Optical Communications from a Fourier Perspective

Fourier Theory and Optical Fiber Devices and Systems

1st Edition - November 17, 2023
Authors: Palle Jeppesen, Bjarne Tromborg
Language: English
Paperback ISBN:
9 7 8 - 0 - 4 4 3 - 2 3 8 0 0 - 0
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 2 3 8 0 1 - 7

Optical Communications from a Fourier Perspective: Fourier Theory and Optical Fiber Devices and Systems covers Fourier theory and signal analysis over photonic component… Read more

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Optical Communications from a Fourier Perspective: Fourier Theory and Optical Fiber Devices and Systems covers Fourier theory and signal analysis over photonic components, including time lenses in optical communication. Sections cover wave propagation in optical waveguides based on Maxwell equations and the nonlinear Schrödinger equation. Optical Fourier transform in the form of time lens is covered, for example in modulation format conversion and spectrum magnification, and couplers and their use for optical discrete Fourier transformation are also discussed. Other important subjects are discussed such as shot noise, thermal noise and also the basics of four wave mixing in relation to time lenses. Detailed derivations and a deeper background for the chapters are provided in appendices where appropriate. Some of the theory is more generally applicable beyond optical communication and is of relevance also for communications engineering. The Fourier theory dimension of the book presents the relationship between Fourier series and Fourier integrals and also the related Laplace transform.

Cover image
Title page
Table of Contents
Copyright
Preface
Acknowledgments
Bibliography
Chapter 1: The Dirac delta function and Heaviside step function
Abstract
1.1. Definition of the Dirac delta function
1.2. Dirichlet integrals
1.3. Properties of the delta function
1.4. The Heaviside step function
1.5. Summary
Bibliography
Chapter 2: Fourier series, Parseval's theorem, FFT and Cooley–Tukey algorithm
Abstract
2.1. Fourier series
2.2. Train of rectangular pulses
2.3. Train of delta pulses and the comb function
2.4. Parseval's power theorem for periodic signals
2.5. The discrete Fourier transform
2.6. FFT based on Cooley–Tukey algorithm
2.7. Summary
Bibliography
Chapter 3: Fourier integrals and series
Abstract
3.1. The Fourier transformation and the Fourier integral theorem
3.2. From Fourier series to integrals
3.3. From Fourier integrals to series
3.4. Notation
3.5. Summary
Chapter 4: Properties of the Fourier transform and Heaviside's step function
Abstract
4.1. Important properties
4.2. Important examples of Fourier transforms
4.3. Relation between pulse and spectral widths
4.4. Fourier transform of the Heaviside step function
4.5. Integration theorem and other examples involving the step function
4.6. Sampling theorem for aperiodic functions
4.7. Numerical calculation of the Fourier transform
4.8. Numerical calculation of the inverse Fourier transform
4.9. Summary
Bibliography
Chapter 5: Complex signal, complex envelope, and Hilbert transform
Abstract
5.1. Complex signal
5.2. Complex envelope of a bandpass signal
5.3. IQ modulation and demodulation
5.4. The complex signal and Hilbert transform
5.5. Summary
Bibliography
Chapter 6: Correlation functions, spectral density, Wiener–Khinchine theorem
Abstract
6.1. Correlation functions and Wiener–Khinchine theorem for deterministic energy signals
6.2. Correlation functions and Wiener–Khinchine theorem for deterministic power signals
6.3. Random processes, ensemble and probability density averages, stationary and ergodic processes
6.4. Autocorrelation function and Wiener–Khinchine theorem for wide-sense stationary random process, effective bandwidth, and noise considerations
6.5. Shot noise
6.6. Thermal noise
6.7. Summary
Bibliography
Chapter 7: Linear, time-invariant systems
Abstract
7.1. Impulse response and transfer functions
7.2. Linear electrical circuits
7.3. Convolution and RMS widths
7.4. Summary
Bibliography
Chapter 8: Transfer matrices and frequency filters
Abstract
8.1. Transfer functions, transfer matrices, and quadrature filter
8.2. Frequency filters
8.3. Summary
Chapter 9: Laplace transforms, transfer functions, Nyquist criterion
Abstract
9.1. The Laplace transform and its inverse
9.2. Important properties of Laplace transform
9.3. Important examples of Laplace transforms
9.4. Transfer functions for passive and active LTI systems
9.5. Laplace-method applied on series resonant circuit
9.6. Transimpedance front end
9.7. Nyquist criterion
9.8. Summary
Bibliography
Chapter 10: Maxwell's equations, optical waveguides, and Poynting vector
Abstract
10.1. Maxwell's equations and Helmholtz equation
10.2. Wave propagation in planar optical waveguide
10.3. Wave propagation in a step-index optical fiber
10.4. Wave propagation in a graded-index optical fiber
10.5. Derivation of the wave equation for both rectangular and circular waveguide
10.6. Poynting vector and photodetection
10.7. Summary
Bibliography
Chapter 11: Pulse propagation in optical fibers
Abstract
11.1. Fundamental equations for a general pulse shape
11.2. Introduction of a new time reference
11.3. Fiber impulse response
11.4. Introduction of a Gaussian input pulse
11.5. Propagation of a Gaussian pulse
11.6. Material and waveguide dispersion
11.7. Pulse propagation in a dispersion compensating fiber
11.8. Summary
Bibliography
Chapter 12: Split-step Fourier method and nonlinear Schrödinger equation
Abstract
12.1. The nonlinear Schrödinger equation
12.2. Operators for dispersion and nonlinearity
12.3. Split-step Fourier method
12.4. Fundamental soliton solution to the NLSE
12.5. Summary
Bibliography
Chapter 13: Introduction to modulation formats
Abstract
13.1. Intensity modulation and direct detection
13.2. ASK, PSK, and FSK binary modulation
13.3. Multilevel modulation formats
13.4. Differential binary and quadrature PSK
13.5. Short introduction to PDM and WDM
13.6. Summary
Bibliography
Chapter 14: Required bandwidth for heterodyne and homodyne detection
Abstract
14.1. Baseband signal
14.2. Modulated transmitter
14.3. Heterodyne detection
14.4. Homodyne detection
14.5. Summary
Bibliography
Chapter 15: Bandpass noise
Abstract
15.1. Introduction
15.2. Gaussian bandpass noise
15.3. Formulas for autocorrelation functions and power spectral densities
15.4. Autocorrelation and cross-correlation functions of quadrature components
15.5. Power spectral density functions of quadrature components
15.6. Proof of statistical independence of quadrature components
15.7. Bandlimited white noise
15.8. Summary
Bibliography
Chapter 16: Bit error rate
Abstract
16.1. Receiver noise in direct ASK detection
16.2. Bit error rate for ASK direct detection
16.3. ASK direct detection and the quantum limit
16.4. Synchronous ASK heterodyne detection
16.5. Synchronous ASK homodyne detection
16.6. Synchronous BPSK heterodyne detection
16.7. Synchronous BPSK homodyne detection
16.8. Summary
Bibliography
Chapter 17: Pulse shaping using optical Fourier transform
Abstract
17.1. Output pulse shaping using Cd configuration
17.2. Output spectrum shaping using a dC configuration
17.3. OTDM-to-WDM conversion using a dC configuration
17.4. OTDM-to-WDM conversion in WDM-PON
17.5. WDM-to-OTDM conversion using a dC conversion
17.6. WDM-to-OTDM conversion using a complete dCd configuration
17.7. WDM-to-OTDM conversion using a Cd configuration
17.8. WDM-to-OTDM conversion using a complete CdC configuration
17.9. Optical Nyquist transmission using time-domain optical Fourier transformation
17.10. Summary
Bibliography
Chapter 18: Spectrum magnification
Abstract
18.1. Gaussian input pulse
18.2. Rectangular input pulse
18.3. Spectrum magnification of WDM signals
18.4. Summary
Bibliography
Chapter 19: Optical Fourier transformation, dispersion compensation, jitter suppression
Abstract
19.1. Dispersion compensation for general input pulse shape
19.2. Time lens used for jitter suppression in direct detection receiver
19.3. Trends and opportunities for time lenses
19.4. Summary
Bibliography
Chapter 20: Regeneration of WDM phase-modulated signals
Abstract
20.1. FWM basics
20.2. Nonlinear equations for nonstatic fields
20.3. Degenerate FWM
20.4. Degenerate signal and idler. Phase sensitive amplifier
20.5. Time lenses and WDM phase regeneration for BPSK modulation
20.6. Summary
Bibliography
Chapter 21: Time–space duality, dispersion and diffraction, time lens
Abstract
21.1. Gaussian pulse propagating in a dispersive fiber
21.2. Introduction to the Gaussian beam
21.3. Relations between dispersion of a narrow pulse and diffraction of a narrow beam
21.4. Comparison with thin optical cell with quadratic index variation
21.5. Comparison with thin optical lens
21.6. Summary
Bibliography
Chapter 22: Couplers and their use for optical DFT
Abstract
22.1. Scattering matrices
22.2. Optical DFT using an N×N coupler
22.3. Summary
Bibliography
Chapter 23: Multicarrier modulation, OFDM, DFT, Nyquist modulation
Abstract
23.1. Multicarrier modulation and OFDM
23.2. OFDM using IDFT and DFT
23.3. Nyquist modulation format
23.4. Conversion of DWDM signals to Nyquist channel based on complete CdC optical Fourier transformation
23.5. OFDM-to-Nyquist-WDM conversion based on complete CdC optical Fourier transformation
23.6. Summary
Bibliography
Chapter 24: Optical orthogonal frequency division modulation
Abstract
24.1. “On-the-fly” inverse Fourier transformation
24.2. Transfer function of combined serial-to-parallel converter and OIFT circuit
24.3. OIFT based on a circuit synthesized of delay interferometers
24.4. Transmission system based on OIFT circuits
24.5. Summary
Bibliography
Appendix A: Acronyms
Appendix B: Analytic complex functions, Cauchy's integral theorem, and integral formula
B.1. Simple application of the Cauchy integral theorem
B.2. Application of Cauchy integral theorem and integral formula
B.3. The principal value integral of an analytic function
B.4. Analyticity and causality
B.5. Example of inverse Fourier of H(ω)
B.6. Cauchy's argument principle
Bibliography
Appendix C: Fourier series expansion of the comb function
Appendix D: Calculation of the integral of a Gaussian function
Appendix E: Time–bandwidth product
Appendix F: Examples of MATLAB® programs
F.1. The Fourier transforms in Fig. 4.8
F.2. Fourier transforms using window functions
F.3. Solution of the nonlinear Schrödinger equation (NLSE) by the split step Fourier method (SSFM)
Appendix G: Proof of autocorrelation going to zero for large values of τ
Appendix H: Autocorrelation of digital signal
Appendix I: Ensemble averaging and pdf averaging
Appendix J: Probability density functions and jointly Gaussian random variables
J.1. Probability density functions
J.2. Jointly Gaussian random variables
Bibliography
Appendix K: Poisson statistics
K.1. Poisson statistics derived from differential equations
K.2. Poisson statistics derived from binomial distribution
Appendix L: RMS widths of convoluted signals
Appendix M: Example of Fourier–Mellin integral determined by Cauchy integral theorem
Appendix N: Proof of Thévenin's and Norton's theorems
Appendix O: Propagation of a Gaussian pulse in a single-mode fiber
O.1. Introduction of new time reference
O.2. Ae(z,t) for Gaussian input
O.3. Pulse width of output power
O.4. Minimum pulse width
Appendix P: Carrier envelope offset
Bibliography
Appendix Q: Bandwidth of random ASK and PSK signals
Q.1. Binary ASK
Q.2. Binary PSK
Q.3. Summary
Bibliography
Appendix R: Solution of coupled wave equations
Appendix S: Matched filter
Bibliography
Bibliography
Bibliography
Index

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Optical Communications from a Fourier Perspective

Fourier Theory and Optical Fiber Devices and Systems

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Palle Jeppesen

Bjarne Tromborg