# OPTICAL COMMUNICATIONS

## Learning outcomes of the course unit

The course aims to provide the main tools to analyze and design modern

optical communication systems. Strictly speaking, 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 trasmission/amplification/detection of an optical

signal.

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

Applying the knowledge and the understanding mentioned above, the

student should be able to:

- analyze the main distortions of an optical link.

- analyze the main sources of noise that impact the bit error rate of an

optical digital transmission.

- find strategies to cope with the distortions

- describe the optical channel by theoretical models in different cases.

- write a numerical algorithm simulating the propagation of a signal

within an optical fiber.

## Prerequisites

suggested basic knowledge of Digital Communications and Signal Processing.

## Course contents summary

Introduction, motivations, state of the art.

Brief introduction of single mode fibers.

Group velocity dispersion.

Optical Transmitters.

Optical Amplifiers.

Principles of Photodetection.

Performance Evaluation.

Nonlinear Schroedinger Equation.

Self phase modulation.

Cross phase modulation.

Four wave mixing.

Optical Solitons.

Raman Effect.

Parametric gain and modulation instability.

Polarization Mode Dispersion.

Advanced modulation formats and optical coherent detection.

## Course contents

Lecture 1

Introduction, presentation of the course, motivations. Brief history of

optical communications.

Lecture 2

Ray optics. Fermat's principle. Snell's law. Total reflection. Numerical

aperture of an optical fiber. Multi-mode fibers. Problems of multi-mode

fibers. Single-mode fibers (overview). V-number (overview). Systems

theory approach to the optical fiber. Phase delay and group delay. Group

velocity dispersion (GVD). Propagation constant beta. Delay between two

frequencies induced by GVD. Conversion from beta2 to dispersion

coefficient D.

Lecture 3

GVD: examples. Waveguide and material dispersion. Rigorous proof of

GVD using Maxwell's equations.

Lecture 4

Attenuation. Group delay. Impact of GVD over a Gaussian pulse.

Dispersion length. Anomalous and normal dispersion. GVD in presence of

signal's chirp. Instantaneous frequency.

Lecture 5

GVD in presence of signal chirp. Best chirp using Heisenberg's principle.

Matched filter interpretation of GVD with chirp. Third order dispersion.

Eye closure penalty in presence of GVD.

Lecture 6

Chen's formula for the GVD induced eye closure penalty. Fourier

transform induced by strong GVD. de Bruijn sequences. Memory of GVD.

Lecture 7

Erbium doped fiber amplifier (EDFA). Cross sections. Propagation

equation for the photon flux over distance. Rate equation in time.

Reservoir. State model interpretation of reservoir. Small signal gain. Gain

saturation.

Lecture 8

Propagation equation with gain saturation. Fixed output power of an

EDFA in saturation. Reservoir dynamics with modulated signals. Amplified

spontaneous emission (ASE) noise. Noise figure of an EDFA: definition.

Lecture 9

Friis's formula. Excess noise figure. Dual stage amplification: evaluation

of noise figure.

Photo-detectors: photo-diode. Quantum efficiency. Responsivity. Reasons

for photo-current: electron-holes contributions to current. P-i-n junction.

Junction capacity. Photo-diode bandwidth.

Lecture 10

Avalanche photo-diode (APD).

Poisson statistics. Poisson counting process. Shot noise. Campbell's

theorem with proof. Power spectral density (PSD) of shot noise. PSD with

A P D .

Lecture 11

Optical receivers. Matched filter. Amplifiers for the photo-current: low

impedance, high impedance, trans-impedance. Bit error rate (BER) for onoff

keying (OOK) transmission. Quantum limit. Sensitivity power. Thermal

noise. Gaussian approximation and Personick's formula.

Lecture 12

Gaussian approximation. Q-factor. Gaussian approximation with APD.

Optimal multiplication factor with APD. Power budget.

Lecture 13

Relation between Sensitivity penalty and Eye closure penalty for PIN and

APD. Case with GVD using Chen's formula. Exercise regarding the

amount of chirp yielding a given sensitivity penalty. Pre-amplified

receivers. Signal to spontaneous and spontaneous to spontaneous noise

beat.

Lecture 14

BER with ASE noise: Gaussian approximation. Isserlis's formula. Average

and variance of signal/spontaneous, spontaneous/spontaneous, shot,

thermal noise. Comparison of noise variances.

Lecture 15

Optical signal to noise ratio (OSNR). Comparison signal/spontaneous,

spontaneous/spontaneous. Marcuse's formula. Pre-amplified receivers:

comparison with quantum limit. Exercises.

Bergano's method to estimate BER. Threshold error using the Gaussian

approximation.

Lecture 16

Nonlinear Schroedinger equation (NLSE). Reasons for the cubic nonlinear

effect. Self Phase Modulation (SPM). Comparison between temporal

interpretation of SPM and frequency interpretation of GVD.

Lecture 17

Comparison between temporal interpretation of SPM and frequency

interpretation of GVD. SPM with sinusoidal power. Bandwidth

enlargement induced by SPM. Wave breaking (WB). Impact of chirp

induced by SPM and GVD over a Gaussian pulse.

Lecture 18

Noise figure of optical amplifiers measured in the electrical domain.

OSNR budget. Distributed amplification. Amplifier chains: limitations of

ASE noise and nonlinear Kerr effect. Inhomogeneous amplifier chains.

Lagrange

## Recommended readings

Part of the course is written in lecture notes.

Reading of the following books is suggested:

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

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

Further scientific papers will be indicated during the course.

## Teaching methods

Lessons mainly with blackboard but also by a video projector. In particular, some examples of numerical simulation will be provided.

## Assessment methods and criteria

The exam consists in an oral examination and in an individual project (4

pages) regarding the study of an optical link by simulation. The project is

evaluated in terms of correctness, completeness, clarity of exposition,

bibliography.

## Other informations

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