Preface

Ray, wave and quantum concepts are central to diverse and seemingly incompatible models of light. Each model particularizes a specific “manifestation” of light, and then corresponds to adequate physical assumptions and formal approximations, whose domain of applicability is well established. Accordingly each model comprises its own set of geometrical and dynamical postulates with the pertinent mathematical means.

Geometrical optics models the light field as an aggregate of incoherent light rays, naïvely perceived as the trace of the motion of the “luminous corpuscles”, which, emitted by the source, move through space in obedience to the usual laws of mechanics. It treats light rays as lines in 3-space dimensions and is accordingly concerned with the dynamical laws establishing how the rays bend when propagating in inhomogeneous media described by the refractive index function. Geometrical optics is not suited to explain interference, diffraction and quantum coherence effects, but, in contrast, it provides a particularly convenient means for the design of optical systems, which is based on the purely geometrical rules of ray tracing.

Geometrical optics has developed its own mathematical framework, which can remarkably be brought into correspondence with that of the Hamiltonian mechanics of point-particles, with “time” corresponding to the arc-length along the ray path and the mechanical “potential” to the refractive index of the optical medium.

Wave optics accounts for the wave characteristics of light. Originating directly from the classical electromagnetic theory, it shares with this theory the same system of theoretical principles and methods, which can notably be put in correspondence with those proper to relativistic quantum mechanics. Then, the geometry of light rays is replaced by the geometry of “luminous” waves, whose propagation is geometrically pictured as the transfer of the interference shaped vibrations from one portion of the medium to the contiguous one.

Wave optics treats the light waves as complex functions of position in 3-space dimensions and is accordingly concerned with the dynamical laws establishing how the wave function changes as the optical wave propagates through inhomogeneous media.

Quantum optics recovers the grainy view of geometrical optics, picturing the light ray as a stream of particle-like entities, the photons. Whereas geometrical optics deals with the influence at a macroscopic level of the medium on the trajectory of the photon streams, quantum optics is typically concerned with the wave-like question relevant to the coherence properties of the photon beams and to the relevance of those properties on the interaction of light with matter, which can correspondingly be treated quantum mechanically. Coherent and squeezed states of light are the building concepts of quantum optics.

Wigner optics bridges between ray and wave optics. It offers the optical phase space as the ambience and the Wigner function based technique as the mathematical machinery to accommodate between the two opposite extremes of light representation: the localized ray of geometrical optics and the unlocalized wave function of wave optics.

Notably quantum optics finds a convenient formulation in the proper phase space with the consequent geometrical view of coherent and squeezed states as circles and ellipses. The Wigner function methods can suitably be applied to quantum optics as well to enable effective analytical means for calculating expectation values and transition probabilities for the aforementioned states.

The purpose of the book is to introduce the reader to the optical phase-space and to the approaches to optics based on the Wigner distribution function, that have been developed over the past 25 years or so in several scientific titles. These yield the formal context, where concepts and methods of both ray and wave optics coalesce into a unifying formalism. In this respect, emphasis is given to the Lie algebra representation of optical systems and accordingly to the Lie algebra view of light propagation through optical systems.

The book is made as self-contained as possible. Chapter 1 presents the Hamiltonian equations of motion, which are basic to the development of both the transfer matrix formalism, appropriate to paraxial ray optics (Chapters 2 and 3), and the transfer operator formalism, suited to paraxial wave optics (Chapters 4 and 5). The relation of both formalisms to the Lie algebra methods is gently displayed.

Chapter 6 introduces the Wigner distribution function, elucidating its origin taken in quantum mechanics and illustrating its properties. A host of diverse optical signals are considered and the relevant Wigner distribution functions are analytically evaluated and graphically shown to help the intuitive perception of the simultaneous account of the signal in the space and spatial frequency domains, conveyed by the Wigner distribution function. Chapter 7 frames the Wigner distribution function within the broad realm of the phase-space signal representations, and illustrates the procedure, and the relevant optical architectures, for displaying the Wigner distribution function of a given signal. In Chapter 8 the laws for the transfer of the Wigner distribution function through linear optical systems are derived. Attention is drawn to the relation between the Wigner distribution function and the fractional Fourier transform which is a revealing and effective tool for the space-frequency representation of signals (optical or not). Chapter 9 is concerned with the moments of the Wigner distribution function and their propagation laws.

The Wigner representation is presented on the fascinating border-line between quantum mechanics and signal theory.

Chapters are made as self-consistent as possible. Indeed, the Introduction to each chapter is conceived as a summary of the basic results of previous chapters, which are central to those that are going to be presented. A basic role is assigned to the diagrams, which illustrate the syllabus of each chapter, and the figures, which confer physical reality to conceptual architectures. A wide bibliography is given in relation to topics both carefully investigated and briefly mentioned.

Throughout the book the calculations are kept at an accessible level; most mathematical steps are justified. Difficulties might be encountered in connection with the algebra of operators, which do not obey the familiar rules of the algebra of scalars. Careful and illustrative comments on the peculiar behavior of operators are provided in § 1.4.1 in order to help the readers who are not acquainted with the operator algebra.

It is my hope to give the flavour of the fascinating feature of optics that enables a visible account of abstract mathematical entities, like, for instance, symplectic matrices and metaplectic operators, represented through integral transforms. Symplectic matrices and integral transforms, which essentially provide the formal structures for the considerations developed in Chapters 1 to 5 , are intimately related, being indeed different representations of the same Sp(2, ) ∼ Mp(2, ) group element. Firstly recognized within a purely quantum mechanical context, this relation has been applied in optics in connection with the fractional Fourier transform. The link between ray matrices and transfer operators from the alternative viewpoint of linear canonical transformations and relevant representations, is elucidated in § 5.6. This is an example of those parallel paths, that, explicitly illustrated or implicitly suggested in the text or in the problems, are intended to improve the feeling for the specific topic under consideration and to gain some insight and intuition for unforeseen correspondences and analogies between totally different physical problems.

I am pleased to express my deep gratitude to Professor W.A.B. Evans, whose stimulating discussions, critical comments and technical suggestions have been precious to the completion of the book. I am greatly indebted to Dr. A. De Angelis for his enlightening suggestions, and to Professor A. Reale and Professor A. Scafati for their helpful comments. It is dutiful of me to thank Dr. G. Dattoli, who introduced me to the Lie algebra theory during the stage of our collaboration on the quantum picture of the Free Electron Laser dynamics. I am grateful to Dr. S. Bollanti, Dr. F. Flora and Dr. L. Mezi for their useful comments, and to Mrs. G. Gili, Mr. S. Lupini, Mrs. G. Martoriati, Mrs. M.T. Paolini, Mrs. L. Santonato, Dr. S. Palmerio, Dr. B. Robouch, Dr. N. Sacchetti and Dr. V. Violante for their constant and invaluable sympathy. I thank our librarians, Mrs. C. De Palo and Mrs. M. Liberati, who at certain periods have patiently accepted the role of “my” librarians. It is a pleasure to thank the Optical Society of America for kindly giving me the permission to reproduce material from Applied Optics and Optics Letters, and Einaudi for permitting me to reproduce the lines from Merini’s poem, which opened this Preface, my literal translation of which now closes it (below). I express my appreciation to Professor A. Lohmann for his prompt and kind response to my request of...