Note that Jones calculus is only applicable to light that is already fully polarized.
The Jones vector describes the polarization of light in free space or another homogeneous isotropic non-attenuating medium, where the light can be properly described as transverse waves.
Then the electric and magnetic fields E and H are orthogonal to k at each point; they both lie in the plane "transverse" to the direction of motion.
Furthermore, H is determined from E by 90-degree rotation and a fixed multiplier depending on the wave impedance of the medium.
So the polarization of the light can be determined by studying E. The complex amplitude of E is written: Note that the physical E field is the real part of this vector; the complex multiplier serves up the phase information.
The sum of the squares of the absolute values of the two components of Jones vectors is proportional to the intensity of light.
It is also common to constrain the first component of the Jones vectors to be a real number.
This discards the overall phase information that would be needed for calculation of interference with other beams.
Note that all Jones vectors and matrices in this article employ the convention that the phase of the light wave is given by
Also, Collet and Jones follow different conventions for the definitions of handedness of circular polarization.
The reader should be wary of the choice of convention when consulting references on the Jones calculus.
The following table gives the 6 common examples of normalized Jones vectors.
A general vector that points to any place on the surface is written as a ket
These matrices are implemented by various optical elements such as lenses, beam splitters, mirrors, etc.
Each matrix represents projection onto a one-dimensional complex subspace of the Jones vectors.
[3] Mathematically, using kets to represent Jones vectors, this means that the action of a phase retarder is to transform light with polarization to where
Linear phase retarders are usually made out of birefringent uniaxial crystals such as calcite, MgF2 or quartz.
Other commercially available linear phase retarders exist and are used in more specialized applications.
Any linear phase retarder with its fast axis defined as the x- or y-axis has zero off-diagonal terms and thus can be conveniently expressed as where
The Jones matrix for an arbitrary birefringent material is the most general form of a polarization transformation in the Jones calculus; it can represent any polarization transformation.
To see this, one can show The above matrix is a general parametrization for the elements of SU(2), using the convention where the overline denotes complex conjugation.
can be expressed as it becomes clear that the Jones matrix for an arbitrary birefringent material represents any unitary transformation, up to a phase factor
However, in the Jones calculus, such phase factors do not change the represented polarization of a Jones vector, so are either considered arbitrary or imposed ad hoc to conform to a set convention.
Assume an optical element has its optic axis[clarification needed] perpendicular to the surface vector for the plane of incidence[clarification needed] and is rotated about this surface vector by angle θ/2 (i.e., the principal plane through which the optic axis passes,[clarification needed] makes angle θ/2 with respect to the plane of polarization of the electric field[clarification needed] of the incident TE wave).
Therefore, the Jones matrix for the rotated polarization state, M(θ), is This agrees with the expression for a half-wave plate in the table above.
The reflected and transmitted components acquire a phase θr and θt, respectively.
In the following notation α, β and γ are the yaw, pitch, and roll angles (rotation about the z-, y-, and x-axes, with x being the direction of propagation), respectively.
Using the above, for any base Jones matrix J, you can find the rotated state J(α, β, γ) using:
[5] The simplest case, where the Jones matrix is for an ideal linear horizontal polarizer, reduces then to:
where ci and si represent the cosine or sine of a given angle "i", respectively.