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Book Review — Optical Communications From a Fourier Perspective


by Palle Jeppesen and Bjarne Tromborg, Elsevier 2023

Reviewed by Klaus Petermann, Technische Universität Berlin, Germany

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There exists already a large variety of books on optical communications so that one might ask what might be the role of a new book on a similar subject. The present book by Palle Jeppesen and Bjarne Tromborg is indeed quite different and may be considered to be complementary to the existing literature.

The leitmotif of the present book is the Fourier transform, which indeed represents an important basis for optical communications and beyond and also for communications in a wider sense. The first half of the book presents a very thorough analysis of the Fourier transform whereas the second half is related to applications in optical communications with a special emphasis on all-optical signal processing.

The analysis starts with basic discussions on the delta- and step-functions followed by the discrete Fourier transform and the FFT (fast Fourier transform) algorithm. The Fourier transform as well as the Laplace transform are discussed in detail including discussions on correlation functions on deterministic as well as on stochastic signals. The appendix of the book also contains a detailed discussion of the probability density function and its Fourier transform, the characteristic function.

The pulse response as well as the transfer function and transfer matrices are given for very general linear networks including a short discussion on filters where in particular the Butterworth filter is discussed.

The discussion on the Fourier transform is very general and by no means restricted to optical communications. Therefore, the very detailed analysis in this book may also be valuable for anyone even outside of the field of optical communications who is interested in a deeper understanding of the Fourier transform.

With respect to optical communications the book contains a short discussion on the basic relations for the wave propagation in optical fibers but a detailed discussion of the specific propagating modes is outside the scope of this book.  On the other hand, the pulse propagation in single-mode fibers with their chromatic dispersion is discussed in detail. In particular, chirped Gaussian pulses are considered. Even though the optical fiber is mostly treated to be linear, the nonlinear propagation including the Kerr effect via the nonlinear Schrödinger equation is also shortly discussed along with introducing the split-step Fourier method for its solution.

The book also contains a discussion on modulation formats for optical communications with some emphasis for phase modulated formats. Both heterodyne and homodyne receivers are discussed including a discussion on their receiver sensitivity for ideal receivers.

An important focus of the book is related to all-optical signal processing. In this respect, the concept of a time lens is discussed in detail. A time lens consists of an optical element, e.g. a fiber, with chromatic dispersion and a subsequent phase modulation applying a linear chirp to the signal, or vice versa. This is equivalent to the generation of an image, considering the propagation of a beam in free space with a subsequent classical lens as outlined in Chapter 21 of the book. Like in imaging, where a lens may provide a spatial Fourier transform, a time lens may also provide a Fourier transform of the optical signal.

Several applications for a time lens are discussed in the book, e.g. the conversion of an OTDM (optical time domain multiplex) signal to a WDM (wavelength division multiplex) signal, which may be of interest for example for PON (passive optical network) distribution networks as outlined in Chapter 17. Other applications concern the spectrum magnification (like the magnification in imaging with classical lenses) and the optical Fourier transform as also discussed in the book.

Even though the time lens represents a very elegant way for all-optical signal processing, the required linear chirp may be very challenging since many applications require a chirp over a wide frequency range, which require a phase modulator with an extremely large bandwidth. Therefore, an alternative consists of generating a suitably chirped pulse, which is then transferred to the signal by four-wave mixing, which is also explained in the book.

The final chapters of the book deal with the dualism between OFDM (orthogonal frequency division multiplex) and Nyquist WDM. It is nicely described how a time lens can again be used for the conversion between these modulation formats. The optical Fourier transform may also be realized by an NxN coupler with suitable delay lines, which can be successfully used for the detection of OFDM signals with extremely high bit rates, as explained in the final chapter.

This book covers a vast amount of material and it impresses by a very thorough analysis with extensive appendices. In order to better appreciate the methodology of the book it may be helpful if the reader has already some basic ideas of the Fourier transform for the first part and optical communications for the second part of the book. The book is thus very well suited for an advanced undergraduate student with respect to the first part of the book and for a graduate student with respect to the second part of the book. It is also very helpful that each chapter is concluded with a short summary at its end.

In summary, the present book represents a valuable extension to the existing literature providing the reader with a very deep knowledge for better understanding advanced optical communication systems.