bentley field guide to lens design.pdf

Lens DesignField Guide to

Julie BentleyCraig Olson

SPIE Field GuidesVolume FG27

John E. Greivenkamp, Series Editor

Bellingham, Washington USA

FG27 covers and title.indd 3FG27 covers and title.indd 3 2/24/12 8:18 AM2/24/12 8:18 AM

Library of Congress Cataloging-in-Publication Data

Bentley, Julie (Julie L.)Field guide to lens design / Julie Bentley, Craig Olson.

pages cm. – (The field guide series)Includes bibliographical references and index.ISBN 978-0-8194-9164-01. Lenses–Design and construction. I. Olson, Craig 1971-

II. Title.QC385.B43 2012681′.423–dc23

2012035700

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SPIEP.O. Box 10Bellingham, Washington 98227-0010 USAPhone: +1.360.676.3290Fax: +1.360.647.1445Email: [email protected]: http://spie.org

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All rights reserved. No part of this publication may be re-produced or distributed in any form or by any means with-out written permission of the publisher.

The content of this book reflects the work and thought ofthe author. Every effort has been made to publish reliableand accurate information herein, but the publisher is notresponsible for the validity of the information or for anyoutcomes resulting from reliance thereon. For the latestupdates about this title, please visit the book’s page on ourwebsite.

Printed in the United States of America.First printing

Introduction to the Series

Welcome to the SPIE Field Guides—a series of publica-tions written directly for the practicing engineer or sci-entist. Many textbooks and professional reference bookscover optical principles and techniques in depth. The aimof the SPIE Field Guides is to distill this information,providing readers with a handy desk or briefcase refer-ence that provides basic, essential information about op-tical principles, techniques, or phenomena, including def-initions and descriptions, key equations, illustrations, ap-plication examples, design considerations, and additionalresources. A significant effort will be made to provide aconsistent notation and style between volumes in the se-ries.

Each SPIE Field Guide addresses a major field of opticalscience and technology. The concept of these Field Guidesis a format-intensive presentation based on figures andequations supplemented by concise explanations. In mostcases, this modular approach places a single topic on apage, and provides full coverage of that topic on that page.Highlights, insights, and rules of thumb are displayed insidebars to the main text. The appendices at the end ofeach Field Guide provide additional information such asrelated material outside the main scope of the volume,key mathematical relationships, and alternative methods.While complete in their coverage, the concise presentationmay not be appropriate for those new to the field.

The SPIE Field Guides are intended to be livingdocuments. The modular page-based presentation formatallows them to be easily updated and expanded. We areinterested in your suggestions for new Field Guide topicsas well as what material should be added to an individualvolume to make these Field Guides more useful to you.Please contact us at [email protected].

John E. Greivenkamp, Series EditorOptical Sciences Center

The University of Arizona

Field Guide to Lens Design

The Field Guide Series

Keep information at your fingertips with all of the titles inthe Field Guide Series:

Adaptive Optics, Second Edition, Robert Tyson &Benjamin Frazier

Atmospheric Optics, Larry AndrewsBinoculars and Scopes, Paul Yoder, Jr. &

Daniel VukobratovichDiffractive Optics, Yakov SoskindGeometrical Optics, John GreivenkampIllumination, Angelo Arecchi, Tahar Messadi, &

John KoshelImage Processing, Khan M. Iftekharuddin &

Abdul AwwalInfrared Systems, Detectors, and FPAs, Second Edition,

Arnold DanielsInterferometric Optical Testing, Eric Goodwin &

Jim WyantLaser Pulse Generation, Rüdiger PaschottaLasers, Rüdiger PaschottaMicroscopy, Tomasz TkaczykOptical Fabrication, Ray WilliamsonOptical Fiber Technology, Rüdiger PaschottaOptical Lithography, Chris MackOptical Thin Films, Ronald WilleyOptomechanical Design and Analysis, Katie Schwertz

& James BurgePolarization, Edward CollettProbability, Random Processes, and Random Data

Analysis, Larry AndrewsRadiometry, Barbara GrantSpecial Functions for Engineers, Larry AndrewsSpectroscopy, David BallTerahertz Sources, Detectors, and Optics,

Créidhe O’Sullivan & J. Anthony MurphyVisual and Ophthalmic Optics, Jim Schwiegerling



Optical design has a long and storied history, from themagnifiers of antiquity, to the telescopes of Galileo andNewton at the onset of modern science, to the ubiquityof modern advanced optics. The process for designinglenses is often considered both an art and a science. Whileadvancements in the field over the past two centurieshave done much to transform it from the former categoryto the latter, much of the lens design process remainsencapsulated in the experience and knowledge of industryveterans. This Field Guide provides a working referencefor practicing physicists, engineers, and scientists fordeciphering the nuances of basic lens design. Because theoptical design process is historically (and quite practically)closely related to ray optics, this book is intended asa companion to the Field Guide to Geometrical Optics,in which first-order optics, thin lenses, and basic opticalsystems are treated in more detail. Note that this compactreference is not a substitute for a comprehensive technicallibrary or the experience gained by sitting down anddesigning lenses.

This material was developed over the course of severalyears for undergraduate and graduate lens design classestaught at the University of Rochester. It begins withan outline of the general lens design process beforedelving into aberrations, basic lens design forms, andoptimization. An entire section is devoted to techniquesfor improving lens performance. Sections on tolerancing,stray light, and optical systems are followed by anappendix covering related topics such as optical materials,nonimaging concepts, designing for sampled imaging, andray tracing fundamentals, among others.

Thanks to both of our families—Danielle, Alison, Ben,Sarah, Julia, and especially our spouses, Jon and Kelly.The cats will now get fed, and all soccer parents beware!

Julie BentleyUniversity of Rochester

Craig OlsonL-3 Communications


vii

Table of Contents

Glossary of Symbols and Acronyms xi

Fundamentals of Optical Design 1Sign Conventions 1Basic Concepts 2Optical Design Process 3Aperture and Wavelength Specifications 4Resolution and Field of View 5Packaging and Environment 6Wave Aberration Function 7Third-Order Aberration Theory 8Spot Diagram and Encircled Energy 9Transverse Ray Plot 10Wavefront or OPD Plots 11Point Spread Function and Strehl Ratio 12MTF Basics 13Using MTF in Lens Design 14Defocus 15Wavefront Tilt 16Spherical Aberration 17Coma 18Field Curvature 19Petzval Curvature 20Astigmatism 21Distortion 22Primary Color and Secondary Color 23Lateral Color and Spherochromatism 24Higher-Order Aberrations 25Intrinsic and Induced Aberrations 26

Design Forms 27Selecting a Design Form: Refractive 27Selecting a Design Form: Reflective 28Singlets 29Achromatic Doublets 30Airspaced Doublets 31Cooke Triplet 32Double Gauss 33Petzval Lens 34Telephoto Lenses 35Retrofocus and Wide-Angle Lenses 36Refractive versus Reflective Systems 37Obscurations 38Newtonian and Cassegrain 39


viii

Table of Contents

Gregorian and Schwarzschild 40Catadioptric Telescope Objectives 41Unobscured Systems: Aperture Clearance 42Unobscured Systems: Field Clearance 43Three-Mirror Anastigmat 44Reflective Triplet 45Wide-Field Reflective Design Forms 46Zoom Lens Fundamentals 47Zoom Lens Design and Optimization 48

Improving a Design 49Techniques for Improving an Optical Design 49Angle of Incidence and Aplanatic Surfaces 50Splitting and Compounding 51Diffraction-Limited Performance 52Thin Lens Layout 53Lens Bending 54Material Selection 55Controlling the Petzval Sum 56Stop Shift and Stop Symmetry 57Telecentricity 58Vignetting 59Pupil Aberrations 60Aspheres: Design 61Aspheres: Fabrication 62Gradient Index Materials 63Diffractive Optics 64

Optimization 65Optimization 65Damped Least Squares 66Global Optimization 67Merit Function Construction 68Choosing Effective Variables 69Solves and Pickups 70Defining Field Points 71Pupil Sampling 72

Tolerancing 73Tolerancing 73Design Margin and Performance Budgets 74Optical Prints 75Radius of Curvature Tolerances 76


ix

Table of Contents

Surface Irregularity 77Center Thickness and Wedge Tolerances 78Material and Cosmetic Tolerances 79Lens Assembly Methods 80Assembly Tolerances 81Compensators 82Probability Distributions 83Sensitivity Analysis 84Performance Prediction 85Monte Carlo Analysis 86Environmental Analysis 87Athermalization 88

Stray Light 89Stray Light Analysis 89Stray Light Reduction 90Antireflection (AR) Coatings 91Ghost Analysis 92Cold Stop and Narcissus 93Nonsequential Ray Tracing 94Scattering and BSDF 95

Optical Systems 96Photographic Lenses: Fundamentals 96Photographic Lenses: Design Constraints 97Visual Instruments and the Eye 98Eyepiece Fundamentals 99Eyepiece Design Forms 100Telescopes 101Microscopes 102Microscope Objectives 103Relays 104

Appendix: Optical Fundamentals 105Index of Refraction and Dispersion 105Optical Materials: Glasses 106Optical Materials: Polymers/Plastics 107Optical Materials: Ultraviolet and Infrared 108Snell’s Law and Ray Tracing 109Focal Length, Power, and Magnification 110Aperture Stop and Field Stop 111Entrance and Exit Pupils 112Marginal and Chief Rays 113


x

Table of Contents

Zernike Polynomials 114Conic Sections 115Diffraction Gratings 116Optical Cements and Coatings 117Detectors: Sampling 118Detectors: Resolution 119The Lagrange Invariant and Étendue 120Illumination Design 121

Equation Summary 122Bibliography 127Index 129


xi

Glossary of Symbols and Acronyms

A AreaAOI Angle of incidenceAR AntireflectionBBAR Broadband antireflection coatingBFL Back focal lengthBFS Best fit sphereBRDF Bidirectional reflectance distribution

functionBSDF Bidirectional scattering distribution

functionBTDF Bidirectional transmittance distribution

functionc Surface curvatureC Lens conjugate factorCA Clear apertureCCD Charge-coupled deviceCDF Cumulative distribution functionCGH Computer-generated hologramCMOS Complementary metal-oxide

semiconductorCRA Chief ray angleCT Center thicknessCTE Coefficient of thermal expansionCTF Contrast transfer functiond Airspaced ThicknessDLS Damped least squaresdn/dT Thermo-optic coefficientDOE Diffractive optical elementEFL Effective focal lengthEPD Entrance pupil diameterESF Edge-spread functionETD Edge thickness differencef Focal lengthf /# f -number or relative apertureFEA Finite-element analysis


xii


FFL Front focal lengthFFOV Full field of viewFFT Fast Fourier transformFOV Field of viewGQ Gaussian quadratureGRIN Gradient indexh, h′ Object/image heightH Lagrange invariantH Normalized field coordinateHFOV Half field of viewHO Higher orderHOE Holographic optical elementHR High-reflectioni, i′ Angle of incidence w.r.t. surface normali, ia Marginal ray angle w.r.t. surface normali, ib Chief ray angle w.r.t. surface normalID Inner diameter of a lens barrel or mountIR InfraredL RadianceLOS Line of sightLR Limiting resolutionLSF Line spread functionLWIR Long-wave infraredm Diffraction orderm MagnificationMP Magnifying power (magnifier or

telescope)MTF Modulation transfer functionMWIR Midwave infraredn, n′ Index of refractionn(z), n(r) Gradient index profile functionNA Numerical apertureNITD Narcissus-induced temperature

differenceNRT Nonsequential ray tracingNUC Nonuniformity correction


xiii


OAP Off-axis parabolaOAR Off-axis rejectionOD Outer diameter (of a lens)OPD Optical path differenceOTF Optical transfer functionp Pixel pitch in sampled detector arraysP Partial dispersionPDF Probability distribution functionPSF Point spread functionPSNIT Point-source normalized irradiance

transmittancePST Point-source transmittanceP–V Peak to valleyQ Sampling ratior Radial surface coordinateR, ROC Radius of curvatureRI Relative illuminationRMS Root mean squareRSS Root sum squareRT Reflective triplets, s′ Object/image distanceSA Spherical aberrationSLR Single-lens reflext Thickness or airspaceT TemperatureTIR Total indicator runoutTIR Total internal reflectionTIS Total integrated scatterTMA Three-mirror anastigmatTML Three mirror longu, u′ Paraxial ray angles w.r.t. optical axisu, ua Marginal ray angle w.r.t. optical axisu,ub Chief ray angle w.r.t. optical axisUV UltravioletV Abbe numberW Wave aberration function


xiv


Wi jk Wavefront aberration coefficientWD Working distancey, ya Marginal ray height at a surfacey, yb Chief ray height at a surfacez Optical axisz(r) Surface sag/profile functionZn Zernike polynomial coefficientβ Lens shape factor∆λ Wavelength range or bandwidthδz Defocusε Obscuration ratioε,εx,εy Transverse ray errorθ,θ′ Angle of incidence/refractionθ Half field of viewθ Pupil azimuthal coordinateκ Conic constantλ Wavelengthλ0 Center wavelengthρ,ρx, ρy Normalized radial pupil coordinateΦ System powerφ Element or surface powerφ Merit or penalty function


Fundamentals of Optical Design 1

Sign Conventions

Throughout this Field Guide, a set of fully consistent signconventions is utilized.

• The z axis is the optical axis or the axis of symmetry ofa rotationally symmetric optical system.

• The y axis is in the plane of the paper, perpendicular tothe z axis, and the x axis points directly into the paper,forming a right-handed coordinate system.

• All distances are measured relative to a reference point,line, or plane in a Cartesian sense: directed distancesabove or to the right are positive; below or to the left arenegative.

• Angles u are measured from the optical axis to the raywith the smallest of the two angles chosen, while angleof incidence i are measured from the normal of thesurface to the incident ray.

• Counter-clockwise angles are positive, and clockwiseangles are negative.

• The radius of curvature of a surface is defined asthe directed distance from its vertex to its center ofcurvature and is therefore positive when the center ofcurvature is to the right of the surface.

• Light travels from left to right (from −z to +z) unlessreflected.

• A ′ (prime) symbol following a variable denotes a valueafter refraction at a surface.

• The signs of all indices of refraction following a reflectionare reversed.

• To aid in the use of these conventions, all directeddistances and angles are identified by arrows with thetail of the arrow at the reference point, line, or plane.


2 Fundamentals of Optical Design

Basic Concepts

A lens is often described as a device that creates an imageof an object on a detector. More generally, it can be thoughtof as any system that tries to collect and distribute light ina specified way. Although a lens can consist of a single lenselement, it is often composed of multiple lens elements ofdifferent shapes and sizes, referred to as a lens system oran optical system.

Optical design or lens design is the process used tofind the best set of lens construction parameters (e.g.,radii of curvature, thicknesses, airspaces, and materials)that optimizes the overall performance (including themanufacturability) of an optical system.

A specification document lists the requirements neededto design a lens system. It defines basic specificationssuch as object and image location, aperture, field ofview, and wavelength range. It contains image qualitymeasures such as spot size, resolution, and distortion. Italso includes packaging and environmental requirementssuch as diameter, length, and temperature.

Optical design software is computer code that aidsin the selection and optimization of lens constructionparameters and performance evaluation. Given a startingpoint and a set of variables and constraints, the computerprogram drives an optical design to a local optimum, asdefined by a merit function.

A lens designer (or “ray bender”) finds the variations inlens parameters that yield the greatest improvements inoptical system performance. Computer optimization andreal ray tracing (use of Snell’s law to find the positionand direction of a ray after refraction or reflection ateach lens surface) are key components of the process.A lens designer is also often responsible for developinga specification document (and resolving any conflicts)and identifying engineering trades between performance,packaging constraints, tolerances, cost, and schedule.



Optical Design Process

The basic optical design process is as follows:• A specification document for the optical system is

established, and the appropriate type of lens form (e.g.,microscope, telescope, camera lens) is selected.

• A starting-point design is chosen from previous projects,historical databases, or patent literature (primarilybased on the aperture, field, wavelength, and packagingrequirements).

• If an appropriate starting-point design does not exist,designers can choose to start from a first-ordersolution that identifies focal lengths and object/image/pupil locations and sizes using thin lenses and paraxialrays. The first-order solution can be converted to a third-order thin lens solution where thin lens parameterssuch as radius of curvature and index of refractionare chosen to minimize third-order aberrations.Thickness is then introduced into the thin lenses toarrive at a thick lens starting point.

• The starting-point design is assessed by tracing realrays at multiple wavelengths through the system toanalyze the chromatic and higher-order aberrations.Compliance to performance requirements (e.g., imagequality, packaging restrictions, fabrication and assemblytolerances, cost) is evaluated.

• If the performance is not acceptable, optimization isused to improve the design. A set of variables andconstraints is defined, and computer software is used tofind design solutions as defined by a merit function.

• If the solution is still unacceptable, designers mustreturn to a previous step and iterate until the designmeets specifications. This process may require theselection of a different starting-point design and/orrequesting a significant change in one or more of thespecifications.

• The manufacturability of the design is just as importantas the final image quality. Therefore, a completedesign process and performance evaluation includestolerancing, stray light analysis, and thermal andother environmental analyses.



Aperture and Wavelength Specifications

The aperture specification deter-mines the amount of light collectedfrom an object and establishes a res-olution limit to the system. The threemost common ways to specify an aper-ture for a lens system are entrance pupil diameter(EPD), f /#, and numerical aperture (NA).

The entrance pupil diameter defines the size of the beamin object space and is a common way to specify theaperture of lens systems during computer-aided design.The ratio of the focal length of a lens to its entrance pupildiameter is called the relative aperture or f /#.

f /#= fEPD

N A = n′ sinu′.

Numerical aperture is the index of refraction of the imagespace media times the sine of the half-angle cone of light.f /# is typically used to specify systems with distant objects(e.g., camera lenses or telescopes), whereas NA is used tospecify systems that work at finite conjugates (e.g., relaysystems or microscope objectives).

For infinite conjugate systems, f /# and NA are relatedby f /# = 1/(2N A). Although this equation only holds forsystems with infinite object distances, the same quantityis often calculated for finite conjugate systems and isreferred to as the working f /#.

The spectral range of an optical system is identified bya center wavelength λ0 and a bandwidth ∆λ. Relativebandwidth is the ratio of the total bandwidth tothe center wavelength. Narrow-bandwidth systems arecalled monochromatic (e.g., laser collimators), whereaslarger-bandwidth systems are called polychromatic(e.g., camera objectives). The optical materials areideally chosen for both high transmission at the systemwavelengths and good chromatic correction over the fullbandwidth.



Resolution and Field of View

Diffraction from a finite circular aperture results in anAiry disc pattern. The Rayleigh criterion determinesthe smallest lateral separation ∆X between two Airydisc patterns for which two distinct points can still beresolved. Therefore, the fundamental resolution of arotationally symmetric optical system is a function of bothits wavelength and its numerical aperture. To increaseresolution, systems move to shorter wavelengths andhigher numerical apertures.

The field of view (FOV) of an optical system is specifiedas an object height, object angle, or image height. Theimage height can be directly correlated to the size of thedetector (e.g., film or CCD) butcan also represent the size ofthe field over which the lens iswell corrected. The half fieldof view (HFOV) is the radiusof the object height h or imageheight h′, whereas the full fieldof view (FFOV) indicates adiameter measure.

For lenses with infinite object distances, one can quicklyconvert from angular HFOV θ to image height h′ using

h′ = f tanθwhere f is the focal length of the system. This equationis also very useful for converting detector resolution ∆h′to angular resolution ∆θ for a focal length f .



Packaging and Environment

The achievable imaging performance of a lens designis often limited by packaging and interface require-ments. For example, the maximum number of elementsthat can be used in a design may be dictated by a maxi-mum lens envelope (diameter and length) and/or weightconstraint. Other interface requirements that may di-rectly or indirectly drive the design include working dis-tance, pupil location and size, filter/window/fold mirrorposition and size, and center-of-gravity location and shift(for lenses with movable el-ements). Special access formodularity (e.g., an eye-piece) or field service mayalso be required.

Most lens systems have a working distance requirementto keep the lens from crashing into an object or a sensor.It is important to clarify if a specification references thevertex of the lens element (back focal length or BFL),the closest mechanical surface point to the sensor (imageclearance), or a mechanical flange surface (flange-to-focus).

Environmental conditions and outside influences such astemperature, altitude, and vibration place additionalconstraints on a lens design. Moisture or humidityeffects may induce coating delamination, absorptionand scattering from condensed water droplets, andglass staining. Shock requirements dictate single-eventextreme conditions that the lens must survive (e.g., athermal shock, drop, or sudden pressure loss that canoccur during shipping). Extreme environments, such ashighly corrosive salt water, or use in a vacuum chambercan further restrict design freedom. Lenses operating inionizing radiation require special radiation-hardenedglass choices because many optical glasses will “fog” orbecome opaque over time in such an environment.



Wave Aberration Function

Geometrical optics describes the properties of lenseswhen the effects of diffraction and interference areignored (in the limit where the wavelength approacheszero). In this case, an object consists of a collection ofindependently radiating point sources. Each point sourceemits a spherical wavefront (surface of constant opticalpath length from a point on the object), and rays are tracedperpendicular to the wavefront.

First-order optics is the study of perfect optical systemswithout aberrations. In a perfect system, an incidentspherical wave is mapped to a perfect spherical wave inthe exit pupil, converging on the ideal image location.

Aberrations are imperfections in the image formationof an optical system. They are measured either by thedeformation of the wavefront in the exit pupil or bythe lateral distance ε, by which the rays miss the idealimage point. The distance ε is called the transverse rayerror, whereas the wave aberration function W is theoptical length, measured along a ray, from the aberratedwavefront to the reference sphere.

W and ε are related by a partial derivative with respect topupil coordinate ρ, where u′

a is the angle of the marginalray in image space:

ε′y =1

n′u′a

∂W∂ρy

ε′x =1

n′u′a

∂W∂ρx



Third-Order Aberration Theory

In a rotationally symmetric optical system, one can specifyany ray in the system with only three coordinates: its1D position in the field plane (typically along the y axis),denoted by the normalized field coordinate H, and its2D position in the pupil plane, denoted by the normalizedpupil coordinates ρx(ρsinθ) and ρy(ρcosθ).

Monochromatic aberrations are described by expand-ing the wave aberration function W in a power series ofaperture and field coordinates, ρ,θ, and H:

WIJK ⇒ H IρJ cosK θ

W(H,ρ,θ)=W020ρ2 +W111Hρcosθ+W040ρ

4 +W131Hρ3 cosθ+W222H2ρ2 cos2θ+W220H2ρ2 +W311H3ρcosθ+O(6)

where the other terms in the power series are forbiddenby rotational symmetry. In third-order aberrationtheory (because of the derivative dependence, third-ordertransverse aberrations have a fourth-order wavefrontdependence), the higher-order terms O(6) in the waveexpansion are dropped. The coefficients for the remainingterms are traditionally defined as

W020: Defocus W222: AstigmatismW111: Wavefront tilt W220: Field curvatureW040: Spherical aberration W311: DistortionW131: Coma

The value of each polynomial coefficient (WIJK ) gives themagnitude of the aberration and is a function of lensparameters (e.g., index of refraction, thickness, and radiusof curvature), and marginal/chief ray heights and angles.



Spot Diagram and Encircled Energy

A spot diagram or scatter plot shows the transverseray error (εy versus εx) for a given object height H. Raysare traced into the entrance pupil from a single objectpoint. A “spot” is plotted where each ray hits the imageplane with respect to the paraxial image point; severalhundred rays at different wavelengths are usually plotted.

Spot diagrams are good for final system evaluation,especially for systems with aberrations much larger thanthe diffraction limit.

The RMS spot diameter or RMS spot size is a commonmetric for the quality of an image point. Many rays aretraced from a single object point, and the centroid or“center of gravity” of all of the image plane intersectionsis determined. The RMS spot size is then calculated bytaking the square root of the sum of the squares of eachray’s distance from the centroid, divided by the totalnumber of rays.

Encircled energy is an im-age quality measure for a singlefield point, using the intensitydistribution of a spot to calcu-late the percentage of transmit-ted energy through an openingof variable radius. Ensquaredenergy assumes that the open-ing has a square shape.



Transverse Ray Plot

Transverse ray aberration plots or ray interceptplots are plots of transverse ray error versus the relativeposition of the ray in the pupil (εy versus ρy, and εx versusρx) for different field points H. They are similar to spotdiagrams, but instead of tracing a 2D grid of rays in thepupil, they use a tangential (Y ) and a sagittal (X ) sliceof the pupil. A “perfect” lens has a straight line along thehorizontal axis in both plots.

Transverse ray plots and spot diagrams are typicallyplotted for three different field points: on-axis, 0.7 field,and full-field. The shape of the plots at different valuesof H yield a lot of information about which aberrationsare present in the lens. An estimate of peak-to-valley spotdiameter (∆) can be easily calculated from the transverseray plot by drawing a rectangle around the total error.



Wavefront or OPD Plots

The departure of an aberrated wavefront from an idealspherical wavefront is called the optical path difference(OPD) and is typically measured in units of wavelength.Wavefront or OPD plots are plots of OPD versus ρy

and OPD versus ρx for different field points H. Wavefrontplots are proportional to the integral of the transverseray error plot and also indicate what aberrations arepresent. OPD plots are used instead of transverse ray plotsfor well-corrected systems with nearly diffraction-limitedperformance.

Wavefront plots show slices of the full wavefront errorof an optical system that would be measured by a testinterferometer at a given field point.

Peak-to-valley (P–V)OPD is the differencebetween the wavefrontmaximum and the wave-front minimum, and is agood measure of the im-age quality of an optical system if the wavefront error issmooth with small changes in slope. Most real wavefrontsare not smooth, and the RMS OPD is generally a bettermeasure of image quality. The RMS OPD is calculated bytaking the square root of the mean of the squared OPDvalues sampled over the pupil.

The Rayleigh quarter-wave criterion states that ifthe total wavefront deviation is less than a quarter of awavelength (P–V OPD < λ/4), then the image quality is“good” or nearly diffraction limited. Although the exactrelationship between P–V OPD and RMS varies with theshape of the wavefront, a common rule of thumb is RMSOPD = (P–V OPD/5). This is then used to define the RMSlimit, RMS OPD < λ/20 (0.05), for the image quality of a“good” lens.



Point Spread Function and Strehl Ratio

The point spread func-tion (PSF) is the im-pulse response function ofa lens and includes the ef-fects of both aberrationsand diffraction. For a sin-gle point object, a PSFplot shows the full 3D in-tensity distribution in theimage plane. In practice,a line spread function (LSF) or edge spread function(ESF) gives the intensity distribution of a line or edge ob-ject and is a more-convenient measure of the lens perfor-mance.

The Strehl ratio is defined as the ratio of the peakintensity of an aberrated lens to the peak intensity of an

aberration-free lens. TheStrehl ratio is a good single-number, image-quality de-sign measure for well-corrected systems but is lessmeaningful for highly aber-rated systems.

The Maréchal criterion for image quality is a Strehlratio of 0.82, which is equivalent to an RMS OPD of ∼ λ/14or 0.071λ RMS. A Strehl ratio of 0.8 corresponds closelyto a P–V OPD of λ/4 for a lens with only defocus and/orspherical aberration. Both criteria are commonly referredto as diffraction limited.

For a lens with a relatively small OPD, the relationshipbetween the Strehl ratio and the RMS OPD isapproximated by

Strehl ∼=(1−2π2ω2

)ω= RMSOPD



MTF Basics

The modulation transfer function (MTF) is themagnitude of the complex optical transfer function(OTF) and is a measure of how well a lens relays contrastfrom object to image. MTF is the ratio of the image sinewave modulation to that of the object as a function ofthe spatial frequency, typically specified at the imagein cycles/mm. A cycle is one full period (peak and trough)of the sine wave. For distant objects, the spatial frequencycan be expressed in cycles/mrad.

The optical cut-off frequency is the spatial frequencybeyond which an optical system cannot transmit informa-tion. It equals 2NA/λ or 1/(λ f /#) for incoherent light.

In a perfect image, the MTF is unity at all frequencies.In a real image, contrast is degraded by both diffractionand aberrations. The OTF is equal to either the Fouriertransform of the incoherent PSF or the autocorrelation ofthe OPD (pupil function). In lenses with large aberrations,the OTF can be negative, leading to MTF bounce, whichrepresents a phase reversal of light and dark features(spurious resolution).

The contrast transfer function (CTF) uses a square-wave modulation to determine constrast and is specifiedin line pairs/mm. A line pair consists of one black lineand one white line. MTF specifications often refer to linepairs/mm when cycles/mm is intended because squarewave targets are easier to generate for measurementpurposes.



Using MTF in Lens Design

A complete MTF plot includes performance curvesfor several different field points and a diffraction-limited reference line. For off-axis field points, theMTF depends on the orientation of the “lines” inthe object and is therefore plotted for both spoke-likeradial modulation (sagittal S) and wheel-like azimuthalmodulation (tangential T), where the modulationdirection is orthogonal to the feature direction. The on-axis MTF is typically a single curve but will be separatedinto sagittal and tangential components in lenses withdecentered or tilted elements.

To aid in developing system performance budgets, theMTF can be indirectly linked to wave aberrations or OPDusing the Hopkins ratio, which describes the ratio ofthe aberrated MTF to the diffraction-limited MTF as afunction of RMS OPD.

Imaging performance is quantified by specifying an MTFvalue at a particular spatial frequency. For digital sensors,this frequency is usually associated with the sensor’sNyquist frequency. Although the MTF plots do notdirectly tell a designer which aberrations are present inthe lens, MTF is useful for specifying system performancebecause it is easily measured and allows the cascading ofMTFs using linear system theory (where the camera,atmosphere, lens, and other MTFs are multiplied to derivethe overall system MTF). However, optimizing a lensdirectly for MTF is computationally intensive, and phasereversals in the MTF can cause local minima in the meritfunction and stagnated optimization.



Defocus

In an optical system, focus is the act of changing therelative position of the sensor with respect to the elements(usually to improve the sharpness of an image).

The hyperfocal distance is the object distance for alens that maintains the depth of focus criteria at theimage plane for all object distances between half thehyperfocal distance and infinity (without refocusing).

Defocus is the amount of aberration introduced whenmoving the image plane axially from the paraxial imageplane and is directly proportional to the amount ofmovement δz. Defocus is used to balance the blur of otheraberrations and improve the image quality of the system.It is not typically considered a “true” aberration.

A wavefront plot for defocus shows a quadratic dependenceon aperture coordinate ρ, whereas a transverse ray plotshows a linear dependence on ρ that has the same slope inboth x and y and is independent of field coordinate H.



Wavefront Tilt

Wavefront tilt is the difference between the paraxialmagnification and the actual magnification of the system.There is no tilt on axis (H = 0); for objects located along they axis, there is no tilt in x. A wavefront plot for tilt showsa linear dependence on aperture coordinate ρ, whereas atransverse ray plot shows a constant dependence on ρ thatincreases in magnitude with field coordinate H. The spotdiagram shows a perfect point image, but it is not locatedat the expected paraxial image point.

The depth of focus of a lens is the longitudinal shiftof the image sensor that produces an image degradationthat is still acceptable for the application. For example,the Rayleigh quarter-wave criterion for OPD results in adepth of focus equal to ±2λ( f /#)2 or ±λ/(2N A2). Similarly,the depth of field of a lens is the allowable shift of theobject position and is related to the depth of focus by thelongitudinal magnification.



Spherical Aberration

Spherical aberration is a variation in focus positionbetween the paraxial rays through the center of the lensaperture and those through the edge of the aperture.

Spherical aberration is the only one of the five primarythird-order aberrations to appear on axis (H = 0), and ithas the same magnitude for any field coordinate H. Awavefront plot for spherical aberration shows a fourth-order dependence on aperture coordinate ρ, whereas atransverse ray plot shows a cubic dependence on ρ thatis the same in both x and y.

Spherical aberration is the first aberration to appearin most symmetrical lens tolerancing budgets. Becauseerrors in the airspaces between lens components alsoresult in spherical aberration, a space adjust procedurecan be devised whereby adjustable spacers are used in themost sensitive airspaces of a design as compensators tofix the as-built performance.



Coma

For symmetrical optical systems, coma is an off-axisaberration where annular zones of the aperture havedifferent magnifications.

There is no coma on axis (H = 0), and for objects locatedalong the y axis, there is no coma in x when ρy equalszero. A wavefront plot for coma shows a cubic dependenceon aperture coordinate ρ, whereas a transverse ray plotshows a quadratic dependence on ρ that increases inmagnitude with field coordinate H. The spot diagramshows that the resulting image of a point object looks likea comet.

Coma is the first aberration to appear in mostasymmetrical lens tolerancing budgets. For example,errors in the tilt or decenter of lens components result incoma. Correction schemes can be devised that include anadjustable “push-around” element as a compensator tofix the as-built performance.



Field Curvature

A symmetrical optical system with spherical surfacesnaturally forms an image on a curved surface. Fieldcurvature is an off-axis aberration that describes thedeparture of the image surface from a flat surface. Fieldcurvature looks similar to defocus; however, there isno field curvature on axis (H = 0), and its magnitudeincreases with field coordinate H. A wavefront plot forfield curvature shows a quadratic dependence on aperturecoordinate ρ, whereas a transverse ray plot shows a lineardependence on ρ that has the same slope in both x and y.

A field flattener is a lens (usually negative) placed closeto the image plane where the marginal ray height issmall. These lenses are used to reduce the field curvaturewithout significantly changing the image size or addingspherical aberration.



Petzval Curvature

W220 = 12

W∗222 +W220p

The field curvature coefficientW220 can be separated into twocomponents, where W∗

222 is a term whose functionaldependence is equivalent to the astigmatism coefficient(W222), and W220p is the portion of field curvature due toPetzval curvature.

The separation of field curvature into two distinct terms isuseful during lens design, as the techniques for correctingthe two aberrations are significantly different. Theamount of astigmatism introduced by a lens element is afunction of the power and shape of the lens and its distancefrom the aperture stop, whereas the amount of Petzvalcurvature introduced by that same element is purely afunction of lens power and index of refraction. Petzvalcurvature is the only aberration completely independentof ray quantities (angles and heights); this makes Petzvalcurvature one of the most difficult aberrations to correctonce the first-order power balance in the design hasbeen established. The key to fixing the Petzval curvature(thereby reducing the overall field curvature) in opticaldesign is to change the first-order power balance betweenpositive and negative power.

In most positive-power refracting lenses, the Petzvalsurface will be curved toward the lens (known as aninward-curving field). If there is no astigmatism in thelens, the field curvature reduces to the Petzval curvature.If Petzval curvature cannot be corrected, overcorrectingastigmatism can be introduced to artificially flatten thetangential field, leaving the sagittal field to fall betweenthe image plane and the Petzval surface, and yielding thesmallest amount of residual field curvature.

Some optical systems use a curved image surface (e.g.,the retina) to significantly reduce the effects of Petzval.Modern-day “curved” digital sensors have the potential todramatically simplify lens systems by removing the needto correct Petzval in the design.



Astigmatism

Astigmatism is the aber-ration that causes a tan-gential fan of rays to fo-cus at a different loca-tion than a sagittal rayfan. Fundamentally, astig-matism looks like field cur-vature, but the magnitude of the aberration is differentbetween x and y. In the wavefront expansion, astigmatismis defined in terms of a tangential (y axis) component only.However, a portion of the field curvature of a lens is alsoproportional to the astigmatism coefficient, adding a sagit-tal (x axis) component; this term is also included in theplots.

There is no astigmatism on axis (H = 0), and its magnitudeincreases with field coordinate H. A wavefront plot forastigmatism shows a quadratic dependence on aperturecoordinate ρ, whereas a transverse ray plot shows a lineardependence on ρ that has a different slope in x and y. Aspot diagram shows the image of a point to be an ellipse.

Design programs plot the longitudinalastigmatic field curves on a separategraph. When there is no astigmatism,the sagittal and tangential imagesurfaces coincide with each other andlie on the Petzval surface. If thereis astigmatism, then the tangentialimage surface lies three times asfar from the Petzval surface as thesagittal image surface.



Distortion

Distortion is an aberration in which the magnificationvaries over the image height. Object points are imagedto perfect points with no blur, but straight lines in theobject become curved lines in the image. Design programsdo not include distortion as part of their wavefront ortransverse ray plots. They instead plot distortion as eithera percentage versus field coordinate H or as a deviationfrom a grid of points. The percent of distortion is usuallycalculated from the difference in height between a realand a paraxial chief ray for a given field point; however,a centroid distortion (using either the geometric spotor PSF centroid location instead) may be a more relevantevaluation for a particular lens/application.

• Pincushion distortion occurs if the magnificationincreases toward the edge of the field.

• Barrel distortion occurs if the magnification decreasestoward the edge of the field.

• Keystone distortion produces a trapezoidal image of arectangular object and can result from nonparallel objectand image planes.



Primary Color and Secondary Color

Chromatic aberrations arise because the refractiveindex of a lens material changes with wavelength(characterized by the material dispersion). Light froma single object point at different wavelengths is not

focused to the samepoint in the image (e.g.,for a simple lens, bluelight from an axial,white-light point source

is focused before red light). For polychromatic lenses, per-formance evaluations such as transverse ray plots arecomputed for at least three wavelengths that span the sys-tem bandwidth (one at each end of the spectrum and one inthe middle). For the visible spectrum, these wavelengthsare typically C (red), d (yellow), and F (blue).

The difference between the red and blue focus isknown as primary color (also referred to as axialcolor, longitudinal color, or longitudinal chromaticaberration). Similar todefocus, the transverse rayplot shows a linear depen-dence on ρ that has thesame slope in both x and yand is independent of fieldcoordinate H; but the slopeof the line changes withwavelength.

If the primary color of the lens has been corrected, thelens can still have secondary color, which is a differencebetween the common red/blue focus position and theyellow focus position (also indicated by a change in slope

on the transverse ray plot).When the d line is also imagedto the same focus as the Fand C lines, the system is saidto be corrected for secondaryspectrum.



Lateral Color and Spherochromatism

Lateral color (or transverse chromatic aberration)is an aberration in which the magnification of the systemis wavelength dependent (the image height varies withwavelength). For a simple visible lens with the aperturestop after the lens, the blue image appears to be largerthan the red image.

There is no lateral color onaxis (H = 0), and for objectslocated along the y axis, thereis no lateral color in x. Atransverse ray plot showsa constant dependence on ρ

that increases in magnitudewith field coordinate H (similar to tilt); however, theintercept value changes with wavelength. Unlike primarycolor, the magnitude of lateral color is highly dependent onthe stop location.

Color correction equations developed for the visiblespectrum (d, F, and C wavelengths) can be used outside ofthe visible spectrum if a custom Abbe numer V is definedfor a material over the band of interest, with the indicesat the long, short, and middle wavelengths.

Spherochromatism is a variation of spherical aber-ration with wavelength. In some lenses, there isvery little spherochromatism. In others, the spher-ical aberration can switch sign (typically blue is

overcorrected and red is un-dercorrected), and the sphe-rochromatism must be cor-rected or balanced with axialcolor to reduce image blur.



Higher-Order Aberrations

Lens systems with large apertures (< f /4) and widefields of view (>50-deg FFOV) will show evidence ofhigher-order (HO) aberrations. The five third-orderaberrations have fifth-order counterparts (e.g., fifth-order spherical aberration looks very similar to third-order spherical aberration, but its dependence on aperturecoordinate ρ increases in order). There are also two “new”fifth-order aberrations: elliptical coma and obliquespherical aberration.

During optimization, the best per-formance is typically achieved bybalancing low-order aberrationswith high-order aberrations. Theresidual errors then will show aninflection in the transverse ray, dis-tortion, or field curves, indicating asign change between the low-orderand high-order aberration contri-butions.



Intrinsic and Induced Aberrations

Intrinsic aberrations are aberrations associated witha single surface in an optical system and can becalculated solely from paraxial ray angles and heightsat a given surface. In contrast, induced aberrationsare aberrations generated from the interaction betweenaberrated rays coming into a surface and the intrinsicaberrations of that surface.

A simple example of induced spherochromatism can beconstructed by placing a concave spherical mirror in theconverging cone created by a positive-powered singlet. Thelens sends uncorrected axial color into the mirror surfaceso that the marginal ray height at the surface varies withwavelength; the axial color interacts with the mirror’sintrinsic spherical aberration resulting in a differentamount of spherical aberration for each wavelength. Themirror itself is not dispersive, so the aberration is notintrinsic.

Intrinsic and induced aberrations can be used to offseteach other to improve performance. The functional formof an induced aberration can be identical to an intrinsicaberration, but it is usually harder to correct the inducedaberrations in a design. Complex, well-corrected designsare often limited by higher-order induced aberrations(specifically, higher-order Petzval curvature and sagittaloblique spherical aberration).


Design Forms 27

Selecting a Design Form: Refractive

Aperture and field size are the two key systemspecifications that determine the difficulty/complexityof a design and can therefore be used to identify anappropriate starting design form. If the aperture or fieldsize of a design form is increased outside of its workingregion, the performance drops off; if the aperture orfield size is decreased, the design form is likely to meetthe performance requirement but is more complex/costlythan necessary. The classic design-form regions indicatedbelow are approximate and do not necessarily representdiffraction-limited performance. Most assume sphericalsurfaces and are valid over the visible band. Moderndesigns build on these basic forms, often incorporatingaspheres to reduce package size, improve performance,and/or increase the aperture or field size.


28 Design Forms

Selecting a Design Form: Reflective

Similar to refractive design form selection, apertureand field size are the two key metrics for choosing areflective design form. It is important to first select aform that can support the FOV requirements. Then theaperture specification is a strong driver for the numberof mirrors and their asphericity; aspheres are effectiveat correcting spherical aberration when increasing theaperture but are less effective at increasing the fieldcoverage. The system size requirements will often filterout most of the candidate design forms. For example, ifthe envelope diameter must be less than 2–3× the EPD,it is unlikely that a system with a negative primarymirror that performs well over a wide field can be used.If an accessible field stop and/or cold stop is required, thisimplies that a reimaging system is needed and excludesnon-reimaging forms. Furthermore, if a Lyot stop isneeded with high-quality pupil imaging, more complexityin the mirrors (e.g., tilts and/or higher-order aspherics)might be necessary.


Design Forms 29

Singlets

A singlet is a lens usedwithout any other poweredelements; the stop is typicallylocated at the lens. Singlets

have relatively poor image quality, limited by largechromatic aberrations and field curvature. They are oftenused as a simple magnifying glass or photon collector.

An aspheric collimator is a singlet withat least one aspheric surface. The as-pheric surface allows for an increased nu-merical aperture and increased resolutionover a very narrow wavelength range and small field.These lenses are commonly found in laser collimating ap-plications such as fiber coupling, laser diodes, optical datastorage, and barcode scanners.

Landscape photography requires wide-FOV lenses thatcapture as much of the scenery as possible. Shifting thestop in a singlet away from the lens increases its FOV andresults in a simple landscape lens. As the stop is moved,the lens shape changes and is generally bent around thestop to reduce the ray angles of incidence and the off-axisaberrations. The stop shift also introduces astigmatismthat helps to flatten the tangential field curvature.A single-element landscape lens still has sphericalaberration and chromatic aberrations and is thereforerestricted to a small NA. For better correction, thesinglet may be aspherized or compounded into a doublet.

The farther the aperture stopis shifted from the lens, thelarger the lens diameter. Dis-posable “box” cameras gen-erally use simple landscapelenses.


30 Design Forms

Achromatic Doublets

An achromatic doubletor achromat is composedof two singlets designed incombination to correct pri-mary color at two wave-

lengths (typically red and blue). The individual elementpowers of the doublet are determined by the total powerand the individual Abbe numbers of the materials. Forstandard refracting materials with positive Abbe num-bers, the two elements will have opposite powers, and eachhas stronger absolute power than the total doublet power.The shape of the elements is chosen to minimize sphericalaberration and coma.

The on-axis spot size of an f /5 visible-band achromat is∼25 times smaller than a singlet of the same aperture.

Doublet elements can be either cemented together orseparated by an airspace. The cemented doublet isoften preferred for ease of fabrication and alignment. AFraunhofer doublet is a positive crown followed by anegative flint, whereas a Steinheil doublet is a flint-firstdoublet. The Fraunhofer form has a shallower curvatureat the cemented interface and is usually the preferreddesign form.

Achromats are limited in field to less than a few degreesby residual Petzval and in f /# to f /4 or larger by higher-order spherical aberration. Fast achromats (< f /4) are rarebecause the V ratio that determines element power isfundamentally limited by existing glass choices. Commondoublet applications include broadband collimating lensesand telescope objectives. Doublets are also critical buildingblocks for complex broadband lenses.

Lenses corrected for axial color at three and four wave-lengths are called apochromats and super-achromats,respectively, and usually require more than two ele-ments/materials.


Design Forms 31

Airspaced Doublets

The speed of a Fraunhoferdoublet can be increased to∼ f /2.5 by separating the ce-mented surface to create anairspaced doublet. The ex-tra curvature and airspace areused to control higher-orderspherical aberration and sphe-rochromatism. However, the ray angles of incidence in theairspace between the two elements can become very large,making the alignment tolerances of the two elements verytight. A second type of airspaced doublet is a Gauss dou-blet, which has much steeper curves and significantlylarger higher-order aberrations. As a result, Gauss dou-blets are not as fast as Fraunhofer doublets and are oftenfound in stop-symmetric anastigmatic lenses.

Two separated positive and neg-ative elements (or groups of ele-ments) with a relatively large airgap between them is known as adialyte. Although the performanceof a simple two-element dialyte is

usually insufficient (primarily due to its inability to cor-rect lateral color), it is the basis for more-complex tele-photo (positive/negative) and reverse telephoto (nega-tive/positive) multi-element designs.

A Schupmann lens is a special airspaced doublet that iscorrected for primary color but uses the same material inboth elements. Because the lens has a virtual image, it isseldom used alone but rather is often part of more-complexlens systems.


32 Design Forms

Cooke Triplet

The Cooke tripletis the simplest de-sign form withenough degrees offreedom to correctall first- and third-order aberrations and hold first-order imaging constraintssuch as EFL and BFL. The outer two elements are pos-itive crowns, and the middle element is a negative flintwith the aperture stop located on either side of the middleelement. The symmetry of the design helps control coma,distortion, and lateral color. Because it is only correctedto third order, the Cooke triplet is limited by higher-orderaberrations and is thus capable of relatively slow aper-tures (∼ f /6) and moderate fields (10–15 deg). Up to 50%vignetting may be needed in systems with larger fields.

The Cooke triplet is the most basic form of an anastigmatlens. Strictly speaking, the term anastigmat means “freeof astigmatism;” however, it is also used to describeoptical systems with reduced astigmatism and relativelyflat fields. In these designs, the field is flattened byseparating positive and negative power along the opticalaxis (between surfaces, elements, or even multi-elementgroups of components). PLAN microscope objectivesuse this principal to flatten the field curvature.

Many photographic objective design forms are derivedfrom the Cooke triplet by splitting, compounding,or otherwise modifying elements of the basic triplet.For example, in the Tessar lens, the last elementof a Cooke triplet is compounded into a doublet.This design allows for speedsup to f /4 and/or largerfields of view than the ba-sic triplet with reduced vi-gnetting.


Design Forms 33

Double Gauss

The double Gauss (or Biotar orPlanar) is a powerful anastigmatdesign form used in a wide rangeof applications, including fixed-focuscamera objectives, projection lenses,and high-resolution microscope objectives. The classicsix-element design is nearly symmetric about the stop(reducing coma, distortion, and lateral color), with outerpositive singlets and inner cemented thick-meniscusdoublets (to minimize Petzval). Astigmatism is controlledby adjusting the distance of the elements from the stop, asthis separation has little effect on the other aberrations.Design improvements to the basic form have been studiedextensively in the literature; allowing the system to departfrom symmetry and/or adding elements (by splitting orcompounding) allows one to achieve speeds up to f /1 orFFOVs larger than 50 deg.

Fast anastigmats with small fields tend to have relativelyshort vertex lengths, whereas slow-speed wide-anglelenses tend to have longer vertex lengths. In Cooketriplets the vertex length can be controlled by glasschoice (particularly the difference in V between positiveand negative elements).

Many other lens forms use stop symmetry and powerseparation to achieve wide fields of view. The simplestis the Hypergon; two identical meniscus elementssymmetric about the stop achieving an impressive 135-deg FFOV. Petzval is reduced by the separation of positiveand negative power in the elements themselves whereeach element has a positive and anegative surface with radii differing byless than 7%. However, Hypergons areslow (∼ f /20) as there is no remainingdegree of freedom to correct the sphericalaberration. More-complex designs (e.g.,Topogon, Hologon, and Biogon) addelements to increase their speed.


34 Design Forms

Petzval Lens

The Petzval or Petzval portrait lens is a designform useful for high-speed applications that requirehigh-quality imaging over relatively small fields (e.g.,aerial reconnaissance). Originally designed for indoorphotography, Petzval lenses are an order of magnitudefaster than landscape lenses. The Petzval lens is the basisof many microscope objectives where aplanatic elementsare added to the short conjugate for increased NA.

Unlike many of the other ba-sic design forms, the Petzvallens has very little symme-try. The basic layout uses twoseparated, positively powered

achromats with the stop at or near the front lens. Negativeastigmatism from the first doublet is balanced with posi-tive astigmatism from the second doublet. Because thereare no lens groups with net negative power, the designform is fundamentally limited by Petzval, restricting itsfield size. Decreasing the airspace between the two dou-blets can help reduce the Petzval at the expense of in-creased astigmatism.Given the focal length f of a Petzval lens, the first-orderthin lens layout can be calculated as follows:

• Front element focal length: 2 f

• Rear element focal length: f

• Element separation: f

• Back focal length: f /2

Improvements to the basic Petzval lens include adding astrong negative field lens near the image plane to correctPetzval and increase its field (although this significantly

decreases the working distance).The Petzval lens is also the basedesign form of many extremelyfast (∼ f /1) lenses, including veryhigh-speed, 6–8-element Petzvalprojection lenses.


Design Forms 35

Telephoto Lenses

A basic telephoto lens con-sists of two lens groups—apositive group followed by anegative group. The stop isat or near the front group.

The front principal plane is pushed out in front of thelens so that the EFL of a telephoto lens is greater thanits physical length (measured from the first lens surfaceto the image plane). The ratio of the lens length to EFL iscalled the telephoto ratio. Typical telephoto ratios rangefrom 0.6 to 0.9; the smaller the ratio is, the more compactthe system. The asymmetry of the design form limits itsfield (<10-deg HFOV). Moderate-speed (∼ f /6) lenses canbe built with two cemented doublets, whereas faster lensescan require the doublets to be air spaced and/or an elementto be added to the first or second lens group.

The focal lengths of any two-element thin lens systemcan be calculated given the system focal length f , theirseparation d, and the back focal length BFL:

fa = d ff −BFL

fb = dBFLf −BFL−d

For zero-Petzval sum:

fa =− fb = f −BFL d = ( f −BFL)2

f

Both lens groups are usually separately achromatized asprimary color and secondary spectrum limit the imagequality of telephoto lenses. The telephoto ratio cannotbe arbitrarily reduced without increasing Petzval (whichlimits the field coverage of the design). Faster lenses andsmaller telephoto ratios require more-complex designs.Long-focal-length telephotos are particularly sensitiveto small changes in object distance, and if used forphotography, require an additional floating element thatadjusts the focus as the object position is changed.


36 Design Forms

Retrofocus and Wide-Angle Lenses

A retrofocus lens (or re-verse telephoto) is com-posed of a negative lensgroup followed by a posi-tive lens group producinga BFL much greater than

its EFL (the opposite of a telephoto lens). This formis extremely useful for short-focal-length lenses thatneed a long working distance (e.g., for the placementof an SLR viewfinder mirroror a color-separation prism).This design form is also thebasis for most wide-anglelenses covering large fields.Extremely wide-angle or fish-eye lenses can cover morethan 180-deg full field.

Retrofocus lenses are large in both diameter and length(can be 10–20× their focal length). Many telephoto designtechniques are also valid for retrofocus lenses (e.g.,component achromatization and Petzval reduction). Alarge airspace between the component groups helps reducetheir individual power and aids in correction. The lack ofsymmetry makes correcting secondary lateral color andchromatic distortion a challenge in these designs. A conicasphere on the inner surface of the front element can helpthe overall correction of a wide-angle lens.

Wide-angle lenses suffer from significant third-orderbarrel distortion that is difficult to correct and is oftenbalanced with a fifth-order pincushion distortion. Fish-eyelenses have an inherently large barrel distortion that isnot necessarily considered to be an aberration but rathera necessity for imaging a 180-deg hemispherical sceneonto a flat image plane. The standard h = f tanθ definitionof focal length no longer holds and may be substitutedwith h = f θ or h = f sinθ distortion constraint. The barreldistortion also offsets the illumination fall-off that resultsfrom the cosine-fourth-power rule at large field angles.


Design Forms 37

Refractive versus Reflective Systems

Optical systems can be refractive (lenses), reflective(mirrors), or catadioptric (both lenses and mirrors).Because many refractive design forms have analogousreflective forms (e.g., the telephoto and the Cassegrain,the retrofocus and the Schwarzchild, or the Cooke tripletand the reflective triplet), it is useful to understand theadvantages and disadvantages of one type over the other.

Advantages of reflective systems:• No chromatic aberration (a single system can then be

“shared” among many spectral bands/channels)

• Spherical aberration for a spherical mirror is 1/8 of thatfor a lens with a glass index 1.5 and an equivalent f /#

• Opposite sign of Petzval compared to refractive lenses

• Better performance over a wider thermal range

• Fewer surfaces needed

• Integrated mounting and alignment features

• Elements can be independently shaped to reduce weight

• Design can be folded to meet packaging constraints.

Disadvantages of reflective systems:• Limited FOV

• Aspheres needed for high-quality performance

• More sensitive to irregularity and alignment errors

• Per-element fabrication cost and schedule

• Stray light is harder to manage

• Complex off-axis configurations needed to get mirrors inthe light path without obscuring the incoming beam.

In very wide-angle lenses, rays perpendicular to theoptical axis can fail to trace because their directioncosines are zero. Imaging the stop through the frontgroup of negative elements also yields an anamorphicentrance pupil that needs to move forward, shift laterally,and tilt to get light into the lens as the field angleincreases. In software, ray aiming or a “wide-angle mode”(uses real chief rays versus paraxial rays) is needed toovercome these ray failures.


38 Design Forms

Obscurations

In reflective systems, obscurations are objects that blocka portion of the incident light bundle. For example,the image from a single on-axis parabolic mirror islocated directly in the incoming beam path. If a detectoris placed at the image plane, it will obscure theincoming beam. Common on-axis astronomical telescopes(e.g., Cassegrain or Gregorian) have secondary mirrorobscurations. Although obscurations help reduce packagevolume for systems with space constraints, they causean overall transmission or throughput loss. The straylight from obscurations also requires critical attention(especially in the infrared where the thermal effects ofblackbody radiation from obscuring features must beconsidered).

It is important to clarify if the obscuration percentageis specified as a linear obscuration (% of the apertureradius) or an area obscuration (% of the aperture area);linear specifications are more common. A 20% linearobscuration is ∼4% by area.

Obscurations modify the diffraction pattern of an objectby moving energy from the central Airy disc to the outerrings. Central obscurations (blocking light in the cen-ter of the pupil) are the most common. Central obscura-tions primarily reduce the image contrast (MTF) at themid-spatial frequencies. The drop in MTF becomes notice-able around 15% obscuration (measured linearly) and in-creases as the size of the obscuration increases. A sec-ond common type of obscuration is a structural supportmount or spider used to hold mirrors in place. Each spi-der vane produces a diffraction spike in the PSF at a

right angle to the direction ofthe vane. Obscurations can alsoinduce vignetting and, if near theimage plane, can directly alterthe shape of the final image (e.g.,Hubble Space telescope WFPC).


Design Forms 39

Newtonian and Cassegrain

A single parabolic mirrorused with a fold mirror iscalled a Newtonian tele-scope. It has zero spheri-cal aberration on axis for

an infinitely distant object but is quickly limited off-axisby coma. A Cassegrain telescope adds a negativelypowered hyperbolic secondary mirror to a parabolic pri-mary. Both mirrors share a common focus point, form-ing a mathematically perfect, aberration-free or stig-matic image of an infinite on-axis object. The field islimited by coma and astigmatism to less than a degree,but the compact package and small telephoto ratio makethe form extremely attractive formany portable applications. Shift-ing the mirrors axially from stig-matic alignment balancescoma with spherical, improving theperformance in the field.

In a first-order layout, the mirror curvatures of any two-mirror system are given by

c1 = BFL− f2d f

c2 = BFL+d− f2dBFL

where f is the effective focal length, d is the separationbetween the two mirrors, and BFL is the distance fromthe second mirror to the image.

The Ritchey–Chrétien telescope uses a hyperbolicprimary and hyperbolic secondary to correct spherical andcoma exactly to third order. The field coverage is slightlylarger than that of a Cassegrain; however, the primarymirror can no longer be tested with an on-axis divergingspherical beam on the finite-conjugate side, requiring theuse of a null lens. Refractive corrector lenses canalso be inserted to improve field performance, but theyadd chromatic aberration, limiting the telescope’s spectralregion.


40 Design Forms

Gregorian and Schwarzschild

A Gregorian telescope has aparabolic primary and positivelypowered elliptical secondary. Byaligning the focus of each conic sec-tion, a perfect on-axis stigmatic im-age is formed. Many early telescopes were Gregorian; his-torically, positive concave mirrors were easier to fabricate.However, the Gregorian with two positive mirrors hasmore field curvature than a Cassegrain. Modern mirror-fabrication techniques have made the Cassegrain morecommon. The performance of a Gregorian telescope can beimproved by departing from the stigmatic imaging condi-tion and changing the primary mirror to an ellipse.

The 200 ′′ Hale Telescope at Mt. Palomar is aCassegrain; the Hubble Space Telescope (HST) is aRitchey–Chrétien; and the Large Binocular Telescope(LBT) in Arizona is a Gregorian design.

A Schwarzschild objective has asmall negative spherical primary anda large positive spherical secondary.If the mirrors are concentric with thestop located at their common center ofcurvature, the system can be exactly corrected for third-order spherical, coma, and astigmatism, and the imagelies on a curved surface whose radius equals the focallength. The mirror separation d and the mirror curvaturesc1 and c2 are given by

d = 2 f c1 =(p

5−1)

f c2 =(p

5+1)

f

for a system focal length f . The size of the secondarymirror and large obscuration ratio (>40%) dramaticallylimit the use of a Schwarzschild as a telescope objective,but its low f /# and long working distance make it anattractive form for reflective microscope objectives. Inpractice, the mirrors may be aspherized and/or lensesadded to the design to increase the NA and field coverage.


Design Forms 41

Catadioptric Telescope Objectives

Catadioptric telescope objectives use weak refractiveelements to correct the aberration of high-power sphericalmirrors without adding significant chromatic aberration.

The simplest example is a Manginmirror or back-surface reflective el-ement with two powered surfaces. Ina Schmidt telescope, the aperturestop is placed at the center of cur-vature of a spherical mirror, elimi-

nating coma, astigmatism, and distortion. A thin correc-tor plate with one aspheric surface is then located atthe stop, correcting all orders of spherical while addingonly a small amount of chromatic aberration. The finalimage plane is curved with a radius equal to the fo-cal length. A field lens can be added near the imageplane to flatten the image surface, if needed. In practice,ghost reflections from the corrector plate can be a prob-lem. Spherical airspaced doublet or triplet corrector platesyield Houghton or Buchroeder systems with improvedachromatization without the use of aspheres.

The Maksutov or Bouwers telescope has a sphericalprimary mirror and a weak-meniscus, concentric spher-ical corrector plate (historically easier to fabricate thanthe aspheric Schmidt plate). Nominally, all three sur-faces are concentric to the stop; however, the correctorplate can reduce axial color with a slight departure fromconcentricity at the expense of slightly increased spher-ical. Both the Schmidt and Maksutov can be combinedwith other telescope objective forms. For example, theSchmidt–Cassegrain and Maksutov–Cassegrain arecommon commercial telescopes for amateur astronomythat combine the corrector optics with a Cassegrain sys-tem.


42 Design Forms

Unobscured Systems: Aperture Clearance

One method to eliminate the obscu-ration in a reflective system involvesshifting the on-axis beam so that it is nolonger centered on the optical axis (usu-ally at the expense of increased pack-age size). For example, in an off-axisparabola (OAP), the aperture stop issimply decentered until the image is

clear of the beam. It is important to note that the object,image, and mirrors still share a common optical axis in theunobscured system. The aperture may be made noncir-cular to keep the package size small, but this will changethe shape of the PSF.

Tips for designing complex multi-mirror designs:• The design process typically starts with definition of

the size and speed of the primary mirror.• On-axis conic sections, when used off-axis, suffer

mainly from coma, and suffer from spherical aberrationat nonideal conjugates.

• An analysis of the obscured system with nonaperturedparent mirrors helps determine the aberrationlimitations of the unobscured design.

• Many unobscured systems use both aperture and fieldclearance to eliminate obscurations.

Unobscured two-mirror afo-cal systems can be con-structed using two off-axisparabolas. These systemshave perfect on-axis imageryand no coma or astigmatism (Petzval dominates the field).When set up as two positive confocal parabolas with aninternal image plane, the exit pupil is real and accessi-ble. A Mersenne configuration with a negative mirror hasless Petzval and a smaller package size, but a virtual exitpupil. Both configurations are typically used as attach-ments to increase the aperture and focal length of a sys-tem or, conversely, shorten its focal length and increase itsFOV.


Design Forms 43

Unobscured Systems: Field Clearance

For unobscured infinite-conjugate systems, field clear-ance is achieved by tilting the entire system with respectto the incoming axial beam (this is equivalent to using anangular field definition that is not centered on the opticalaxis). For unobscured finite-conjugate systems, field clear-ance is achieved by eliminating the on-axis field point andusing the system “off-axis in field,” where the center of thefield is decentered until the beam is clear of all obscura-tions (often at the expense of performance and packagingvolume). Although it is often not obvious, the object, im-age, and mirrors share a common optical axis. If needed,the image plane and/or mirrors can be slightly tilted ordecentered to improve performance; however, this changesignificantly increases the alignment complexity. As mostdetectors are rectangular, the smaller dimension of fieldis located in the plane of the field decenter to minimizepackage size. It is important to make sure that the detec-tor is oriented correctly in the system, as the mirrors areoptimized and sized for one image orientation only.

The Offner relay is a classicexample of an unobscuredfinite-conjugate all-reflectivesystem. It is a 1:1 systemcomposed of two concentricspherical mirrors with the object (and image) plane attheir common center of curvature. The stop is locatedat the secondary mirror, and the system is doublytelecentric. The Offner design form eliminates all third-order aberrations exactly and is limited by higher-orderastigmatism (but can be improved by deviating fromperfect concentricity). Early high-performance, 1:1 ring-field lithography systems were based on the Offner designform. This type of design is also the preferred formfor many spectrometers, where the secondary mirror isreplaced by a reflective grating.


44 Design Forms

Three-Mirror Anastigmat

The Cassegrain telescope is a promis-ing design form, as it can fit a longfocal length in a short package whilebalancing Petzval with a positivemirror and a negative mirror. How-ever, similar to many two-mirror de-sign forms, the field coverage of aCassegrain is limited to ∼1 deg by

astigmatism. Adding a third mirror to a Cassegrain pro-vides the degrees of freedom necessary to correct the astig-matism across the field. The result is a three-mirroranastigmat (TMA) design useful for applications withHFOVs of 2–5 deg. In a focal TMA design, the tertiarymirror is an ellipse; in an afocal TMA design, the tertiarymirror is a parabola. Both obscured and unobscured (usu-ally using aperture clearance) designs are possible. Thekey feature and main advantage of the TMA over otherthree-mirror design forms is that it is a reimaging de-sign with an accessible internal image plane and an ac-cessible exit pupil. This allows the placement of field stops,Lyot stops, and cold stops (for infraredsystems) for improved stray lightsuppression. Other TMA variationsinclude designs with the internal focusafter the primary rather than after thesecondary, or designs with remote exitpupils for test setups.

Adding a fourth “corrector” mirror near the internalfocus of a TMA improves the wavefront and reduces thedistortion within a given FOV. The additional mirror alsohelps extend the field size in the direction of the mirroroffsets to allow a square or circular FOV. The correctormirror can be oriented coaxially with the other threemirrors or highly tilted to fold the design into a ball-shaped enclosure for pod-mounted applications.


Design Forms 45

Reflective Triplet

The non-reimaging TMA or reflec-tive triplet (RT) is the reflectiveequivalent of the refractive Cooketriplet. The stop is at or near a neg-ative secondary mirror with a posi-tively powered mirror on each side ofthe stop. All three mirrors are conicsand nominally share a common opti-

cal axis (the mirrors may be slightly decentered or tiltedto improve performance). Because there is no intermediateimage, the RT design form can support a larger field thanthe reimaging TMA design form; however, an RT design isnot appropriate for thermally cooled systems, as the exitpupil is virtual and therefore inaccessible for the place-ment of a cold stop. The RT design form is recommendedfor moderate-FOV (up to 10-deg HFOV) applications thatdo not require reimaging. RTs perform best with the aper-ture stop at secondary (distortion is easier to correct due tothe symmetry), but the design form also supports an aper-ture stop in front of the system, if needed. The beam clear-ance in an unobscured RT design is provided through bothan off-axis aperture and an off-axis FOV. Systems that areoff-axis in field will tilt and decenter the image plane tocompensate for residual field curvature.

Distortion in an optical system is de-fined as the difference in image heightbetween a real ray and a paraxial ray.For systems with centered FOVs thespecification can be given by a singlenumber at the edge of the field (e.g., 4% distortion).However, for unobscured designs with off-axis FOVs, theparaxial region may not even be in the FOV, and a singlespecification is not adequate. In this situation, a series ofnew terms such as “smile,” tow/keystone, expansion,and anamorphism are needed to describe and quantifythe distortion.


46 Design Forms

Wide-Field Reflective Design Forms

Wide-angle mirror de-signs with more thana 10-deg HFOV use adesign principle simi-lar to wide-angle in-verted telephoto refrac-tive lenses. A front neg-ative mirror is followed by a positive focusing group. Theaperture stop is located inside the lens (usually near thepositive focusing group). However, wide-angle reflectivedesigns tend to be large (many times larger than the en-trance pupil diameter). The simplest example is an off-axis Schwarzschild, which can also be converted to areimaging form for stray light suppression by adding aconcave relay mirror on the other side of focus in thesame way that a Cassegrain is converted to a TMA. Theadditional mirror extends the FOV beyond what a stan-dard Schwarzschild can achieve (e.g., up to 20 × 20 deg or30 × 30 deg).

Afocal versions of 3- and 4-mirror reflective designs areoften used as beam reducers for optical systems needingboth large apertures (to collect light from distant objects)and limited diameter filters, scan mirrors, diffractiongratings, or beamsplitters that need to be placed in acollimated beam.

Extremely large (by reflective standards > 20 deg), wide-angle reflective designs are based on the design conceptthat when the entrance pupil is at one focus of anegative hyperbolic primary mirror, the reflection is freeof astigmatism. An aspheric secondary mirror is placed atthe aperture stop to control spherical aberration, and aconcave tertiary mirror is used to form the image. Thisconfiguration has been referred to as the three mirrorlong (TML), as opposed to the three mirror compact,another name for the reflective triplet. Variations onthe TML concept include forms with an aerial aperturestop or WALRUS. The TML family provides the widestfield coverage of the reflective design forms. The maindisadvantage of this form is its very large envelope.


Design Forms 47

Zoom Lens Fundamentals

A zoom lens is a lens whose focal length can be changedeither continuously or in discrete steps. Continuouszooms use zoom groups with fixed power that movealong the optical axis. Discrete zooms laterally switchdifferent fixed-power groups into (or out of) the opti-cal path. Some zoom lenses (e.g., liquid lenses) can dy-namically alter element powers without motion. “True”zoom lenses have a fixed image-plane location. Varifocal

lenses change fo-cal length but allowthe image plane tomove, requiring re-focus.

The zoom ratio isthe ratio of the twoextreme focallengths (or, in thecase of a finite con-jugate zoom, the twoextreme magnifications). For example, a 25–200-mm zoomlens is an 8× zoom. Zoom lenses are now standard for mostconsumer cameras, including “point and shoot” cameras,SLR cameras, and video camcorders, where the primarytrade-off is large zoom range versus compact package size.

The f /# of most modern camera zoom lenses varies acrossthe zoom range. At long focal lengths the aberrations tendto be larger, and the f /# is slower. A smaller aperture alsoreduces the physical size of the outer lens components.Distortion and lateral color will often change sign throughzoom. The relative illumination and vignetting can changesignificantly through zoom.


48 Design Forms

Zoom Lens Design and Optimization

Zoom lenses can be classified according to their distribu-tion of group power (e.g., + − + +). A simple mechani-cally compensated zoom lens has two moving groups:the variator changes magnification, and the compen-sator moves to maintain image plane focus. The mo-tion of the compensator group is typically nonlinear;the zoom groups are linked with a mechanical cam.The two zoom groups form a zoomkernel that relays a fixed objectdistance to a fixed image plane.In photographic objectives, a frontfocusing group is used to providea fixed image position for the zoom kernel and toaccommodate changes to the object distance. More-complex zooms add a fourth (fixed) prime group to therear of the zoom kernel and/or more moving groups.

Optically compensated zooms move linked groups usinga linear motion but do not have a perfectly fixed imageplane. Many discrete zooms use a form of two-positionoptically compensated designs.

Zoom lens optimization tips:

• Start with a thin lens layout (e.g., first-order zoom lenslayouts can be designed with Gaussian brackets) ora predefined starting point; global optimization is notas useful for finding zoom lens starting points as it isfor fixed focal length lenses.

• Independently achromatize each zoom group overmultiple conjugates.

• Optimize over at least 3–5 zoom positions simultane-ously and then verify the lens performance between theoptimized zoom positions.

• Check that moving groups do not collide with eachother or fixed groups during their zoom motion.

• More than two moving groups may be needed tomaintain f /#, aberration control, or telecentricityacross zoom, or for thermal compensation. Adding moremoving groups also helps reduce the package size forzoom lenses with large zoom ratios.


Improving a Design 49

Techniques for Improving an Optical Design

Many techniques exist either to improve an existinglens with inadequate performance or to extend anexisting design over new requirements without changingthe basic design form. The most common scenarioarises when a designer must “push” a legacy designto increased aperture and/or larger field (for betterresolution/throughput). Such requests often come with achallenging expectation of no significant increase in thephysical size of the design. Options available to a designerinclude:• Split a lens • Raise (or lower) index• Compound a lens • Raise (or lower) V• Aspherize a surface • Use an abnormal partial• Use a diffractive surface dispersion materialEach of the above techniques seeks to reduce the aber-ration contribution of a surface/component. A Seidel orPegel diagram shows the surface-by-surface aberrationcontribution and is a key tool in identifying elements orsurfaces to target for improvement. It can also be veryuseful for comparing the sensitivity of different design so-lutions.

Increasing the aperture or field of an existing designcan lead to ray failures during lens evaluationand optimization. Examples include total internalreflection (TIR) or ray intercept failures (where raysphysically miss a surface). The solution is to slowly“walk” up to the desired aperture or field in smallincrements, optimizing the lens at each step.


50 Improving a Design

Angle of Incidence and Aplanatic Surfaces

Ray angles of incidence (AOIs)are measured relative to the sur-face normal at the ray intercept.A large AOI indicates significantdeparture from paraxial refrac-

tion, where θ 6= sinθ. Ray AOI is therefore an importantindicator of both the aberration contributions (aberrationcoefficients depend on the chief and/or marginal ray AOI)and the tolerance sensitivity of a surface. During the im-provement process, elements with large ray angles aregood candidates for modification (e.g., splitting into multi-ple elements). Examining a lens layout and assessing theray angle distribution gives a quick insight into the perfor-mance, manufacturability, and improvement potential fora design. Several “design for tolerance” optimization met-rics have been proposed that minimize ray AOIs through-out a design.Three conditions exist where a single surface can havezero third-order spherical aberration:• The object or the image is concentric with the surface so

that rays do not refract (zero AOI);• The marginal ray height is zero at the surface (useful for

field lenses); or• The surface is aplanatic where the marginal ray AOI

(i) equals the negative of the angle that the ray makeswith the optical axis after refraction (u′).

The final aplanatic condition is apowerful one, as it allows opticalpower and large AOIs withoutadding spherical aberration (orcoma and astigmatism).

An aplanatic lens exploits two of the three conditions forzero spherical at a surface: the first surface is aplanatic,and the second surface is concentric with the imageplane. Microscope objectives and interferometer referencespheres use aplanatic lenses to increase their speedwithout introducing significant aberrations.



Splitting and Compounding

Splitting a lens element into two (ormore) elements of the same material whilemaintaining the total power is a highlysuccessful design technique for improvingan optical system. It

• Adds degrees of freedom for optimization.• Reduces the ray AOIs by reducing the individual surface

powers and is therefore very effective at improvingspherical, astigmatism, coma, and distortion.

• Reduces spherical aberration when splitting positiveelements in high-NA lenses.

• Reduces astigmatism, distortion, and coma whensplitting negative elements in wide-angle lenses.

• Does not improve Petzval or primary color since the totalpower contribution does not change.

The exact method to split a lens (e.g., literally cuttingthe lens in half or using two equi-convex elements) isdesign dependent. If the change in element shape is notsignificant enough, the lens will revert to its originallocal minimum (after splitting, the result will be twolens elements with a narrow crack-like airspace betweenthem whose outer shape mimics the shape of the originalelement). However, if the change is too large, ray failurescan occur during optimization.

Compounding a lens element into a dou-blet or triplet (while holding total power)is another powerful design improvementtechnique. It

• Uses two (or more) materials to simu-late a nonexistent but desirable material type (e.g., amaterial with a high index, large V , or unusual partialdispersion).

• Primarily reduces axial color, lateral color, and Petzval,although the extra degrees of freedom can also be usedto correct other aberrations.



Diffraction-Limited Performance

During the design improvement process, performancetargets are useful for guiding the process and ultimatelyknowing when to stop optimizing. Many optical systemsuse diffraction to set the performance limit, and the lensrequirements may state “diffraction-limited by design.”However, different definitions of diffraction-limited exist,so a designer should clarify the intent of the statement.

Strictly speaking, a diffraction-limited lens is lim-ited solely by diffraction and refers to a zero-wave RMSOPD; however, these criteria are rarely the intent ofa “diffraction-limited” specification. Common approxima-tions for diffraction-limited performance include:

• a peak-to-valley OPD< λ/4 (Rayleigh criterion),

• an RMS OPD< 0.070λ (Maréchal criterion), or

• a Strehl ratio > 0.8.

Reversing a lens can help a stagnated design duringoptimization. Swapping two surfaces of an element inthe lens design program often requires adjusting thesurrounding thicknesses so that the principal planes andfirst-order properties remain constant and so that rayfailures do not occur. Changes in ray intercept heightsand AOIs will alter the aberration distribution.



Thin Lens Layout

For a lens whose thickness is much smaller than its focallength, the marginal ray height is approximately the sameat both surfaces. The ray transfer contribution throughthe element can be ignored and the lens treated as a thinlens with zero thickness. A thin lens layout simplifiesa complex optical system by making each element (orlens group) a thin lens with optical power. This layouthighlights the balance of positive and negative powerbetween lens elements and/or lens groups and can be avital aid to understanding the performance limitations ofan optical design, especially if the design is limited byPetzval or color. A thin lens layout is also a useful toolfor starting an optical design (essential for zoom lenses) orjoining independent lens modules together while matchingtheir pupils in size and location.

β= c1 + c2

c1 − c2C = ua +u′

a

ua −u′a

For a thin lens, the third-order aberration equations canbe simplified and written as a function of a lens bendingparameter or shape factor β that describes whatthe lens looks like (defined by the two curvatures), aconjugate factor C that describes the magnification atwhich the lens is used (defined by the input and outputmarginal ray angles), and its index of refraction. TheKingslake G-sums are similar thin lens aberrationexpressions, where each aberration coefficient includesa “G” coefficient that depends solely on index of refraction.

A y-ybar diagram is a plot ofthe marginal ray height y versusthe chief ray height y, whereeach point on the plot representsa lens element (or surface) inthe system. The plot quicklyidentifies the locations of objectsand images (y = 0) and stops and pupils ( y = 0) and theirsizes, and is an alternative to a thin lens layout.



Lens Bending

A key variable when optimizing a design is lens radius.For a specific index of refraction and thickness, there arean infinite number of combinations of radii R1 and R2 thatwill yield a lens with a given focal length. The result is thatfor a fixed focal length, a lens may take on any number ofdifferent shapes or “lens bendings,” and the aberrationsof the lens (primarily spherical aberration and coma) willchange as the shape is changed.

The thin lens aberrationequations can be usedto show that for a sin-gle thin lens with anobject at infinity (conju-gate factor C = −1) andthe stop at the lens, abending parameter β ex-ists either for minimum(but nonzero) sphericalor for zero coma that de-pends only on the in-dex of refraction of thelens, whereas astigma-tism, Petzval, and distor-tion do not depend on theshape of the lens.

The shape factor for a single thin lens (object at infinity)for minimum spherical aberration changes from nearconvex-plano to meniscus as the index of refraction isincreased from 1.5 to 4.0. This behavior helps explainwhy very high-index infrared lenses “look” different fromvisible lenses.



Material Selection

Material selection is one of the most critical choices inlens design. Changing a lens material during the opti-mization of a design can significantly alter its perfor-mance. In general, raising the index of refraction ofall elements is an effective way of reducing the lenscurvatures for given lens powers, minimizing ray AOIsand therefore aberrations at each surface. Primary chro-matic aberrations can be corrected by selecting mate-rials with the appropriate dispersion (V). In a thinlens achromatic doublet, the V dif-ference between the two materials de-termines the individual element pow-ers. Increasing the positive elementV and decreasing the negative ele-ment V will reduce the componentpowers, the ray AOIs, and the corresponding aberrations.“Old” achromats use “lead-line” glasses with a posi-tive low-index crown and a negative high-index flint (e.g.,BK7/SF2). “New” achromats use a high-index positivecrown and a lower-index negative flint (e.g., LaKN17/F2).A new achromat has ∼4× less Petzval than an old achro-mat, but the small V separation due to material limita-tions restricts it to about half the speed of an old achro-mat. New achromats are useful in designs driven by fieldperformance (e.g., wide-angle lenses).

The secondary spectrum of a thin lens achromatic doubletis proportional to the ∆P/∆V ratio of the two materials,where P is the partial dispersion of a material.Ideally, a pair of glasses with the same P but significantdifference in V is needed to correct secondary spectrum.Unfortunately, for most glass doublet combinations, the∆P/∆V ratio is nearly a constant. Only a few materials(e.g., CaF2) exist with anomalous partial dispersionthat helps reduce secondary spectrum. A triplet achromatcan correct secondary spectrum by using two of thematerials in the triplet to “synthesize” a material that canbe matched in partial dispersion with the third material.



Controlling the Petzval Sum

∑

j

φ j

n j

In order to have an optical design with areasonably flat field, its Petzval sum must besmall. The Petzval sum is directly proportionalto the ratio of each element’s power to its index ofrefraction. The reciprocal of the Petzval sum is thePetzval radius, which equals the radius of the idealimage surface. The ratio of the Petzval radius to the focallength of the lens is the Petzval ratio.

Designing a lens to have zero Petzval is difficult, aspositive focal length lenses want to have primarilypositive-powered elements. A key strategy to control thePetzval sum is to separate positive power from negativepower:

Φ=φ1 +φ2 −tn

φ1 φ2.

For example, if the radii are equal in a single lens (ameniscus lens), the surface powers are equal but oppositesign, resulting in a lens with zero Petzval and a powerthat is directly proportional to the thickness of the lens.In more-complex lenses, where positive lens groups areseparated in space from negative lens groups.

In most positive-power refracting lenses, the Petzvalsurface will be curved toward the lens (inward-curvingfield). In catadioptric designs, reflective surfaces canbe used to balance the Petzval from positive lensesbecause a positive mirror has an index of refraction of –1, yielding an overcorrected (backward-curving) Petzvalcontribution equal to its optical power.

Petzval curvature is perhaps the hardest aberration tocorrect in an optical system. Once the first-order powerbalance has been established, the index of refraction isthe only variable available to control the Petzval sum.In positive elements, a higher index reduces the inward-curving Petzval; however, it is not so clearly defined innegative elements because a lower index increases theovercorrecting Petzval, but a higher index reduces thesurface curvature, reducing other aberrations.



Stop Shift and Stop Symmetry

Finding the best position for the stop is an importantpart of the design process. A stop shift (moving thestop location and scaling its size to maintain thef /#) forces the chief ray through different parts of alens. Aberrations that depend on the chief ray (coma,astigmatism, distortion, and lateral color) will change,whereas spherical, Petzval, and axial color are unchangedwith stop shift. For a thin lens with nonzero sphericalaberration, there is always a stop position that exactlyeliminates coma (it also minimizes the tangential fieldcurvature). This position is called the natural stopposition.

If the best stop location is not obvious, a floating stopcan be utilized during optimization. The first surface inthe lens is designated as a dummy “stop” surface witha variable negative distance. This surface is really theentrance pupil location and allows the aperture stopsurface to “float” to its best location. Before finalizing thedesign, the aperture stop surface designation should bemoved to the correct location to account for the effects ofany pupil aberrations.

Distortion, coma, and lateral colorcan be eliminated in lenses withstop symmetry. These aberra-tions all have an odd-order depen-dence on the chief ray height (or

angle), so lenses on one side of the stop have an aberra-tion contribution equal and opposite to lenses on the otherside. In practice, even lenses with stop symmetry still havesome residual aberrations because the object/image conju-gates are not usually symmetric. Stop symmetry is a keystrategy for reducing distortion (similar to Petzval, distor-tion is a notoriously difficult aberration to correct).

Unlike film, digital sensors cannot accept large rayAOIs and typically have chief-ray-angle (CRA) matchingrequirements. This trait can restrict the positioning ofthe aperture stop in the system.



Telecentricity

A telecentric lens is a lens with either the entrance pupil(object-space telecentric) or exit pupil (image-spacetelecentric) at infinity. The degree to which the chief rayis parallel to the optical axis at either the object or imageplane is called telecentricity. The simplest way to makea lens telecentric is to placethe aperture stop at a focalpoint of the lens.

In a design code, the object space is easily defined to betelecentric; however, the chief ray must be constrained togo through the center of the stop during optimization. Atelecentric image space can be attained by targeting theinverse of the exit pupil distance to zero in the meritfunction or with a CRA solve or CRA constraint (usuallyat several field points).

The primary benefit of telecentricity is that the systemmagnification is insensitive to slight changes in imagedistance (focus). The incident cone angle is also constantover the entire field, improving relative illumination.This is important for microlithography, microscopes,machine vision systems, and other types of metrologylenses. However, the lens elements must be at least aslarge as the image, making telecentric lenses larger thantheir nontelecentric counterparts.

A 4f optical system is a doubly telecentric finite-conjugate relay (also known as an afocal relaytelescope) with an accessible pupil plane for Fourierfiltering. Each of the two lenses has a focal length of fso that the total distance from object to image is 4 f .



Vignetting

By definition, the aperture stop limitsthe size of the imaging bundle foran on-axis object point. Vignettingoccurs when another surface (optical ormechanical) in the system clips image-forming rays for off-axis field points.The amount of vignetting is measuredas a percentage of the width of thepupil. Vignetting reduces the amountof light reaching the image plane foroff-axis field points. Some lenses (e.g.,lithographic) cannot tolerate any lightloss or drop in relative illuminationacross the field, while others (e.g.,camera lenses) allow as much as 75%vignetting.

Optical design codes simulate vignetting by approximat-ing the vignetted pupil shape as an ellipse. Vignettingfactors relate the scaling and offset of the elliptical pupilto the unvignetted pupil. (Note that some programs usea percentage and an offset, and others define both anupper and a lower percentage.) Pupil aberrations maylead to confusing “negative” vignetting factors (even in theabsence of physical vignetting) because the paraxial en-trance pupil must be overfilled to evenly fill the aperturestop surface.

The minimum clear aperture for zero vignetting at anysurface is the sum of the absolute value of its marginaland chief ray heights (|ya| + |yb|). If allowed, vignettingis a powerful technique to improve the image quality ofa design because the pupil rim rays at the edge of thefield typically cause the largest spot flare. Vignetting alsoreduces element diameters that are too large to meetpackaging and/or weight constraints. Any clipping surfaceshould be far enough away from the stop so as to not clipthe axial beam during optimization. Stop shifts can helpincrease (or decrease) the amount of vignetting.



Pupil Aberrations

The pupils in an optical designare not necessarily perfectlyimaged to each other. To derivethird-order equations for pupilaberrations, the entrance andexit pupil planes are treated asconjugate planes (similar to the object/image surfaces),and the marginal and chief rays in the system areinterchanged in the third-order field aberration equations.Although pupil aberrations do not directly impact imagequality, they can strongly influence other critical lensproperties. For example, large pupil aberrations makeit hard to pupil match two optical systems (especiallyobjectives and eyepieces) together.

Pupil spherical is the most commonly encountered pupilaberration. In wide-FOV eyepieces, pupil spherical causesthe exit beams from different field points to not crossat a well-defined axial location for an observer’s eye.The result is that eyepieces with large pupil sphericalmay have a shadowing or “kidney-bean” effect in theapparent brightness of off-axis fields. Pupil spherical canalso cause ray tracing ray failures if there is a largeshift between the paraxial entrance pupil location (usedfor the initial chief ray aiming) and the full-field pupillocation. Design codes can calculate pupil aberration rayfans and/or help determine if ray aiming (which forcesexplicit x/y coordinates for the chief ray on a given surface)is required to continue with the design analysis.

Distortion is less sensitive to changes in object positionif the pupil spherical in a lens is small—an importanttrait in photoreducers or magnifiers that require well-corrected distortion over variable conjugates.



Aspheres: Design

By definition, aspheres are “nonsphe-rical” surfaces. Standard aspheres arerotationally symmetric, but off-axisconic sections and freeform surfacesare also referred to as aspheres. Manyqualitative terms exist that help describe an asphericshape (e.g., mild, weak, steep, or gull wing). In opticaldesign, aspheres are used to increase performance orreduce the size and weight of a system. Near a pupil,aspheres can be used to correct all orders of sphericalaberration. Near an image plane (e.g., on a field lens), theycan be used to correct astigmatism or distortion, or achievetelecentricity. Early applications for aspheres includedilluminators, infrared systems, and single-wavelength,high-NA, small-field systems like DVD objectives or laserdiode collimators.

When optimizing with aspheres, a designer shouldsignificantly increase the pupil ray density and, if theasphere is near an image plane, add more field points.

The sag of a surface is the displacement (along z) of thesurface from its vertex as a function of lateral position(x, y, or radial coordinate r). The traditional asphericsag equation combines a conic expression with aneven-order polynomial, yielding a rotationally symmetricsurface with no vertex discontinuity:

sag = z = cr2

1+√

1− (κ+1) c2r2+dr4 + er6 + f r8 + gr10 +·· ·

Both the conic constant κ and the r4 coefficient willaffect third-order spherical, whereas the higher-ordercoefficients are useful for correcting higher orders ofspherical aberration (in high-NA systems). However,unneeded high-order terms can create oscillatory surfacesand optimization convergence problems. Alternatively, aQ-type (Forbes) polynomial aspheric expansion is anorthogonal basis set where the coefficients are linked toaspheric departure and slope, making it more practical fortolerancing and manufacturing.



Aspheres: Fabrication

Fabrication advances have made high-quality aspherespractical for almost any design. The aspheric departureis a measure of the surface deviation from a purelyspherical surface and is one of the most critical fabricationmetrics. A lens drawing typically specifies both a base orvertex radius and a best-fit sphere (BFS). The BFSminimizes the amount of material to be removed. Severaldifferent techniques exist for fabricating aspheres (eachwith their own pros and cons).

Technique Material CommentsConventionalpolishing

Glass / Crystals High-quality surfaces; longcycle time

Ion milling Glass High-precision removal; slowremoval rates and limiteddepartures

Deterministicpolishing

Glass / Crystals High-quality surfaces;mid-spatial “ripple” residuals

Molding Glass / Plastic Low-cost, high-volumereplication (high NRE formolds); limited materials andlower surface quality

Diamondturning

Plastic / Crystals Wide range of surface shapes;serial throughput;scattering/diffraction fromtool marks

Maximum slope error (the slope is the first derivativeof surface departure) is another metric that drives boththe fabrication and the testing of an asphere. Slightlyoversizing the aspheric clear aperture during the designwill keep the large slope errors that tend to occur at theedge of the part outside of the critical aperture; however,the total departure and part diameter will increase.Aspheric testing can contribute significantly to theoverall part cost. Standard interferometric testing can beused with small-departure surfaces, but other techniquesare required for large-departure surfaces.

Test Method CommentsSurface profilometry 1D low-resolution scan; probe contacts

part (risk of damage)Null lenses andcomputer-generatedholograms (CGHs)

High accuracy; costly and specific to asingle surface; expensive and long leadtime; precision alignment required

Subaperture stitching Useful for a wide range of aspheres;full 2D surface coverage; limited slope



Gradient Index Materials

A gradient index or GRIN material has a spatiallyvarying index of refraction typically described by an in-dex polynomial. However, the index polynomial is not di-rectly related to the manufacturing parameters, leading tolengthy development cycles. Axial gradients have planesof constant index perpendicular to the optical axis. Theirindex equation is convergent; only the linear term N01is needed to correct all orders of spherical aberrationwhen combined with a spheri-cal surface. The result is thataxial gradients make fast col-limating lenses (similar to as-pheres).

Radial gradients have cylin-ders of constant index cen-tered on the optical axis. Ra-dial gradients have the uniqueability to focus light with flatsurfaces. A thin radial gradi-ent is called a Wood lens, andits power is proportional to its thickness times its N10 co-efficient. Radial gradients are used in fiber couplers and1:1 relay systems such as endoscopes, boroscopes, anddesktop copiers (replacing complex lens assemblies witha smaller, less-expensive single component). A quarter-pitch rod focuses incoming collimated light at its backsurface, whereas a full-pitch rod relays an upright imagefrom its front surface to its back.

GRIN aberrations are split into a surface contributioninto the medium and a transfer contribution withinthe medium. Axial gradients have large surface andsmall transfer contributions, whereas radial gradientshave large transfer and small surface contributions. Theprimary manufacturing process for GRIN materials ision exchange. Although convenient for modeling andaberration analysis, the index polynomial is not directlyrelated to manufacturing parameters, leading to lengthymaterial development.



Diffractive Optics

A diffractive optical element (DOE) uses wavelength-scale surface or phase structure to redirect an opticalbeam using diffraction rather than refraction or reflection.Diffraction efficiency is the fraction of light diffractedinto the desired order. Low diffraction efficiencies cancause significant stray light problems. DOEs work withboth refractive and reflective surfaces and, similarto conventional elements, they have a focal length,dispersion, and aberrations. Similar to aspheres, DOEscan be used for aberration correction or system size/weightreduction. DOEs can also generate optical power fromflat surfaces or be used for athermalization, but they areusually more expensive than aspheres.

Diffractive surfaces have a constant negative V (equal to−3.45) that can be used to correct primary color. However,their abnormal partial dispersion yields a much largersecondary spectrum than a standard doublet.

Kinoforms are smoothly varying sur-face-relief DOEs typically made by di-amond turning a curved substrate. Ki-noforms can be replicated in high vol-ume and are common in infrared sys-tems as hybrid diffractive achromats,where the DOE replaces the negativeflint element. A binary optic is a “stair-step” DOEwith a discrete number of steps in either the surfaceor phase profile and is formed by multiple lithogra-phy steps (the diffraction efficiency is directly propor-tional to the number of steps). A holographic opti-cal element (HOE) is formed by interfering two wave-fronts. A computer-generated hologram (CGH) is aHOE formed by directly writing a phase profile on a

substrate. Both types of HOEs canrepresent arbitrary phase profiles.CGHs are often used as null lensesto test large departure and/or off-axis aspheric surfaces.


Optimization 65

Optimization

Given an optical design starting point, modern lens-designprograms use numerical optimization to find designsolutions with improved performance by minimizing apredefined merit function. Also called a cost or penaltyfunction, the merit function uses a single number torepresent total lens performance; in general, smallernumbers denote better design solutions.

Designers first identify the lens variables (e.g., radii,thicknesses, airspaces, and material parameters) thatare allowed to vary during optimization; all other lensparameters are kept frozen. Performance defects (oroperands) that depend on the lens variables are thendefined and used to construct the merit function. Exampleoperands include image-quality measures (e.g., spot size)and first-order properties (focal length). Constraints(e.g., minimum lens thickness) can also be applied thatlimit the amount of change in a variable.

Because the operands in the meritfunction usually outnumber thevariables in the optical design,optimization algorithms attempt tofind merit function minima, notexact solutions. The merit functiontopology and solution complexitydepend on the number of variables.

A lens with only two variables has a 2D topographicalsolution space, whereas systems with more lens variableswill have multiple minima in a multi-dimensional meritfunction. Commercial software programs use proprietarylocal and global optimization algorithms to find minima inthe merit function.

Local optimizers use gradient methods to find the “near-est” minimum. The starting variable values determinewhich local minimum is found. Global optimizers areallowed to break out of local minima to find a more-comprehensive solution set.


66 Optimization

Damped Least Squares

The most common method of local optimization isdamped least squares (DLS), which minimizes themerit function using linear regression approximations.In DLS, each variable is changed by a (small) stepsize that balances good numerical accuracy with fastconvergence time. Calculating the change in each meritfunction operand for each variable step change produces aderivative or gradient matrix and defines a local slopethat allows the optimizer to move “downhill” and find abetter solution using a numerical gradient search.

An optimizer can stagnate before reaching a gooddesign solution when the merit function changes verylittle with the variable changes. Manually checking themerit function for overconstrained variables, changingthe operand weighting, or dramatically altering oneor two variable values (e.g., radius of curvature orairspace) can often overcome stagnation. In somesoftware programs the variable step sizes (also calledderivative increments) and/or the damping factorcan be modified to prevent stagnation and improve theconvergence of the optimizer.

Standard least-squares algorithms assume linear de-pendences between targets and variables. However,in lens design, the relationship between merit func-tion operands and lens variables is highly nonlin-ear. DLS also approximates the functional dependencebetween operands and variables as linear but its

damping factor adds a weightingfactor to the merit functionthat heavily penalizes any largechanges in the variables. Ifthe changes are small, then anonlinear function approaches alinear approximation. Althoughthe damping factor may slow theoptimization, it is more robustfor highly nonlinear targets.


Optimization 67

Global Optimization

Local optimization uses gra-dient search to find thenearest merit function min-imum and moves “down-hill.” Global optimizationattempts to find the globalminimum by allowing bothuphill and downhill move-ment in the merit function.However, global optimiza-tion can require extensivecomputation time.

In many cases, the global merit function has a reducedset of image performance operands and more physicalconstraints (e.g., maximum ray AOI) than a local meritfunction in order to save time but keep the designsolutions realistic. The result is that global optimizersdo not generally find the absolute best merit functionminima. Global search designs should be followed by amore-comprehensive DLS local optimization to find theoptimum design solution.

Adaptive simulated annealing algorithms mimic thephysical process of cooling excited atoms and molecules,where the system has a nonzero probability of reaching ahigher energy state (moving uphill). Genetic algorithmsencode lens variables in a numeric sequence andselectively choose “genes” from best-performing “parents”to create a new system.

Global optimizers are most often used as global searchalgorithms to rapidly scan parameter space for multiplesolutions. Global searches require only a minimal startingdesign—a series of plane parallel plates with a properlydefined merit function can yield good results. However,there is no guarantee that global optimization will everfind the absolute global minimum for a given problem.


68 Optimization

Merit Function Construction

In optimization, a merit function is a single numberthat captures all aspects of desired lens performance. Themerit function φ is constructed by taking the RMS of allidentified operands, where

φ=m∑

i=1w2

i (ci − ti)2m =number of operandswi =weighting factor for operand ici = current value for operand iti = target value for operand i

Squaring each operand serves to magnify the operandswith the worst performance and ensure that positiveand negative operand values do not offset each other inthe sum. Individual operands are relatively weightedto emphasize their desired contribution to overallperformance. The target value for most operands is zero.

Most design programs have a default merit functionthat automatically defines a set of image qualityoperands and includes the appropriate pupil sampling andweighting for fields and wavelengths.

Merit function operands consist of both image qualitymeasures (e.g., RMS spot diameter) and constructionparameters (e.g., focal length or magnification). Rayposition errors and/or OPD values are used to calculatethe image quality measures. For example, a simpleimage quality operand is the distance in the imageplane of a single ray from the centroid (or “centerof mass”) of a large number of rays, all traced fromthe same object point. Construction parameters canbe included in the merit function as heavily weightedoperands (to ensure that they are met), but this approachcan significantly slow down improvements in imagequality. Lagrange multipliers are an alternative toheavily weighted merit function operands and place theconstruction parameter constraints outside of the meritfunction, allowing for more stable optimization; however,each Lagrange multiplier constraint requires one freevariable.


Optimization 69

Choosing Effective Variables

Certain variables are more effective at improving lensperformance than others. Radii of curvature andairspaces are almost always left free to vary duringoptimization. Packaging or manufacturing constraints(e.g., requirements for a flat surface, equal radii, orworking distance) may restrict their range of values.

In contrast, lens thicknesses are much less effectivevariables (a change in a lens thickness is often equivalentto a change in a nearby airspace). Element thicknesses areoften kept frozen during a majority of the design process.However, in some cases (e.g., thick-meniscus, Petzval-correcting lenses), the thickness of a lens is an essentialvariable in improving image quality; thicknesses may alsoneed to be released to meet edge thickness constraints. Ifallowed to vary, optimizers tend to make lenses very thick(or very thin), slightly improving Petzval with only a smallimprovement in the merit function.

A lens element whose thickness is chosen to maintain adiameter-to-thickness or aspect ratio between 10:1 and5:1 is often easier and less costly to manufacture.

“Glass” choice is a critical variable for color correction asboth the index of refraction and dispersion are dependenton the material choice. Glass differs from most variablesas it is a discrete rather than a continuous variable.Model or fictitious glasses allow continuous index andAbbe number for optimization, but a real material musteventually be substituted from a glass map.

Aspheric coefficients are important variables foraberration correction in modern designs. If possible,aspheric coefficient variables should be enabled duringoptimization one at a time (starting with the lowestorders), one surface at a time. It is not necessary in mostdesigns to vary the conic constant simultaneously with thefourth-order polynomial coefficient; rather, vary the conicon strong surfaces and the fourth-order term on flattersurfaces.


70 Optimization

Solves and Pickups

Solves define a surface curvature or thickness to bea function of a given system requirement. As the lenschanges, a solve modifies the selected parameter tomaintain the system requirement. Solves are typicallyused in the initial setup of a lens, but some lensrequirements such as focal length can be constrainedusing solves rather than being defined as meritfunction operands. This method decreases the numberof independent variables and corresponding constraintsduring optimization, reducing computing time. Examplesinclude:• A marginal ray angle solve or f /# solve on the

curvature of the last surface can be used to maintainthe system focal length for a fixed entrance pupil size.

• A marginal ray height solve on the thickness of thelast surface can be used to locate the image plane atparaxial focus. If an internal image exists, this solve canalso be used to find and hold its location.

• A chief ray angle solve on the curvature of the lastsurface can force image-space telecentricity.

• Aplanatic solves force the aplanatic condition on asurface by changing its curvature.

• Thickness solves can be used to control lens edgethicknesses or the total distance between separatedsurfaces (useful in zoom or multi-configuration designs).

• A magnification solve adjusts the object (or image)distance to maintain a system magnification.

Many marginal-ray and chief-ray solves require rayinformation based on the entrance pupil and cannot beplaced on surfaces before the aperture stop.

A pickup is a special type of solve that “picks up” or usesdata from another surface. It maintains a set relationshipbetween two lens parameters as one of them is varied.Pickups are very useful for optimizing double-pass orsymmetrical systems (e.g., to allow a lens to change itsradius yet maintain symmetry, the first radius is “pickedup” by the second radius as its negative value).


Optimization 71

Defining Field Points

A lens should be optimized at multiple field points. Thenumber and location of the field points heavily influencethe optimization process and the final design result.To ensure that the lens performance does not oscillatebetween field points, a final design should be evaluatedat more field points than were used for optimization,especially if aspheric surfaces are used on field lenses.

For a rotationally symmetric lens, thefield is traditionally divided into cir-cles with equal areas. The normalizedradius of each circle is the normal-ized field coordinate H. In lenseswith moderate field (<25-deg FFOV)and no aspheres, three fields (H =0,0.7, and 1.0) are generally sufficientfor optimization. For systems with a rectangular sensor,it is common to circumscribe a circle around the sen-sor whose diameter equals the diagonal of the sensor,and this is used to define the field. Asymmetric systems

(e.g., TMAs) require fully definedrectangular fields in all four quad-rants because no rotational symmetryexists. In some cases, lateral or reflec-tion symmetry reduces the number offields required for optimization.

Common field definitions are object height, objectangle, and image height. In complex systems, usingimage height fields often requires ray aiming ofreference rays to avoid ray trace failures.

Fields can also be given field weights to balanceor emphasize on- or off-axis performance for a givenapplication. While projection and lithography lensesrequire uniform image quality over the entire field, visualand photographic systems often allow some performanceloss toward the edge of the field.


72 Optimization

Pupil Sampling

Pupil sampling defines the number and the distributionof the rays traced through the pupil and is criticalto both optimization and analysis. Tracing fewer raysleads to faster optimization but also to reduced accuracy.Obscurations and vignetting will influence the requiredpupil sampling and the algorithm used to generate rays.Pupil sampling methods may weight rays differentlyaccording to their pupil coordinate. The pupil samplingray density should be increased for systems with asphericsurfaces or large aberrations.

Common pupil sampling methods include:

• Rectangular grid sampling assumes no symmetryand often requires a high-density grid 10×10 to obtainaccurate sampling. Rays are generally evenly weighted.

• Polar or hexapolar sampling uses periodically spacedradial and azimuthal rays, and generally requires fewerrays than rectangular sampling. Rays may be weightedby ring (concentric circle) or arm (radial line).

• Gaussian quadrature (GQ) sampling uses a verysmall number of skew rays at very specific pupilcoordinates and weightings. GQ sampling returns amathematically exact integral of the pupil with fewerrays and provides higher sampling near the edge. GQ isthe fastest sampling for the majority of cases.

GQ sampling works best with circular or elliptical pupilsand may not give accurate results with highly obscured orvery noncircular pupils (unless specifically modified in thedesign code for this special case).


Tolerancing 73

Tolerancing

Tolerancing is a statistical process that determinesthe allowable change or tolerance in each of thelens construction parameters. All lenses must betoleranced before being built. The toleranced design needsto simultaneously satisfy the performance requirements,minimize manufacturing costs, and minimize alignmentcomplexity. These goals tend to be contradictory, so a largepart of tolerancing is finding the best compromise amongthem.

Tolerances include both optical print values (e.g., radius ofcurvature, center thickness, index variation, wedge) andmechanical print and assembly values (e.g., tilt, decenter,axial spacing, subassembly alignment). Compensatorsare parameters that can be adjusted during the lensbuild (e.g., focus, airspace changes, active elementcentering) to recover performance losses caused byother tolerances. Tolerances are often separated intosymmetric (compensated by focus and space adjust)and asymmetric tolerances (compensated with imagetilt and push-arounds). Tolerances that cannot be easilymodeled (e.g., material inhomogeneity or complex surfacefigure errors) form part of an unallocated performancebudget, requiring extra design margin for the unknownvariations. Many different tolerance methods exist, butthey generally follow the same basic process:1.Choose initial tolerance values for all parameters.2.Define the performance metrics (e.g., MTF, RMS spot

diameter, boresight error) and the requirements.3.Run a sensitivity analysis to determine the impact on

performance from each tolerance, and identify sensitiveand cost-driving tolerances.

4.Define compensators and their allowable ranges.5.Run appropriate statistical analyses (e.g., Monte

Carlo analysis) and evaluate the expected as-builtperformance and manufacturing yield.

6.Adjust tolerances and compensators until cost andperformance goals are met or a redesign is needed.


74 Tolerancing

Design Margin and Performance Budgets

Design margin is thedifference between therequired as-built per-formance and the nom-inal design performa-nce. The design marginshould be large enoughto allow for all man-ufacturing errors andenvironmental changes. However, the best design for man-ufacturing is not always the one with the largest designmargin. A lens with slightly worse nominal performanceis usually less sensitive to manufacturing errors, result-ing in a much higher yield and lower cost of production.Designers determine the most manufacturable design bycomparing both tolerances and predicted as-built perfor-mance of each design solution.

The design margin for a single top-level requirementcan be allocated to several categories in a performancebudget. The allocations in each category are theneither summed or root-sum-squared (RSS) to a top-levelnumber. In addition to tolerancing, the budget valuesare used for design targets, mechanical structure design(including thermal management and resonance), andother environmental requirements. There may be more

than one performancebudget necessary (e.g.,RMS OPD for imagequality and pointing ac-curacy for boresight).Complicated lenses withmany subassembliesmay also require a per-formance budget whereeach subassembly isbroken out separately.


Tolerancing 75

Optical Prints

An optical drawing or optical print consists of all ofthe design and tolerance information necessary for thefabrication of the part, including:

• Material index of refraction, dispersion, and quality• Radii and surface figure (power and irregularity)• Center thickness (CT) and edge thickness• Special surface parameters (e.g., aspheres)• Surface finish (roughness) and quality (scratch-dig)• Outer diameter (OD), surface sag, and edge beveling• Wedge or total indicator runout (TIR)• Clear aperture (CA) per surface• Antireflection (AR) or other coating requirements

Talk with lens fabricators early in the design processto discuss tolerance requirements, as drawing formatsand achievable tolerances vary from vendor to vendor.It is also important to make sure that the vendor’smeasurement methods support your specifications.

Optical prints typically isolate singlets or cementeddoublets or triplets.Higher-level assemblytolerances (e.g., groupor barrel axial dis-placement, and grouptilt and decenter toler-ances) are listed on as-sembly prints or as-sembly drawings.

ISO 10110 is the in-ternational standardfor optical drawings,although many orga-nizations develop theirown customized draw-ing formats.


76 Tolerancing

Radius of Curvature Tolerances

Radius of curvature (ROC) tolerances refer to thechange in vertex curvature of a surface and arespecified differently depending on how the lens is tobe measured. Some designs tolerance ROC directly,assuming a measurement of ROC with either aspherometer (contact measurement) or a distance-measuring interferometer (noncontact measurement).Other designs use test plate fringes to indirectlytolerance ROC. In this case, it is common to list twotolerances for ROC, a radius tolerance and a test platefringe tolerance, which can appear confusing. The radiusvalue tells the fabricator to use a test plate within acertain radius range, and the number of fringes gives theallowable part deviation from the test plate.

To compare the sensitivity of different radii in a design,do not apply a “uniform” radius tolerance in lengthunits across the lens; instead, use a uniform toleranceof curvature (c = 1/R) or percentage of radius (%R).

Test plates are precision sur-faces used during polishing tomonitor both ROC and surfaceirregularity. They are placed incontact with the lens and viewedunder monochromatic light (typ-ically a 546.1-nm mercury lamp),forming fringes (or “Newton’srings”). The ROC can be calculated from the part diame-ter and either the number of fringes (flat test plate) or thecurvature of the fringes (spherical test plate). Test platefringes represent a double-pass measurement, where onefringe corresponds to a half-wave of surface deviation. De-signs are usually test-plate fit to a specific supplier’s cat-alog of test plates to avoid large test plate tooling costs.Each radius in the design is individually altered to matchthe closest catalog test plate value, and the entire lens isre-optimized after each change.


Tolerancing 77

Surface Irregularity

Surface irregularity refers to anymeasured deviation of an optical sur-face from its intended shape. During astandard polishing process, irregularityis monitored with a test plate by assess-ing the number of fringes and/or their ir-regularity. For more-accurate, noncontact measurements,a surface-measuring interferometer is used to analyze thesurface irregularity.

Thin elements can “spring” when removed from apolishing block, resulting in two surfaces with large butapproximately equal and opposite cylinder error. If theclear aperture on each surface is about the same, theeffect of such cylinder error on lens performance nearlycancels when used in transmission.

The wide range of possible “shapes” for surface irregular-ity makes it difficult to model and tolerance deterministi-cally, and it is often simply specified in fringes or waves ofpeak-to-valley (P–V) error. Power and cylinder (astig-matism) are common low-order errors produced by stan-dard polishing processes, whereas higher-order errors in-clude an “edge roll” or “edge rip.” Irregularity specificationand measurement are sensitive to the exact CA called out.Allowing extra room between the CA and the outside partdiameter makes it easier to achieve high-quality surfaces.

Simple tolerance models assume that the irregularityerror is pure cylinder. Compensation schemes may includerotating or clocking an element. More-complex tolerancemodels use a combination of Zernike polynomials,either toleranced individually or as an aggregate RMSvalue of all coefficients up to a certain order. Irregularityfrom sources such as mounting stress must also beconsidered, especially for mirrors. Mechanically inducedsurface deformation often has three-way symmetry (“3-point” or trefoil) from kinematic mount designs.


78 Tolerancing

Center Thickness and Wedge Tolerances

A center thickness (CT) tolerance specifiesthe allowable error in the vertex thicknessof an element. Fabricators typically leaveCT on the thick side of the tolerance rangeto allow for removal of cosmetic defects orrework during final polishing. The sensitivityof a design to a CT error is often similar inmagnitude to the sensitivity of the design toan airspace error on either side of the element.

“Thin” lenses with diameter-to-CT ratios greater thanabout 15:1 typically require special handling duringfabrication, increasing their manufacturing cost.

A wedge tolerance in a spherical el-ement indicates the amount of al-lowable tilt between the optical axisof its two surfaces (defined by con-necting their centers of curvature)and a mechanical axis [usually theoutside diameter (OD) of the ele-ment]. Wedge is measured on a spin-ning chuck, either by optical methods such as angularbeam deviation or by mechanical methods such as totalindicator runout (TIR) or edge thickness difference(ETD). The fabrication process of centering or edging alens reduces element wedge and sets the final part diame-ter. Aspherical elements require special treatment (bothfor tolerance modeling and fabrication) because they donot have a clearly defined optical axis between their twosurfaces.

Meniscus lenses with nearly concentric radii areextremely sensitive to wedge errors. They can also bedifficult to center because it is hard to remove the wedgewithout significantly reducing the part diameter andencroaching on the CA.


Tolerancing 79

Material and Cosmetic Tolerances

The various parameters required to specify opticalmaterials must also be toleranced in a lens design. Thetwo most common tolerances are index of refractionn and dispersion V . A variation of n ± 0.001 andV ±0.8% is standard for most commercial glasses. Tightertolerances are available at higher cost. High-performancedesigns must also consider index inhomogeneity andbirefringence. Inhomogeneity is the P–V variation inbulk index throughout the lens volume and is typicallyless than ±5×10−6 for commercial glass. Birefringence isdefined as the maximum OPD between polarization statesover a standard path length (typically less than 20 nm/cmfor commercial glass). The complexity of modeling theinhomogeneity and birefringence errors often push theminto part of the unallocated performance budget, but sucherrors can be toleranced effectively as surface or wavefrontirregularity.

Many lenses are overspecified for cosmetic errors, leadingto reduced yield and additional cost. If the lens surfaceis not near an image or visible to the user, demandingcosmetic specifications may not be required.

Cosmetic defects include scratches, digs, pits, andstaining. Such defects generally have little influence onimaging performance, with a few exceptions:• Defects in lenses sensitive to stray light will scatter light

onto the focal plane and may add background noise. Theamount of light intercepted by the defect is proportionalto the ratio of the defect area to the beam area and isgenerally scattered over 4π sr.

• Defects in lenses for high-power laser applications mayact as damage centers or stress intensifiers.

• Defects near an image surface or subtending a largefraction of the beam footprint on a given surface mayappear in the image.

Cosmetic defects and bulk material defects (e.g., bubbles,inclusions, and striae) are hard to tolerance accuratelyand are often part of the unallocated performance budget.


80 Tolerancing

Lens Assembly Methods

To properly tolerance a lens assembly, it is importantto anticipate how the lens elements will be mountedand aligned. In general, as the required lens centrationand axial spacing tolerances get tighter, the mechanicalcomplexity, lens assembly time, and cost increase.Adhesives and flexures can be used to reduce mounting-induced surface deformation. Common assembly methodsinclude (from low to high precision):• “Drop-in” assembly: Elements are “dropped” into a

lens barrel, separated by spacers, and held in placewith a mechanical retainer. The assembly tolerances aredriven by the achievable fit between the element OD andthe barrel inner diameter (ID). These assemblies workbest with low-cost targets in high-volume applicationswhere ease of assembly is paramount.

• Shim-centered assembly: Elements are centeredradially in a lens barrel using shims and then bonded inplace. This method is used for lenses that require bothtight centration tolerances and large radial gaps to allowfor thermal expansion.

• “Poker-chip” assembly: Each element is accuratelyaligned to the reference surface of an individual lens celland bonded in place. Cell and spacers are inserted as astack into a precision-machined lens barrel.

• Bolt-together assembly: Each element is accuratelyaligned to the reference surface of a precision-machinedlens cell and then held in place on a seat with eitheradhesive and/or flexures. The lens cells are precision-aligned to each other either optically or mechanicallyand then bolted together.


Tolerancing 81

Assembly Tolerances

Assembly tolerances, such asdecenter, tilt, roll, and axialspacing, represent positioningerrors of optical surfaces in alens system. Assembly toler-ances may be applied to lenssurfaces, lens elements, and/orelement subassemblies. Accu-rate tolerancing requires knowledge of the optomechanicaldesign and mechanical part tolerances, and a well-definedassembly plan.

Multiple decenter errors can be added together andtoleranced as a single motion. However, multiple tilterrors with their own individual pivot points must betoleranced as separate motions.

Most assembly tolerances are not derived from a singlepart error but must be calculated from multiple part errorsusing a tolerance stack-up. For example, in a simplemount, a single airspace tolerance can depend on fiveseparate part tolerances: two lens surface or bevel sags,a spacer thickness and diameter, and a lens CT. Theimpact of a CT error on an airspace also depends on howthe element is mounted. Elements separated by spacersshift all subsequent surfaces by their CT errors, thusmaintaining the airspaces between elements, whereasthe CT error in an element mounted on an independentmechanical shoulder in a lens barrel also changes theneighboring air space value but may not affect the positionof other elements.

Tilt and decenter tolerances break rotational symmetryintroducing boresight error and field tilt as well asaxial coma, axial astigmatism, and axial lateral color. Toproperly apply a tilt tolerance, it is important to includethe pivot point for the tilt motion. For cemented elements,the tilt and decenter of one element with respect to theother is specified as roll about the cemented interface.


82 Tolerancing

Compensators

Manufacturing errors that have a similar effect onperformance can often be corrected by modifying a singlelens parameter. Compensators are lens variables thatsimulate adjustments made to the lens during assemblyand test to improve as-built performance. The mostsensitive tolerance parameters are usually selected ascompensators. Compensators loosen tolerances and reducepart cost but add time (and cost) to assembly andtest. Without compensators, most tolerances would beunreasonably tight. For each compensator, the toleranceprocess must define the adjustment range and itsrequired resolution and accuracy. The optomechanicaldesign typically constrains the compensator range ofmotion.

Compensators should be limited in number, independent(two compensators should not affect/correct the sameaberration), and practical (e.g., glass dispersion is not arealistic compensator).

• Focus and image plane tilt are almost universallyused as image plane compensators to loosen tolerances.

• A custom spacer may be designated as a post-buildairspace compensator whenever focus is insufficientto control the spherical aberration introduced bysymmetrical tolerance errors.

• Asymmetrical tolerance errors typically cause axialcoma. This can be corrected during test by decentering asingle lens element known as a push-around.

• In complex lenses, several airspaces can be optimizedprior to assembly using the as-built measured data(radii, CT, airspaces, and index) in a space adjust.

• Tight index-of-refraction or dispersion requirementsmay require a melt-recompensation process, in whichthe indices of refraction of the actual pieces of materialto be used in the assembly are measured. The design isthen re-optimized (usually only the airspaces are varied)using the measured glass material data. This methodcan help loosen material tolerances.


Tolerancing 83

Probability Distributions

The tolerance extremes for each lens variable limit onlythe maximum and minimum values. The probabilitydistribution function (PDF) defines the likelihoodof obtaining an as-built value between the extremes.The tolerance PDF depends on the fabrication and/oralignment process (which may even be specific to a givensupplier) and should be defined for each tolerance in aMonte Carlo tolerance analysis.

CT tolerances often have skewed PDFs because polisherswill leave lenses on the thicker side of nominal to allowfor possible rework due to cosmetic errors.

• Uniform distributions are constantover the tolerance range with equalprobability between the extremes.

• Normal or Gaussian distributionshave a defined mean and standarddeviation, where the mean is thenominal design value and is the mostlikely value to occur.

• Because normal distributions allowinfinite range, a truncated Gaussianis often used, where the probability isdefined to be zero beyond the toleranceextremes. This distribution representsa supplier yield where parts out ofspecification are not used.

• Skewed Gaussian distributions havea nonzero higher-order moment, wherethe mean or nominal value is not themost likely value.

• Spiked or parabolic distributionsare heavily weighted toward the toler-ance extremes, where the nominal design value is theleast likely. They can also simulate binary tolerances.

• Most design codes permit the use of custom distribu-tions to incorporate measured statistical data.


84 Tolerancing

Sensitivity Analysis

A sensitivity analysis determinesthe sensitivity of a lens to manufac-turing errors by calculating the changein each performance metric (e.g., RMSwavefront or MTF) for both a plus anda minus change (equal to the toler-ance value) in each tolerance param-eter. Performance changes may not besymmetric about the nominal design value and often varywith field point. The analysis is initially performed un-compensated (although focus is regularly included in theinitial run). If needed, a second compensated run (wherethe compensator motion is optimized to improve perfor-mance) helps gauge the effectiveness of the compensatorand identifies tolerances that drive large compensator mo-tions. During the compensated run, constraints are oftenremoved from the merit function (e.g., focal length) thatcan interfere with performance improvement.

An inverse sensitivity analysis determines the toler-ance value that produces a given change in performance,with the goal of having each tolerance contribute equallyto the performance degradation of the system.

It is often useful to apply the same tolerance valueto all surfaces (or elements) for common toleranceparameters (e.g., CT ±0.025 mm or index ±0.001)and then group the results into tables and/or charts.

A sensitivity table orchart lets you identify at aglance the tolerances thatwill drive the performanceloss and highlight possi-ble compensators. The ta-bles and charts are alsovery useful in comparingthe sensitivities of differentdesign solutions to possiblemanufacturing errors.


Tolerancing 85

Performance Prediction

A basic sensitivity analysis considers the effects on systemperformance for each tolerance individually. In order topredict the image quality of an as-built system, it isnecessary to simulate the effect on performance of allof the tolerances simultaneously. Most design softwarepackages compute the as-built performance predictionusing a root-sum-square (RSS) calculation, displayingsensitivity analysis results in tabular or graphical form.This approach works well if the performance losses fromall of the tolerances are of similar form.

The RSS of the performance change from each tolerancegives an estimate of overall predicted performance;however, it generally overestimates the performance losswhen compared to a Monte Carlo analysis with theappropriate PDFs and tolerance interactions (e.g., a lensradius may have a different sensitivity if the radius on theother side of the lens also changes).

In some cases, part tolerances are divided into symmetric(surface-centered) or asymmetric (non-surface-centered)tolerances for separate tolerance runs:

Symmetric Asymmetric• Curvature (radius) • Surface tilt/decenter• Center thickness • Lens tilt/decenter• Airspace • Group tilt/decenter• Index of refraction • Surface irregularity• Dispersion • Index inhomogeneity

Wavefront differential or derivative tolerancinggenerates a multivariable differential tolerance sensi-tivity matrix containing first-order (linear), second-order (quadratic), and mixed partial derivatives or crossterms. It requires a one-time analysis of the nominalsystem and can be hundreds of times faster than a fullMonte Carlo analysis.


86 Tolerancing

Monte Carlo Analysis

A Monte Carlo analysis is an alternative tolerancingapproach to predict as-built performance. It simulatesa manufacturing environment by generating a series oflenses or trials. Each trial run randomly perturbs alltolerances within its defined extremes according to itsPDFs, and then evaluates the image performance criteria.By considering all applicable tolerances simultaneouslyand exactly (no approximations are made), a highlyaccurate simulation of expected performance is possible.If compensators are needed, each trial can be reoptimizedwith only the compensators as variables. If allowed,relaxing exact constraints (such as focal length or workingdistance) in the merit function during compensation cansignificantly loosen the symmetric tolerance values.

A rule of thumb for a Monte Carlo analysis is that thenumber of trials should exceed the square of the numberof tolerances for good statistical validity.

The result of a Monte Carlo analysis is a probabil-ity distribution table and/or cumulative distribu-tion function (CDF) or yield curve showing the per-centage of lenses that meet or exceed a given per-formance criterion. The number of random trials eval-uated determines the maximum statistical confidencelevel. Because tolerancing isstatistical, a designer needsto decide in advance what isan acceptable manufactur-ing yield (e.g., 98% pass and2% fail). Individual trials canbe saved for forensic analysisor to investigate poorly per-forming outliers.

Monte Carlo analyses generally require significantcomputation time and can only give statistical results,which may not be valid for one-off builds or very smallproduction runs.


Tolerancing 87

Environmental Analysis

A complete tolerance analysis must include the perfor-mance impact on a lens due to changes in its environment.Environmental requirements such as temperature orpressure can consist of multiple specification ranges. Theoperating range dictates the total extent over which thelens must meet all performance requirements. A survivalrange gives the extent over which the lens need not meetperformance but cannot break or become nonfunctional.The sensitivity of the lens to environmental perturbationsmust be assessed and allocated as part of the performancebudget using an environmental analysis.

In addition to ambient temperature changes, heatsources such as electronics, motors, light absorption, andsolar radiation can all induce thermal effects in lenssystems. Two key material properties are needed for anaccurate thermal model: The coefficient of thermalexpansion (CTE) gives the percentage change in lineardimension per unit temperature change, and the glassthermo-optic coefficient gives the absolute change inrefractive index per unit temperature change. Simpletolerance models specify the temperature change asa thermal soak or steady-state effect; however, more-complex models can include both spatial and temporaltemperature gradients. Pressure or altitude changesprimarily affect the refractive index of the gas materialsurrounding the lens elements (typically air) but can alsoalter the airspaces between lenses or change the shape ofoptical surfaces in sealed assemblies.

Most design codes provide the ability to perturb the lenstemperature and/or pressure and evaluate performancechanges as long as the appropriate material data isentered. However, complex environmental influences needfinite element analysis (FEA) to help predict changesin performance [e.g., vibrations can disrupt componentalignment and need FEA to model the vibration effects online-of-sight (LOS) stability and image jitter].


88 Tolerancing

Athermalization

As the temperature of an optical system changes, thelens elements and housing physically expand/contract.In addition, the glass refractive index will also change.The primary result of both changes is a focus (orimage plane) shift, although aberrations and pointingerrors (from lateral component shifts) are also possible.An athermalized optomechanical design minimizesperformance variation with temperature. For example,focus shift can be actively athermalized by measuringthe temperature and then physically moving either thelens or the sensor. This approach is not the ideal solutionin many situations, and passive athermalization(no motor-driven component movements) is employedwith the use of dissimilar materials (e.g., opticalmaterials with different thermo-optic coefficients or metalmounting materials with different CTEs). Most infraredmaterials have large thermo-optic coefficients, makingathermalization mandatory for large temperature ranges.

A reflective system can be passively athermalizedagainst isotropic thermal soak conditions by making themirrors and the mounting structure out of the samematerial. The system then scales in dimension as thetemperature changes but maintains overall focus andimaging performance.


Stray Light 89

Stray Light Analysis

Stray light is any unwantedlight that reduces performanceby striking the image plane of anoptical system. Primary sourcesof stray light include reflectionsfrom uncoated or poorly coatedsurfaces, surface or bulk scat-tering, thermal emission, brightout-of-field objects, glints frommechanical structures, diffractedlight from edges, and unwanted diffraction orders fromgratings.

Stray light can be divided into two main categories: ghostimages and veiling glare. Ghost images have structure(often with sharp edges) and create local image-qualitydisruptions. Veiling glare produces a diffuse backgroundhaze on the image plane, resulting primarily in a loss ofoverall image contrast.

Tracing rays backwards from the detector (or “puttingyour eye at the detector”) is a highly efficient techniquefor isolating potential stray light paths and is much moreeffective than tracing from a source to a detector.

A stray light analysis determines both the amount ofstray light at the detector and the different paths thatit took to get there. If the stray light is large enough tocause a problem, the analysis should also identify anydesign changes needed to reduce the stray light. Somekey metrics used to quantify stray light include off-axisrejection (OAR), point-source transmittance (PST),point-source normalized irradiance transmittance(PSNIT), and Narcissus-induced temperature differ-ence (NITD) in infrared systems. The accuracy of a straylight analysis is limited by the availability of source andscatter data, the completeness of the computer model, andthe available computer time.


90 Stray Light

Stray Light Reduction

Identifying potential stray light paths during the designphase is essential to avoiding costly errors in hardware.A critical object is any surface that can be seen by thedetector, including surfaces that are imaged through lenselements. An illuminated object receives power directlyfrom the source and depends on the source location. Theoverlap between the list of critical and illuminated objectshighlight key stray light paths. An object on both listsis called a first-order stray light path, which is amajor contributor to stray light and should be addressed.Second-order stray light paths exist anywhere thatlight can propagate (transmit, reflect, scatter, or diffract)from an illuminated object to a critical object.

Field stops reduce the number of illuminated objects in adesign because objects behind the field stop cannot be seenby the source—only objects in front of the field stop can beilluminated. Similarly, placing the aperture stop as closeas possible to the detector reduces the number of criticalobjects in a design since objects in front of the aperturestop cannot be seen by the detector. This approach mayrequire that the aperture stop be relayed from the frontof the lens system to the rear because the farther theaperture stop is from a lens, the larger the front elementsbecome in diameter. Lyot or glare stops are aperturesplaced in conjugate pupil planes; such stops might need toduplicate any aperture stop obscurations from mountingfeatures for extreme stray light control.

Identified stray light paths can beeliminated by blocking light withbaffles, vanes, and knife edges,or by relocating an object so thatit is no longer seen by the detector

or illuminated by a source. Mechanical surfaces andedges of optical elements can be treated to absorb orscatter light by blackening or roughening. Rifling orthreading barrels and mechanical surfaces is a commonand economical way of reducing stray light.


Stray Light 91

Antireflection (AR) Coatings

Optical surfaces require antireflection (AR) coatingsto reduce stray light and improve transmission. For anuncoated surface, the amount of Fresnel reflection lossdepends on the angle of incidence of the light and theindex of the lens material. For light normally incident on asurface in air, the reflection loss is ∼4% for a visible glasswith an index of 1.5. In the infrared spectrum, the loss peruncoated surface can be as much as 36% due to the highindex of infrared materials.

Large changes in ray AOIs across a lens surface canlead to highly nonuniform pupil transmission and pupilpolarization in high-NA systems. Steep ray angles ofincidence on an AR-coated surface will also blue-shiftthe spectral region of a coating, especially coatings withsharp band edges.

The ideal single-layer AR-coating material for a glass-to-air interface has a quar-ter-wave of optical thicknessand an index of refractionequal to the square rootof the lens material index.This condition is difficult to achieve in practice. The closestmaterial choice for the visible spectrum is a single quarter-wave layer of MgF2 (index 1.38), which will reduce thesurface reflection from ∼4% to ∼1% at a single wavelength.More-complex coatings are made of alternating layers ofhigh and low index, each with a thickness equal to aquarter-wave optical thickness. Two- and three-layer V-coats can eliminate the reflection at a single wavelength,whereas more-complex broadband AR (BBAR) coatingsreduce the reflection loss to <0.5% over an extendedwavelength band. A variety of coating design optimizationsoftware packages exist (very similar to lens design). Mostlens design codes will model (but not optimize) thin filmstacks for accurate transmission calculations.


92 Stray Light

Ghost Analysis

Ghosts are a form of stray light from unwantedreflections. Traditional ghosts are double-image artifactsformed by two lens surfaces (two-bounce ghosts) withimperfect coatings. However, ghosts can also result fromreflections from object, filter, detector, and mechanicalsurfaces. In interferometer optics, single-reflection ghostscreate extra “reference” beams that complicate fringeanalysis. In systems with high-power lasers, ghosts canproduce an internal focus inside a lens material or ona lens surface that may actually damage the lens. Inthermal infrared lenses, Narcissus reflections from cooleddetector features cause image artifacts. A pupil ghost is aspecial ghost image that echoes the shape of the aperturestop and is typically seen in camera lenses when a brightsource (such as the sun) is inside the object field.

Windows, beamsplitters, concentric mensicus lenses, andoptical surfaces with near-normal incident rays are theprimary sources of in-focus ghost images.

Most of the potential ghostimages in a lens systemare highly aberrated andout of focus. If the ghostimage is significantly de-focused from the imageplane, it produces a gen-eral haze on the detector,reducing image contrast. Aghost analysis is primarily a graphical techniquewhereby sequential ray trace routines generate graphicalghost paths that analyze every potential two-bounce re-flectance in the optical system and then report the “imagequality” of each ghost in terms of spot size, defocus, mag-nification, and f /#. The relative intensity of a ghost imagecan be estimated from its number of reflections; however,nonsequential ray tracing is required for a quantitativeradiometric ghost analysis.


Stray Light 93

Cold Stop and Narcissus

Many infrared systems requirea cyrogenically cooled detectorthat sits in a thermally insulatedvacuum dewar, reducing back-ground noise and increasing sen-sitivity. Because the system com-ponents also emit the same wave-lengths of thermal radiation as

those trying to be imaged by the detector, a cold shield isneeded to minimize the number of “warm” surfaces (out-side of the object scene) that the detector can see. For opti-mum stray light control, the lens exit pupil should matchthe detector cold shield in size and location. Imaging thelens stop onto the cold shield forms a cold stop, and ifimaged perfectly, the system is said to have 100% coldshield efficiency.

Surfaces inside the dewar (e.g.,the focal plane and the cold stop)are at significantly different tem-peratures than the scene imagery,and their reflections from an opti-cal surface will form “cold” Narcissus ghost images. Suchimages are either diffuse background noise or sharp imageartifacts varying in field or time. Retroreflection due to asmall marginal ray angle-of-incidence at a surface or im-age reflection due to a small marginal ray height are pri-mary sources of Narcissus. The YNI product combinesthese into a paraxial quantity that can be used to opti-mize the lens or locate baffles. Staring detectors allow anonuniformity correction (NUC) to calibrate out staticreflections, but any lens motion or environmental changesafter NUC will induce Narcissus. The NITD is the sum ofreflected energy from all optical surfaces.

Techniques for minimizing Narcissus include anti-Narcissus coatings, baffles, tilting flat windows and filters,and designing with Narcissus ray constraints.


94 Stray Light

Nonsequential Ray Tracing

Nonsequential ray tracing (NRT) is an extremelypowerful analysis technique for evaluating stray light.Unlike standard sequential ray tracing in which raysintercept surfaces only once and in a given order, NRTallows a ray to “see” all of the surfaces in a systemsimultaneously and the ray interacts with the first surfaceit meets. It is possible for a ray to intersect a single surfacemultiple times. NRT is also widely used to design andoptimize illumination and other nonimaging systems.

For accurate results, many orders of magnitude more raysneed to be traced with NRT than sequential ray traces. Asa result, stray light analysis can be time consuming andcomputationally expensive.

A stray light analysis withNRT requires three components:sources that generate rays, ob-jects that interact with rays,and receivers that collect therays and quantify the resultingirradiance or intensity distribu-tions. Key features that can be enabled depending on com-putation limitations are ray scattering and ray split-ting, which split energy between a parent ray and multi-ple child rays. The sum of the transmittance, reflectance,scattering, and absorption on a surface should equal unity.Ray databases that include all ray-surface data can besaved for subsequent filtering and analysis.

NRT models are typically much more complex thansequential optical surface design models and can include:

• multiple sources with rays propagating in any direction;

• mechanical features such as lens mounts, barrels,apertures, baffles, and vanes;

• absorption and scattering from bulk materials; and

• absorption and scattering from mechanical and opticalsurfaces (ground, polished, coated, or painted).


Stray Light 95

Scattering and BSDF

Optical and mechanicalsurfaces have a finite sur-face roughness that caninduce light scattering.The direction and amount of scattered light depend on theincoming ray wavelength, polarization, and angle of inci-dence. Because the scattered light is a potential source ofstray light, a complete stray light analysis requires accu-rate scatter models for each surface. Ideal scatter mod-els assume that the scatter is either all concentrated inthe specular direction (reflected angle governed entirelyby Snell’s law) or is perfectly Lambertian (diffuse sur-face with the same reflected radiance over all viewing an-gles). Real surface scatter usually lies somewhere betweenthese two extremes, often with most of the light scattered“around” the specular reflection direction.

The bidirectional scattering distribution function(BSDF) indicates how specular or diffuse a surface is as afunction of both the 2D incident angle (θi,φi) and the 2Dscattering angle (θo,φo):

BSDF (θi,φi;θo,φo)= LE

(sr−1

).

The BSDF is defined as exiting radiance divided byincident irradiance and is centered around the specularangle defined by θi = θr. A Lambertian surface hasa constant BSDF for all angles of incidence. Thereflective and transmissive versions of the BSDF arethe BRDF and BTDF, respectively. A scatterometer isused to measure angle-resolved BSDF, BRDF, and BTDF.Isotropic surfaces, such as pitch-polished lenses, scatterlight equally in all azimuthal directions, simplifyingthe BSDF function. Anisotropic surfaces, such asdiffraction gratings, diamond-turned mirrors, or brushed-metal surfaces, have inherently azimuthal directionality.Total integrated scatter (TIS) is the integral of theBSDF and is a common scalar metric for total surfacescattering without any angular resolution information.


96 Optical Systems

Photographic Lenses: Fundamentals

A photographic objective (or camera lens) is a lensassembly that forms a real image of a real object.Photographic lenses were historically designed toimage on film, but modern digital cameras use CCDsor CMOS detectors. Camera objectives are typicallyoptimized for distant objects, incorporating floatingelements to refocus the image onto a fixed plane as theuser changes the object distance.

The expressions fast, slow, and speed of a lensoriginate from photography, where a camera lens witha smaller f /# results in a wider aperture and more lightcollected, allowing a user to set a faster shutter speed forthe same exposure time.

A wide variety of applications for camera lenses exist inconsumer (e.g., point-and-shoot cameras, mobile phones,camcorders), commercial (professional movie cameras),industrial (quality control), and scientific markets. Mostmodern photographic objectives have zoom capability(to change magnification), and use aspheres and plasticmaterials for both size and cost reduction. Camerastypically operate in the visible, but they can be designedfor any wavelength band where a detector is available.

Camera lenses are classified by focal length and imageformat size (e.g., 100-mm telephoto for 35-mm format).The focal length for a given image format determinesthe angular FOV. A “normal” lens has a focal lengthapproximately equal to the image diagonal and a FOVof about 50 deg. A long-focus lens is a narrow-FOVlens with a long focal length; the most common typeof long-focus lens is a telephoto lens. A wide-anglelens is a lens with a FOV wider than 60 deg and afocal length shorter than normal. Fish-eye lenses haveextremely large FOVs (>180 deg). Macro lenses arespecial objectives designed for objects that are smallerthan the image sensor (>1:1 reproduction ratio).


Optical Systems 97

Photographic Lenses: Design Constraints

Most cameras need to be handheld, restricting the sizeand weight of a photographic lens. A single-lens reflex(SLR) camera lens has an additional working distancerequirement. A diagonal mirror is located between theSLR lens and the image plane to allow the user to viewthrough the lens itself (the mirror flips out of the way justbefore exposure). In contrast, a viewfinder camera hasa separate viewfinder so that lens elements can be muchcloser to the image plane. However, the viewfinder FOVdoes not perfectly overlap the lens FOV, and “what yousee” is not necessarily “what you get.”

MTF (in cycles/mm) is used to specify and measure cameralens performance. Unlike telescope and microscopeobjectives, which are usually diffraction limited at fullaperture, photographic objectives are aberration limitedat full aperture and are rarely used “wide open.” Stoppingthe system down decreases the spot size blur due toaberrations and increases the depth of focus of system. Aspeed of ∼ f /8 balances diffraction and aberration effectsin SLR photographic lenses.

Photographic lenses have re-laxed requirements for rela-tive illumination (RI), theratio of on-axis image plane ir-radiance to the minimum ir-radiance over the full imagefield. This permits designersto use vignetting to reduceaberrations and increase the aperture and/or FOV. Forwide-FOV designs, a 50% RI specification is not uncom-mon. Distortion requirements for lenses with standardFOVs range from <1% to 4%. However, in wide-FOV lenses(especially fish-eye lenses), correcting distortion so thatimage height is proportional to f tan(θ) is very difficult;it becomes easier to control f -θ distortion, where equalobject-angle increments map to equal image-height incre-ments.


98 Optical Systems

Visual Instruments and the Eye

The human eye is a dynamic optical system that canaccommodate varying object distances by changing theshape and refractive power of its lens. The refractivepower is typically measured in diopters, where 1 diopter= 1/meter. The eye forms its image on the retina, a curvedsurface containing a series of rods and cones that serveas detecting units. The resolution of the eye varies overits FOV, with the best resolution found in the centralfoveal region where the cone density is a maximum.The fovea subtends about a 1-deg FOV with roughly 1-arcmin angular resolution. The resolution drops outsidethis region (e.g., it is ∼5 arcmin for a 30-deg off-axis fieldpoint).

There is a practical limit on the maximum usefulmagnification for visual instruments. At some point,the image becomes so large that the details in theimage exceed the resolution of the eye. This is referredto as empty magnification. In applications whereresolved detail is important (e.g., microscopy), this canlead to artifacts in the viewed image. However, inapplications where the boresight of the image outlineis more important (e.g., riflescopes), some amount ofempty magnification can be beneficial for relievingeye strain. As a rule, to avoid empty magnification,the visual instrument magnification should not exceed∼11× the aperture diameter (in inches) in telescopes and∼225× the NA in microscopes.

Visual instruments, such as telescopes, microscopes,riflescopes, endoscopes, periscopes, and binoculars, formimages that can be viewed with the human eye. Theimage is typically magnified and placed at a convenientlocation for viewing, ranging from 25 cm (10 in) to infinity,depending on the application. Simple visual instrumentsconsist of an objective and an eyepiece, while morecomplex systems add image inverters and/or relays.Visual instruments should also have a focus adjustmentto accommodate differences between individual users.


Optical Systems 99

Eyepiece Fundamentals

Eyepieces are essential components of visual systems,magnifying the image created by an objective and placingit at a suitable location for viewing with the human eye.The eyepiece focal length determines its magnification. Aneyepiece is usually designed “backwards,” starting fromthe infinite conjugate (eye side) to the short conjugate(objective side). To avoid vignetting, eyepieces should bepupil-matched (in both size and location) to the humaniris; this requires that the aperture stop be completelyoutside the lens so that the eye can be positioned acomfortable distance away from the eyepiece.

High-power microscope objectives often have uncor-rectable lateral color that can be offset with a compen-sating eyepiece.

The distance from the aperture stop to the first lenselement (or first mechanical surface) is called the eyerelief. Eye relief requirements vary from as little as10 mm to simply clear the eyelashes, to 20 mm foreyeglass clearance, to as much as 5 in. for protection fromhigh-power rifle recoil. For maximum light collection, theaperture stop diameter should be matched to the eye’s irisdiameter. Eyepieces are typically designed for a 3 to 4-mmdiameter pupil, but this value may vary between 2 mm (forbright scenes) and 10 mm (for use in moving vehicles).

Due to the external aperture stop requirement, eyepiecedesigns have little symmetry, resulting in significantastigmatism, lateral color, and distortion (e.g., 8–10%distortion is not unusual). The off-axis image quality ofeyepieces can be relatively poor—the eye dynamicallyscans the image during its normal operation, employing itslow-quality peripheral vision in static situations. Simpleeyepieces with positive elements have large uncorrectedPetzval, whereas more-complex eyepieces use a negativefield lens to flatten the field. The residual field curvature istypically evaluated in diopters of required accommodation(defocus).


100 Optical Systems

Eyepiece Design Forms

Eyepiece design forms can be classified by number ofelements, apparent FOV, and eye relief. Complex designswith more elements have improved performance, largerFOVs, and/or longer eye reliefs, but with added cost,size, and weight (important for handheld or head-mountedinstruments). Modern eyepieces incorporate aspheresand diffractive elements to reduce size and weight. A keydistinction among eyepiece design forms is the presence ofan accessible image plane, which can be used to overlay areticle onto the final image.

Lens distortion changes sign if the optical systemis reversed. This fact is useful when balancing thedistortion between an eyepiece and an objective if theeyepiece is designed in the opposite direction.

The Huygens eyepiece is a simple two-element designwith a virtual image, while the three-element Kellnereyepiece has a real image plane. With two identicaldoublets, the Plossl design is a low-cost, medium-FOV eyepiece. The orthoscopic eyepiece (known for itslow distortion) is common in astronomical and medicaldevices. The Erfle is a standard wide-FOV militaryeyepiece with long eye relief but short working distance.The wide-FOV Nagler design uses a strong negative fieldlens group to correct field curvature over a virtual imageplane, but it suffers from strong spherical aberrationof the pupil, leading to a “kidney bean effect” whereinportions of the field become alternately light and dark(vignetted) as an observer’s eye position changes.


Optical Systems 101

Telescopes

Traditional afocal tele-scopes magnify distantobjects by enlarging theirapparent angular extentat the eye. They consist

of a long-focal-length, narrow-FOV objective and a short-focal-length, wide-FOV eyepiece. The angular magnifi-cation of the telescope is defined by the ratio of the objec-tive focal length to the eyepiece focal length. Telescopestypically have interchangeable eyepieces with differentfocal lengths to change magnification. Modern imagingtelescopes remove the eyepiece and use the telescopeobjective to image directly onto a sensor. In terrestrial(spotting) telescopes, an erector is placed between theobjective and eyepiece to form an upright image.

Afocal telescopes and beam expanders are specified bytheir magnification (e.g., 4×), whereas imaging telescopesare characterized by their aperture diameter (e.g., 12 in.).

Astronomical telescopes require large apertures togather as much light as possible from distant objects.Astronomical telescopes larger than 12 in. are usuallyreflective. In contrast, surveillance spotting scopesneed to be compact and portable and are usuallyrefractive. A riflescope is a specialized afocal telescoperequiring extended eye relief for recoil. Binoculars aredual co-boresighted telescopes with upright images.

Beam expanders are afocal telescopes used “in reverse”to expand a laser beam diameter and reduce itsdivergence. A Keplerian telescope (with a positivelypowered eyepiece) has an internal focus, allowing apossible spatial filter location. A Galilean telescope(with a negatively powered eyepiece) is shorter and workswell for high-power lasers because there is no internalfocus. A collimator is a reversed telescope objective.


102 Optical Systems

Microscopes

A microscope is an instrument used to magnify andview objects that are too small for the naked eye.Applications include biological and medical research,metallurgy, and industrial inspection. A basic compoundmicroscope requires an objective and an eyepiece,where the magnification is the product of the individualmagnifications and is defined relative to the angle theobject subtends at the eye at a distance of 250 mm. Thetube length is the distance from the objective’s rear focalpoint to the intermediate image.

A modern microscope is designed with an infinity-corrected objective and a tube lens (placed betweenthe objective and eyepiece). This form allows the insertionof tilted components (e.g., beamsplitters, filters, andpolarizers) in the collimated space between the objectiveand tube lens. The tube lens can also focus theimage directly onto a detector. Microscopes need to bemodular (e.g., interchangeable objectives to adjust systemmagnification and resolution). The objective flange-to-focus distance is standardized at either 45 or 65 mm. Theintermediate image diameter ranges from 20 to 28 mm.Standard tube lengths include 160, 180, 200, and 210 mm.

The design of the illumination system can have asignificant impact on the contrast and resolution ofa microscope. Trans-illumination uses a condenserto illuminate the sample (light is collected by theobjective on the other side), whereas epi-illuminationuses the objective to illuminate the sample. Dark fieldillumination (the opposite of bright field illumination)is a technique used to enhance contrast where thecondenser has a central obscuration that is larger than thecollecting aperture of the objective lens.


Optical Systems 103

Microscope Objectives

Microscope objectives are small-FOV lenses thatare typically diffraction limited on-axis, telecentric inobject space, and unvignetted. The objective housing isimprinted with a magnification for a fixed tube lensfocal length and NA (e.g., 20× /0.5). Although there is norigid relationship between magnification and NA, if themagnification is increased, the NA is increased to avoidempty magnification. Biological microscope objectivesneed to correct the spherical and chromatic aberrationintroduced by a thin cover slip or microscope slide.

Microscope objectives are typically designed “backwards”from the long conjugate to the short conjugate. Low-powerobjectives are ordinary achromatic doublets dominatedby field curvature. A medium-power Lister objectiveconsists of two achromats in a Petzval-like arrangement.To increase the NA, aplanatic lenses (as in the Amici)are added to the object end of the lens, decreasing theworking distance. Medium- to high- power objectives arelimited by lateral color, which can be offset by a tube lensor eyepiece. Inverted telephoto and double-Gauss designcan be turned into microscope objectives with long workingdistances. An immersion objective has an extremelyhigh NA (>1) and is designed to interface with objects influid (index > 1). Objectives are classified by their fieldflatness and color correction capability:• PLAN: corrected over a flat field (digital focal planes

require flatter fields than visual instruments).• APO (apochromatic): broad-spectrum color correction.• PLAN–APO: corrected for both field curvature and

chromatic aberration for a high level of performance.


104 Optical Systems

Relays

Finite-conjugate relay lenses image a real object to a realimage, typically with a magnification close to one. Theycan be grouped into single-stage reproduction lensesand multi-stage unity-magnification relays, where eachadditional stage inverts the image.

Reproduction lenses range from simple document copy/scan lenses to complex lithography projection lenses.Their magnification varies between 1:1 and 20:1 (eitherreducing or enlarging) and may be adjustable. Sincethe applications for these lenses require a high levelof distortion correction, design forms with inherent stopsymmetry (e.g. the double-Gauss) are frequently used.

F-theta lenses are designed to have deliberate barreldistortion where the image height follows an h′ = f θ

rather than the standard h′ = f tanθ relationship. F-θlenses are useful in scanning systems where the f -θdistortion creates a scanned spot with a constant velocityacross the field, producing a uniform exposure.

Unity-magnification relays are found in periscopes,endoscopes, and boroscopes, where light must beguided down a long tube. Minimizing the lens diameter insuch long-distance relays is critical; this requires multiplestages, each with a primary imaging lens (usually adoublet or triplet for color correction). Field lenses areplaced near the internal image planes of the relay toconverge the oblique beams so that they pass through asmaller clear aperture. Due to their proximity to an imageplane, field lenses must have excellent surface quality andvery few defects. The relay is typically only one part of theoptical system and is coupled to a high-resolution objectiveon one end and an eyepiece or camera lens for viewing theimage on the other end.


Appendix: Optical Fundamentals 105

Index of Refraction and Dispersion

The index of refraction (or index) of a material is theratio of the speed of light in vacuum to the speed of lightinside the material (typical values range from 1.45 to 4.0).The change in index as a function of wavelength is calledmaterial dispersion. Normaldispersion occurs when in-dex decreases with wavelength.Anomalous dispersion oc-curs near material absorptionpeaks where index increaseswith wavelength. The disper-sion of a material over agiven spectral band is quanti-fied by an Abbe number orV-number (V), which is calcu-lated with the index at threewavelengths—the two pointsnear the ends of the spectralband and a midpoint—which in the visible corresponds tothe F, d, and C spectral lines. A high V corresponds to lowdispersion, and a low V indicates high dispersion.

Custom Abbe numbers can be defined over any band ofinterest when designing outside of the visible spectrum(e.g., the g line is used instead of the F line whencorrection at shorter wavelengths is needed). Thedispersion of a material can be vastly different in twodifferent spectral bands (e.g., visible and MWIR).

Sellmeier

n (λ)=√

1+ c1λ2

λ2 − c4+ c2λ2

λ2 − c5+ c3λ2

λ2 − c6

Schott

n (λ)=√

c1 + c2λ2 + c3

λ2 + c4

λ4 + c5

λ6 + c6

λ8

Two common disper-sion models are theSellmeier andSchott index equa-tions. Discrete mea-sured data can be fitto one of these equa-

tions for use in optical design. However, using dispersionequations outside of their intended wavelength region canlead to significant extrapolation errors.


106 Appendix: Optical Fundamentals

Optical Materials: Glasses

Common optical ma-terials used in thevisible spectrum in-clude glasses, crys-tals, and plastics.Of the three, glasshas the most exten-sive historical indus-trial development. Theprimary optical prop-erties of a glass are illustrated on a glass map, whichplots the index of refraction of the glass versus its disper-sion (typically plotted in decreasing V ). Each dot on theglass map represents a different material. Low-dispersioncrown glasses have a V > 55, and high-dispersion flintglasses have a V < 55. Glass makers further divide theglass map into regions based on proprietary chemistry.Historically, lead was used to increase the index of glassduring production. The increased lead content also in-creased the dispersion so that the main sequence of legacyglasses lie on a “lead line.” Glasses far from the lead linetend to be more expensive.

The manufacturer is the ultimate authority on an opticalmaterial. Designs with extreme material sensitivityshould confirm the optical properties with an actualmeasurement or melt certification.

Major glass manufacturers now produce environmentallyfriendly, lead-free glasses and have preferred glassesthat are in high production and have lower cost thanstandard glasses. Other important glass properties notshown on a glass map include transmission, partialdispersion, coefficient of thermal expansion, thermo-opticcoefficient (dn/dT), birefringence, and glass transitiontemperature.



Optical Materials: Polymers/Plastics

Plastics or optical poly-mers are common in high-volume lens systems. In-jection molding is thepreferred method for fabri-cating low-cost lenses, butpolymers can also bediamond-turned or conven-tionally fabricated. Common materials include acrylic(polymethlymethacrylate/PMMA), polycarbonate (whichcan be used as a shatter-resistant material), polystyrene,and cyclic-olefin copolymer (COC/Zeonex). Plastics havematerial properties that are significantly different fromoptical glasses or UV/IR crystals.Advantages• Inexpensive, lightweight material allows extremely

high-volume and repeatable production.• Injection molding allows precision optomechanical

mounting features to be integrated with optical surfaces.• Absorbing dopants or dyes can be included in the optical

material, allowing integrated spectral filtering.• Optical designs can easily incorporate diffractive and

highly aspheric (with inflection points) surfaces.

Many glasses (e.g., Schott B270) can now be injection-molded and are cost-effective complements to polymers.

Disadvantages• Material selection is limited compared to optical glass.• Injection molding requires large tooling costs.• Very low heat tolerance (<100–200 °C maximum).• Spectral bandpass is generally only 0.4–1.6 µm.• High thermal sensitivity: 10× larger CTE and 50× larger

dn/dT than optical glass.• Optical coatings can be difficult to apply.• Very large stress birefringence.• Sensitivity to abrasion and chemical exposure.



Optical Materials: Ultraviolet and Infrared

The selection of high-quality ultraviolet (UV) materialsfor UV applications (e.g., lithography, semiconductorinspection, broadband spectrometers, and fluorescencemicroscopy) is very limited. Most optical materials do nothave good transmission below 400 nm and have increasedbulk scattering. Fused silica (SiO2) and calcium fluoride(CaF2) are the two most-common UV materials. Othermaterials with high UV transmission (e.g., crystal quartz,sapphire, lithium fluoride, barium fluoride, and potassiumbromide) have undesirable qualities for lens fabrication(soft/hard, birefringent, or hygroscopic). Materials withlow visible dispersion are more dispersive in the UV. Manybroadband deep-UV systems are reflective due to the lackof materials for color correction.

Most visible glasses are opaque (have a long-wavetransmission cutoff) above 2.7 µm. Most IR materials areopaque (have a short-wave cutoff) below 1 µm.

Infrared (IR) materials are less limited in selection thanUV materials. Many IR materials are crystalline, anda large fraction are semiconductors. The two main IRspectral regions are the mid-wave (MWIR) band from3–5 µm and the long-wave (LWIR) band from 8–12 µm.Common IR materials are silicon, germanium, CaF2,zinc selenide, and zinc sulfide. AMTIR, gallium arsenide,magnesium fluoride, and barium fluoride are also used. IRmaterials generally have much higher indices (as largeas 4) than visible glasses. Dispersion can be very low,with V numbers as high as 1000 in the LWIR. Thelonger wavelengths reduce surface scatter and roughnessrequirements, making IR materials attractive for cost-effective aspheres. However, the material itself can be verybrittle or soft.

IR materials have much higher thermal expansionand thermo-optic coefficients than visible glasses,and athermalization is required to maintain the lensperformance over large temperature changes.



Snell’s Law and Ray Tracing

Snell’s law (of refraction) is theprimary equation used in ray tracing.It governs the change in direction of aray at a dielectric interface between twomaterials. Snell’s law relates two anglesin one plane of incidence; a full 3D raytrace typically uses direction cosines, the ray’s unitvector projection onto each coordinate axis. Because ofthe sine functions in Snell’s law, ray tracing calculationsthrough multiple surfaces become highly nonlinear forlarge angles. Snell’s law of reflection can also be applied toa reflecting surface if the sign of the index of the incidentmaterial is reversed after the reflection.

Once the ray is refracted or reflected at a surface,it must be translated (or propagated) to the nextsurface, changing its height (or location in 3D) withoutchanging its angle. Paraxial ray tracing assumes thatfor small angles, sin(θ) is approximately equal to θ; thissignificantly simplifies ray tracing equations [replacingthe nsin(θ) with nθ in Snell’slaw]. The resulting paraxialrefraction and a paraxialtransfer equation allow vi-sual brick (or ynu) diagramsor matrix math to be set upquickly to calculate the first-order properties of an opticalsystem.

Ray failures generally occurwhen either a ray fails to interceptthe next sequential surface orthe ray undergoes total internalreflection (TIR) at a surface. Op-tical design codes have significantdifficulty evaluating a lens systemif reference rays (e.g., the chiefray) fail to trace.



Focal Length, Power, and Magnification

Focal length is a funda-mental first-order prop-erty of a lens. For a thinlens, given the object dis-tance and focal length,the Lens maker’s equa-tion can be used toquickly find the imagedistance. The focal lengthalso determines the mag-nification (relative size) between the object and image fora given object position. The power of a lens is the recipro-cal of its focal length. For a thin lens, the power is relatedsolely to its two curvatures and its index. For a thick lens,an effective focal length (EFL) is defined that equalsthe distance between its focal and principal planes ineither object or image space.

The powered surface ofa plano-convex singletmade from glass (n ∼ 1.5)has a convex radius ap-proximately equal to halfits focal length. For thebest performance, the sur-face should be oriented withthe convex side facing thecollimated light or laserbeam.

Objects and images form conjugateplanes. A finite conjugate system hasboth a finite object and a finite imagedistance. An infinite conjugate systemhas either the object or the image locatedat infinity. In an afocal system, both theobject and image are located at infinity.The performance of a lens designed for aspecific conjugate usually degrades whenused at other conjugates.



Aperture Stop and Field Stop

The aperture stop (oftenshortened to just stop) orstop surface is the singleaperture in the lens systemthat limits the bundle of light

that propagates through the system from the axial objectpoint. In the absence of vignetting, the ray bundles fromother object points also fill the aperture stop (“pivoting”about the center of the stop). The aperture stop is typicallya mechanical surface (for good edge definition) and canbe located anywhere in the optical system except at theobject, the image, or an internal image. The size of theaperture stop sets the system f /# and NA. For objects atinfinity, the aperture stop limits the input beam diameter.

A ray fan is a “slice” ofrays that emerge from asingle field point and passthrough a line in the pupil.The tangential fan (Y-fanor meridional fan) passesthrough the pupil’s y axis,and the sagittal fan (orX-fan) passes through thepupil’s x axis.

Field stops also limitthe amount of light thatcan propagate through thesystem from the object

plane to the image plane by setting the maximum objectheight from which a ray bundle can emerge and still passthrough the lens. Field stops are typically the image (orthe object) surface; however, if the system has an internalimage plane, a field stop may be located there (usually forstray light control), and because this plane is conjugatewith the image, reticles, crosshairs, or linear scales areplaced at the internal field stop to be simultaneouslyimaged on the focal plane.



Entrance and Exit Pupils

Wavefront performance for imaging lenses is oftenmeasured at the exit pupil. The exit pupil and theimage plane form a Fourier transform pair underFresnel propagation. Optical design codes calculatepoint spread functions (PSFs) and other diffraction-based metrics using a fast-Fourier transform (FFT) ofthe exit pupil wavefront.

The physical stop sur-face is often buried ina lens system and dif-ficult to access, lead-ing to the definitionof pupils. The imagesof the aperture stopformed by the lens el-ements before and af-ter the stop surface arecalled the entranceand exit pupils, re-spectively. If the stop isat the first (last) sur-face, it is also the entrance (exit) pupil. The entrance (exit)pupil appears to limit the incoming (outgoing) bundle ofrays in object (image) space. Pupil matching is a criti-cal consideration when combining two independently de-signed optical systems. The entrance pupil of the secondsystem should have the same position and diameter as theexit pupil of the first system to avoid losing light and re-ducing the FOV through vignetting. The entrance and exitpupils are images of each other forming conjugate planes,and, consequently, they can have pupil aberrations.

Similar to entrance and exit pupils, internal field stops canbe imaged by the lens elements on either side of the fieldstop, forming entrance and exit windows. However, notall lenses have internal images and, therefore, entranceand exit windows.



Marginal and Chief Rays

Given the surface curvatures, material indices, andelement thicknesses, the marginal and chief rays arethe only two reference rays that need to be tracedto calculate all first-order properties and third-orderaberrations of a lens system.

The marginal ray starts at the base of the object (theon-axis, H = 0, field point) and passes through the edgeof the aperture stop. The marginal ray locates all imagepositions, including any intermediate images; it alsodefines the edge of the ray bundle, and, therefore, themaximum sizes of theentrance and exit pupils.

The chief ray origi-nates from the edgeof the object (the full-field, H = 1, field point)and passes through thecenter of the aperturestop. The chief ray de-fines the maximum FOVof the lens system.Pupils exist wherever the chief ray crosses the optical axis(the entrance and exit pupil are typically virtual; the chiefray needs to be extended “back” until it crosses the opticalaxis). The image size is determined by the height of thechief ray at the image plane.

The chief ray intercept on the focal plane may not definethe center of the energy distribution for a given fieldpoint. In this case, the spot centroid or “center of mass”is a better metric for determining true field position at animage plane. Optical design programs allow a designerto reference various performance metrics (e.g., RMS spotdiameter, distortion) from either the chief ray location orthe centroid location.



Zernike Polynomials

Zernike polynomials are a set of circularly symmetricorthogonal basis functions defined over a unit circle.They are 2D functions of both radial and azimuthalcoordinates. In optical design, Zernikes are used todescribe either surface irregularity or system wavefront(measured in the pupil). Fringe Zernikes are definedsuch that each polynomial has a value of unity at theedge of the unit circle; typically, only a limited 37-termsubset is used, starting at Z1. Standard Zernikes arenumbered differently (starting with Z0) and are usedwhen many more coefficients are needed (e.g., simulationsof turbulence or time-varying statistical phase surfaces).However, either type is capable of representing the samedeformation. Standard Zernikes are normalized such thateach polynomial term has an RMS value of 1 over theunit circle, allowing coefficients to add in quadrature todetermine the overall RMS value. Both types of Zernikescan be renormalized so that they are orthogonal over anannular or otherwise noncircular region.

In tolerancing, Zernikes are useful for modeling complexsurface irregularity (beyond cylinder and spherical)from polishing residuals or mounting effects.

When used to model wavefront errorin the pupil, Zernikes appear identicalto the traditional Seidel aberrations.However, Zernikes only represent thepupil wavefront at one field and arenot true expansions of the point char-acteristic aberration function. SpecialZernike expansions that are functions of both pupil andfield have been developed for lithography applications.



Conic Sections

A surface of rotation formed from a conic section isa special type of asphere. Conic surfaces are definedusing the sag equation and parameterized by the conicconstant κ. Spheres, paraboloids, hyperboloids, andellipsoids are all surfaces formed from conic sections.

Conic surfaces have stigmatic imaging in which onepoint is perfectly focused to another point. In a sphere,the two focal points coincide. In a parabola, one point is atinfinity. Hyperboloids have one real and one virtual focus,whereas ellipsoids have two real foci. The deterministiccorrespondence of foci makes testing conic surfaces easierthan testing a general aspheric surface.

Prolate or “pointy” ellipsoids (football-shaped) areellipses rotated about their major axis and have a negativeκ. Oblate or “squashed” ellipsoids (pumpkin-shaped) areellipses rotated about their minor axis with a positive κ.Oblate ellipsoids are more difficult to test.

Conic surfaces have closed-form analytic solutions for allthird-order aberrations. Conicsurfaces used off-axis are lim-ited by coma and, when usedat nonideal conjugates, suffermainly from spherical aberra-tion. Conic surfaces are very common in multiple-mirrorreflective telescopes.



Diffraction Gratings

A diffraction grating splits in-cident light into different anglesbased on wavelength and gratingspacing d. For a given incident an-gle, the exiting angle for order mcan be calculated with the grat-ing equation. Because neither in-cident nor output angles can be greater than 90 deg, thehighest possible grating order is d/λ. Gratings can betransmissive or reflective. A Littrow grating is a retrore-flective grating with an output angle equal to the incidentangle for one particular diffraction order and wavelength.

Tooling marks on diamond-turned surfaces can bemodeled as gratings for stray light analysis.

Linear gratings can be manufactured by etching orscribing grooves on a glass/metal surface, photo-reducinga regular pattern of lines on a glass plate, or by stampinga master form into a substrate. Curved gratings can alsoact as powered optical elements in complex optical systemsbut are harder to fabricate and are usually reflective. Aholographic grating is formed by interfering two opticalbeams in a photosensitive material.

Light in nondesired diffracted orders can cause significantstray light. Diffraction efficiency is the ratio of powerin a given order to the total incident power. It is primarilya function of grating order and grating “tooth” shape(e.g., continuous or stepped). Blazed gratings have ahigh diffraction efficiency in one order at the design orblaze wavelength. Grating dispersion is the derivativeof output angle with respect to wavelength and is anothermeasure of grating efficiency. Increasing the order m willincrease dispersion but reduce the diffraction efficiency.Free spectral range is thetotal range of wavelengths ina given order that do notangularly overlap those ofanother order.



Optical Cements and Coatings

Achromatic doublets can be airspaced or cemented(where an optical cement is used to bond the achromatat the common internal surface). Airspaced doubletshave significantly increased sensitivity to manufacturingtolerances, so cementing is the preferred option if atransparent cement exists for the wavelength range(e.g., cements that work well with low absorption inthe UV and IR are difficult to find). Canada balsam(n = 1.53 to 1.55) and UV-cured synthetic adhesives arecommon in the visible and work over large temperatureranges. The cement layer must accommodate elementCTE mismatches over the full temperature range thatthe lens is expected to see. A typical thickness for thecement layer is 10–20 µm and is usually not included inthe optical design; however, very high-performing lensesmay need to be optimized with the cement layer in place.Cemented surfaces do not need AR coatings unless theindex difference between the glasses is large.

High-reflection (HR) coatings are needed on reflectiveoptics to improve the base metal reflectivity (e.g., anenhanced dielectric coating can boost the reflectanceof an aluminum mirror from 90% to over 97%).Dichroics are short- or long-wave-pass filters usedfor color balancing or color splitting. Cold and hotmirrors are special dichroics that transmit or reflect,respectively, thermal (long-wave) radiation to protectthermally sensitive optics. Cube beamsplitters require AR

coatings on the entrance and exit facesas well as an internal coating to dividethe beam. Polarizing beamsplitterssplit unpolarized incident light into twoorthogonally polarized beams.

Reflective dielectric coatings generally have very lowabsorption, whereas metal coatings have more absorption.Some reflectors use all-dielectric mirrors to entirelyavoid the absorption from the metal surfaces. Suchmirrors may also provide spectral selectivity.



Detectors: Sampling

Modern lenses are used with periodically sampleddigital staring arrays. The Shannon sampling theoremrequires a minimum of two samples per cycle to perfectlyreconstruct a sampled signal. The Nyquist limit definesthe maximum frequency at which features can be sampledand corresponds to one full white–dark cycle spanning twopixels. Therefore, the highest spatial frequency that canbe detected equals the reciprocal of twice the pixel pitchp. Real systems reliably resolve about 70% of the Nyquistfrequency (roughly 2.8 pixels/cycle). Image content above

the sampling frequency (1/p)will alias and appear as lower-frequency content. An approx-imate sinc-function samplingMTF is often used for systemmodeling and assumes spa-tially averaged random-phasefrequency content.

Not all detectors have rectangular geometry. Fiber opticfaceplates have hexagonal spacing, leading to a samplingMTF with non-Cartesian symmetry.

For the best-performing opticalsystem, the lens and the detec-tor should be “matched” to eachother. A common design tradeis to optimize the ratio of thediffraction-limited optical blurspot to the sampling size. Thisratio is referred to as the Q parameter, which is a com-parison of the Nyquist frequency with the lens cutofffrequency. A Q value of 1 corresponds roughly to 2 pix-els per Airy disc diameter. Q values range from 1–2 formost sensors; higher values of Q correspond to lens-limited(oversampled) imaging, and Q values under 1 correspondto pixel-limited (undersampled) imaging.



Detectors: Resolution

Limiting resolution (LR) is the spatial (or angular)frequency at which an MTF degrades to a desired contrastlevel. The lens is only one part of a complex opticalsystem; besides the diffraction limit and lens aberrations,the detector is often the most significant contributorto the system MTF and limiting resolution. In CCDsfor high-performance systems, a charge-diffusion MTFrepresents a blurring function caused by photoelectronsdiffusing into neighboring pixels. Linear systems theoryallows multiplication of cascaded component MTFs toobtain the final system MTF.

Limiting resolution is often measured subjectively usinga square-wave bar target in units of line-pairs/mm(lp/mm). LR objectively derived from an MTF curve hasunits of cy/mm. Although square-waves and sinusoidsare related by an infinite Fourier series, cycles/mm andlp/mm should not be used interchangeably.

Both periodically sampled detectors (CCD/CMOS arrays)and film have MTF curves with finite resolution. Classicfilm LR (based on chemistry and grain size) can range from80 cycles/mm for panchromatic film to over 2000 cy/mmfor scientific-grade spectroscopic plates. LR for a digitalarray is typically defined by the Nyquist frequency,equal to the reciprocal of twice the pixel pitch. The Nyquistfrequency for a 3-µm pixel is 167 cycles/mm, comparable tohigh-performance photographic film.

Limiting resolution isinsufficient to determineoverall system imagequality because two verydifferent system MTFscan have the same LR. Acomplete MTF curve is amuch better performancepredictor.



The Lagrange Invariant and Étendue

In an optical system, the Lagrange invariant (alsocalled the optical invariant) is a constant that can becalculated at any surface within the lens. At the imagesurface, it equals the product of the image space index,the image height, and the marginal ray (convergence)angle. At the object surface, it equals the product of theobject space index, the object height, and the marginalray (divergence) angle. At other surfaces, it is given by acombination of marginal and chief ray angles and heights:

H = nuy−nuy n2 AΩ=π2H2.

For a fixed image size and f /#, the object size cannot bereduced (increased) without illuminating the object overa larger (smaller) NA.

Étendue (or throughput) is the 3D optical invariantand equals the product of the image (object) area timesthe solid angle of collection (illumination). Étendue isproportional to the square of the Lagrange invariant; it is

a geometric measure of the to-tal brightness of an opticalsystem. The conservation of ra-diance states that the étendueof an optical bundle can onlyremain constant (or increase)through an optical system—itcannot decrease and still obey

energy conservation. Étendue is purely geometric and hasno inherent relation to imaging performance, representingonly the boundaries of the light bundle and not the actualdistribution. An aberrated blur spot will increase étenduebecause the beam area is larger for the same solid angle.

For a flat surface with a uniform solid angle, the étendueequals πA sin2(θ), where A is the area of the surface, andθ is the half angle of the marginal ray.



Illumination Design

Many imaging systems also require optimizedilluminators (e.g., projectors, microscopes, and machinevision systems). The primary goal of illumination designis to create a desired spatial and/or angular distribution ofenergy at a given surface. Illumination systems are alsofound in automobile headlights, spotlights, commercial

lighting, and displays.Collection or concentratorsystems maximize radiationtransfer between a brightsource and a target surface,such as a simple parabolicor elliptical concentrator.

Creating a uniform distribution from a complex sourcerequires homogenization, which can be done bytailoring a known source distribution or by usingsuperposition of many source images. Automotiveheadlights and streetlights use wedged parabolas astailored homogenizers. Light pipes and fly’s eye arraysare examples of superposition homogenizers.

Critical (or Abbe) illumination images the sourcedirectly onto the image, increasing brightness butpotentially adding unwanted image structure (e.g., thelamp filament). In contrast, Köhler illumination imagesthe source into the imaging system pupil for a moreuniform illumination but requires more lenses.

In a microscope, bright field illumination fills the imag-ing pupil either through the objective (epi-illumination)or from underneath the sample (trans-illumination).Dark field illumination puts no light in the imagingpupil—the imaging system collects scattered light, high-lighting edges. Lithography illuminators fill the pupil withcomplex ring or dipole illumination. The illumination co-herence can play a critical role in high-performance sys-tems by shifting the performance limits (partially coherentlight has a slightly different “diffraction-limited” spot sizethan fully coherent light).


122

Equation Summary

Numerical aperture and f /#:

N A = n′ sinu′ f /#= fEPD

Rayleigh criterion:

∆X = 0.61λN A

Image height as a function of field angle:

h′ = f tanθ

Transverse ray error:

ε′y =1

n′u′a

∂W∂ρy

ε′x =1

n′u′a

∂W∂ρx

Strehl ratio:

Strehl∼=(1−2π2ω2

)ω= RMSOPD

Wavefront aberration polynomial:

WIJK ⇒ H IρJ cosK θ

W(H,ρ,θ)=W020ρ2 +W111Hρcosθ+W040ρ

4 +W131Hρ3 cosθ+W222H2ρ2 cos2θ+W220H2ρ2 +W311H3ρcosθ+O (6)

Contrast:

Contrast= Imax − Imin

Imax + Imin


123

Equation Summary

Focal lengths of any two thin lens system:

fa = d ff −BFL

fb = d BFLf −BFL−d

Zero-Petzval solution for two thin lenses:

fa =− fb = f −BFL d = ( f −BFL)2

f

Two-mirror solution:

c1 = BFL− f2d f

c2 = BFL+d− f2d BFL

Schwarzchild solution:

d = 2 f c1 =(p

5−1)

f c2 =(p

5+1)

f

Aplanatic condition:

i =−u′

Bending and shape factors:

β= c1 + c2

c1 − c2C = ua +u′

a

ua −u′a

Thin lens bending for minimum spherical:

c2

c1= 2n2 −n−4

n (2n+1)


124

Equation Summary

Thin lens bending for minimum coma:

c2

c1= (n2 −n−1)

n2

Achromatic doublet:

Φ=φ1 +φ2 φ1 =Φ V1

V1 −V2φ2 =−Φ V2

V1 −V2

Petzval sum:

∑j

φ j

n j

Thick lens power:

φ=φ1 +φ2 −tn

(φ1φ2)

Minimum clear aperture for no vignetting:

CAmin = |ya|+ |yb|

Aspheric sag equation:

sag = z (r)= c r2

1+√

1− (κ+1)(c r)2+d r4 + e r6 + f r8 + g r10 + . . .

Gradient index profiles:

n (r)= N00 +N10 r2 +N20 r4 + . . .

n (z)= N00 +N01 z+N02 z2 + . . .


125

Equation Summary

Merit function:

φ =m

i= 1w2

i (ci − ti)2

Athermalization condition:

d fLensdT

= CTE1d1 CTE2d2

Bireflectance scattering distribution function:

BSDF (θ i,φ i;θo,φ o) =LE

sr−1

Sellmeier dispersion:

n (λ) = 1+c1λ2

λ2 − c4+

c2λ2

λ2 − c5+

c3λ2

λ2 − c6

Schott dispersion:

n (λ) = c1 + c2λ2 +c3λ2 +

c4λ4 +

c5λ6 +

c6λ8

Snell’s law:

nsinθ = n′ sinθ′

Paraxial ray tracing:

n′u′ = nu− yφ

y′ = y + n′u′dn′

φ = c n′ − n


√

√

( )

126

Equation Summary

Lens maker’s equation and linear magnification:

1s′

= 1f+ 1

sm = h′

h= f

s+ f

Thin lens power:

Φ= 1f= (c1 − c2)(n−1)

Diffraction gratings:

mλ= d [sin(θm)−sin(θi)] λblaze = 2d sinα

Sampling ratio:

Q = λ( f /#)pixel pitch

Lagrange invariant and étendue:

H = n uy−n u y n2 AΩ=π2H2

Etendue=Ï

sur f ace

dA aΩ


127

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129

Index

3-point symmetry, 774 f optical system, 58

Abbe illumination, 121Abbe number, 105aberrations, 7absorption, 94achromat, 30achromatic doublet, 30, 55actively athermalized, 88adaptive simulated

annealing, 67adjustment range, 82afocal relay telescope, 58afocal system, 42, 110afocal telescope, 101airspace, 69airspace compensator, 82airspace tolerance, 81airspaced doublet, 31Airy disc, 5alias, 118all-dielectric mirror, 117altitude, 6, 87Amici lens, 103anamorphism, 45anastigmat, 32angle of incidence (AOI),

1, 50angular magnification,

101anisotropic surface, 95anomalous dispersion,

105anomalous partial

dispersion, 55anti-reflection (AR)

coating, 91aperture, 27

aperture specification, 4aperture stop, 90, 111aplanatic lens, 50, 103aplanatic solve, 70apochromats, 30area obscuration, 38arm, 72aspect ratio, 69asphere, 61, 100, 115aspheric coefficient, 69aspheric collimator, 29aspheric departure, 62aspheric sag equation, 61aspheric testing, 62assembly drawing, 75assembly plan, 81assembly print, 75assembly tolerance, 81astigmatism, 21astronomical telescope,

101asymmetrical tolerance,

73, 82, 85athermalization, 108athermalized, 88axial color, 23axial gradient, 63

back focal length (BFL), 6baffle, 90balsam, 117bandwidth, 4barrel distortion, 22base radius, 62beam expander, 101best-fit sphere (BFS), 62bidirectional reflectance

distribution function(BRDF), 95


130

Index

bidirectional scatteringdistribution function(BSDF), 95

bidirectionaltransmittancedistribution function(BTDF), 95

binary optic, 64binoculars, 101biogon, 33Biotar, 33birefringence, 79blaze wavelength, 116blazed grating, 116blue shift, 91bolt-together assembly, 80boresight error, 81boroscope, 104Bouwers telescope, 41brick diagram, 109bright field illumination,

102, 121brightness, 120broadband anti-reflection

(BBAR) coating, 91bubbles, 79Buchroeder system, 41

cam, 48camera lens, 96camera objective, 96Cassegrain telescope, 39catadioptric, 37catadioptric design, 56catadioptric telescope

objectives, 41cemented doublet, 30center thickness (CT), 78centering, 78

central obscuration, 38centroid, 9, 68, 113centroid distortion, 22charge-coupled device

(CCD), 96charge-diffusion

modulation transferfunction (MTF), 119

chief ray, 113chief ray angle solve, 70child ray, 94chromatic aberration, 23clocking, 77coefficient of thermal

expansion (CTE), 87coherence, 121cold mirror, 117cold shield, 93cold shield efficiency, 93cold stop, 93collection system, 121collimator, 101coma, 18compensated, 84compensating eyepiece, 99compensator, 17, 48, 73,

82complementary

metal-oxidesemiconductor(CMOS) detector, 96

compound microscope,102

compounding, 32, 51computer-generated

hologram (CGH), 64concentrator system, 121condenser, 102conic, 61


131

Index

conic constant, 61conic section, 115conjugate factor, 53conjugate plane, 110constraint, 65construction parameter,

73continuous variable, 69continuous zoom, 47contrast transfer function

(CTF), 13Cooke triplet, 32corrector lens, 39cosmetic defect, 79cost function, 65cover slip, 103critical illumination, 121critical object, 90cross term, 85crown glass, 106crystal, 106crystalline, 108cumulative distribution

function (CDF), 86curved grating, 116custom distribution, 83cutoff frequency, 118cycles/mm, 13cylinder, 77

damped least squares(DLS), 66

damping factor, 66dark field illumination,

102, 121decenter, 81default merit function, 68defect, 65defocus, 15

depth of field, 16depth of focus, 16derivative increments, 66derivative matrix, 66derivative tolerancing, 85design margin, 73, 74dewar, 93dialyte, 31dichroic, 117diffraction efficiency, 64,

116diffraction grating, 116diffraction limited, 12, 14diffraction-limited lens,

52diffractive, 100diffractive optical element

(DOE), 64digital camera, 96digs, 79diopter, 98directed distance, 1direction cosine, 109discrete variable, 69discrete zoom, 47dispersion, 23, 55, 79,

105, 116distance-measuring

interferometer, 76distortion, 22double Gauss, 33doubly telecentric, 58drop-in assembly, 80

edge spread function(ESF), 12

edge thickness difference(ETD), 78

edging, 78


132

Index

effective focal length(EFL), 110

ellipsoid, 115elliptical coma, 25empty magnification, 98,

103encircled energy, 9endoscope, 104enhanced dielectric

coating, 117ensquared energy, 9entrance pupil, 112entrance pupil diameter

(EPD), 4entrance window, 112envelope, 6environmental analysis,

3, 87environmental

requirement, 87epi-illumination, 102, 121Erfle eyepiece, 100étendue, 120exit pupil, 112exit window, 112expansion, 45eye relief, 99eyepiece, 98, 99eyepiece design form, 100

fast-Fourier transform(FFT), 112

fictitious glass, 69field clearance, 43field curvature, 19field curve, 21field flattener, 19field lens, 34, 41, 104field of view (FOV), 5

field point, 71field size, 27field stop, 90, 111field tilt, 81field weight, 71fifth-order spherical

aberration, 25finite conjugate system,

110first-order optics, 7first-order solution, 3first-order stray light

path, 90fish-eye lens, 36, 96flange-to-focus, 6, 102flint glass, 106floating element, 35, 96floating stop, 57fly’s eye array, 121focal length, 110focal plane, 110focus, 15, 82focusing, 48Fourier filtering, 58fovea, 98foveal region, 98Fraunhofer doublet, 30,

31free spectral range, 116freeform surface, 61Fresnel reflection, 91fringe Zernike

polynomials, 114full field of view (FFOV), 5full-pitch, 63f -theta lens, 104f -θ distortion, 97f /#, 4f /# solve, 70


133

Index

G-sums, 53Galilean telescope, 101Gauss doublet, 31Gaussian bracket, 48Gaussian distribution, 83Gaussian quadrature

(GQ) sampling, 72genetic algorithm, 67geometrical optics, 7ghost, 92ghost analysis, 92ghost image, 89, 92ghost reflection, 41glare stop, 90glass, 106glass map, 69, 106global optimization, 65, 67global search, 67gradient index (GRIN), 63gradient matrix, 66grating equation, 116grating spacing, 116Gregorian telescope, 40gull wing, 61

Hale Telescope, 40half field of view (HFOV),

5hexapolar sampling, 72high-reflector (HR)

coating, 117higher-order (HO)

aberration, 25hologon, 33holographic grating, 116holographic optical

element (HOE), 64homogenization, 121Hopkins ratio, 14

hot mirror, 117Houghton system, 41Hubble Space Telescope

(HST), 40human eye, 98humidity, 6Huygens eyepiece, 100hyperboloid, 115hyperfocal distance, 15hypergon, 33

illuminated object, 90illumination, 94illumination design, 121illuminator, 121image clearance, 6image format, 96image height, 71image inverter, 98image jitter, 87image plane tilt, 82image-space telecentric,

58imaging telescope, 101immersion objective, 103inclusions, 79index of refraction, 55, 79,

105induced aberration, 26infinite conjugate system,

110infinity-corrected

objective, 102infrared (IR), 108inhomogeneity, 79injection molding, 107interface requirement, 6intrinsic aberration, 26


134

Index

inverse sensitivityanalysis, 84

inward-curving field, 20ISO 10110, 75isotropic surface, 95

Kellner eyepiece, 100Keplerian telescope, 101keystone distortion, 22kidney-bean effect, 60,

100kinoforms, 64knife edge, 90Köhler illumination, 121

Lagrange invariant, 120Lagrange multiplier, 68Lambertian, 95landscape lens, 29Large Binocular Telescope

(LBT), 40lateral color, 24lead line, 106legacy design, 49lens, 2lens bending, 54lens bending parameter,

53lens design, 2Lens maker’s equation,

110lens system, 2lens thickness, 69light pipe, 121limiting resolution (LR),

119line spread function

(LSF), 12line-of-sight (LOS), 87

linear grating, 116linear obscuration, 38linear system theory, 14Lister objective, 103Littrow grating, 116local optimization, 65, 66long-focus lens, 96long-wave infrared

(LWIR), 108longitudinal chromatic

aberration, 23longitudinal color, 23Lyot stop, 90

macro lens, 96magnification, 110magnification solve, 70Maksutov telescope, 41Maksutov–Cassegrain

telescope, 41Mangin mirror, 41manufacturing yield, 86Maréchal criterion, 12marginal ray, 113marginal ray angle solve,

70marginal ray height solve,

70mechanically

compensated, 48melt certification, 106melt-recompensation

process, 82meridional fan, 111merit function, 3, 65, 68Mersenne configuration,

42microscope, 102microscope objective, 103


135

Index

microscope slide, 103mid-wave infrared

(MWIR), 108model glass, 69modulation transfer

function (MTF), 13modulation transfer

function (MTF)bounce, 13

moisture, 6monochromatic, 4monochromatic

aberration, 8Monte Carlo analysis, 73,

86Monte Carlo tolerance, 83

Narcissus, 92, 93Narcissus-induced

temperaturedifference (NITD), 89,93

natural stop position, 57new achromat, 55Newton’s rings, 76Newtonian telescope, 39non-reimaging

three-mirroranastigmat (TMA), 45

nonimaging system, 94nonsequential ray tracing

(NRT), 92, 94nonuniformity correction

(NUC), 93normal dispersion, 105normal distribution, 83normalized field

coordinate, 8, 71

normalized pupilcoordinate, 8

null lens, 39, 64numerical aperture (NA),

4numerical optimization,

65Nyquist frequency, 14,

118, 119Nyquist limit, 118

object angle, 71object height, 71object-space telecentric,

58objective, 98oblate, 115oblique spherical

aberration, 25obscuration, 38off-axis parabola (OAP),

42off-axis rejection (OAR),

89off-axis Schwarzschild

mirror, 46Offner Relay, 43old achromat, 55operand, 65operating range, 87optical axis, 1optical cement, 117optical cut-off frequency,

13optical design, 2optical design process, 3optical design software, 2optical drawing, 75optical invariant, 120


136

Index

optical path difference(OPD), 11

optical polymer, 107optical print, 75optical system, 2optical transfer function

(OTF), 13optically compensated, 48optimization, 3orthoscopic eyepiece, 100outside diameter (OD), 78overcorrected, 24oversampled, 118

packaging requirements,6

parabolic distribution, 83paraboloid, 115paraxial ray tracing, 109paraxial refraction, 109paraxial transfer

equation, 109parent mirror, 42parent ray, 94partial dispersion, 55passive athermalization,

88peak-to-valley (P–V), 77peak-to-valley optical

path difference (P–VOPD), 11

Pegel diagram, 49penalty function, 65performance budget, 73,

74performance prediction,

85periscope, 104Petzval curvature, 20

Petzval lens, 34Petzval portrait lens, 34Petzval projection lenses,

34Petzval radius, 56Petzval ratio, 56Petzval sum, 56phase reversal, 13photographic lens, 96photographic objective,

32, 96pickup solve, 70pincushion distortion, 22pits, 79pixel pitch, 118PLAN microscope

objectives, 32Planar, 33plano-convex singlet, 110plastic, 106, 107Plossl eyepiece, 100point characteristic, 114point spread function

(PSF), 12, 112point-source normalized

irradiancetransmittance(PSNIT), 89

point-sourcetransmittance (PST),89

poker-chip assembly, 80polar sampling, 72polarizing beamsplitter,

117polychromatic, 4power, 77, 110preferred glass, 106pressure, 87


137

Index

primary color, 23prime group, 48principal plane, 110probability distribution

function (PDF), 83probability distribution

table, 86prolate, 115propagation, 109pupil aberration, 59, 60,

112pupil ghost, 92pupil match, 60pupil matching, 112pupil sampling, 72pupil spherical, 60pupil-matched, 99push-around, 18, 82

Q parameter, 118Q-type (Forbes)

polynomial, 61quarter-pitch, 63

radial gradient, 63radiation-hardened, 6radius of curvature

(ROC), 1, 69, 76range of motion, 82ray aiming, 60, 71ray failure, 49, 60, 109ray fan, 111ray intercept failure, 49ray intercept plot, 10ray scattering, 94ray splitting, 94ray tracing, 2Rayleigh criterion, 5

Rayleigh quarter-wavecriterion, 11

receiver, 94rectangular grid

sampling, 72reference ray, 71, 113reflection, 109reflective, 37reflective triplet (RT), 45refraction, 109refractive, 37reimaging, 44relative aperture, 4relative bandwidth, 4relative illumination (RI),

58, 59, 97relay, 98relay lens, 104reproduction lens, 104reproduction ratio, 96resolution, 5retina, 98retrofocus lens, 36reverse telephoto, 31, 36reversing, 52riflescope, 101rifling, 90ring, 72Ritchey–Chrétien

telescope, 39roll, 81root mean square (RMS)

spot diameter, 9root mean square (RMS)

spot size, 9root mean square optical

path difference (RMSOPD), 11

root-sum-square (RSS), 85


138

Index

sag, 61sag equation, 115sagittal, 10, 14sagittal fan, 111sampling frequency, 118sampling modulation

transfer function(MTF), 118

scatter plot, 9scattering, 94, 95scatterometer, 95Schmidt telescope, 41Schmidt–Cassegrain

telescope, 41Schott index equation,

105Schupmann lens, 31Schwarzschild objective,

40scratches, 79second-order stray light

path, 90secondary color, 23secondary spectrum, 23Seidel aberration, 114Seidel diagram, 49Sellmeier index equation,

105semiconductor, 108sensitivity analysis, 84sensitivity matrix, 85sensitivity table/chart, 84sequential ray tracing, 94Shannon sampling

theorem, 118shape factor, 53shim-centered assembly,

80shock requirement, 6

single-lens reflex (SLR),97

singlet, 29skew ray, 72skewed Gaussian

distributions, 83slope error, 62smile, 45Snell’s law, 109solve, 70source, 94space adjust, 17, 82spatial frequency, 13specification document, 2,

3specular direction, 95sphere, 115spherical aberration, 17spherical aberration of

the pupil, 100spherochromatism, 24spherometer, 76spider, 38spiked distribution, 83splitting, 32, 51spot diagram, 9spurious resolution, 13staining, 79standard Zernike

polynomials, 114Steinheil doublet, 30step size, 66stigmatic, 39stigmatic imaging, 115stop, 111stop shift, 57stop symmetry, 57stray light, 89stray light analysis, 3, 89


139

Index

stray light suppression,44

Strehl ratio, 12striae, 79super achromats, 30superposition, 121surface contribution, 63surface irregularity, 77,

114surface roughness, 95surveillance spotting

scope, 101survival range, 87symmetrical tolerance,

73, 82, 85

tailoring, 121tangential, 10, 14tangential fan, 111telecentric lens, 58telecentricity, 58telephoto, 31telephoto lens, 35, 96telephoto ratio, 35temperature, 6, 87terrestrial (spotting)

telescope, 101Tessar, 32test plate, 76test plate fit, 76test plate fringes, 76thermal analysis, 3thermal expansion, 108thermo-optic coefficient,

87, 108thick lens, 3thickness solve, 70thin lens, 3, 53, 110

thin lens aberrationexpression, 53

third-order aberration, 3third-order aberration

theory, 8threading, 90three mirror long (TML),

46three-mirror anastigmat

(TMA), 44three-mirror compact, 46tilt, 81tolerance stack-up, 81tolerancing, 3, 73topogon, 33total indicator runout

(TIR), 78total integrated scatter

(TIS), 95total internal reflection

(TIR), 49tow/keystone, 45trans-illumination, 102,

121transfer contribution, 63translation, 109transverse chromatic

aberration, 24transverse ray aberration

plot, 10transverse ray error, 7trefoil symmetry, 77trial, 86truncated Gaussian, 83tube length, 102tube lens, 102two-bounce ghost, 92

ultraviolet (UV), 108


140

Index

uncompensated, 84undercorrected, 24undersampled, 118uniform distribution, 83unity-magnification

relays, 104unobscured system, 42

V-number, 105V-coat, 91vane, 90variable, 65variator, 48varifocal, 47veiling glare, 89vertex radius, 62vibration, 6, 87viewfinder camera, 97vignetting, 59vignetting factor, 59visual instrument, 98

WALRUS, 46wave aberration function,

7

wavefront, 7wavefront differential, 85wavefront plot, 11wavefront tilt, 16wedge, 78weighted, 68wide-angle lens, 36, 96Wood lens, 63working f /#, 4working distance, 6

y-bar diagram, 53yield, 74yield curve, 86YNI product, 93ynu diagram, 109

Zernike polynomial, 77Zernike polynomials, 114zoom group, 47zoom kernel, 48zoom lens, 47zoom ratio, 47


Julie Bentley is an Associate Pro-fessor at The Institute of Optics,University of Rochester and hasbeen teaching courses in geometri-cal optics, optical design, and prod-uct design for more than 15 years.She received her B.S., M.S., andPh.D. in Optics from the Universityof Rochester. After graduating, shespent two years at Hughes AircraftCo. in California and then twelveyears at Corning Tropel in New

York. She has experience designing a wide variety of op-tical systems for both commercial and military markets,from the UV to the IR, in both refractive and reflectiveconfigurations. She now owns Bentley Optical Design, anoptical design consulting company. Dr. Bentley is a Mem-ber of OSA and an SPIE Fellow.

Craig Olson is a Principal Engi-neer at L-3 Communications. Hereceived a B.S. in Electrical En-gineering from the Georgia Insti-tute of Technology and a Master’sand Ph.D. in Optics from the Insti-tute of Optics at the University ofRochester. He previously worked inthe telecommunications componentsector and the commercial productsectors at JDS Uniphase. In ad-dition to developing sensor system

models, he currently designs, builds, and tests active andpassive multispectral sensor payloads for airborne andground-based imaging systems—occasionally finding timeto design a lens. His current research interests involve thephysics of image formation throughout the complete imag-ing chain. Dr. Olson is a Member of OSA and IEEE, and aSenior Member of SPIE.

bentley field guide to lens design.pdf

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