Tutorial Texts Series Optical Design Applying the Fundamentals, Max J. Riedl, Vol. TT84
Infrared Optics and Zoom Lenses, Second Edition, Allen Mann, Vol. TT83 Optical Engineering Fundamentals, Second Edition, Bruce H. Walker, Vol. TT82 Fundamentals of Polarimetric Remote Sensing, John Schott, Vol. TT81 The Design of Plastic Optical Systems, Michael Schaub, Vol. TT80 Radiation Thermometry Fundamentals and Applications in the Petrochemical Industry, Peter Saunders, Vol. TT78
Matrix Methods for Optical Layout, Gerhard Kloos, Vol. TT77 Fundamentals of Infrared Detector Materials, Michael A. Kinch, Vol. TT76 Practical Applications of Infrared Thermal Sensing and Imaging Equipment, Third Edition, Herbert Kaplan, Vol. TT75 Bioluminescence for Food and Environmental Microbiological Safety, Lubov Y. Brovko, Vol. TT74 Introduction to Image Stabilization, Scott W. Teare, Sergio R. Restaino, Vol. TT73 Logic-based Nonlinear Image Processing, Stephen Marshall, Vol. TT72 The Physics and Engineering of Solid State Lasers, Yehoshua Kalisky, Vol. TT71 Thermal Infrared Characterization of Ground Targets and Backgrounds, Second Edition, Pieter A. Jacobs, Vol. TT70
Introduction to Confocal Fluorescence Microscopy, Michiel Müller, Vol. TT69 Artificial Neural Networks An Introduction, Kevin L. Priddy and Paul E. Keller, Vol. TT68 Basics of Code Division Multiple Access (CDMA), Raghuveer Rao and Sohail Dianat, Vol. TT67 Optical Imaging in Projection Microlithography, Alfred Kwok-Kit Wong, Vol. TT66
Metrics for High-Quality Specular Surfaces, Lionel R. Baker, Vol. TT65 Field Mathematics for Electromagnetics, Photonics, and Materials Science, Bernard Maxum, Vol. TT64 High-Fidelity Medical Imaging Displays, Aldo Badano, Michael J. Flynn, and Jerzy Kanicki, Vol. TT63 Diffractive Optics–Design, Fabrication, and Test, Donald C. O’Shea, Thomas J. Suleski, Alan D. Kathman, and Dennis W. Prather, Vol. TT62 Fourier-Transform Spectroscopy Instrumentation Engineering, Vidi Saptari, Vol. TT61 The Power- and Energy-Handling Capability of Optical Materials, Components, and Systems, Roger M. Wood, Vol. TT60 Hands-on Morphological Image Processing, Edward R. Dougherty, Roberto A. Lotufo, Vol. TT59 Integrated Optomechanical Analysis, Keith B. Doyle, Victor L. Genberg, Gregory J. Michels, Vol. TT58 Thin-Film Design Modulated Thickness and Other Stopband Design Methods, Bruce Perilloux, Vol. TT57
Optische Grundlagen für Infrarotsysteme, Max J. Riedl, Vol. TT56 An Engineering Introduction to Biotechnology, J. Patrick Fitch, Vol. TT55 Image Performance in CRT Displays, Kenneth Compton, Vol. TT54 Introduction to Laser Diode-Pumped Solid State Lasers, Richard Scheps, Vol. TT53 Modulation Transfer Function in Optical and Electro-Optical Systems, Glenn D. Boreman, Vol. TT52
Uncooled Thermal Imaging Arrays, Systems, and Applications, Paul W. Kruse, Vol. TT51 Fundamentals of Antennas, Christos G. Christodoulou and Parveen Wahid, Vol. TT50 Basics of Spectroscopy, David W. Ball, Vol. TT49 Optical Design Fundamentals for Infrared Systems, Second Edition, Max J. Riedl, Vol. TT48 Resolution Enhancement Techniques in Optical Lithography, Alfred Kwok-Kit Wong, Vol. TT47
Copper Interconnect Technology, Christoph Steinbrüchel and Barry L. Chin, Vol. TT46 Optical Design for Visual Systems, Bruce H. Walker, Vol. TT45
For a complete listing of Tutorial Texts, visit http://spie.org/tutorialtexts.xml
Tutorial Texts in Optical Engineering Volume TT84
Bellingham, Washington USA
Library of Congress Cataloging-in-Publication Data Riedl, Max J. Optical design : applying the fundamentals / Max J. Riedl. p. cm. -- (Tutorial texts series) Includes bibliographical references and index. ISBN 978-0-8194-7799-6 1. Optical instruments--Design and construction. I. Title. QC372.2.D4R54 2009 681'.4--dc22 2009022688
Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: 360.676.3290 Fax: 360.647.1445 Email:
[email protected] www.spie.org
Copyright © 2009 Society of Photo-Optical Instrumentation Engineers All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the thought of the author(s). Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America.
I dedicate this book to Warren J. Smith, my friend and colleague for more than 50 years. He has influenced my life more than anyone else.
Introduction to the Series Since its inception in 1989, the Tutorial Texts (TT) series has grown to more than 80 titles covering many diverse fields of science and engineering. The initial idea for the series was to make material presented in SPIE short courses available to those who could not attend and to provide a reference text for those who could. Thus, many of the texts in this series are generated by augmenting course notes with descriptive text that further illuminates the subject. In this way, the TT becomes an excellent standalone reference that finds a much wider audience than only short course attendees. Tutorial Texts have grown in popularity and in the scope of material covered since 1989. They no longer necessarily stem from short courses; rather, they are often generated by experts in the field. They are popular because they provide a ready reference to those wishing to learn about emerging technologies or the latest information within their field. The topics within the series have grown from the initial areas of geometrical optics, optical detectors, and image processing to include the emerging fields of nanotechnology, biomedical optics, fiber optics, and laser technologies. Authors contributing to the TT series are instructed to provide introductory material so that those new to the field may use the book as a starting point to get a basic grasp of the material. It is hoped that some readers may develop sufficient interest to take a short course by the author or pursue further research in more advanced books to delve deeper into the subject. The books in this series are distinguished from other technical monographs and textbooks in the way in which the material is presented. In keeping with the tutorial nature of the series, there is an emphasis on the use of graphical and illustrative material to better elucidate basic and advanced concepts. There is also heavy use of tabular reference data and numerous examples to further explain the concepts presented. The publishing time for the books is kept to a minimum so that the books will be as timely and up-to-date as possible. Furthermore, these introductory books are competitively priced compared to more traditional books on the same subject. When a proposal for a text is received, each proposal is evaluated to determine the relevance of the proposed topic. This initial reviewing process has been very helpful to authors in identifying, early in the writing process, the need for additional material or other changes in approach that would serve to strengthen the text. Once a manuscript is completed, it is peer reviewed to ensure that chapters communicate accurately the essential ingredients of the science and technologies under discussion. It is my goal to maintain the style and quality of books in the series and to further expand the topic areas to include new emerging fields as they become of interest to our reading audience. James A. Harrington Rutgers University vii
Contents Preface…………….. .......................................................................... xv Chapter 1 1.1 1.2 1.3 1.4 1.5
Chapter 2 2.1 2.2 2.3
2.4 Chapter 3 3.1 3.2 3.3 3.4 3.5
Law of Refraction: The Foundation of Geometrical Optics .................... 1 Introduction ............................................................................ 1 Fermat’s Principle .................................................................. 1 1.2.1 Historic remarks ......................................................... 1 1.2.2 Derivation ................................................................... 1 Snell and the Lens .................................................................. 3 Graphical Ray Tracing ........................................................... 4 Paraxial Ray Tracing.............................................................. 5 1.5.1 Equations, symbols, and sign conventions .................. 6 Best Shape for a Thin Lens ....................................... 11 Concept of Thin Lens............................................................. 11 Object at Any Position ........................................................... 12 Object at Infinity with Added Field of View ......................... 16 2.3.1 Spherical aberration .................................................... 16 2.3.2 Chromatic aberration .................................................. 18 2.3.3 Coma .......................................................................... 19 2.3.4 Astigmatism................................................................ 19 2.3.5 Total blur-spot size ..................................................... 21 Special Case ........................................................................... 21 Best Shapes for Multiple Thin Lenses, Aspherizing, and the Natural Stop Position ............. 23 Introduction ............................................................................ 23 Best Shapes for Minimum Spherical Aberration ................... 23 Aspherizing a Singlet to Eliminate Spherical Aberration ...... 26 Correcting Coma and Spherical Aberration ........................... 28 3.4.1 Eliminating coma........................................................ 28 3.4.2 Eliminating spherical aberration................................. 29 Natural Stop Position ............................................................. 31
ix
Chapter 4 4.1 4.2
4.3 Chapter 5 5.1 5.2
5.3
5.4 5.5 5.6
Chapter 6 6.1 6.2
6.3
Transition from a Thin Lens to a Thick Lens ........... 33 Introduction ............................................................................ 33 Adding a Thickness and Changing the Second Surface Radius..................................................... 33 4.2.1 VIS singlet with n = 1.5.............................................. 34 4.2.2 MWIR singlet with n = 3.4 ......................................... 35 4.2.3 LWIR singlet with n = 4 ............................................. 35 Change of Spherical Aberration with Added Thickness ........ 35 Achromats .................................................................. 39 Introduction ............................................................................ 39 Thin Achromat for the VIS Spectrum, On-Axis Performance ............................................................ 39 5.2.1 Adding a field to the on-axis corrected achromat....... 43 5.2.2 Optimizing .................................................................. 45 Smith’s Method of Determining the Surface Shapes ............. 45 5.3.1 Curvatures and inverse object distances ..................... 45 5.3.2 The K-functions .......................................................... 46 5.3.3 The calculations .......................................................... 47 Achromat for the MWIR Region ........................................... 48 Achromat for the LWIR Region ............................................ 50 Diamond-Turned Hybrid ....................................................... 50 5.6.1 Hybrid for the MWIR region ...................................... 51 5.6.1.1 Basic lens shape ..................................................... 51 5.6.1.2 Aspherizing............................................................ 52 5.6.1.3 The diffractive phase profile.................................. 52 5.6.1.4 Comments .............................................................. 53 5.6.2 Useful nomograms...................................................... 55 Systems with Two Separated Components ............. 57 Introduction ............................................................................ 57 Dialyte—An Air-Spaced Achromat ....................................... 57 6.2.1 Example for the MWIR region .................................. 59 6.2.1.1 Aspheric deformation coefficients of surface 3.............................................. 61 6.2.1.2 Phase coefficients .................................................. 61 6.2.1.3 Step height at the zone transition ........................... 62 Telephoto and Reversed Telephoto ....................................... 62 6.3.1 Examples for the MWIR and LWIR regions .............. 63
x
Chapter 7 7.1 7.2 7.3 7.4 Chapter 8 8.1 8.2 8.3 Chapter 9 9.1 9.2 9.3 9.4 9.5
9.6 9.7 9.8 9.9
From an Air-Spaced Doublet to a Triplet .................. 65 Introduction ............................................................................ 65 Chromatic Aberration ............................................................ 66 Example, a Conventional Triplet ........................................... 66 Hybrid Petzval Objective with a Cold Stop ........................... 67 A Hybrid for Two Wavelengths ................................. 69 Introduction ............................................................................ 69 Basic Lens Shape for the Long Wavelength .......................... 69 Required Diffractive Profile .................................................. 70 Athermats ................................................................... 73 Introduction ............................................................................ 73 Focus Shift of a Refractive Element ...................................... 73 Athermalization with a Doublet ............................................. 74 Focus Shift of a Diffractive Lens ........................................... 74 Design Examples ................................................................... 78 9.5.1 Athermat with two elements in an aluminum housing ............................................. 78 9.5.2 Hybrid athermat in an aluminum housing .................. 79 Impact of Housing Material .................................................. 80 Athermat for the CO2 Laser Line ........................................... 81 Athermalized Achromat ......................................................... 84 9.8.1 Three-element athermat in an aluminum housing ...... 85 9.8.2 Two-element athermat in an aluminum housing ........ 86 Effect of Quarter-Wave Limit without Athermalization........ 87
Chapter 10 The Ball Lens.............................................................. 89 10.1 10.2 10.3 10.4
Introduction ............................................................................ 89 Spherical Aberration .............................................................. 90 Coma 91 Astigmatism ........................................................................... 92
Chapter 11 Seidel and the Pegel Diagrams ................................. 95 11.1 11.2
Introduction ............................................................................ 95 Triplet for the LWIR Region ................................................. 95 11.2.1 Surface contributions ................................................ 96 11.2.2 Pegel diagram for the triplet ..................................... 96 11.2.3 Remarks to field curvature ........................................ 97
xi
11.3
Cassegrain Version with a Maksutov-Mangin Mirror Combination for the LWIR Region ....................................... 98 11.3.1 Pegel diagram ........................................................... 98
Chapter 12 The Single-Imaging Mirror .........................................101 12.1 12.2 12.3 12.4
12.5 12.6 12.7
Introduction ............................................................................ 101 Spherical Mirror ..................................................................... 101 Toroidal Mirrors .................................................................... 104 Examples ................................................................................ 104 12.4.1 Spherical mirror ........................................................ 106 12.4.1.1 Additional comments ............................................. 106 12.4.2 Toroidal mirror ......................................................... 107 Parabolic Mirror ..................................................................... 107 Manufacturing Remarks......................................................... 109 Mangin Mirror ....................................................................... 109
Chapter 13 Eight Single Optical Elements as Imaging Objectives ............................................... 113 13.1 13.2 13.3 13.4 13.5 13.6
Introduction ............................................................................ 113 Diffraction Limit .................................................................... 113 Eight Chosen Configurations ................................................. 113 Shapes of the Elements .......................................................... 115 Aberrations............................................................................. 116 Examples ................................................................................ 117
Chapter 14 A Progression of Performance with an Increase in Lens Complexity ................................ 119 14.1
Objectives .............................................................................. 119
Chapter 15 Two-Mirror Systems as Telescope and Microscope Objectives .......................................121 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9
Introduction ............................................................................ 121 Basic Cassegrain Telescope Layout....................................... 122 15.2.1 Equations ................................................................. 123 Cassegrain with Two Spherical Mirrors ................................ 123 Classic Cassegrain System..................................................... 124 Dall-Kirkham Arrangement ................................................... 124 Ritchey–Chretien Configuration ............................................ 125 Examples ................................................................................ 125 Cassegrain with Mangin as a Secondary Reflector ................ 126 Gregorian Telescope .............................................................. 127
xii
15.10 Gregorian Microscope Objective ........................................... 130 15.11 Two Schwarzschild Objectives .............................................. 131 15.11.1 Prescription for the classic configuration for an object at infinity .......................................... 133 15.11.1.1 Brief historic remark about the golden ratio ........ 134 15.11.2 Prescription for the inverted configuration for an object at infinity .......................................... 134 15.12 Solid Microscope Objective ................................................... 136 Chapter 16 The Plane-Parallel Plate .............................................137 16.1 16.2 16.3 16.4 16.5
Introduction ............................................................................ 137 Aberrations............................................................................. 138 16.2.1 Examples .................................................................... 140 Shift of Image ........................................................................ 141 Tilted Plate ............................................................................. 143 16.4.1 Lateral displacement ................................................... 144 Two Tilted Plates ................................................................... 144
Chapter 17 MTF, Limits, and Pixel Sizes ..................................... 147 17.1 17.2 17.3
Introduction ............................................................................ 147 Optical Modulation Transfer Function .................................. 147 Focal Plane Array .................................................................. 148
Chapter 18 Details of a Hybrid Lens ............................................151 18.1 18.2 18.3
Introduction ............................................................................ 151 Hybrid .................................................................................... 151 Coordinates of the Combined Surface ................................... 153
Chapter 19 From the Höegh Meniscus to Double Anastigmats ..............................................157 19.1 19.2 19.3 19.4 19.5
Introduction ............................................................................ 157 Höegh Meniscus .................................................................... 157 19.2.1 Approach and design method.................................... 158 Hypergon Lens ....................................................................... 160 Achromatic Double Lens ....................................................... 161 19.4.1 Approach and design method.................................... 162 Double Anastigmats ............................................................... 165 19.5.1 Approach and design method.................................... 165
Index………………… ........................................................................169
xiii
Preface This book is written for engineers and scientists who have some experience in the field of optics and want to know more about the details and derivations of equations used in optical design. Such knowledge is especially valuable in the layout stages of an optical system, when the question is “Where shall I begin?” The other question may be “How come?” instead of just using a given equation. The book begins with the derivation of the fundamental law of geometrical optics, Snell’s law of refraction, and states the paraxial ray trace equations. The following discussions are organized by subjects, starting with a thin lens and progressing to increasingly more sophisticated components and multi-element systems. Each subject is covered in depth to provide a good understanding for performance and limitations. The often ignored effects of plane-parallel plates are included as a separate subject. While the text is based on general optical laws, special emphasis has been placed on the two major infrared regions, the mid-wave (MWIR) and the longwave (LWIR). This is particularly apparent in the discussion about diffractive hybrids, which have found their place in these long wavelengths areas for the correction of chromatic aberrations and athermalization. Comments relating to single-point diamond turning have been included because this process is predominantly used to produce optical elements for the infrared regions. The final subject gradually leads the reader from a single element as an imaging objective for the visible spectrum, the historic Höegh, to a four-element anastigmat. This is done by successively adding and shaping elements and selecting suitable glasses for aberration reduction. Finally, I thank Tim Lamkins, Scott Schrum, and Gwen Weerts of SPIE for their special support and editorial assistance. Max J. Riedl July 2009
xv
Chapter 1 Law of Refraction: The Foundation of Geometrical Optics 1.1 Introduction Snell’s law of refraction is the fundamental law that governs geometrical optics. We begin, therefore, with the proof of this basic rule, as it has been verified by Fermat. We then demonstrate how this surprisingly simple law can be applied to graphical ray tracing. With the equations for paraxial ray tracing, we provide the tool required for the initial optical design phase. These equations are sufficient to determine the third-order aberrations, which will be used throughout the book.
1.2 Fermat’s Principle1 1.2.1 Historic remarks Pierre de Fermat was a jurist and mathematician. He pursued his mathematical avocation mostly for his own enjoyment. He formulated his famous theorem in 1657. It declares that light takes the path that requires the least time. His reasoning led to the proof of the law of refraction, which Wilibrord Snel van Royen found experimentally some 20 years earlier. This law of refraction is the foundation of geometrical optics and is stated by n' sin i' n sin i ,
(1.1)
where n and n′ are the indices of refraction of the media before and after refraction. The angle of incidence is i, and the angle of the ray is i′, relative to the normal, after refraction.
1.2.2 Derivation Travel time with reference to Fig. 1.1 is t
PO OP' v1 v2
and b2 a x h2 x 2 , v1 v2 2
t
1
2
Chapter 1
Figure 1.1 Schematic for Fermat’s theory.
where v1 is the speed of light in medium 1 before refraction, and v2 is the speed in medium 2 after refraction. We differentiate this expression with respect to x and set the result equal to zero. dt x a x 0, dx v1 h 2 x 2 v b 2 a x 2 2
which means x a x . v1 PO v2 PO'
Law of Refraction: The Foundation of Geometric Optics
3
With x sin i, and PO
a x OP'
sin i' , we get
sin i sin i' . v1 v2
Since the index of refraction is defined as n'
vvacuum v1 , vmedium v2
we finally obtain sin i'
1 sin i , n'
which is, of course, Snell’s law with n = 1 (air).
1.3 Snell and the Lens To apply Snell’s law to the refraction on a lens surface, we simply place a tangent at point P where the ray meets the first lens surface. The angles before and after refraction are then i and i′ as indicated in Fig. 1.2. This simple procedure covers flat, spherical, and aspheric surfaces. It also serves for calculations of chromatic aberrations because the index of refraction is a function of wavelength.
Figure 1.2 Snell’s law applied to a curved lens surface.
4
Chapter 1
1.4 Graphical Ray Tracing The procedure for graphical ray tracing is demonstrated in Fig. 1.3, where a meridional ray is traced through a planoconvex lens with a refractive index of 1.5. The object is located at infinity. Therefore, the ray entering the lens is parallel to the optical axis. Draw line 1 from center C (center of the first surface radius) through point P1, the point where the entering ray meets the first surface. Draw arcs 2 and 3 around point P1. The scales of the radii are chosen in proportion to the media indices before and after the refraction. Our lens is surrounded by air with n1 1 , and the index of the lens material is n'1 n2 1.5 . The ratio between the radii 2 and 3 is therefore 1 to 1.5. Draw line 4 parallel to line 1 through the point where arc 2 intersects the entering ray. From the intersection of arc 3 and line 4, draw a line through point P1 and extend it until it crosses the second lens surface. This point is marked P2 in the figure. Repeat the steps for the second lens surface to find the direction of the exiting ray. The focus location F of the lens can be established by tracing a ray near the optical axis, the paraxial region. The lateral distance from the optical axis and the point where the marginal ray passes through the focal plane is the transverse spherical aberration. The distance from the crossover of the ray on the optical axis is the longitudinal spherical aberration. While this approach may not appear to be too practical, it gives a good insight and demonstrates the principle of refraction at lens surfaces. Figure 1.4 serves for additional explanation of the relationships.
Figure 1.3 Procedure of graphical ray tracing.
Law of Refraction: The Foundation of Geometric Optics
5
Figure 1.4 Principle of graphical ray tracing, which is also suitable for aspheric surfaces.
1.5 Paraxial Ray Tracing Snell’s law was stated with Eq. (1.1) as n 'sin i ' n sin i . The sine of an angle can be expressed by sin i i
i3 i5 i7 ...... . 3! 5! 7!
(1.2)
sin i i ,
(1.3)
n'i' ni .
(1.4)
For small angles,
and Snell’s law reduces to
This is accurate to the sixth decimal place if the angles are not larger than about 41 minutes of arc. Ray tracing based on, but not limited to, this narrow region is referred to as paraxial ray tracing. The simple paraxial ray-trace equations are exceedingly useful in lens design. The image location is determined with these equations. Deviations from the paraxial image position are the measures for aberrations. By including the second term of the sine series, the so-called thirdorder aberrations, known as Seidel aberrations, can be determined. This will be demonstrated throughout the following chapters.
6
Chapter 1
Figure 1.5 Symbols and sign conventions for paraxial ray-trace equations: (a) an axial ray and (b) an oblique ray.
1.5.1 Equations, symbols, and sign conventions The symbols and sign convention used in the equations are identified in Figs. 1.5(a) and 1.5(b). The sign for a surface radius is positive if its center lies to the right. If it lies to the left, it is negative.
Opening The distance of an object from the first lens surface is l1. For an axial ray, u1
y1 , l1
(1.5)
Law of Refraction: The Foundation of Geometric Optics
7
for an oblique ray, y1 h l1u1 ,
(1.6)
and for an object located at infinity, u1 0, and y1 h .
(1.7)
Axial ray [Figure 1.5(a)] If the object is located 100 units to the left of the lens, l1 100 . With a ray height of 10 units at the first surface of the lens, y1 10 , according to Eq. (1.5), the slope of the axial ray is u1
y1 10 0.1 . l1 100
Oblique ray [Figure 1.5(b)] If the object height h 15 units, and the slope of this oblique ray u1 0.075 , the ray height at the first surface of the lens is y1 h l1u1 15 (100) (0.075) 7.5 units.
Transfer to the next surface n'i u'i ni ui yi ci n'i ni with ci yi1 yi t 'i u 'n
1 Ri
(1.8) (1.9)
Equations (1.8) and (1.9) are repeatedly applied for the i surfaces in the optical train.
Closing For the axial ray, l'i
yi , u 'i
(1.10)
for the oblique ray, h' yi l'i u'i ,
and for magnification,
(1.11)
8
Chapter 1
m
u1 h' . u'i h
(1.12)
For mirror surfaces n 1 (reversal of ray direction).
Example Given: A lens with the surface radii R1 = 50 (c1 =1/50 = 0.02) and R2 = −50 (c2 = −0.02). t'1 10 . n1 n2' 1 (air), n'1 n2 1.5, and l 200
Trace an axial ray with y1 20, and an oblique ray with h 12 and y1 10 .
1. Axial ray u1
y1 20 0.1 l1 200
n'1u'1 n1u1 y1c1 n'1 n1 1 0.1 20 0.02 1.5 1 0.1
u'1
0.1 0.066667 1.5
y2 y1 t'1u'1 20 10 0.066667 19.333333 n'2u'2 n2u2 y2 c2 n'2 n2 1.5 0.066667 19.333333 0.02 1 1.5 0.293333 u'2 l'2
y2 19.333333 65.909091 u'2 0.293333
m
u1 0.1 0.340909 u'2 0.293333
2. Oblique ray u1
h y1 25 10 0.075 l1 200
n'1u'1 n1u1 y1c1 n'1 n1 1 0.075 10 0.02 1.5 1 0.175
Law of Refraction: The Foundation of Geometric Optics
u'1
9
0.175 0.116667 1.5
y2 y1 t'1u'1 10 10 0.116667 8.833333 n'2u'2 n2u2 y2 c2 n'2 n2 1.5 0.116667 8.833333 0.02 1 1.5 0.263334 u'2 h' mh 0.340909 25 8.522727
An oblique ray from the edge of the object through the center of the aperture stop is called the chief, or principle, ray. An axial ray from the axial point of the object through the edge of the aperture stop is the marginal axial ray. To determine the third-order aberration contributions, only these two paraxial rays have to be traced. Ray-trace equations in addition to the ones presented here can be found in Ref. 2.
References 1. J. R. Meyer-Arendt, Introduction to Classical and Mirror Optics, Prentice Hall, Inc., Englewood Cliffs, New Jersey (1984). 2. W. J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York (2008).
Chapter 2 Best Shape for a Thin Lens 2.1 Concept of Thin Lens The “thin lens” concept is an extremely useful one for quick preliminary layout calculations and initial performance analyses. “Thin” means that the thickness of the lens is ignored in the calculations. The power of a lens is the reciprocal of its focal length f. The reciprocal of a surface radius R is called the surface curvature c. With t being the lens thickness and n the index of refraction of the lens material, we can state n 1 c c t . 1 n 1 c1 c2 1 2 f n
(2.1)
Neglecting the thickness t yields the power of the thin lens n 1 c1 c2 n 1 c .
(2.2)
In this equation, c is called the net curvature and is the difference between the front and rear surface curvatures, i.e., c c1 c2 . This indicates that
1
Figure 2.1 Bending a lens affects its spherical aberration but not its focal length. Shown is the effect for a germanium lens with an index of n = 4. 11
12
Chapter 2
the power of a thin lens remains the same as long as the net curvature is kept constant. Changing the curvatures but maintaining their difference is called lens bending. This is a powerful tool because the shape of a lens primarily influences its spherical aberration, as indicated in Fig. 2.1. The relation between lens shape and spherical aberration of a thin germanium lens for an object located at infinity is plotted. The ratio of the first curvature c1 and the net curvature c is known as the shape factor K. We plot spherical aberration as a function of the lens shape K and identify the surface radii of the lens. Because germanium has an index of refraction of n = 4 at a wavelength = 10 µm, the result for the best shape is amazingly simple and easy to remember. The front surface radius is equal to the focal length of the lens, and the radius of the rear surface is 1.5 times the focal length.
2.2 Object at Any Position Figure 2.2 shows the case where the object is located at a finite distance from the lens and identifies object and image distances, l and l′, the semi-aperture y of the lens, and the slope u′ of the exiting ray.
Figure 2.2 Thin lens with object located at a finite distance.
Figure 2.3 The marginal ray penetrates the image plane a distance away from the optical axis. This lateral distance is called the transverse spherical aberration TSC. See also Figure 2.4.
Best Shape for a Thin Lens
13
To derive the general equations for the best shape surface radii, when the object is located at any position, we begin the derivation with the expression for the third-order thin-lens transverse spherical-aberration contribution for any conjugate2: TSC
y4 G1c 3 G2c 2c1 G3c 2v G4cc12 G5cc1v G6cv 2 . u'k
(2.3)
In Eq. (2.3), the G-sums are functions of the optical material’s index of refraction n. They are G1 G4
n 2 n 1
,
2 n 2 n 1 2n G7
G2
2n 1 n 1 ,
, G5
2 2 n 1 n 1
2n 1 n 1 2n
n
G3 , G6
, and G8
3n 1 n 1 2 3n 2 n 1
n n 1 2
2n .
The reciprocal object distance v is defined in Eq. (2.3) as v (1 / l ) , where l is the object’s distance from the lens. Differentiating Eq. (2.3) with respect to the first curvature c1 yields dTSC y 4 G2c 2 2G4cc1 G5cv . dc1 u'k
To find the optimum, we set this expression to zero and solve for c1, c1
G2c G5v . 2G4
(2.4)
The net curvature c c1 c2 n 1 , which is derived from the thin lens power equation: n 1 c1 c2 n 1 c . Inserting this and the G-sums into Eq. (2.4) leads to c1
n 2n 1
2 n 2 n 1
2 n 1
n 2
v,
(2.5)
14
Chapter 2
where c1 1 / R1 , the reciprocal first surface radius; 1 / f , the power or reciprocal focal length of the lens; and v 1 / l , the reciprocal object distance from the lens, as stated above.
It is interesting to observe that the first term of Eq. (2.5) states the surface curvature c1 for an object at infinity because, for that case, v = 0. To find the curvature for the second surface, we use the relation c2 c1 n 1 . n 2n 1 2 n 1 c2 v , 2 n 2 n 1 n 2 n 1
which reduces to n 2n 1 4 2 n 1 c2 v. 2 n 2 n 1 n 2
(2.6)
Here, too, the first term is for an object at infinity. Considering that the inverse object distance can be stated by m / (m 1) , where m is the magnification, we insert this relation into Eqs. (2.5) and (2.6), to form n 2n 1 2 n 1 m c1 2 n 2 n 1 n 2 m 1
(2.7)
n 2n 1 4 2 n 1 m c2 . 2 n 2 n 1 n 2 m 1
(2.8)
and
To demonstrate the usefulness of these expressions, we analyze a lens with a focal length of 100 mm 1/f 0.01 mm -1 , made from germanium with n = 4. We choose the magnification to be m = –2. The first surface curvature is then 4 2 4 1 2 4 1 2 c1 0.01 4 2 2 1 2 4 2 4 1
and c1 1 1.666667 0.666667 0.01 0.001111 .
Best Shape for a Thin Lens
15
The first surface radius is R1
1 1 900 mm . c1 0.001111
The second surface curvature, according to Eq. (2.8), is 4 2 4 1 4 2 4 1 2 c2 0.01 4 2 2 1 2 4 2 4 1
and c2 0.666667 1.666667 0.666667 0.01 0.0044444 .
The second surface radius is R2
1 1 225 mm . c2 0.0044444
The G-sums for n = 4 are G1 24.00, G2 13.50, G3 19.50, G4 2.25, G5 7.50, and G6 5.25.
The net curvature c c1 c2 0.0011111 0.0044444 0.0033333 and the inverse object distance are m 2 v 0.01 0.0066667 . m 1 2 1
For a relative aperture of f/2, the semi-aperture y = 25 mm. The final slope angle u'k v y 0.01 0.0066667 25 0.0833333 . Now, we can insert all of these numbers into Eq. (2.3) and find the minimum transverse third-order spherical aberration for this thin lens to be 254 [24 0.0033333 13.5 0.0033332 (0.001111) 0.083333 19.5 0.0033332 0.006667 2.25 0.003333 (0.001111)2
TSC
7.5 0.003333 (0.001111) 0.006667 5.25 0.003333 0.006667 2 ] = 0.994859. TSC 1 mm .
16
Chapter 2
Expressing the transverse spherical aberration directly as a function of the index n, the semiaperture y, the power of the lens , the inverse object distance v, and the G-sums transforms Eq. (2.3) into TSC
y 3n
4n 1 2 4 n 12 v v . 8 n 1 n 2 v 2
(2.9)
For our case, this results in 253 0.01 4 TSC 2 8 4 1 4 2 0.01 0.006666 2 4 4 1 0.012 4 4 1 0.01 0.006666 0.006666 ,
and TSC 434.02691 0.0015 0.0008 0.998 mm 1 mm.
2.3 Object at Infinity with Added Field of View In this section, we apply the third-order thin-lens aberration equations to determine the blur spots contributed by spherical and chromatic aberrations, as well as coma and astigmatism. Comparing the results with an actual ray trace will indicate how valuable these thin-lens aberration expressions are. Of great interest in this discussion is also the best image (detector) location.
2.3.1 Spherical aberration For an object located at infinity, the image is formed at the focal plane of the lens. As mentioned above, the equations for a singlet, shaped for minimum spherical aberration, become simpler.1 The first surface radius is 2 n 2 n 1 R1 f, n 2n 1
(2.10)
and the second surface radius is 2 n 2 n 1 R2 f. n 2n 1 4
Since v = 0, Eq. (2.9) reduces to
(2.11)
Best Shape for a Thin Lens
TSC
17
y 3 n 4n 1 2 8 n 1 n 2 2
n 4n 1 f 64 n 1 n 2 f / # 2
3
.
(2.12)
The blur spot is half the size of TSC after refocusing from the paraxial focus by 3 / 2 TSC f / # . Applied to our f/2, 100-mm focal-length germanium lens set for infinity, we obtain 2 n 2 n 1 2 4 2 4 1 36 R1 100 100 100 mm f 4 2 4 1 36 n 2n 1
and 2 n 2 n 1 2 4 2 4 1 36 R2 100 100 150 mm .* f 4 2 4 1 4 24 n 2n 1 4 TSC
n 4n 1 f 64 n 1 n 2 f /# 2
3
4 15 100 0.217014 mm . 64 9 6 23
The minimum blur spot size is Bmin TSC 2 0.217 2 0.1085 mm when refocused by 3 2 TSC f /# 1.5 0.217 2 0.651 mm. The best image location is, therefore, 100 – 0.651 = 99.349 mm from the last lens surface. Details are shown in Fig. 2.4.
Figure 2.4 Location and size of the minimum blur spot due to spherical aberration, which is called “the circle of least confusion.”
*
Also see Fig. 2.1.
18
Chapter 2
2.3.2 Chromatic aberration The blur spot size due to axial chromatic aberration for a thin lens is expressed by Bchrom
f , 2V f / #
(2.13)
where V is the inverse relative dispersion of the lens material (also known as Abbe number, named after Ernest Abbe, who introduced this concept). In the visible spectrum, V nd 1 nF nC , where the indices of refraction refer to the Fraunhofer absorption lines. In the infrared regions, V nM 1 nS nL , with reference to the middle, short, and long wavelengths of the covered spectral band. For our germanium lens in the LWIR region (8–12 µm), V ≈ 800. Therefore, Bchrom 100 2 800 2 0.03125 mm for our lens. Note how mild the chromatic aberration is with such a large Abbe number. In the visible spectrum, the V values are lower by more than a factor of 10. As will be discussed in Chapter 4, this attribute is the main justification for correcting chromatic aberration in the infrared regions with diffractive phase profiles. The expected on-axis blur spot, caused by spherical and chromatic aberrations, according to the thin-lens aberration equations, is Baxial Bspher Bchrom 0.10851 0.03125 0.140 mm. Figure 2.5 shows the spot diagram of a thick lens, which was obtained with a computer ray-trace program.
Figure 2.5 Computer ray trace confirms validity of thin-lens calculations for onaxis aberrations (spherical and chromatic).
Best Shape for a Thin Lens
19
Figure 2.6 Thin lens with added field for object at infinity. The principal ray is an oblique ray that goes through the center of the aperture stop, which is located here at the lens.
2.3.3 Coma Figure 2.6 shows the layout of a thin lens for an object located at infinity and identifies the image height h′ and the half-field angle up. The thin-lens expression for sagittal coma is2 CC h' y 2 c 0.25G5c1 G8 c .
(2.14)
If we insert the curvatures c and c1 from the thin lens (shaped for minimum spherical aberration) and the expressions for the G-sums G5 and G8, we get CC
up f 16 n 2 f/#
2
.
(2.15)
With a half field angle of up = 0.1 radian, the sagittal coma blur for our lens becomes Bcoma CC
0.1 100 0.026042 mm . 16 4 2 22
2.3.4 Astigmatism Another off-axis aberration is astigmatism, which introduces a blur spot size of 1: Bastig 2TAC
u 2p f
2 f / #
.
(2.16)
20
Chapter 2
Figure 2.7 Comparison of predicted blur spots with third-order thin-lens aberration equations and the actual sizes for the thick germanium lens.
Figure 2.8 Balanced image position for equal blur spot sizes of the f/2, 100-mm focal-length germanium lens for the LWIR spectrum with a half field of 5.73 deg.
Best Shape for a Thin Lens
21
The location of this blur (circle of least confusion) is located halfway between the sagittal and tangential image shells. This distance is 0.5u 2p f . The blur spot size due to astigmatism for our example is Bastig
0.12 100 0.250 mm. 2 2
2.3.5 Total blur-spot size We add all four effects together and get the approximate size of the total blur at the best location: Btotal Bspher Bchrom Bcoma Bastig Btotal 0.10851 0.03125 0.02604 0.25000 0.42 mm.
This does not take into account the effects of field curvature. This can be seen in Fig. 2.7, which is from a real ray trace of the thick lens. The off-axis blur is somewhat larger than the predicted 0.42 mm. By slightly refocusing, the total blur size is about 0.460 mm in diameter. The choice of best position depends on the application. A 0.460-mm square detector element is large enough to accept all the energy from a point source within the stated field of view. For completeness, the best focus position relative to the paraxial focal plane for the example is illustrated in Fig. 2.8.
2.4 Special Case A closer look at Eq. (2.11) reveals that the second surface of a lens becomes a flat when n 2n 1 4 . This is the case when n = 1.686141. An optical material with an index of refraction very close to this value is sapphire at a wavelength of 3.7 µm with n = 1.687. In the visible spectrum there are the glasses SF5 (with nd = 1.673) and LAKN13 with (nd = 1.694).
References 1. M. J. Riedl, Optical Design Fundamentals for Infrared Systems, SPIE Press, Bellingham, Washington (2001). 2. W. J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York (2008).
Chapter 3 Best Shapes for Multiple Thin Lenses, Aspherizing, and the Natural Stop Position 3.1 Introduction Expanding on the previous case for a single lens in Chapter 2, we develop expressions for elements of a multiple lens arrangement, made from the same material and individually bent for minimum spherical aberration. We then demonstrate the benefit of an aspheric surface. This is especially of interest for elements used in the infrared spectrum because they can be routinely machined with the process of single-point diamond turning. Coma can be eliminated by properly shaping the single element or by placing the aperture stop at a specific position, which is referred to as the natural stop position. Details will be covered under this Chapter.
3.2 Best Shapes for Minimum Spherical Aberration To expand on the general case for a single lens with the object located at any position as discussed under Chapter 2, one can develop the following expressions if all elements of a multiple lens arrangement are made from the same material and individually bent for minimum spherical aberration. The surface curvatures of the elements are as follows: For the first surface (front surface) of element j, curvature is c j1
n 2n 1 4 n 2 1 j 1 2i n 2 n 1
2 n 1
n 2
v.
(3.1)
In this equation and in the following, j is the element number, and i is the total number of elements in the set. For the second surface (rear surface) of element j, curvature is
23
24
Chapter 3
c j2
n 2n 1 4 4 n 2 1 j 1 2i n 2 n 1
2 n 1
n 2
v.
(3.2)
The minimum spherical aberration is TSCi
y 3n 8i 2 n 1 n 2 v 2
2 j i 2 4n 1 2 4i 2 n 1 v v 3 j j 1 . i j 1
(3.3)
These expressions reduce to the forms for a single lens if we set j = i = 1. With v = 0, the lens will be focused for an object at infinity. Table 3.1 lists the radii and the spherical aberrations for an f/2, 100-mmfocal-length lens, made from germanium with 1, 2, and 3 elements for an object at infinity. Notice the change in shapes with the increase of elements for an object at infinity.
Table 3.1 Radii and resulting transverse spherical aberrations of configurations for an object at infinity. i 1 2 3
R11 100 200 300
R12 150 300 450
R21
R22
R31
R32
75 112.5
85.714 128.571
69.231
75
TSC -0.217 +0.076 +0.130
Best Shapes for Multiple Thin Lenses, Aspherizing, and the Natural Stop Position
25
From Table 3.1, it can be seen that the single element is afflicted with a negative TSC, which means that it has undercorrected spherical aberration. (The marginal ray crosses the optical axis ahead of the paraxial image point.) With two elements, the sign changes and the lens has overcorrected spherical aberration. (The marginal ray crosses the optical axis behind the paraxial image). Three elements amplify this trend. This means that the high index of germanium “overshoots” the correction with only two elements. Let us find the index for which the two-element configuration is zero, when the object is located at infinity. For those conditions, Eq. (3.3) reduces to TSC2
y 3 n2
4n 1 16 n 12 1 . 4 32 n 1 n 2 2
(3.4)
Setting this expression to zero leads to 4n 1 4 n 1 , for which the solution is n = 2.5. That is close to the index for amorphous material transmitting infrared radiation (AMTIR-1) for the mid-wave infrared (MWIR) with n4 2.514 , and for the long-wave infrared (LWIR) where n10 2.498 . Also close to the desired value is zinc selenide with n4 2.433 in the MWIR region, and n10 2.406 for the LWIR band. 2
To find a material for which the TSC is zero when the system consists of three elements, we proceed as above and get 4n 1 36 n 1 8 / 27 . This leads to n = 1.75. That is the index for the rare earth glass LAFN7, for example, with nd 1.7495 . For the LWIR spectrum, cesium iodide (CsI) has n10 1.74 , which is close to the desired value. 2
Table 3.2 on the next page shows the radii and the spherical aberrations for lenses with a magnification of m 2 . The negative sign indicates that the image is reversed, as demonstrated in Fig. 1.5(b). Notice how the shapes change for an object at a finite distance. It is worthwhile to point out here the advantage of an aspheric surface, especially for the infrared spectrum, where the surface can be shaped by diamond turning. It is an easy task to find the conic constant for the surface to be used to eliminate (third-order) spherical aberration. We shall demonstrate the process for a thin singlet, by aspherizing first the front surface and then the rear surface.
26
Chapter 3
Table 3.2 Radii and resulting transverse spherical aberrations of configurations for finite conjugates (magnification m = –2). i 1 2 3
R11 -900 -163.636 -128.574
R12 -225 -128.571 -112.5
R21
R22
R31
R32
450 -450
1,800 -300
300
450
TSC -0.998 -0,119 -0.043
3.3 Aspherizing a Singlet to Eliminate Spherical Aberration With reference to Eq. (2.3), the transverse spherical aberration of a thin lens is TSC
y 4c G1c 2 G2cc1 G4c12 , u'
(3.5)
when the object is located at infinity. With c1
n 2n 1
2 n 2 n 1 c
,
, n 1
u ' y ,
and G1 , G2 , G4 , as identified in Sec. 2.2, and the object is at any position, Eq. (3.5) becomes
Best Shapes for Multiple Thin Lenses, Aspherizing, and the Natural Stop Position
TSCmin spher
y 3 n 4n 1 2 8 n 1 n 2 2
27
.
(3.6)
To get rid of spherical aberration, we aspherize one surface of the lens and set the contribution from the asphere equal to the spherical aberration contribution from the lens with spherical surfaces, but opposite in sign. In other words, we write TSCasphere TSClens best shape .
If we limit the correction to changing one spherical surface to a conic section, the spherical aberration contribution of such a surface can be expressed by TSCasphere
n ' n fy 3 c3 ,
(3.7)
2
where is the conic constant, n and n′ are the indices of the media before and after refraction, f is the focal length of the element, y is the semi-diameter of the lens, and c is the curvature of the surface to be aspherized. This leads to a conic constant for the first surface of 1
2 R13TSC . n 1 f y13
(3.8)
If the second surface is to be aspherized, the conic constant for that surface is 2
2 R23TSC . n 1 f y23
(3.9)
Example For a 100-mm-focal-length lens (made from silicon) with an f/2 speed, with n = 3.4, TSCmin spher
y 3 n 4n 1 2 8 n 1 n 2 2
253 3.4 4 3.4 1 0.012 8 3.4 1 3.4 2 2
0.269007.
The curvature of the first surface is c1
n 2n 1
2 n 2 n 1
3.4 2 3.4 1 0.01 2 3.4 2 3.4 1
0.010231.
28
Chapter 3
For the second surface, c2 c1
0.01 0.010231 0.006065 . n 1 3.4 1
The radii are therefore R1 1 / c1 1 / 0.010231 97.742156 mm and R2 1 / c2 1 / 0.006065 164.885496 mm. Aspherizing the first surface leads to a conic constant of 1
2 97.7421563 0.269007 2 R13TSC 0.13397 . n 1 f y13 3.4 1 100 253
If we decide to leave the first surface a sphere and aspherize the second, we obtain a conic constant for that surface of 2 164.8854963 0.269007 2 R23TSC 2 0.643146 . n 1 f y23 3.4 1 100 253
In the first case, the aspheric surface is a prolate ellipsoid, in the second it is an oblate ellipsoid.
3.4 Correcting Coma and Spherical Aberration If spherical aberration and coma are to be eliminated, the procedure is to remove coma first by selecting the proper curvatures of the lens, and then to aspherize one of the two surfaces.
3.4.1 Eliminating coma For an object located at infinity, sagittal coma of a thin lens is stated by Eq. (2.14) and repeated here: CC h' y 2 c 0.25G5c1 G8 c .
(3.10)
Setting this expression to zero yields the curvature of the first surface as c1
4G8c . G5
(3.11)
Best Shapes for Multiple Thin Lenses, Aspherizing, and the Natural Stop Position
29
The G-sums have been identified previously, and c n 1 . After insertion, Eq. (3.11) changes to c1
n
n2 2
.
1
(3.12)
The radius is the reciprocal of the curvature, i.e., R1
n
2
1
f.
n2
(3.13)
The second surface curvature is n n 1 1 c2 c1 c n2 1
(3.14)
and R2
n
2
1
n n 1 1
f .
(3.15)
3.4.2 Eliminating spherical aberration As we know from Eq. (3.5), the transverse spherical aberration of a thin lens for an object located at infinity is TSC
y 4c G1c 2 G2cc1 G4c12 . u'
(3.16)
When c1 n 2 n 2 1 from Eq. (3.12), c n 1 , u' y , and the G-sums are inserted, Eq. (3.16) turns into TSCcoma=0
y 3 n3 2 n 2 1 f 2 2
.
(3.17)
To apply this formula to our silicon lens with y = 25, n = 3.4, and f = 100, we get TSCcoma=0
253 3.43 2 3.42 1 1002 2
0.275359 .
30
Chapter 3
We eliminate spherical aberration again with an asphere. This time we have to set TSCasphere TSCcoma=0 0.275359 .
The first surface radius is R1
n
1
2
n
2
f
3.4
2
1
3.42
100 91.349481 ,
and the conic constant for the first surface is then κ1
2 91.3494813 0.275359
3.4 1 100 253
0.111948 .
This equation indicates the shape of a prolate ellipsoid. The second surface radius is R2
n
2
1
n n 1 1
f
3.4
2
1
3.4 3.4 1 1
100 147.486034 .
If we desire to aspherize the second surface instead of the first, the conic constant for that surface is κ2
2 147.4860343 0.275359 0.471140 , 3.4 1 100 253
which identifies again the shape of an oblate ellipsoid.
Remark The same exercises can be repeated for the other monochromatic aberrations and extended to include the effects of the aperture stop location. Such exercises give much insight into the behavior of optical elements. We limit ourselves to the case of a single lens, shaped for minimum spherical aberration, and determine the stop location for which coma vanishes. This stop location is called the natural stop position.
Best Shapes for Multiple Thin Lenses, Aspherizing, and the Natural Stop Position
31
3.5 Natural Stop Position We begin with the so-called stop-shift equation, which leads to the elimination of coma. If the aperture stop is not located at the lens, the thin-lens, third-order sagittal coma is CC* CC
yp y
TSC ,
(3.18)
whereby it is interesting to note that coma cannot be eliminated if there is no spherical aberration. As we know, the thin-lens, third-order sagittal coma is expressed by CC h' y 2 c 0.25G5c1 G8 c ,
(3.19)
with h' u p f , and c 1/ n 1 f . The curvature of the first surface for a best-shaped lens (one with minimum spherical aberration) is c1
n 2n 1
2 n 2 n 1 f
.
Figure 3.1 Stop position.
32
Chapter 3
With that, the sagittal coma for an object located at infinity is CC
up f 16 n 2 f / #
2
.
(3.20)
The transverse spherical aberration is for that case TSC
n 4n 1 f 64 n 1 n 2 f / # 2
3
.
(3.21)
From Fig. 3.1, it can be seen that y p u p lstop . (lstop is defined as negative because it is measured from the first lens surface). Eliminating coma requires that CC y pTSC y 0 . If the lens has been shaped for minimum spherical aberration, inserting the relations from above leads to the simple expression for the natural stop position, 4 n 1 y f / # 2
lstop
n 4n 1
.
(3.22)
This reduces for our three examples with y = 25 and f/# = 2 to lstop
200 n 1 n 4n 1
2
.
Table 3.3 lists the results. Table 3.3 Natural stop position for three thin lenses shaped for minimum spherical aberration, at which third-order coma vanishes. Index n Stop position lstop
1.5 (VIS) -6.666 mm
3.4 (MWIR) -26.891 mm
4.0 (LWIR) -30.000 mm
Chapter 4 Transition from a Thin Lens to a Thick Lens 4.1 Introduction It was shown under Chapters 2 and 3 that the “thin lens” concept is an extremely useful one for preliminary calculations and analysis as well as a general design tool. Determining the difference in performance between that fictitious thin lens and a realistic thick lens is an interesting exercise which confirms the statement above. We begin our exercise with a thin lens shaped for minimum spherical aberration, add a reasonable thickness, and modify only the rear surface radius to maintain the focal length of the lens. We then calculate the spherical aberration for both cases and compare the results for three f/2, 100-mm-focal-length lenses. The first lens is made from glass with an index of refraction of n = 1.5 for the visible spectrum (VIS). The second is for the infrared mid-wave region (MWIR, 3–5 µm), made from silicon with n = 3.4. The third singlet is made from germanium, n = 4, for the application in the infrared long-wave region (LWIR, 8–12 µm).
4.2 Adding a Thickness and Changing the Second Surface Radius The focal length for a thin lens (one for which its thickness is assumed to be zero) is expressed by 1 1 1 n 1 . f R1 R2
(4.1)
For a thick lens, where the thickness of the lens is accounted for, the equation is stated as 1 1 1 n 1 t n 1 , f R1 R2 nR1R2
with t being the thickness of the lens. Solving for R2 results in
33
(4.2)
34
Chapter 4
R2
n 1 n 1 t nR1 f n R1 n 1 f
.
(4.3)
We recall that the radius of a thin lens, shaped for minimum spherical aberration, is 2 n 2 n 1 R1 f. n 2n 1
(4.4)
Inserting Eq. (4.4) into Eq. (4.3) leads to the rear radius of a thick lens, shaped for minimum spherical aberration: R2
n 1 2 n 2 f 2n 1 t . n 2n 1 2 n 2
(4.5)
4.2.1 VIS singlet with n = 1.5 The radii for the thin lens are 2 n 2 n 1 2 1.5 2 1.5 1 R1 f 100 58.3333 mm 1.5 2 1.5 1 n 2n 1
and 2 n 2 n 1 2 1.5 2 1.5 1 R2 f 100 350mm . n 2n 1 4 1.5 2 1.5 1 4
We add a thickness of 10 mm and solve for the second surface radius using Eq. (4.5):
n 1 2 n 2 f 2n 1 t n 2n 1 2 n 2 1.5 1 2 1.5 2 100 2 1.5 1 10 330.00 mm. 1.5 2 1.5 1 2 1.5 2 R2
These results can be confirmed by using the ray trace equations [Eqs. (1.7) and (1.8)] and solving for the curvature c2 . We demonstrate this for the thick lens.
Transition from a Thin Lens to a Thick Lens
35
With reference to Fig. 1.5(a), u1 0 because the object is located at infinity, and n1 n '2 1 because the lens is surrounded by air, for which the index of refraction is 1. We assume a ray height at the first surface of 10 units, i.e., y1 10 . The index of the lens material is n '1 n2 1.5 , and the front surface curvature is c1 1 R1 1 58.333..... 0.017142857 . With that, n'1u'1 n1u1 y1c1 n'1 n1 10 0.017142857 1.5 1 = 0.085714286,
and the angle after refraction at the first surface is u'1 u2
0.085714286 0.05714285 . 1.5
With the lens thickness of t '1 10 , the ray height at the second surface is y2 y1 t'1u'1 10 10 0.057142857 9.428571429 .
Recognizing that u'2 y1 f 10 100 0.1 , we rearrange Eq. (1.7) and solve for the curvature of the second surface: c2
n2u2 n'2 u'2 1.5 0.057142857 1 0.1 0.003030303 . 9.428571429 1 1.5 y2 n'2 n2
The radius R2 1/ c2 1/ 0.003030303 330.00 , which confirms the earlier obtained result.
4.2.2 MWIR singlet with n = 3.4 For the thin lens, R1 = 97.737557 mm and R2 = 164.885496 mm. Adding a 5-mm thickness to the element changes the second surface radius to R2 = 158.931299 mm.
4.2.3 LWIR singlet with n = 4 The thin-lens front surface radius R1 = 100.00 mm, and the rear surface radius R2 = 150.00 mm. Again, adding a 5-mm thickness to the element changes the second surface radius to R2 = 144.375 mm.
4.3 Change of Spherical Aberration with Added Thickness We know from Eq. (2.12) that the third-order transverse spherical aberration for a thin lens is
36
Chapter 4
TSCthin lens
n 4n 1 f 64 n 1 n 2 f / # 2
3
.
(4.6)
The contributions for our three lenses are summarized in Table 4.1. Determining the spherical aberration of a thick lens is more elaborate and based on the individual surface contributions.1 We follow the described procedure and begin with n2
TSC B i h' , n 1
2 n n
(4.7)
whereby B
n n' 1 2n' Inv
y u' i .
(4.8)
The optical invariant Inv n'2u'2 h' 0.25h' for our f/2 lenses. This changes Eq. (4.8) to B
From
the
paraxial
ray
2n n' 1
trace
n'
y u' i .
equations
nn' un' un nn yn cc nn' nn ,
yn 1 yn tn' un' , and in cn yn un , we find the ray height y2 and the angles u′ and i. Table 4.2 lists these values and the resulting spherical surface aberration contributions. Table 4.1 Spherical aberrations for three thin singlets in different spectral regions. Spectral region VIS MWIR LWIR
Transverse spherical aberration TSC (mm) -1.6741 -0.2690 -0.2170
Table 4.2 Ray trace and third-order surface aberration contributions for the VISlens with n = 1.5 Surface 1 2
y 25 23.57143
t′ 10
u′ -0.142857 -0.250000
i 0.428575 -0.214278
B -4.761967 -16.415545
TSC -0.8747 -0.7537
Transition from a Thin Lens to a Thick Lens
37
Figure 4.1 Undercorrected and overcorrected spherical aberration.
The total transverse spherical aberration of the thick lens is therefore
TSC TSC
1
TSC2 1.6284 mm .
We compare this result with the thin-lens aberration and find TSC TSCthick TSCthin 1.6284 1.6741 0.0457 mm .
As expected, the spherical aberration for the thick lens is somewhat less because the added thickness contributes positive spherical aberration. Positive, or overcorrected spherical aberration, exists when the marginal ray intercepts the optical axis beyond the paraxial focal plane. Spherical aberration is called negative, or undercorrected, when the marginal ray crosses the optical axis ahead of the paraxial focal plane. This occurrence is illustrated in Fig. 4.1. We follow the same procedure for the two IR lenses. The results are listed in Table 4.3. The errors for these lenses are even smaller. This indicates that with the higher indices of refraction for silicon and germanium, and the longer wavelengths, the thin-lens aberration expressions are quite accurate in the infrared regions. Table 4.3 Transverse spherical aberrations for two singlets in the infrared regions. Spectral region MWIR LWIR
TSC Thin lens -0.26901 -0.21701
TSC Thick lens -0.26557 -0.21437
TSC (mm) 0.00344 0.00264
TSC (%) 1.2 1.2
Reference 1. W. J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York (2008).
Chapter 5 Achromats 5.1 Introduction The progression of an achromat from the basic conventional form to a diamondturned aspheric hybrid for the infrared spectrum will be demonstrated in this Chapter. We start with a cemented doublet for the visible spectrum, then change to the mid-wave and long-wave infrared regions, and proceed with the rest of the improvements relating to aberration reductions in the long-wavelength spectral band. For this exercise, we elect a 100-mm focal length with an f/4 relative aperture and add a field of ±2 deg.
5.2 Thin Achromat for the VIS Spectrum, OnAxis Performance A conventional achromat consist of two elements. One has positive power and a low relative dispersion (high Abbe number), the other has negative power and a high relative dispersion (low Abbe number). The elements’ powers, required for chromatic aberration correction, are a
Va for the front element, Va Vb
(5.1)
b
Vb for the rear element. Vb Va
(5.2)
and
Va and Vb are the mentioned Abbe numbers for the two elements.
With a na 1 ca and b nb 1 cb , Eqs. (5.1) and (5.2) change to
na 1 ca
Va and Va Vb
nb 1 cb
Vb , Vb Va
from which we can extract the net curvature for the front element a:
39
40
Chapter 5
ca
Va . Va Vb na 1
(5.3)
The net curvature of the rear element b is cb
Vb . Vb Va nb 1
(5.4)
Since the shapes of the elements in an achromat do not influence the chromatic aberration, we make the first element equi-convex, i.e., ca1 ca 2 0.5ca . Further, we give the first surface of element b the same curvature as the second surface of element a, i.e., cb1 ca 2 . With that, ca1
Va ca 2 . 2 Va Vb na 1
(5.5)
Furthermore, since cb1 ca 2 and cb
Vb
Vb Va nb 1
cb 2 ca 2 cb ca1
,
Vb . Vb Va nb 1
(5.6)
For the front element a (crown) we take glass BK7 with na 1.517 and Va 64.17. The rear element b (flint) is made from glass F2 with nb 1.620 and Vb 36.37 . With that we get ca1
64.17 0.01 0.022324 , 2 64.17 36.37 1.517 1
or Ra1 1 / ca1 44.795387 mm and Ra 2 44.795387 mm. With cb1 ca 2 0.022324, cb 2 0.022324
36.37 0.01 0.001223 , 36.37 64.17 1.62 1
and Rb 2 1 / cb 2 817.966784 mm.
Achromats
41
Figure 5.1 Cemented achromat for the VIS spectrum (corrected for on-axis only) after adding thicknesses and optimizing.
Figure 5.2 Blur-spot size of the optimized achromat for the visible spectrum.
42
Chapter 5
Figure 5.3 The encircled energy with reference to the diffraction limit.
Figure 5.4 Modulation transfer function of the achromat.
Achromats
43
This choice of curvatures results in a respectable small spherical aberration of 4
TSC 0.0225 mm . 1
We add reasonable thicknesses of 5 mm and 2.5 mm and optimize. The chromatic aberration remains corrected and the transverse spherical aberration reduces to –0.0156 mm. We summarize our findings for the expected performance with a 4-plot report shown in Figs. 5.1, 5.2, 5.3, and 5.4. These selected plots are from the ZEMAX lens design program, which was used for the optimization of the achromat.
5.2.1 Adding a field to the on-axis corrected achromat Adding a ±2-deg field of view to the on-axis corrected lens shows that there is a relatively large amount of coma present. This is indicated in Fig. 5.5. To also correct for coma, the shapes of the elements need to be changed. We perform this task first by an optimization process with the computer, and then show for comparison Smith’s analytical method.1 This is done to provide more insight into the method of aberration correction.
Figure 5.5 The presence of coma indicates the need for further correction. This is also indicated in the modulation transfer function shown in Fig. 5.6.
44
Chapter 5
Figure 5.6 The modulation transfer function drops off drastically for the 2-deg offaxis point.
Figure 5.7 Improvement of blur spot after optimization.
Achromats
45
5.2.2 Optimizing We include the off-axis aberrations and computer-optimize the configuration from above. The result is shown in Fig. 5.7.
5.3 Smith’s Method1 of Determining the Surface Shapes In addition to satisfying the previously stated requirements of Eqs. (5.1) and (5.2), there are two expressions that lead to the elimination of coma and thin-lens, third-order spherical aberration. These expressions are K1ca1 K 2 K 3cb1 K 4 0 ,
(5.7)
K 7 ca21 K8ca1 K9 K10 cb21 K11cb1 K12 0 .
(5.8)
and
As before, the curvatures are identified as follows: ca1 and ca 2 refer to the first and second surfaces of the front element a, and cb1 and cb 2 classify the curvatures of first and second surfaces of the rear element b. With K5 K3 / K3 , and K 6 K 2 K 4 / K 3 , the first radius of element b (rear element) can be found with cb1 K 5ca1 K 6 .
(5.9)
The Ks are functions of the element curvatures, the inverse object distances, and the familiar G-sums, which are, in turn, functions of the index of refraction of the optical materials used for the elements, as we already know.
5.3.1 Curvatures and inverse object distances The net curvatures for the two elements are ca a na 1 and cb b nb 1 . We assume in our example calculation that the object is located at infinity. For this case, the inverse object distance va for element a = 0. For element b, the inverse object distance vb 1 / f a a .
46
Chapter 5
5.3.2 The K-functions K1 0.25G5a ca
K 7 G4a ca
K 2 G c K3 0.25G5b cb
K8 G2a ca2
2 8a a
K 4 G7b cb vb G8b cb2 K5 K1 / K3 K6 K2 K4 / K3
K9 G1ca3 K10 G4b cb K11 G2b cb2 G5b cb vb K12 G1b cb2 vb G3b cb2vb G6b vb2
Substituting Eq. (5.9) into Eq. (5.8) leads to the quadratic equation
K
7
K10 K 52 ca21 K 8 2 K10 K 5 K 6 K11 K 5 ca1
( K 9 K10 K 62 K11K 6 K12 ) 0
or Aca21 Bca1 C 0 .
To solve for the first curvature of element a, we apply the standard form for a quadratic equation: ca1
B B 2 4 AC , 2A
(5.10)
where A K 7 K10 K 52 , B K 8 2 K10 K 5 K 6 K11K 5
and C K 9 K10 K 62 K11K 6 K12 .
We now relate all these expressions to our example, where the focal length is 100 mm, and the field is ±2 deg. The indices of refraction are na 1.517 and nb 1.620 , and the Abbe numbers are Va 64.17 and Vb 36.37 . Notice, the relative aperture (f/#) does not have to be known for these calculations.
Achromats
47
5.3.3 The calculations a b
64.17 0.01 0.023083 64.17 36.37
36.37 0.01 0.013083 36.37 64.17 ca cb
Element a G1 = 0.5945 G2 = 1.0425 G3 = 1.4342 G4 = 0.5991 G5 = 1.7150 G6 = 1.1159 G7 = 0.6872 G8 = 0.3919
0.023083 0.44647 1.517 1
0.013083 0.021101 1.620 1
K1 = 0.019149 K2 = –0.000782 K3 = –0.010578 K4 = 0.000172 K5 = 1.810242 K6 = –0.057667 K7 = 0.026758 K8 = –0.002079 K9 = 0.000053 K10 = –0.014615 K11 = 0.000392 K12 = –0.000003731
Element b G1 = 0.8137 G2 = 1.3145 G3 = 1.8168 G4 = 0.6928 G5 = 2.0055 G6 = 1.3128 G7 = 0.8114 G8 = 0.5022
The coefficients for the quadratic Eq. (5.10) are A = –0.021136329, B = 0.001681137, and C = –0.000021951. With that, ca1 0.016465574 . Inserting this into Eq. (5.9) results in cb1 1.810267 0.016466 0.057667 0.027859 . The remaining two ca 2 ca1 ca 0.016466 0.04461 0.028195, and curvatures are cb 2 cb1 cb 0.027866 (0.021097) 0.006769. Finally, we get from these curvatures the radii of the elements.
48
Chapter 5
Radii Ra1 = 60.7691mm, Ra2 = –35.4673mm, Rb1 = –35.8950mm, and Rb2 = –147.7323mm We use this prescription, add thicknesses to the elements, and optimize. There is no difference in comparison with the more direct method applied earlier. If one prefers this approach using the K-factors, it is highly recommended to create a program for the lengthy calculations.
5.4 Achromat for the MWIR Region We repeat the calculation for an achromat used in the MWIR (3–5 µm) region, and select silicon for the front element and germanium for the rear element. The indices are na =3.4253 (Si) and nb = 4.0251 (Ge), with the Abbe numbers Va = 236.5 and Vb = 107.3. The powers for the achromat are a
Va 236.5 0.01 0.018305, Va Vb 236.5 107.3 fa
b
1 1 54.6300 mm, a 0.018305
Vb 107.3 0.01 0.008305, Vb Va 107.3 236.5
and fb
1 1 120.4101 mm, b 0.008305
and the thin-lens radii for the silicon element a are R1 = 264.971 mm, R2 = R3 = –R1 = –264.971 mm, and R4 = –971.817 mm. Since elements for the MWIR spectrum cannot be cemented (the cement does not transmit in that region), we have two additional freedoms with the option of making R3 different from R2 and choosing a suitable spacing between the elements. (This freedom of enlarging the spacing between the elements will be discussed later). The lens shown in Fig. 5.8 has been optimized after adding appropriate thicknesses to the elements.
Achromats
49
Figure 5.8 Blur spots for the MWIR achromat with a ±2-deg field.
Figure 5.9 This LWIR achromat is diffraction limited. The circles indicate the size of the Airy disk. This indicates that the f/# could be reduced without losing image quality.
50
Chapter 5
5.5 Achromat for the LWIR Region For completeness, we design an achromat using the same procedure with the same focal length and field coverage for the LWIR (8–12 µm) region. The front element is made from AMTIR-1, with na 2.4975 and Va 113.4 . For the rear element we use zinc sulfide with nb 2.2003 and Vb 22.7. The result is a thin lens with the following radii: Ra1 = 91.8029, Ra2 = 393.0097, Rb1 = −780.7366, and Rb2 =1,243.5363. Adding thicknesses leads to a diffractionlimited lens whose spot sizes are shown in Fig. 5.9.
5.6 Diamond-Turned Hybrid2 Combining refractive and diffractive powers in one optical element is referred to as a hybrid lens. It is frequently used to correct chromatic aberration, and by aspherizing at least one surface, one can also control the spherical aberration. Because most of the infrared materials have a mild spectral dispersion (high Abbe number), the required number of zones for the diffractive phase profile is low. Both the aspheric shape and the diffractive structure can be conveniently generated with the process of single-point diamond turning on an ultra-precision lathe. For protective reasons, the preferred arrangement is to place the asphere with the diffractive profile on the second surface. Figure 5.10 schematically shows the cross section of a hybrid element with the diffractive phase profile, highly exaggerated. We will design such hybrids for both IR regions.
Figure 5.10 Hybrid lens with refractive and diffractive powers.
Achromats
51
5.6.1 Hybrid for the MWIR region
A suitable material for this region is silicon with n 3.4253 . The refractive Abbe number VR 236.5 . The diffractive Abbe number is VD
0 4 2 . S L 3 5
This relation indicates that the diffractive phase profile is independent of the lens material. It also indicates by the negative sign that the chromatic aberration is in the opposite direction of the aberration from the refractive portion of the lens. The required refractive focal length for the chromatic aberration correction is V f R 1 D f , VR
(5.11)
V f D 1 R VD
(5.12)
and the diffractive focal length is f .
The radius of the first diffractive zone is r1 2λ 0 f D ,
(5.13)
and the total number of zones is itotal
D , 2r1
(5.14)
where D is the lens diameter. The step height of the profile is d max
λ0 . n0 1
5.6.1.1 Basic lens shape According to Eq. (5.11), the required refractive focal length is V 2 f R 1 D f 1 100 100.845667 mm . 236.5 VR
(5.15)
52
Chapter 5
From Chapter 2 we apply Eqs. (2.10), (2.11), and (2.12) for a “best-shaped” lens (minimum spherical aberration) and get for the surfaces the radii R1
2 3.4253 2 3.4253 1 3.4253 2 3.4253 1
100.845666 98.690419 mm
and R2
2 3.4253 2 3.4253 1 3.4253 2 3.4253 1 4
100.845667 165.451409 mm .
The transverse spherical aberration of this best-shaped lens, with a 100-mm focal length and an f/4 relative aperture, is TSC
3.4253 4 3.4253 1 100.845667 64 3.4253 1 3.4253 2 43 2
0.033565 mm .
5.6.1.2 Aspherizing To eliminate this spherical aberration, we aspherize the second surface with the method described in Sec. 3.1.2. The conic constant for the aspheric second surface is κ2
2 165.4514093 0.033565 2 R23TSC 0.636465 . n 1 f y23 3.4253 1 100.845667 12.53
5.6.1.3 The diffractive phase profile To correct for chromatic aberration, we design the necessary phase profile and begin with the requirement for the diffractive focal length, V f D 1 R VD
236.5 f 1 100.845667 12, 025.84579 mm . 2
The first zone radius r1 2λ 0 f D 2 0.004 12,025.84579 9.809 mm and the total number of zones needed across the aspheric surface is itotal D 2r1 25 2 9.809 1.274 , which means one full and a second partial zone. The step height d max 0.004 3.4253 1 0.00165 mm.
Achromats
53
5.6.1.4 Comments The phase equation is expressed by r
2π S 2 r 2 S 4 r 4 S6 r 6 ...Sn r n , λ
(5.16)
and the quadratic coefficient for the diffractive surface can be calculated from S2
1 . 2 fD
(5.17)
In the ZEMAX program, the quadratic phase coefficient is expressed as P2
2π 2 R0 S 2 , λ0
(5.18)
where R0 is called the normalizing radius and is usually chosen to be 100. The phase coefficient is S2
1 1 4.158 105 . 2 f D 2 12025.8
When converted into ZEMAX format, it is 2π 2 5 P2 100 4.158 10 653.137 . 0.004
We add a reasonable thickness of 2.5 mm and optimize this hybrid with a 4-deg total field. The result is summarized in Fig. 5.6. As a reference, it is mentioned that the diffraction limit for this lens is BAiry 2.44 f/# 2.44 4 4 39 µm. The size of the square detector element is shown in Fig. 5.11 with 40 µm. Besides using a diffractive profile for chromatic aberration correction, it can also be applied for athermalization of a lens. This will be discussed in Chapter 9.
54
Chapter 5
Figure 5.11 Blur-spot sizes of this optimized MWIR hybrid are equal to the size of the diffraction-limited blur (Airy disk), which is also the size shown for the square detector element (pixel).
Figure 5.12 Nomogram for hybrids in the MWIR region (object at infinity).
Achromats
55
Figure 5.13 Nomogram for hybrids in the LWIR region (object at infinity).
5.6.2 Useful nomograms Two nomograms are inserted here to provide quick answers for the zone numbers required to correct chromatic aberration in the two infrared regions. Figure 5.12 refers to the MWIR region, and Fig. 5.13 to the LWIR band. The example in Fig. 5.12 shows that the first zone radius of an f/1.5 lens with f = 50 mm, made from zinc selenide, is 6 mm, and there are eight zones.
56
Chapter 5
In the LWIR region (8–12 µm), germanium is a superior material with its high Abbe number of about 800. This is indicated by the large separation from the other materials on the lens material scale in Fig. 5.13. Example 1 shows an f/1 lens with a 50-mm focal length. The first zone radius is found to be 6.7 mm, leading to a total of 14 zones across the lens diameter. Example 2 refers to the 100 mm, f/1 germanium lens discussed above. The first zone radius is already quite large with 28.3 mm. This germanium lens requires only three steps (four zones) across its 100 mm diameter.
References 1. W. J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York (2008). 2. M. J. Riedl, Optical Design Fundamentals for Infrared Systems, SPIE Press, Bellingham, Washington (2001).
Chapter 6 Systems with Two Separated Components 6.1 Introduction The telephoto, the reversed telephoto, and the dialyte (which is an air-spaced achromat) belong in this category, and there are others. We limit our discussion to the case where the object is located at infinity. The basic layout of such a system is shown in Fig. 6.1.
6.2 Dialyte—an Air-Spaced Achromat The relations for an air-spaced achromat are the focal length of front element are V f f A 1 B f , VA b
(6.1)
V b f B 1 A VB f
(6.2)
the focal length of rear element b ,
Figure 6.1 General layout of system with two separated components for an object located at infinity.
57
58
Chapter 6
and the separation of the two elements b d 1 f A . f
(6.3)
It can be seen from these expressions that the achromatic doublet is covered as a special case with d = 0, which occurs when b = f. Equations (6.1) and (6.2) then take on the form of the equations presented in Chapter 5, Achromats. There is a maximum useful separation for the elements, and the separation is dependent on the materials used. We can derive this limit by first inserting Eq. (6.1) into Eq. (6.3) and then differentiating the new expression with respect to b. Setting the result to zero brings us the maximum separation d for the dialyte under investigation. Let us demonstrate: b b V f d 1 f A 1 1 B f . f f VA b
(6.4)
We differentiate Eq. (6.4) and find dd VB 2 2 f b 1. db VA
Setting this result to zero yields b
VB f . VA
With that result, Eq. (6.4) changes to the simple expression 2
d max
V 1 B f . VA
(6.5)
At this maximum distance, the focal lengths are equal but, of course, opposite in sign, i.e., f B f A . Typical values for the three spectral windows, VIS, MWIR, and LWIR are listed in Table 6.1. Table 6.1 Maximum possible spacings for dialytes. Region VIS MWIR LWIR
Material Front lens Rear lens BK7 S2 Silicon Germanium AMTIR-1 Zinc sulfide
Abbe number Front lens Rear lens 64.17 36.37 236.5 107.3 113.4 22.7
Spacing dmax 0.06109 f 0.10656 f 0.30536 f
Systems with Two Separated Components
59
Figure 6.2 Ranges of element separations for dialytes with a 100-mm focal length.
Figure 6.2 shows a plot of the element separation d as a function of the back focal length b for dialytes with a 100-mm focal length. The materials of the elements are called out in Table 6.1. It is interesting to observe how the spacing limits increase with wavelength.
6.2.1 Example for the MWIR region For the 100-mm-focal-length objective, we decide on a 10-mm separation of the Si/Ge elements, which is a bit less than the limit called out in Table 6.1. Rearranging Eq. (6.4) leads to a quadratic equation for the back focal length b: b2 d
V V f 1 B b B f 2 0 . VA VA
(6.6)
For our example, this becomes 107.3 107.3 b 2 10 100 1 1002 0 , b 236.5 236.5
b 0.5 135.369979 135.3699792 4 4,536.997886 , b1 74.3378 mm, and b2 61.0322 mm .†
We add reasonable thicknesses to the elements and optimize both configurations with a ±2-deg field. The first form with b = 74.3378 mm leads to a slightly better solution. The performance plots of this preferred configuration are shown in Fig. 6.3. It is preferred because of better performance and compactness. †
Also see Fig. 6.2.
60
Chapter 6
Figure 6.3 Dialyte performance of a configuration, derived from thin-lens expressions.
As an interesting exercise, we use silicon for both elements and make the rear element an aspheric hybrid. The basic prescription for dialyte is listed in Table 6.2, and Fig. 6.4 summarizes the performance of this air-spaced achromat with a hybrid element. EFL = f = 99.998 mm, BFL = b = 94.188 mm, f/4. Table 6.2 Basic prescription of dialyte with a hybrid element.
Surface # 1
Radius 154.2073
2
478.3504
3
−25.7727
4
Thickness
Material
Index n
V-Number
3
silicon
3.4253
236.5
10
air
3
silicon
3.4253
236.5
93.99151
air
−27.9023
Systems with Two Separated Components
61
Figure 6.4 Performance of an air-spaced achromat with a hybrid element.
6.2.1.1 Aspheric deformation coefficients of surface 3 A4 = 4.44547×10–8
A6 = –2.62939×10–10
6.2.1.2 Phase coefficients OSLO and CODE 5 S2 = –6.5155×10–5 ZEMAX with R0 = 100 mm P2 = –1,023.4466 Table 6.3 Diffractive Zones on Surface 3 (for the first diffraction order) Zone # Radius r
1 7.8353
2 11.0808
3 13.5712
62
Chapter 6
6.2.1.3 Step height at the zone transition d max
λ0 4 1.649 µm nb 1 3.4253 1
It can be seen that the performance is somewhat better. An advantage of this design is that it avoids the higher-priced germanium as material for the second element.
6.3 Telephoto and Reversed Telephoto Figures 6.5 and 6.6 show the telephoto and reversed telephoto arrangements. The focal lengths for the elements are the same for both configurations, namely df f b
(6.7)
db . d b f
(6.8)
fA
and fB
Figure 6.5 The basic telephoto system.
Figure 6.6 The basic reversed telephoto system.
Systems with Two Separated Components
63
6.3.1 Examples for the MWIR and LWIR regions We present a reversed-telephoto system that was derived from the basic layout shown in Fig. 6.6. The curved surfaces have been strongly aspherized. For its specific application, a cold stop and a spectral bandpass filter have been placed inside a Dewar housing. The objective covers a large field of ±13.5 deg and has a relative aperture of 1.5. The front element is made from zinc sulphide. The material for the rear element and the Dewar window, as well as the filter substrate, is silicon. The next example of a reverse telephoto for the LWIR region, as shown in Fig. 6.8, uses only germanium for all elements including the filter. It has a 50mm focal length and a relative aperture of f/1.5. The field coverage is ±10 deg. The front element is a hybrid with the phase profile on the aspherized second surface. The second surface of the rear element is also aspherized. The aperture stop is located at the front surface of the first element.
Figure 6.7 Reversed telephoto for the MWIR window, after Aldrich.
64
Chapter 6
Figure 6.8 Reversed telephoto for the LWIR with a hybrid front element.
Chapter 7 From an Air-Spaced Doublet to a Triplet 7.1 Introduction As an informative exercise, we begin with a suggested formulation1 for a Petzval objective as shown in Fig. 7.1.
Figure 7.1 Basic Petzval objective as a convenient starting configuration.
Figure 7.2 The proposed triplet. 65
66
Chapter 7
To correct chromatic aberration, we split the front element to form an achromatic triplet.
7.2 Chromatic Aberration The thin-lens third-order transverse chromatic aberration of the system is expressed by c
TAchC a
y 2 . Vu 'k
(7.1)
For an object at infinity, u'k y . With that, Eq. (7.1) changes to TAchC
1 ya total
y 2 a V . c
(7.2)
To eliminate chromatic aberration, we have to satisfy the condition y 2 ya2 a yb2 b yc2c a V V V V 0. a b c c
(7.3)
We know that a b 0.5tot , and therefore b 0.5tot a . Further, c tot , ya yb y, and yc 0.5 y . After substitution and solving for the power of the first element, we get a
Vc Vb / 2 Va . tot 2 Va Vb Vc
(7.4)
By choosing the same material for elements a and c, Eq. (7.4) simplifies to read a
Va Vb / 2 . tot 2 Va Vb
(7.5)
7.3 Example, a Conventional Triplet We design an objective, again for the MWIR region, maintaining our 100-mm focal length. However, we reduce the relative aperture from f/4 to f/2 to collect more energy. We also expand the field of view and double it to ±4 deg.
From an Air-Spaced Doublet to a Triplet
67
Elements a and c are made from silicon with na = 3.4253 and Va = 236.5. The material for element b is germanium with nb = 4.0251 and Vb = 107.3. The power of the front element is therefore a
236.5 107.3 / 2 0.01 0.011229, and f 1 a 2 236.5 107.3 a
89.057 mm .
b 0.5 0.01 0.011229 0.006229, and f b 160.539 mm.
The power of the rear element is c tot 0.01 . Therefore, f c f tot 100 mm. Assuming the shapes of the positive elements a and c to be convex/plano, and the negative element c plano/concave, we find the surface curvatures to be ca1
a 0.011229 0.004630 , na 1 3.4253 1
cb 2
b 0.006229 0.002059 , 1 nb 1 4.0251
cc1
c 0.01 0.004123 . nc 1 3.4253 1
and
The radii are therefore: Ra1 = 215.987 mm, Ra2 = infinity, Rb1 = infinity, Rb2 = 485.673 mm, Rc1 = 242.542 mm, and Rc2 = infinity. We add thicknesses to the elements and adjust the spacings before we optimize. The final result is shown in Fig. 7.3.
7.4 Hybrid Petzval Objective with a Cold Stop2 We replace the front doublet of the conventional configuration shown in Fig. 7.3 with an aspheric hybrid element and position the aperture stop inside a Dewar housing where the stop will be kept cold so that the detector will only receive energy from the scene. Included in the optical design is also the Dewar window and the spectral filter, which is also located inside the Dewar housing. This configuration, as shown in Fig. 7.4, is frequently used for IR imaging systems.
68
Chapter 7
Figure 7.3 Conventional triplet, derived from the basic Petzval form.
Figure 7.4 Hybrid Petzval objective for MWIR with a cold stop.
Reference 1. W. J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York (2008). 2. M. J. Riedl, “The design of an IR Petzval objective using an aspheric and diffractive element: an exercise with MOE for beginners,” Proc. SPIE 5865, 586501 (2005).
Chapter 8 A Hybrid for Two Wavelengths 8.1 Introduction A single element that is corrected for spherical aberration for two chosen wavelengths can be advantageous for alignment purposes in the visible spectrum when the application is in the infrared region. We design such an element for the helium-neon and the carbon dioxide lines. The wavelength for a HeNe laser is 0.6328µm, and is 10.6 µm for a CO2 laser. The design principle is to determine the shape of the lens for the longer wavelength first and then place a diffractive structure on one side of the lens to add the required power for the shorter wavelength to maintain focus position. Since the wavelength ratio is large, 10.6/0.6328 ≈ 17, the step height of the diffractive structure for the short wavelength is d
λ 0.6328 0.4 µm . n 1 2.5907 1
This is in the domain of surface roughness for the infrared region. First, we have to pick a material that is fitting for both wavelengths. We choose ZnSe with an index of refraction of n0 6328 2.5907 and n10 6 2.4027 . We elect for our design example 100 mm as a focal length and a relative aperture of f/# = 2, and begin with a thin-lens pre-design.
8.2 Basic Lens Shape for the Long Wavelength Considering that it is much easier to place a diffractive structure on a flat surface when the number of zones is high, which will be the case here; we opt for an aspheric plano-convex shape and aspherize the first surface to eliminate spherical aberration. The front radius is R1 n10 6 1 f CO2 2.4027 1 100 140.27 mm . This shape leads to a focal length for the shorter wavelength of f HeNe
R1 n0 6328 1
140.27 88.1054 mm . 2.59207 1
To maintain a common focal length for both wavelengths, a negative diffractive power has to be added to the flat surface. That diffractive power is determined with D CO2 HeNe . The required focal length for the diffractive optical element (DOE) is, therefore, 69
70
Chapter 8
fD
f
f HeNe f CO2 HeNe
f CO2
88.1054 100 740.72 mm . 88.1054 100
8.3 Required Diffractive Profile The first zone radius of the diffractive structure is1 r1 2λ HeNe f D 2 0.0006328 740.72 0.9682 mm ,
and the total number of zones required is 2
itotal
2
D 50 667 . 2r1 2 0.968
As hinted before, this is a large number of zones, which is most likely more cost effective to etch than to diamond-turn. The individual zone radii can be obtained by ri r1 i 0.9682 i . From this, one can see that at the edge of the lens the zones are only 19 µm apart. We add a thickness of 6 mm to the lens, aspherize the front surface slightly in the optimization process, and obtain a lens that is within the quarter-wave Raleigh limit for both wavelengths, as can be seen from the plots in Fig. 8.1.
Remark If we consider the zone step height as the P-V surface roughness for the long wavelength application, and take a ratio of 2 between the P-V and RMS values, we have δ 0.5 P V 0.5 0.4 0.2 µm .
Figure 8.1 The quarter-wave Raleigh limit has been maintained for both cases.
A Hybrid for Two Wavelengths
71
The total integrated scatter of an optical surface is expressed by TIS 4πδ λ .1 For our case this amounts to TIS 4π 0.2 10.6 5.6% , a relatively low amount of the energy that will become stray radiation and reduce the contrast ratio (MTF) somewhat. 2
2
Reference 1. M. J. Riedl, Optical Design Fundamentals for Infrared Systems, SPIE Press, Bellingham, Washington (2001).
Chapter 9 Athermats 9.1 Introduction Athermats are lenses that are designed to compensate for the focus shift that occurs with temperature excursions. The changing parameters of an optical element are the radius, the thickness, and the index of refraction. The spacing of the lens from the detector also changes and is a function of the coefficient of expansion of the housing material.1
9.2 Focus Shift of a Refractive Element The power (reciprocal of the focal length f) of a thin lens is given by 1 1 n 1 , R1 R2
(9.1)
where n = index of refraction, and R1 and R2 are the surface radii. Differentiation of Eq. (9.1) with respect to temperature yields 1 dR 1 dR 1 d 1 dn n 1 2 1 2 2 . dt R1 dt R2 dt R1 R2 dt
(9.2)
We recognize in this equation that 1 R1 dR1 dt 1 R2 dR2 dt α L , the thermal coefficient of the lens material. Therefore, 1 1 d 1 1 dn n 1 α L α L . dt R2 R1 R2 dt R1
Rearranging and making use of Eq. (9.1) results in d dn . dt n 1 dt
(9.3)
Since 1 f , and therefore d dt 1 f 2 df dt , for a finite temperature change of t we get a change in focal length of 73
74
Chapter 9
dn / dt f f α L t . n 1
dn / dt / n 1 L
(9.4)
is called the thermal glass constant TR. With that, the
focus shift of a thin lens is simply f TR f t .
(9.5)
9.3 Athermalization with a Doublet For two elements, defocusing due to a temperature change is given by T T f f f a b α h t . f a f b
(9.6)
Element a is the front element and b is the rear element. h is the thermal coefficient of the housing in which the lens is mounted. Setting f = 0 to eliminate any focal shift requires that the focal length of the front element a is T T fa b a f . Tb α h
(9.7)
The focal length of the rear element b is then T T fb a b Ta α h
f , or
fa f . fa f
(9.8)
9.4 Focus Shift of a Diffractive Lens Based on the quadratic term of the general phase equation, the first zone radius of a purely diffractive element is r1 2λ 0 f D .
Therefore, the focal length is
(9.10)
Athermats
75
fD
r12 . 2λ 0
(9.11)
The zone radius r1 as a function of the temperature change t is r1 ( t ) r1 1 α L t ,
(9.12)
where L is the thermal coefficient of expansion of the lens material and t is the temperature change from the design temperature t0. The diffractive focal length at the changed temperature t can be expressed by f D t
r12 t r12 2 1 2α L α 2L t , 2λ 0 2λ 0
and by neglecting the term α 2L t , 2
f D t
r12 1 2α 2L t . 2λ 0
(9.13)
Since f D t f D f D , the change in focal length due to a change in temperature of t is f D 2α L f D t ,
(9.14)
which means the thermal glass constant of a diffractive lens is simply TD 2α L .
(9.15)
This indicates that the change of a diffractive element due to a change in temperature is independent of the index of refraction of the element material. Due to the different behavior of a diffractive element, the equations for the elements’ focal lengths (to eliminate any focal shift caused by a change in temperature) are T TD fR R f . TD α h
(9.16)
fR is the refractive focal length of the element, and f D is the diffractive focal length of the element.
76
Chapter 9
T T fR f . f D R D f or f D fR f TR α h
(9.17)
A positive temperature change of the refractive portion of the hybrid reduces the focal length [see Eq. (9.5)]. As the zone radii r of the diffractive portion (phase profile) increases with a temperature increase, the diffractive focal length also increases [see Eq. (9.14)]. It is important to remember that the diffractive focal length is positive when the steps at the zone transition are negative. The focal length is negative when the steps are cut in the opposite direction. The direction of the transition steps is illustrated in Fig. 9.1. The sign of the focal length is indicated in the phase equation r
2π S 2 r 2 S4 r 4 S6 r 6 ... , λ0
(9.18)
Figure 9.1 Direction of phase profile and effect of sign on focal length.
Athermats
77
where the first coefficient S2
1 . 2 fD
(9.19)
The size of the steps at the zone transition is d
λ0 . n0 1
(9.20)
The change in step height with a temperature change is minimal and results in a very slight shift of the wavelength at which the theoretical diffraction efficiency is 1. This change can be neglected and can be confirmed with λ λ 0 d 1 α L t n dn / dt t 1 .
(9.21)
The symbols used in this equation have been defined above. Tables 9.1 and 9.2 list the properties of the most frequently selected materials for optical elements for the infrared spectra.
Table 9.1 Selected materials for the 3–5 µm spectral band (MWIR). Optical material Ge Si ZnS ZnSe AMTIR-1 MgF2 CaF2 BaF2 As2S3 Al2O3 GASIR-1 GASIR-2
Refractive index n (at 4 m) 4.025 3.425 2.253 2.433 2.514 1.349 1.410 1.457 2.412 1.675 2.510 2.606
Abbe Index number change / C V (at 3–5 m) dn/dt (10–6) 103 396 236 150 110 43.3 179 60 198 72 13.5 2 21.7 –7.5 45.1 –16 181 9.3 7.65 3.4 196 67 170 70
Coefficient of expansion / C L (10–6) 5.7 2.6 6.6 7.1 12 12 19 18 21 6 17 16
Thermal glass constant –6 TR (10 ) 125.2 59.3 28.0 34.8 35.6 –6.3 –37.3 –53.0 –14.4 –1.0 27.4 27.6
78
Chapter 9
Table 9.2 Selected materials for the 8–12 m spectral band (LWIR). Optical material Ge ZnS ZnSe AMTIR-1 GaAs CaTe NaCl CsI GASIR-1 GASIR-2
Refractive index n(10 m) 4.003 2.200 2.406 2.498 3.278 2.672 1.495 1.740 2.496 2.586
Abbe number V(8–12 m) 1000 23 58 110 110 170 19 230 120 101
Index change / C –6 dn/dt (10 ) 396 39 61 72 185 97 –25 –89 67 70
Coefficient of expansion / C L (10–6) 5.7 6.6 7.1 12 5 5 44 50 17 16
Thermal glass constant –6 TR (10 ) 126.2 25.9 36.3 36.1 76.2 53.0 –94.5 –170.3 27.8 28.1
9.5 Design Examples We examine two f/4 lenses with 100-mm focal lengths used in the LWIR region at 10 µm. The first one consists of two elements, and the second one is a diffractive hybrid. The thermal conditions are the same for both cases. The design temperature is 20°C, and the temperature extremes are –20°C and +60°C. This means t = ±40°C. Both lenses are mounted in aluminum housings.
9.5.1 Athermat with two elements in an aluminum housing
For the front element we choose germanium with Ta = 126.2×10–6. The rear element is made from zinc sulfide with Tb = 25.9×10–6 . The thermal coefficient of expansion of the aluminum housing ah = 23×10–6. Equation (9.7) yields 25.9 126.2 fa 100 205.112 mm . 25.9 23
This leads, according to Eq. (9.8), to fb
fa f 205.112 100 67.225 mm . f a f 205.112 100
For simplicity, we use a plano-concave and a plano-convex lens, for which the radii are Ra1 na 1 f a 4.003 1 205.112 615.953 mm and Rb1 nb 1 f b 2.200 1 67.225 80.670 mm. Ra2 and Rb2 are infinite.
Athermats
79
9.5.2 Hybrid athermat in an aluminum housing We use zinc selenide as the lens material, with n = 2.406 at 10 µm. TR = 36.3 and TD =14.2. Using Eqs. (9.16) and (9.17), we get T TD 36.3 14.2 fR R f 100 573.86 mm 14.2 23 TD h
and fD
573.86 100 85.16 mm . fR f f R f 573.86 100
For the plano-concave lens, the front surface radius R1 n 1 f R 2.406 1 573.86 806.472 mm. R2 is infinite.
is
To determine the details of the phase profile for the flat rear surface of the lens, we begin by using Eq. (9.19) and get the required phase coefficient, S2
1 1 0.0058713 . 2 fD 2 85.16
For reference, we translate S2 into ZEMAX format, which is P2
2πRn2 2π 1002 S2 0.0058713 36,890.5 . λ0 0.01
The first zone radius is found with Eq. (9.10): r1 2λ 0 f D = 2 0.01 85.16 =1.305067 mm
and r2 r1 2 1.305067 2 1.84564 mm.
The other zone radii n can be found with ri r1 i . The total number of zones required is i D 2r1 25 2 1.305067 92 . D is the lens diameter. It is 25 2
2
80
Chapter 9
mm for the f/4, 100-mm focal-length objective. The step size at the zone transition is d λ 0 n0 1 10 2.406 1 7.11 µm.
9.6 Impact of Housing Material We analyze how different housing materials impact the focus shift of a hybrid lens. The design focal length is 100 mm, and the optical material is zinc selenide with n = 2.406 and L = 7.1×10–6, which means that the thermal glass coefficients are TR = 36.3×10–6 (see Table 9.2) and TD 2 L 2 7.1 106 14.2 106 . The second lens surface is chosen to be a plano for all cases. h is the thermal coefficient of the housing material, fR is the refractive focal length of the hybrid, fD is the diffractive focal length, R1 is the first surface radius of the element, r1 the first diffractive zone radius, and i the total number of required zones across the f/4 lens diameter. Table 9.3 Change of parameters for a hybrid lens as a function of housing material. Material Alum Steel Invar
–6
h(×10 ) 23 12 1.3
fR –573.86 2,295.45 391.47
fD 85.16 104.55 134.09
R1 (mm) –806.847 3,227.403 550.407
r1 (mm) 1.30507 1.44603 1.63954
i 92 75 59
Remark It can be seen from Eq. (9.16) that the lens will be a plane-parallel plate with a diffractive structure on the second surface, if the thermal expansion coefficient of the housing h is equal to the thermal glass coefficient TD, which is 2L, i.e., twice the thermal coefficient of expansion on the lens material. AMTIR-1 (L = 12 ×10–6) and aluminium 6061 (h = 23×10–6) are such a matching combination. We investigate the lens at =10 µm with f = 100 mm, and D = 25 mm (f/4): T T 36.1 24 fR R D f 100 infinite , 24 24 TD α h T T 36.1 24 fD R D f 100 100 f , 36.1 24 TR α h
and S2
1 1 0.005 . 2 fD 2 100
Athermats
81
Figure 9.2 Flat AMTIR-1 LWIR athermat in an aluminium housing.
The ZEMAX phase coefficient is P2
2πRn2 2π 1002 S2 0.005 31, 415.927 . λ0 0.01
r1 2λ 0 f D 2 0.01 100 2 1.4142 .
The total number of zones across the 25-mm lens diameter is 2
2
D 25 i 79 . 2r1 2 2
Figure 9.2 shows the blur spots of this flat, purely diffractive element at –20°C, + 20°C (the design temperature), and at + 60°C.
9.7 Athermat for the CO2 Laser Line We analyze an f/4 lens with a 100-mm focal length made from ZnSe. The thermal index variation for ZnSe is dn/dt = 60×10–6, the index of refraction at 10.6 µm (CO2-line) is n = 2.40266, and the thermal coefficient of the element is L = 7.6×10–6. The diffractive glass constant TD 2 7.1 106 14.2 106 , and the refractive one is 60 7.1 106 35.676 106 . TR 2.40266 1
82
Chapter 9
With the refractive 100-mm focal length we get 35.676 fD 100 251.1239 . 14.2
From Eq. (9.9) we obtain the quadratic phase coefficient S2
1 1 0.001991 . 2 fD 2 251.1239
The first zone radius r1 2λ 0 f D 2 0.0106 251.1239 2.307342 mm .
For the full lens diameter of D = 100/4 = 25 mm, the total number of required zones is itotal D 2r1 25 / 2 2.307342 30 . 2
2
Remark ZEMAX applies a normalized radius of R0 =100 mm for the phase coefficient P. With that, P2
2πR02 2π1002 S2 0.001991 11,801 . λ0 0.0106
We put the numbers from above into the ZEMAX program, add a suitable thickness to the lens, and optimize. The result is shown in Fig. 9.3. The top row shows the increase of the blur spot for a purely refractive singlet. The lower row indicates that the blur spot size stays within the diffraction limit over the entire 120°C temperature excursion.
Danger! To demonstrate the effect of machining the phase profile in the wrong direction (which has happened), we show in Fig. 9.4 how the slopes of the exiting rays are changed. The focal length of our example lens would increase from 100 mm to 485 mm at the design temperature of 20°C.
Athermats
83
Figure 9.3 Change of blur-spot size due to defocusing caused by a temperature change of ±60°C from the 20°C design temperature without the compensating diffractive structure.
Figure 9.4 Devastating effects with the wrong (reversed) phase-profile direction. See Fig. 9.1 for direction definition.
84
Chapter 9
9.8 Athermalized Achromat To achromatize and athermalize an objective, we apply the following equations: total power, j
i ,
(9.22)
i1
athermalization, j
T α , i
i 1
i
h
(9.23)
and a chromatization, j
i
V 0 . i 1
i
(9.24)
As previously stated, V is the Abbe number. VR is for a refractive element, and VD is for a diffractive one. For the infrared spectrum VR
nmiddle 1 nshort nlong
VD
λ middle . λ short λ long
and
The required powers (inverse focal lengths) for the three elements are 1 / V1 T1 h 1 1 / V3 1 / V1 T3 T1 2 , f 2 1 / V1 1 / V2 T1 T2 1 / V3 1 / V1 T3 T1 3
1 1 / V1 1 / V2 1 / V1 2 , f 3 1 / V3 1 / V1 1 / V3 1 / V1
(9.25)
(9.26)
Athermats
85
and 1
1 2 3 . f1
(9.27)
We demonstrate the use of these seemingly cumbersome equations with two examples of an achromat for the MWIR region. First, we use three elements and then two with an added diffractive phase profile on the rear surface of the second element. Both objectives are mounted in aluminum housings.
9.8.1 Three-element athermat in an aluminum housing As lens material, we elect for the elements of the 100-mm focal-length f/4 lens‡:
Front element 1: Silicon with V1 = 236, and T1 = 59.3×10–6, Center element 2: Germanium with V2 = 103, and T2 = 125.2×10–6, and Rear element 3: Zinc sulphide with V3 = 110, and T3 = 28.0×10–6
The thermal coefficient of expansion for aluminium is h = 23×10–6. With Eqs. (9.25), (9.26), and (9.27), we obtain the elements’ focal lengths f1 = 57.6334 mm, f2 = –92.3000 mm, and f3 = 287.0930 mm. After adding appropriate thicknesses to the elements and maintaining the proper powers of the elements, we optimize and get the configuration shown in Fig. 9.5. We also added an angular field of ±2 deg.
Figure 9.5 Performance of an athermalized, 100-mm, three-element achromat in an aluminium housing. There is no change over a temperature excursion of ±40°C. ‡
Also see Table 9.1
86
Chapter 9
9.8.2 Two-element athermat in an aluminum housing We reduce the number of elements but add a diffractive profile on the second surface of the rear element. The front element is again made from silicon, and for the rear element we use germanium. The elements’ refractive parameters remain V1 = 236, T1 = 59.3×10–6, and V2 = 103, T2 = 125.2×10–6. The Abbe number for the diffractive surface is V3
λm 4 2 λ s λl 3 5
and T3 2α 3l 2 5.7 106 11.4 106 .
We recall that the thermal coefficient of expansion for the aluminum housing is α h 23 106 . Equations (9.25), (9.26), and (9.27) lead to the elements’ focal lengths. They are: f1 = 44.291 mm, f2 = –79.832 mm, and the diffractive focal length f3 = –19,273.3 mm. This long focal length indicates that the chromatic aberration to be corrected with the phase profile is very small. Only two diffractive zones are required. Figure 9.6 is included for comparison with the three-element design. Note, there is very little difference. The advantage, of course, is the savings of the cost of one element, weight, and space.
Figure 9.6 Performance of an athermalized 100-mm focal-length achromat with two elements and a diffractive surface in an aluminum housing.
Athermats
87
9.9 Effect of Quarter-Wave Limit without Athermalization Before adding the cost of optical system athermalization, it is wise to determine the temperature range over which the required performance is acceptable. We demonstrate the procedure with a singlet mounted in a housing with the same length as the focal length of the lens. For a single element, Eq. (9.6) reduces to f f TR ah t .
(9.28)
We assume that the lens is diffraction limited and allow defocusing of f 2 f / # , which represents the well-known Rayleigh quarter-wave criterion. § This changes Eq. (9.28) to 2
t
2 f / #
2
f TR h
.
(9.29)
Table 9.4 lists the allowable temperature depatures from the design temperature for three diffraction-limited singlets mounted in an aluminium housing. The chosen focal length is again 100 mm, and the relative aperture is f/4. Table 9.4 Allowable temperature excursions for quarter-wave defocusing. Spectral region VIS MWIR LWIR
(μm)
n
0.5876 4 10
1.5168 3.4253 4.0043
Lens material BK7 Si Ge
TR –6 (x10 ) –4.0137 61.7189 131.7705
Δt (°C) ±9.4 ±15.1 ±20.7
It is interesting to note that the focal length of the BK7 lens increases with the allowable increased temperature of 9.4°C from 100 mm to 100.004 mm, while the focal length of the silicon lens reduces from 100 mm to 99.908 mm. For the germanium lens, the focal length shortens to 99.729 mm.
§
The Rayleigh criterion states that a wavefront error of a quarter wave does not noticeably affect the image quality.
88
Chapter 9
Figure 9.7 Performance plots for a diffraction-limited germanium lens in an aluminium housing at a design temperature of 20°C.
Figure 9.8. Performance plots for the lens at 40.7°C. The quarter wave optical path difference (OPD) is indicated.
The performance plots of the germanium lens at the design temperature of 20°C and at the elevated temperature of 40.7°C are shown in Fig. 9.7 and Fig. 9.8, respectively.
Reference 1. M. J. Riedl, Optical Design Fundamentals for Infrared Systems, SPIE Press, Bellingham, Washington (2001).
Chapter 10 The Ball Lens 10.1 Introduction A ball lens is an interesting optical element worth a closer look. Figure 10.1 indicates the optical behaviour of such a lens. It can be seen that the principal planes fall together and are located at the center of the sphere. For the aberrations, we only consider here the case where the object is located at infinity. Some remarks will be made for other situations. The focal length is f
Rn , 2 n 1
(10.1)
and the back focal length is bfl f R
2 n R , 2 n 1
(10.2)
Figure 10.1 Ball lens. The white lines suggest that the lens could be made into a cylinder for easier mounting.
89
90
Chapter 10
where R is the radius of the sphere, and n is the refractive index of the lens material. Deriving the third-order aberrations is straightforward and instructive. We will do that here and limit ourselves to the discussion of the spherical aberration, coma, and astigmatism.
10.2 Spherical Aberration The contribution of the first surface is TSC1
n1 n'1 n1 y1 u'1 i1 i12 , 2n'1n'2u'2
(10.3)
recognizing that n1 n'2 1, and calling n'1 n , we find by paraxial ray tracing,
u'1 n 1 y1 nR , i1 y1 R , and u'2 2 n 1 y1 nR . Inserting all these relations into Eq. (10.3) results in the simple expression TSC1
y3 , 4nR 2
(10.4)
where we substituted y for y1 . For the second surface we go about it in the same way. The surface contribution is TSC2
n2 n'2 n2 2 y2 u'2 i2 i22 . 2n'2 n'2u'2
The substitutes are n2 n and n'2 1 ,
y2
2 n y ,
i2
n
y , Rn
and as before, u'2
2 n 1 y . nR
(10.5)
The Ball Lens
91
With that we get TSC2
n 2 2n 1 y 3 .
(10.6)
4n 2 R 2
Adding these two surface contributions together leads to the straightforward expression for the third-order transverse spherical aberration of the ball lens. We remember that y is the ray height at the first surface as indicated in Fig. 10.1. Transverse spherical aberration is n n 3 1 y 3 TSCball . 2n 2 R 2 lens
(10.7)
We recall that the blur spot size due to spherical aberration is Bspher TSC 2 , or angularly expressed βspher ball
n n 3 1 128 n 1
2
f / #
3
.
In this format, we recognize that the relative aperture f/# f
(10.8)
2y .
10.3 Coma To get the expression for coma, all we have to do is to multiply the spherical aberration surface contributions with the factor i p1 i1 i p 2 i2 u p R y . The symbol u p is the half-field angle. This leads to CCball
lens
n n 3 1 y 2u p . 2n 2 R
(10.9)
Angularly expressed, with Bcoma CC , the extent is n n 3 1 u p f β coma . 2 ball 16n n 1 f / #
(10.10)
92
Chapter 10
10.4 Astigmatism By multiplying the coma expression once more with the factor i p1 i1 i p 2 i2 u p R y , we get the third-order astigmatic contribution. The result is n n 3 1 yu 2p . TAC 2n 2
(10.11)
The minimum astigmatic blur size is Bastig 2TAC . The angular measure is then ball
n n 3 1 u 2p β astig . 2n 2 f / # ball
(10.12)
Example After that somewhat laborious preparation, we demonstrate the use of these equations by designing a ball lens, made from BK7, for use with a HeNe laser ( = 0.6328µm). The index of refraction of BK7 is, for that wavelength, n 1.5151 . We employ a relative aperture of f/2.5, and a rather large half-field angle of u p 0.05 rad 2.8645 deg. The ball radius R = 2.5 mm. The results are: Transverse spherical aberration
TSC = −0.0173 mm
Blur-spot size, linear
Bspher 0.0087 mm
Blur-spot size, angular
βspher 2.35 mrad
Coma
CC = −0.0029 mm
Blur-spot size, linear
Bcoma 0.0029 mm
Blur-spot size, angular
β coma 0.80 mrad
Transverse astigmatism
TAC = −0.0005 mm
Blur-spot size, linear
Bastig 0.001 mm
Blur-spot size, angular
β astig 0.272 mrad
The focal length f = 3.6767 mm, and the back focal length bfl = 1.1767 mm.
The Ball Lens
93
Figure 10.2 Image locations inside a ball lens, a function of the refractive index.
From Eq. (10.2) it can be seen that if n > 2, the focus falls inside the lens. Therefore, a ball lens is not suitable for normal use if it is made from zinc selenide, silicon, or germanium when the object is located at infinity. This is indicated in Fig. 10.2. The image locations can be derived by using Eqs. (1.7) and (1.9) from Chapter 1. With n being the index of the ball material, and choosing the ray height y = 1, Eq (1.7) changes to nu' n 1 R , and from Eq. (1.9) we get u' 1 / l' . This leads to l' n / n 1 R . Table 10.1 Image distances inside the ball lens (n>2) for an object at infinity. Material LASF35 Zinc Selenide Silicon Germanium
Index of refraction n 2 2.4 3.4 4.0
Image distance l′ 2R (rear surface) 1.7143 R 1.4167 R 1.3333 R
For the image to fall on the backside of the ball lens, the object has to be placed at a distance l in front of the lens. This distance is a function of the lens material and is expressed by l
2R . n 2
(10.13)
Table 10.2 lists this distance for selected materials as a function of the ball radius. Figure 10.3 shows the relative positions for the different materials.
94
Chapter 10
Table 10.2 Object distances for image to fall on the rear side of a ball lens. Material Glass Zinc Selenide Silicon Germanium
Index of refraction n 1.5 2.4 3.4 4.0
Object distance l -4R (virtual) 5R 1.4286R R
Figure 10.3 Limits of object distances for different materials.
Chapter 11 Seidel and the Pegel Diagrams 11.1 Introduction In 1856, Ludwig von Seidel published his famous formulae for the primary aberrations. They are based on the truncation of the series expansion for the sine of an angle to sin x x x3 / 6 . That is why these calculations are also referred to as third-order aberration calculations. This method has been applied in evaluating the performance of the presented examples in the previous chapters. Besides providing relatively quick answers, one additional important feature of these calculations is that they point out the individual surface aberration contributions. This, in turn, indicates the sensitivity to radii, thickness, and spacing tolerances. The desire is to have the “work” equally divided among the surfaces; meaning one likes to avoid wild swings between under correction and overcorrection from surface to surface. While the sheer numbers of these surface contributions give the full information, a graphic depiction, which is called a Pegel diagram, is even more telling (Pegel is the German word for level). Another name for this form of representation may be “aberration distribution diagram.” Of course, the ideal case would be if all surfaces would be free of aberrations. Since this is not the case, aberration balancing rather than aberration correction more accurately describes the effort to improve the image quality.
11.2 Triplet for the LWIR Region We shall demonstrate aberration balancing by analyzing a triplet with a cold stop for the LWIR. The layout of the objective is shown in Fig. 11.1.
Figure 11.1 An f/1.4 triplet, with a 100-mm focal length and a field of ±3 deg. 95
96
Chapter 11
11.2.1 Surface contributions We limit the presentation to spherical aberration, but remember that the other Seidel aberrations can be shown in the same manner. The surface contributions for spherical aberration are listed in Table 11.1. Table 11.1 Seidel surface contributions in mm. Surface TSC Surface TSC
1 +0.008224 6 +0.404949
2 +0.149244 7 –0.582071
3 –0.629787 8 +0.571206
4 +0.088954 9 –0.307748
5 –0.004288 10 +0.305024
The sum of the surface contributions is 10
TSCtotal TSC 0.003707 mm . 1
11.2.2 Pegel diagram for the triplet The diagram in Fig. 11.2 shows the summation of the spherical aberration from surface to surface. After the last surface, the size of the remaining system aberration is the above stated 0.003707 mm.
Figure 11.2 This diagram clearly shows the aberration contributions of the individual lens surfaces.
Seidel and the Pegel Diagrams
97
11.2.3 Remarks to field curvature Astigmatism is well corrected for this lens, so that the field curvature is basically represented by the Petzval curvature. Josef Max Petzval, an Hungarian mathematician, is known for this theorem, which states that in absence of astigmatism there is a field curvature, expressed by the radius ρ
h' 2 , 2 PC
(11.1)
where h′ is the image height, and PC is the sum of the third-order longitudinal Petzval contribution. For our example lens with a 100-mm focal length and a 3-deg half-field angle, h' f tan u p 100 tan 3deg 5.241 mm . PC is determined by the same method as the other third-order aberration contributions. We find PC to be –0.04357. This leads to a Petzval radius of
ρ h' 2 2 PC 5.2412 2 0.04353 315.5 mm . The negative sign indicates that the curvature is bent inwards, i.e., towards the lens. This inward bending of the field curvature was especially pronounced in the earlier cameras where the objectives were made up of but a few elements, or even just one. To compensate for this defocusing effect at the edge of the image, photographers at the time used the trick in group pictures to align the assembly in an arc. This is demonstrated in Fig. 11.3.
Figure 11.3 Petzval curvature “applied” for the object. (Shown in the picture are some of the author’s professors from the Academy of Applied Sciences in Munich.)
98
Chapter 11
Figure 11.4 Maksutov-Mangin combination for the LWIR region.
11.3 Cassegrain Version with a MaksutovMangin Mirror Combination for the LWIR Region A more complex design example is shown in Fig. 11.4. The focal length of the f/2.15 system is 554 mm. The field coverage is ±0.5 deg. The track length (overall length from the first surface to the focal plane) is 384 mm. All elements are germanium with the exception of one, which is made from zinc selenide. The center portion of the inside Maksutov window is the secondary mirror of the basic Cassegrain configuration. There are no aspheric surfaces in the system.
11.3.1 Pegel diagram We elected this more elaborate system to show how a Pegel diagram provides a quick overview of the aberration contributions. We limit ourselves again to the display of the spherical aberration. It can be seen at a glance how strong a contributor the second surface is, and how the third one overshoots with its compensation. At the end, the sum is exceedingly small, with mm. The plus sign indicates overcorrection.
TSC = +0.01616
The individual surface contributions are listed in Table 11.3, and the Pegel diagram is shown in Fig. 11.5.
Seidel and the Pegel Diagrams
99
Table 11.3 Third-order transverse spherical aberration surface contributions. Surface TSC Surface TSC Surface TSC Surface TSC
1 –0.040580 6 –0.526075 11 +0.784353 16 +0.166109
2 +2.838347 7 –0.046702 12 +0.001294 17 –0.177425
3 –4.117814 8 +0.046419 13 +0.127990 18 +0.174250
4 +0.858031 9 +0.259682 14 +0.026294 TSC +0.016161
5 +0.417371 10 –0.671524 15 –0.103859
Figure 11.5 Pegel diagram for the Cassegrain version depicted in Fig. 11.4.
Chapter 12 The Single-Imaging Mirror 12.1 Introduction Included in this discussion are mirrors with spherical and aspheric surfaces, which are single-surface reflectors, and the Mangin mirror, which is a catadioptric element. The advantage of a pure reflector is that it is free from chromatic aberration and therefore suitable for applications over the entire optical spectrum. This property allows alignments of IR systems in the visible spectrum because there is no focus shift.
12.2 Spherical Mirror We apply again the third-order aberration equations. With the object at infinity and the aperture stop at the mirror, the aberrations are spherical aberration, TSC
y3 f , 2 2 8f 64 f / #
(12.1)
sagittal coma, CC
y 2u p
4f
up f 16 f / #
,
2
(12.2)
astigmatism, TAC
yu 2p 2
u 2p f
4 f / #
,
(12.3)
Petzval curvature, TPC
yu 2p 4
and distortion is zero. 101
u 2p f
4 f / #
,
(12.4)
102
Chapter 12
The blur-spot sizes are again expressed as before with spherical aberration, Bspher
TSC f , 3 2 128 f / #
(12.5)
coma, Bcoma CC
up f 16 f / #
2
,
(12.6)
,
(12.7)
astigmatism, Bastig 2TAC
u 2p f
2 f / #
and Petzval radius, 2u 2p f 2 h' 2 h' 2 ρ f . 2 PC 2u 'TPC 2u 2p f
(12.8)
Figure 12.1 Sagittal focus moves along the tangent of the circle as a function of the half-field angle up.
The Single-Imaging Mirror
103
For a better understanding of how astigmatism occurs at the spherical mirror we derive the measure of the blur spot due to this aberration in another way. Figure 12.1 identifies the locations of the sagittal and tangential foci. The difference in their location is twice the longitudinal astigmatic contribution 2AC.1 One can see that AC
1 f 1 cos u p . f s ft 2 2 cos u p
The transverse astigmatic contribution TAC ACu'K . With u'
1 , 2 f / #
and TAC
AC , 2 f / #
therefore, TAC
1 f cos u p . 4 f / # cos u p
For small angles, cos u p 1
u 2p 2
.
With this substitution, 2 2 u 2p f TAC 4 f / # 2 u 2p 2
fu 2p . 4 f / #
(12.9)
104
Chapter 12
Figure 12.2 Three-dimensional view of the forming of astigmatism at a concave spherical mirror for an object at infinity.
The smallest blur spot is Bastig 2TAC fu 2p 2 f / # , which is, of course, the same as derived with the third-order aberrations equation. Figure 12.2 gives a 3D view of the condition.
12.3 Toroidal Mirrors Figure 12.3 illustrates the generation of toroidal mirrors. It can be seen that a spherical surface is a special case in which the sagittal and tangential radii are equal. While the sketches indicate mirrors, the shapes are also suitable and used for lenses.
12.4 Examples We analyze the image from a spherical mirror, modify the shape of the mirror to a toroid, and compare the results. To keep the spherical aberration small, we use a slow system of f/10 with a 100-mm focal length. The half-field angle up = 0.05 rad, which is 2.864789 deg. Figures 12.1 and 12.2 show that the tangential radius Rt has been shortened, and
The Single-Imaging Mirror
105
Figure 12.3 Generation of toric surfaces.
the sagittal radius Rs has to be increased to make the two focal lengths fs and ft equal to each other and to f. The relations are Rt 2 f s
2f cos u p
(12.10)
and Rs 2 ft 2 f cos u p .
(12.11)
106
Chapter 12
Figure 12.4 Geometry of the toroidal surface for our example.
Figure 12.4 shows the shape of a square toroidal mirror and indicates again the generation of it.
12.4.1 Spherical mirror Bspher
Bcoma
f 128 f / #
3
up f 16 f / #
Bastig
u 2p f
2
2 f / #
100 0.000781 mm 128 103
0.05 100 0.003125 mm 16 102
0.052 100 0.0125 mm 2 10
The Petzval radius ρ f 100 mm .
12.4.1.1 Additional comments To remove coma and astigmatism, one needs to place the aperture stop at the center of curvature of the spherical mirror.1 Spherical aberration remains.
The Single-Imaging Mirror
107
Figure 12.5 Coma, the remaining aberration.
12.4.2 Toroidal mirror The tangential radius Rt
2 f 2 100 200.250261 mm , cos u p cos(2.864789deg)
and the sagittal radius Rs 2 f cos u p 2 100 cos(2.864789 deg) 199.750052 mm .
With astigmatism corrected (third-order), a very slow relative aperture (f/10), and a small field, the remaining aberration is essentially only coma. According to Eq. (12.6), it is Bcoma
up f 16 f / #
2
0.05 100 0.003125 mm . 16 102
This is the size of the sagittal coma. The tangential coma blur is three times as large, i.e., 0.009375 mm. This is confirmed in Fig. 12.5, which is the result of a real ray trace.
12.5 Parabolic Mirror A typical application for a single-imaging mirror is a collimator. Figure 12.6 shows an arrangement frequently found in an optical laboratory for testing the
108
Chapter 12
performance of an infrared system. The imaging mirror is a section of an paraboloid. The only purpose of the flat-fold mirror is to achieve a more convenient overall arrangement. In such a device, the focal length is made rather long, the f/# large, and the pinhole small. Spherical aberration is corrected by the parabolic shape of the imaging mirror. The off-axis aberrations can be ignored due to the very narrow field. The pinhole should be as small as possible to represent a point source, but sufficiently large enough to provide the required energy from the black body for reliable performance measurements of the system under test.
Figure 12.6 Typical collimator arrangement for infrared system measurements.
Figure 12.7 Oscillating Slow Tool Servo generating a saddle-shaped reflector on a single-point diamond turning ultra-precision lathe. Movement of cutting tool z = f(x,).
The Single-Imaging Mirror
109
12.6 Manufacturing Remarks Generating toroidal mirrors and off-axis conic sections is costly and difficult at best. For many applications, such mirrors can be fabricated by single-point diamond turning if the chosen substrate material is suitable for the process. A typical candidate is electroless nickel-plated aluminium. With the introduction of a special mechanism, called a Slow Tool Servo (STS), such free-form surfaces can be readily machined on-axis. The trick is that the cutting tool moves in the zaxis (axis of rotation) as a function of the lateral ordinate, the x direction, and rotation angle Figure 12.7 points out the meaning of these symbols and illustrates the cutting of a saddle-shaped surface by this method.
12.7 Mangin Mirror In 1876, the French Army officer A. Mangin invented this catadioptric element to avoid the manufacturing difficulties associated with an aspheric surface. It can be seen from Fig. 12.8 that the element is a negative meniscus with a reflecting second surface. The principle of correcting spherical aberration is as follows: a spherical mirror has undercorrected spherical aberration, and a negative lens is overcorrected. By proper choice of the radii and the substrate material, spherical aberration can be corrected. While the original application was intended for illumination (search lights), the Mangin mirror has since been employed in other areas. One example is the objective for a radiometer. It also can serve as a secondary mirror in a modified Cassegrain telescope. This will be discussed as the next chapter.
Figure 12.8 Mangin mirror, with the object at infinity.
110
Chapter 12
The power (reciprocal focal length) of the Mangin mirror is 1 2 n 1 c1 2nc2 , f
(12.12)
with the curvatures (reciprocal radii) c1 1 / R1 and c2 1 / R2 . The refractive index of the substrate material is n. Expressing the second curvature as a function of the first surface, which is of course also the third one, and assuming zero thickness for the element, leads to c2
2 n 1 c1 2n
.
(12.13)
The second radius is then R2
1 2nR1 f . c2 2 n 1 f R1
(12.14)
Finding the curvature c1 for which spherical aberration vanishes leads to the cubic equation 3 2 2 c1 n 3 c1 4n 5 c1 4n 3 0. 2 4 8 n 1
(12.15)
The derivation of the aberration expressions and the analytical solution of Eq. (12.5) are quite cumbersome2 but enlightening. Matters become much easier if we set the curvature of the first surface equal, but opposite in sign, to the mirror’s power, i.e., c1 , which means making the first surface radius R1 f . With this postulation for the thin Mangin mirror, we get R2
2nR1 2nf . 2n 1 2n 1
(12.16)
The angular blur spots due to the different aberrations for this shape are spherical aberration, βspher
2n 3 , 3 128n 2 f / #
(12.17)
The Single-Imaging Mirror
111
coma, β coma
up 16n 2 f / #
2
,
(12.18)
astigmatism, β astig
u 2p
2 f / #
,
(12.19)
chromatic, β chrom
n 1 , 2nV f / #
(12.20)
and Petzval radius, ρ
n2 f
2n
2
1
.
(12.21)
V is the Abbe number, and up is the half-field angle in radians.
Remark An interesting case exists when n = 1.5. From Eq. (12.17) we see that with n = 1.5, spherical aberration vanishes. This was recognized by Mangin. Table 12.1 lists the radii required for different materials to eliminate spherical aberration of the Mangin mirror when the object is located at infinity. The results are obtained with Eqs. (12.14) and (12.15). Table 12.1 Surface radii for Mangin mirrors, corrected for third-order spherical aberration. Material Glass Zinc Selenide Silicon Germanium
n 1.5 2.4 3.4 4.0
R1 −1.0 f −1.1506 f −1.1790 f −1.1823 f
R2 −1.5 f −1.3980 f −1.3409 f −1.3169 f
112
Chapter 12
References 1. W. J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York (2008). 2. M. J. Riedl, “The Mangin mirror and its primary aberrations,” Appl. Opt., 13(7), 1690–1694 (1974).
Chapter 13 Eight Single Optical Elements as Imaging Objectives 13.1 Introduction A single optical element is often adequate to meet the image quality expected, especially from an infrared system, if the field to be covered is moderate. To provide a reference for a quick evaluation and comparison, eight different singlet configurations are presented in the form of a summary matrix of the previous subjects covered. This matrix will indicate whether a single element is sufficient for the task at hand or if a multi-element objective is required. To make matters manageable and simple, it has been assumed that the object is located at infinity and the aperture stop is placed at the objective. To avoid unpleasant surprises, it is recommended to always check first the size of the diffraction blur formed by the element. This is especially important for applications in the infrared region where the wavelengths are five to ten times longer than in the visible spectrum.
13.2 Diffraction Limit The diameter of the diffraction blur, which contains 83.9% of the energy of an imaged object point, is Bdiffr 2.44 λ f/# ,
(13.1)
where is the wavelength and f/# the relative aperture or “speed” of the objective. This diffraction blur is also known as Airy disk, named after Lord Airy, the British mathematician and astronomer.
13.3 Eight Chosen Configurations The eight chosen single elements, as possible imaging objectives, are pictured in Fig. 13.1. Aspheric lens #1 has a concentric second surface with respect to the focal point. Therefore, rays entering the lens parallel to the optical axis are not refracted at that surface. Aspheric lens #2 has been shaped to eliminate spherical aberration and coma.
113
114
Chapter 13
Figure 13.1 Eight selected shapes for a single imaging element.
Figure 13.2 Basic relations for a thin lens (object at infinity).
Eight Single Optical Elements as Imaging Objectives
115
Before making any statement about aberrations, Fig. 13.2 identifies the symbols and relations used in the expressions. Table 13.1 Symbol definitions. 2y f f/(2y) n
V nM 1 / nS nL R1 and R2 h′ up u′
free aperture diameter focal length f/# = relative aperture or f-number index of refraction of the lens material the inverse relative dispersion or Abbe number for the infrared spectra (M = middle, S = short, and L = long wavelength). first and second surface radii respectively image height half field angle final slope angle of axial ray
Table 13.2 Shapes of the elements. Conf. 1 2 3
Radius R1
n 1 f 2 n 2 n 1 f n 2n 1 n 1 f
4
n 2 n 1
5
n
n2 2
1
n2
6
f
7
2 f 2 f
8
f
f
Radius R2 plano
2 n 2 n 1 n 2n 1 4
2
n
2
1
n n 1 1 1
n n 1 1
Conic 2 0
0
0
1 n2
0
f
f
n
Conic 1 0
f
f
0
0
n3 n 1 n n 1 1
3
n3 n 1 n n 1 1
2n f 1 2n
0
0
-
0
-
-
–1
-
3
13.4 Shapes of the Elements Table 13.2 lists the surface shapes of the eight elements. For the diffractive hybrid (configuration 5), there is a phase profile superimposed on the second surface. It consists of a number of diffraction zones (see Chapters 3 and 14). The
116
Chapter 13
radius of the first zone r1 2λf d , with being the wavelength, and fd being the focal length of the diffractive portion of the lens. The other zone radii of the profile can be obtained with ri r1 i . The step size at the transition from one
zone to the next is d max λ / n 1 .
13.5 Aberrations1 Remarks
1. The aberrations are stated as angular-blur spot sizes , in radians. The linear blur-spot sizes are B f . 2. The blur size, due to astigmatism, remains the same for all eight configurations, namely β astig u 2p 2 f / # . Table 13.3 Angular aberration blur spot sizes of the elements in radians. Config.
spher
chrom
n n 2 2 2
1
32n n 1
2
f / #
3
n 4n 1
n n 1 1 u p 2 8n n 1 f / #
1 2V f / #
up
2
128 n 1 n 2 f / #
3
0
1 2V f / #
4
0
1 2V f / #
5
0
0
6
2
3
2n 3 3 128n 2 f / # 1
7
128 f / #
8
0
3
coma
1 2V f / #
n 1 2nV f / # 0 0
16 n 2 f / # up 8 n 1 f / # 0 0
up 16n 2 f / # up 16 f / #
2
up 16 f / #
2
2
2
2
Eight Single Optical Elements as Imaging Objectives
117
13.6 Examples We determine the shapes and aberration blur-spot sizes for an element with f = 100 mm, f/# = 1.5, M = 4 m, S = 3 m, and L = 5 m. The half field angle u p = 0.05 rad. With silicon as the chosen lens material, the index of refraction n = 3.425, and the Abbe number V = 236. For a point of reference, we determine first the diffraction blur size. Bdiff = 2.44 0.004 1.5 0.015 mm, which is angularly expressed as diff Bdiff / f = 0.15 mrad. Analyzing all eight configurations yields an informative comparison matrix. For the definitions of R1, R2, and , see Tables 13.1 and 13.2. The angular blur-spot functions are stated in Table 13.3. Table 13.4 Comparison Matrix. Linear dimensions are in mm. Angular dimensions are in mrad. #
R1
R2
sph
ch
c
a
1
242.50
0
8.60
1.41
2.44
0.83
13.28
2
97.861
164.074
0
3.16
1.41
0.26
0.83
5.66
3
70.803
100.00
−0.08525
0
1.41
1.15
0.83
3.39
4
91.475
146.882
0.45595
0
1.41
0
0.83
2.24
5
91.475
146.882
0.45595
0
0
0
0.83
0.83
6
−100.0
−117.094
0
0.76
1.00
0.12
0.83
2.71
7
−200.0
-
0
2.32
0
1.39
0.83
4.54
8
−200.0
-
-1
0
0
1.39
0.83
2.22
118
Chapter 13
There are 12 zones for the diffractive hybrid, Element 5. Their radii are listed below. Table 13.5 Radii for the 12 diffractive hybrid zones. Radius # 1 2 3 4 5 6
r (mm) 9.757 13.799 16.900 19.514 21.817 23.900
Radius # 7 8 9 10 11 12
r (mm) 25.815 27.597 29.271 30.855 32.360 33.799
The maximum step dimension at the transition radii is d max = 0.00165 mm.
Reference 1. M. J. Riedl, Optical Design Fundamentals for Infrared Systems, SPIE Press, Bellingham, Washington (2001).
Chapter 14 A Progression of Performance with an Increase in Lens Complexity 14.1 Objectives We apply the knowledge gained from the preceding sections by analyzing six different objectives for the MWIR spectrum with the same focal length (50 mm), the same f/# (1.5), and the same square field coverage (4 deg × 4 deg). We start with the simplest form of a lens and reach (in five steps) an objective that is frequently found in thermal imaging systems. The six configurations are: 1. 2. 3. 4. 5. 6.
plano-convex singlet best shaped singlet aspheric singlet aspheric hybrid Petzval objective with one aspheric element Petzval objective with one aspheric hybrid
The plot in Fig. 14.1 is based on using square sensor elements (pixels) that contain 80% of the received energy from a point source at infinity. It is clear that the simple singlets are not suitable for IR cameras, but are sufficient for radiometers (pyrometers) if the target is large enough to resolve the projected sensor element. The single aspheric hybrid with a 70-µm pixel size over a linear field of view of 3.5 mm in the horizontal and vertical directions may begin to get interesting for thermal imaging. The total number of pixels amounts to (3.5 / 0.07)2 2500 , and the angular resolution is 0.07/50 =1.4 mrad. At the end of the list is the frequently employed Petzval objective, which was discussed in detail in Sec. 7.4. With a 20-µm pixel size, the focal plane array contains more than 30,000 sensing elements. The angular resolution is 0.4 mrad.
Remarks 1. For the off-axis calculations, the field was increased by the factor 2 to cover the pixels located at the extreme of the field’s diagonal. 2. All lenses are made from silicon with the exception of the aspheric hybrid, which is made from GASIR-1, a material that lends itself to molding. Its refractive index is 2.5116 at 4 µm, and its Abbe number in the MWIR region is 197.3.
119
120
Chapter 14
3. For reference, notice that the Airy disk, containing 84% of the energy from an object point, is 15 µm for the stated wavelength of 4 µm and the assumed relative aperture of f/# 1.5 . 4. For an increased field of view and a lower f/#, elements have to be added to these basic designs with a choice of making use of the properties of different materials.
Figure 14.1 Performance improvement with increasing complexity of the objective.
Chapter 15 Two-Mirror Systems as Telescope and Microscope Objectives 15.1 Introduction In 1663, the Scottish mathematician and astronomer James Gregory suggested a two-mirror system with a parabolic primary and an elliptical secondary mirror. The configuration is shown in Fig. 15.1. At that time, he was unable to find an optician to grind and polish the elliptical mirror. Nine years later, in 1672, the Frenchman Guillaume Cassegrain came up with the idea to intercept the reflected rays from the primary mirror before they come to a focus. For that purpose, a hyperbolic secondary mirror is needed. Cassegrain’s system is much shorter and has found its place in many applications—from astronomical telescopes to thermal imaging devices. It is interesting to note that Isaac Newton, the great British scientist, never gave Cassegrain any credit for his invention. He referred to it as “just an obvious extension” of Gregory’s concept. If the field of view is narrow, only spherical aberration has to be corrected. Dutchmen Horace Dall and Allan Kirkham conceived of the idea to replace the hyperbolic secondary mirror with an easierto-manufacture convex spherical mirror. To keep the system free from spherical aberration, the primary mirror becomes, in that case, an ellipsoid. This increases coma somewhat, but that was not of much concern to the two astronomers. In the 1920s, the American George Ritchey and the Frenchman Henri Chretien modified the basic Cassegrain arrangement so that it is corrected for spherical aberration and third-order coma. This Ritchey-Chretien configuration is now the most widely used combination. The famous Hubble Space Telescope is of this
Figure 15.1 Gregorian telescope. 121
122
Chapter 15
type. We shall discuss the details of all these configurations, including the use of a Mangin mirror as a secondary reflector. Additionally, we will present the classic Schwarzschild objective and a special configuration, the inverted Schwarzschild, which is corrected for spherical aberration, coma, astigmatism, and field curvature.
15.2 Basic Cassegrain Telescope Layout A convenient way of identifying the layout of a Cassegrain system is by choosing a desired focal length f, the spacing d of the two mirrors, and the back focal distance b. The separation d and the back focal distance b depend on the application. If, for example, a 25% obstruction ratio (in area) from the secondary mirror is acceptable, d will be half of the primary focal length f or a quarter of the primary mirror radius R1. Another factor to consider is whether there is a need to insert, for example, filters or a folding reflector behind the primary mirror, which would influence the length of the back focal distance b. Figure 15.2 shows the typical Cassegrain arrangement and identifies the symbols used in the subsequently presented equations. For reference and comparison, the location and size of the secondary mirror of an equivalent Gregorian telescope is indicated with reference to Newton’s remark.
Figure 15.2 Layout of a Cassegrain telescope.
Two-Mirror Systems as Telescope and Microscope Objectives
123
15.2.1 Equations1 From the geometry of the layout and a paraxial ray trace one finds the radii of the mirrors. They are 2df (primary mirror) b f
(15.1)
2db (secondary mirror). b d f
(15.2)
R1
and R2
15.3 Cassegrain with Two Spherical Mirrors With spherical surfaces for both mirrors, the angular third-order aberration blur spots are spherical aberration, f b f b f d b f d b 3
βspher
128 fd 3 f /#
3
2
,
(15.3)
coma,
β coma
2 f b f 2 f d b f d b d f b u p , 2 32d 2 f f /#
(15.4)
and astigmatism,
β astig
4bf b f f d b d f b 2 u p 2 . 8 fbd f /#
(15.5)
The Petzval radius is ρ Petzval
bfd fd b f
2
.
(15.6)
124
Chapter 15
15.4 Classic Cassegrain System To correct spherical aberration, the surface of the primary becomes a paraboloid, for which the conic constant κ1 1 . The conic constant for the hyperbolic secondary mirror is 2
f d b κ2 . bd f
(15.7)
Coma for this configuration is expressed by β coma
up 16 f /#
2
,
(15.8)
.
(15.9)
and astigmatism is
d f u 2p β astig 2b f /# The Petzval radius remains the same with ρ Petzval
bfd fd b f
2
.
15.5 Dall-Kirkham Arrangement We recall that this form has a spherical secondary mirror. Spherical aberration correction is achieved with an aspheric primary mirror. The conic constant is f f b b f d b f d b 3
κ1
f b f
3
2
.
(15.10)
Coma is
β coma
2 u p 2 f b f f d b f d b d f b , 2 2 32d f f /#
and astigmatism is
(15.11)
Two-Mirror Systems as Telescope and Microscope Objectives
β astig
125
2 u 2p 4bf b f f d b d f b . 8dbf f /#
(15.12)
15.6 Ritchey–Chretien Configuration As stated earlier, spherical aberration and third-order coma are corrected in the Ritchey–Chretien configuration. The required conic constants are, for the primary mirror κ1
2bd 2
b f
3
1 ,
(15.13)
and for the secondary mirror 2 f b f 2 f d b f d b d f b . κ2 3 b d f
(15.14)
Left is astigmatism with β astig
u 2p 2 f d 4b f /#
.
(15.15)
15.7 Examples For performance comparison, we analyze several different telescope systems with the identical focal length of f = 500 mm, a separation of the elements of d = 100 mm, and back focal distance of b = 150 mm. The relative aperture is f/5, and the half-field angle up = 0.01 radians. The mirror radii are, according to Eqs. (15.1) and (15.2), R1
2df 2 100 500 285.714286 mm b f 150 500
and R2
2db 2 100 150 120.0000 mm . b d f 150 100 500
The Petzval radius is the same for all configurations, namely Petzval
bfd 150 500 100 103.448 mm. 2 fd (b f ) 500 100 (150 500) 2
126
Chapter 15
Table 15.1 Conic constants and angular blur-spot sizes for chosen two-mirror systems. Configuration Basic Cassegrain Classic Cassegrain Dall-Kirkham Ritchy-Chretien
0 –1 –0.645775 –1.069971
0 –3.24 0 –388
spher 1.73 0 0 0
coma 0.152 0.025 0.152 0
astig 0.01 0.027 0.01 0.03
The angular blur spot sizes spher , coma , and astig are in milliradians. We analyze now a Cassegrain system that uses a Mangin mirror as a secondary reflector. The focal length, spacing, back focal distance, f/#, and the covered field are the same as in the previous examples.
15.8 Cassegrain with Mangin as a Secondary Reflector This configuration uses only spherical surfaces. The third-order spherical aberration is corrected. Due to the refractive secondary element, chromatic aberration is introduced. An analytical approach to solve this task is not practical. Therefore, a solution was created with ZEMAX. Figure 15.3 shows the on-axis blur spot size, which is due to chromatic aberration. The system was designed for the MWIR region with silicon as the material for the Mangin mirror. As mentioned, spherical aberration has been corrected with the Mangin mirror. The blur-spot shape on the right side clearly shows that the predominant contributor to the blur is coma. The data are for a half-field angle of up = 0.01 radians.
Figure 15.3 Blur spots from a Cassegrain system with a Mangin mirror as a secondary reflector.
Two-Mirror Systems as Telescope and Microscope Objectives
127
One can see what improvement a Mangin mirror brings if the blur-spot sizes are compared with the ones from a two-sphere mirror system, where the spherical aberration blur alone is 0.865 mm when refocused to the best position. This can be confirmed with Eqs. (15.3), (15.4), and (15.5).
15.9 Gregorian Telescope All the equations presented for the Cassegrain and Ritchey–Chretien systems remain valid for the Gregorien telescope, with the requirement that the focal length f is inserted as a negative number. Maintaining the focal length f = 500 mm, the f/# with 5, and the image height h′ = 5 mm, we determine first with the help of Fig. 15.4 the required spacing d of the two elements and the back focal distance b. From the ratios, one finds d = 220 mm, b = 270 mm, and y2 = 27 mm. We now can calculate the radii of the mirrors with Eqs. (15.1) and (15.2), and the conic constants, for which we apply Eqs. (15.13) and (15.14) to correct spherical aberration and coma. Recall that changing the sign of the focal length for the equations leads to R1
2 220 500 2df 285.714286 mm b f 270 500
and R2
2db 2 220 270 120.0 mm. b d f 270 220 500
Figure 15.4 Change in d, b, and y as shown transforms a Cassegrain system to Gregory system. The sketch is not to scale.
128
Chapter 15
The conic constants are 2 270 2202
κ1
270 500
3
1 0.942751 ,
and
2 500 270 500 990 550 450 0.358524. 2
κ2
270 220 500
3
Astigmatism is astig
u 2p (2 f d ) 4b( f /#)
0.012 2 (500) 200 4 270 5
0.000022 rad.
While this astigmatic blur is very small, the Petzval curvature is steep with a radius of ρ Petzval
bfd fd b f
2
270 500 220
500 220 270 500
2
42.2535 mm .
For the classical Cassegrain and R/C systems, with the same focal length and f/#, the Petzval radius is –103.4483 mm. After refocusing by a small amount, the geometrical blur spots are still smaller than the diffraction limit because of the relatively slow speed of f/5 and the wavelength of 4 µm that we had assumed. The Airy disk is BAiry 2.44λ f /# 2.44 0.004 5 0.0488 mm . We expand on this system by describing a Gregorian system that was cleverly shortened by inserting a flat folding mirror to place the secondary mirror at the location of the primary mirror. The system was developed by the fSONA Systems Corporation in Canada. Figure 15.5 shows the arrangement. The primary and secondary mirrors form a monolithic element, which was fabricated by single-point diamond turning on an ultra-precision lathe. In Fig. 15.6 we show the ray traces, based on the Gregorian system we designed above for the MWIR region.
Two-Mirror Systems as Telescope and Microscope Objectives
129
Figure 15.5 Folded Gregorian telescope.
Figure 15.6 Folded Gregorian system, designed with the provided equations. (f = 500 mm, d = 220 mm, b = 270, folded to d/2 = 110 mm.)
130
Chapter 15
15.10 Gregorian Microscope Objective It is rewarding to use the same equations to design an inverted Gregorian system that can serve as a microscope objective, for which as before, spherical aberration and coma are corrected. Let us demonstrate with f = 20 mm, d = 55 mm, and b = 60 mm.
Figure 15.7 Inverted Gregorian as a microscope objective.
Figure 15.8 Blur-spot sizes for the inverted 20-mm-focal-length, f/1.5 Gregorian objective. The reference circle is the size of the Airy disk.
Two-Mirror Systems as Telescope and Microscope Objectives
131
Figure 15.9 Compact folded Gregorian microscope objective.
The calculations yield surface radii of R1 27.5 mm and R2 48.888 mm . The conic constants are κ1 0.29106 and κ 2 0.083473. Figure 15.7 shows the layout, and Fig. 15.8 depicts the linear blur-spot sizes, compared with the diffraction limit in the visible spectrum. We now present a folded variation of this objective. For this case, the effective focal length is 10 mm, and the relative aperture f/# =1.5. The folding mirror has been slightly aspherized to aid in the correction of the off-axis aberrations. A possible layout, as a so-called snap-together arrangement, suitable for diamond turning, is shown in Fig. 15.9. Notice the large free-working distance.
15.11 Two Schwarzschild Objectives The classic Schwarzschild objective has been discussed in many textbooks and is presented here for comparison with the lesser known second configuration. The first kind uses only spherical surfaces, while for the second kind, two aspheres are required. Until recently, aspheres have been very expensive to manufacture. This may have been a reason to avoid this second configuration. The two types shown in Fig. 15.10 have the same focal length and the same relative aperture.
132
Chapter 15
Figure 15.10 Two Schwarzschild configurations.
Two-Mirror Systems as Telescope and Microscope Objectives
133
The first striking difference is the size. Another one that is not quite as obvious is the fact that in the second form, the “inverted” configuration, there is no need for a spider to hold the secondary mirror in place as is the case with the classic one. The first kind has a large free-working distance which is frequently required. On the other hand, the short free-working distance of the second kind is no shortcoming for applications in, for example, photolithography.
15.11.1 Prescription for the classic configuration for an object at infinity The conditions required to correct spherical aberration, coma, and astigmatism for this configuration are:
d 2f
b
R1
5 1 f
R2
5 1 f
y1
f 2 f /#
y2
Fractional area obscuration ε 1 / 5 20%
52 f
5 2 y1
52 f 2 f /#
Note that both mirror surfaces are spherical and concentric to each other. It is an interesting exercise to demonstrate that the third-order spherical aberration, coma, and astigmatism vanish for this configuration. To eliminate the Petzval curvature and distortion, one can position a field flattener in the focal plane with a front radius of R3
n2 1 f 1 n n 1
and a rear radius of R4 n 2 1 f . It is also remarkable to note that in this configuration, the mirror radii can be derived from the “golden ratio.” Let us demonstrate with the help of Fig. 15.11.
134
Chapter 15
Figure 15.11 Relations of the “golden ratio.”
The diagram shows the relations of the “golden ratio,” which says, 2f x . x 2 f x
The root of this equation is x1 f ( 5 1) , which is equal to the radius of the first mirror. The radius of the second sphere can then be easily derived since it is longer by 2f and concentric to R1. Therefore, R2 R1 d R1 2 f
5 1 f 2 f
5 1 f . **
15.11.1.1 Brief historic remark about the golden ratio2 As early as 1202, the Italian mathematician Leonardo Fiboncci discussed in his book Liber abacihis a series that leads to the ratio stated above. This ratio was rediscovered toward the end of the 19th century and found especially pleasing for its harmonic proportions.
Example For a system with f = 10 mm and a relative aperture of f/1, we get d = 20 mm, b = 42.3607 mm, R1 = 12.3607 mm, R2 = 32.3607 mm, y1 = 5 mm, and y2 = 21.1803 mm.
15.11.2 Prescription for the inverted configuration for an object at infinity In addition to correcting spherical aberration, coma, and astigmatism, field curvature is also eliminated in this configuration: **
See the fourth equation on page 133, under 15.11.1.
Two-Mirror Systems as Telescope and Microscope Objectives
d 2 f (Same as above)
b 1 2 f
R1 R2 2 2 f
κ1 1 2
κ2 1 2
y1
y2
y2 1 2 y1
Fractional area obscuration ε
135
2
= 5.828427 (Conic constant for first surface)
2
0.171573 (The reciprocal of κ1 )
f 2 f /#
1 2 f 2 f /#
2
2 1 1 / 5.8 17.3%
Note that both surfaces’ radii are equal. By aspherizing, both become oblate ellipsoids.
Figure 15.12 Inverted Schwarzschild objective. The layout suggests two diamond-turned components with a spidered disk inserted to block the central energy from flooding the focal plane.
136
Chapter 15
Figure 15.13 Solid catadioptric microscope objective, suggested by D.D. Maksutov. The chosen material is CaF2, which can be diamond turned.
Here is another example for the same focal length and f/#. The system is shown in Fig 15.12. d = 20 mm as before. b = 24.1421 mm, and R1 = R2 = 28.2843 mm. The conic constants are independent of focal length and f/#. They are 1 = 5.828427 and 2 = 1/1 = 0.171573.
15.12 Solid Microscope Objective To close this chapter, we present a solid microscope objective as suggested by Maksutov3 and shown in Fig. 15.13. With calcium fluoride as the material, the objective can be used in the visible spectrum as well as in the MWIR region. The large radius is aspherized with two higher-order deformation terms. The entrance and exit surface radii are concentric to the axial object and image points to eliminate refraction.
References 1. W. J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York (2008). 2. Bibliographisches Institut & F. A. Brockhaus AG, Germany (2007). 3. K. Mütze, ABC der Optik, Verlag Werner Dausien, Hanau, Germany (1961).
Chapter 16 The Plane-Parallel Plate 16.1 Introduction There are two aspects that have to be dealt with when using a plane-parallel plate inserted in a converging or diverging beam. First, there are the aberrations, and second there is a relocation of the image. This relocation is a simple longitudinal shift if the plate is perpendicular to the optical axis. If the plate is tilted, as is the case with most beam splitters, there is a longitudinal and a lateral displacement, which is not too easy to assess because they interact. We shall address both cases. It should also be clear that most prisms behave like thick plane-parallel plates. This is indicated in Fig. 16.2 with a number of examples. The “unfolding” of the prism into an equivalent plane parallel plate is called the tunnel diagram. The first two examples are equivalent to plane-parallel plates inserted perpendicularly to the optical axis. The third one, the Dove prism, corresponds to a tilted plate, which of course introduces astigmatism and coma as will be pointed out further down in the text. For several reasons, such as cost and absorption, prisms are not much used in the infrared spectrum. However, wedges, which are thin prisms with a small apex angle, are employed in this region as beam-steering elements. The deviation of a ray exiting such a wedge is n 1 , where n is the index of refraction, and α is the wedge angle. Details are shown in Fig. 16.1. A pair of counter-rotating wedges, known as Risley prisms, is used as a linear-scanning device.
Figure 16.1 Thin prism (wedge) and the deviation of an incoming ray.
137
138
Chapter 16
16.2 Aberrations1 It is frequently overlooked that a plane-parallel plate also contributes aberrations when it is inserted in a converging or diverging beam. Using the third-order surface-contribution equations again, we demonstrate step by step the derivation for the spherical aberration. The symbols used in the derivation are identified in Fig. 16.3 The transverse spherical aberration surface contribution is TSC Bi 2 h' ,
(16.1)
where B
n n' n 2n'n'k Inv
y u' i ,
where i cy u , and h′ is the image height. The total transverse spherical aberration is the sum of the two surface contributions
2
TSC TSC
1
TSC2 . We make things easier if we set y1 0 ,
1
because then TSC1 0, and
2
TSC TSC
2
. Equation (16.1) can now be restated
1
as B
n 1 y 2nInv
2
u'2 i2 i22 h' .
For the optical invariant Inv we set h'n'2u'2 h' 2 f /# and y2 y1 tu'1 t 2n f /# , with y1 0, and u'1 u1 n 1 2n f /# . With
i22 1 / 2n f /#
2
1 / 4n 2 f /#
2
and
recalling that
the
minimum blur size, when refocused, is half the transverse spherical aberration, we finally get the blur size of a plane-parallel plate with thickness t and index n. It is simply Bspher
n
2
1 t
32n3 f /#
3
.
(16.2)
The Plane-Parallel Plate
139
Figure 16.2 Tunnel diagrams of three prisms show the relation to a plane-parallel plate.
Figure 16.3 The plane-parallel plate.
140
Chapter 16
The blur-spot sizes due to coma, astigmatism, axial chromatic, and lateral color can be derived by the same procedure. They are coma, Bcoma
n
Bastig
n
2
1 tu p
,
(16.3)
,
(16.4)
n 1 t , 2n 2V f /#
(16.5)
8n3 f /#
2
astigmatism, 2
1 tu 2p
2n3 f /#
axial chromatic, Bchrom
and lateral color, Blat color
n 1 tu p n 2V
,
(16.6)
where up is the half field angle, and V is the Abbe number. These effects need to be taken into account, especially if filters with different thicknesses and indices are inserted in the converging beam. The Dewar window of a cooled detector package is another example of that. The impact is amplified when the plane-parallel plate is used as a beam splitter under 45 deg.
16.2.1 Examples In Table 16.1, we compare the performance of three 2-mm-thick beam splitters made from different materials, inserted in a converging f/2 cone. For the visible spectrum we use BK7, for the MWIR range from 3 µm to 5 µm, we contrast silicon against sapphire.
The Plane-Parallel Plate
141
Figure 16.4 Blur spot sizes of a 2-mm beam splitter, tilted about 45 deg. The circles indicate the calculated blur-spot sizes. Table 16.1 Blur spot sizes as a function of material and spectral range. Material BK7 Silicon Sapphire
Region VIS MWIR MWIR
Bspher 2.9 2.1 3.0
Bcoma 18.3 13.1 18.9
Bastig 115.5 82.2 118.5
Bchrom 1.8 0.4 15.8
Blat color 5.5 1.4 49.7
Btotal 143.5 µm 99.2 µm 205.9 µm
The calculated sizes, using the developed equations, are compared with the results from a full computer analysis in Fig. 16.4. The scaled blur-spot diagrams from the computer confirm the validity of the simple third-order expressions. The plot for the sapphire substrate shows clearly the impact of the very high dispersion for that material in the MWIR. Especially pronounced is the lateral color, which is already indicated in the table.
16.3 Shift of Image Figure 16.5 shows a plate perpendicular to the optical axis and indicates the longitudinal displacement of the image. This shift is a function of the substrate material and the thickness of the plate. The longitudinal displacement is 1 sin 2 u L 1 2 n sin 2 u
t .
(16.7)
For small angles this reduces u to l
n 1 t . n
(16.8)
142
Chapter 16
Figure 16.5 Longitudinal image shift caused by a plane-parallel plate.
To assess the error introduced by substituting Eq. (16.8) for (16.7), we compare the results for different relative apertures (f/#). From the relation tan u D 2 f 1 2 f /# , we get the slope angle u for the insertion in Eq. (16.7). Table 16.2 lists the errors in percent for a glass plate with a refractive index n = 1.5. Table 16.3 indicates that the difference is much smaller for silicon, for which the index n = 3.4 in the MWIR region. Table 16.2 Error resulting from substituting Eq. (16.8) for small angles u. n =1.5 (glass) Relative aperture f/# 4 3 2 1.5 1
Slope angle u in deg
Exact image displacement L
7.125016 9.462322 14.036243 18.434949 26.565051
0.336208 t 0.338419 t 0.344614 t 0.353003 t 0.375305 t
For small angles l 0.333333 t 0.333333 t 0.333333 t 0.333333 t 0.333333 t
Error in %
L l L 0.86 1.50 3.27 5.57 11.18
100
The Plane-Parallel Plate
143
Table 16.3 Error resulting from substituting Eq. (16.8) for small angles u. n = 3.4 (silicon). Relative aperture f/# 4 3 2 1.5 1
Slope angle u in deg
Exact image displacement L
7.125016 9.462322 14.036243 18.434949 26.565051
0.707959 t 0.709544 t 0.713935 t 0.719761 t 0.734628 t
For small angles l
Error in %
0.705882 t 0.705882 t 0.705882 t 0.705882 t 0.705882 t
L l L
100
0.29 0.52 1.13 1.93 3.91
16.4 Tilted Plate Figure 16.6 indicates the effect of a tilted plate. In addition to the longitudinal shift, there is also a lateral displacement of the image.
Figure 16.6 Displacements caused by a tilted plate.
144
Chapter 16
Figure 16.7 Image displacements due to tilt and thickness of a plane-parallel plate.
16.4.1 Lateral displacement The lateral or transverse displacement is 1 sin 2 ut T 1 2 n sin 2 ut
t sin ut .
(16.9)
ut is the tilt angle, as indicated in Fig 16.6. The combined effect of transverse displacements and longitudinal shifts as a function of the tilt angle is shown in Fig. 16.7. The adjustments for locating the minimum blur spots are indicated as obtained from actual retracings. A 20-mmthick plate has been chosen to magnify the excursions.
16.5 Two Tilted Plates To correct for the astigmatism introduced by a tilted plate, one can insert a second plate, made from the same material, with the same thickness and the same tilt but rotated by 90 deg, as shown in Fig. 16.8. The offset of the optical axis as stated in Eq. (16.9) will then be the same in the x-axis as in the y-axis. We demonstrate the situation with two germanium plates tilted by 45 deg. The first serves as a beam splitter, and the second is inserted to correct astigmatism. The
The Plane-Parallel Plate
145
Figure 16.8 The arrangement for correcting astigmatism.
Figure 16.9 Blur-spot comparison with one and two plates arranged as shown in Fig. 16.8. Also indicated is the best longitudinal image position.
plates are 2 mm thick and are inserted in an f/2.5 cone. The remaining aberrations are negligible, except coma, which is approximately 20 µm [according to Eq. (16.3)] for the two plates. The results are compared against the performance of a beam splitter without a compensator. Figure 16.9 compares the blur-spot shapes and sizes of a single plate with a two-plate arrangement.
Reference 1. M. J. Riedl, Optical Design Fundamentals for Infrared Systems, SPIE Press, Bellingham, Washington (2001).
Chapter 17 MTF, Limits, and Pixel Sizes 17.1 Introduction The optical modulation transfer function (MTF) is a well-known concept for evaluating the performance of a lens.1 It contains valuable information about the resolving power of the lens and the contrast in the image. Modern infrared imaging systems use sensor arrays that are made up of small individual sensing elements, the pixels. The practical minimum size of such a pixel depends on the wavelength of the collected radiation and sets the spatial resolution limit of the optical system, referred to as the Nyquist frequency.
17.2 Optical Modulation Transfer Function The MTF for diffraction limited systems is expressed by 1,2
MTFdiffr ( )
2 2 arccos 1 , π 0 0 0
(17.1)
where v is the spatial frequency of interest, 0 is the cut-off frequency, and / 0 is the normalized frequency. Figure 17.1 shows the graph of the diffraction-limited MTF as a function of the normalized frequency / 0 . In Fig. 17.1 there is also a remark about the Nyquist frequency, which will be discussed in the following section. The cut-off frequency is expressed by 0
1 λ f /#
(17.2)
This indicates that for the visible spectrum with = 0.5 µm, the cut-off frequency in line pair/mm v0VIS 2000 / f /# . In the MWIR region, with = 4 µm,
0MWIR 250 f /# . For the LWIR spectrum, with = 10 µm,
0LWIR 100 f / # . These numbers are reminders of the sizeable differences
among the different spectral regions.
147
148
Chapter 17
Figure 17.1 Diffraction-limited MTF.
17.3 Focal Plane Array Figure 17.2 schematically shows the geometry of a focal plane array (FPA) with square pixels. With a common postulation that the pixel size should be not smaller than the Airy disk, we can state that p 2.44λ f /# . The Nyquist frequency is then Ny
1 . 5λ f /#
(17.3)
Forming the normalized cut-off frequency, relating to Nyquist, we can write 0Ny 0
1 / 5λ f /# 1 / λ f /#
1 . 5
In other words, the normalized Nyquist cut-off frequency is five times lower than the optical one. This is marked in Fig. 17.1. Referring to the different spectral regions, Eq. (17.3) states that the limiting Nyquist frequency, expressed in line pairs per millimeter, is 50/(f/#) for the MWIR region and 20/(f/#) for the LWIR window.
MTF, Limits, and Pixel Sizes
149
Figure 17.2 Focal plane array. Columns 3 and 4 form a line pair 2p wide. The spatial frequency is the Nyquist frequency Ny 1 / (2 p) .
References 1. W. J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York (2008). 2. M. J. Riedl, Optical Design Fundamentals for Infrared Systems, SPIE Press, Bellingham, Washington (2001).
Chapter 18 Details of a Hybrid Lens 18.1 Introduction Hybrid lenses are being frequently used in infrared systems, primarily for color correction. We shall analyze one such lens, which is the third element of the MWIR objective shown in Fig. 18.1.
18.2 Hybrid1 The prescription of the diffractive surface (front surface) is as follows: 1. Radius R1 = 21.246406 mm 2. Conic constant = –0.151513 (prolate ellipse) 3. Phase coefficient P2 = −5975.55048. 4.
S2
0.004 5975.550548 λP2 3.804 104 2 2πR0 2π 1002
5. Free aperture D = 11.84 mm For reference, we state the general phase equation of a diffractive surface as
Figure 18.1 25-mm focal-length wide-angle objective with a cold stop for the MWIR region. The relative aperture is f/1.4, and the field coverage is ±12 deg. The third lens is a germanium hybrid, which will be analyzed. 151
152
Chapter 18
r
2π S 2 r 2 S4 r 4 S6 r 6 ... . λ0
(18.1)
Note that, for this design, only the quadratic term of the phase function has been used, and the asphere is a conic section without additional higher deformation coefficients. This has been done to simplify the analysis without impacting the correctness of the general approach. For this equation, r is the zone radius of the diffractive profile and λ 0 is the design wavelength, which in our example is 4 µm. The first zone radius can be obtained from the relation r1
λ0 . S2
(18.2)
The other zone radii are found with ri r1 i .
(18.3)
The total number of zones required over a free lens aperture is 2
itotal
D . 2r1
(18.4)
The step height of the profile at the zone transition is d max
λ0 . n 1
(18.5)
We now apply all these relations to our example and find r1
0.004 3.42657 mm . 3.804 104
The other radii are r2 r1 2 4.5858 mm , r3 = 5.6164 mm, r4 =6.4853 mm, and r5 =7.2508 mm. Equation (18.4) states that
Details of a Hybrid Lens
153
2
itotal
2
D 13.6 4.397 , 2r1 2 3.24657
which means the fifth zone is not a full zone. Only a fraction of it will be used. The zone transition step size is d max
λ0 0.004 0.001322 mm . n 1 4.02506 1
18.3 Coordinates of the Combined Surface To generate the profile of the aspheric/diffractive surface with an ultra-precision lathe, one needs to know the axial tool position z as a function of the lateral cross position r. Details for this statement are given in Fig. 18.2. The conic asphere is expressed by zasph
cr 2 1 1 κ 1 c 2 r 2
,
(18.6)
with c = 1/R, R = vertex radius, and = conic constant. The diffractive profile is represented by zdiff
S2 2 1 2 S 2 r λ 0 Int r . λ0 n0 1
(18.7)
Figure 18.2 Coordinates for a diffractive profile on an aspheric surface.
154
Chapter 18
Let us apply this to a point at r = 5 mm: zasph
zdiff
1
1/ 21.24606 52 0.000669 mm 2 1 0.151513 1 1/ 21.24606 52
3.804 104 2 1 4 2 3.804 10 5 0.004 Int 5 0.004 4.02506 1
zdiff
1 0.009510 0.004 2 0.000499 mm 3.02506
ztotal zasph zdiff = 0.000669 - 0.000449 = 0.00022 mm
A good check is to insert these equations at a known transition radius. We demonstrate with r4 = 6.4853. zdiff
1 4.02506 1
3.804 104 2 4 3.804 10 6.4853 0.004 Int 6.48532 0.004
If we insert the third radius with 5.616400 mm in the integer (Int) bracket, we get zdiff 1 4.02506 1 0.016 0.012 0.001322 mm . This is the maximum step height as stated with Eq. (18.5). Figure 18.3 shows performance plots of this objective.
Details of a Hybrid Lens
155
Figure 18.3 25-mm MWIR objective with germanium hybrid lens.
Reference 1. M. J. Riedl, Optical Design Fundamentals for Infrared Systems, SPIE Press, Bellingham, Washington (2001).
Chapter 19 From the Höegh Meniscus to Double Anastigmats 19.1 Introduction This chapter outlines the design steps for several photo objectives in an evolutionary way. Beginning with a single element, the Höegh meniscus, we progress by adding elements. Two elements lead to the Hypergon, a symmetrically arranged doublet. In the next step, the two lenses are shaped into cemented achromats. Finally, these cemented achromats are split into airspaced achromats, which leads to an arrangement referred to as a double anastigmat. In the final optimization, we break the symmetry and change the radii of the second achromat. The configuration is an excellent general-purpose lens with a relative wide angle coverage.
19.2 Höegh Meniscus Emil von Höegh reasoned that if both radii of a meniscus are equal, the Petzval curvature must be zero. Furthermore, if the stop is placed at the proper position, astigmatism can be eliminated. This arrangement is shown in Figure 19.1.
Figure 19.1 Höegh Meniscus with aperture stop placed in front.
157
158
Chapter 19
19.2.1 Approach and design method Third-order transverse Petzval surface contributions of the two meniscus surfaces are TPC1
n1 n1' c1h'Inv 2n1n1'
and TPC2
n2 n'2 c2 h'Inv 2n2 n'2
.
With c2 c1 c, n1 n2' 1, and n2 n1' n , we find TPC1
1 n ch'Inv 2n
and TPC2
Therefore,
n 1 ch'Inv 2n
TPC1 .
TPC 0. (All symbols have been identified in Chapter 4).
The focal length of this special meniscus can be easily derived from equation (2.1), which is n 1 c c t . 1 n 1 c1 c2 1 2 f n
By inserting c2 = c1, we get 2
r n f . n 1 t
For a given focal length, we solve for the thickness t and obtain 2
r n . t n 1 f
From the Höegh Meniscus to Double Anastigmats
159
We select for our example r = −10 mm, n = 1.5, and f = 100 mm. This results in a thickness of t = 6 mm. To correct astigmatism, the aperture stop has to be placed so that TAC 0. To find that proper place in front of the lens, as indicated in Fig. 19.1, is a somewhat cumbersome (but interesting) exercise. Assuming a relative aperture of f/30, and a half-field angle of 40 deg, we trace axial and principal rays through the lens with an estimate of the stop position. We chose a first stop position at l p 5 mm and then one at l p 6 mm . u p1 tan 40 deg 0.8391 . The third-order transverse astigmatism
TAC is 0.1842 mm if the stop is located at lstop 5 mm , and −0.1190 mm if it is at lstop 6 mm . By interpolation we determine that TAC is 0 when lstop 5.627 mm . All third-order aberrations as a function of the stop position are summarized in Table 19.1. We add the plots showing the astigmatism for the three stop positions, which are graphically demonstrated in Figure 19.2. For a better understanding of the impact of the material selection, we repeat the calculations with a glass that has a higher index of refraction and a higher Abbe number. Our choice is Schott glass LAKN13 with an index of n = 1.6935 and an Abbe number of V = 53.33. The high index leads to a thinner lens (reduced from 6 mm to 3.5 mm) and a lower lateral color as well, even though the Abbe number is not as large as that used in the first calculation, where n was assumed to be 1.5 and V was 62.5. With a somewhat lower Abbe number, the axial chromatic aberration is slightly larger. All this can be seen from the data in Table 19.2. Table 19.1 The third-order aberrations. lp −5.000 −6.000 −5.627
TSC −0.0998 −0.0998 −0.0998
CC −0.2759 −0.3262 −0.3074
TAC 0.1842 −0.1190 0
TPC 0 0 0
DC −7.7747 −7.7192 −7.6860
TAchC −0.0498 −0.0498 −0.0498
TchC −0.3579 −0.3829 −0.3736
Table 19.2 Impact on third-order aberrations by changing the lens material. Glass 5:625 LAKN13
TSC −0.0998 −0.0840
CC −0.3074 −0.2133
TAC 0 0
TPC 0 0
DC −7.6860 −6.0809
TAchC −0.0498 −0.0542
TchC −0.3736 −0.3595
160
Chapter 19
Figure 19.2 Astigmatism as a function of stop position.
19.3 Hypergon Lens Two Höegh lenses mounted symmetrically about the aperture stop in a double meniscus arrangement is known as the Hypergon lens, as shown in Fig. 19.3. For comparison, we maintain the focal length, the f/#, and the field coverage for our calculations. The principle of symmetry results in coma, lateral color, and distortion being corrected as well but only at unit magnification. Plots of the OPDs shown in Figure 19.4 indicate that this objective performs rather well. The third-order and chromatic aberrations are tabulated below. They indicate that the f/# has to be kept large due to a large spherical aberration. Table 19.3 Third-order and chromatic aberrations. TSC −0.08026
CC 0.00564
TAC −0.00019
TPC 0
DC −0.01661
TAchC −0.04732
Tch −0.01896
From the Höegh Meniscus to Double Anastigmats
161
Figure 19.3 Hypergon lens (double meniscus).
Figure 19.4 Performance of a Hypergon lens.
19.4 Achromatic Double Lens For an achromatic double lens, we maintain the symmetry but replace the singlets with cemented achromats. For variety, we reduce the f/# from f/30 to f/20, and the field from ±40 deg to ±20 deg.
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Figure 19.5 Symmetrically spaced achromats.
19.4.1 Approach and design method We keep the two achromatic lenses identical and mount them symmetrically about the aperture stop as shown in Figure 19.5. With fictitious indices and Abbe numbers, and chosen surface radii, lens thicknesses, and stop position, the starting layout prescription is summarized in Table 19.4. We use a substrate with a high index of refraction and a low dispersion for the positive front element A, and a material with a high dispersion and low index material for the negative rear element B. Such a combination is known as the “new achromat”. Table 19.4 Prescription of starting layout. Radii and thicknesses are given in millimeters. Notice that the outside radii are all the same, namely 26 mm. Surface 1
Radius 26
2
37
3
26
4 (stop)
5
−26
6
−37
7
−26
Thickness
n
V
2
1.9
40
2
1.6
35
2
1.6
35
2
1.9
40
6 6
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163
With the help of ZEMAX, we vary the radii and substitute real glasses. For element A, we select Schott glass N-LASF31, with nA = 1.880609 and VA = 41.0098. For element B, we pick TIFN5, with nB = 1.593555 and VB = 35.5145. Finally, we scale the lens so that we obtain the desired 100-mm focal length. The result is summarized in Table 19.5. Table 19.5 Radii and spacings of the scaled lens. Surface 1
Radius 23.4103
2
45.9339
3
21.0019
4 (stop)
5
-21.0019
6
-45.9339
7
-23.4103
Thickness 2.6617 2.0709 14.2406 14.2406 2.0709 2.6617
Optimizing the input data from Table 19.5 yields the configuration shown in Figure 19.6.
Figure 19.6 Optimized achromatic double lens.
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The performance plots for the achromatic double lens are shown in Figure 19.7 in the form of OPDs.
Figure 19.7 Performance of an achromatic double lens.
Figure 19.8 Double Anastigmat (Celor).
From the Höegh Meniscus to Double Anastigmats
165
19.5 Double Anastigmats Splitting the cemented achromats and providing airspace between elements A and B provides additional degrees of freedom for aberration correction. Such a configuration goes back more than 100 years, when it was designed by Höegh at the Goerz Company and offered under the name Celor.1 The lens consists of two positive outer elements and two negative inner elements. The aperture stop is located between the negative elements, as shown in Fig. 19.8. For this exercise, we decide on a 50-mm focal length and a relative aperture of f/5 with a half-field coverage of 20 deg.
19.5.1 Approach and design method We start with a system consisting of two thin achromats, each containing half of the system’s power. For the positive element A we pick SSKN5 as the material, and we select LF5 for the negative element B. The indices and V values of the chosen materials are nA = 1.658 and VA = 50.9, and nB = 1.581 and VB = 40.9, respectively. This combination is again a “new achromat”. Since the total system power is 1 / f 1 / 50 0.02 mm 1 , the power for one achromat is AB / 2 0.02 / 2 0.01mm 1 . Neglecting the spacing between element A and B, and assuming thin lenses, the requirements for the achromat are A
VA 50.9 AB 0.01 0.0509 mm 1 VA VB 50.9 40.9
and B
VB 40.9 A 0.0509 0.0409 mm 1 . 50.9 VA
Therefore, fA
1 1 19.646365 mm A 0.0509
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and fB
1 1 24.449878 mm . B 0.0409
For an equi-convex element A, the radii are RA1 2 nA 1 f A 2 1.658 1 19.646365 25.854616 mm
and RA2 25.854616 mm . Similarly, we calculate the radii for the equiconcave element B and find RB1 RB 2 28.410758 mm . We put this prescription into the computer and add a thickness of 2 mm for the positive elements and 1 mm for the negative elements. The stop is placed symmetrically between the two negative elements, which are 2 mm apart. After setting the half-fields for 14 deg and 20 deg, we make the radii variables but maintain symmetry (i.e., the radii of the elements after the stop are identified as pick-ups). For the first optimization, we set the operand for the focal length and use the default merit function. The result after this first optimization is indicated in the graphs in Figures 19.9 and 19.10.
Figure 19.9 MTF after first optimization.
From the Höegh Meniscus to Double Anastigmats
Figure 19.10 Ray intercept curves after the first optimization.
Figure 19.11 MTF after second optimization.
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Figure 19.12 Ray intercept curves after the second optimization
For the second optimization, we vary all radii, which breaks the symmetry. This asymmetric configuration is now called a Dogmar.2 The improvement is shown in Figures 19.11 and 19.12. We end this chapter with the understanding that adding elements and varying curvatures, thicknesses, and spacings, as well as lens element materials, improve performance even more.
References 1. R. Kingslake, Lens Design Fundamentals, Academic Press, New York (1978), page 236. 2. W. J. Smith, Modern Optical Engineering, 4th Edition , McGraw-Hill, New York (2008), page 486.
Index focus shift of a diffractive lens, 74 of a refractive element, 73 frequency cut-off, 147 normalized, 147 Nyquist, 149 spatial, 147
Abbe number, 18 for the infrared spectra, 115 aberrations, 116 chromatic, 18 achromats, 39 air-spaced doublet, 65 triplet, 65 Airy disk, 113 AMTIR-1, 58 aspherizing, 23, 52 a singlet, 26 astigmatism, 19, 104 athermats, 73
Gregorian microscope objective, 130 telescope, 127 G-sums, 15 half field angle, 115 hybrid, 69 lens, 151 petzval objective, 67
ball lens, 89 bending a lens, 11 best shape, 11 for minimum spherical aberration, 23
image height, 115 impact of housing material, 80
Cassegrain basic telescope layout, 122 classic system, 124 with two spherical mirrors, 123 chromatic aberration, 18 cold stop, 67 coma, 19 curvature, 11
limits, 147 lens thick, 34 thin, 11 LWIR region, 50
Dall-Kirkham arrangement, 124 detector, 16, 21 Dewar housing, 67 dialyte, 57 diamond-turned hybrid, 50 diffraction limit, 113 diffractive phase profile, 52
manufacturing remarks, 109 materials for the 3–5 µm spectral band (MWIR), 77 materials for the 8–12 m spectral band (LWIR), 78 microscope objectives, 121 minimum blur, 17 multiple lens arrangement, 23 MWIR region, 48
f/#, 115 focal plane array, 148
n, 12 net curvature, 11
169
170
Newton, Isaac, 121 nomograms, 55 Nyquist frequency, 149 optical modulation transfer function, 147 parabolic mirror, 107 paraxial focal plane, 21 Pegel diagrams, 95, 98 performance plots, 59 Petzval curvature, 101 objective, 65 phase coefficients, 61 equation, 76 pixel sizes, 147, 148 plane-parallel plate, 137 power, 11 prisms, 137 relative aperture, 113 reversed telephoto, 62 Ritchey-Chretien configuration, 125 sapphire, 141
Index
Seidel diagrams, 95 surface contributions, 96 shift of image, 141 silicon, 141 single optical elements, 113 single-imaging mirror, 101 single-point diamond turning, 108, 109, 128 slow tool servo, 108 Smith’s method, 45 spherical aberration, 16 telephoto, 62 telescope objectives, 121 thermal glass constant, 74 thick lens, 34 thin lens, 11 concept, 33 toroidal mirrors, 104 transverse spherical-aberration contribution, 12 third-order, 35 tunnel diagrams, 139 two Schwarzschild objectives, 131 two separated components, 57 two-mirror systems, 121
About the Author Max J. Riedl was born in Kempten, Germany, where he was educated at the Mathematisch Mechanisches Institut. He also graduated from the Akademie für angewandte Technik in Munich, where he studied precision mechanics and optics. He has worked in the field of mathematical and optical instruments for many years and holds numerous patents for mechanical, optical, and electro-optical devices. He held technical as well as business leadership positions during the 45 years he lived in the United States. He has published and presented many papers on fundamental lens design as well as on replicating optical elements and diamond turning diffractive optical components for applications in the infrared spectrum. He regularly teaches short courses on these subjects. He is the author of the well-received tutorial text Optical Design Fundamentals for Infrared Systems (SPIE Press, Vol. TT48), which is now in its second edition. He has also translated that textbook into German. He is a Fellow of SPIE. Presently, he resides in Bavaria, Germany.
