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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

Lecture VII

  • The Theory of Polarization
  • ch. 14 – part 5 “Notes”

Statistical Models in Optical Communications

vector algebra in dirac notation3
Vector algebra in Dirac notation

Column:

Row:

OuterProduct:

InnerProduct:

Coherency matrix

math background square bra ket notation for column and row vectors i
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
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
Introduction to polarization (I)

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
Introduction to polarization (II)

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 matrix17
The Coherency Matrix

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

sop descriptions

phasor of 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
The four Stokes parameters

Total power

SAME SOP

Power

imbalance

Interferometric

terms

many  one

Jones vector

one one

coherency matrix stokes parameters d 2
Coherency matrixStokes parameters (D=2)

Stokes parameters

…in terms of

coherency matrix

Jones vector

Coherency matrix in terms of Jones vector elements

the poincare sphere51

Poincare sphere

cc circ. pol.

135lin-pol.

y-pol.

ellipt. pol.

45lin-pol.

x-pol.

ccc circ. pol.

The Poincare sphere
the poincare sphere52

cc circ. pol.

135lin-pol.

y-pol.

ellipt. pol.

45lin-pol.

x-pol.

ccc circ. pol.

The Poincare sphere
the poincare sphere53

cc circ. pol.

135lin-pol.

y-pol.

ellipt. pol.

45lin-pol.

x-pol.

ccc circ. pol.

The Poincare sphere
partially polarized sops inside the poincare sphere
Partially polarized SOPs – inside the Poincare sphere
  • 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

The Stokes Parameters and the degree of polarization

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

S0=

S1=

S2=

S3=

S0=

S1=

S2=

S3=

slide60

S0=

S1=

S2=

S3=

a dot product of matrices the trace inner product
A dot product of matrices: the trace-inner product

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 product72
Inner product = Trace of outer product

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
Inner product = Trace of outer product
  • Quadratic form as trace inner product:
  • Squared envelope as trace inner product (or quadratic form):
slide79

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
Jones Vectors, Coherency Matrices, Stokes Vectors

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

multi dimensional generalized stokes parameters an overview
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
Coherency matrixStokes parameters (D=2)

Stokes parameters

…in terms of

coherency matrix

Jones vector

Coherency matrix in terms of Jones vector elements

slide84

Examine D=2 construction of Stokes parameters…

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

Generalize construction of Stokes parameters to D=4

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
The D2 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
What about D-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
Example: Generalized Pauli base for D=3

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
Gen. Stokes Parameters extractor for D=2

Diagonal Elements

generation

Stokes

vector

Mutual

Intensity

Jones

vector

Intensities

  • Coherency matrix
  • extraction stage
gen stokes parameters extractor for d 3
Gen. Stokes Parameters extractor for D=3

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
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
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:

linear transformation in jones space
Linear transformation in Jones space

Linear

Coherency matrices domain

Non-linear

=

=

Stokes space

Linear

Constructed

generalized Mueller matrix

(for any dimension D):

New result:

constructing the generalized mueller matrix
Constructing the Generalized Mueller Matrix

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
Special case: The classical Mueller matrixof 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
The Mueller matrix of a polarization retarder

Relativephaseshift

in Jones space

-rotation around

in Stokes space

the mueller matrix of a polarization rotator
The Mueller matrix of a polarization rotator

-rotation

in Jones space

-rotation around

in Stokes space

slide112
Motivation: The coherency matrix is expanded in the Pauli basis with coefficients given by the Stokes parameters
slide122

(

)

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)

ad