Jones calculus
Encyclopedia
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
.
The x and y components of the complex amplitude of the electric field of light travel along z-direction,
and 
, are represented as
.
Here
is the Jones vector (
is the imaginary unit
with
).
Thus, the Jones vector represents (relative) amplitude and (relative) phase of electric field in 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 phase information needed for calculation of interference with other beams. Note that all Jones vectors and matrices on this page assumes that the phase of the light wave is
, which is used by Hecht. In this definition, increase in 
(or 
) indicates retardation (delay) in phase, while decrease indicates advance in phase. For example, a Jones vectors component of 
(
) indicates retardation by 
(or 90 degree) compared to 1 (
). Collett uses the opposite definition (
). The reader should be wary when consulting references on Jones calculus.
The following table gives the 6 common examples of normalized Jones vectors.
When applied to the Poincare sphere
(also known as the Bloch sphere
), the basis kets (
and 
) must be assigned to opposing (antipodal) pairs of the kets listed above. For example, one might assign 
= 
and 
= 
. These assignments are arbitrary. Opposing pairs are
The
ket
is a general vector that points to any place on the surface. Any point not in the table above and not on the circle that passes through
is collectively known as elliptical polarization
.
, MgF2 or quartz
. Uniaxial crystals have one crystal axis that is different from the other two crystal axes (i.e., ni ≠ nj = nk). This unique axis is called the extraordinary axis and is also referred to as the optic axis
. An optic axis can be the fast or the slow axis for the crystal depending on the crystal at hand. Light travels with a higher phase velocity through an axis that has the smallest refractive index
and this axis is called the fast axis. Similarly, an axis which has the highest refractive index is called a slow axis since the phase velocity
of light is the lowest along this axis. Negative uniaxial crystals (e.g., calcite
CaCO3, ruby
Al2O3) have ne < no so for these crystals, the extraordinary axis (optic axis) is the fast axis whereas for positive uniaxial crystals (e.g., quartz
SiO2, magnesium fluoride
MgF2, rutile
TiO2), ne > n o and thus the extraordinary axis (optic axis) is the slow axis.
Any phase retarder with fast axis vertical or horizontal has zero off-diagonal terms and thus can be conveniently expressed as
where,
and 
are the phases of the electric fields in 
and 
directions respectively. In the phase convention 
, the relative phase between the two waves when represented as 
suggests that a positive 
(i.e., 
> 
) means that 
doesn't attain the same value as 
until a later time i.e., 
leads 
. Similarly, if 
i.e., 
> 
, 
leads 
.
For e.g., if the fast axis of a quarter wave plate is horizontal, this suggests that the phase velocity along the horizontal direction is faster than that in the vertical direction i.e.,
leads 
. Thus, 
which for a quarter wave plate suggests that 
.
In the opposite convention
, the relative phase when defined as 
suggests that a positive 
means that 
doesn't attain the same value as 
until a later time i.e., 
leads 
.
The special expressions for the phase retarders can be obtained by using the general expression for a birefringent material. In the above expression:
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, polarized light can be described using the Jones calculus, invented by R. C. Jones
Robert Clark Jones
R. Clark Jones was an American physicist working inthe field of Optics.He studied at Harvard University and received his PhD in 1941.Until 1944 he worked at the Bell Labs, later until 1982 with the Polaroid Corporation....
in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. 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
Mueller calculus
Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of incoherent light. It was developed in 1943 by Hans Mueller, then a professor of physics at the Massachusetts Institute of Technology...
.
Jones vectors
The Jones vector describes the polarization of light.The x and y components of the complex amplitude of the electric field of light travel along z-direction,
and , are represented as.Here
is the Jones vector ( is the imaginary unitImaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
with
).Thus, the Jones vector represents (relative) amplitude and (relative) phase of electric field in 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
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
. This discards the phase information needed for calculation of interference with other beams. Note that all Jones vectors and matrices on this page assumes that the phase of the light wave is
, which is used by Hecht. In this definition, increase in (or ) indicates retardation (delay) in phase, while decrease indicates advance in phase. For example, a Jones vectors component of () indicates retardation by (or 90 degree) compared to 1 (). Collett uses the opposite definition (). The reader should be wary when consulting references on Jones calculus.The following table gives the 6 common examples of normalized Jones vectors.
| Polarization | Corresponding Jones vector | Typical ket Bra-ket notation Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics... Notation |
| Linear polarized in the x-direction Typically called 'Horizontal' |
 |
|
| Linear polarized in the y-direction Typically called 'Vertical' |
 |
|
| Linear polarized at 45° from the x-axis Typically called 'Diagonal' L+45 |
 |
|
| Linear polarized at −45° from the x-axis Typically called 'Anti-Diagonal' L-45 |
 |
|
| Right Hand Circular Polarized Typically called RCP or RHCP |
 |
|
| Left Hand Circular Polarized Typically called LCP or LHCP |
 |
When applied to the Poincare sphere
Poincaré sphere
Poincaré sphere, often spelled Poincare sphere may refer to:* Poincaré sphere, in optics, is a graphical tool for visualizing different types of polarized light...
(also known as the Bloch sphere
Bloch sphere
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....
), the basis kets (
and ) must be assigned to opposing (antipodal) pairs of the kets listed above. For example, one might assign = and = . These assignments are arbitrary. Opposing pairs are-
and 
-
and 
-
and 
The
ketBra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
is a general vector that points to any place on the surface. Any point not in the table above and not on the circle that passes through
is collectively known as elliptical polarizationElliptical polarization
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation...
.
Jones matrices
The Jones matrices are the operators that act on the Jones Vectors as listed 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:| Optical element | Corresponding Jones matrix |
| Linear polarizer Polarizer A polarizer is an optical filter that passes light of a specific polarization and blocks waves of other polarizations. It can convert a beam of light of undefined or mixed polarization into a beam with well-defined polarization. The common types of polarizers are linear polarizers and circular... with axis of transmission horizontal |
 |
| Linear polarizer with axis of transmission vertical |  |
| Linear polarizer with axis of transmission at 45° with the horizontal |  |
| Linear polarizer with axis of transmission at -45° with the horizontal |  |
| Right circular polarizer |  |
| Left circular polarizer |  |
Linear polarizer with axis of transmission at angle  with the horizontal. (Shown construction from rotating up from the horizontal into the polarizing element, the polarizing element, and then rotating back down into the horizontal.) |
  |
Phase retarders
Phase retarders introduce a phase shift between the vertical and horizontal component of the field and thus change the polarization of the beam. Phase retarders are usually made out of birefringent uniaxial crystals such as calciteCalcite
Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate . The other polymorphs are the minerals aragonite and vaterite. Aragonite will change to calcite at 380-470°C, and vaterite is even less stable.-Properties:...
, MgF2 or quartz
Quartz
Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...
. Uniaxial crystals have one crystal axis that is different from the other two crystal axes (i.e., ni ≠ nj = nk). This unique axis is called the extraordinary axis and is also referred to as the optic axis
Optic axis of a crystal
The optic axis of a crystal is the direction in which a ray of transmitted light suffers no birefringence . Due to the internal structure of the crystal , light propagates along the optical axis differently than in other directions...
. An optic axis can be the fast or the slow axis for the crystal depending on the crystal at hand. Light travels with a higher phase velocity through an axis that has the smallest refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
and this axis is called the fast axis. Similarly, an axis which has the highest refractive index is called a slow axis since the phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
of light is the lowest along this axis. Negative uniaxial crystals (e.g., calcite
Calcite
Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate . The other polymorphs are the minerals aragonite and vaterite. Aragonite will change to calcite at 380-470°C, and vaterite is even less stable.-Properties:...
CaCO3, ruby
Ruby
A ruby is a pink to blood-red colored gemstone, a variety of the mineral corundum . The red color is caused mainly by the presence of the element chromium. Its name comes from ruber, Latin for red. Other varieties of gem-quality corundum are called sapphires...
Al2O3) have ne < no so for these crystals, the extraordinary axis (optic axis) is the fast axis whereas for positive uniaxial crystals (e.g., quartz
Quartz
Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...
SiO2, magnesium fluoride
Magnesium fluoride
Magnesium fluoride is an inorganic compound with the formula MgF2. The compound is a white crystalline salt and is transparent over a wide range of wavelengths, with commercial uses in optics.-Production and structure:...
MgF2, rutile
Rutile
Rutile is a mineral composed primarily of titanium dioxide, TiO2.Rutile is the most common natural form of TiO2. Two rarer polymorphs of TiO2 are known:...
TiO2), ne > n o and thus the extraordinary axis (optic axis) is the slow axis.
Any phase retarder with fast axis vertical or horizontal has zero off-diagonal terms and thus can be conveniently expressed as
where,
and are the phases of the electric fields in and directions respectively. In the phase convention , the relative phase between the two waves when represented as suggests that a positive (i.e., > ) means that doesn't attain the same value as until a later time i.e., leads . Similarly, if i.e., > , leads .For e.g., if the fast axis of a quarter wave plate is horizontal, this suggests that the phase velocity along the horizontal direction is faster than that in the vertical direction i.e.,
leads . Thus, which for a quarter wave plate suggests that .In the opposite convention
, the relative phase when defined as suggests that a positive means that doesn't attain the same value as until a later time i.e., leads .| Phase retarders | Corresponding Jones matrix |
|---|---|
| Quarter-wave plate Wave plate A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.- Operation :A wave plate works by shifting the phase between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent crystal... with fast axis vertical |
 |
| Quarter-wave plate Wave plate A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.- Operation :A wave plate works by shifting the phase between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent crystal... with fast axis horizontal |
 |
| Half-wave plate Wave plate A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.- Operation :A wave plate works by shifting the phase between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent crystal... with fast axis at angle  w.r.t the horizontal axis |
 |
| Any birefringent material (phase retarder) |  |
The special expressions for the phase retarders can be obtained by using the general expression for a birefringent material. In the above expression:
- Phase retardation induced between
and 
by a birefringent material is given by 
is the orientation of the fast axis with respect to the x-axis.
is the circularity (For linear retarders, 
= 0 and for circular retarders, 
= ± 
/2. For elliptical retarders, it takes on values between - 
/2 and 
/2).
Rotated elements
If an optical element is rotated about the optical axis by angle θ, the Jones matrix for the rotated element, M(θ), is constructed from the matrix for the unrotated element, M, by the transformation
- where
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