The fundamental objective of optical design is to findthe best system that satisfies the requirements asso-ciated with a given application. Many factors candetermine the best system, but they typically involvethe first-order image properties, including the speedand the field of view as well as the higher-order ab-errations. Often the standard is a figure of meritdescribing the average amount of wave-front error atthe exit pupil or geometrical spot size at the image.Other design criteria, however, can be more difficultto quantify. Consider the packaging of an opticalinstrument, i.e., the size and the location of the indi-vidual optical elements with respect to other compo-nents in the overall system. Generally, someexternal knowledge beyond the optical surfaces isnecessary for performing such an evaluation. Inspace optics applications, for example, numerous in-struments and subsystems compete for limited roomaboard the spacecraft, often making packaging con-straints a critical issue. Self-obstruction is anotherform of a packaging constraint, and its removal isoften a challenge during the design of reflective sys-tems. For these issues and other criteria that are
J. M. Howard [email protected]! is with the Na-tional Research Council, NASA Goddard Space Flight Center, Op-tics Branch ~Code 551!, Greenbelt, Maryland 20771.
Received 29 September 2000; revised manuscript received 19March 2001.
difficult to quantify, the most straightforward way toprovide an initial evaluation is by actually viewing adrawing of the system itself and letting experienceguide the designer. This paper discusses this visualdesign process, in which systems are modified andgraphics presented in real time to help the designerdetermine the best system for the application athand.
The methods presented here employ the followingdesign paradigm. First, image requirements are de-fined in accordance with the application, such as thespeed and the field of view. Then a general class ofsystem is chosen. This selection establishes thenumber of elements, whether they are reflective orrefractive ~or both!; their glass types ~if applicable!;nd whether aspheres, diffractive surfaces, or evenolographic elements will be used. An example of ahosen class of system would be a telescope composedf three spherical mirrors. After these big-picturetems have been settled, constraints are determinedn individual construction parameters to ensure thatome or all the defined image requirements are met.hese constraints reduce the available degrees of
reedom, which make for a more efficient search. Fi-ally, a detailed investigation commences. This it-rative process starts by determining all constructionarameters in the system ~both variable and con-trained!, such as the curvatures and separations ofhe surfaces, as well as the parameters associatedith any aspheres or unconventional elements. The
ystem is analyzed numerically according to somerror function ~a.k.a. merit function!, and a systemrawing and other plots are rendered. The graphics
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are then evaluated visually by the designer to seewhether the system is the best for the given applica-tion. If not, the procedure is repeated by use of anew set of construction parameters until the bestsystem ~or one that is good enough! is found.
An essential element to the efficiency of this designparadigm is the reduction of the number of construc-tion parameters to the minimum possible. For sys-tems with rotational symmetry there are standardmethods, or tricks, that are typically used. Theseinclude placing a “solve” on a particular curvature orthickness, which positions the image plane at ~or atleast near! the paraxial image location or enforces adesired magnification or focal length. Note that aparameter with a solve placed on it is no longer avail-able to the designer as a degree of freedom. Ad-vanced techniques include surrendering variables toeliminate some combination of the higher-order ab-errations ~e.g., choosing an aspheric coefficient toeliminate spherical aberration!. To summarize,some subset of system parameters is constrained toensure that a given set of appropriate first- ~and pos-ibly higher-! order image properties is obtained.
This effectively prevents any system with the wrongset of image properties from ever being consideredand reduces the degrees of freedom with which thedesigner is confronted. Imaging constraints also al-low the designer to explore the neighboring configu-ration space for new systems with identical imageproperties.
Once the number of degrees of freedom is mini-mized, many search methods are available to find thebest system. The most thorough is to perform a sys-tematic global search over all available variables.1A second and less time-consuming approach is to con-sider a previously known system and then modify itlocally in the configuration space to meet the imme-diate needs. For brevity, only the second approachis used in the examples presented here.
Note that an underlying assumption in using thisdesign paradigm is that an optimized system accord-ing to some standard numerical error function maynot the most desirable. This is often the case forsystems with packaging constraints or other consid-erations not specifically accounted for in the errorfunction. As such, the designer is included in theoptimization process through real-time feedback ofgraphics and performance data with minimal mo-tions necessary to change the input parameters forthe system. This automation removes the designerfrom the repetitive details of using design softwarewhile allowing for efficient observation and control ofthe evolving design itself.
Three examples are presented. The first is in Sec-tion 2 and considers the design of a rotationally sym-metric microscope objective composed of twoachromatic doublets. In Section 3 the second exam-ple investigates an unobstructed telescope composedof three spherical mirrors, and in Section 4 a unitmagnification relay is modified. Concluding re-marks are given in Section 5.
The macro routines presented in this paper are
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written for OSLO lens design software. In eachexample the degrees of freedom are assigned to agraphical user interface referred to as a slider. Thedesign parameters are modified by means of movingthe sliders to the right and the left, which raises andlowers their value, respectively. After each slidermovement, a callback routine is executed that auto-matically updates the system and graphical output.A copy of the macros demonstrated in this paper canbe obtained by contacting the author.
2. Microscope Objective
This student exercise and accompanying macro weredeveloped by B. D. Stone as an educational tool for asummer course in geometrical optics.3 It illustratesthe design and analysis of a microscope objective us-ing two achromatic doublets. One of the goals of theexercise is to give the student intuition in designingan optical system by use of actual lens design soft-ware but without the burden of having to learn manyof the details in using the software itself. In Sub-section 2.A the design problem is presented accordingto the paradigm outlined above. This is followed byillustrations and discussion of one of the student ex-ercises in Subsection 2.B.
A. Generate Imaging Constraints
The microscope objective is specified to have the fol-lowing requirements: tube length of 200 mm, mag-nification of 10.0, object size of 1 mm, and numericalaperture ~N.A.! in object space of 0.25. As typicallydone for microscope objectives, the system is designedin reverse ~i.e., object and image planes are inter-hanged!. These requirements equate to a systemith a focal length of 20 mm, or a total power of 0.05.The chosen class of system is an objective composed
f two cemented achromatic doublets using BK7 andF5 glass ~each in that order!. As such, this system
nitially presents 14 degrees of freedom to the de-igner: the locations of the stop, the doublets, andhe image; and the three curvatures and two thick-esses associated with each doublet. Four con-traints are built into the system to ensure that themage requirements are met and to reduce the totalumber of degrees of freedom. The constraints areummarized as follows ~details can be found in Ref.!. First, the power of one of the doublets can bepecified, whereas the power of the other doublet var-es to keep the total power of the objective fixed.his removes 1 degree of freedom, where the power ofach doublet ~whether chosen or determined! is en-ured by means of constraining a single curvature inhat doublet. Next, two degrees of freedom are re-oved by means of constraining an additional curva-
ure in each doublet to ensure that it is achromatici.e., has zero longitudinal color!. Finally, since theube length and magnification of the system are sety the design requirements, the front focal point ofhe objective is fixed as well. This determines thehysical location of the objective itself, hence con-training the thickness of the airspace between thebject and the first surface of the system. Thus ten
degrees of freedom remain: the curvatures of the firstsurface of each doublet, the thickness of each glass inthe doublets ~totaling four!, the power of the firstdoublet, the spacing between the doublets, the stoplocation, and the amount of defocus from the paraxialimage location.
B. Optimize with Computer Graphics
Using this macro, the student is able to adjust theavailable free construction parameters as well as thedesign specifications such as the system magnifica-tion, a wavelength range for evaluation, and a choiceof illustrating spot diagrams or transverse ray-errorplots. Each of these parameters is changed bymovement of the associated slider illustrated in Fig.1. This action runs a callback routine that automat-ically recalculates the constrained parameters of thesystem, and then it updates the drawing of the sys-tem and the plots. Note that the details of operatingthe lens design software are buried into the macrolanguage, leaving the student with simple point-and-click activities that let them concentrate on the lessonat hand.
In one exercise the student starts from the systemshown in Fig. 1 and must find the best system bychanging the curvature of the first surface of the firstdoublet ~i.e., moving the fourth slider from the top!and then add defocus if beneficial. After eachchange of the slider value, the system drawing andthe transverse ray-error plots are updated, showingthe user in real time how this change affects systemperformance. Figure 2 is a static illustration of thisexercise showing four transverse ray-error plots:
Fig. 1. Sliders, system drawing, and transverse ray-aberration paberration plots are automatically updated when the sliders are m
the first three after each change in curvature and thefourth with some defocus added. Note the smallerscale in plots 3 and 4.
As stated above, a significant advantage of thisteaching tool is that the student need not be familiarwith the lens design program and all the commandsneeded to enter a lens and produce the plots. Allthat is needed are instructions to start the programand the macro routine, and then how to operate thesliders. Hence the emphasis is on the lens designconcepts revealed by illustration of how the perfor-mance of the optical system changes in a dynamic,interactive environment. Note that the added di-mension of moving pictures emphasizes these chang-ing trends more readily than does a slide show ofindividual plots, such as those illustrated here in this
or an achromatic microscope objective. The system drawing and.
Fig. 2. Optimization of microscope objective with computergraphics. Ray-error plots for on-axis meridian rays are shown forchanging curvature of the first doublet. The horizontal axes arenormalized pupil coordinates, and the vertical axes are ray errormeasured in millimeters. Three wavelengths ~red, green, andblue! are included in each plot. In plot 4, defocus is added.
lots foved
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static journal paper. Although only a single-parameter graphical optimization was shown here,many other lens design and geometrical optics con-cepts can be taught with educational tools such asthese. As an aside, student response has been verypositive.
3. Unobstructed Three-Spherical-Mirror Telescope
The usefulness of visual design is not limited to ed-ucation alone. In this example the design for a tele-scope from a study presented in the journal paper“Imaging with three spherical mirrors”4 is altered.For the purposes of this example the starting designhas the desired image quality, but it lacks the work-ing space necessary for the mounting structure be-hind the third mirror. To alleviate this packagingproblem, we modify the system by exploring the localconstrained configuration space while visually ana-lyzing the changes in the system.
A. Generate Imaging Constraints
The design paradigm outlined in Section 1 is similarto the one used in Ref. 4. In this case, only planesymmetry is assumed, and the systems are con-strained to provide ƒy10 imaging with three sphericalmirrors ~e.g., with an entrance beam diameter of 10.0cm and a focal length of 100.0 cm!. Nine construc-tion parameters define the system: three mirrorcurvatures ~c1, c2, c3!, three mirror tilt angles ~t1, t2,3!, two mirror separations ~d1, d2!, and the imageistance ~d3!. ~For the purposes of this example, themage tilt is fixed, and the stop is fixed at the first
irror.! These parameters are measured with re-pect to a central ray passing through the system—he base ray. To ensure an ƒy10 imaging system,our construction parameters ~c2, c3, d2, d3! are re-
oved as degrees of freedom—two associated withhe focal lengths and two associated with the first-rder blur.5 The constraints are determined by
Fig. 3. Sliders and system drawing for a three-mirror telescopeincoming beam.
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means of expanding the image properties in a Taylorseries about the base ray and then equating the an-alytic expressions for the first-order coefficients to thedesired first-order image properties ~ƒy10 focalengths and zero blur!. These four equations arehen solved simultaneously for c2, c3, d2, and d3,esulting in expressions for the constrained parame-ers as explicit functions of the five remaining de-rees of freedom ~c1, d1, t1, t2, t3!. As an aside,rst-order design about a base ray with arbitraryymmetry is referred to as parabasal optics and isnalogous to paraxial methods used for rotationallyymmetric systems.
B. Modify System with Computer Graphics
Once the constraints have been determined, a macroroutine is written for the three-mirror system thatautomatically calculates the dependent parameters,updates the lens data, plots the system, and deter-mines the root-mean-square ~rms! spot radius of thebasal image point. The sliders for this macro and aplot of the starting system are illustrated in Fig. 3.Note that the packaging of this system is not ideal,because the third mirror is too close to the incomingbeam to provide sufficient working space for a mount.As such, a manual search is performed for a bettersystem with sufficient working space and good imagequality. This is done in real time by means of mov-ing the sliders to change the parameters while ana-lyzing the systems by eye as the system evolves.
To illustrate this search, systems A, B, and C inFig. 4 show how the system evolves by changing thevalue of t3 a few clicks on the slider from 218.0 to216.0 and then to 214.0. Recall that changing theindependent parameter t3 also changes the values ofthe four dependent parameters ~c2, c3, d2, and d3!.In this case changing t3 causes d2 to lengthen suchthat the third mirror initially obstructs the incomingbeam in system B and then clears it on the other side
ote the limited space for mounting the third mirror due to the
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in system C. An evaluation by eye clearly showsthat the final system has a desirable configurationwith regard to the working space for the third mirror.Although the image quality slightly degrades, as seen
Fig. 4. Modification of three-mirror telescope with computergraphics. The tilt on the third mirror t3 is adjusted from 218.0 to216.0 to 214.0 as illustrated in systems A, B, and C, respectively.As a result, imaging constraints on the system move the third mirrorabove the incoming beam, providing the necessary space for mount-ing the optic. Although the numbers representing the rms spotradius of the basal image point show a reduction in image quality,optimizing the three curvatures of system C more than compensatesfor this loss with little change in system configuration.
Fig. 5. Sliders and system drawing for a concentric three-mirror rthe second mirror.
by the increase in the rms spot radius of the basalimage point shown next to each figure, it turns outthat downhill optimization of the three mirror curva-tures easily makes up for this loss.
Note that, by performing this manual search, adesigner is able to cross a region in the constrainedconfiguration space that a downhill optimizer ~withbstruction penalized! would not. This demon-
strates that new systems can be found and exploredby use of this visual design technique. In somesense one can call this a regional optimizationmethod, or a regional explorer.
4. Three Concentric Spherical Mirrors
In this final example the design of a unit magnifica-tion relay is modified to meet the packaging require-ments associated with an internal stop. Thisresearch was performed for a design study on thenear-infrared camera ~NIRCAM! for NASA’s Next
eneration Space Telescope ~NGST!.6 The base de-sign for the camera is the Offner relay,7 which iscomposed of three concentric mirrors configured suchthat the first and third are concave and have twicethe radius of curvature of the second mirror, which isconvex. Note that the two concave mirrors can berealized by a single mirror surface, as illustrated inFig. 5. One packaging requirement of NIRCAM isthat access be available to the internal stop to placecomponents such as filters and a pupil mask. How-ever, in an Offner relay that is telecentric in objectspace, the internal stop is inaccessible, since it islocated next to the surface of the second mirror. Toalleviate this dilemma, the radii of curvature of themirrors are changed while the underlying concentricnature of the system is maintained.
. Note that the internal pupil is inaccessible since it is located on
elay
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A. Generate Imaging Constraints
For the purposes of this example the NIRCAM designis specified to have a 100 mm 3 100 mm object, anf-number of 24.0, and a telecentric entrance pupil. Abase ray is defined for the system, which originates atthe center of the object and traverses the center of theentrance pupil. For simplicity, the object and theimage are assumed to be planar and normal tothe base ray. Since the three mirrors are concentric,the system can be modeled as rotationally symmetricbut with an off-axis object and pupil. The axis isdefined as the line passing through the centers ofcurvature and parallel to the base ray in object space.
Six construction parameters are necessary to de-fine this system: the radii of curvature of the threemirrors ~r1, r2, and r3!, the object distance ~d0!, themage distance ~d3!, and the height ~h! measuring the
inimum distance from the basal object point to thexis. Two construction parameters are constrained,eaving four available degrees of freedom. In therst constraint the radius of the third mirror r3 isnalytically solved to give zero Petzval curvature forhe system. In the second the image distance d3 isownhill optimized to minimize the spot radius of theasal image point. Figure 5 shows the starting val-es for the remaining degrees of freedom ~r1, r2, d0,nd h!. These values were chosen as a typicalffner design with the approximate dimension de-
ired for NIRCAM. Note that each of the parameteralues on the sliders is scaled to r1.
B. Unobstruct Stop with Computer Graphics
In system D illustrated in Fig. 6 the stop is initiallylocated at the second mirror, and the common centerof curvature of the mirrors is labeled O. The num-ber next to the image is the maximum rms spot ra-dius of nine image points sampled over the entireimage. Since the camera is required to have accessto the internal stop, the radius of the second mirror r2is initially reduced to remove the obstruction. Be-cause the system stays concentric, this action movesthe mirror to the left toward O, as illustrated by thearrows shown in system E. Since r3 is constrainedto zero the Petzval curvature, it will change as well.After further reduction of r2, access to the stop isdetermined by eye as seen in system F. With theobstruction eliminated, a final adjustment is made tothe object distance to compensate for the degradedimage quality over the field. The result is that sys-tem G has less than half the maximum rms spotradius as system D.
The smoothness of the constrained configurationspace in this example is largely due to the con-straint on r3 to zero the Petzval curvature. Had r3been free to vary, the chosen figure of merit ~max-imum rms spot radius! would have quickly de-graded because of the image curvature introduced.After each adjustment of r2 a similar adjustment ofr1 or r3 would have been necessary to flatten themage over the field. This illustrates that placing
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onstraints such as these on the construction pa-ameters is an essential element to the effective-ess of visual design. Without them, the sliderserform only a sort of tolerance analysis, making itifficult for the designer to find new and viableystems.
Fig. 6. Modification of three-mirror relay with computer graph-ics. The point labeled O is the common center of curvature, andhe maximum rms spot radius over the field is included in eachystem. To make the internal pupil accessible, the radius of theecond mirror ~slider r2yr1! is reduced from 0.50 to 0.46 to 0.41, ashown by systems D, E, and F, respectively. During this process,he radius of the third mirror r3 also reduces since it is constrainedo zero the Petzval curvature. A final adjustment is made in thebject distance in system G to improve image quality.
5. Concluding Remarks
In the examples presented here only one degree offreedom could be altered at a time because of theone-dimensional nature of the graphical slider tool.The natural follow-on to the graphical slider would bea graphical rectangle, or a “rink,” in which the posi-tion in the rink would provide changing two-dimensional input for the visual design macro. Thiswould give the designer the ability to explore in realtime a two-dimensional slice in a multidimensionaldesign space. Going even further, a hand-held“wand” with three-dimensional position sensingcould provide additional input to the visual designprocess.
The automation of visual design is not limited tojust system drawings and analysis plots. One ex-tension would be to include downhill optimizationover several of the construction parameters, whichwould effectively break the system out of its con-strained configuration space. That is, the systemwould first be defined by its degrees of freedom andconstraint equations; then some set of variableswould be assigned and the system optimized. Withappropriate variables the overall performance of thesystem can improve significantly without fundamen-tally altering the smoothness of the design space asdefined by the original degrees of freedom. Thisidea was incorporated in the macro for the three-mirror relay, in which an automatic focus feature ofthe design software was employed to optimize theimage distance. Another application of this ideawould be to vary the curvatures of the three-mirrortelescope to optimize the performance over the field,as discussed in the caption of Fig. 4.
The visual design methods discussed here arebased on analytical approaches to the design of opti-cal systems. Fortunately, computer algebra soft-ware programs are available to assist in solving thesometimes-lengthy constraint equations. With theever-increasing power of modern machines, sophisti-cated macros could be developed to design entireclasses of optical systems. They would range fromsimple laboratory configurations, such as imaging aninterferometer onto a CCD, to more-complicated ap-plications, such as the design of a telephoto cameralens. Reflective systems are especially suitable for
these methods, since they typically consist of rela-tively few surfaces, which can simplify the necessaryprogramming.
These graphical tools are not universal. Althoughmost lens design software has macro language sup-port, not all ~at this time! have the ability to providean interactive user environment with associated call-back routines. Just as computer operating systemshave evolved into primarily graphical user interfaces,optical design software should follow suit in a timelymanner. This includes incorporating not only fea-tures such as the sliders presented here but othergraphical interfaces as they appear elsewhere in thecomputer-aided-design software industry.
I am thankful to the National Research Council forsupporting this research. Additional support wasprovided by the Optics Branch of NASA GoddardSpace Flight Center and the Next Generation SpaceTelescope project. Personal thanks go to Bryan D.Stone and Timo T. Saha for their discussions andinput.
References and Notes1. For an example of a systematic global search, see J. M. Howard
and B. D. Stone, “Imaging a point with two spherical mirrors,”J. Opt. Soc. Am. A 15, 3045–3056 ~1998!.
2. OSLO is a registered trademark of Lambda Research Corpo-ration, 80 Taylor Street, P.O. Box 1400, Littleton, Mass. 01460.
3. B. D. Stone, “Supplement to lecture notes: lens design exam-ple,” in Geometrical Optics, The Institute of Optics SummerCourse Series ~University of Rochester, Rochester N.Y., 1999!.
4. J. M. Howard and B. D. Stone, “Imaging with three sphericalmirrors,” Appl. Opt. 39, 3216–3231 ~2000!.
5. In a plane symmetric system there are two focal lengths: onefor rays in the plane of symmetry and one for rays outside ofthe plane of symmetry. Also, it is possible for rays in theplane of symmetry to be defocused from rays out of the planeof symmetry, causing what I refer to as first-order blur. Thusfour constraints are needed to control each of these first-orderproperties.
6. For a summary of the yardstick design of the NIRCAM on theNGST, see Chap. 7 of The Next Generation Space Telescope:Visiting a Time When Galaxies Were Young, H. S. Stockman,ed., 2nd ed. ~Association of Universities for Research in As-tronomy, Washington, D.C., 1998!.
7. A. Offner, “Unit power imaging catoptric anastigmat,” U.S.patent 3,748,015 ~24 July 1973!.
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