## Learning objectives

- Knowledge of the techniques and formalisms used for representing the polarization of light.

- Understanding linear propagation in fiber optics, with an emphasis on polarization related phenomena.

- Applying mathematical/geometrical tools for the description of light polarization, within the telecommunications context or other technological contexts.

- Ability to apply the techniques of propagation of polarized light, to evaluate distortions and penalties in telecom systems.

## Prerequisites

There are no strict prerequisites, to follow this Coursem besides some basic knowledge in Electromagnetics and Linear Algebra, assumed to have been acquired during the Bachelor Degree.

However, despite this course appears in the first year of the Course Catalogue, it is perfectly adequate to take it in parallel with the "Optical Communications" course, placed at the second (and last) year.

## Course unit content

- Polarization of light.

- Fiber-optic propagation of polarized light.

- Polarization Mode Dispersion (PMD) and PMD compensation techniques.

- Formalisms to represent polarized optical signals and polarization-sensitive optical systems.

## Full programme

A detailed outline of each single lecture follows (an asterisk is placed before some 'keywords' of this course):

PFOT-lecture01: General informations on this course, bibliographic references, teaching and assessment methods, course outline.

*Polarization of light.

Use of EM polarization in radio and oprtical communications (hints).

Historical perspective on fiber optic transmission, in the last thirty years (brief hints): transmission speed, main impairments and enabling techniologies.

PFOT-lecture02: Uniform plane wave (TEM) approximation. Vectorial representation of the Electromagnetic (EM) field (hints): optical frequencies, optical bandwidth and complex envelope.

Polarization representation: case of a continuous monochromatic wave (CW). Lissajous curves.

*Jones vectors and polarization ellipses:

implicit and parametric forms; phase-difference "phi"; degenerate cases; handedness: right- and left-handed polarizations; conventions.

PFOT-lecture03: Special States Of Polarization (SOP): LH, LV, L+45, L-45, RHC, LHC; unitmagnitude Jones vectors and corresponding angles (\chi, \phi). Inner product in C2: norm, orthogonality. More general polarizations: linear SOPs and elliptic SOPs with zero azimuth.

*Orthonormality between states of polarization.

Azimuth and ellipticity of a SOP (\theta, \epsilon) formalism.

Software “Polarization Tutor” (HP): use and check of theoretical results.

PFOT-lecture04: *Fiber-optic propagation of polarized light.

The fiber as a (2x2) MIMO system: transfer matrix. The Engineer's approach to the "fiber-system". Delay: variations of refractive index; chromatic dispersion and birefringence; causes. Attenuation (Loss): measurements in dB. Experimental results on attenuation: linear dependence on fiber length; propagation windows and historical perspective; independence of SOP.

*Hints on Polarization Dependent Gain/Loss (PDG/PDL):

optical amplifiers.

PFOT-lecture05: Transfer matrix of a "birefringent" lossy fiber.

Propagation in a "homogeneous fiber with H/V birefringence" (diagonal \beta): example with a monochromatic diagonal input SOP. SOP evolution:

*Beat Length.

Polarization eigenstates (for a homogeneous H/V fiber).

*Polarization Mode Dispersion (PMD):

pulse broadening and intersymbol interference.

*Differential Group Delay (DGD).

Birefringence axes (and birefringence "strength"). Generalization of results for any homogeneous fiber.

Factorization of losses: lossless transfer matrix: adjpoint matrix; unitarity.

PFOT-lecture06: Different approaches to optical propagation: geometric optics, wave optics, Maxwell's equations, quantum optics. Examples of application. Physics approach to optical transmission: Maxwell's equations and constituent relations of matter. The case of Silica (SiO2): structure and aggregation forms. Differential operators ("nabla") and vector properties. Helmholtz equation: uniform planar wave (TEM) and monochromatic solution; wavefronts. Electric susceptibility in optical fibers: physical assumptions and mathematical consequences; peculiarities of the PFOT course (inhomogeneous matrix vs homogeneous scalar).

PFOT-lecture07: Formulation in the "omega" domain, with "narrowband" modulated fields (complex envelopes). Slowly Varying Envelope Approximation (SVEA). SVEA hypothesis and paraxial optics.

Passband propagation equation and effects of electric susceptibility: attenuation and phase/polarization distortion. System matrices: general, diagonal and degenerate cases.

*Vectorial Linear Schroedinger Equation (VLSE):

solution in the homogeneous case and corresponding transfer matrix.

VLSE for a lossless fiber: conservation of energy. "beta" (local) matrix and "T" (global) matrix.

*System matrix: Hermitian and unitary matrices.

PFOT-lecture08: *The Polarimeter: hardware scheme;

the photodiode.

*Ideal polarizers: the projector;

Hermitian and idempotent transfer matrix. SOP measurement and field intensity. Output currents from a Polarimenter. Pauli matrixes and identity matrix as the sum of orthonormal projectors.

The Polarimeter: simplified hardware scheme; mechanical (rotating filter) and opto-electronic (birefringent waveplates) constructive technologies. Commercial Polarimeters (HP, Thorlabs series PAX, Thorlabs series IPM/DPC): technologies, costs, performance (measurement bandwidth).

PFOT-lecture09: *Stokes parameters and Pauli matrices.

*Stokes vectors (3D),

polar coordinates. Unit-magnitude Stokes vectors and states of polarization (SOP). Spherical coordinates:

*the Poincaré sphere.

Special SOPs on the Poincaré sphere: circular SOPs and linear SOPs; emispheres (handedness). Orthogonal SOPs. relationship with azimuth/ellipticity and "gepgraphical coordinates".

PFOT-lecture10: *The Degree Of Polarization (DOP):

measurement bandwidth of a Polarimeter and averaged Stokes components. Definition of DOP and limit values. "Fully polarized light", "partially polarized light" and "natural (depolarized) light": corresponding geometrical, mathematical and physical conditions.

Coherency matrix and DOP. Decomposition of the Projector.

SOP evolution on the Poincaré sphere: (known) case of homogeneous fiber with linear H/V birefringence. Solution (Jones) and visualization (Stokes): L_B on the sphere.

PFOT-lecture11: Schroedinger Equation (VLSE) for a homogeneous fiber with general axes. Diagonalization of the propagation matrix: (real) eigenvalues and (orthogonal) eigenvectors for Hermitian matrices.

Monochromatic (fully polarized) light.

Spectral decomposition (diagonalization): "common mode" propagation constant. CLSE solution: check (derivative); exponential matrices and their derivatives. Amplitude, phase and polarization distorsion: unitary matrix T and unit-determinant matrix U.

*Exponential forms for homogeneous fibers:

calculation of the matrix exponential U and its determinant; det[T]. Eigenvalues/vectors of U: local and global axes of birefringence.

PFOT-lecture12: Polarization Maintaining Fibers (PMF): physical reasons. PMF fibers: propagation of a single polarized pulse:

*Photodetected current and "ghost pulses".

DGD and Inter-Simbol Interference. Power-splitting coefficients: (homework) completeness of power-splitting coeffs.

*PMD compensation through “PSP transmission”.

PFOT-lecture13: System matrix for a general inhomogeneous fiber: unitary (lossless) matrix and U matrix.

Properties of unitary matrices and their spectral decomposition. Trigonometric form and exponential form for U; its eigenvalues and eigenvectors; det[U]=1.

*Relationship between Jones and Stokes vectors: the spin-vector.

Basic properties of the 'spin-vector': hermitianity of Pauli matrices, their powers and products (permutator symbol); square of a hermitian matrix.

Summary of results found for Jones matrices of homogeneous and inhomogeneous fibers.

PFOT-lecture14: solution of homework: completeness of power-splitting coeffs. Completeness of an orthonormal basis in C2.

Mueller matrix M: input-output relationship in the Stokes domain; 3x3 real matrix. Tensor relationship between U and M. Explicit calculation of M as a function of the Pauli coordinates of U; (sanity-check) 3x3 real.

Spin-Vector algebra: tensors as multidimensional arrays. Properties of Pauli matrices and of the Projector. Further properties of the spin-vector; the vector-product matrix [hx]: properties.

PFOT-lecture15: Explicit calculation of the Mueller matrix M; alternative forms, with dyad. Reference to [hx] matrix and its properties; paper [Gordon e Kogelnik, PNAS 2000]. Exponential form: demonstration. Geometrical construction of the multiplication by M: rotation on the Poincaré sphere. Implications: cw and ccw rotations; rotation angle and rotation axis, polarization eigenstates; unit determinant[M]; inverse=transpose matrix..

*Pauli decomposition:

“Pauli coordinates” of U.

Polarization evolution for homogeneous fibers; PMF case: SOPin rotation and Beat Length.

*Vectorial propagation: the "birefringence vector" W.

PFOT-lecture16: Applications of birefringence: avionics, mechanics, robotics and computer-graphics: Euler angles, axial rotations, general roations. Arts: "Polages".

*Polarization-sensitive technological (and non-tech.) contexts;

*Equations of motion in Stokes space

(its demonstration). VLSE (Jones); paper [Frigo, JQE 1986]. Analysis of the terms (loss, phase distortion, PMD and PDL). Mueller matrix as the solution of the Equation of motion (PMF case). Frequency evolution of SOP (PMF case): depolarization trace.

PFOT-lecture17: “RWM Model” for real fibers: PMF waveplates and

*Correlation Length:

system matrix. Depolarization trace of a SOP (Stokes) and "time of flight" of a Dirac pulse (Jones): maximum delay and intensity profile.

*The PMD vector "Omega".

Dependence of the output SOP on frequency:

*Photodetected current and depolarization: depolarization trace;

SOP evolution: N-matrix, Omega vector, DGD and

Evoluzione del SOP: matrice N, vettore Omega, DGD e

*Principal States of Polarization (PSP).

*"1st-order" PMD and "higher-order" PMD.

Jones/Mueller matrices (1st ord.). Paper by Poole-Wagner [Eectron. Lett.1986].

PFOT-lecture18: Binary transmission system (OOK), with modulator and (optical and elctrical) filters: evaluation of the Eye Opening. Chen hypothesis (“101010...” transmission) and consequent evaluation of (filtered) field and photodetected current, both in back-to-back and with a 1st-order PMD fiber.

*Eye Closure Penalty (ECP): generalized Chen's formula;

expression, limit values, plot; comparison with ECP from simulations.

PFOT19-simulation laboratory: Laboratory: Work environment and Matlab simulations: brief hints and summary. Optical simulation software Optilux: installation form repository; functional structure; examples. Ex_02 Eye diagram.

PFOT20-simulation laboratory: Optilux: Simulation of optical instruments and devices: DOP-meter (Ex_21, with measurement of OSNR from DOP) and Polarization Scrambler (Ex_22). Simulation of a randomly birefringent fiber: RWM; Jones matrix; numerical analysis of eigenstates and retardation (rotation angle); PMD vector analysis (DGD, PSP).

PFOT21-measurement laboratory ( Optical communications Lab. "Optiklab"): Electrical, electro-optical and optical intruments and devices. Tunable laser (Tunics); intensity measurements with (handheld) power-meter. Fiber patchcords: connectors FC/PC-APC-SMC. Microscope for inspection: ferrule/core/cladding. Optical Spectrum Analyzer (OSA): parameters and measurements. Manual and engineered (HP 11896A) Polarization Controllers (PC). Polarimeter (Thorlabs DPC5500): measurements. Polarization Scrambling and DOP: times.

## Bibliography

- Alberto Bononi, Armando Vannucci, "PMD: a Math Primer", technical report 14 july 2001, rev. 18/12/2008, available at the Faculty copying facility.

- Jay N. Damask, "Polarization Optics in Telecommunications", Ed. Springer (New York, USA), 2005, ISBN: 0-387-22493-9. Available at the Biblioteca Politecnica (reference ELE2/731).

- Serge Huard, "Polarization of Light", Ed. J.Wiley&sons, 1997, ISBN: 0-471-96536-7. Available at the Biblioteca Politecnica (reference BIE2/446).

- Andrea Galtarossa, Curtis R. Menyuk, (Eds.), "Polarization Mode Dispersion", Ed. Springer (New York, USA), 2005, ISBN-10: 0-387-23193-5. Available at the Biblioteca Politecnica (reference ELE2/730).

## Teaching methods

- class lectures (36h), given by the teacher, with the aid of blackboard and overhead projector/PC (for showing software applications, figures, web pages).

- simulation laboratory (4h), using the open source simulator Optilux (University of Parma) for signal propagation in fiber optics.

- measurement laboratory (2h), using hardware instruments and devices.

## Assessment methods and criteria

oral exam:

with reference to the contents of the lectures given during the course, the understanding level is evaluated, as well as the capability of analyzing and presenting the topics.

At the Student discretion, a project/case-study can be undertaken by the Student alone or within a small group, to deepen the analysis of a specific topic, agreed with the Teacher.

No homeworks or classworks are foreseen, during the course.

A midterm multiple-choice test is foreseen during the 'spring session' (right after Easter Holidays).

## Other information

For this course, informations and materials are available on the Elly/Moodle web-learning platform ( http://elly.dii.unipr.it/2015/course/view.php?id=48 )

## 2030 agenda goals for sustainable development

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