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**Statistical Models in Optical Communications**

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

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**Vector algebra in Dirac notation**

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**Vector algebra in Dirac notation**

Column: Row: Inner Product: Outer Product: Coherency matrix

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**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

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**Math background - square bra-ket notation for column and row vectors (II)**

A “ket-bra” is an outer product Unit vectors

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

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**Jones polarization calculus**

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**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

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**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.

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**Jones polarization vectors and matrices**

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**Jones polarization vectors and matrices (II)**

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**Jones polarization vectors and matrices (III)**

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**Jones polarization vectors and matrices(IV)**

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**Jones polarization vectors and matrices (V)**

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**Jones polarization vectors and matrices(VI)**

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The coherency matrix

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**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

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**The coherency matrix (II)**

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**The coherency matrix (III)**

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**The coherency matrix (IV)**

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**The coherency matrix (V)**

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**The coherency matrix (VI)**

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**The coherency matrix(VII)**

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**The coherency matrix(VIII)**

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**The coherency matrix (IX)**

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**The coherency matrix (X)**

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**The coherency matrix(XII)**

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**The coherency matrix (XII)**

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**The degree of polarization**

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**The degree of polarization (I)**

correlation coeff.

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**The degree of polarization (II)**

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**The degree of polarization (III)**

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**The degree of polarization (IV)**

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**The degree of polarization (V)**

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**The degree of polarization (VI)**

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**The degree of polarization (VII)**

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**The degree of polarization (VIII)**

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**The degree of polarization (IX)**

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**The degree of polarization (X)**

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**The degree of polarization (XI)**

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**The degree of polarization (XII)**

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The Stokes parameters

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**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.

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**The four Stokes parameters**

Total power SAME SOP Power imbalance Interferometric terms many one Jones vector one one

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**Coherency matrixStokes parameters (D=2)**

…in terms of coherency matrix Jones vector Coherency matrix in terms of Jones vector elements

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**The Stokes Parameters vs. the coherency matrix**

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**The Poincare sphere (I)**

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**The Poincare sphere (II)**

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**The Poincare sphere (III)**

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The Poincare sphere radius

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**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.

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**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.

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**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.

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The Poincare sphere

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**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)

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**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.

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**Measuring the Stokes parameters**

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S0= S1= S2= S3= S0= S1= S2= S3=

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S0= S1= S2= S3=

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**Measuring Stokes parameters**

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**Quadratic detection in the Dirac formalism**

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**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!!

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**Inner product = Trace of outer product**

Animation…

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**Inner product = Trace of outer product**

Animation… Application:

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**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):

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**Inner product = Trace of outer product**

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

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**Generalizing the Stokes parameters**

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**Generalized Pauli bases and Generalized Stokes parameters**

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**Generalized Pauli bases and Generalized Stokes parameters**

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**Generalized Pauli bases and Generalized Stokes parameters**

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**Generalized Pauli bases and Generalized Stokes parameters**

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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

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**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

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**Constructing Generalized Pauli Bases and Generalized Stokes Parameters**

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**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

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**Coherency matrixStokes parameters (D=2)**

…in terms of coherency matrix Jones vector Coherency matrix in terms of Jones vector elements

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**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

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**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

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**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

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**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

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**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…

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**Gen. Stokes Parameters extractor for D=2**

Diagonal Elements generation Stokes vector Mutual Intensity Jones vector Intensities Coherency matrix extraction stage

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**Gen. Stokes Parameters extractor for D=3**

vector Jones vector lin. comb. of intensities Mutual Intensities Intensities Coherency matrix extraction stage

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**Quadratic constraints on the generalized Stokes parameters - the Poincare hyper-sphere**

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**The Poincare hypershere**

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**The Poincare hypershere**

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**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…

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**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:

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**Maximum Stokes space distance (and angle)**

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**Evolution of the generalized Stokes vector - generalized Mueller matrices**

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**Linear transformation in Jones space**

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

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**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

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**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

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**The Mueller matrix of a polarization retarder**

Relative phaseshift in Jones space -rotation around in Stokes space

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**The Mueller matrix of a polarization rotator**

-rotation in Jones space -rotation around in Stokes space

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**The 4x4 Mueller matrix (old)**

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**The Stokes Parameters and the Mueller matrix (II)**

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

Propagation: where we used:

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**The Stokes Parameters and the Mueller matrix (IV)**

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**The Stokes Parameters and the Mueller matrix (V)**

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**The Stokes Parameters and the Mueller matrix (VI)**

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**The Stokes Parameters and the Mueller matrix (VII)**

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**The Stokes Parameters and the Mueller matrix (X)**

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**The Pauli spin matrices formalism (PMD background)**

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

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices (II)**

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**Some properties of the four Pauli spin matrices (III)**

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**Some properties of the four Pauli spin matrices (IV)**

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**Some properties of the four Pauli spin matrices (V)**

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**Some properties of the four Pauli spin matrices (IV)**

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**Some properties of the four Pauli spin matrices (VII)**

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**Some properties of the four Pauli spin matrices (VIII)**

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**Some properties of the four Pauli spin matrices (IX)**

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**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)

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

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

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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**Some properties of the four Pauli spin matrices**

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

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

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