Learning objectives
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 unit content
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.
Full programme
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).
Bibliography
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.
Teaching methods
Lessons mainly with blackboard but also by a video projector. There will be some lessons in the computer lab.
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 information
During the course a numerical simulator of optical links will be introduced
