# Linear Ray and Wave Optics in Phase Space

## 1st Edition

### Bridging Ray and Wave Optics via the Wigner Phase-Space Picture

**Authors:**Amalia Torre Amalia Torre

**Paperback ISBN:**9780444517999

**eBook ISBN:**9780080535531

**Imprint:**Elsevier Science

**Published Date:**11th November 2005

**Page Count:**540

## Description

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 domains of applicability are well-established. Accordingly each model comprises its own set of geometric and dynamic postulates with the pertinent mathematical means.

At a basic level, the book is a complete introduction to the Wigner optics, which bridges between ray and wave optics, offering 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.

At a parallel level, the analogies with other branches of both classical and quantum physics, like classical and quantum mechanics, quantum optics, signal theory as well as magnetic optics, are evidenced by pertinent comments and/or rigorous mathematics. So, the Lie algebra and group methods are introduced and explained through the elementary optical systems within both the ray and wave optics contexts, the former being related to the symplectic group and the latter to the metaplectic group. In a like manner, the Wigner function is introduced by following the original issue to individualize a phase space representation of quantum mechanics, which is mirrored by the issue to individualize a local frequency spectrum within the signal theory context.

The basic analogy with the optics of charged particles inherently underlying the ray-optics picture in phase space is also evidenced within the wave-optics picture in the Wigner phase space.

## Key Features

· amalgamation of a great deal of contributions having witnessed the phase space picture of optics over the past 30 years
· introduces abstract concepts through concrete systems
· hosts of figures and logical diagrams to favour intuition and to introduce mathematics
· emphasis on the interrelations with quantum optics, signal theory and magnetic optics

· feeds a feeling for genuine issues in higher mathematics and theoretical physics

## Readership

Researchers in optics, graduate and undergraduate students

## Table of Contents

- Dedication
- Preface
- 1: Hamiltonian Picture of Light Optics. First-Order Ray Optics
- 1.1 Introduction
- 1.2 Hamiltonian picture of light-ray propagation
- 1.3 Hamiltonian picture of light-ray propagation: formal settings
- 1.4 Hamilton’s equations for the light-ray
- 1.5 Lie transformations in the optical phase space
- 1.6 Linear ray optics and quadratic Hamiltonian functions
- 1.7 Planar model of first-order optical systems
- 1.8 ABCD matrix and focal, principal and nodal planes
- 1.9 Summary
- Problems

- 2: 1D First-Order Optical Systems: The Ray-Transfer Matrix
- 2.1 Introduction
- 2.2 Ray-ensemble description of light propagation
- 2.3 Quadratic monomials and symplectic matrices
- 2.4 Quadratic monomials and first-order optical systems
- 2.5 Quadratic monomials in phase space
- 2.6 Summary
- Problems

- 3: The Group of 1D First-Order Optical Systems
- 3.1 Introduction
- 3.2 Ray matrix of composite optical systems
- 3.3 The subgroup of free propagation and thin lens matrices
- 3.4 Optical matrices factorized in terms of free propagation sections and thin lenses
- 3.5 Wei-Norman representation of optical elements: LST synthesis
- 3.6 Rotations and squeezes in the phase plane
- 3.7 Iwasawa representation of optical elements: LSFα synthesis
- 3.8 Canonical and noncanonical representations of symplectic matrices
- 3.9 Integrating the equation for the ray transfer matrix
- 3.10 Summary
- Problems

- 4: Wave-Optical Picture of First-Order Optical Systems
- 4.1 Introduction
- 4.2 Essentials of the scalar wave model of light. The paraxial wave equation in a quadratic medium
- 4.3 Ray and wave optics
- 4.4 From the ray-optical matrix to the wave-optical operator
- 4.5 Eigenfunctions of Q^ and P^: point-like and spatial harmonic waveforms
- 4.6 Spatial Fourier representation of optical wave fields
- 4.7 Summary
- Problems

- 5: 1D First-Order Optical Systems: The Huygens-Fresnel Integral
- 5.1 Introduction
- 5.2 Quadratic Hamiltonians and metaplectic Lie algebra
- 5.3 Wave-optical transfer relations for an ABCD system
- 5.4 The optical Fourier transform
- 5.5 Recovering the ray-optical description
- 5.6 Wave-optical propagators as unitary representations of linear canonical transformations
- 5.7 Summary
- Problems

- 6: The Wigner Distribution Function: Analytical Evaluation
- 6.1 Introduction
- 6.2 The optical Wigner distribution function: basic concepts
- 6.3 The Wigner distribution function: basic properties
- 6.4 The Wigner distribution function of light signals: further examples
- 6.5 Summary
- Problems

- 7: The Wigner Distribution Function: Optical Production
- 7.1 Introduction
- 7.2 The sliding-window Fourier transform
- 7.3 The Wigner distribution function and the general class of space-frequency signal representations
- 7.4 The ambiguity function
- 7.5 Understanding the Wigner and ambiguity functions from the viewpoint of the mutual intensity function
- 7.6 Optical production of the Wigner distribution function: general considerations
- 7.7 Wigner processor for 1
*D*real signals: basic configurations - 7.8 Wigner processor for 1
*D*complex signals: basic configurations - 7.9 The smoothed Wigner distribution function and the cross-ambiguity function: optical production
- 7.10 Summary
- Problems

- 8: 1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function
- 8.1 Introduction
- 8.2 From the wave function to the phase space representation
- 8.3 First-order optical systems: propagation law for the Wigner distribution function
- 8.4 The Wigner distribution function and the optical Fourier transform: linking Fourier optics to Wigner optics
- 8.5 Transport equation for the Wigner distribution function
- 8.6 Summary
- Problems

- 9: 1D First-Order Optical Systems: Moments of the Wigner Distribution Function
- 9.1 Introduction
- 9.2 Basic notions on moments
- 9.3 Preliminaries to the calculation of the moments of the Wigner distribution function
- 9.4 Wigner distribution function: local and global moments
- 9.5 Gaussian Wigner distribution functions: the variance matrix and its evolution
- 9.6 Propagation laws for the moments of the Wigner distribution function in first-order optical systems
- 9.7 Higher-order moments of the Wigner distribution function
- 9.8 Summary
- Problems

- A: Lie Algebras and Lie Groups: Basic Notions
- Index

## Details

- No. of pages:
- 540

- Language:
- English

- Copyright:
- © Elsevier Science 2005

- Published:
- 11th November 2005

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080535531

- Paperback ISBN:
- 9780444517999

## About the Author

### Amalia Torre

Amalia Torre received her degree in physics at the University of Naples. She is a member of the Theory Group of the Department of Applied Physics at the ENEA Research Center of Frascati, Rome. Her research interests include Free Electron Laser, optics and special functions. She is presently involved in two research projects concerning the Extreme Ultra-Violet Lithography (FIRB project) and the X-ray production via Free Electron Laser operation (SPARC project).

### Affiliations and Expertise

ENEA UTAPRAD-MAT, Frascati, Rome, Italy

### Amalia Torre

Amalia Torre received her degree in physics at the University of Naples. She is a member of the Theory Group of the Department of Applied Physics at the ENEA Research Center of Frascati, Rome. Her research interests include Free Electron Laser, optics and special functions. She is presently involved in two research projects concerning the Extreme Ultra-Violet Lithography (FIRB project) and the X-ray production via Free Electron Laser operation (SPARC project).

### Affiliations and Expertise

ENEA UTAPRAD-MAT, Frascati, Rome, Italy