A concise, comprehensive reference text covering electro-optical systems, optical system design, optical physics, holography, Fourier optics, and optical metrology. It emphasizes physical insight aimed at engineering applications. This book is suitable as an advanced undergraduate or graduate level text; problems and solutions are included.

- Preface xi
- 1 Introduction
- 2 Review of electromagnetic wave propagation
- 2.1 Wave fronts ....................................8
- 2.2 Phase velocity and the speed of ligh...........10
- 2.3 Power and intensity ...........................12
- 2.4 Reflection and transmission at a boundary .....14
- 2.5 Stratified layers..............................17
- 2.6 Problems ......................................18
- 3 Linear systems theory
- 3.1 Linear systems.................................19
- 3.2 Fourier transformation.........................20
- 3.3 Singular functions ............................21
- 3.4 Fourier transform theorems ....................23
- 3.5 Frequently used functions and their Fourier transforms..26
- 3.6 Linear system response ........................28
- 3.7 Problems ......................................29
- 4 Wavefront transformations
- 4.1 Free-space propagation ........................32
- 4.1.1 The paraxial approximation ..................34
- 4.1.2 The free-space propagation operator .........36
- 4.2 Operator relations ............................37
- 4.3 Discussion ....................................39
- 4.4 Refraction in dielectric materials ............40
- 4.5 Thin optical elements .........................43
- 4.5.1 The transparency.............................43
- 4.5.2 The thin dielectric slab.....................44
- 4.5.3 The thin prism...............................46
- 4.5.4 The thin lens................................48
- 4.5.5 Gratings.....................................51
- 4.5.6 Mirrors as optical elements .................53
- 4.5.7 Discussion ..................................54
- 4.6 One-dimensional operator definitions ..........54
- 4.7 Cylindrical lens operators.....................56
- 4.7.1 Transformations with the C operator..........58
- 4.8 The Gaussian beam and its transformations .....58
- 4.8.1 Free-space propagation of Gaussian beams ....59
- 4.8.2 Lens transformations of Gaussian beams ......61
- 4.9 Operator algebra--discussion...................63
- 4.10 Problems .....................................63
- 5 Basic optical systems
- 5.1 Imaging with a thin lens.......................68
- 5.2 Fourier transformation with a thin lens........70
- 5.3 Some aspects of geometrical optics.............73
- 5.4 Applications of single lens systems............76
- 5.4.1 The single lens image projector .............76
- 5.4.2 The magnifying glass ........................78
- 5.4.3 Applications of a single Fourier transforming system ..79
- 5.5 Two lenses in free space.......................80
- 5.5.1 Bonnet spheres and field flattening .........80
- 5.5.2 Microscope and some of its characteristics ..83
- 5.5.3 The double Fourier transforming system ......85
- 5.5.4 The telescope................................85
- 5.5.5 An invariance property of the two-lens system..87
- 5.6 Spatial filtering and optical correlation .....89
- 5.6.1 The joint transform correlator JTC ..........92
- 5.6.2 The matched filter ..........................95
- 5.6.3 Bandwidth consideration .....................98
- 5.7 Space-variant and space-invariant systems ....100
- 5.8 Problems .....................................101
- 6 Non-ideal optical systems
- 6.1 Optical systems of finite extent .............106
- 6.1.1 Apertured imaging system ...................108
- 6.1.2 Apertured Fourier transforming system ......114
- 6.1.3 Depth of focus..............................117
- 6.2 Real optical elements ........................118
- 6.2.1 Aberrations.................................119
- 6.2.2 Real lenses.................................122
- 6.3 Problems .....................................123
- 7 Statistical aspects of light
- 7.1 Interference..................................127
- 7.2 Mutual coherence .............................129
- 7.3 Self coherence ...............................130
- 7.4 Temporal coherence............................131
- 7.5 The Michelson interferometer..................132
- 7.6 Spatial coherence and spatial correlation ....135
- 7.7 Propagation of the coherence function ........137
- 7.8 Spatial coherence from incoherent sources ....138
- 7.9 Speckle patterns .............................141
- 7.9.1 Correlation function model of speckle patterns..143
- 7.9.2 Rigid translation...........................145
- 7.9.3 Free space observation......................145
- 7.9.4 Discussion .................................151
- 7.10 Problems ....................................153
- 8 Interference and interferometers
- 8.1 Interference fringes..........................156
- 8.2 Dynamic interference fringes .................157
- 8.2.1 Interference of two plane waves ............159
- 8.2.2 Interference between a plane wave and a spherical wave..161
- 8.3 Interferometry ...............................161
- 8.4 Interferometers and energy conservation.......162
- 8.5 The Michelson interferometer..................163
- 8.5.1 Interferometric displacement measurement ...163
- 8.5.2 Interferometric velocity measurement .......165
- 8.5.3 Interferometric profile and phase analysis .166
- 8.6 Other double-beam interferometers ............168
- 8.6.1 The Mach Zender interferometer .............168
- 8.6.2 Ring interferometer ........................171
- 8.6.3 The Jamin interferometer ...................174
- 8.6.4 Beam splitters .............................174
- 8.6.5 The Kosters prism interferometer............176
- 8.7 Using corner cubes............................176
- 8.8 Advanced interferometric procedures ..........178
- 8.8.1 Amplitude modulation interferometry ........178
- 8.8.2 Phase shifting interferometry ..............180
- 8.8.3 Heterodyne interferometry ..................180
- 8.8.4 Multiwavelength interferometry .............181
- 8.8.5 Coherence interferometer ...................183
- 8.9 The laser Doppler velocimeter ................183
- 8.10 Multibeam interferometers ...................188
- 8.10.1 Elementary diffraction gratings ...........188
- 8.10.2 Generalized diffraction gratings ..........190
- 8.10.3 The grating spectroscope ..................192
- 8.10.4 The Fabry Perot interferometer ............194
- 8.11 Self-referencing interferometers ............197
- 8.11.1 Phase visualization by spatial filtering...198
- 8.12 Problems ....................................199
- 9 Polarization
- 9.1 Polarization of plane waves...................201
- 9.2 Superposition of polarized waves .............203
- 9.2.1 Superposition of two plane polarized waves .204
- 9.2.2 Superposition of two circularly polarized waves..205
- 9.3 Propagation in an isotropic media.............206
- 9.3.1 Maxwell's equations in anisotropic media ...207
- 9.3.2 The index ellipsoid ........................208
- 9.3.3 Birefringence...............................209
- 9.4 Basic polarization components ................211
- 9.4.1 The polarizer...............................211
- 9.4.2 The retardation plate.......................214
- 9.4.3 Optical isolator............................215
- 9.5 Electro-optic modulation .....................216
- 9.6 The Jones matrix representation...............219
- 9.7 Circular birefringence........................222
- 9.8 Polarization aberrations .....................224
- 9.9 Problems .....................................225
- 10 Spatial light modulation
- 10.1 Intensity response of a recording material ..227
- 10.2 Spatial frequency response of recording materials ..229
- 10.3 Diffractive optical elements.................231
- 10.4 Electronic recording.........................232
- 10.5 Acousto-optic modulation ....................235
- 10.6 Two-dimensional spatial light modulators ....240
- 10.6.1 Controllable birefringence ................241
- 10.6.2 Deformable mirrors ........................242
- 10.6.3 Semiconductor modulators ..................242
- 10.7 Problems ....................................243
- 11 Holography
- 11.1 The holographic process......................245
- 11.2 Hologram recording with plane reference wave.249
- 11.3 Spherical wave recording magnification ......250
- 11.4 Wavelength changes in holography ............253
- 11.5 Phase conjugation ...........................255
- 11.6 Classification of holograms: conditions and properties..257
- 11.6.1 On-axis and off-axis holography ...........257
- 11.6.2 Transmission and reflection holograms .....258
- 11.6.3 Object wave configurations ................261
- 11.7 Hologram recording conditions................262
- 11.7.1 Coherence and stability conditions ........263
- 11.7.2 Recording medium consideration ............264
- 11.8 Phase holograms..............................264
- 11.8.1 Thermoplastic films .......................265
- 11.8.2 Surface relief recording...................266
- 11.8.3 Photopolymers .............................267
- 11.8.4 Photorefractive materials .................267
- 11.9 Synthetic holograms..........................268
- 11.10 Electronic recording........................269
- 11.11 Holographic interferometry..................269
- 11.11.1 Time average holographic interferometry ..269
- 11.11.2 Real-time holographic interferometry .....272
- 11.11.3 Double exposure holographic interferometry..275
- 11.11.4 Phase conjugate interferometry ...........276
- 11.12 Generalized treatment of the holographic process..278
- 11.13 Problems ...................................284
- 12 Advanced operator algebra
- 12.1 Ray transfer matrix of optical systems ......287
- 12.2 The canonical operator ......................289
- 12.3 Integral representation of canonical operators..291
- 12.4 Wave optics and geometrical ray matrices.....293
- 12.5 Canonical operator relations ................296
- 12.6 Real lenses .................................297
- 12.7 Gaussian beam transformations................298
- 12.8 Roots and powers of optical systems .........300
- 12.8.1 Matrix calculus............................300
- 12.8.2 Roots and powers of specific optical systems..303
- 12.9 Problems ....................................310
- 13 Optical information processing
- 13.1 Electro-optic pattern recognition ...........311
- 13.2 DOE design as an optimization problem........314
- 13.2.1 Optimization algorithms an overview........316
- 13.2.2 Cost function in filter design ............322
- 13.3 Transformations with cylindrical lenses .....326
- 13.3.1 The variable focal length astigmatic system..326
- 13.3.2 Imaging and Fourier transformation with astigmatic systems..327
- 13.4 One-dimensional signal processing............329
- 13.4.1 The vector matrix multiplier ..............329
- 13.4.2 Optical interconnection network ...........330
- 13.4.3 Scale and coordinate transformation .......331
- 13.4.4 The ambiguity function ....................332
- 13.4.5 Wavelet transform..........................333
- 13.4.6 Space-variant convolution .................335
- 13.4.7 Convolution of 1D functions using the C operator..338
- 13.5 Matrix matrix multiplication ................340
- 13.6 Problems ....................................343
- A Summary of operator relations
- A.1 Definition of basic operators ................345
- A.2 Commutation rules among the basic operators ..346
- A.2.1 Operations of the quadratic phase factor ...346
- A.2.2 Operations by the linear phase factor ......347
- A.2.3 Operations of the scaling operator .........347
- A.2.4 Operations of the shift operator ...........348
- A.2.5 Operations by the FT operator ..............348
- A.2.6 Operations by the FPO.......................348
- A.2.7 Other useful relations .....................349
- A.3 Normalized operator relations.................349
- B Bibliography 351
- C Problems and solutions
- C.1 The solution manual...........................359
- C.2 Chapter2......................................359
- C.3 Chapter3......................................361
- C.4 Chapter4......................................367
- C.5 Chapter5......................................371
- C.6 Chapter6......................................377
- C.7 Chapter7......................................378
- C.8 Chapter8......................................380
- C.9 Chapter9......................................381
- C.10 Chapter10 ...................................382
- C.11 Chapter11 ...................................383
- C.12 Chapter12 ...................................384
- C.13 Chapter13 ...................................385
- D Index

### Preface

Many good books are available on optics in general, and on specific subjects such as optical signal processing, holography, interferometry and other areas. The question should be asked: why do we need yet another book?

The incentive to write this book is rooted in many years of teaching electro-optics related subjects, particularly, a one-term course on electro-optical systems for senior undergraduate and graduate students of electrical engineering. The objective of this course is to provide a broad foundation and deep understanding of fundamental physical processes related to optics and optical systems. This groundwork should serve as a sound basis for more specialized study.

The students taking this course are supposed to have some background on the basics of optics and lasers but they should be able to follow the course even if this background is not complete. They are also expected to possess some knowledge about electromagnetic fields, Fourier analysis and linear systems theory.

Existing texts with the required breadth and depth tend to engulf the reader in heavy mathematical rigor which masks the physical insight and obscures engineering aspects. These texts are also too long for a one-term course. Other texts that are, in principle, suitable for the objectives of this course are too specialized and several of them are needed to cover the whole scope of the course. As a matter of fact, the desired curriculum cannot be covered within the allocated time if conventional procedures are used.

This book was started as a unique answer to the requirements of the course. However, as it proceeded it became broader and, in its present form, it covers a significant fraction of the field of optics. Nevertheless, it still remains concise due to a new approach and it can be used as a text and reference also for many other courses, such as optical physics, Fourier optics, optical system design and analysis, optical signal processing, optical metrology, holography and optical nondestructive evaluation. The book also fills up possible gaps in the background of the students by providing an overview of linear systems theory, electromagnetic fields, and additional information which is embedded in the main text.

Although initially planned as a text for one term, and the book can be used as such, it contains adequate material for more than 100 lecture hours. If used as a text for a one-term course, a set of chapters can be selected according to the main line of interest for that specific course. This can be done since the various subjects of the book are presented with minimal cross references although the sequence of the chapters are representative of an optimal curriculum. Some sections in the selected chapters may be skipped as well. For example, in a one-term course it is not necessary to discuss all the instruments treated in chapter 5 and it is adequate to study the basic concepts of interferometry without getting into the details of various architectures and procedures.

As indicated above, the emphasis of this book is on physical understanding toward engineering applications and, therefore, some of the mathematical rigor is sacrificed in favor of clarity and physical insight. Nevertheless, most mathematical steps are justified and all approximations involved in any procedure are carefully considered to avoid any misinterpretation of the final results. This is particularly important in view of the practical aspects considered throughout the book.

The book is made as self-contained as possible without unnecessarily inflating its volume. Accordingly, the book starts with two standard overview chapters on electromagnetic wave theory and linear systems theory with focus on Fourier analysis.

Optics really starts in chapter 4. This chapter introduces diffraction theory from a linear systems point of view using a group of linear operators. The operator algebra is the main innovation of this book. Historically, the operator algebra was initiated as a shorthand for the integral calculus involved in Fourier optics. However, it is much more than that in this book, first-order diffraction theory of optical systems is derived from linear systems theory. Physical entities (i.e. free space, lenses, etc.) are represented by linear operators, a cascade of which describe the operation of a complete optical system. Due to the physical meaning of each operator in an operator expression, much physical insight can be derived just by inspection. Furthermore, since these operators satisfy certain group theoretical relations, they provide an exceptionally powerful mathematical tool. The main result of this chapter is the derivation of Fourier optics in its operator form which can be translated into integral expressions at any stage.

In its simple form, as presented in chapter 4, the operator algebra is constructed from a few simple rules based on elementary Fourier analysis. Readers will realize very quickly that the small effort invested in learning those simple rules will be enormously rewarded by an unprecedented simplification of their work for the rest of the course and, probably, for their whole professional career. Due to its concise nature, the operator algebra allows coverage of the whole subject of conventional Fourier optics in chapters 4 and 5. This material can be covered in about fifteen lecture hours, leaving plenty of time for additional subjects. Moreover, by avoiding the tedious diffraction integrals, complicated systems can be easily analyzed, providing deep understanding and physical insight that are not masked by the calculations. Nevertheless, readers who are more comfortable with integral expressions can easily translate the operator expressions into integrals. Some examples are provided in the text and problem sections.

As an aid to the reader and also as a reference for future work, the basic rules of the operator algebra are summarized in appendix A.

After completing the above chapters the student has a significant basis to understand the rest of the book and also other texts. The other chapters are quite self contained and they may be selected for study according to the interest of any specific course. A limited number of cross-references are provided when some specific terms are borrowed from a different chapter but this does not mean that the student must learn the whole chapter.

The standard approach is also modified in the presentation of coherence theory (chapter 7) and interference (chapter 8). This whole subject is presented from an observational point of view. That is, parameters and characteristics of wave fields are defined and investigated in terms of observables. Accordingly, some of the conventional definitions of coherence (for example, spatial coherence) are slightly modified to suit coherent radiation in contrast to thermal radiation which was the basis for traditional coherence theory. Interference is presented in three dimensions leading to interference surfaces rather than interference fringes. The shape of these surfaces is discussed together with their dynamic characteristics when the two interfering waves have different frequencies. The treatment of heterodyne interferometry, laser Doppler velocimetry and other interferometric applications follow naturally from these fundamental discussions.

The chapter on holography (chapter 11) contains a comprehensive treatment of the subject and can be studied immediately after chapter 5 although it is better to study them in the order presented. The same is true for the chapter on polarization (chapter 9). However, since polarization effects are closely related to coherence effects it is better to study polarization after the basic concepts of coherence theory are understood.

The operator algebra is put into a more rigorous mathematical framework in chapter 12, which is again enhanced by application examples including the implementation of root (fractional) Fourier transforms. Although this framework of the operator algebra could have been used from the beginning, the intuitive form is more convenient for simple applications. Therefore, the linear systems approach is maintained throughout the book up to this chapter.

Various applications for signal processing are contained in the last chapter, 13, which is on a slightly more advanced level and is directed mainly to students working in this field.

Joseph Shamir

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