# 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 above problems

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

- implement numerical algorithms for the analysis of nonlinear systems.

## 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.

## RECOMMENDED READINGS

Slides of the course are available.

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.

## 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.

## TEACHING METHODS

Lessons mainly with blackboard but also by a video projector. There will be some lessons in the computer lab.

## FURTHER INFORMATIONS

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

## 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). 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 multipliers method.

Lecture 19

Best amplifers gain in inhomogeneous chains.

Solitons. Proof of fundamental soliton. Higher order solitons. Notes on

Dark solitons.

Lecture 20

Solitons: from dimensionless to standard units. Collision length and

symbol rate of solitons. Scaling laws of solitons. Perturbation of solitons:

solitons of non-integer order, impact of chirp. Solitons in amplified

systems: impact of losses. Notes on impact of ASE noise: sliding filters.

Lecture 21

Numerical examples of soliton propagation: 3rd order soliton, dark

soliton, soliton of non-integer order, interaction of solitons.

Wavelength division multiplexing (WDM) systems. NLSE with separate

fields. Cross-phase modulation (XPM) and four wave mixing (FWM).

Intraand

inter-channel GVD.

Lecture 22

XPM with inter-channel GVD: probe/pump case. XPM filter for single fiber.

Walk-off coefficient. Bandwidth of XPM filter.

Small-signal model of GVD.

Lecture 23

XPM filter for multi-span systems in absence of intra-channel GVD. XPM

filter with intra-channel GVD. Numerical results. Example: hybrid

OOK/DQPSK system.

Split-step Fourier method (SSFM). Formal solution using operators.

Lecture 24

Non commutative operators. SSFM with symmetrized and asymmetric

step: accuracy. Choice of the step: constant step, step based on the

nonlinear phase criterion, step based on the local error. Richardson

extrapolation.

Lecture 25

local error method: choice of the step size. Block diagram of the local

error method.

The Matlab programming language.

Lecture 26

Software Optilux. Examples. Discretization of a signal in the time and

frequency domain.

Lecture 27

Pills on how to write a scientific report.

Unique and separate fields: numerical cost comparison.

Four wave mixing (FWM). Regular perturbation (RP) method to

approximate the solution of the NLSE.

Lecture 28

FWM with CW signals. FWM efficiency. Phase matching coefficient.

Gaussian Nonlinear (GN) model. Best power using the GN model.

Lecture 29

Application of the GN model: best SNR, scaling of SNR. Exercise: getting

the entire SNR curve by two measurements. Constrained performance:

scaling of nonlinear asymptote with the number of spans.

From SSFM to the first order perturbation model.

Modulation instability (MI): linearized NLSE.

Lecture 30

Modulation instability: solution in absence of attenuation. Eigenvalues of

MI.

Optical parametric amplifier (OPA). Bandwidth and frequency of

maximum gain of an OPA. Two pumps OPA. Quantum noise in an OPA.

Lecture 31

Noise figure of an OPA.

Raman amplification. Motivations (distributed amplification, co- and

counter-propagating pump). Memory induced by Raman effect. SPM, XPM

and FWM in presence of Raman. Raman impact on XPM. Raman

amplification: pump-signal case.

Lecture 32

Notes on the amplified spontaneous Raman scattering and Rayleigh back

scattering.

Polarization of light. Birefringence. Jones formalism. Ellipse of

polarization. Polarimeter.

Lecture 33

Stokes space. Poincaré sphere. Degree of polarization (DOP).

Input/output relation with birefringence. Unitary matrices. Local behavior

of birefringence. Hermitian matrices. Eigenvalues and eigenvectors of

Hermitian matrices.

Lecture 34

Polarization mode dispersion (PMD). Motion in omega. Differential group

delay (DGD). First order PMD.

Manakov equation. Cross polarization modulation (XPolM). Memoryless

XPolM.

Lecture 35

Advanced modulation formats: motivations. Phase modulator and Mach

Zehnder (MZ) modulator. Return to zero(RZ) pulses and its variants

(carrier-suppressed (CS-RZ), chirped-RZ (CRZ), alternate phase-RZ

(APRZ)). Duobinary transmission. Differential phase shift keying (DPSK).

Generation and detection of DPSK. Nonlinear phase noise. Differential

quadrature phase shift keying (DQPSK). Generation of M-ary PSK.

Lecture 36

Coherent detection: motivations. Historical background. Optical hybrid.

Detection of in-phase and quadrature components. Polarization division

multiplexing (PDM). Polarization diversity receiver. Digital signal

processing (DSP). Analog to digital conversion (ADC): choice of the

number of samples per symbols. Electronic dispersion compensation of

GVD. Electronic dispersion compensation of PMD: constant modulus

algorithm (CMA). Phase estimation: Viterbi & Viterbi algorithm. Numerical

and experimental results. Interaction of PMD and nonlinear Kerr effect.

Cross polarization modulation (XpolM): impact of channel walk-off.

Nonlinear threshold (NLT) of optical links. Digital back-propagation (DBP)

algorithm. Polarization switched quadrature phase shift keying (PS-QPSK).