In optics, polarized light can be described using the Jones calculus, invented by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light. Note that Jones calculus is only applicable to light that is already fully polarized. Light which is randomly polarized, partially polarized, or incoherent must be treated using Mueller calculus.
Contents
Jones vector
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. Suppose that a monochromatic plane wave of light is travelling in the positive z-direction, with angular frequency ω and wave vector k = (0,0,k), where the wavenumber k = ω/c. 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:
(
E
x
(
t
)
E
y
(
t
)
0
)
=
(
E
0
x
e
i
(
k
z
−
ω
t
+
ϕ
x
)
E
0
y
e
i
(
k
z
−
ω
t
+
ϕ
y
)
0
)
=
(
E
0
x
e
i
ϕ
x
E
0
y
e
i
ϕ
y
0
)
e
i
(
k
z
−
ω
t
)
.
{\displaystyle {\begin{pmatrix}E_{x}(t)\\E_{y}(t)\\0\end{pmatrix}}={\begin{pmatrix}E_{0x}e^{i(kz-\omega t+\phi _{x})}\\E_{0y}e^{i(kz-\omega t+\phi _{y})}\\0\end{pmatrix}}={\begin{pmatrix}E_{0x}e^{i\phi _{x}}\\E_{0y}e^{i\phi _{y}}\\0\end{pmatrix}}e^{i(kz-\omega t)}.}
Note that the physical E field is the real part of this vector; the complex multiplier serves up the phase information. Here
i
{\displaystyle i}
is the imaginary unit with
i
2
=
−
1
{\displaystyle i^{2}=-1}
.
The Jones vector is
(
E
0
x
e
i
ϕ
x
E
0
y
e
i
ϕ
y
)
.
{\displaystyle {\begin{pmatrix}E_{0x}e^{i\phi _{x}}\\E_{0y}e^{i\phi _{y}}\end{pmatrix}}.}
Thus, the Jones vector represents the amplitude and phase of the electric field in the x and y directions.
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 common to normalize it to 1 at the starting point of calculation for simplification. 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
ϕ
=
k
z
−
ω
t
{\displaystyle \phi =kz-\omega t}
, a convention used by Eugene Hecht. Under this convention, increase in
ϕ
x
{\displaystyle \phi _{x}}
(or
ϕ
y
{\displaystyle \phi _{y}}
) indicates retardation (delay) in phase, while decrease indicates advance in phase. For example, a Jones vectors component of
i
{\displaystyle i}
(
=
e
i
π
/
2
{\displaystyle =e^{i\pi /2}}
) indicates retardation by
π
/
2
{\displaystyle \pi /2}
(or 90 degrees) compared to 1 (
=
e
0
{\displaystyle =e^{0}}
). Collett uses the opposite definition for the phase (
ϕ
=
ω
t
−
k
z
{\displaystyle \phi =\omega t-kz}
). Also, Collet and Jones follow different conventions for the definitions of handedness of circular polarization. Jones's convention is called: "From the point of view of the receiver", while Collett's convention is called: "From the point of view of the source". 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
|
ψ
⟩
{\displaystyle |\psi \rangle }
. When employing the Poincaré sphere (also known as the Bloch sphere), the basis kets (
|
0
⟩
{\displaystyle |0\rangle }
and
|
1
⟩
{\displaystyle |1\rangle }
) must be assigned to opposing (antipodal) pairs of the kets listed above. For example, one might assign
|
0
⟩
{\displaystyle |0\rangle }
=
|
H
⟩
{\displaystyle |H\rangle }
and
|
1
⟩
{\displaystyle |1\rangle }
=
|
V
⟩
{\displaystyle |V\rangle }
. These assignments are arbitrary. Opposing pairs are
|
H
⟩
{\displaystyle |H\rangle }
and
|
V
⟩
{\displaystyle |V\rangle }
|
D
⟩
{\displaystyle |D\rangle }
and
|
A
⟩
{\displaystyle |A\rangle }
|
R
⟩
{\displaystyle |R\rangle }
and
|
L
⟩
{\displaystyle |L\rangle }
The polarization of any point not equal to
|
R
⟩
{\displaystyle |R\rangle }
or
|
L
⟩
{\displaystyle |L\rangle }
and not on the circle that passes through
|
H
⟩
,
|
D
⟩
,
|
V
⟩
,
|
A
⟩
{\displaystyle |H\rangle ,|D\rangle ,|V\rangle ,|A\rangle }
is known as elliptical polarization.
Jones matrices
Jones calculus is a matrix calculus developed in 1941 by Henry Hurwitz Jr. and R. Clark Jones and published in the Journal of the Optical Society of America.
The Jones matrices are operators that act on the Jones vectors defined above. These matrices are implemented by various optical elements such as lenses, beam splitters, mirrors, etc.
The following table gives examples of Jones matrices for polarizers. Each of these matrices represents a projection onto a one-dimensional complex subspace of the Jones vectors.
Phase retarders
A phase retarder is an optical element that produces a phase difference between two orthogonal polarization components of a monochromatic polarized beam of light. Mathematically, using kets to represent Jones vectors, this means that the action of a phase retarder is to transform light with polarization
|
P
⟩
=
c
1
|
1
⟩
+
c
2
|
2
⟩
{\displaystyle |P\rangle =c_{1}|1\rangle +c_{2}|2\rangle }
to
|
P
′
⟩
=
c
1
e
i
η
/
2
|
1
Axially rotated elements
Assume an optical element has its optic axis perpendicular to the surface vector for the plane of incidence and is rotated about this surface vector by angle θ/2 (i.e., the principal plane through which the optic axis passes, makes angle θ/2 with respect to the plane of polarization of the electric field of the incident TE wave). Recall that a half-wave plate rotates polarization as twice the angle between incident polarization and optic axis (principal plane). Therefore, the Jones matrix for the rotated polarization state, M(θ), is
M
(
θ
)
=
R
(
−
θ
)
M
R
(
θ
)
,
{\displaystyle M(\theta )=R(-\theta )\,M\,R(\theta ),}
where
R
(
θ
)
=
(
cos
θ
sin
Arbitrarily rotated elements
Finding the Jones matrix, J(α, β, γ), for an arbitrary rotation involves a three-dimensional rotation matrix. 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. The full combination of the 3-dimensional rotation matrices is the following:
R
3
D
(
θ
)
=
[
cos
α
cos
β
cos
α
sin
β
sin
γ
−
sin
α
cos
γ
cos