2013 THOSS Media DE GRUYTERD0110.1515/aot-2012.0068Adv.0pt.Techn.2013:2(10:53-62Review ArticleDavid Shafer*Lens designs with extreme image quality featuresAbstract:In order to best assess the importance of new2 Perfect and nearly perfecttechnologies to optical design,it is useful to considerwhat the limits are to what can be done with'old'techno-designslogies.That may show where something new is neededto overcome the limitations of existing optical designs.Figure 1 shows the Luneburg lens from 1944 [1],a gradi-This article will give a survey of some remarkable high-ent index ball with a radially symmetric index gradient.Itperformance designs,some of which are extremely sim-forms a perfect monochromatic image of collimated lightple,and most of which only use technology that has on the back surface of the ball.The light rays are curvedalready been around for decades.Each of these designsinside the ball,and the perfect focus occurs for any fieldhas some limitation that would be nice to overcome.Oneangle incident on the ball.The index of refraction is 1414new technology that will probably revolutionize optical(square root of 2)at the center and 1.0 at the rim of the ball.design will be curved surfaces on image chips.This is not physically realizable but can be approximatedwith structures in the microwave region.Figure 2 shows aKeywords:aberrations;aspherics;optical design.related design,the Maxwell fisheye lens from 1854 [2].Itis also a perfect system,monochromatically,and focusesany point on the surface of the spherical ball onto theopposite side of the ball at unit magnification.The ray*Corresponding author:David Shafer,56 Drake Lane,Fairfield,curves inside the ball are arcs of circles.The refractiveCT06430,USA,e-mail:shaferlens@sbcglobal.netindex in the center is 3.0,and at the rim of the ball,it is 1.5.It is hard to imagine that there could be a simple two-element design that is nearly perfect and with no indexgradient,but Figure 3 shows just such a design [3].1 IntroductionIt is a monocentric catadioptricsystem,and allfour sur-faces,one of them reflecting,have the same center of cur-A survey of some unusual designs,which have extremelyvature.There are only four design variables:two-elementhigh performance under certain conditions,such as a thicknesses and an airspace on either side of the commonsingle wavelength or a curved image surface,can show center of curvature.With these four variables,it is possiblewhat can be done with existing technologies.These to correct for the focal length and third-,fifth-,and seventh-designs can then indicate what areas of technology need order spherical aberration.Higher orders beyond that areto be developed to further extend the limits of what is extremely small.The glass can be any typical optical glass.possible.There are very few designs that are perfect orIf the aperture stop is at the common center of curva-nearly perfect,and that is a good place to start this short ture of this monocentric design,then,there are no fieldsurvey.Here,'perfect'means just that-it is an optical aberrations.The image is curved and also has the samesystem,like a flat mirror,which has no optical aber-center of curvature as all the surfaces.Now,here is arations at all.A flat mirror,however,makes a perfect remarkable feature ofthis design.The higher-order spher-virtual image of a real object,and that is an example of ical aberration is so small that this design works very wellthe kind of limitation that we would like to overcome.It at 0.99 NA.When optimized for the wavefront,a 099-NAis much more desirable to have a perfect real image of a design with a 20-mm focal length has an r.m.s.OPD ofreal object.0.0035 wave at 0.55 u,assuming a glass index of about1.6.The performance goes to about 0.006 waves r.m.s.When the design is reoptimized for the lower glass indexof n=1.5.If the NA is extended to 0999 NA,the wavefronthardly changes.Because of that,this design has the verywww.degruyter.com/aotDownload Date 8/13/13 3:01 AM54 -D.Shafer:Lens designs with extreme features2013 THOSS Media DE GRUYTERto the pupildiameter and its curved image surface.The varia-tion of the orders of spherical aberration with these very fewmonocentric design parameters is extremely nonlinear,andit is quite difficult to fully optimize this design.There is a new perfect optical design [4,which is theultimate in simplicity-a single optical element-and alsothe ultimate in image quality,with no monochromaticaberrations.It is shown in Figure 4,and it has perfectimage quality at NA=1.0,in air,and at NA=3.0 with animmersion focal surface.This design is simply a gradientindex Maxwell's fisheye lens that has been cut in half andthen given a reflective outer surface.By means of a verysimple geometrical construction,it is possible to provethat this new design,just y2 of Maxwell's design and witha reflective outer surface,is a perfect design with no geo-metrical aberrations of any order.It images perfectly,toFigure 1 Luneburg gradient index lensa point,any point on the flat surface to another point onthat same surface.The radial gradientindex goes from 3.0unusual feature of not needing an aperture stop.Totalat the center to 1.5 at the outer surface.intemal reflection at the second surface as the design NAPrior to this new design,there were only four knownapproaches 1.0 will effectively define an aperture stop,perfect optical systems.The Luneburg gradient indexand reflection losses will apodize the pupil near its rim solens and the Maxwell fisheye lens both form real imagesthat there is very little energy beyond 0.99 NA.on a curved surface.Another perfect system is the singleThe Figure 3 picture shows the design with a 20 field.surfaceaplanatic surface,a spherical surface betweentwoThe obscuration due tothe image surface is very small,about different index of refraction materials,with an object and15%diameter for a 20 full field 099 NA design.The perfor-image conjugate ratio that is the same as the ratio of themance is the same for any field angle.This is a monochro-refractive index on opposite sidesof the aplanatic surface.matic design,but it could be achromatized by adding twoThis gives perfect imagery,but both the object and imageburiedsurfaces'to thedesign.The main practicallimitations are curved,and one conjugate is virtual.The last of theof this remarkable system are that its size is large compared well-known perfect optical systems is a flat miror.This isthe only one where both object andimage can be flat,withperfect imagery,but one has to be virtual.This new design here is the first new perfect opticalsystem in many decades and has both the object and imageflat and real,with no virtual conjugate.Of course,there isno practical way right now to make this large change in theindex gradient,but the design shows that gradient index isa very powerful design tool,and more complicated designsmay not require such large index differences.The design isonly perfect monochromatically,but there is a partial equiv-alent to this system in the Wynne-Dyson design [5],whichdoes have color correction and does not use gradient index.Figure 5 shows how two of theseelements can bejoinedso that the input and output are in opposite directions.3 Curved image designsAllowing a design's image to be curved can greatly sim-Figure 2 Maxwell Fisheye gradientindex lens.plify the design complexity required to meet variousDownload Date 8/13/13 3:01 AM2013 THOSS Media DE GRUYTERD.Shafer:Lens designs with extreme features -55Monocentric designUNITS:MFOCAL LENGTH =-20 NA 0.99DES:Shafer27.5Figure 3 A nearly perfect 0.99-NA monocentric design.performance goals.Figure 6 shows a design that is f/1.0Flat objectFlat imageis 1.0 arc second over the whole 20 field with no vignet-ting,so the design is diffraction-limited for a 150-mm focallength at 0.55 u.The best performance happens when thelast lens surface is right up against the curved image,but this design here is still good with a 2.0-mm workingdistance.lens,much less glass path,correction for axial and lateralcolor,and the same monochromatic correction as before.Designs like this have worked in the past with fiber-opticFigure 4 Perfect 1.0X gradient index catadioptric design.field flatteners and image tubes,for night vision appli-focal length.The example shown is f/1.0 and a 140 fieldcations.This design,however,has a very much higherwith no vignetting and nearly uniform illumination.Therecorrection level than prior designs and would be a greatmatch to a curved surface detector array like that recentlyare some very aspheric elements.The use of designs withan intermediate image is a growing trend and led to somediscussed by Dumas et al.7.remarkable designs,like a 300 to 1 zoom ratio system,among others.4 Extremely large NA and fieldangle combinationsThe closest design to a perfect system that does not usea gradient index and which does not require a curvedobject or image is probably one of the examples given byBrian Caldwell [8]that combines high NA and a large fieldshows one example,and as Caldwell points out,this isonly possible because of having an intermediate imagein the design.His remarkable designs achieve this greatcombination of high NA and field angle at the expenseof a very,very large system size compared to the systemFigure 5 Opposite input and output directions.Download Date 8/13/13 3:01 AM