Theory of Mueller Matrices and Optical Components
Optical devices such as polarizers and wave retarders are often used to manipulate the polarization of radiation. These devices can be modeled by multiplying the Stokes vector of the incident ray si by a Mueller matrix M,
As explained in The Stokes Parameters, the Stokes vector requires a set of orthogonal coordinate axes to be defined at the location of a propagating ray. The x-axis of this local coordinate system, e1, is the first principal curvature direction in 3D models and the out-of-plane direction in 2D models.
Similarly, the Mueller matrix of an optical device is defined with respect to a local coordinate system. This system need not coincide with the coordinate system that defines the Stokes vector of the incident ray. Therefore it is usually necessary to apply a rotation to one of these local coordinate systems before applying the Mueller matrix:
where R is the rotation matrix
where θ is the angle by which the local coordinate system of the ray must be rotated so that the x-axes of the two local coordinate systems both lie in the plane of incidence and the y-axes are parallel.
Mueller matrices of some common optical devices are tabulated in Ref. 11.
Linear Polarizer Theory
A Linear Polarizer is an optical device that produces linearly polarized light. An ideal linear polarizer transmits radiation that is polarized in one direction, indicated by the transmission axis T, while completely preventing the transmission of radiation polarized in the orthogonal direction.
In a local coordinate system in which the x-axis is parallel to the transmission axis, the Mueller matrix of the ideal linear polarizer is
Linear Wave Retarder Theory
A Linear Wave Retarder is an optical device that separates a ray into two rays with orthogonal linear polarizations, retards the phase of one of these rays with respect to the other, and then recombines the two rays. The direction of polarization that undergoes this retardation is called the slow axis, and the orthogonal direction is called the fast axis F. The phase delay between the two orthogonal components is the retardance δ of the device.
In a local coordinate system in which the x-axis is parallel to the fast axis, the Mueller matrix of an ideal linear wave retarder is
For example, a quarter-wave retarder (δ = π/2) converts circularly polarized light to linearly polarized light, whereas a half-wave retarder (δ = π) converts left-handed circularly polarized light to right-handed circularly polarized light.
Circular Wave Retarder Theory
A Circular Wave Retarder, or polarization rotator, is an optical device that separates a ray into two rays with left-handed and right-handed circular polarizations, retards the phase of one of these rays with respect to the other, and recombines the two rays. A right circular polarization rotator is represented by the Mueller matrix
The circular retarder rotates the polarization ellipse of the incident ray by an angle δ/2 about the origin.
Ideal Depolarizer Theory
The Ideal Depolarizer converts radiation of any polarization to completely unpolarized radiation. The Mueller matrix of an ideal depolarizer is