In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
,
polarized light
Polarization (also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the ...
can be described using the Jones calculus, discovered by
R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by ''Jones
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
''. 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 is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller. In this technique, the effect of a particular optical element is represented by a Mueller ...
.
Jones vector
The Jones vector describes the polarization of light in free space or another
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
non-attenuating medium, where the light can be properly described as
transverse waves
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
. Suppose that a monochromatic
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
of light is travelling in the positive ''z''-direction, with angular frequency ''ω'' and
wave vector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
k = (0,0,''k''), where the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
''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 The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic ...
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. Here
is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
with
.
The Jones vector is
:
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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. This discards the overall phase information that would be needed for calculation of
interference
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extr ...
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
, a convention used by Hecht. Under this convention, 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 (
). Circular polarization described under Jones' convention is called : "From the point of view of the receiver." Collett uses the opposite definition for the phase (
). Circular polarization described under 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 . When employing the
Poincaré sphere Poincaré sphere may refer to:
* Poincaré sphere (optics), a graphical tool for visualizing different types of polarized light
** Bloch sphere, a related tool for representing states of a two-level quantum mechanical system
* Poincaré homology sph ...
(also known as the
Bloch sphere
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
), 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 polarization of any point not equal to
or
and not on the circle that passes through
is known as
elliptical 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. An elli ...
.
Jones matrices
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. Each matrix represents projection onto a one-dimensional complex subspace of the Jones vectors. The following table gives examples of Jones matrices for polarizers:
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
Kets (russian: Кеты; Ket: Ostygan) are a tribe of Yeniseian speaking people in Siberia. During the Russian Empire, they were known as Ostyaks, without differentiating them from several other Siberian people. Later, they became known as ''Ye ...
to represent Jones vectors, this means that the action of a phase retarder is to transform light with polarization
:
to
:
where
are orthogonal polarization components (i.e.
) that are determined by the physical nature of the phase retarder. In general, the orthogonal components could be any two basis vectors. For example, the action of the circular phase retarder is such that
:
However, linear phase retarders, for which
are linear polarizations, are more commonly encountered in discussion and in practice. In fact, sometimes the term "phase retarder" is used to refer specifically to linear phase retarders.
Linear phase retarders are usually made out of
birefringent
Birefringence is the optics, optical property of a material having a refractive index that depends on the Polarization (waves), polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or ...
uniaxial crystal
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefrin ...
s such as
calcite
Calcite is a Carbonate minerals, carbonate mineral and the most stable Polymorphism (materials science), polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on ...
, MgF
2 or
quartz
Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical form ...
. Plates made of these materials for this purpose are referred to as
waveplates. Uniaxial crystals have one crystal axis that is different from the other two crystal axes (i.e., ''n
i'' ≠ ''n
j'' = ''n
k''). This unique axis is called the extraordinary axis and is also referred to as the
optic axis
An optical axis is a line along which there is some degree of rotational symmetry in an optics, optical system such as a camera lens, microscope or telescopic sight.
The optical axis is an imaginary line that defines the path along which light ...
. 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 along an axis that has the smallest
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
and this axis is called the fast axis. Similarly, an axis which has the largest refractive index is called a slow axis since the
phase velocity
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
of light is the lowest along this axis. "Negative" uniaxial crystals (e.g.,
calcite
Calcite is a Carbonate minerals, carbonate mineral and the most stable Polymorphism (materials science), polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on ...
CaCO
3,
sapphire
Sapphire is a precious gemstone, a variety of the mineral corundum, consisting of aluminium oxide () with trace amounts of elements such as iron, titanium, chromium, vanadium, or magnesium. The name sapphire is derived via the Latin "sapphir ...
Al
2O
3) have ''n
e'' < ''n
o'' so for these crystals, the extraordinary axis (optic axis) is the fast axis, whereas for "positive" uniaxial crystals (e.g.,
quartz
Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical form ...
SiO
2,
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 that are also used in space telescopes. It occurs natur ...
MgF
2,
rutile
Rutile is an oxide mineral composed of titanium dioxide (TiO2), the most common natural form of TiO2. Rarer Polymorphism (materials science), polymorphs of TiO2 are known, including anatase, akaogiite, and brookite.
Rutile has one of the highest ...
TiO
2), ''n
e'' > ''n
o'' and thus the extraordinary axis (optic axis) is the slow axis. Other commercially available linear phase retarders exist and are used in more specialized applications. The
Fresnel rhombs is one such alternative.
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
and
are the phase offsets of the electric fields in
and
directions respectively. In the phase convention
, define the relative phase between the two waves as
. Then a positive
(i.e.
>
) means that
doesn't attain the same value as
until a later time, i.e.
leads
. Similarly, if
, then
leads
.
For example, if the fast axis of a quarter waveplate is horizontal, then the phase velocity along the horizontal direction is ahead of the vertical direction i.e.,
leads
. Thus,
which for a quarter waveplate yields
.
In the opposite convention
, define the relative phase as
. Then
means that
doesn't attain the same value as
until a later time, i.e.
leads
.
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)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
, using the convention
:
where the overline denotes
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.
Finally, recognizing that the set of
unitary transformations on
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
. Therefore, for appropriate choice of
,
, and
, a transformation between any two Jones vectors can be found, 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.
The special expressions for the phase retarders can be obtained by taking suitable parameter values in the general expression for a birefringent material.
In the general expression:
*The relative phase retardation induced between the fast axis and the slow axis is given by
*
is the orientation of the fast axis with respect to the x-axis.
*
is the circularity.
Note that for linear retarders,
= 0 and for circular retarders,
= ±
/2,
=
/4. In general for elliptical retarders,
takes on values between -
/2 and
/2.
Axially rotated elements
Assume an optical element has its optic axis perpendicular to the surface vector for the
plane of incidence
In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave optic ...
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
:
: where
This agrees with the expression for a half-wave plate in the table above. These rotations are identical to beam unitary splitter transformation in optical physics given by
:
where the primed and unprimed coefficients represent beams incident from opposite sides of the beam splitter. The reflected and transmitted components acquire a phase ''θ
r'' and ''θ
t'', respectively. The requirements for a valid representation of the element are
:
and
:Both of these representations are unitary matrices fitting these requirements; and as such, are both valid.
Arbitrarily rotated elements
This would involve a three-dimensional
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end ...
. See Russell A. Chipman and Garam Yun for work done on this.
See also
*
Polarization
*
Scattering parameters
Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals.
The parameters are useful f ...
*
Stokes parameters
The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes in 1852, as a mathematically convenient alternative to the more common description of incoher ...
*
Mueller calculus
Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller. In this technique, the effect of a particular optical element is represented by a Mueller ...
*
Photon polarization
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon
can be described as having right or left circular polarization, or a superposition of the two. Equi ...
Notes
References
Further reading
* E. Collett, ''Field Guide to Polarization'', SPIE Field Guides vol. FG05, SPIE (2005). .
* D. Goldstein and E. Collett, ''Polarized Light'', 2nd ed., CRC Press (2003). .
* E. Hecht, ''Optics'', 2nd ed., Addison-Wesley (1987). .
* Frank L. Pedrotti, S.J. Leno S. Pedrotti, ''Introduction to Optics'', 2nd ed., Prentice Hall (1993).
* A. Gerald and J.M. Burch, ''Introduction to Matrix Methods in Optics'',1st ed., John Wiley & Sons(1975).
*
*
*
*
*
*
*
*
*
*
*
*
*
*
William Shurcliff William Asahel Shurcliff (March 27, 1909 – June 20, 2006) was an American physicist.
Biography
He received his BA cum laude in 1930, a PhD in Physics in 1934, and a degree in Business Administration in 1935, all from Harvard University. In the 1 ...
(1966) ''Polarized Light: Production and Use'', chapter 8 Mueller Calculus and Jones Calculus, page 109,
Harvard University Press
Harvard University Press (HUP) is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing. It is a member of the Association of American University Presses. After the retirem ...
.
External links
''Jones Calculus written by E. Collett on Optipedia''
{{DEFAULTSORT:Jones Calculus
Optics
Polarization (waves)
Matrices