x x x
y y y
ω β (1.10)
The variables ηx(t) and ηy(t) can be arranged as the components of
the following 2 × 1 complex vector:
η η η
y i t t
(1.11)
Measurable quantities as, for instance, the Stokes parameters
(Stokes 1852; Fano 1953), which will be con- sidered later, involve
necessarily time or ensemble averaging of second-order products
such as η ηi jt t( ) ( )∗ (or even higher-order products) taken at
a fixed point z so that, as it has been done for the real
representation in Equation 1.1, the global phase factor can be
removed in the description of polarization states in terms of
observables. Consequently, the instantaneous Jones vector is
defined as
ee t A t e
A t e x
2 (1.12)
Note that ε(t) is defined up to a nonmeasurable global phase factor
eiφ. In the general case of a polychro- matic wave, the
instantaneous Jones vector has slow time dependence with respect to
the coherence time, so that for time intervals shorter than the
coherence time, the polarization ellipse can be considered
constant. For time intervals larger than the coherence time of the
electromagnetic wave, the instantaneous Jones vector can vary,
resulting in partial polarization.
The instantaneous Jones vector includes all measurable information
relative to the temporal evolution of the electric field. As for
the polarization ellipse and for the intensity, ε(t) is called
instantaneous in the sense that the possible time dependence of the
amplitudes and relative phase is considered.
Let us now consider the particular case where the quantities
Ay(t)/Ax(t) and δ(t) remain constant in time, and consequently, the
shape of the polarization ellipse remains fixed during the
measurement time. The cor- responding state of polarization is
described by means of the Jones vector (Jones 1941),
ee ≡
2 (1.13)
where ax and ay are respectively given by the averages a Ax x 2 2≡
and a Ay y
2 2≡ of the respective square of the amplitudes Ax(t) and Ay(t)
over the measurement time T.
Leaving aside a global phase factor, the Jones vector can also be
expressed in terms of the intensity I, the azimuth φ, and the
ellipticity angle χ as follows:
ee = − +
cos sin si
nn cos
i (1.14)
which can be interpreted in the following manner (from right to
left): the rightmost vector, a function of χ, represents the Jones
vector of an elliptic state whose semiaxes are aligned with the
reference laboratory axes
8 Polarized electromagnetic waves
X and Y; the matrix, a function of φ, is a rotation matrix that
rotates the said Jones vector by the angle φ; and the scalar factor
I represents the overall amplitude (i.e., the square root of the
intensity of the state).
Thus, a totally polarized state is fully described by its
corresponding Jones vector ε, which provides com- plete information
about the characteristic quantities of the polarization ellipse, as
well as the intensity. The definition (1.13) of the Jones vector is
consistent with the fact that total polarization is compatible with
inten- sity fluctuations. In fact, totally polarized waves maintain
the azimuth and ellipticity of the polarization ellipse fixed,
whereas the size of the ellipse fluctuates, resulting in a mean
intensity over the measurement time. Moreover, slow time variations
of the Jones vector with respect to the measurement time can be
repre- sented by this model (Gil 2007).
Jones vectors have been defined in Equation 1.13 with respect
to a XY reference frame in plane Π (Figure 1.3) in such a
manner that a generic Jones vector ε can be written as
ee = + ≡
ε εx x y y x ye e e e
1 0
0 1 (1.15)
where the basis vectors ex and ey represent respective linearly
polarized states whose electric fields lie along the axes X and
Y.
′ = ( ) ( ) = −
cos sin sin cos (1.16)
where the orthogonal matrix Q corresponds to a proper
counterclockwise rotation, by the angle θ about the axis Z, from
the original reference frame XY to the new axes X′Y′ (Figure
1.8).
Moreover, in general, any pair of complex vectors (e1, e2)
satisfying
e e e e e e1 2 1 1 2 20 1† † †= = = (1.17)
where the superscript † denotes conjugate transpose, constitutes a
generalized orthonormal basis. Thus, pairs of mutually orthogonal
linear, elliptical, or circular states can be used as generalized
reference bases by trans- forming the canonical basis (ex, ey)
through unitary transformations like
′ = =( )−ee eeU U U† 1 (1.18)
Y
X'Y'
Xθ
Figure 1.8 A change of coordinate frame from XY to X′Y′ for the
representation of Jones vectors is performed to an orthogonal
transformation of the form ε′ = Q(θ)ε, where Q corresponds to a
proper counterclockwise rotation by the angle θ, around the axis Z,
from the original reference frame XY to the new axes X′Y′.
1.3 Analytic signal representation and the Jones vector
9
Particularly interesting alternative bases are the linear +45° and
linear –45° (e+π/4, e−π/4) defined by the basis vectors
e e U Q+ −≡
(1.19)
and the right-handed and left-handed circular (er, el), defined by
the basis vectors
e e Ur li i i i ≡
≡
−
=
−
1 1 2
1 1 2
1 1 (1.20)
Despite the fact that, unless otherwise stated, in this book the
polarization states are described with respect to the basis (ex,
ey), the generic notation ε = ε1e1 + ε2e2 is used in order to
indicate the validity of the mathematical expressions regardless of
the particular basis chosen.
Generalized bases containing complex components are very useful for
some purposes, for example, rep- resenting a pure state as a
coherent superposition of a right-handed and a left-handed
circularly polarized state. However, such types of generalized
bases involving imaginary parameters, while being algebraically
acceptable, are not physically realizable as laboratory reference
frames. In fact, only orthogonal transforma- tions of the form ei′
= Qei (i = x, y), where Q is a 2 × 2 orthogonal matrix (i.e., a
unitary matrix that is real), are admissible for generating
physically realizable laboratory reference frames.
The scalar product of two Jones vectors μ and ν is defined as
mm nn† ,= ( )
µ ν µ ν1 2 1
2 1 1 2 2 (1.21)
and the squared absolute value |ε|2 = ε†ε = I of a given Jones
vector ε is precisely the intensity of the corre- sponding state of
polarization. Two pure states represented by respective Jones
vectors μ and ν are said to be orthogonal when μ†ν = 0. Moreover,
the product of a Jones vector ε by a complex number t produces a
new Jones vector ε′ = tε.
To complete this brief survey of the algebraic properties of Jones
vectors, let us now consider the coherent superposition, at a given
point r, of two totally polarized waves whose respective
polarization ellipses lie in a common plane Π (Figure 1.9).
The composed wave at point r is totally polarized, and its Jones
vector is given by the addition of the Jones vectors of the
mutually coherent components
ee ee ee= +1 2 (1.22)
Due to the very definition of the Jones vector, it cannot represent
partially polarized states. The use of Jones vectors is restricted
to totally polarized (or pure) states. In the case of partially
polarized states, the
Coherent
Y Π
X Z
Y Π
X Z
Y Π
Figure 1.9 Coherent superposition, at a given point r, of two
totally polarized waves whose respective polar- ization ellipses
lie in a common plane Π. The composed wave at point r is totally
polarized, and its Jones vector is given by the addition of the
Jones vectors of the mutually coherent components.
10 Polarized electromagnetic waves
azimuth or the ellipticity of the polarization ellipse varies
during the measurement time and a different math- ematical
description, different from that used in the Jones approach, is
necessary in order to take into account all parameters that
characterize completely the state of polarization.
1.4 COHERENCY MATRIX AND STOKES VECTOR
The analytic signals of the components of the electric field of the
wave are zero-mean variables that can be considered ergodic
stochastic processes whose complete statistical description,
equivalent to the bivariate joint probability distribution function
for the two real components of the electric field of the wave,
requires in general the knowledge of all their n-order moments. In
the particular case of waves with a Gaussian spectral profile, as
is the case of thermal light, the second-order moments are
sufficient (Brosseau 1998). Nevertheless, there are important cases
in practice where higher-order moments play an important role,
especially for radi- ation emitted by certain artificial sources,
as well as in the quantum domain. In the second-order approach,
polarization refers to the second-order moments of the zero-mean
analytic signals (ε1, ε2) at a given fixed point in space.
A proper description of the second-order polarization properties of
electromagnetic waves relies on the concept of the coherency
matrix. This mathematical formulation is applicable regardless of
the particular spectral range of the electromagnetic spectrum
considered.
1.4.1 2D coherency matrix
The 2D coherency matrix (or polarization matrix) Φ (Wiener 1930;
Wolf 1959; Barakat 1963), is defined as
FF ee ee= ( )⊗ ( ) = ( ) ( ) ( ) ( )
(1.23)
where ε is the instantaneous Jones vector whose two components are
the analytic signals of the electric field of the wave, ⊗ stands
for the Kronecker product, and the brackets indicate time averaging
over the measure- ment time
x t T
0
(1.24)
As a result of this definition, Φ is a 2 × 2 covariance matrix
(i.e., a positive semidefinite Hermitian matrix) that contains all
second-order measurable information about the 2D state of
polarization (includ- ing intensity). Under the assumption that the
stochastic processes (ε1, ε2) are stationary and ergodic, the
brackets can alternatively be considered ensemble averaging of
ε⊗ε†, where ε(t) in Equation 1.23 are simple
realizations.
The statistical definition of Φ as a covariance matrix entails the
fact that its two eigenvalues are nonnega- tive. These constraints
constitute a complete set of necessary and sufficient conditions
for a Hermitian matrix Φ to be a coherency matrix, that is, to
represent a particular 2D state of polarization of an
electromagnetic wave at a given point in space.
The elements ij (i.j = 1, 2) of Φ can be written as follows in
terms of the corresponding standard deviations σ1, σ2 and the
complex degree of mutual coherence μ:
FF =
1 2
1.4 Coherency matrix and Stokes vector 11
where
σ φ ε σ φ ε µ φ σ σ
φ φ φ
≡ = ≡ = = =( ) ( )t t (1.26)
For some purposes, it is useful to consider the normalized
coherency matrix (or polarization density matrix)
ˆ tr
≡ (1.27)
which in turn can be interpreted as the density matrix containing
complete information about the popula- tions and coherences of the
polarization states (Fano 1953, 1957).
Coherency matrices inherit, as an underlying reference basis, the
generalized reference basis e1, e2 used for representing the
analytic signals constitutive of the two components of Jones
vectors. Unless otherwise stated, Φ will be considered as described
with respect to the underlying canonical basis constituted by the
orthonormal set of column vectors (1, 0)T, (0, 1)T (where the
superscript T stands for transposition). Moreover, regardless of
the underlying reference basis considered, the coherency matrix Φ
can be expressed as a linear expansion, with real coefficients, on
the following matrix basis constituted by the three Pauli matrices
plus the identity matrix
ss ss ss ss0 1 2 3 1 0 0 1
1 0 0 1
0 1 1 0
i (1.28)
Note that the notations σ1, σ2 (plain letters) are used for the
variances in Equation 1.25 and σi (bold letters) are used for
the Pauli matrices (in order to preserve the common notations used
in related works), but this should not lead to confusion because σi
are matrices, while the variances σ1, σ2 are scalar
quantities.
These well-known linearly independent matrices σi have interesting
properties as hermiticity ss ssi i= † and trace-orthogonality
tr(σiσj) = 2δij (δij being the Kronecker delta) and satisfy
ssi
2 2= I , (I2 being the 2 × 2 identity
matrix). Therefore, σi are also unitary and, except for σ0, are
traceless.
1.4.2 StokeS vector
As mentioned above, it is straightforward to show that Φ always
admits the following linear expansion (Falkoff and Macdonald 1951;
Fano 1953):
FF ss= = ∑1
s ii i= ( ) =( )tr FFss 0 1 2 3, , , (1.30)
or, in the explicit form,
s t t t t
s t t
1 11 22 1 1
= + = ( ) ( ) + ( ) ( )
= − = ( ) ( ) −
∗ ∗
∗
2 12 21 1 2 2 1
3 12 21
s i
φ φ ε ε ε ε
φ φ == ( ) ( ) − ( ) ( )( )∗ ∗i t t t tε ε ε ε1 2 2 1 (1.31)
12 Polarized electromagnetic waves
The quantities s0, s1, s2, s3 are the so-called Stokes parameters
(Stokes 1852) and constitute a complete set of measurable
parameters, which allow for expressing Φ in the following
manner:
FF = + − + −
s s s i s s i s s s
(1.32)
The nonnegativity of Φ (i.e., the positive semidefiniteness of Φ)
entails the following constraints, which constitute a pair of
necessary and sufficient conditions for a Hermitian matrix to be a
covariance matrix:
tr detFF FF= ≥ = − − − ≥s s s s s0 0 2
1 2
2 2
3 20 4 0 (1.33)
Consequently, any set of four parameters s0, s1, s2, s3 satisfying
conditions (1.33) can be considered a physi- cally realizable set
of Stokes parameters. Even though the indicated notation is
commonly used in many related works, it is important to warn the
reader that the Stokes parameters are also frequently noted as I,
Q, U, V.
The Stokes parameters are usually arranged as a 4 × 1 Stokes vector
s:
s ≡
0
1
2
3
(1.34)
Let us note now that the relation between Φ and s can also be
expressed as
s =Ljj (1.35a)
L = −
−
1 0 0 1 1 0 0 1 0 1 1 0 0 0i i
(1.35b)
and the coherency vector φ is defined as the column vector whose
components are the elements of the coher- ency matrix arranged in
the following manner:
jj ≡
ee ee (1.35c)
When appropriate, Stokes vectors and other column vectors are
expressed in the horizontal notation as s ≡ (s0, s1, s2, s3)T.
Equation 1.31 shows that, obviously, the information contained
in s (or φ) is completely equiv- alent to that provided by Φ.
It should be noted that the term vector is used here in a very wide
sense as referring to s as the indicated 4-tuple. The
multiplication of a Stokes vector s by a real scalar c produces a
Stokes vector s′ = cs = sc if and only if c ≥ 0. The resultant
Stokes vector s′ represents the same state of polarization as s up
to a positive scale factor that only affects the intensity, I(s′) =
cI(s). Moreover, in general, the addition of two Stokes vectors s1
and s2 only has physical meaning in the form s = s1 + s2, and not
as a subtraction s1 − s2. In fact, the addition
1.4 Coherency matrix and Stokes vector 13
represents the incoherent superposition of two 2D states whose
respective polarization ellipses lie in a com- mon plane. The
intensity of the resultant Stokes vector is given by the sum of the
intensities of the superposed states I(s) = (s1) + (s2). However,
there are situations where the subtraction can be physically
admissible; if we consider the superposition represented by s = s1
+ s2, then the Stokes vector s1 can be considered the result of the
polarimetric subtraction s1 = s − s2 in the sense that s2 is a
Stokes vector that, added to s1, gives a Stokes vector s = s1 + s2,
which represents the incoherent superposition of s1 and s2. The
polarimetric subtraction is a relevant concept in polarimetry that
will be dealt with in later sections.
Obviously, since negative intensities do not have physical meaning,
given a Stokes vector s ≠ 0, it does not have an inverse Stokes
vector with respect to the + operation. Consequently, the set of
Stokes vectors
s s s s s s s s s T
0 1 2 3 0 0 2
1 2
2 2
3 20, , , ,( ) ≥ ≥ + +{ }, together with the product (.) by a
nonnegative scalar and the sum (+),
constitutes a semiring algebraic structure, and not a vector space.
Even though the use of generalized bases for the representation of
Jones vectors is not unusual, Stokes vec-
tors are always considered to refer to an underlying real basis
(ex, ey), that is, to a laboratory reference frame XYZ, Z being the
direction of propagation.
′ = ( ) ( ) ≡ −
θ θ θ θ
1 0 0 0 0 2 2 0 0 2 2 0 0 0 0 1
cos sin sin cos
(1.36)
where the orthogonal matrix MG(θ) corresponds to a proper
counterclockwise rotation by the angle θ about the axis Z, from the
original X reference axis to X′.
A Stokes vector sp satisfying
s Gs Gp T
p p p p ps s s s= − − − = ≡ − − −( )0 2
1 2
2 2
3 2 0 1 1 1 1diag , , , (1.37)
corresponds to a totally polarized state and is said to be a
totally polarized or pure Stokes vector. The matrix G represents
the Minkowskian metric.
Two pure Stokes vectors s1, s2 are said to be mutually orthogonal
when their corresponding Jones vectors ε1, ε2 are mutually
orthogonal ee ee1 2 0† =( ), so that the mutual orthogonality of
s1, s2 is expressed by the fact that the scalar product of s1 and
s2 is zero, s s1 2 0T = . In other words, a pure Stokes vector s1 =
(s0, s1, s2, s3)T is said to be orthogonal to another pure Stokes
vector s2 when s Gs2 1 0 1 2 3
T T T s s s s= = − − −( ), , , .
Multiplication by a positive scalar and additive compositions
translate directly from the space of Stokes vectors to the space of
2D coherency matrices, and consequently, both formalisms are
completely equivalent with regard to their physical
interpretation.
A Stokes vector can always be expressed as
s = + +( ) + − + +( )s s s s s s s s s s T T
1 2
2 2
3 2
2 2
3 2 0 0 0, , , , , , (1.38)
so that s can be interpreted as an incoherent superposition of a
pure state (first addend, hereafter called the characteristic
component) and an unpolarized state (hereafter called the 2D
unpolarized component). The characteristic component defines
the average polarization ellipse (or characteristic polarization
ellipse) of the whole state s. Furthermore, s can be
parameterized as
s =
2
sin
χ
where
I = s0 is the intensity, or power density flux through the
reference plane Π containing the polarization ellipse.
P ≡ + +s s s s1 2
2 2
3 2
0 is the degree of polarization of the 2D state represented by s. P
is a dimensionless quantity whose values are restricted to 0 ≤ P ≤
1. The maximum P = 1 corresponds to totally polarized states that
therefore can also be represented by respective Jones vectors.
Intermediate values 0 < P < 1 correspond to partially
polarized states; the higher the value of the degree of
polarization P, the higher is the correlation (or mutual coherence)
of the field components. The minimum P = 0 corresponds to
unpolarized states, that is, to states with a completely random
temporal distribution of the polarization ellipse or, in other
words, to states with zero correlation between the field
components.
The azimuth φ, with 0 ≤ φ < π, is that of the direction of the
major semiaxis of the characteristic polar- ization ellipse with
respect to the given reference axis X.
The ellipticity angle χ, with −π/4 ≤ χ ≤ π/4, is that of the
characteristic polarization ellipse.
The above parameterization provides an interpretation of 2D states
of polarization in terms of meaningful physical quantities. By
taking into account the above analysis, the Stokes parameters can
also be interpreted as follows:
s0 is the intensity, given by the sum of the intensities associated
with the components of the electric field with respect to any
orthonormal generalized basis: s I I I I I I I Ix y r l0 45 45 1 2=
+ = + = + = ++ ° − ° e e .
s1 is the difference between the respective intensities
corresponding to the components of the electric field with respect
to the canonical basis (ex, ey) (see Equation 1.15), s1 = Ix
−Iy. s1 = 1 (φ = 0, χ = 0) for lin- early x-polarized states; s1 =
−1 (φ = π/2, χ = 0) for linearly y-polarized states. A simple
procedure for the measurement of the parameter s1 of a plane wave
consists of two consecutive intensity measurements by placing a
linear polarizer (usually called analyzer) at 0° and 90° before the
detector.
s2 is the difference between the respective intensities
corresponding to the components of the elec- tric field with
respect to the basis (e+π/4, e−π/4) (see Equation 1.19), s2 =
I+π/4 − I−π/4. s2 = 1 (φ = π/4, χ = 0) for linearly +45°-polarized
states; s2 = −1 (φ = 3π/4, χ = 0) for linearly –45°-polarized
states. A simple procedure for the measurement of the parameter s2
of a plane wave consists of two consecutive intensity measurements
by placing a linear analyzer at +45° and –45° before the
detector.
s3 is the difference between the respective intensities
corresponding to the components of the elec- tric field with
respect to the basis (er, el) (see Equation 1.20), s3 = Ir−Il.
s3 = 1 (χ = π/4) for right-handed circular polarized states; s3 =
−1 (χ = −π/4) for left-handed circularly polarized states. A simple
pro- cedure for the measurement of the parameter s3 of a plane wave
consists of two consecutive intensity measurements by placing a
right-circular analyzer and a left-circular analyzer before the
detector. Note that a circular polarizer (analyzer) can be achieved
by the serial combination of a linear total polarizer and a
quarter-wave plate whose respective eigenaxes make an angle of
45°.
At this point, it is worth summarizing some preliminary conclusions
derived from the above analysis:
Two-dimensional unpolarized states entail the equality of the
intensities associated with the respec- tive pair of orthogonal
components of the electric field with respect to the bases (ex,
ey), 0 = s1 = Iy − Ix; (e+45°, e−45°), 0 = s2 = I+45° −I−45°, and
(er, el), 0 = s3 = Ir−Il. In fact, the fulfillment of these three
equalities implies necessarily the equality I1 = I2 of the
intensities I1, I2 associated with the respective pair of orthog-
onal components of the electric field with respect to any
generalized orthonormal basis e1, e2. As pointed out in the seminal
works of Stokes (1852) and Verdet (1869), the indicated invariance
is an essential and characteristic property of unpolarized states.
There are various random distributions that correspond to
unpolarized waves. As Ellis and Dogariu (2004a) have shown, the
measurement of the correlations of the Stokes parameters allows for
distinguishing between the different types of unpolarized
states.
Any 2D polarization state s can be considered the result of the
incoherent superposition of the characteristic component s p
Ts s s s s s≡ + +( , , , )1 2
2 2
3 2
Ts s s s≡ − + +( , , , )0 1 2
2 2
3 2 0 0 0 . That is, a 2D state represented by a given Stokes
vector s ≡ ( , , , )I s s s T
1 2 3
1.4 Coherency matrix and Stokes vector 15
is polarimetrically indistinguishable from an incoherent
combination of two states propagating in the same direction,
namely, a pure state sp with intensity I s s s Ip = + + =1
2 2 2
with intensity I I s s s Iu = − + + = −1 2
2 2
3 2 1( )P . For pure states that are characterized by the equality
P = 1,
the total intensity is associated with the pure contribution
(characteristic component) and the shape of the polarization
ellipse is constant for time intervals larger than the measurement
time. For 2D unpo- larized states that propagate in a well-defined
direction and satisfy the equality P = 0, the total intensity is
associated with the unpolarized contribution where the shape of the
polarization ellipse fluctuates in a completely random manner
during the measurement time.
The degree of polarization P is just the ratio of the intensity I s
s s Ip = + + =1 2
2 2
3 2 P of the characteristic
component to the intensity I = Ip + Iu of the entire state
(Figure 1.10). Thus, P is a dimensionless and nonnegative
quantity limited by the double inequality 0 ≤ P ≤ 1. Moreover, P is
invariant with respect to any rotation of the underlying reference
frame XY about the direction of propagation Z. Furthermore, from a
more general point of view, P is invariant with respect to any
change of the generalized underly- ing reference basis (e1, e2),
that is, with respect to any unitary transformation of the basis
vectors (e1, e2).
An alternative formulation of the polarimetric purity of a 2D state
of polarization is given by the randomness, or degree of
depolarization, defined as D ≡ s GsT I= ( )2 2tr trFF FF = −1 P2 ,
which is a measure of the randomness of the polarization ellipse.
Obviously, D2 + P2 = 1, and therefore 0 ≤ D ≤ 1; D = 0 for totally
polarized states, while D = 1 for unpolarized states. Note that the
quantity s GsT I= 2 2D (which is intensity dependent and has the
same dimension as I2) was called the mean randomness by Barakat
(1987a).
The characteristic component determines the corresponding
characteristic polarization ellipse with semiaxes
a s s s s s b s s s s s= + + + +( ) = + + − +( )1 2
1 21
A s I= =π π χ 4 4
23 2 2 2 2P sin (1.41)
s3 is a measure of twice the magnitude n of the angular momentum n
of the state s (Figure 1.11). The vector n lies along the axis
Z (i.e., along the direction of propagation at the point r
considered). Since the Stokes vector s associated with a given
state of polarization is scaled by the intensity I = s0, the
normalized angular momentum n k≡ ( )s s3 0/2 (where k is the unit
vector along the positive Z direction) provides an appropriate way
to represent this property regardless of the value of I. Therefore,
the scalar value n s s≡ 3 02 of the normalized angular momentum is
restricted by − ≤ ≤1 2 1 2n . Thus, n = +1 2 corresponds to
right-handed circularly polarized pure states, n = −1 2 corresponds
to left-handed circularly polarized pure states, while the minimum
n = 0 is reached when s3 = 0, that is, for states whose
characteristic polarization ellipse has zero ellipticity, including
the particular cases of linearly polarized
Ip
Y
Z
Π
Figure 1.10 The degree of polarization P of a 2D state is defined
as the ratio of the intensity Ip of the totally polarized component
to the intensity I = Ip + Iu of the whole state.
16 Polarized electromagnetic waves
pure states, as well as unpolarized states. For states whose
characteristic polarization ellipse has positive ellipticity (i.e.,
with right-handedness), n is parallel to k, while n is antiparallel
to k for states whose characteristic polarization ellipse has
negative ellipticity (i.e., left-handedness). Since the state of
polar- ization refers to a given point r in space, all previous
comments about the angular momentum refer to the spin or intrinsic
angular momentum, despite the possibility of considering a complete
or partial spatial region of the wavefront and its associated
orbital angular momentum (Gori et al. 1998). Thus, the total
angular momentum of the electromagnetic wave can often be separated
into two parts, the spin angular momentum (wave polarization) and
the orbital angular momentum, which is determined by the spatial
variation in intensity and phase (Van Enk and Nienhuis 1992; Gori
et al. 1998). The orbital angu- lar momentum can, in turn, be
further decomposed into (1) an origin-independent angular momentum
that is associated with the helical or twisted properties of the
shape of the wavefront (internal orbital angular momentum) and (2)
an origin-dependent angular momentum given by the vector product of
the position vector of the center of the electromagnetic beam and
its total linear momentum (external orbital angular
momentum).
1.5 2D SPACE–TIME AND SPACE–FREQUENCY REPRESENTATIONS OF COHERENCE
AND POLARIZATION
Electromagnetic waves exhibit randomness due to the random
fluctuations associated with the spontaneous or stimulated emission
of photons by matter, as well as to random fluctuations in the
propagation medium. The degree of correlation of the emission
processes caused by myriads of atoms or molecules of the source
material located closely to each other leads to a certai
