix
$$J=\begin{bmatrix} E(u_{x}u_{x}^{\ast})&E(u_{x}u_{y}^{\ast})\\ E(u_{y}u_{x}^{\ast})&E(u_{y}u_{y}^{\ast}) \end{bmatrix}=\begin{bmatrix} 2E(aa)&E(ac)+E(bd)+i(E(bc)-E(ad))\\ E(ac)+E(bd)-i(E(bc)-E(ad))&2E(cc) \end{bmatrix}$$
which has only $4$ free parameters because two pairs of free parameters of $C$ are combined in two free parameters in $J$. So we lost 2 degrees of freedom. Does this mean that $E(ac)=E(bd)$ and $E(bc)=-E(ad)$ or does this mean that the coherency matrix doesn't contain all information on the polarization state?

Andy Resnick
Your notion is slightly different than what I am familiar with, but you seem to be (re)discovering the difference between the Stokes vector/Mueller matrix form of optics, which is based on stochastic equations, and the Jones form of optics, which is based on deterministic equations. That is, the Jones calculus is good for highly coherent light (monochromatic, pure polarization state, etc), while the Mueller calculus is valid for partially polarized light.

You can always transform the Jones calculus to the Mueller calculus, but cannot always do the converse: randomly polarized light cannot be expressed in the Jones calculus.

Does that help?

Thanks for your suggestion. However, the coherency matrix definitely treats polarized, unpolarized and partially polarized radiation, just as the Mueller matrix does (and unlike the Jones matrix). For example the coherency matrix of unpolarized radiation is
$$J=\begin{bmatrix} \frac{I}{2}&0\\ 0&\frac{I}{2} \end{bmatrix}$$
I'm not sure whether the Mueller matrix contains the same information as the coherency matrix or as the covariance matrix, as I'm not that familiar with Mueller calculus. Any idea?

As for stochastic vs. deterministic equations: the expression for the electric field contains random processes, so that makes it stochastic.

Andy Resnick