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Each lecture is 2 hours. Download the page in pdf.
Introduction, presentation of the course, motivations. Brief history of optical communications.
Notes: The slides of the presentation can be found in LeA.
Supplementary reading: An interesting analysis of the future of optical communications can be found in [Des06]. An introduction to the history of optical communications can be found in [Agrbase]. A very good analysis of the main effects in optical communications with emphasis to channel capacity can be found in [Ess10]. A list of the main companies in optical communications is in [Ecocex]. A list of the main Italian companies is in [ListITA].
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.
Notes: The principles of ray optics can be found in [Alb05] and [Agrbase]. A rigorous proof of GVD can be found in [AgrNL,Agrbase].
Supplementary reading: A list of optical fibers can be found in [Cisco]. The carrier and envelope delay theory can be found in ``Carrier and envelope delay'' in [Carlson].
GVD: examples. Waveguide and material dispersion. Rigorous proof of GVD using Maxwell's equations.
Notes: For the waveguide and material dispersion see [Alb05] and [Agrbase]. For the rigorous proof of GVD see [AgrNL].
Supplementary reading: for a general description of Maxwell's equation see [Saleh]. A more rigorous proof of GVD (and nonlinear Kerr effect) using the multiple scales approach can be found in [Men99,Men06]. A list of the main optical fibers can be found in [Cisco].
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.
Notes: GVD is described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details can be found in [Saleh].
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.
Notes: details are described in [Alb05]. The impact of chirp can also be found in [Agrbase,AgrNL].
Supplementary reading: For the matched filter interpretation in the frequency domain see [Proakis]. For the definition of the variance of a signal and the Heisenberg's principle see the appendix of [Saleh].
Chen's formula for the GVD induced eye closure penalty. Fourier transform induced by strong GVD. de Bruijn sequences. Memory of GVD.
Notes: details can be found in [Alb05].
Supplementary reading: Chen's formula was introduced in [Chen99] regarding polarization mode dispersion (PMD), but the idea still works for GVD. The GVD induced Fourier transform is similar to the Fresnel diffraction effect [Saleh].
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.
Notes: details can be found in [Alb05]. A detailed description of the reservoir is in [Alb98].
Supplementary reading: A discussion about the cross section is in [Saleh]. The models of the EDFA are in [Sal90,Sun96].
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.
Notes: details can be found in [Alb05]. A detailed description of the reservoir is in [Alb98].
Supplementary reading: For the noise figure definition of an EDFA see [Hau98].
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.
Notes: details can be found in [Alb05]. For the noise figure definition see [Hau98]. The property that one photon contributes to one charge to the net current is described in [Saleh] in section 17.1 ``Properties of semiconductor photodetectors''.
Supplementary reading: a more general discussion about photo-diodes can be found in [Alexander] and [Agrbase].
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 APD.
Notes: details can be found in [Alb05].
Supplementary reading: An alternative proof of Campbell's theorem can be found in [Saleh].
Optical receivers. Matched filter. Amplifiers for the photo-current: low impedance, high impedance, trans-impedance. Bit error rate (BER) for on-off keying (OOK) transmission. Quantum limit. Sensitivity power. Thermal noise. Gaussian approximation and Personick's formula.
Notes: details can be found in [Alb05]. Quantum limit is also discussed in [Agrbase].
Supplementary reading: other details can be found in [Saleh].
Gaussian approximation. Q-factor. Gaussian approximation with APD. Optimal multiplication factor with APD. Power budget.
Notes: details can be found in [Alb05].
Supplementary reading: general discussions about power budget can be found in [Saleh].
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.
Notes: details can be found in [Alb05].
Supplementary reading: general discussions can be found in [Agrbase]. For the Rice representation of a bandpass stochastic process see [Papoulis,Carlson].
BER with ASE noise: Gaussian approximation. Isserlis's formula. Average and variance of signal/spontaneous, spontaneous/spontaneous, shot, thermal noise. Comparison of noise variances.
Notes: details can be found in [Alb05].
Supplementary reading: general discussions can be found in [Agrbase]. Isserlis's formula is well described in wikipedia.
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.
Notes: details can be found in [Alb05].
Supplementary reading: Better methods to evaluate the BER without the Gaussian approximation can be found in [Hum91,For00].
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.
Notes: for the NLSE see [AgrNL]. For the reasons of the cubic nonlinearity see [Iizuka] at p. 523.
Supplementary reading: A rigorous proof of NLSE can be found in [Men06,Men99]. A tutorial about NLSE is in [Agr11].
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.
Notes: SPM e WB are described in [AgrNL]. The physical interpretation of the chirp is described in [And92].
Supplementary reading: Further information on WB can be found in [And93].
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.
Notes: details can be found in [Alb05]. For the the noise figure see [Agrbase].
Supplementary reading: For the inhomogeneous amplifier chains see [Mec98].
Best amplifers gain in inhomogeneous chains.
Solitons. Proof of fundamental soliton. Higher order solitons. Notes on Dark solitons.
Notes: details can be found in [AgrNL].
Supplementary reading: Further details on solitons can be found in [Agr11]. A good book on solitons in the Unipr library is [Hasegawa].
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.
Notes: details can be found in [AgrNL].
Supplementary reading: Further details on solitons can be found in [Agr11]. A good book on solitons in the Unipr library is [Hasegawa].
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). Intra- and inter-channel GVD.
Notes: For cross-channel nonlinear effects see [AgrNL].
Supplementary reading: The numerical examples about solitons are taken form [AgrNL].
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.
Notes: For cross-channel effects see [AgrNL]. XPM without intra-channel GVD is discussed in [Kaz96]. Small-signal model of GVD has been introduced in [Pet92].
Supplementary reading: XPM impact on ASE noise without intra-channel GVD is discussed in [Ho04] with a model close to the one analyzed in the lesson.
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.
Notes: for the XPM filter see [Bel98,BelVar98]. Slides of the lesson available at LeA. The basic idea of SSFM is discussed in [AgrNL].
Supplementary reading: An alternative derivation of the XPM filter can be found in [Car99]. XPM in hybrid systems is discussed in [Alb09].
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.
Notes: The basic idea of SSFM is discussed in [AgrNL]. The step size is analyzed in [Sin03]. The method based on the local error is described in [Feldman].
Supplementary reading: An extension of the method of the nonlinear phase including GVD is described in [Zha08]. An analysis of the spurious resonances induced by a constant step is shown in [Bos00].
local error method: choice of the step size. Block diagram of the local error method.
The Matlab programming language.
Notes: The slides about Matlab can be found in LeA.
Supplementary reading: Open-source versions of Matlab are Octave and SciLab. With Octave a graphical user interface may be useful, e.g., gnuplot (in wikipedia at the voice Octave there is an exhaustive list of alternatives). The basic toolbox of Matlab are available in Octave at Octave-Forge. A Matlab primer is [Sig93]. An advanced tutorial of Matlab is [Ack03]. A collection of general purpose Matlab functions can be found in Matlab-File-Exchange.
Some useful code lines regarding saving the body of a Matlab file into a variable can be found in the forum of the Optical Communication course at LeA, date 9 May 2012.
Software Optilux. Examples. Discretization of a signal in the time and frequency domain.
Notes: Optilux can be downloaded at OptiluX.
Supplementary reading:
I suggest to keep update Optilux with
Subversion. In
Linux, the command is the following:
<svn co https://optilux.svn.sourceforge.net/svnroot/optilux>.
This commands creates a copy of the global Optilux repository, such
that updating the files to the latest version can be done (e.g., in
the bash of Linux) by typing <svn up> in the directory of
the repository.
I suggest to compile the mex files inside Optilux (e.g.,
run the function comp_mex.m in the Optilux directory). In
this case, under Linux may be useful the installation of the package
build-essential.
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.
Notes: For the unique and separate fields approaches see OptiluX documentation. FWM can be found in [AgrNL].
Supplementary reading: an excellent reference about writing a scientific report can be found in [Whi04]. I suggest to read even [Abd08]. The RP method is discussed in [Van02].
FWM with CW signals. FWM efficiency. Phase matching coefficient.
Gaussian Nonlinear (GN) model. Best power using the GN model.
Notes: FWM is discussed in [AgrNL]. The nonlinear Gaussian model for the signal to noise ratio can be found in [Alb11].
Supplementary reading: Further details on the GN model can be found in [Pog11] and [Car12].
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.
Notes: The nonlinear Gaussian model for the signal to noise ratio can be found in [Alb11]. Modulation instability is discussed in [AgrNL].
Supplementary reading: The scaling laws of the SNR can be found in [Alb12].
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.
Notes: Modulation instability is discussed in [AgrNL]. A good tutorial about OPA is [Han02].
Supplementary reading: Two pumps OPA is described in [MKi02].
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.
Notes: Noise figure of an OPA is discussed in [Kyl04]. The pump/signal model of Raman is discussed in [AgrNL].
Supplementary reading: A book on Raman amplification is [AgrRaman].
Notes on the amplified spontaneous Raman scattering and Rayleigh back scattering.
Polarization of light. Birefringence. Jones formalism. Ellipse of polarization. Polarimeter.
Notes: A good book on polarization of light is [Damask]. A good description of linear algebra is in [Gor00]. A software to examine the state of polarization of light is in LeA.
Supplementary reading: Advanced details on linear algebra are in [Gor00].
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.
Notes: A good book on polarization of light is [Damask]. A good description of linear algebra is in [Gor00].
Supplementary reading: Advanced details on linear algebra are in [Gor00].
Polarization mode dispersion (PMD). Motion in omega. Differential group delay (DGD). First order PMD.
Manakov equation. Cross polarization modulation (XPolM). Memoryless XPolM.
Notes: A good book on polarization of light is [Damask]. A good description of linear algebra is in [Gor00].
Supplementary reading: An introduction on XPolM is in [Kar06]. A rigorous proof of the Manakov equation is in [Wai96].
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.
Notes: The slides are available at LeA. A good tutorial of DPSK is [Win06].
Supplementary reading: A detailed bibliography can be found into the slides.
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).
Notes: The slides are available at LeA. Good tutorials on coherent detection are [Cha08,Kik10,Sav10].
Supplementary reading: A detailed bibliography can be found into the slides.
Paolo 2018-02-21