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This book provides all the essential and best elements of Kidger's many courses taught worldwide on lens and optical design. It is written in a direct style that is compact, logical, and to the point--a tutorial in the best sense of the word.

*"I read my copy late last year and read it straight through, cover to cover. In fact, I read it no less than three times. Its elegant expositions, valuable insights, and up-front espousal of pre-design theory make it an outstanding work. It's in the same league with Conrady and Kingslake."*--Warren Smith

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- Foreword / xiii
- Preface / xvii
- List of symbols / xix
- Chapter 1 Geometrical Optics / 1
- 1.1 Coordinate system and notation / 1
- 1.2 The rectilinear propagation of light / 2
- 1.3 Snell's law / 2
- 1.4 Fermat's principle / 4
- 1.5 Rays and wavefronts-the theorem of Malus / 5
- 1.6 Stops and pupils / 6
- 1.6.1 Marginal and chief rays / 7
- 1.6.2 Entrance and exit pupils / 7
- 1.6.3 Field stops / 8
- 1.7 Surfaces / 8
- 1.7.1 Spheres / 8
- 1.7.2 Quadrics of revolution (paraboloids, ellipsoids, hyperboloids) / 10
- 1.7.3 Oblate ellipsoid / 12
- 1.7.4 The hyperbola / 13
- 1.7.5 Axicon / 14
- Chapter 2 Paraxial Optics / 17
- 2.1 Paraxial rays / 17
- 2.1.1 The sign convention / 17
- 2.1.2 The paraxial region / 18
- 2.2 The cardinal points / 18
- 2.2.1 Principal points / 19
- 2.2.2 Nodal points / 20
- 2.3 Paraxial properties of a single surface / 21
- 2.4 Paraxial ray tracing / 23
- 2.4.1 Discussion of the use of paraxial ray trace equations / 25
- 2.5 The Lagrange invariant / 25
- 2.5.1 Transverse (lateral) magnification / 27
- 2.5.2 Afocal systems and angular magnification / 28
- 2.6 Newton's conjugate distance equation / 30
- 2.7 Further discussion of the cardinal points / 32
- 2.7.1 The combination of two lenses / 34
- 2.7.2 The thick lens / 35
- 2.7.3 System of several elements / 38
- 2.8 The refraction invariant, A / 39
- 2.8.1 Other expressions for the Lagrange invariant / 40
- 2.9 The eccentricity, E / 41
- 2.9.1 The determination of E / 42
- References / 44
- Chapter 3 Ray Tracing / 45
- 3.1 Introduction / 45
- 3.2 A simple trigonometric method of tracing meridian rays / 46
- 3.3 The vector form of Snell's law / 48
- 3.3.1 Definition of direction cosines / 50
- 3.4 Ray tracing (algebraic method) / 51
- 3.4.1 Precision / 54
- 3.5 Calculation of wavefront aberration (optical path difference) / 55
- 3.6 Ray tracing through aspheric and toroidal surfaces / 57
- 3.7 Decentered and tilted surfaces / 60
- 3.8 Ray tracing at reflecting surfaces / 61
- References / 62
- Chapter 4 Aberrations / 63
- 4.1 The relationship between transverse and wavefront aberrations / 63
- 4.2 Ray aberration plots / 65
- 4.3 Spot diagrams / 69
- 4.4 Aberrations of centered optical systems / 70
- 4.4.1 First-order aberrations / 73
- 4.4.1.1 Defocus/ 73
- 4.4.1.2 Lateral image shift / 74
- 4.4.2 The five monochromatic third-order (Seidel) aberrations / 74
- 4.4.2.1 Spherical aberration / 74
- 4.4.2.2 Coma / 76
- 4.4.2.3 Astigmatism and field curvature / 77
- 4.4.2.4 Distortion /79
- 4.4.2.4.1 The finite conjugate case / 79
- 4.4.2.4.2 The infinite conjugate case / 80
- 4.4.2.4.3 The afocal case / 81
- 4.4.2.4.4 Effect of pupil aberrations and defocus on distortion / 81
- 4.4.2.4.5 F-theta lenses / 81
- 4.4.2.4.6 Effect of a curved object on distortion / 82
- 4.4.3 Higher-order aberrations / 82
- 4.4.3.1 Balancing spherical aberration / 82
- 4.4.3.2 Balancing coma / 83
- 4.4.3.3 Balancing astigmatism and field curvature / 85
- 4.4.3.4 Balancing distortion / 86
- 4.5 Modulation transfer function (MTF) / 86
- 4.5.1 Theory / 87
- 4.5.2 The geometrical approximation / 88
- 4.5.3 Practical calculation / 88
- 4.5.4 The diffraction limit / 89
- References / 90
- Chapter 5 Chromatic Aberration / 91
- 5.1 Variation of refractive index-dispersion / 91
- 5.1.1 Longitudinal chromatic aberration (axial color) of a thin lens / 92
- 5.1.2 The Abbe V-value / 93
- 5.1.3 Secondary spectrum / 94
- 5.1.4 Transverse chromatic aberration (lateral color) / 97
- 5.2 The Conrady method for calculation of chromatic aberration / 97
- 5.3 Chromatic variation of aberrations / 100
- Chapter 6 Seidel Aberrations / 101
- 6.1 Introduction / 101
- 6.2 Seidel surface contributions / 101
- 6.2.1 Spherical aberration / 102
- 6.2.2 Off-axis Seidel aberrations / 107
- 6.2.3 Alternative formula for distortion / 108
- 6.2.4 Aberrations of a plano-convex singlet / 109
- 6.2.5 First-order axial color and lateral color / 111
- 6.2.6 Summary of the Seidel surface coefficients / 112
- 6.2.7 A numerical example / 113
- 6.3 Stop-shift effects / 115
- 6.3.1 Derivation of the Seidel stop-shift equations / 116
- 6.4 Dependence of the Seidel aberrations on surface curvature / 120
- 6.5 The aplanatic surface / 122
- 6.5.1 An example-the classical oil-immersion microscope
- objective / 125
- 6.6 Zero Seidel conditions / 126
- 6.7 "Undercorrected" and "overcorrected" aberrations / 128
- 6.8 Seidel aberrations of spherical mirrors / 129
- 6.9 Seidel aberration relationships / 130
- 6.9.1 Wavefront aberrations / 130
- 6.9.2 Transverse ray aberrations / 131
- 6.9.3 The Petzval sum and the Petzval surface / 132
- 6.9.4 The Petzval surface and astigmatic image surfaces / 133
- 6.10 Pupil aberrations / 135
- 6.11 Conjugate-shift effects / 136
- References / 137
- Chapter 7 Principles of Lens Design / 139
- 7.1 Thin lenses / 139
- 7.2 Thin lens at the stop / 142
- 7.2.1 Spherical aberration / 142
- 7.2.2 Coma / 142
- 7.2.3 Astigmatism / 142
- 7.2.4 Field curvature / 143
- 7.2.5 Distortion / 144
- 7.2.6 Axial color / 145
- 7.2.7 Lateral color / 146
- 7.3 Discussion of the thin-lens Seidel aberrations / 146
- 7.3.1 Spherical aberration / 148
- 7.3.1.1 Bending for minimum spherical aberration / 148
- 7.3.1.2 Effect of refractive index / 149
- 7.3.1.3 Effect of change of conjugates / 150
- 7.3.1.4 Correction of spherical aberration with two positive lenses / 150
- 7.3.1.5 Correction of spherical aberration with positive and negative lenses / 151
- 7.3.1.6 Seidel aberrations of thin lenses not at the stop / 152
- 7.3.2 Correction of coma / 152
- 7.3.3 Correction of astigmatism / 153
- 7.3.4 Correction of field curvature / 153
- 7.3.4.1 Different refractive indices / 154
- 7.3.4.2 Separated lenses / 154
- 7.3.4.3 Thick meniscus lens / 155
- 7.3.5 Reduction of aberrations by splitting lenses into two / 156
- 7.3.6 Seidel aberrations of a thin lens that is not at the stop / 157
- 7.3.7 Correction of axial and lateral color / 157
- 7.4 Shape-dependent and shape-independent aberrations / 158
- 7.5 Aspheric surfaces / 159
- 7.5.1 Third-order off-axis aberrations of an aspheric plate / 161
- 7.5.2 Chromatic effects / 162
- 7.6 The sine condition / 162
- 7.6.1 Sine condition in the finite conjugate case / 162
- 7.6.2 The sine condition with the object at infinity / 163
- 7.6.3 The sine condition for the afocal case / 164
- 7.7 Other design strategies / 164
- 7.7.1 Monocentric systems / 165
- 7.7.2 Use of front-to-back symmetry / 165
- Chapter 8 Achromatic Doublet Objectives / 167
- 8.1 Seidel analysis / 167
- 8.1.1 Correction of chromatic aberration / 167
- 8.1.2 Astigmatism and field curvature / 168
- 8.1.3 Comparison with the actual aberrations of a doublet / 168
- 8.1.4 Correcting both Petzval sum and axial color in doublets / 169
- 8.1.5 Possibilities of aberration correction in doublets / 170
- 8.2 The cemented doublet / 170
- 8.2.1 Optimization of cemented doublets / 171
- 8.2.2 Crown-first doublet / 172
- 8.2.3 Flint-first doublet / 174
- 8.3 The split doublet / 177
- 8.3.1 The split Fraunhofer doublet / 177
- 8.3.2 The split Gauss doublet / 179
- 8.4 General limitations of doublets / 182
- Chapter 9 Petzval Lenses and Telephoto Objectives / 183
- 9.1 Seidel analysis / 184
- 9.1.1 Calculation of predicted transverse aberrations from Seidel coefficients / 185
- 9.2 Optimization / 186
- 9.3 Examples / 186
- 9.3.1 Simple Petzval lens with two doublets / 186
- 9.3.2 Petzval lens with curved image surface / 189
- 9.3.3 Petzval lens with field flattener / 191
- 9.4 The telephoto lens / 193
- Chapter 10 Triplets / 199
- 10.1 Seidel theory / 199
- 10.2 Example of an optimized triplet / 202
- 10.3 Glass choice / 204
- 10.4 Vignetting / 206
- Chapter 11 Eyepieces and Afocal Systems / 209
- 11.1 Eyepieces-design considerations / 209
- 11.1.1 Specification of an eyepiece / 210
- 11.1.1.1 Focal length / 210
- 11.1.1.2 Field angle / 210
- 11.1.1.3 Pupil diameter / 210
- 11.1.1.4 Exit pupil position ("eye relief") / 211
- 11.1.2 Aberration considerations / 211
- 11.1.2.1 Prism aberrations / 211
- 11.1.2.2 Pupil spherical aberration / 211
- 11.1.2.3 Distortion / 212
- 11.1.2.4 Field curvature / 212
- 11.1.2.5 Special factors in optimization / 212
- 11.1.2.6 General comments on eyepieces / 212
- 11.2 Simple eyepiece types / 213
- 11.2.1 The Ramsden eyepiece / 213
- 11.2.2 The achromatized Ramsden, or Kellner, eyepiece / 214
- 11.2.3 The Ploessl eyepiece / 216
- 11.2.4 The Erfle eyepiece / 217
- 11.3 Afocal systems for the visible waveband / 219
- 11.3.1 Simple example of a complete telescopic system / 220
- 11.3.2 More complex example of a telescopic system / 222
- 11.3.3 Gallilean telescopes / 224
- 11.3.4 Magnifiers / 226
- References / 229
- Chapter 12 Thermal Imaging Lenses / 231
- 12.1 Photon detection / 231
- 12.1.1 8- to 13- (m waveband / 232
- 12.1.2 3- to 5- (m waveband / 233
- 12.2 Single-material lenses / 233
- 12.2.1 Single germanium lens / 234
- 12.2.2 Germanium doublets / 236
- 12.2.2.1 Plus-minus germanium doublet solution / 236
- 12.2.2.2 Plus-plus germanium doublet solution / 238
- 12.2.3 Germanium Petzval lens / 240
- 12.2.4 Germanium triplet / 242
- 12.3 Multiple-material lenses / 244
- 12.4 Infrared afocal systems / 247
- 12.4.1 The objective / 247
- 12.4.2 The eyepiece / 247
- 12.4.3 Optimization and analysis / 249
- 12.5 Other aspects of thermal imaging / 249
- 12.5.1 Narcissus effect / 249
- 12.5.2 Thermal effects / 250
- 12.5.3 Special optical surfaces / 250
- References / 250
- Chapter 13 Catadioptric Systems / 253
- 13.1 General considerations / 253
- 13.1.1 Reminder of Seidel theory-spherical aberration, S1 / 253
- 13.1.2 Correction of field curvature, S4 / 254
- 13.1.3 General topics relating to computations with cadioptric systems / 255
- 13.1.4 Baffles / 255
- 13.2 Simple examples / 255
- 13.2.1 Cassegrain telescope / 255
- 13.2.2 Field corrector for a Cassegrain telescope / 257
- 13.2.3 Coma corrector for a paraboloidal mirror / 259
- 13.2.4 Field corrector for a paraboloidal mirror / 260
- 13.2.5 The Ritchey-Chr?tien telescope / 262
- 13.2.6 Field corrector for a Ritchey-Chretien telescope / 263
- 13.2.7 Field corrector for a hyperbolic mirror / 265
- 13.2.8 Schmidt camera / 269
- 13.2.9 The achromatized Schmidt camera / 271
- 13.2.10 The field-flattened Schmidt camera / 272
- 13.2.11 The Maksutov-Bouwers Cassegrain system / 274
- 13.2.12 A simple Mangin mirror system by Wiedemann / 276
- 13.3 More complex examples / 279
- 13.3.1 Canzek Mangin system / 279
- 13.3.2 Mirror telephoto lens / 281
- References / 284
- Index / 287

### Preface

This volume is based on Michael Kidger's short course for SPIE entitled
"Fundamental Optical Design." It reviews basic geometrical optics and third-order
aberration theory, using the nomenclature and sign conventions of the Optical
Design Group at Imperial College in the 1960s given in W.T. Welford's book,
*Aberrations of Optical Systems* (Adam Hilger, 1986).

Michael's courses for SPIE were abbreviated forms of workshops that he taught for Kidger Optics. In these short courses, Michael concentrated on the application of this theory to the design of a variety of simple optical systems, with students spending about half of the course time working on these lenses with Michael's optical design code, SIGMA, under his supervision. This book attempts to re-create, for the reader, the teaching style and practical work that made these courses so popular with students all around the world. The design examples include prescriptions and aberration data generated by SIGMA, although the interested reader will find sufficient data to further explore the designs with any available optical design software.

With the advent of the PC in the 1980s, such software has become much more accessible to engineers and scientists without formal training in optics. Michael's short courses were aimed at such newcomers to what can seem, at first, a very daunting field. He always emphasized the need to understand why a particular lens works (or, more commonly, does not work!), rather than to blindly hope that the optimization code will find a miraculous, practical solution. In fact, the earlier Imperial College courses on optical design did not encourage access to an optimization program (in the 1960s and 1970s, residing on mainframe computers) until well after the student had a thorough understanding of geometrical optics and third-order aberration theory. This more academic approach, as part of a master's course in applied optics, encouraged a certain intangible "feel" for the subject that had the potential to develop into a more intuitive approach as experience was gained with a wider variety of optical systems, some of which might be novel or innovative.

In spite of the enormous improvements in computing power, optical design remains a discipline that is best developed in apprenticeship to a master practitioner. Traditionally, this takes years, but Michael gave many students a brief taste of this mysterious process of osmosis for a few days or hours. In the fast-paced modern world, this is becoming increasingly rare and precious, and it is hoped that this book at least provides a glimpse of it for posterity.

One of the reasons optical design is not easily learned from a textbook is that, in many cases, it is a necessary but not sufficient condition that the third-order aberrations are correctable. When fifth- and higher-order aberrations dominate, as they do at the larger apertures and field sizes often required, analytical dissection of the problem starts to fail, and optimization codes, experience and intuition become the designer's principal "tools of the trade." In the commercial world, there is also the need to find a design that is manufacturable, often with conflicting requirements of low cost and high performance. A second volume, Intermediate Optical Design, will explore some of these issues-applied to more complex designs-but the intermediate material will remain firmly grounded in the foundations of this first volume.

Most of the material in these volumes originates either directly from Michael's course notes, or from the unfinished book that he was working on. The editing process has included, for completeness, the addition of some material, while at the same time trying to retain Michael's original intent and style. In the first volume, this includes a brief discussion of pupil aberrations, some additional visual optical designs, as well as some catadioptric astronomical telescopes. Our hope is that Michael would be pleased with the result!

David M. Williamson

July 2001

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