Meaning of circularity in phase retarder definition

Question

A phase retarder is generally some birefringent material used to modulate the phase of polarized light. A common example is a half-wave plate, which rotates linearly polarized light symmetrically about its fast axis.

Using Jones calculus, we can describe the Jones matrix associated with an arbitrarily-oriented retarder. According to the Wikipedia page on Jones calculus (https://en.wikipedia.org/wiki/Jones_calculus, see 'Arbitrary birefringent material'), the Jones matrix is:

$J_{retarder} = e^{-i\eta/2}\begin{pmatrix} \cos^2\theta+e^{i\eta}\sin^2\theta & (1-e^{i\eta})e^{-i\phi}\cos\theta\sin\theta \\ (1-e^{i\eta})e^{i\phi}\cos\theta\sin\theta & \sin^2\theta+e^{i\eta}\cos^2\theta \end{pmatrix}$

Here $\theta$ is the angle of the fast axis relative to the horizontal, $\eta$ is the relative phase retardation induced between the fast and slow axis, and $\phi$ is the 'circularity', which for linear retarders is $0$, but can take any value between $-\pi/2$ and $\pi/2$ in general (for elliptical retarders).

My confusion: the circularity term seems redundant, after considering the presence of $\eta$. Since the behavior of the retarder (its impact on the ouptput polarization state) depends on the relative phase change between field components along the principal axes, doesn't $\eta$ govern the 'circularity'? How is the term $\phi$ distinct from $\eta$?

Answer 1

The circularity is not redundant, because retarding the relative phase between components along two spacial axes is not the most general thing that can happen in these materials.

A more general situation is that the eigenvectors of the Jones matrix don't correspond to linearly polarized light (which is the case for linear retarders), but to some generic elliptically polarized light. For example, there can be materials that leave circularly polarized light alone if it's clockwise (one eigenvector), and add some kind of a phase if it's counterclockwise (the other eigenvector). That would be an example of a circular retarder, and you couldn't represent it with a Jones matrix if you set the circularity to zero.

You can also see that you need three parameters and not two from the following considerations. You need two parameters to specify one eigenvector up to an overall phase and normalization. The second eigenvector will be determined automatically, by orthogonality. And you need a third parameter to specify the relative phase retardation between those eigenvectors.