Jones matrix (${\mathbf{J}}_{\mathrm{out}}=M{\mathbf{J}}_{\mathrm{in}}$)

A Jones matrix is a complex two-by-two linear transformation of a Jones vector through a deterministic nondepolarizing optical element. Successive elements multiply in reverse reading order. An ideal linear polarizer along a real transverse unit vector $\mathbf{a}$ has matrix $\mathbf{a}{\mathbf{a}}^{\mathsf{T}}$. A common nonzero scalar phase does not change the output polarization of waves.

For fixed circular eigenvectors ${\mathbf{u}}_{\pm }=(1,\mp i{)}^{\mathsf{T}}/\sqrt{2}$, let $P_{\pm }={\mathbf{u}}_{\pm }{\mathbf{u}}_{\pm }^{†}$. A uniform optical medium with amplitude propagation coefficients ${\gamma }^{\pm }$ has the displayed Jones matrix. Put $\gamma =({\gamma }^{+}+{\gamma }^{-})/2$ and $\delta =({\gamma }^{+}-{\gamma }^{-})/2$. ThenThe sum of the two modal exponentials supplies a hyperbolic cosine, not its reciprocal. The determinant is $e^{-2\gamma z}$, and $\mathcal{J}(z_1+z_2)=\mathcal{J}(z_1)\mathcal{J}(z_2)$. Real modal coefficients give circular dichroism; purely imaginary coefficients give lossless circular birefringence. Complex modal labels do not have an intrinsic numerical ordering.