1. Statistical Models in Optical Communications
Lecture VII
Statistical Models in Optical Communications
The Theory of Polarization
ch. 14 – part 5 “Notes”

2. Vector algebra in Dirac notation

3. Vector algebra in Dirac notation
Column:
Row:
Inner Product:
Outer Product:
Coherency matrix

4. Math background - square bra-ket notation for column and row vectors (I)
is a
column
A “bra”
is a
row -
(the complex transpose of the corresponding ket)
A “bra-ket”
is an
inner product

5. Math background - square bra-ket notation for column and row vectors (II)
A “ket-bra”
is an
outer product
Unit vectors

6. Math background - square bra-ket notation for column and row vectors (III)

7. Jones polarization calculus

8. 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

9. Introduction to polarization (II)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.

10. Jones polarization vectors and matrices

11. Jones polarization vectors and matrices (II)

12. Jones polarization vectors and matrices (III)

13. Jones polarization vectors and matrices(IV)

14. Jones polarization vectors and matrices (V)

15. Jones polarization vectors and matrices(VI)1

16. The coherency matrix

17. The Coherency Matrix E E Coherency matrix (D=2) E E E Jones vector
MUTUAL INTENSITIES
INTENSITIES
Coherency matrix (D=2)
(optical polarization theory)
Correlation/covariance matrix (statistics)
Density matrix (quantum mechanics)
Coherency matrix E E E

18. The coherency matrix (II)

19. The coherency matrix (III)

20. The coherency matrix (IV)

21. The coherency matrix (V)

22. The coherency matrix (VI)

23. The coherency matrix(VII)

24. The coherency matrix(VIII)

25. The coherency matrix (IX)

26. The coherency matrix (X)

27. The coherency matrix(XII)

28. The coherency matrix (XII)

29. The degree of polarization

30. The degree of polarization (I)
correlation coeff.

31. The degree of polarization (II)

32. The degree of polarization (III)

33. The degree of polarization (IV)

34. The degree of polarization (V)

35. The degree of polarization (VI)

36. The degree of polarization (VII)

37. The degree of polarization (VIII)

38. The degree of polarization (IX)

39. The degree of polarization (X)

40. The degree of polarization (XI)

41. The degree of polarization (XII)

42. The Stokes parameters

43. SOP descriptions Polarization ellipse Poincare sphere
cc circ. pol.
135 lin-pol.
y-pol.
ellipt. pol.
Jones polarization vector
45 lin-pol.
phasor of x-pol.
phasor of y-pol.
x-pol.
ccc circ. pol.

44. The four Stokes parameters
Total power
SAME SOP
Power
imbalance
Interferometric
terms
many  one
Jones vector
one  one

45. Coherency matrixStokes parameters (D=2)
…in terms of
coherency matrix
Jones vector
Coherency matrix in terms of Jones vector elements

46. The Stokes Parameters vs. the coherency matrix

47. The Poincare sphere (I)

48. The Poincare sphere (II)

49. The Poincare sphere (III)

50. The Poincare sphere
radius

51. The Poincare sphere Poincare sphere cc circ. pol. 135 lin-pol. y-pol.
x-pol.
y-pol.
45 lin-pol.
135 lin-pol.
ccc circ. pol.
cc circ. pol.
ellipt. pol.

52. The Poincare sphere cc circ. pol. 135 lin-pol. y-pol. ellipt. pol.
x-pol.
y-pol.
45 lin-pol.
135 lin-pol.
ccc circ. pol.
cc circ. pol.
ellipt. pol.

53. The Poincare sphere cc circ. pol. 135 lin-pol. y-pol. ellipt. pol.
x-pol.
y-pol.
45 lin-pol.
135 lin-pol.
ccc circ. pol.
cc circ. pol.
ellipt. pol.

54. The Poincare sphere

55. 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)

56. 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.

57. Measuring the Stokes parameters

59. S0=
S1=
S2=
S3=
S0=
S1=
S2=
S3=

60. S0=
S1=
S2=
S3=

67. Measuring Stokes parameters

68. Quadratic detection in the Dirac formalism

69. 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!!

70. Inner product = Trace of outer product
Animation…

71. Inner product = Trace of outer product
Animation…
Application:

72. 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):

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

74. Generalizing the Stokes parameters

75. Generalized Pauli bases and Generalized Stokes parameters

76. Generalized Pauli bases and Generalized Stokes parameters

77. Generalized Pauli bases and Generalized Stokes parameters

78. Generalized Pauli bases and Generalized Stokes parameters

79. 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
Stokes parameters
(of )
2x2 Coherency matrix

80. 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

81. Constructing Generalized Pauli Bases and Generalized Stokes Parameters

82. 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

83. Coherency matrixStokes parameters (D=2)
…in terms of
coherency matrix
Jones vector
Coherency matrix in terms of Jones vector elements

84. Examine D=2 construction of Stokes parameters…
Stokes parameters array….in terms coherency matrix elements 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

85. Generalize construction of Stokes parameters to D=4
Stokes array:
Coherency matrix:
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 order D=4

86. The D2 generalized Pauli matrices for D=4
The diagonals of are
the rows of a Hadamard matrix
of order D=4:
These matrices
are diagonal
Scaled
unity matrix
Note: All matrices but are traceless

87. 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

88. 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…

89. Gen. Stokes Parameters extractor for D=2
Diagonal Elements
generation
Stokes
vector
Mutual
Intensity
Jones
vector
Intensities
Coherency matrix
extraction stage

90. Gen. Stokes Parameters extractor for D=3
vector
Jones
vector
lin. comb. of
intensities
Mutual
Intensities
Intensities
Coherency matrix
extraction stage

91. Quadratic constraints on the generalized Stokes parameters - the Poincare hyper-sphere

92. The Poincare hypershere

93. The Poincare hypershere

94. 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…

95. Special case – the Poincare Sphere [1890]:
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:

96. Maximum Stokes space distance (and angle)

97. Evolution of the generalized Stokes vector - generalized Mueller matrices

98. Linear transformation in Jones space
Coherency matrices domain
Non-linear = =
Stokes space
Linear
Constructed
generalized Mueller matrix
(for any dimension D):
New result:

99. 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

100. Special case: The classical Mueller matrix of 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

101. The Mueller matrix of a polarization retarder
Relative phaseshift
in Jones space
-rotation around
in Stokes space

102. The Mueller matrix of a polarization rotator
-rotation
in Jones space
-rotation around
in Stokes space

103. The 4x4 Mueller matrix (old)

104. The Stokes Parameters and the Mueller matrix (II)

105. The Stokes Parameters and the Mueller matrix (III)
Propagation:
where we used:

106. The Stokes Parameters and the Mueller matrix (IV)

107. The Stokes Parameters and the Mueller matrix (V)

108. The Stokes Parameters and the Mueller matrix (VI)

109. The Stokes Parameters and the Mueller matrix (VII)

110. The Stokes Parameters and the Mueller matrix (X)

111. The Pauli spin matrices formalism (PMD background)

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

113. Some properties of the four Pauli spin matrices

114. Some properties of the four Pauli spin matrices (II)

115. Some properties of the four Pauli spin matrices (III)

116. Some properties of the four Pauli spin matrices (IV)

117. Some properties of the four Pauli spin matrices (V)

118. Some properties of the four Pauli spin matrices (IV)

119. Some properties of the four Pauli spin matrices (VII)

120. Some properties of the four Pauli spin matrices (VIII)

121. Some properties of the four Pauli spin matrices (IX)

122. 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)

123. Pauli spin matrices representations of coherency matrices and Stokes parameters (II)

124. Pauli spin matrices representations of coherency matrices and Stokes parameters (III)

125. Some properties of the four Pauli spin matrices

126. Some properties of the four Pauli spin matrices

127. Some properties of the four Pauli spin matrices

128. Some properties of the four Pauli spin matrices

129. Some properties of the four Pauli spin matrices

130. Some properties of the four Pauli spin matrices

131. Some properties of the four Pauli spin matrices

132. Some properties of the four Pauli spin matrices

133. Some properties of the four Pauli spin matrices

134. Some properties of the four Pauli spin matrices

135. Some properties of the four Pauli spin matrices

136. Some properties of the four Pauli spin matrices

137. Some properties of the four Pauli spin matrices

138. Some properties of the four Pauli spin matrices

139. Some properties of the four Pauli spin matrices

140. Some properties of the four Pauli spin matrices

141. Pauli spin matrices representations of coherency matrices and Stokes parameters (VI)

142. Pauli spin matrices representations of coherency matrices and Stokes parameters (VII)