# Meaning of circularity in phase retarder definition

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 $$\eta$$?

• I just encountered this same confusion. Commented Sep 6, 2020 at 0:07