Next: Bibliography Up: Lezioni Previous: 2019
Each lecture is 2 hours.
Introduction, presentation of the course, motivations. A Brief history of optical communications.
Notes: References can be found within the slides, available at Elly.
Supplementary reading: An interesting analysis of the market of optical communications can be found in [Win17,Des06]. An introduction to the history of optical communications can be found in [Agrbase]. A very good analysis of 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]. Seldom, conferences on optical communications have a special web-page for job positions. See for instance [OFCjobs].
Notation.
Optical Modulators. ITU-T grid. Terminology: Super-channel, elastic networking, Nyquist transmission. Review of fundamentals of digital communications. Real lasers. Phase Modulator. Mach Zehnder modulator. Non-return to zero (NRZ) on-off keying (OOK) modulation. Return to zero (RZ) carving. Carrier suppressed RZ. Phase shift keying. Binary phase shift keying with phase modulator or Mach Zehnder modulator. I/Q modulators. Quadrature phase shift keying (QPSK) and quadrature amplitude modulation (QAM). Polarization division multiplexing (PDM).
Notes: The optical modulators are described in the book [Seimetz]. A good comparison of modulation formats is in [Win06,Big04]. Other references are included in the slides.
Supplementary reading: Other information on optical modulators can be found in [Binh].
Models of light. Ray optics. Fermat's principle. Snell's law. Total reflection. The 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).
Notes: The principles of ray optics can be found in [Alb05] and [Saleh,Agrbase]. A rigorous proof of GVD can be found in [AgrNL,Agrbase].
Supplementary reading: The carrier and envelope delay theory can be found in ``Carrier and envelope delay'' in [Carlson] and in Wikipedia.
The rigorous derivation of propagation equation by using Maxwell's equations. Analysis of the approximations made in the derivation.
Notes: 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].
Linear Schrödinger equation. Unit of measure of electric field. Attenuation. Group delay. Dispersion length. Impact of GVD over a Gaussian pulse. Power and phase of Gaussian pulse impaired by GVD.
Notes: GVD is described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details can be found in [Saleh]. I suggest to revise the main relationships of Fourier transform, for instance by looking at the Wikipedia page of Fourier transform.
Impact of GVD over a Gaussian pulse. Instantaneous frequency. Anomalous and normal dispersion. Dispersion Management (DM). GVD in presence of signal's chirp.
Notes: GVD and GVD-chirp interaction are described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details about GVD can be found in [Saleh].
GVD in presence of signal's chirp.
Third order dispersion. Eye closure penalty (ECP) in presence of GVD. Fourier transform induced by strong GVD.
Notes: details can be found in [Alb05]. The Fourier transform induced by GVD is an approximation also called by the alternative names: stationary phase approximation, far field approximation, saddlepoint method, Fresnel diffraction.
Supplementary reading: Fresnel diffraction is discussed in [Saleh].
Fourier transform induced by strong GVD in presence of signal's chirp. Memory of GVD. Erbium doped fiber amplifier (EDFA). Cross sections. Propagation equation for the photon flux over distance.
Notes: details can be found in [Alb05]. A good book on EDFA is [Becker].
Supplementary reading: A discussion about the cross sections is in [Saleh]. The models of the EDFA are in [Sal90,Sun96].
Reservoir. Rate equations. Gain saturation. Amplified spontaneous emission (ASE) noise.
Notes: details can be found in [Alb05]. A detailed description of the reservoir is in [Alb98].
Supplementary reading: A discussion about the cross sections is in [Saleh]. The models of the EDFA are in [Sal90,Sun96].
Amplified spontaneous emission (ASE) noise. Noise figure of an EDFA: definition. Excess noise figure. Optical signal to noise ratio (OSNR). OSNR in transparent links. Friis's formula. Dual stage amplification: evaluation of noise figure.
Notes: details can be found in [Alb05]. For the noise figure definition see [Hau98].
Supplementary reading: For the zero-point energy and the Casimir effect take a look at Wikipedia. General discussions on OSNR can be found in [Agrbase].
Dual stage amplification: evaluation of noise figure. Comparison of single-, dual-stage amplification and dispersion uncompensated. Exercises on OSNR evaluation.
Notes: details can be found in [Alb05].
Supplementary reading: General discussions on OSNR can be found in [Agrbase].
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. Avalanche photo-diode (APD). Poisson statistics. Poisson counting process.
Notes: details can be found in [Alb05]. The property that one photon contributes to one charge to the net current is described in [Saleh] in section 17.1 ``Properties of semiconductor photo-detectors''.
Supplementary reading: A more general discussion about photo-diodes can be found in [Alexander] and [Agrbase].
Poisson statistics. Shot noise. Power spectral density (PSD) of shot noise.
Bit error rate (BER) for on-off keying (OOK) transmission. Optical receivers. Matched filter with Poisson arrivals. Quantum limit.
Notes: Details can be found in [Alb05]. Quantum limit is also discussed in [Agrbase].
Supplementary reading: An alternative proof of Campbell's theorem can be found in [Saleh]. Other details about BER can be found in [Saleh].
Bit error rate (BER) for on-off keying (OOK) transmission. Thermal noise. Gaussian approximation. Comparison between shot noise and thermal noise. Sensitivity power. BER with ASE noise.
Notes: details can be found in [Alb05] and [Agrbase].
Supplementary reading: other details about BER can be found in [Saleh].
BER with ASE noise. Variance of signal/spontaneous beating noise. Spontaneous-spontaneous beat noise: motivations for Gaussian approximation. Comparison between true probability density function (PDF) and Gaussian approximation and motivations for the match.
Notes: details can be found in [Alb05]. The BER in direct detection is also discussed in [Agrbase].
Supplementary reading: More complete methods to evaluate the BER without the Gaussian approximation can be found in [Hum91,For00].
Marcuse's formula. Noise figure revised. Exercises on IM-DD detection.
Polarization of light: ellipse of polarization, Jones formalism and Stokes formalism. Poincaré sphere. Examples of Stokes vectors over the Poincarè sphere.
Notes: A good book on PMD is [Damask].
Supplementary reading: Basic ideas of birefringence are discussed in [AgrNL].
Polarization maintaining fiber (PMF). Motion along distance. Motion along frequency. Polarization mode dispersion (PMD). Differential group delay (DGD). Principal states of polarization (PSP).
Notes: A good book on PMD is [Damask].
Supplementary reading: A nice tutorial on PMD is [Gor00]. Many other details on polarization effects are discussed in the course ``Polarized fiber optic transmission''.
Overview of statistics of differential group delay (DGD). Outage probability.
Proof of NLSE. Reasons of the nonlinear term.
Notes: The reference book for nonlinear effects is [AgrNL].
Supplementary reading: A nice tutorial on PMD is [Gor00]. Many other details on polarization effects are discussed in the course ``Polarized fiber optic transmission''. A rigorous proof of NLSE can be found in [Men06,Men99].
Self phase modulation (SPM). Comparison between temporal interpretation of SPM and frequency interpretation of GVD. SPM in frequency domain. SPM with sinusoidal power. Bandwidth enlargement due to SPM.
Notes: SPM is well described in [AgrNL].
Supplementary reading: SPM induces chirp, see [And92].
Inhomogeneous amplifier chains. Lagrange multipliers method. Best amplifiers gain in inhomogeneous chains.
Solitons. Proof of fundamental soliton.
Notes: details about solitons can be found in [AgrNL].
Supplementary reading: Further details on solitons can be found in [Agr11]. A reference book on solitons in the Unipr library is [Hasegawa]. The inhomogeneous amplifier chains are described in [Mec98].
Solitons. Proof of fundamental soliton. Scaling properties of solitons. Pills on dark solitons and problems of solitons.
Wavelength division multiplexing (WDM). WDM propagation in linear regime.
Notes: For WDM see [AgrNL].
Supplementary reading: I suggest to review the superposition principle at Wikipedia.
NLSE with separate fields. Cross-phase modulation (XPM) and four wave mixing (FWM). Intra- and inter-channel GVD. XPM with inter-channel GVD: probe/pump case.
Notes: For cross-channel effects see [AgrNL]. XPM without intra-channel GVD is discussed in [Kaz96].
Supplementary reading: An improved version of the XPM filter is discussed in [Bel98,BelVar98]. XPM impact on ASE noise without intra-channel GVD is discussed in [Ho04]. XPM in hybrid systems is discussed in [Alb09]. Unique and separate fields are described in [Ros15].
XPM with inter-channel GVD: probe/pump case. XPM filter for single fiber. Walk-off coefficient. Frequency response of XPM filter. XPM with multiple spans.
Notes: For cross-channel effects see [AgrNL]. XPM without intra-channel GVD is discussed in [Kaz96].
Supplementary reading: An alternative derivation of the XPM filter can be found in [Car99].
Four wave mixing (FWM). Regular perturbation (RP) method to approximate the solution of the NLSE. FWM with CW signals. FWM efficiency. Phase matching coefficient. Examples of FWM. FWM in dual polarization (hints).
Notes: FWM can be found in [AgrNL].
Supplementary reading: An alternative derivation of the XPM filter can be found in [Car99]. The RP method is discussed in [Van02].
Introduction to the lab experience about FWM.
Split-step Fourier method (SSFM). Formal solution using operators.
Notes: FWM can be found in [AgrNL]. The RP method is discussed in [Van02]. SSFM can be found in [AgrNL].
Supplementary reading:
The reasons for the 
factor in
FWM with dual polarization is related to the Manakov equation and
discussed in [Wai96,Men06]. The reasons for the
factor in FWM with dual polarization are discussed in [Inoue92].
The Matlab programming language.
Notes: A Matlab primer is [Sig93]. Free versions of Matlab are Octave and SciLab.
Supplementary reading: An advanced tutorial of Matlab is [Ack03]. A collection of general purpose Matlab functions can be found at Matlab-File-Exchange.
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 toolboxes of Matlab are available in Octave at Octave-Forge.
Matlab Exercises. Brief overview of software Optilux.
Notes: Optilux can be downloaded at OptiluX. The discretization is discussed in the slides about Matlab exercises at Elly.
Supplementary reading:
I suggest to keep update Optilux with
Subversion, both
for Windows and Linux. In Linux, the command is the following:
<svn checkout svn://svn.code.sf.net/p/optilux/code/trunk optilux-code>.
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. In any case, the Optilux repository can be found at
Sourceforge.
I suggest to compile the mex files inside Optilux (e.g.,
run the function comp_mex.m in the Optilux_files
directory). In this case, under Linux it may be useful the installation
of the package build-essential.
Asymmetric and symmetrized SSFM. Choice of the SSFM step: constant step, step based on the nonlinear phase criterion. Richardson extrapolation. Block diagrams of SSFM. Block diagrams of RP1 algorithm.
Gaussian Noise (GN) model. Motivations for the additive white Gaussian noise (AWGN) assumption.
Notes: The Gaussian noise 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].
Gaussian Noise (GN) model. Best power using the GN model. Application of the GN model: best SNR, scaling of SNR. Constrained Performance. Constrained performance: scaling of nonlinear asymptote with the number of spans. Reach.
Notes: The Gaussian noise model for the signal to noise ratio can be found in [Alb11]. The scaling laws of the SNR can be found in [Alb12].
Supplementary reading: Further details on the GN model can be found in [Pog11] and [Car12].
Software Optilux.
Notes: Optilux can be downloaded at OptiluX. The discretization of a signal is discussed in the slides about Matlab exercises at Elly.
Supplementary reading:
I suggest to keep update Optilux with
Subversion, both
for Windows and Linux. In Linux, the command is the following:
<svn checkout svn://svn.code.sf.net/p/optilux/code/trunk optilux-code>.
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. In any case, the Optilux repository can be found at
Sourceforge.
I suggest to compile the mex files inside Optilux (e.g.,
run the function comp_mex.m in the Optilux_files
directory). In this case, under Linux it may be useful the installation
of the package build-essential.
Maximum reach with GN model. Tolerance to errors and implications in optical link design. Reconfigurable optical add and drop multiplexer (ROADM). Power setup in a network.
Coherent detection: motivations. 3 dB optical coupler. Differentially coherent detection: differential phase shift keying (DPSK). Differential quadrature phase shift keying (DQPSK). Comparison OOK/DPSK, DPSK/DQPSK.
Historical background of coherent detection. Comparison between coherent and differentially coherent detection. Optical hybrid. Detection of in-phase and quadrature components. Polarization division multiplexing (PDM). Polarization diversity receiver.
Notes: The Gaussian noise model for the signal to noise ratio can be found in [Alb11]. The scaling laws of the SNR can be found in [Alb12]. Parametric amplification is discussed in [AgrNL].
A detailed bibliography of coherent detection can be found into the slides.
Supplementary reading: Further details on the GN model can be found in [Pog11] and [Car12].
Coherent detection is an hot topic, hence I suggest to take a look at the most recent scientific articles.
Digital signal Processing (DSP). Analog to digital converter (ADC). Timing recovery: Gardner algorithm. Static channel equalization: finite impulse response (FIR) and infinite impulse response (IIR) filters. GVD memory time and number of taps. Dynamic channel equalization: data-aided and blind algorithms. Constant modulus algorithm (CMA). Carrier phase estimation. Frequency estimation. Example of real experiment.
Notes: A detailed bibliography can be found into the slides.
Supplementary reading: Digital signal processing for equalization of optical impairments is the most hot topic today, hence I suggest to take a look at the most recent scientific articles.
Seminar on submarine networks.
Notes: -
Supplementary reading: -
Exercises on Optilux.
Notes: see previous notes at lecture 29.
Supplementary reading: see previous notes at lecture 29.
Each lecture is 2 hours. Lecture slides are available at Elly.
Introduction, presentation of the course, motivations. Brief history of optical communications.
Notes: References can be found within the slides, available at Elly.
Supplementary reading: An interesting analysis of the market 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 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]. Seldom, conferences on optical communications have a special web-page for job positions. See for instance [OFCjobs].
Optical Modulators. ITU-T grid. Terminology: Super-channel, elastic networking, Nyquist transmission. Review of fundamentals of digital communications. Real lasers. Phase Modulator. Mach Zehnder modulator. Non-return to zero (NRZ) on-off keying (OOK) modulation. Return to zero (RZ) carving. Carrier suppressed RZ. Phase shift keying. Binary phase shift keying with phase modulator or Mach Zehnder modulator. I/Q modulators. Quadrature phase shift keying (QPSK) and quadrature amplitude modulation (QAM). Polarization division multiplexing (PDM).
Notes: The optical modulators are described in the book [Seimetz]. A good comparison of modulation formats is in [Win06,Big04]. Other references are included in the slides.
Supplementary reading: Other information on optical modulators can be found in [Binh].
Models of light. 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).
Notes: The principles of ray optics can be found in [Alb05] and [Saleh,Agrbase]. A rigorous proof of GVD can be found in [AgrNL,Agrbase].
Supplementary reading: The carrier and envelope delay theory can be found in ``Carrier and envelope delay'' in [Carlson] and in Wikipedia.
Rigorous derivation of propagation equation by using Maxwell's equations. Analysis of the approximations made in the derivation.
Notes: 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].
Paraxial wave approximation. Linear Schrödinger equation. Unit of measure of electric field. Attenuation. Group delay. Dispersion length. Impact of GVD over a Gaussian pulse.
Notes: GVD is described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details can be found in [Saleh]. I suggest to revise the main relationships of Fourier transform, for instance by looking at the Wikipedia page of Fourier transform.
Impact of GVD over a Gaussian pulse. Instantaneous frequency. Anomalous and normal dispersion. Dispersion Management (DM). GVD in presence of signal's chirp.
Notes: GVD and GVD-chirp interaction are described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details about GVD can be found in [Saleh].
Third order dispersion. Eye closure penalty (ECP) in presence of GVD. Fourier transform induced by strong GVD.
Notes: details can be found in [Alb05]. The Fourier transform induced by GVD is an approximation also called by the alternative names: stationary phase approximation, far field approximation, saddlepoint method, Fresnel diffraction.
Supplementary reading: Fresnel diffraction is discussed in [Saleh].
Fourier transform induced by strong GVD in presence of signal's chirp. Memory of GVD. Erbium doped fiber amplifier (EDFA). Cross sections. Propagation equation for the photon flux over distance.
Notes: details can be found in [Alb05]. A good book on EDFA is [Becker].
Supplementary reading: A discussion about the cross sections is in [Saleh]. The models of the EDFA are in [Sal90,Sun96].
Reservoir. Rate equations. Gain saturation. Amplified spontaneous emission (ASE) noise.
Notes: details can be found in [Alb05]. A detailed description of the reservoir is in [Alb98].
Supplementary reading: A discussion about the cross sections is in [Saleh]. The models of the EDFA are in [Sal90,Sun96]. For the zero-point energy and the Casimir effect take a look at Wikipedia.
Amplified spontaneous emission (ASE) noise. Noise figure of an EDFA: definition. Excess noise figure. Optical signal to noise ratio (OSNR). OSNR in transparent links. Friis's formula. Dual stage amplification: evaluation of noise figure.
Notes: details can be found in [Alb05]. For the noise figure definition see [Hau98].
Supplementary reading: For the zero-point energy and the Casimir effect take a look at Wikipedia. General discussions on OSNR can be found in [Agrbase].
Dual stage amplification: evaluation of noise figure. Comparison of single-, dual-stage amplification and dispersion uncompensated. Exercises on OSNR evaluation.
Notes: details can be found in [Alb05].
Supplementary reading: General discussions on OSNR can be found in [Agrbase].
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. Avalanche photo-diode (APD). Poisson statistics. Poisson counting process.
Notes: details can be found in [Alb05]. The property that one photon contributes to one charge to the net current is described in [Saleh] in section 17.1 ``Properties of semiconductor photo-detectors''.
Supplementary reading: A more general discussion about photo-diodes can be found in [Alexander] and [Agrbase].
Shot noise. Power spectral density (PSD) of shot noise.
Bit error rate (BER) for on-off keying (OOK) transmission. Optical receivers. Matched filter with Poisson arrivals. Quantum limit.
Notes: Details can be found in [Alb05]. Quantum limit is also discussed in [Agrbase].
Supplementary reading: An alternative proof of Campbell's theorem can be found in [Saleh]. Other details about BER can be found in [Saleh].
Bit error rate (BER) for on-off keying (OOK) transmission. Thermal noise. Gaussian approximation. Comparison between shot noise and thermal noise. Sensitivity power. Relation between Sensitivity penalty and Eye closure penalty for PIN and APD.
Notes: details can be found in [Alb05] and [Agrbase].
Supplementary reading: other details about BER can be found in [Saleh].
BER with ASE noise. Variance of signal/spontaneous beating noise. Spontaneous-spontaneous beat noise: motivations for Gaussian approximation. Marcuse's formula. Comparison between true probability density function (PDF) and Gaussian approximation and motivations for the match.
Noise figure revised.
Notes: details can be found in [Alb05]. The BER in direct detection is also discussed in [Agrbase].
Supplementary reading: More complete methods to evaluate the BER without the Gaussian approximation can be found in [Hum91,For00].
Exercises on direct detection: uncompensated GVD, sensitivity penalty from experimental results, multi-carrier transmission.
Notes: The BER in direct detection is also discussed in [Agrbase].
Supplementary reading: More complete methods to evaluate the BER without the Gaussian approximation can be found in [Hum91,For00].
Polarization of light: ellipse of polarization, Jones formalism and Stokes formalism. Poincaré sphere. Examples of Stokes vectors over the Poincarè sphere. Brief review of properties of unitary and hermitian matrices. Polarization maintaining fiber (PMF). Motion along distance.
Notes: A good book on PMD is [Damask].
Supplementary reading: Basic ideas of birefringence are discussed in [AgrNL].
Polarization maintaining fiber (PMF). Motion along distance. Motion along frequency. Polarization mode dispersion (PMD). Differential group delay (DGD). Principal states of polarization (PSP).
Proof of NLSE. Reasons of the nonlinear term.
Notes: The reference book for nonlinear effects is [AgrNL].
Supplementary reading: A nice tutorial on PMD is [Gor00]. Many other details on polarization effects are discussed in the course ``Polarized fiber optic transmission''.
Proof of NLSE. Dimensional units of the electric field.
Self phase modulation (SPM). Comparison between temporal interpretation of SPM and frequency interpretation of GVD.
Notes: SPM is well described in [AgrNL]. The physical interpretation of the chirp is described in [And92].
Supplementary reading: A rigorous proof of NLSE can be found in [Men06,Men99].
SPM in frequency domain. SPM with sinusoidal power. Bandwidth enlargement due to SPM. Chirp superposition with GVD and SPM.
Inhomogeneous amplifier chains. Lagrange multipliers method. Best amplifiers gain in inhomogeneous chains.
Notes: The superposition of chirps is described in [And92].
Supplementary reading: Additional notes on chirp supersposition are described in [AgrNL] and [And93]. For the inhomogeneous amplifier chains see [Mec98].
Coherent detection: motivations. 3 dB optical coupler. Differentially coherent detection: differential phase shift keying (DPSK). Differential quadrature phase shift keying (DQPSK). Comparison OOK/DPSK, DPSK/DQPSK.
Historical background of coherent detection. Comparison between coherent and differentially coherent detection. Optical hybrid. Detection of in-phase and quadrature components. Polarization division multiplexing (PDM). Polarization diversity receiver.
Notes: A detailed bibliography can be found into the slides.
Supplementary reading: Coherent detection is an hot topic, hence I suggest to take a look at the most recent scientific articles.
Digital signal Processing (DSP). Analog to digital converter (ADC). Timing recovery: Gardner algorithm. Static channel equalization: finite impulse response (FIR) and infinite impulse response (IIR) filters. GVD memory time and number of taps. Dynamic channel equalization: data-aided and blind algorithms. Constant modulus algorithm (CMA).
Notes: A detailed bibliography can be found into the slides.
Supplementary reading: Digital signal processing for equalization of optical impairments is the most hot topic today, hence I suggest to take a look at the most recent scientific articles.
Carrier phase estimation. Frequency estimation. Example of real experiment.
Solitons. Proof of fundamental soliton.
Notes: details about solitons can be found in [AgrNL].
Supplementary reading: Further details on solitons can be found in [Agr11]. A reference book on solitons in the Unipr library is [Hasegawa].
Solitons. Scaling properties of solitons. Pills on dark solitons and problems of solitons.
Wavelength division multiplexing. Unique field vs. separate fields approach.
Notes: For cross-channel effects see [AgrNL].
Supplementary reading: I suggest to think about the motivations for using the Fourier transform in linear system theory.
NLSE with separate fields. Cross-phase modulation (XPM) and four wave mixing (FWM). Intra- and inter-channel GVD. XPM with inter-channel GVD: probe/pump case. XPM filter for single fiber. Walk-off coefficient. Frequency response of XPM filter.
Notes: For cross-channel effects see [AgrNL]. XPM without intra-channel GVD is discussed in [Kaz96].
Supplementary reading: An improved version of the XPM filter is discussed in [Bel98,BelVar98]. XPM impact on ASE noise without intra-channel GVD is discussed in [Ho04]. XPM in hybrid systems is discussed in [Alb09]. Unique and separate fields are described in [Ros15].
XPM with multiple spans.
Four wave mixing (FWM). Regular perturbation (RP) method to approximate the solution of the NLSE. FWM with CW signals. FWM efficiency.
Notes: FWM can be found in [AgrNL].
Supplementary reading: An alternative derivation of the XPM filter can be found in [Car99]. The RP method is discussed in [Van02].
Phase matching coefficient. Examples of FWM. FWM in dual polarization (hints).
Split-step Fourier method (SSFM). Formal solution using operators.
Notes: FWM can be found in [AgrNL]. The RP method is discussed in [Van02]. SSFM can be found in [AgrNL].
Supplementary reading:
The reasons for the factor in
FWM with dual polarization is related to the Manakov equation and
discussed in [Wai96,Men06]. The reasons for the
factor in FWM with dual polarization are discussed in [Inoue92].
SSFM: reason of the accuracy. Asymmetric and symmetrized SSFM. Choice of the SSFM step: constant step, step based on the nonlinear phase criterion, step based on the local error. Richardson extrapolation. Block diagrams of SSFM.
Lab experience: measure of fiber attenuation.
Notes: The basic idea of SSFM is discussed in [AgrNL]. The step size is analyzed in [Sin03]. The adaptive 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].
The Matlab programming language.
Notes: A Matlab primer is [Sig93]. Free versions of Matlab are Octave and SciLab.
Supplementary reading: An advanced tutorial of Matlab is [Ack03]. A collection of general purpose Matlab functions can be found at Matlab-File-Exchange.
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 toolboxes of Matlab are available in Octave at Octave-Forge.
Matlab Exercises. Software Optilux. Discretization of a signal in the time and frequency domain.
Notes: Optilux can be downloaded at OptiluX. The discretization is discussed in the slides about Matlab exercises at Elly.
Supplementary reading:
I suggest to keep update Optilux with
Subversion, both
for Windows and Linux. In Linux, the command is the following:
<svn checkout svn://svn.code.sf.net/p/optilux/code/trunk optilux-code>.
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. In any case, the Optilux repository can be found at
Sourceforge.
I suggest to compile the mex files inside Optilux (e.g.,
run the function comp_mex.m in the Optilux_files
directory). In this case, under Linux it may be useful the installation
of the package build-essential.
Block diagrams of RP1 solution.
Gaussian Noise (GN) model. Best power using the GN model. Application of the GN model: best SNR, scaling of SNR. Constrained Performance.
Notes: The Gaussian noise 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].
Software Optilux. Examples.
Notes: Optilux can be downloaded at OptiluX.
Supplementary reading: I suggest to keep update Optilux with Subversion, see Elly for more details.
Constrained performance: scaling of nonlinear asymptote with the number of spans. Tolerance to errors and implications in optical link design. Reconfigurable optical add and drop multiplexer (ROADM). Power setup in a network. System setup with the AWGN channel information rate point of view.
Notes: The Gaussian noise model for the signal to noise ratio can be found in [Alb11]. The scaling laws of the SNR can be found in [Alb12]. Parametric amplification is discussed in [AgrNL].
Supplementary reading: Further details on the GN model can be found in [Pog11] and [Car12].
Pills on Raman effect. Motivations for distributed amplification.
Cross-polarization modulation (XPolM). Geometric interpretation of XPolM. XPolM with walk-off. Numerical comparison of SPM, XPM and XPolM. Interleaved return to zero (iRZ).
Digital back-propagation (DBP). RP interpretation of DBP. Zero forcing properties of DBP. Impact of nonlinear signal noise interaction (NSNI) on DBP performance.
Notes: A detailed bibliography about XPolM and DBP can be found into the slides.
Supplementary reading: A book on Raman amplification is [AgrRaman]. Basics about Raman are in [Blo89].
Suggestions on how to write a report.
Notes: Very useful suggestions about writing a scientific report can be found in [Whi04]. An excellent book on scientific English is [Wallwork].
Supplementary reading: other useful details can be found in [Abd08].
Each lecture is 2 hours. Lecture slides are available at Elly.
Introduction, presentation of the course, motivations. Brief history of optical communications.
Notes: Slides regarding the history of optical communications can be found at Elly.
Supplementary reading: An interesting analysis of the market 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 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]. Seldom, conferences on optical communications have a special web-page for job positions. See for instance [OFCjobs].
Optical Modulators. ITU-T grid. review of fundamentals of digital communications. Real lasers. Phase Modulator. Mach Zehnder modulator. Non-return to zero (NRZ) on-off keying (OOK) modulation. Return to zero (RZ) carving. Carrier suppressed RZ. Phase shift keying. Binary phase shift keying with phase modulator or Mach Zehnder modulator.
Notes: The optical modulators are described in the book [Seimetz]. A good comparison of modulation formats is in [Win06,Big04].
Supplementary reading: Other information on optical modulators can be found in [Binh].
I/Q modulators. QPSK and QAM modulator.
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).
Notes: The principles of ray optics can be found in [Alb05] and [Saleh,Agrbase]. A rigorous proof of GVD can be found in [AgrNL,Agrbase].
Supplementary reading: The carrier and envelope delay theory can be found in ``Carrier and envelope delay'' in [Carlson].
Rigorous derivation of propagation equation by using Maxwell's equations. Analysis of the approximations made in derivation.
Notes: 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].
Linear Schrödinger equation. Unit of measure of electric field. Attenuation. Group delay. Dispersion length. Impact of GVD over a Gaussian pulse.
Notes: GVD is described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details can be found in [Saleh]. I suggest to revise the main relationships of Fourier transform, for instance by looking at the Wikipedia page of Fourier transform.
Instantaneous frequency. Anomalous and normal dispersion. Dispersion Management (DM). GVD in presence of signal's chirp. Third order dispersion. Eye opening. Eye closure penalty (ECP) in presence of GVD.
Notes: GVD and GVD-chirp interaction are described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details about GVD can be found in [Saleh].
Eye closure penalty (ECP) in presence of GVD. Chen's formula for the GVD induced eye closure penalty. Fourier transform induced by strong GVD. Memory of GVD (intro).
Notes: details can be found in [Alb05]. The Fourier transform induced by GVD is an approximation also called by the alternative names: stationary phase approximation, far field approximation, saddlepoint method, Fresnel diffraction.
Supplementary reading: Fresnel diffraction is discussed in [Saleh].
Memory of GVD. Erbium doped fiber amplifier (EDFA). Cross sections. Propagation equation for the photon flux over distance. Rate equations.
Notes: details can be found in [Alb05]. A good book on EDFA is [Becker].
Supplementary reading: A discussion about the cross sections is in [Saleh]. The models of the EDFA are in [Sal90,Sun96].
Reservoir. Rate equations. State model interpretation of reservoir. Amplified spontaneous emission (ASE) noise. Noise figure of an EDFA: definition. Friis's formula. Excess noise figure.
Notes: details can be found in [Alb05]. A detailed description of the reservoir is in [Alb98]. For the noise figure definition see [Hau98].
Supplementary reading: A discussion about the cross sections is in [Saleh]. The models of the EDFA are in [Sal90,Sun96]. For the zero-point energy and the Casimir effect take a look at Wikipedia.
Dual stage amplification: evaluation of noise figure. Comparison of single-stage and dual-stage amplification. Optical signal to noise ratio (OSNR). OSNR in transparent links. Exercise on OSNR evaluation.
Notes: details can be found in [Alb05].
Supplementary reading: general discussions on OSNR can be found in [Agrbase].
Exercises on OSNR evaluation.
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. Avalanche photo-diode (APD).
Notes: details can be found in [Alb05]. The property that one photon contributes to one charge to the net current is described in [Saleh] in section 17.1 ``Properties of semiconductor photo-detectors''.
Supplementary reading: A more general discussion about photo-diodes can be found in [Alexander] and [Agrbase].
Amplifiers for the photo-current: trade-off on impedance value. Trans-impedance amplifier.
Poisson statistics. Poisson counting process. Shot noise. Power spectral density (PSD) of shot noise.
Optical receivers. Matched filter with Poisson arrivals.
Notes: details can be found in [Alb05]. Miller effect is well discussed on Wikipedia.
Supplementary reading: An alternative proof of Campbell's theorem can be found in [Saleh].
Bit error rate (BER) for on-off keying (OOK) transmission. Quantum limit. Sensitivity power. Thermal noise. Gaussian approximation. Comparison between shot noise and thermal noise.
Notes: details can be found in [Alb05]. Quantum limit is also discussed in [Agrbase].
Supplementary reading: other details about BER can be found in [Saleh].
Gaussian approximation with APD. Optimal multiplication factor with APD. Relation between Sensitivity penalty and Eye closure penalty for PIN and APD. ASE noise correlation function. BER with ASE noise: Gaussian approximation. Average and variance of signal/spontaneous beating noise.
Notes: details can be found in [Alb05].
Supplementary reading: other details can be found in [Saleh]. General discussions can be found in [Agrbase]. For the Rice representation of a bandpass stochastic process see [Papoulis,Carlson].
Spontaneous-spontaneous beat noise: motivations for Gaussian approximation. Marcuse's formula. Comparison between true probability density function (PDF) and Gaussian approximation and motivations for the match.
Exercises on direct detection.
Notes: details can be found in [Alb05]. The BER in direct detection is also discussed in [Agrbase].
Supplementary reading: More complete methods to evaluate the BER without the Gaussian approximation can be found in [Hum91,For00].
Exercises on direct detection: uncompensated GVD, sensitivity penalty from experimental results, multi-carrier transmission.
Polarization effects in optical fibers. Birefringence. Polarization mode dispersion (PMD).
Notes: PMD A good book on PMD is [Damask].
Supplementary reading: Basic ideas of birefringence are discussed in [AgrNL].
Polarization of light: Jones formalism and Stokes formalism. Poincaré sphere. Examples of Stokes vectors over the Poincarè sphere. Brief review of properties of unitary and hermitian matrices. Polarization maintaining fiber (PMF). Motion along distance. Motion along frequency. Polarization mode dispersion (PMD). Differential group delay (DGD). Principal states of polarization (PSP).
Notes: PMD A good book on PMD is [Damask].
Supplementary reading: A nice tutorial on PMD is [Gor00]. Many other details on polarization effects are discussed in the course ``Polarized fiber optic transmission''.
Coherent detection: motivations. 3 dB optical coupler. Differentially coherent detection: differential phase shift keying (DPSK). Differential quadrature phase shift keying (DQPSK). Comparison OOK/DPSK, DPSK/DQPSK.
Historical background of coherent detection. Comparison between coherent and differentially coherent detection. Optical hybrid. Detection of in-phase and quadrature components. Polarization division multiplexing (PDM). Polarization diversity receiver.
Notes: A detailed bibliography can be found into the slides.
Supplementary reading: Coherent detection is an hot topic, hence I suggest to take a look at the most recent scientific articles.
Digital signal Processing (DSP). Analog to digital converter (ADC). Static channel equalization: finite impulse response (FIR) and infinite impulse response (IIR) filters. GVD memory time and number of taps. Dynamic channel equalization: data-aided and blind algorithms. Constant modulus algorithm (CMA). Carrier phase estimation. Frequency estimation. Example of real experiment.
Notes: A detailed bibliography can be found into the slides.
Supplementary reading: Digital signal processing for equalization of optical impairments is the most hot topic today, hence I suggest to take a look at the most recent scientific articles.
Proof of NLSE. Dimensional units of the electric field.
Self phase modulation (SPM). Comparison between temporal interpretation of SPM and frequency interpretation of GVD. SPM in frequency domain.
Notes: SPM is well described in [AgrNL]. The physical interpretation of the chirp is described in [And92].
Supplementary reading: A rigorous proof of NLSE can be found in [Men06,Men99].
SPM with sinusoidal power. Bandwidth enlargement due to SPM. Chirp superposition with GVD and SPM. Wave breaking.
Inhomogeneous amplifier chains. Lagrange multipliers method. Best amplifiers gain in inhomogeneous chains.
Notes: Wave breaking is described in [And92].
Supplementary reading: Additional notes on WB are described in [AgrNL] and [And93]. For the inhomogeneous amplifier chains see [Mec98].
Solitons. Proof of fundamental soliton. Solitons: from dimensionless to standard units. Collision length and symbol rate of solitons. Scaling laws of 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].
Higher order solitons. Perturbation of solitons: solitons of non-integer order, interaction of solitons. Numerical examples of solitons propagation. Sliding filters.
Wavelength division multiplexing (WDM). Unique field and separate fields.
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]. Unique and separate fields are described in [Ros15].
NLSE with separate fields. Cross-phase modulation (XPM) and four wave mixing (FWM). Intra- and inter-channel GVD. XPM with inter-channel GVD: probe/pump case. XPM filter for single fiber. Walk-off coefficient. Bandwidth of XPM filter.
Notes: For cross-channel effects see [AgrNL]. XPM without intra-channel GVD is discussed in [Kaz96].
Supplementary reading: XPM impact on ASE noise without intra-channel GVD is discussed in [Ho04].
XPM filter for multi-span systems in absence of intra-channel GVD. Numerical results.
Split-step Fourier method (SSFM). Formal solution using operators. Non commutative operators. SSFM with symmetrized and asymmetric step: accuracy.
Notes: for the XPM filter see [Bel98,BelVar98]. 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].
The Matlab programming language.
Notes: A Matlab primer is [Sig93]. Free versions of Matlab are Octave and SciLab.
Supplementary reading: An advanced tutorial of Matlab is [Ack03]. A collection of general purpose Matlab functions can be found at Matlab-File-Exchange.
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 toolboxes of Matlab are available in Octave at Octave-Forge.
Choice of the SSFM step: constant step, step based on the nonlinear phase criterion, step based on the local error. Richardson extrapolation. Block diagrams of SSFM.
Notes: The basic idea of SSFM is discussed in [AgrNL]. The step size is analyzed in [Sin03]. The adaptive 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].
Matlab Exercises. Software Optilux. 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, both
for Windows and Linux. In Linux, the command is the following:
<svn checkout svn://svn.code.sf.net/p/optilux/code/trunk optilux-code>.
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. In any case, the Optilux repository can be found at
Sourceforge.
I suggest to compile the mex files inside Optilux (e.g.,
run the function comp_mex.m in the Optilux_files
directory). In this case, under Linux it may be useful the installation
of the package build-essential.
Software Optilux. Examples.
Notes: Optilux can be downloaded at OptiluX.
Supplementary reading: I suggest to keep update Optilux with Subversion.
Four wave mixing (FWM). Regular perturbation (RP) method to approximate the solution of the NLSE. FWM with CW signals. FWM efficiency. Phase matching coefficient. FWM with multiple spans.
Notes: FWM can be found in [AgrNL].
Supplementary reading: The RP method is discussed in [Van02].
Gaussian Noise (GN) model. Best power using the GN model. Application of the GN model: best SNR, scaling of SNR. Constrained Performance.
Notes: The Gaussian noise 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].
Constrained performance: scaling of nonlinear asymptote with the number of spans. Tolerance to errors and implications in optical link design. Reconfigurable optical add and drop multiplexer (ROADM). Power setup in a network.
Optical Parametric Amplification (OPA). Small signal NLSE.
Notes: The Gaussian noise model for the signal to noise ratio can be found in [Alb11]. The scaling laws of the SNR can be found in [Alb12]. Parametric amplification is discussed in [AgrNL].
Supplementary reading: Further details on the GN model can be found in [Pog11] and [Car12].
Modulation Instability (MI). eigenvalues of MI. Bandwidth and frequency of maximum gain of an OPA. Notes on two pumps OPA. Phase-sensitive and phase insensitive OPA.
Raman amplification. Motivations (distributed amplification). Memory induced by Raman effect.
Notes: Modulation instability is discussed in [AgrNL]. A good tutorial about OPA is [Kar16,Han02].
Supplementary reading: Two pumps OPA is described in [MKi02]. A book on Raman amplification is [AgrRaman]. Phase-sensitive and phase-insensitive OPA are well described in [Kar16]. Basics about Raman are in [Blo89].
Suggestions on how to write a report. Optilux exercises.
Notes: Optilux and Matlab exercises are available at Elly.
Supplementary reading: Very useful suggestions about writing a scientific report can be found in [Whi04]. An excellent book on scientific English is [Wallwork].
Raman gain: pump-signal case. Notes on double Rayleigh back scattering. Forward vs. backward Raman amplification.
Cross-polarization modulation (XPolM). Geometric interpretation of XPolM. XPolM with walk-off. Numerical comparison of SPM, XPM and XPolM. Interleaved return to zero (iRZ).
Digital back-propagation (DBP). RP interpretation of DBP. Zero forcing properties of DBP. Impact of nonlinear signal noise interaction (NSNI) on DBP performance.
Notes: A detailed bibliography about XPolM and DBP can be found into the slides.
Supplementary reading: A book on Raman amplification is [AgrRaman]. Basics about Raman are in [Blo89].
Each lecture is 2 hours. Slides of the lectures are available at Moodle.
Introduction, presentation of the course, motivations. Brief history of optical communications.
Notes: The slides of the presentation can be found at Moodle.
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).
Notes: The principles of ray optics can be found in [Alb05] and [Saleh,Agrbase]. A rigorous proof of GVD can be found in [AgrNL,Agrbase].
Supplementary reading: The carrier and envelope delay theory can be found in ``Carrier and envelope delay'' in [Carlson].
Group velocity dispersion (GVD). GVD: examples. Rigorous proof of GVD using Maxwell's equations.
Notes: 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].
Linear Schrödinger equation. Unit of measure of electric field. Attenuation. Group delay. Dispersion length.
Notes: GVD is described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details can be found in [Saleh].
Impact of GVD over a Gaussian pulse. Anomalous and normal dispersion. Instantaneous frequency. Dispersion Management (DM). GVD in presence of signal's chirp.
Notes: GVD and GVD-chirp interaction are described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details about GVD can be found in [Saleh].
Third order dispersion. Eye opening. Eye closure penalty (ECP) in presence of GVD. Evaluation of ECP in presence of Gaussian pulses. Chen's formula for the GVD induced eye closure penalty.
Notes: details can be found in [Alb05].
Supplementary reading: Other details about third order dispersion can be found in [AgrNL]. Chen's formula was introduced in [Chen99] regarding polarization mode dispersion (PMD), but the idea still works for GVD.
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]. The Fourier transform induced by GVD is an approximation also called by the alternative names: stationary phase approximation, far field approximation, saddlepoint method, Fresnel diffraction.
Supplementary reading: Fresnel diffraction is discussed in [Saleh].
Erbium doped fiber amplifier (EDFA). Cross sections. Propagation equation for the photon flux over distance. Reservoir. Rate equations. State model interpretation of reservoir.
Notes: details can be found in [Alb05]. A detailed description of the reservoir is in [Alb98].
Supplementary reading: A discussion about the cross sections is in [Saleh]. The models of the EDFA are in [Sal90,Sun96].
Amplified spontaneous emission (ASE) noise. Noise figure of an EDFA: definition. Friis's formula. Excess noise figure.
Notes: details can be found in [Alb05]. For the noise figure definition see [Hau98].
Supplementary reading: For the zero-point energy and the Casimir effect take a look at Wikipedia.
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]. 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].
Poisson statistics. Poisson counting process. Shot noise. Campbell's theorem with proof. Power spectral density (PSD) of shot noise. Avalanche photo-diode (APD). 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.
Notes: details can be found in [Alb05]. Quantum limit is also discussed in [Agrbase].
Supplementary reading: other details about BER can be found in [Saleh].
Gaussian approximation with APD. Optimal multiplication factor with APD. Relation between Sensitivity penalty and Eye closure penalty for PIN and APD. ASE noise correlation function.
Notes: details can be found in [Alb05].
Supplementary reading: other details can be found in [Saleh]. 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. Average and variance of signal/spontaneous, spontaneous/spontaneous, shot, thermal noise. Optical signal to noise ratio (OSNR). Comparison signal/spontaneous, spontaneous/spontaneous.
Notes: details can be found in [Alb05].
Supplementary reading: general discussions can be found in [Agrbase]. Isserlis's theorem is well described in Wikipedia.
Marcuse's formula. Pre-amplified receivers: comparison with quantum limit. Exercises.
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].
Noise figure of optical amplifiers measured in the electrical domain. OSNR budget.
Nonlinear Schroedinger equation (NLSE). Reasons for the cubic nonlinear effect.
Notes: for the NLSE see [AgrNL]. For the reasons of the cubic nonlinearity see [Iizuka] at p. 523. Details about the noise figure measurement can be found in [Alb05]
Supplementary reading: A rigorous proof of NLSE can be found in [Men06,Men99]. A tutorial about NLSE is in [Agr11]. Further details about the noise figure measurement can be found in [Agrbase].
Proof of NLSE. Dimensional units of the electric field.
Self phase modulation (SPM). Comparison between temporal interpretation of SPM and frequency interpretation of GVD. SPM in frequency domain. SPM with sinusoidal power. Bandwidth enlargement induced by SPM.
Notes: SPM is well described in [AgrNL]. The physical interpretation of the chirp is described in [And92].
Supplementary reading: A rigorous proof of NLSE can be found in [Men06,Men99].
Wave breaking (WB). Impact of chirp induced by SPM and GVD over a Gaussian pulse.
Amplifier chains: limitations of ASE noise and nonlinear Kerr effect. Inhomogeneous amplifier chains.
Notes: Wave breaking is described in [And92]. Details about the noise figure measurement can be found in [Alb05]
Supplementary reading: Additional notes on WB are described in [AgrNL] and [And93]. Further details about the noise figure measurement can be found in [Agrbase].
Inhomogeneous amplifier chains. Lagrange multipliers method. Best amplifers gain in inhomogeneous chains.
Solitons. Proof of fundamental soliton.
Notes: For the inhomogeneous amplifier chains see [Mec98]. Details about solitons 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].
Proof of fundamental soliton. Higher order solitons. 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. 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].
Solitons in amplified systems: impact of losses. Numerical examples of soliton propagation: 3rd order soliton, dark soliton, soliton of non-integer order, interaction of solitons.
Wavelength division multiplexing (WDM) systems. Unique and separate fields in linear regime. Separate fields in nonlinear regime.
Notes: For cross-channel nonlinear effects see [AgrNL].
Supplementary reading: The numerical examples about solitons are taken from [AgrNL].
NLSE with separate fields. Cross-phase modulation (XPM) and four wave mixing (FWM). Intra- and inter-channel GVD. XPM with inter-channel GVD: probe/pump case. XPM filter for single fiber. Walk-off coefficient. Bandwidth of XPM filter.
Notes: For cross-channel effects see [AgrNL]. XPM without intra-channel GVD is discussed in [Kaz96].
Supplementary reading: XPM impact on ASE noise without intra-channel GVD is discussed in [Ho04].
XPM filter for multi-span systems in absence of intra-channel GVD. Numerical results. Example: hybrid OOK/DQPSK system.
Split-step Fourier method (SSFM). Formal solution using operators. Non commutative operators. SSFM with symmetrized and asymmetric step: accuracy.
Notes: for the XPM filter see [Bel98,BelVar98]. 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].
Choice of the SSFM step: constant step, step based on the nonlinear phase criterion, step based on the local error. Richardson extrapolation. Block diagrams of SSFM.
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].
The Matlab programming language.
Notes: A Matlab primer is [Sig93]. Free versions of Matlab are Octave and SciLab.
Supplementary reading: An advanced tutorial of Matlab is [Ack03]. A collection of general purpose Matlab functions can be found at Matlab-File-Exchange.
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 toolboxes of Matlab are available in Octave at Octave-Forge. Some useful code lines regarding saving the body of a Matlab file into a variable can be found in the forum of the Optical Communications course at Moodle, (search in the forum for 'thisfile').
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, both
for Windows and Linux. In Linux, the command is the following:
<svn checkout svn://svn.code.sf.net/p/optilux/code/trunk optilux-code>.
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. In any case, the Optilux repository can be found at
Sourceforge.
I suggest to compile the mex files inside Optilux (e.g.,
run the function comp_mex.m in the Optilux_files
directory). In this case, under Linux it may be useful the installation
of the package build-essential.
Software Optilux. Examples.
Notes: Optilux can be downloaded at OptiluX.
Supplementary reading: I suggest to keep update Optilux with Subversion.
Four wave mixing (FWM). Regular perturbation (RP) method to approximate the solution of the NLSE. FWM with CW signals. FWM efficiency. Phase matching coefficient.
Notes: FWM can be found in [AgrNL].
Supplementary reading: The RP method is discussed in [Van02].
Gaussian Noise (GN) model. Best power using the GN model. Application of the GN model: best SNR, scaling of SNR. Constrained Performance.
Notes: The Gaussian noise 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].
Constrained performance: scaling of nonlinear asymptote with the number of spans. Tolerance to errors and implications in optical link design.
Modulation instability (MI): linearized NLSE. Solution in absence of attenuation. Eigenvalues of MI.
Notes: The Gaussian noise model for the signal to noise ratio can be found in [Alb11]. The scaling laws of the SNR can be found in [Alb12]. Modulation instability is discussed in [AgrNL].
Supplementary reading: Further details on the GN model can be found in [Pog11] and [Car12].
Optical parametric amplifier (OPA). Bandwidth and frequency of maximum gain of an OPA. Notes on two pumps OPA.
Raman amplification. Motivations (distributed amplification). Memory induced by Raman effect. SPM, XPM and FWM in presence of Raman. Raman impact on XPM. Forward and backward Raman pumping.
Notes: Modulation instability is discussed in [AgrNL]. A good tutorial about OPA is [Han02].
Supplementary reading: Two pumps OPA is described in [MKi02]. A book on Raman amplification is [AgrRaman]. Basics about Raman are in [Blo89].
Raman gain: pump-signal case. 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: Lecture given by Prof. A. Bononi. The slides are available at Moodle.
Supplementary reading: A detailed bibliography can be found into the slides.
Differential phase shift keying (DPSK). Generation and detection of DPSK. Nonlinear phase noise. Differential quadrature phase shift keying (DQPSK). Generation of M-ary PSK. Coherent Detection: motivations. Historical background. Optical hybrid. Detection of in-phase and quadrature components. Polarization division multiplexing (PDM). Polarization diversity receiver.
Notes: Lecture given by Prof. A. Bononi. The slides are available at Moodle. A good tutorial of DPSK is [Win06].
Supplementary reading: A detailed bibliography can be found into the slides.
Coherent detection. Digital signal processing (DSP). Analog to digital conversion (ADC): choice of the number of samples per symbols. Electronic dispersion compensation of GVD. Brief discussion of polarization mode dispersion (PMD). 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 Moodle. Good tutorials on coherent detection are [Cha08,Kik10,Sav10].
Supplementary reading: A detailed bibliography can be found into the slides.
Each lecture is 2 hours. Slides of the lectures are available at LeA. 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 at 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.
Notes: The principles of ray optics can be found in [Alb05] and [Saleh,Agrbase]. A rigorous proof of GVD can be found in [AgrNL,Agrbase].
Supplementary reading: The carrier and envelope delay theory can be found in ``Carrier and envelope delay'' in [Carlson].
Group velocity dispersion (GVD). GVD: examples. Rigorous proof of GVD using Maxwell's equations.
Notes: 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.
Notes: GVD is described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details can be found in [Saleh].
Anomalous and normal dispersion. GVD in presence of signal's chirp. Instantaneous frequency. Dispersion Management (DM). GVD in presence of signal chirp. Matched filter interpretation of GVD with chirp. Third order dispersion. Eye closure penalty in presence of GVD.
Notes: GVD and GVD-chirp interaction are described in [Alb05] and [Agrbase,AgrNL].
Supplementary reading: Other details about GVD can be found in [Saleh]. 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].
Eye closure penalty in presence of GVD. Chen's formula for the GVD induced eye closure penalty. Fourier transform induced by strong 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].
Memory of GVD.
Erbium doped fiber amplifier (EDFA). Cross sections. Propagation equation for the photon flux over distance. Reservoir.
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].
Reservoir. State model interpretation of reservoir. Small signal gain. Gain saturation. Propagation equation with gain saturation. Fixed output power of an EDFA in saturation. Amplified spontaneous emission (ASE) noise. Noise figure of an EDFA: definition.
Notes: details can be found in [Alb05]. For the noise figure definition see [Hau98].
Supplementary reading: The models of the EDFA are in [Sal90,Sun96].
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.
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].
P-i-n junction. Junction capacity. Photo-diode bandwidth. Avalanche photo-diode (APD).
Poisson statistics. Poisson counting process. Shot noise. Campbell's theorem with proof. Power spectral density (PSD) of shot noise.
Notes: details can be found in [Alb05].
Supplementary reading: An alternative proof of Campbell's theorem can be found in [Saleh].
PSD with APD.
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.
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 and Personick's formula. Gaussian approximation with APD. Optimal multiplication factor with APD.
Notes: details can be found in [Alb05].
Supplementary reading: other details can be found in [Saleh].
Relation between Sensitivity penalty and Eye closure penalty for PIN and APD. 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 theorem. Average and variance of signal/spontaneous, spontaneous/spontaneous, shot, thermal noise. Optical signal to noise ratio (OSNR). Comparison signal/spontaneous, spontaneous/spontaneous.
Notes: details can be found in [Alb05].
Supplementary reading: general discussions can be found in [Agrbase]. Isserlis's theorem is well described in wikipedia.
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. Dimensional units of the electric field.
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].
Self phase modulation (SPM). Comparison between temporal interpretation of SPM and frequency interpretation of GVD. SPM in frequency domain. SPM with sinusoidal power. Bandwidth enlargement induced by SPM. Impact of chirp induced by SPM and GVD over a Gaussian pulse.
Notes: SPM is well described in [AgrNL]. The physical interpretation of the chirp is described in [And92].
Supplementary reading: Further information on the superposition of chirps can be found in [And93].
Wave breaking (WB). Impact of chirp induced by SPM and GVD over a Gaussian pulse.
Noise figure of optical amplifiers measured in the electrical domain. OSNR budget. Amplifier chains: limitations of ASE noise and nonlinear Kerr effect. Inhomogeneous amplifier chains.
Notes: Wave breaking is described in [And92]. Details about the noise figure measurement can be found in [Alb05]
Supplementary reading: Additional notes on WB are described in [AgrNL] and [And93]. Further details about the noise figure measurement can be found in [Agrbase].
Inhomogeneous amplifier chains. Lagrange multipliers method. Best amplifers gain in inhomogeneous chains.
Solitons. Proof of fundamental soliton.
Notes: For the inhomogeneous amplifier chains see [Mec98]. Details about solitons 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].
Proof of fundamental soliton. Higher order solitons. 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: 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. Notes on the impact of ASE noise on solitons: sliding filters.
Wavelength division multiplexing (WDM) systems. Unique and separate fields in linear regime.
Notes: For cross-channel nonlinear effects see [AgrNL].
Supplementary reading: The numerical examples about solitons are taken from [AgrNL].
NLSE with separate fields. Cross-phase modulation (XPM) and four wave mixing (FWM). Intra- and inter-channel GVD. XPM with inter-channel GVD: probe/pump case. XPM filter for single fiber. Walk-off coefficient. Bandwidth of XPM filter. XPM filter for multi-span systems in absence of intra-channel GVD.
Notes: For cross-channel effects see [AgrNL]. XPM without intra-channel GVD is discussed in [Kaz96].
Supplementary reading: XPM impact on ASE noise without intra-channel GVD is discussed in [Ho04].
Numerical results. Example: hybrid OOK/DQPSK system.
Split-step Fourier method (SSFM). Formal solution using operators. Non commutative operators. SSFM with symmetrized and asymmetric step: accuracy.
Notes: for the XPM filter see [Bel98,BelVar98]. 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].
Choice of the SSFM step: constant step, step based on the nonlinear phase criterion, step based on the local error. Richardson extrapolation. Block diagrams of SSFM.
The Matlab programming language.
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]. The slides about Matlab can be found at LeA. A Matlab primer is [Sig93].
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].
Free 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 toolboxes of Matlab are available in Octave at Octave-Forge.
The Matlab programming language.
Notes: The slides about Matlab can be found at LeA. A Matlab primer is [Sig93].
Supplementary reading: An advanced tutorial of Matlab is [Ack03]. A collection of general purpose Matlab functions can be found at 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 Communications 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, both
for Windows and Linux. In Linux, the command is the following:
<svn checkout svn://svn.code.sf.net/p/optilux/code/trunk optilux-code>.
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. In any case, the Optilux repository can be found at
Sourceforge.
I suggest to compile the mex files inside Optilux (e.g.,
run the function comp_mex.m in the Optilux_files
directory). In this case, under Linux it may be useful the installation
of the package build-essential.
Software Optilux. Examples. Unique and separate fields: numerical cost comparison.
Notes: Optilux can be downloaded at OptiluX.
Supplementary reading: I suggest to keep update Optilux with Subversion.
Four wave mixing (FWM). Regular perturbation (RP) method to approximate the solution of the NLSE. FWM with CW signals. FWM efficiency. Phase matching coefficient.
Notes: FWM can be found in [AgrNL].
Supplementary reading: The RP method is discussed in [Van02].
Gaussian Noise (GN) model. Best power using the GN model. Application of the GN model: best SNR, scaling of SNR.
Notes: 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].
Matlab exercises.
Notes: The exercises are available at LeA..
Supplementary reading: see the Matlab documentation for basic exercises.
Exercise: getting the entire SNR curve by two measurements. Constrained performance: scaling of nonlinear asymptote with the number of spans.
Modulation instability (MI): linearized NLSE.
Notes: The Gaussian noise model for the signal to noise ratio can be found in [Alb11]. The scaling laws of the SNR can be found in [Alb12]. Modulation instability is discussed in [AgrNL].
Supplementary reading: Further details on the GN model can be found in [Pog11] and [Car12].
Modulation instability: solution in absence of attenuation. Eigenvalues of MI.
Optical parametric amplifier (OPA). Two pumps OPA.
Raman amplification. Motivations (distributed amplification). Memory induced by Raman effect. SPM, XPM and FWM in presence of Raman. Raman impact on XPM.
Notes: Modulation instability is discussed in [AgrNL]. A good tutorial about OPA is [Han02]. The pump/signal model of Raman is discussed in [AgrNL].
Supplementary reading: Two pumps OPA is described in [MKi02]. A book on Raman amplification is [AgrRaman]. Basics about Raman are in [Blo89].
Raman amplification: pump-signal case. Brief notes about the amplified spontaneous Raman scattering and Rayleigh back scattering. Forward and backward Raman pumping.
Polarization of light. Birefringence. Jones formalism. Ellipse of polarization. Polarimeter. Stokes space. Poincaré sphere. Degree of polarization (DOP).
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].
Unitary matrices. Local behavior of birefringence. Hermitian matrices. Eigenvalues and eigenvectors of Hermitian matrices. 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.
Coherent Detection: motivations. Historical background. Optical hybrid. Detection of in-phase and quadrature components.
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.
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.
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.
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Introduzione, presentazione del corso. Breve storia delle comunicazioni ottiche. Ottica a raggi: postulati. Legge di Snell. Riflessione totale. Apertura numerica di una fibra ottica. Problematiche delle fibre multimodo.
Note: I principi dell'ottica a raggi si trovano in [Alb05]. Le slides della presentazione si trovano in LeA.
Approfondimenti: Un'interessante analisi sul futuro delle comunicazioni ottiche è in [Des06]. Una introduzione alla storia delle comunicazioni ottiche è in [Agrbase]. Un'analisi approfondita sui fondamenti delle comunicazioni ottiche con uno sguardo alla capacità di canale è in [Ess10]. Una lista delle maggiori aziende mondiali nel campo dell'ottica può essere trovata in [Ecocex]. Una lista delle maggiori aziende italiane nel campo dell'ottica si può trovare in [ListITA]. Approfondimenti sull'ottica a raggi sono in [Saleh] nel capitolo ``Ray Optics''.
Richiami sulle fibre singolo modo. V number. Dipendenza del V number dalla lunghezza d'onda. Dipendenza del profilo di campo elettrico dal V number. Approccio sistemistico alla fibra ottica. Costante di propagazione beta. Dispersione di Materiale e di guida d'onda. Fibre DSF e DCF. Dispersione di velocità di gruppo (GVD).
Equazioni di Maxwell. Dimostrazione rigorosa della GVD.
Note: Richiami sulle fibre ottiche si trovano in [Alb05] ed in [Agrbase]. La dimostrazione rigorosa della GVD si trova in [AgrNL,Agrbase].
Approfondimenti: Per le equazioni di Maxwell si veda [Saleh]. Una dimostrazione ancor più rigorosa della GVD (e della non linearità) tramite approccio a scale multiple si trova in [Men99,Men06]. Per una lista delle principali fibre ottiche e delle relative caratteristiche si veda [Cisco]. Per la definizione di ritardo di gruppo e di fase si veda ``Carrier and envelope delay'' in [Carlson].
Attenuazione. Ritardo di gruppo. Impatto della GVD su di un impulso Gaussiano. Lunghezza di dispersione. Dispersione anomala e normale. Impatto di un chirp sulla GVD. Interpretazione del miglior chirp tramite il principio di Heisenberg. GVD come matched filter. Frequenza istantanea di GVD. Dispersione di terzo ordine.
Note: per la GVD si veda [Alb05] ed [Agrbase,AgrNL].
Approfondimenti: Per l'interpretazione del matched filter in frequenza si veda [Proakis]. Per la definizione della varianza di un segnale ed il principio di Heisenberg si veda l'appendice di [Saleh].
Eye closure penalty (ECP) indotta dalla GVD. Formula di Chen. Trasformata di Fourier indotta dalla GVD dopo lunga distanza. Sequenze de Bruijn. Memoria della GVD.
Note: i dettagli si trovano in [Alb05].
Approfondimenti: La formula di Chen è stata introdotta da Chen nel contesto della dispersione modale di polarizzazione in [Chen99], ma l'idea di principio rimane anche per la GVD. Sulla trasformata di Fourier indotta dalla GVD è interessante osservare l'analogia con il fenomeno della diffrazione di Fresnel [Saleh].
Amplificatore ottico. Cross sections di assorbimento ed emissione. Equazione di propagazione del flusso di fotoni in z. Equazione di bilancio (rate equation). Reservoir. Visione sistemistica del reservoir. Guadagno ai piccoli segnali. Saturazione del guadagno. Rumore di emissione spontanea (ASE). Densità spettrale di potenza dell'ASE.
Note: i dettagli si trovano in [Alb05]. Il modello del reservoir si trova in [Alb98].
Approfondimenti: Una discussione sulle cross sections si trova in [Saleh]. I modelli dell'amplificatore sono discussi in [Sal90,Sun96].
Cifra di rumore di un amplificatore ottico. Formula di Friis. Amplificazione dual stage: calcolo della cifra di rumore.
Fotoricevitori: fotodiodo. Efficienza quantica. Responsivity. Ragioni della foto-corrente: cariche generate nella regione di svuotamento. Giunzione P-i-n. Capacità parassite del fotodiodo e banda. Fotodiodo a valanga (APD).
Note: i dettagli si trovano in [Alb05]. Per la definizione della cifra di rumore si legga [Hau98]. La proprietà che un fotone contribuisce con una sola carica alla corrente è in [Saleh] nella sezione 17.1 ``Properties of semiconductor photodetectors''.
Approfondimenti: una discussione generale sui fotodiodi si trova in [Alexander] e in [Agrbase].
Statistiche di Poisson. Processo di conteggio di Poisson. Shot noise. Teorema di Campbell e dimostrazione. Densità spettrale di potenza dello shot noise. Caso APD. Ricevitori ottici. Front end elettronici a bassa impedenza, alta impedenza, transimpedenza. Calcolo dell probabilità di errore (BER) in un sistema ottico on-off keying (OOK). Quantum limit. Potenza di sensitivity. Impatto del rumore termico sulla BER.
Note: i dettagli si trovano in [Alb05]. Il quantum limit è discusso anche in [Agrbase].
Approfondimenti: Una dimostrazione alternativa del teorema di Campbell si trova in [Saleh].
Approssimazione gaussiana della BER. Rumore termico. Approssimazione gaussiana con fotodiodi APD. Valore ottimo della moltiplicazione a valanga in APD. Power budget. Legame Sensitivity penalty e Eye closure penalty (ECP). Caso con GVD.
Note: i dettagli si trovano in [Alb05].
Approfondimenti: discussioni generali sul power budget sono in [Saleh].
Esercizio sul calcolo del chirp dalla sensitivity penalty. Ricevitori pre-amplificati. Rumore di battimento segnale-spontaneo, spontaneo-spontaneo. Approssimazione gaussiana del battimento spontaneo-spontaneo. Formulad di Isserlis. Calcolo della BER con formula di Personick. Confronto varianza shot-noise, rumore termico, battimento segnale-spontaneo.
Note: i dettagli si trovano in [Alb05].
Approfondimenti: La formula di Isserlis è ben descritta in wikipedia. Per la rappresentazione di Rice di un processo passabanda si veda [Papoulis,Carlson].
Confronto rumore segnale-spontaneo, spontaneo-spontaneo. Formula di Marcuse. Ricevitori Pre-amplificati: confronto con quantum limit. Esercizi.
Metodo di Bergano per la misura del Q-factor.
Catene di amplificatori: misura della cifra di rumore. OSNR budget. Amplificazione distribuita.
Note: i dettagli si trovano in [Alb05].
Approfondimenti: per la misura della cifra di rumore si veda [Agrbase]. Metodo più corretti per calcolare la probabilità di errore senza l'ipotesi di rumore Gaussiano si trovano in [Hum91,For00].
Equazione non lineare di Schroedinger (NLSE). Ragioni della non linearità cubica. Self Phase Modulation (SPM). Confronto tra la visione nel tempo del SPM e la visione in frequenza della GVD.
Note: Per la equazione non lineare di Schroedinger si veda [AgrNL]. Per le ragioni della non linearità cubica si veda [Iizuka] a p. 523.
Approfondimenti: La dimostrazione rigorosa della NLSE si trova in [Men06,Men99]. Un interessante tutorial sulla NLSE è in [Agr11].
SPM: caso di segnale sinusoidale. Allargamento di banda indotto dal SPM. Wave breaking (WB) . Impatto del SPM e della GVD sul chirp di un impulso Gaussiano.
Catene di amplificatori: limitazioni imposte dalla non linearità e dal rumore ASE. Catene disomogenee. Metodo dei moltiplicatori di Lagrange.
Note: per il SPM e WB si veda [AgrNL]. Per l'analisi del chirp si veda [And92]. Per le catene di amplificatori si veda [Alb05].
Approfondimenti: Ulteriori approfondimenti sul WB sono in [And93].
Solitoni. Dimostrazione del solitone fondamentale. Proprietà di scalatura dei solitoni. Solitoni di ordine superiore. Perturbazione dei solitoni: energia del continuo.
Dark solitons (cenni). Esempi numerici di propagazione solitonica: solitone di ordine 3, dark soliton, effetto del chirp, effetto della presenza di solitoni vicini.
Note: Per la dimostrazione del solitone si veda [AgrNL].
Approfondimenti: Ulteriori approfondimenti sui solitoni sono in [Agr11]. Un buon libro della biblioteca Unipr dedicato ai solitoni è [Hasegawa].
Solitoni in sistemi amplificati. effetto del rumore ASE sui solitoni: sliding filters. Solitoni: da unità adimensionali a unità dimensionali.
Sistemi wavelength division multiplexing (WDM). NLSE a campi separati. Cross-phase modulation (XPM) e four wave mixing (FWM). GVD intra-canale e inter-canale. XPM in assenza di GVD intra-canale: soluzione in un sistema pompa/segnale.
Note: Per gli effetti cross-canale si veda [AgrNL]. Il XPM senza GVD intra-canale è trattato in [Kaz96].
Approfondimenti: l'impatto del XPM sul rumore ASE in assenza di GVD intra-canale è trattato in [Ho04] con un modello simile nel principio a quello visto a lezione.
Filtro di XPM per singola fibra. Coefficiente di walk-off. Banda 3dB del filtro di XPM.
Filtro di XPM per sistemi multi-span in assenza di GVD intra-canale. Filtro di XPM in presenza di GVD intra-canale. Modello ai piccoli segnali della GVD. Risultati numerici di filtri di XPM. Esempio di applicazione in sistema ibrido OOK/DQPSK.
Note: Per il filtro di XPM si veda [Bel98,BelVar98]. Il modello ai piccoli segnali della GVD è stato introdotto per la prima volta in [Pet92]. Le slides della lezione sono disponibili in LeA.
Approfondimenti: Una derivazione alternativa del modello di XPM con GVD intra-canale si trova in [Car99]. L'analisi dell'impatto del XPM in sistemi ibridi OOK/DQPSK è svolta in [Alb09].
Algoritmo di split-step Fourier method (SSFM). Soluzione formale con operatori. Non commutatività degli operatori. SSFM asimmetrico e simmetrico. Scelta del passo: passo costante, metodo della fase non lineare, metodo dell'errore locale stimato. Estrapolazione di Richardson.
Note: Una introduzione sullo SSFM è in [AgrNL]. La scelta del passo è analizzata in [Sin03]. Il metodo dell'errore locale come spiegato a lezione si trova in [Feldman].
Approfondimenti: Estensioni del metodo della fase non lineare includendo la GVD si trovano in [Zha08]. Un'analisi delle risonanze spurie create dal metodo del passo costante si trova in [Bos00].
Linguaggio Matlab. Linguaggio di programmazione OptiluX.
Note:
Le slides di introduzione sul linguaggio Matlab si trovano
in LeA. Optilux si può scaricare
all'indirizzo OptiluX.
In Linux, per scaricare Optilux con il software Subversion
digitare il comando
<svn co https://optilux.svn.sourceforge.net/svnroot/optilux>.
Per compilare i mex files di Matlab (e.g., usare la funzione
comp_mex.m di Optilux) può essere utile, in sistemi Linux,
installare il pacchetto build-essential. Un esempio di modello
discreto di un segnale PAM si trova in LeA.
Approfondimenti: Versioni open-source di Matlab sono Octave e SciLab. Nel caso di Octave occorre installare anche un programma di grafica, e.g., gnuplot (in wikipedia alla voce Octave c'è una lista esaustiva di alternative). I principali toolbox di Matlab sono presenti separatamente in Octave nel pacchetto Octave-Forge. Un primer di base su Matlab si trova in [Sig93]. Un tutorial avanzato di Matlab si trova in [Ack03]. Una collezione di programmi Matlab di utilità generale si trova in Matlab-File-Exchange.
Software Optilux: ulteriori esempi. Note su come si scrive un articolo scientifico.
Note: Per Optilux si guardi la documentazione allegata al pacchetto OptiluX.
Approfondimenti: Un ottimo riferimento su come si scrive un articolo scientifico si trova in [Whi04]. Suggerisco anche di leggere [Abd08]. Alcuni commenti sulle simulazioni Matlab si trovano nel forum di Comunicazioni Ottiche in LeA alla data 9 maggio 2012.
Esercizio: confronto costo propagazione a campi separati e a campo unico.
Four wave mixing (FWM). Analisi perturbativa della NLSE. Campo di FWM con segnali CW. Efficienza di FWM. Coefficiente di phase matching.
Analisi della distorsione non lineare come rumore distribuito gaussiano. Potenza di soglia.
Note: Il FWM si trova in [AgrNL]. Il modello distribuito gaussiano si trova in [Alb11].
Approfondimenti: Ulteriori dettagli sul modello gaussiano si trovano in [Pog11] e in [Car12].
Ulteriori proprietà dell'SNR non lineare.
Modulation instability. Optical parametric amplifier (OPA). Banda e frequenza di massimo guadagno di un OPA. Rumore negli OPA. Cenni sugli OPA a due pompe.
Note: Per la modulation instability si veda [AgrNL]. Un buon tutorial sugli OPA è [Han02]. Per il rumore degli OPA si veda [Kyl04].
Approfondimenti: L'OPA a due pompe è descritto in [MKi02].
Polarizzazione della luce. Formalismo di Jones e di Stokes. Polarimetro. Sfera di Poincarè. Grado di polarizzazione (DOP). Ellissi di polarizzazione.
Note: Un ottimo libero sulla polarizzazione è [Damask]. Un tutorial si trova in [Gor00]. Il programma per esaminare lo stato di polarizzazione di un segnale si trova in LeA.
Approfondimenti: Approfondimenti si trovano sempre in [Gor00]. Il moto nello spazio di Stokes è ben descritto in [Fri86].
Equazione di moto della polarizzazione lungo la distanza. Fibre polarization maintaning fiber (PMF). Matrici hermitiane. Matrici unitarie. Equazione di moto nello spazio di Stokes. Formalismo di Pauli. Matrice di Mueller.
Polarization mode dispersion (PMD): PMD al primo ordine. Stati principali di polarizzazione (PSP). Corrente ricevuta con PMD al primo ordine.
Note: Un ottimo libero sulla polarizzazione è [Damask]. Un tutorial si trova in [Gor00].
Approfondimenti: Approfondimenti si trovano sempre in [Gor00].
Amplificazione Raman. Memoria introdotta dall'effetto Raman. SPM, XPM e FWM in presenza di Raman. XPM risonante Raman. Amplificazione Raman nel caso pompa-segnale.
Formati di modulazione avanzati: motivazioni. Modulatore di fase e Mach Zehnder (MZ). Impulsi return to zero (RZ) e varianti (carrier-suppressed (CS-RZ), chirped-RZ (CRZ), alternate phase-RZ (APRZ)). Duobinario. Differential phase shift keying (DPSK). Generazione e ricezione di un segnale DPSK. Rumore di fase non lineare. Differential quadrature phase shift keying (DQPSK). Generazione di M-ary PSK.
Note: Il modello pompa/segnale del Raman si trova in [AgrNL]. Le slides della presentazione si trovano in LeA. Un buon tutorial sulla DPSK è [Win06].
Approfondimenti: Le slides della presentazione contengono diversi riferimenti bibliografici sugli specifici argomenti.
Ricezione coerente. Motivazioni. Introduzione storica. Optical hybrid. Recupero della parte in fase e quadratura di un segnale passabanda. Polarization division multiplexing (PDM). Polarization diversity receiver. Digital signal processing (DSP). Analog to digital conversion ADC: scelta del numero di campioni per simbolo. Compensazione elettronica della GVD. Compensazione elettronica della PMD: constant modulus algorithm (CMA). Stima di fase: algoritmo di Viterbi & Viterbi. Risultati numerici e sperimentali. PMD in regime non lineare. Cross polarization modulation (XpolM). Nonlinear threshold (NLT) di link ottici. Algoritmo di digital back-propagation (DBP). Polarization switched quadrature phase shift keying (PS-QPSK).
Note: Le slides della presentazione si trovano in LeA. Ottimi tutorial sulla ricezione coerente sono [Cha08,Kik10,Sav10].
Approfondimenti: Le slides della presentazione contengono diversi riferimenti bibliografici sugli specifici argomenti.
Paolo 2019-09-10