This paper reviews the fundamental concepts and the basic theory of polarization mode dispersion(PMD) in optical fibers. It introduces a unified notation and methodology to link the various views and concepts in jones space and strokes space. The discussion includes the relation between Jones vectors and Strokes vectors and how they are used in formulating the jones matrix by the unitary system matrix.

1. Polarized Fiber Optic Transmission
Report: Jones Matrix and Unitary system matrix
CHIKKPETI Sachidanand
Abstract—This paper reviews the fundamental concepts and
the basic theory of polarization mode dispersion(PMD) in optical
fibers. It introduces a unified notation and methodology to
link the various views and concepts in jones space and strokes
space.The discussion includes the relation between Jones vectors
and Strokes vectors and how they are used in formulating the
jones matrix by the unitary system matrix.
I. INTRODUCTION
We will firstly deal with the concept of Bra-Ket, other
notations, Hermitian and Unitary matrix. Further how they are
used in description of polarization of light in the 3-D space
of Strokes vectors and 2-D space of Jones vectors. Pauli spin
matrices and spin vectors are the key to connect these two
space using the projection operator to form the pauli spin
matrix. Then we try to generalise the propagation equation
of the linear system with homogeneous fiber which proposes
the behaviour of the state of the polarisation expressing the
output current through the use of the fiber eigenmodes.
II. ON NOTATION:
Bra and Ket space are two equivalent vector spaces
that describe the same state space, they are dual of one
another. For a state vector −
→
a ,the ket is written as |a⟩ and
the bra is written as ⟨a| 2-D complex Jones (column) ket
vector, |s⟩ =
sx
sy
. The bra ⟨s| indicates the corresponding
complex conjugate row vector -i.e, ⟨s| = (s∗x, s∗y). The
bra-ket notation is used to distinguish Jones vectors from
Strokes vectors. Our Jones vectors are all of unit magnitude-
i.e,⟨s|s⟩ = s ∗x sx + s ∗y sy = 1
Where ŝ represents 3-D Stokes vector of unit length
indicating the field, and corresponding to |s⟩. The component
of ŝ are the Strokes parameters. We will indicate a complex
vector as à =
Ax
Ay
. A2
= Ã
†
Ã, where † stands for
transpose conjugate. A matrix appears with e.g H. Unit
magnitude vectors will be indicated with r̂, while ⊙ indicate
scalar product.The symbols ℜ(z)andℑ(z) indicate the real
and imaginary parts of a complex number z.
III. HERMITIAN OPERATORS
The defining property of a Hermitian operator is
H†
= H (1)
The eigenvectors of H form a complete orthonormal basis
and the eigenvalues are real as proved from the following
difference.
⟨an|(H†
− H)|am⟩ = (a∗
n − am) ⟨an|am⟩ (2)
= 0
The eigenvectors may be the same or different. Consider first
when the eigenvectors are the same.
Since ⟨an|an⟩ ̸= 0, (a∗
n − an) = 0 and the eigenvalue
is real.Consider when the eigenvectors are different. Unless
am = an, in which case the eigenvectors are not linearly
independent, it must be the case that ⟨an|am⟩ = 0.
⟨an|H†
H|am⟩ = a2
mδm, n (3)
When det(H)̸= 0,H is invertible and the action of H on the
state of a system is reversible. The expansion of H onto its own
basis generates a diagonal eigenvalues matrix. Under operator
relations the expansion yields.
H =
X
n
X
m
|am⟩ ⟨am|H|an⟩ ⟨am|
=
X
am|am⟩⟨am| (4)
When ⟨an|H†
H|am⟩ = a2
mδm, n.The orthonormal expansion
is written in matrix form as H=S ∧ S−
1, where S is square
matrix whose columns are the eigenvectors of H and
V
is a
diagonal matrix whose entries are the associated eigenvalues.
IV. UNITARY OPERATORS
The defining property of a unitary operator is
T†
T = I (5)
Acting on its orthogonal eigenvectors|an⟩, the unitary operator
preserves the unity basis length
⟨am|T†
T|an⟩ = δm, n (6)
Taking the determinant of both sides of equation gives
det(T†
T)=1. Since the determinant of a product is the de-
terminants and the adjoint operator preserves the norm, the
determinant of T must be
det(T) = ejΘ
(7)
Since the determinant is the product of eigenvalues, the eigen-
values of T must themselves be complex exponentials and,

2. accounting for(7), they must have unity magnitude. Therefore
T acting on an eigenvector yields
T|an⟩ = ejαn
|an⟩ (8)
The eigenvalues of T lie on the unit circle in the complex
plane. A special form of T exists where the determinant is
unity. This special form is denoted U and is characterized
by det(U)=+1. To transform from T to U,the common phase
factorβ = exp(jΘ/N) must be extracted from each eigenvalue
of T, when N is the dimensionality of the operator. The T and
U forms are thereby related:
T = ejβ
U (9)
It should be noted that when det(U) = -1,a reflection is present
along an odd number of axes in the basis set of U is
U|an⟩ = e−jϕn
|an⟩ (10)
U expands on its own basis set in the same way H expands
(10).
U =
X
m
e−
jϕm|am⟩⟨am| (11)
This orthonormal expansion has the matrix analogue of U =
Sexp(−j∧)S−
1, where the diagonal matrix is exp(−j∧)=






e−
jϕ1
e−
jϕ2
.
.
e−
jϕN






(12)
Consider as a 2 × 2 unitary complex matrix U can always
be written as U = eiϕ
u1 u2
−u2
∗
u1∗
where the complex
numbers u1 and u2 satisfy |u1|2
+ |u2|2
=1, and ϕ is real.
V. CONNECTION BETWEEN HERMITIAN AND UNITARY
MATRICES:
The connection between Hermitian and Unitary operators is
quite intimate. The eigenvalues of H lies on the real number
line, while those of U lie on the unit circle in the complex
plane. Multiplying the eigenvalues of H by-j and taking the
exponential,one can construct the eigenvalues of U. Note that
the eigenvalues of U are cyclic, so only the real number line
modulo 2Π is significant.Based on the operator expansions of
the preceding sections one has
H = S ∧H S−1
(13)
U = Re−
j∧mR−1
(14)
Since in general exp(S ∧ S−1
)=Sexp(∧)S−1
, the H and U
operators may be connected as
U = e−jH
=⇒ Se−
j∧U S−1
= Se−
j∧HS−1
(15)
For every Hermitian operator H there is a associated unitary
operator U that shares the same basis set and has eigenvalues
related through the complex exponential.
VI. JONES AND STOKES SPACES:-
The preceding paper will concentrate on the study of
polarization which forms the connection between Jones and
Stokes space. From the tangential relation between the electric
fields, the Jones vector can be written in normalized form:
E = E0ejΘ
cosχ
sinχejϕ
(16)
where E0 is real. There are the polar angles in above equa-
tion(16) χandϕ.The phase exp(jΘ) is lost on conversion
to Stokes space. The jones vectors two orthogonal state of
polarisation are combined together or projected to form the
Paulis spin matrices which are further as discussed below.
A. Pauli Spin Matrices:-
The Pauli spin matrices connect Jones to Stokes space
through the projection measurements of the preceding section.
The identity Pauli matrix is
σ0 =
1 0
0 1
(17)
The Pauli spin matrices are
σ1 =
1 0
0 −1
; σ2 =
0 1
1 0
; σ3 =
0 −j
j 0
; (18)
The spin matrices are both Hermitian and Unitary:
σ†
k = σk and σ†
kσk = I (19)
The determinants of the spin matrices are −1 and the traces
zero:
det(σk) = −1 and Tr(σk = 0 (20)
A spin matrix multiplied by itself yields
σkσk = I (21)
and multiplied by other matrices gives
σiσj = −σiσj = jσj (22)
Each Stokes coordinate of a polarization state|s⟩ is calculated
by inserting the associated Pauli matrix into the inner product
⟨s|.|s⟩. The individual Stokes coordinates are
sk = ⟨s|σk|s⟩ (23)
Since the spin matrices are Hermitian, the Stokes coordinates
sK are real, signed quantities, since det(σk) = −1 and the
Jones vector |s⟩ is assumed to be normalised, sk is bounded
by −1 ≤ sk ≤ +1. The norm of ŝ is unity, |s| = 1,

3. B. The Pauli Spin Matrices and its bilateral connection be-
tween the Jones and Stokes Vectors:-
The pauli spin vector condenses further the notation of (23).
The spin vector is defined as
−
→
σ =


σ1
σ2
σ3

 (24)
where −
→
σ is a vector of matrices. The vector of Stokes
coordinates ŝ is derived from the Jones vector |s⟩ using the
spin vector: 

s1
s2
s3

 =


⟨s|σ1|s⟩
⟨s|σ2|s⟩
⟨s|σ3|s⟩

 (25)
More concisely,
ŝ = ⟨s|−
→
σ |s⟩ (26)
The most compact map of Jones vectors to Stokes vectors.
Then the above formulated equation is equated by its vec-
tor form like sx, sy
1 0
0 1
sx
sy
, for all stokes vectors
,Then we get the following form of stokes vectors equation.
s0 = |sx|2
+ |sy|2
s1 = |sx|2
− |sy|2
s2 = 2ℜ|s∗
xsy|
s3 = 2ℑ|s∗
xsy|
and now by taking the projection combination of the bra-ket
vector of |s⟩⟨s| we get the following matrix
=
|sx|2
sxsy
∗
sx
∗
sy |sx|2
incorporating the stokes vectors given above into the following
matrix, First, observe that the spin vector behaves both as
3 × 1 vector and as a 2 × 2 matrix, depending on the
context(25). Above shows the spin vector acting as a 3 × 1
vector. Alternatively, the dot product of ŝ with the spin vector
yields.
ŝ.−
→
σ = 1/2(s0σ0+s1σ1+s2σ2+s3σ3) = 1/2
s0 + s1 s2 − js3
s2 + js3 s0 − s1
(27)
ŝ.−
→
σ in this case is a 2 × 2 Jones matrix and, since the
coefficients sk are real,ŝ.−
→
σ is Hermitian: (ŝ.−
→
σ )†
= (ŝ.−
→
σ ).
The trace operation connects the projection with its inner
product: Tr(|s⟩⟨s|) = ⟨s|s⟩.Since the trace of each Pauli
matrix is zero it is also true that Tr(ŝ.−
→
σ ) = 0 .For a
normalized state vector such that ⟨s|s⟩ = 1, one can construct
the projector for ket |s⟩ in terms of the spin vector:
|s⟩⟨s| = 1/2(I + ŝ.−
→
σ ) (28)
Subsequent multiplication on the right by |s⟩ generates the
eigenvalue equation
ŝ.−
→
σ |s⟩ = |s⟩ (29)
This is the most compact way to map Stokes vectors to Jones
vectors. The eigenvector of ŝ.−
→
σ associated with eigenvalue
+1 generates the Jones vector |s⟩ from Stokes vector ŝ.
VII. THE PROPAGATION EQUATION:
Consider a linear and Homogeneous fiber which is time
invariant and frequency dependent system as shown in the
below figure:
. By the linear system approach in the frequency domain
equation is given as
−
→
E (z, ω) =
−
→
T (z, ω)
−
→
E (0, ω) (30)
The optical fiber can be read as a linear time invariant system
with frequency response
T̃(ω) = e− α
2 z
e−jβ(ω)z
the first term in the equation produces the loses in the fiber
where as the second gives intensity/energy of the system.
By applying the partial differential with respect to the ’z’ for
the whole equation we get
∂
−
→
E
∂z
(z, ω) = −
α
2
−
→
E (z, ω) − jβ(z, ω)
−
→
E (z, ω)
Where α
2 is present always due to the attenuation of the signal
in linear system. By the formulation of the birefringence axes
the wave travel with different propagation constants the x and
y axes are two rays travelling in z axes, x,y independent plane
wave travel through fiber experiences the different propagation
constant due to birefringence axes as given in figure:
.
Considering the diagonal case of the birefringence
βxx 0
0 βyy
by the common propagation β
∆
= βxx + βyy
and the difference propagation constant ∆β
∆
= βxx −βyy then
by degenerate case the vectorial linear Schroedinger equation
in polarization is given by the following
∂
−
→
E (z, ω)
∂z
= −
α
2
−
→
E (z, ω)−j
β
2
(ω)σ0
−
→
E (z, ω)−j
∆β
2
(ω)σ1
−
→
A(z, ω)
(31)

4. We are interested in the propagation of an optical field along
single mode optical fibers (SMF),we will deal with a general
linear optical system.
Let’s begin with the frequency domain description of the
complex field envelop along a linear loseless optical system
with possibly z-varying characteristics the equation given in
(30)
e
E(z, ω) = T(z, ω) e
E(0, ω)
where T is the input/output system matrix, from the input
(z = 0) to the point at coordinate z. Such matrix is known as
the Jones matrix of the system. Matrix T is unitary because of
the lossless requirement, i.e, T−1
= T†1
. This comes from the
fact that E2
(z, ω) = e
E(z, ω)
†
e
E(z, ω) is constant in z, which
by(30) implies T†
T is constant in z, and equals its value at
z=0, i.e. the identity matrix because a zero length fiber does
not alter the field. Properties of unitary and other matrices
widely used in this paper are summarized above. Taking the
derivative with respect to z of both sides of(30) gives.
∂ e
E(z, ω)
∂z
=
∂T(z, ω)
∂z
e
E(0, ω) (32)
and inverting(30) to express e
E(0, ω) in (32) which is always
possible except for the ideal polarizers gives
∂ e
E(z, ω)
∂z
= M(z, ω) e
E(0, ω) (33)
where
M(z, ω)
∆
=
∂T(z, ω)
∂z
T−1
(z, ω) (34)
is the local propagation matrix at coordinate z. The global
matrix T is thus the unique solution to the matrix equation.
∂T(z, ω)
∂z
= M(z, ω)T(z, ω), T(0, ω) = I (35)
Matrix M is skew-hermitian. In fact from(18) and its conjugate
transpose we get:
0 =
∂| e
E(z, ω)|2
∂z
=
∂ e
E
†
∂z
e
E + e
E
† ∂ e
E
∂z
(36)
= e
E
†
M† e
E + e
E
†
M e
E
= e
E
†
(M†
+ M) e
E
which is true for every e
E, hence it must be: M†
= −M. The
eigenmodes of M(z, ω) are called the local birefringence axes
of the system. Using the results in unitary matrix, we spec-
trally decompose M using the unitary matrix L
∆
= [L̂, ˆ
L0]of
its orthonormal birefringence axes, and its associated purely
imaginary eigenvalues −iβ, −iβ2
, being the propagation con-
stants β(z, ω) and β(z, ω) positive numbers, as:
M(z, ω) = −L(z, ω)
iβ(z, ω) 0
0 iβ0(z, ω)
L†
(z, ω)
(37)
and we labelled the two eigenvalues such that the birefringence
strength ∆β(z, ω)
∆
= β − β0 is positive at the frequency.
This amounts to assuming that the first axis L̂ has a lower
propagation speed than the other. We call it the slow
birefringence axis as shown in the figure.
. It is known from system theory that the matrix solution of
(35) satisfies:
det(T(z, ω)) = e
R z
0
T r[M(ζ,ω)]dζ
= e−i
R z
0
(β(ζ,ω)+β0(ζ,ω))dζ
(38)
being Tr(.) is the trace operator, which verifies that the
determinant of T has unity magnitude. It is also known that if
matrices M(z, ω)and
R z
0
M(ζ, ω)dζ commute for all z, then
the Jones matrix T, solution of (35), has the matrix exponential
form:
T(z, ω) = e
R z
0
M(ζ,ω)dζ
T(0, ω) = e
R z
0
M(ζ,ω)dζ
(39)
This is clearly the case when the birefringence axes do not
vary in z, i.e. L(z, ω) = L(ω), which corresponds to a z-
homogeneous fiber which we call a generalized polarization
maintaining fiber(GPMF) as in below figure. In such case
TGP MF
(z, ω) = L(ω)
ei
R z
0
β(ζ,ω)dζ
0
0 ei
R z
0
β0(ζ,ω)dζ
L†
(ω)
(40)
i.e, the global eigenmodes coincide with the birefringence
axes.
. However commercial SMFs are typically markedly z-
homogeneous, due to both the fabrication process and local
stresses in the installed cables, and do not satisfy(40)

5. . In the general (z, ω)- varying case, we can use the properties
of unitary matrices to spectrally decompose the global matrix
T in terms of its unit eigenvalues e−iϕs
.e−iϕf
and of their
associated unit eigenvectors B̂s, ˆ
Bf , as:
T = B
eiϕs
0
0 eiϕf
B†
(41)
where B
∆
= [B̂s, B̂s], and where we defined the common phase
ϕ = (ϕs + ϕf )/2, the differential phase ∆ϕ = (ϕs − ϕf ),
which is positive at the reference frequency, and introduced
the unitary matrix.
U
∆
= B
ei∆ϕ/2
0
0 ei∆ϕ/2
B†
= e
i ∆ϕ
2 B


1 0
0 −1

B†
(42)
which has unit determinant: det[U] = 1. all quantities in
(41)and (42) depends on both z and ω. The second equality
in (42) comes from the definition of the matrix exponential:
e
i ∆ϕ
2 B


1 0
0 −1

B†
∆
= Σ∞
k=0
(−i∆ϕ/2)k
k! {B
1 0
0 −1
B†
}k
by realizing that
{B
1 0
0 −1
B†
}k
= B
1 0
0 −1
k
B†
so that
e
−i
∆ϕ
2
B


1 0
0 −1

B†
= Be
−i
∆ϕ
2


1 0
0 −1


B†
.
[B−
→
σ B†
] = B−
→
σ1B†
B−
→
σ2B†
B−
→
σ3B†
......B−
→
σkB†
= B−
→
σ kB†
= B
1 0
0 (−1)k
B†
and the matrix exponential of a diagonal matrix is the diagonal
matrix whose entries are the exponentials of the entries in the
original matrix.
To make further progress, it is convenient to write explicitly B
in terms of the Stokes unit vector b̂ = [b1.b2, b3]T
associated
with the slow eigenmode B̂s. The Stokes representation of
SOP vectors is introduced in Paulis matrix. Using the repre-
sentation to build B starting from b̂, we obtain
Bσ1B†
=
b1 (b2 − ib3)
(b2 + ib3) −b1
= b̂
K
σ (43)
where σ1, i=1,2,3 are Pauli matrices, which are introduced
in above equation with there properties. The symbols σ =
[σ1, σ2, σ3]T
stands for the formal 3×1 vector whose compo-
nents are the Pauli matrices 1,2,3,so that the scalar products
v⊙σ with a 3×1 vector v is shorthand notation for Σ3
i=1viσi,
and represents a matrix. When the above equation get evalu-
ated with the equation(31) we get the PMD(Polarisation mode
dispersion)equation. We will see that the decomposition of
Jones system matrices on the basis of the Pauli matrices is a
key tool, enabling significant analytical advancements.
From(42) we can now compactly express U as a matrix
exponential as
U = e−i ∆
ϕ 2(b̂⊙σ)
(44)
and, using the method of Exponential Matrix Expansion by
Cayley- Hamilton theorem, we can rewrite U explicitly as:
U = cos(
∆
ϕ
2)σ0 − isin(
∆
ϕ
2)(b̂ ⊙ σ) (45)
VIII. REFERENCE
• Armando Vannucci, “ Polarized Fiber Optic Transmission
Notes”
• Alberto Bononi and Armando Vannucci, “PMD: a Math
Primer”
• J. P. Gordon and H. Kogelnik, “PMD fundamentals:
Polarization mode dispersion in optical fibers”
• Polarization Optics in Telecommunications by Jay N.
Damask
• Handling Polarized light by Mark Shtaif
https://www.eng.tau.ac.il/ shtaif/PolarizationClass.pdf
• https://www.wikipedia.com