## Learning objectives

The course aims to provide the main tools to analyze and design modern fiber optic communication systems. In particular, the course would like to give knowledge and understanding about:

- linear effects in an optical fiber.

- nonlinear effects in an optical fiber.

- investigation of the transmission/amplification/detection of an optical signal.

- design of a fiber-optic communication system.

- the basic principles of a numerical simulation of an optical link.

With such a knowledge the student should be able to:

- analyze the main distortions of an optical link.

- understand and analyze the main sources of noise that impact the bit error rate of a digital transmission by means of fiber optics.

- find strategies to cope with the above problems

- design a fiber optic link.

- implement numerical algorithms for the analysis of nonlinear systems.

- write a scientific report

## Prerequisites

suggested basic knowledge of Digital Communications, Signal Processing and electro-magnetic waves.

## Course unit content

Introduction, motivations, state of the art.

Brief introduction of single mode fibers.

Group velocity dispersion.

Optical Transmitters.

Optical Amplifiers.

Principles of Photo-detection.

Performance Evaluation of optical communication systems.

Birefringence and polarization mode dispersion.

Coherent detection.

Nonlinear effects in optical fibers:

- Self phase modulation.

- Cross phase modulation.

- Four wave mixing.

Numerical simulation of optical communication systems.

Perturbative analysis of communication systems.

Gaussian noise model.

Digital Back-propagation.

## Full programme

Lecture 1

Introduction to the course.

Lecture 2

Optical modulators: Wavelength and frequency. ITU-T grid. Super-channel. Laser. Direct modulation. External modulation. Mach-Zehnder modulation. Return to zero shaping. Differential phase-shift keying. I/Q modulators.

Lecture 3

Polarization-division multiplexing (PDM). I/Q modulators. Probabilistic and geometric shaping. The even-odd trick for experiments.

Geometrical optics. Ray optics postulates. Snell's law. Multi-rays fibers. The numerical aperture of an optical fiber. The problem of multi-ray propagation. Single-mode fiber.

Lecture 4

Wave optics. Wave equation in vacuum.

Wave equation. Polarization field. A linear model of the polarization field. From the wave equation to the linear Schroedinger equation. The paraxial wave approximation.

Lecture 5

The relation between power and electric field.

Attenuation. Loss in a decibel scale. Retarded time frame: proof in the frequency and the time domain.

Group velocity dispersion (GVD). Dispersion length.

Lecture 6

Basic properties of GVD. Effect of GVD on a Gaussian pulse.

Dispersion-management. Dispersion-shifted fibers (DSF). Dispersion-compensating fibers (DCF). Dispersion map.

Compensation of GVD through a laser chirp

Lecture 7

Third-order dispersion.

Eye-closure penalty (ECP) due to GVD.

Highly dispersive regime.

Memory time of GVD.

Lecture 8

Erbium-doped fiber amplifier. Stark-effect. Cross-sections.

Propagation equation. Choice of the pump wavelength.

Lecture 9

General questions on the previous arguments.

Rate equation. Reservoir. Saleh equation of EDFA gain.

The behavior of EDFA at large pump power.

Gain saturation of EDFA.

Semiconductor optical amplifier (SOA).

Amplified spontaneous emission (ASE) noise.

Spontaneous emission factor nsp. Optical signal-to-noise ratio (OSNR).

Lecture 10

Exercises on EDFA and GVD memory time.

Optical signal-to-noise ratio (OSNR).

The noise figure of an EDFA. Optical and excess noise figure.

OSNR of a transparent link.

Exercise: non-homogeneous link.

EDFA in constant-gain mode vs constant-output power mode.

Lecture 11

Exercise: dispersion length, compensation, SNR.

Friis formula.

Dual-stage amplification. Comparison with single-stage and the dispersion-uncompensated case.

Lecture 12

Photo-detection. Quantum efficiency. Responsivity.

Equivalent circuit of the P-N photodiode. PIN photodiode.

The bandwidth of a photodiode: bandwidth due to the parasitic capacitance, bandwidth due to the transit time.

Poisson statistics. Shot-noise.

The variance of shot noise.

Lecture 13

Intensity modulation direct detection (IM-DD).

Quantum limit.

Thermal noise.

ASE noise: signal-spontaneous and spontaneous-spontaneous beating.

Lecture 14

BER with ASE noise. Gaussian approximation.

The formula of Personick's. Minimax threshold. Q-factor.

Sensitivity penalty. Relation between the sensitivity penalty and the eye closure penalty.

The measurement of the noise figure.

Comparison between the Gaussian probability density functions and the true ones.

Birefringence.

The Jones formalism of polarization. The Stokes formalism of polarization.

Lecture 15

Poincaré sphere. Examples.

Linear vectorial propagation. Crosstalk.

Geometrical interpretation of the inner product and the matrix-vector product.

Vectorial propagation in a polarization-maintaining fiber(PMF).

Waveplate model.

First-order PMD model.

PMD at a fixed distance.

Lecture 16

Software MATLAB.

Discrete Fourier transform of an analog signal.

Lecture 17

Outage probability due to PMD.

Nonlinear Kerr effect: reasons for the cubic nonlinearity.

The one-dimensional model of the silica crystal.

The nonlinear Schroedinger equation (NLSE).

The nonlinear coefficient.

Lecture 18

Self-phase modulation (SPM).

Effective length.

A comparison between GVD and SPM.

Spectrum broadening due to SPM.

Optimal design of amplifier gains in the presence of SPM.

Lecture 19

Software OptiluX.

Examples 01 and 02.

Lecture 20

Software OptiluX: examples 03 and 04

Lecture 21

Wavelength-division multiplexing (WDM).

Unique and separate field solution.

Separate field solution in the nonlinear regime: set up of the problem.

Separate-field and unique in the nonlinear regime.

Limits of the separate-field NLSE.

Cross-phase modulation (XPM).

Four-wave mixing (FWM).

Phase matching coefficient.

A comparison between SPM, XPM, and FWM.

Intra- and inter-channel GVD.

XPM: set-up of the problem.

Lecture 22

XPM. Closed-form solution of XPM with two channels and without intra-channel GVD.

XPM filter: impulse and frequency response. Scaling of XPM filter bandwidth.

Examples of XPM interactions.

The NLSE in dual polarization: the Manakov equation.

The split-step Fourier method (SSFM).

Error scaling in the SSFM.

The asymmetric and the symmetrized step.

The modified nonlinear step.

Lecture 23

Software OptiluX: examples 04 and 05. Exercises: exA.m, OX1_4.m.

Lecture 24

SSFM: first-step choice and step updating. The walk-off criterion for the first step choice.

The nonlinear phase criterion for the step choice.

Coherent detection. The problem of phase detection. DPSK detection.

Lecture 25

Coherent detection. Optical hybrid.

In-phase and quadrature component detection.

Heterodyne and homodyne detection.

Analog to digital converter.

Coherent detection: GVD equalization.

Lecture 26

Software OptiluX: examples 6, 7, and 8. Exercise with dark channel.

Lecture 27

Coherent detection: FIR vs IIR filters.

Timing recovery: Gardner algorithm.

Dynamic channel equalization. Steepest descent algorithm. Constant modulus algorithm (CMA).

The problem of decision-directed phase estimation at high speeds.

Carrier phase estimation. Viterbi and Viterbi algorithm.

Lecture 28

Blind phase search: key idea.

Carrier frequency estimation.

Digital back-propagation (DBP).

The regular perturbation (RP) method for the solution of the NLSE.

RP approximation of FWM.

FWM efficiency.

Lecture 29

OptiluX: exercise with the dark receiver. Examples 09, 10, and 11.

Lecture 30

The phase matching coefficient.

FWM combinations: examples.

RP1 with modulated signals. Bandwidth enlargement due to FWM.

Block diagram of RP1 algorithm.

Sampling frequency setup in the presence of FWM.

Lecture 31

FWM: multiple span case. Phased-array term.

Discrete-time channel model of nonlinear interference. Pulse collisions.

Phase and additive noise: discrete-time channel model.

The equivalent AWGN channel.

Frequency regions identified by FWM-like combinations.

Interpretation of the information rate graph of the AWGN channel.

The SNR under perturbative assumptions.

Scaling properties of the SNR. The nonlinear threshold.

Sensitivity penalty.

Lecture 32

OptiluX: example 12, 13, 14, and 15. Exercise on detection optimization of example 15.

Lecture 33

OptiluX: example 16, 17, and 18. Choice of the number of symbols in the presence of GVD.

Lecture 34

The reach of an optical system. Scaling properties of the NLI with distance.

The sensitivity of the reach with system parameters.

Reconfigurable add and drop multiplexers (ROADM). Load aware vs. full-load criterion.

The implication of the Shannon capacity in the design of optical communication systems. The role of the bandwidth.

Lecture 35

OptiluX: examples 19, 20, and 21. Exercise: search the optimal value of the total accumulated dispersion in the nonlinear regime for an OOK transmission.

Lecture 36

General suggestions on the project and about how to write a report.

## Bibliography

Slides of the course are available at Elly web site.

Supporting books are the following:

G. P. Agrawal, "Fiber-optic communication Systems", 3rd ed., Wiley, 2002;

G. P. Agrawal, "Nonlinear Fiber Optics", Academic Press

Supporting scientific papers are indicated at:

http://www.tlc.unipr.it/serena/CO/lezioni.html

## Teaching methods

Theoretical lectures will be provided by blackboard and slides.

Some exercises will be solved during the lectures. Interaction with students is stimulated by open questions.

Some lectures about Matlab will be given in the computer lab.

Seminars from experts in the field of optical communications might be possible.

## Assessment methods and criteria

The exam consists of an oral examination and an individual project (max 4 pages with a template). The project regards the study of an optical link by numerical simulation with the software Optilux. Each student receives an individual topic whose investigation will be reported in the final report of the project. The student can suggest a topic for the project prior to approval by the teacher. The project is evaluated in terms of correctness, completeness, clarity of exposition, and bibliography, with a scale 16-30 in case of approval. The oral exam is based mainly on open questions but also on basic exercises with the aim of testing the student's understanding of the course arguments and his/her skills in solving engineering problems regarding optical communications. The oral exam, if positive, is ranked between 18 and 30. The final grade is the average of the project and the oral, with honors when the maximum score is reached in both tests. A mid-term partial exam will be given in the spring exam session.

## Other information

During the course, a numerical simulator of optical links will be introduced