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Lecture VII. The Theory of Polarization ch. 14 – part 5 “Notes” . Statistical Models in Optical Communications. Vector algebra in Dirac notation. Vector algebra in Dirac notation. Column:. Row:. Outer Product:. Inner Product:. Coherency matrix .

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Lecture vii l.jpg

Lecture VII

  • The Theory of Polarization

  • ch. 14 – part 5 “Notes”

Statistical Models in Optical Communications



Vector algebra in dirac notation3 l.jpg
Vector algebra in Dirac notation

Column:

Row:

OuterProduct:

InnerProduct:

Coherency matrix


Math background square bra ket notation for column and row vectors i l.jpg
Math background - square bra-ket notation for column and row vectors (I)

A “bra”

is a

row -

(the complex transpose of the corresponding ket)

A “ket”

is a

column

A “bra-ket”

is an

inner product


Math background square bra ket notation for column and row vectors ii l.jpg
Math background - square bra-ket notation for column and row vectors (II)

A “ket-bra”

is an

outer product

Unit vectors




Introduction to polarization i l.jpg
Introduction to polarization (I) vectors (III)

Z-propagating beam

For a monochromatic beam

the corresponding

real vector field is

Jones polarization vector

  • The state of polarization may be described in terms of this ellipse as follows:

  • The orientation in space of the plane of the ellipse

  • The orientation of the ellipse in the plane,

  • its shape and the sense in which it is described

  • The size of the ellipse

  • The absolute temporal phase


Introduction to polarization ii l.jpg
Introduction to polarization (II) vectors (III)

a

b

Absolute amplitudes and absolute phases are of secondary interest,

just the amplitude ratio and the phase difference counts.

Hence the relevant information is embedded in the phasors ratio:

Change of basis to e.g. to circular –

corresponds to bilinear transformation in the complex plane.








The coherency matrix l.jpg
The coherency matrix vectors (III)


The coherency matrix17 l.jpg
The Coherency Matrix vectors (III)

Jones

vector

E

E

MUTUAL INTENSITIES

INTENSITIES

Coherency matrix (D=2)

(optical polarization theory)

Correlation/covariance matrix (statistics)

Density matrix (quantum mechanics)

Coherency matrix

E

E

E


The coherency matrix ii l.jpg
The coherency matrix vectors (III)(II)



The coherency matrix iv l.jpg
The coherency matrix vectors (III)(IV)


The coherency matrix v l.jpg
The coherency matrix vectors (III)(V)


The coherency matrix vi l.jpg
The coherency matrix vectors (III)(VI)


The coherency matrix vii l.jpg
The coherency matrix vectors (III)(VII)


The coherency matrix viii l.jpg
The coherency matrix vectors (III)(VIII)


The coherency matrix ix l.jpg
The coherency matrix vectors (III)(IX)


The coherency matrix x l.jpg
The coherency matrix vectors (III)(X)





The degree of polarization i l.jpg
The degree of polarization (I) vectors (III)

correlation coeff.













The stokes parameters l.jpg
The Stokes parameters vectors (III)


Sop descriptions l.jpg

phasor of vectors (III)x-pol.

phasor of y-pol.

SOP descriptions

Polarization ellipse

Poincare sphere

cc circ. pol.

135lin-pol.

y-pol.

ellipt. pol.

Jones polarization vector

45lin-pol.

x-pol.

ccc circ. pol.


The four stokes parameters l.jpg
The four Stokes parameters vectors (III)

Total power

SAME SOP

Power

imbalance

Interferometric

terms

many  one

Jones vector

one one


Coherency matrix stokes parameters d 2 l.jpg
Coherency matrix vectors (III)Stokes parameters (D=2)

Stokes parameters

…in terms of

coherency matrix

Jones vector

Coherency matrix in terms of Jones vector elements






The poincare sphere l.jpg
The Poincare sphere vectors (III)

radius


The poincare sphere51 l.jpg

Poincare sphere vectors (III)

cc circ. pol.

135lin-pol.

y-pol.

ellipt. pol.

45lin-pol.

x-pol.

ccc circ. pol.

The Poincare sphere


The poincare sphere52 l.jpg

cc circ. pol. vectors (III)

135lin-pol.

y-pol.

ellipt. pol.

45lin-pol.

x-pol.

ccc circ. pol.

The Poincare sphere


The poincare sphere53 l.jpg

cc circ. pol. vectors (III)

135lin-pol.

y-pol.

ellipt. pol.

45lin-pol.

x-pol.

ccc circ. pol.

The Poincare sphere


Slide54 l.jpg

The Poincare sphere vectors (III)


Partially polarized sops inside the poincare sphere l.jpg
Partially polarized SOPs – inside the Poincare sphere vectors (III)

  • Convex linear combinations of pure coherency matrices, correspond to convex linear combinations of points on the sphere – taking us inside the sphere

Equality for pure SOPs

For normalized SOPs

(Jones vector of unit average norm)

(Like the density matrix in QM)


Slide56 l.jpg

The Stokes Parameters and the degree of polarization vectors (III)

Sphere

Radius:

The DOP of a partially polarized SOP.

is the radius vector from the center of the sphere,

normalized by the radius of the sphere.



Slide59 l.jpg

S vectors (III)0=

S1=

S2=

S3=

S0=

S1=

S2=

S3=


Slide60 l.jpg

S vectors (III)0=

S1=

S2=

S3=




A dot product of matrices the trace inner product l.jpg
A dot product of matrices: the trace-inner product vectors (III)

A linear space

The hermitian matrices are “abstract vectors”

is a valid inner product in the linear space of hermitian matrices

Trace Inner Product!!


Inner product trace of outer product l.jpg
Inner product = Trace of outer product vectors (III)

Animation…


Inner product trace of outer product71 l.jpg
Inner product = Trace of outer product vectors (III)

Animation…

Application:


Inner product trace of outer product72 l.jpg
Inner product = Trace of outer product vectors (III)

Quadratic form:

Animation…

Coherency

matrix

  • Quadratic form as trace inner product:

  • Squared envelope as trace inner product (or quadratic form):


Inner product trace of outer product73 l.jpg
Inner product = Trace of outer product vectors (III)

  • Quadratic form as trace inner product:

  • Squared envelope as trace inner product (or quadratic form):







Slide79 l.jpg

Expansion of the 2x2 coherency matrix in the basis of the Pauli matrices with the Stokes parameters as coefficients

(trace-normalized) Pauli matrices:

Jones

vector

(of )

Stokes parameters

2x2 Coherency matrix


Jones vectors coherency matrices stokes vectors l.jpg
Jones Vectors, Coherency Matrices, Stokes Vectors Pauli matrices with the Stokes parameters as coefficients

Trace-orthonormal matrix base

“the Generalized Pauli base”:

IT REMAINS TO CONSTRUCT THE BASE…

Generalized

Pauli matrices

…TO ENABLE EXPLICIT CONSTRUCTION OF…

complex-valued

Coherency Matrix

D2 real-valued

Generalized

Stokes

Parameters(GSPs)

D-dimensional

complex-valued

Jones vector

Jones VectorStokes Vector


Constructing generalized pauli bases and generalized stokes parameters l.jpg
Constructing Generalized Pauli Bases Pauli matrices with the Stokes parameters as coefficientsand Generalized Stokes Parameters


Multi dimensional generalized stokes parameters an overview l.jpg
Multi-dimensional generalized Stokes parameters – an overview

  • The 4 classical Stokes parameters (for D=2) were extended to D2 real-valued generalized Stokes parameters (for arbitrary dimension D).

    • Previous generalizations of Stokes parameters in quantum mechanics and polarization optics only applied to D=3 and D= 2r

    • Generalized Stokes Parameters are the expansion coefficients in a new explicitly constructed trace-orthonormal base of D2 matrices called generalized Pauli matrices,

    • For D=2 the Generalized Pauli base reduces to the four conventional Pauli matrices

    • The classical Poincare sphere representation in 3-D (for D=2) was extended to a Poincare hyper-sphere in D2 -1 dimensions

    • A D2 x D2 generalization of the 4x4 Mueller matrix of classical polarization optics was derived

PART II


Coherency matrix stokes parameters d 283 l.jpg
Coherency matrix overviewStokes parameters (D=2)

Stokes parameters

…in terms of

coherency matrix

Jones vector

Coherency matrix in terms of Jones vector elements


Slide84 l.jpg

Examine D=2 construction of Stokes parameters… overview

Stokes parameters array….in terms coherency matrix elements

0

2

3

1

“Diagonally-arrayed” SPs:

Linear combinations of the intensities

“Off-diagonally-arrayed” SPs:

Real/imag. parts of the mutual intensity

Identify a Hadamard matrix


Slide85 l.jpg

Generalize construction of Stokes parameters to D=4 overview

Coherency matrix:

Stokes array:

Hadamard

matrix

“Diagonally-arrayed” SPs:

Linear combinations of the intensities

“Off-diagonally-arrayed” SPs:

Real/imag. parts of the mutual intensities

Above diagonal:

Under the diagonal:

Introduce a Hadamard matrix

of orderD=4


The d 2 generalized pauli matrices for d 4 l.jpg
The D overview2 generalized Pauli matrices for D=4

The diagonals of are

the rows of a Hadamard matrix

of orderD=4:

These matrices

are diagonal

Scaled

unity matrix

Note: All matrices but are traceless


What about d s whereat hadamard matrices are undefined l.jpg
What about overviewD-s whereat Hadamard matrices are undefined?

Definition: A Weak-Sense Hadamard matrix, H, of order D,

is a DxD real-valued matrix satisfying:

  • “Unity initialization”: all elements of top row are 1.

  • All rows are orthogonal and of the same norm D:

  • Give up the requirement that all elements be

Example: D=3

These two rows span the nullspace

of [1, 1, 1]

  • Each of the rows underneath sums up to zero

  • If scaled by the matrix is orthogonal


Example generalized pauli base for d 3 l.jpg
Example: Generalized Pauli base for D=3 overview

Hadamard Matrix of order 3

GENERALIZED PAULI BASE

D=3 (nine matrices)

Physicists might recognize

the SU(3) generators…


Gen stokes parameters extractor for d 2 l.jpg
Gen. Stokes Parameters extractor for D=2 overview

Diagonal Elements

generation

Stokes

vector

Mutual

Intensity

Jones

vector

Intensities

  • Coherency matrix

  • extraction stage


Gen stokes parameters extractor for d 3 l.jpg
Gen. Stokes Parameters extractor for D=3 overview

Stokes

vector

Jones

vector

lin. comb.

of

intensities

Mutual

Intensities

Intensities

  • Coherency matrix

  • extraction stage





A global quadratic constraint on the generalized stokes parameters the poincare hypershere l.jpg
A global quadratic constraint on the generalized Stokes parameters: the Poincare hypershere

(Full) Stokes vector:

D2 parameters

Reduced Stokes vector:

D2-1 parameters

Global quadratic

Constraint:

The equation of a D-dim. sphere:

The Poincare hypersphere

radius:

Note: unlike for D=2, not every point on this sphere is a valid Stokes vector

NOT ALL POINTS OF STOKES SPACE ARE ACCESSIBLE (lattices no good)!!!

THERE ARE ADDITIONAL QUADRATIC CONSTRAINTS, NOT TREATED HERE…


A global quadratic constraint on the generalized stokes parameters the poincare hyper shere l.jpg
A global quadratic constraint on the generalized Stokes parameters: the Poincare hyper-shere

D=2

(Full) Stokes vector:

Reduced Stokes vector:

Special case – the Poincare Sphere [1890]:

Reduced Stokes vector and its squared norm for D=2:

Poincare sphere radius:


Maximum stokes space distance and angle l.jpg
Maximum Stokes space distance parameters: the (and angle)



Linear transformation in jones space l.jpg
Linear transformation in Jones space Mueller matrices

Linear

Coherency matrices domain

Non-linear

=

=

Stokes space

Linear

Constructed

generalized Mueller matrix

(for any dimension D):

New result:


Constructing the generalized mueller matrix l.jpg
Constructing the Generalized Mueller Matrix Mueller matrices

Coherency matrices domain

Non-linear

GEN. PAULI MATRIX

STOKES VECTOR

GEN. MUELLER MATRIX ELEMENTS

The i-th column of the Generalized Mueller Matrix

contains the Stokes vector (with elements labelled by j)

of the i-th transformed generalized Pauli base


Special case the classical mueller matrix of polarization optics l.jpg
Special case: The classical Mueller matrix Mueller matricesof polarization optics

Stokes space lin. transf.

Jones space lin. transf.

If U is unitary then MU is orthogonal (true for any D)

and energy is preserved:

Reduced Mueller matrix:

Rotation/Reflection

of Poincare sphere


The mueller matrix of a polarization retarder l.jpg
The Mueller matrix of a polarization Mueller matricesretarder

Relativephaseshift

in Jones space

-rotation around

in Stokes space


The mueller matrix of a polarization rotator l.jpg
The Mueller matrix of a polarization Mueller matricesrotator

-rotation

in Jones space

-rotation around

in Stokes space




Slide105 l.jpg

The Stokes Parameters and the Mueller matrix (III) Mueller matrices

Propagation:

where we used:







The pauli spin matrices formalism pmd background l.jpg
The Pauli spin matrices formalism Mueller matrices(PMD background)


Slide112 l.jpg
Motivation: The coherency matrix is expanded in the Pauli basis with coefficients given by the Stokes parameters


Some properties of the four pauli spin matrices l.jpg
Some properties of the four Pauli spin matrices basis with coefficients given by the Stokes parameters


Some properties of the four pauli spin matrices ii l.jpg
Some properties of the four Pauli spin matrices basis with coefficients given by the Stokes parameters(II)


Some properties of the four pauli spin matrices iii l.jpg
Some properties of the four Pauli spin matrices (III) basis with coefficients given by the Stokes parameters


Some properties of the four pauli spin matrices iv l.jpg
Some properties of the four Pauli spin matrices (IV) basis with coefficients given by the Stokes parameters


Some properties of the four pauli spin matrices v l.jpg
Some properties of the four Pauli spin matrices (V) basis with coefficients given by the Stokes parameters


Some properties of the four pauli spin matrices iv118 l.jpg
Some properties of the four Pauli spin matrices (IV) basis with coefficients given by the Stokes parameters


Some properties of the four pauli spin matrices vii l.jpg
Some properties of the four Pauli spin matrices (VII) basis with coefficients given by the Stokes parameters


Some properties of the four pauli spin matrices viii l.jpg
Some properties of the four Pauli spin matrices (VIII) basis with coefficients given by the Stokes parameters


Some properties of the four pauli spin matrices ix l.jpg
Some properties of the four Pauli spin matrices (IX) basis with coefficients given by the Stokes parameters


Slide122 l.jpg

( basis with coefficients given by the Stokes parameters

)

It is apparent that this is the most general form of a hermitian (complex symmetric) matrix,

expressed in terms of four independent real parameters, then we have established the first result

Pauli spin matrices representations of coherency matrices and Stokes parameters (I)