C19990 SA/FTA1999Aspheric and diffractive optics extend monochromatic imaging limitsDavid ShaferDavid Shafer Optical Design56 Drake Lane,Fairfield,CT.06430phone 203-259-4929,e-mail:Shaferlens@worldnet.att.netAbstractThe highest possible performance in an opticaldesign occurs when the aberrations of eachelement are as decoupled as possible from eachother.This is achieved when each optical surfaceis aspheric and each element also has diffractivepower in addition to refractive or reflectivepower.Some very simple design examples areshown with amazing performance.Key WordsAspheric,diffractiveFigure 1.Triplet with 3-order aberrations=0.0The reason I am showing such an undesirableIntroductiontype of aberration correction will be clear in amoment.Now this design has astigmatic fieldIn all but very simple optical systems there arecurves shown in Figure 2.The tangential raysgenerally enough design variables to correct all ofthen determined by higher-order aberrations andchromatic aberrations.The higher-orderaberrations which limit performance,in a well-corrected design,are usually higher-order fieldcurvature and oblique spherical aberration.Bothof these higher-order aberrations are usuallypresent at every optical surface in a design,butpartially cancel out to give a small net sum for thewhole system.0The higher-order contributions of each opticalsurface can be broken into two components:Figure 2.Field curves for tripletaberrations which are intrinsic to the surface,andaberrations,called induced aberrations,which arefocus considerably short of the image plane,andthe result of aberrations in the light coming intothe surface.Figure 1 here illustrates thisthe amount quickly grows with larger field angles.difference.It shows a triplet lens that has beenThis is higher-order astigmatism,and we can seespecifically designed to have all the 3"-orderthe large focus error at the edge of the field on theaberrations be exactly zero.Nobody would wantray plot in the lens drawing.The tangential rayssuch a lens,because best performance occursare the ones that lie in the plane of the page here.when the unavoidable higher-order aberrationsNormally this higher-order astigmatism would bebalanced out by non-zero 3"-order astigmatism,tothat occur in such a simple design are partiallybalanced out by deliberately introducing non-zerogive a best average focus over the field.Byvalues of the lower-order (the 3"d-order)making the 3"-order aberrations zero here,weaberrations.have deliberately highlighted the higher-orderproblem.C19990 SA/FTA1999Now in this design we have done a littlethe edge of the field is substantially less than it isexperiment.The middle lens was made theon-axis.Figures 4 and 5 show the ray traces andaperture stop.It tums out that in most highthe field curves for just the first lens by itself.performance systems it is not the higher-orderThe astigmatism and Petzval curvature of thisaberrations that are intrinsic to the surfaces thatsingle lens makes both the tangential and sagittalare most important,but rather those that areinduced in the surface by aberrations coming intothe surfaces.I have shown in Figure 3 what2Figure 4.Field curves of first lens aloneFigure 3.Chief ray behaviorhappens when you aim a series of chief rays,fordifferent field angles,at the paraxial entrancepupil.3"-order aberration theory assumes thechief ray goes through the paraxial entrance pupilfor any field angle.But look what actuallyhappens.There is aberration of the chief ray,andit doesn't hit the middle lens at the center exceptfor small field angles.The ray aberration alsobends the chief ray around to be a steeper anglethan what is assumed by 3"-order theory.Thediscrepancy between the actual ray path and thatFigure 5.Rays for first lens aloneassumed by 3M-order theory diverges even moreby the time we reach the last lens.Is it anyrays focus short,compared to the on-axis focuswonder,then that the aberrations of the middleposition.This causes the beam diameter in bothlens and especially the last lens are different fromthe tangential and sagittal directions to bewhat is assumed by 3-order theory-and thereduced,at the middle negative lens,whendiscrepancy increases with larger field angles.compared to the on-axis beam.But this is the nature of induced higher-orderFigures 6 and 7 show the actual beamaberrations.footprints on the middle lens,when seen head-onThe last two lenses do have some higher-orderfor the on-axis case and also at the edge of theaberrations,intrinsic ones,that occur even if thefield.Now this middle lens is the only lens of theray paths were somehow made to be just what 3d.threetriplet components which puts inorder theory assumes.But in this example,theovercorrected spherical aberration to correct formain thing of importance is rather the effectthe undercorrected spherical aberration from theinduced on the lenses by pupil aberration (chieftwo outer positive lenses (most of it coming fromray aberration)coming into the surfaces not bythe first lens).Since the beam diameter is less onany higher-order aberrations that are intrinsic tothat middle lens off-axis,especially for thethe surfaces.tangential rays,we should expect to see that theOne more point before we move on to somenegative lens puts in an insufficient amount ofnew designs.Look at the rays in Figure 1 on thespherical aberration correction for the off-axismiddle lens for the on-axis beam and the off-axisfield points.That is exactly what is observed,inbeam.Clearly the beam diameter at that lens forfact,and is called oblique spherical aberration-aC19990 SA/FTA1999involves both the Petzval curvature of the elementas well as its power.Unfortunately,the Petzvalcurvature of an element is normally very closelyrelated to its power.Aspherics have no effect onthis.What we need is some way to make thepower and Petzval curvature of an optical elementbe completely independent.Then we could obtainthe maximum amount of control that istheoretically possible over induced higher-orderaberrations,and greatly improvedesignFigure 6.On-axis beam on middle lensperformance.It tums out that there is just such a way ofdecoupling the power of an optical element fromits Petzval curvature,and that is to put adiffractive surface on the element.By choosingthe right ratio,for a lens,of refractive power todiffractive power we can make the Petzvalcurvature be anything we want,including havingit be either positive or negative,while notchanging the total power.The catch,and it is apretty big one,is that this normally puts in a lot ofchromatic aberration and may limit the resultingdesigns to nearly monochromatic use.Figure 7.Off-axis beam on middle lensNew Designshigher-order aberration.Clearly it is mainly aninduced aberration here.It is not so much due toWhen diffractive surfaces are added to a design-any intrinsic properties of the middle lens,butand specifically used in the manner I haverather due to the effect of the aberrations,mostlyindicated great performance improvements areastigmatism here,in the light coming into the lenspossible.The design configurations whichfrom the previous lens.A close study of Figure 7achieve the highest levels will not generally bealso shows the presence of some coma of the off-anything like the triplet of Figure 1,which startsaxis pupil,causing the bottom of the ray footprintoff bad,but rather those lens configurations whichto be flatter than the top,which is an effect due toalready have good performance.I described athe previous lens.The result will naturally besome odd looking imbalance in the aberrationOptical Design Conference.By using asphericitycorrection.What we want to do is to somehow geton the surfaces as well as diffractive power,thecontrol over these aberrations inside the designmost performance possible can be squeezed out ofand thereby get control over these inducedthis simple design,without going to moreaberrations that limit the performance of theelements.This design is diffraction-limited,atsystem.What we would ideally like to do is to.6328u,over a 70 degree diameter flat field at f/.9,have the aberrations of each optical element befor a focal length of 10 mm.There is nomade to be whatever we choose.We would like,vignetting,and distortion is well-corrected.Everyfor example,to be able to make the pupilsurface is aspheric,and each of the threeaberrations,like spherical aberration of the chiefelements has an additional superimposedray that we saw in Figure 3,be made for eachdiffractive overlay.It tums out,however,thatelement to be independent of the imagemost of the performance can be achieved with justaberrations of that same element.It tums out,one diffractive surface in the system,on the frontunfortunately,to be theoretically impossible to doof the last element.It would take a great manythat.additional elements to eliminate the asphericsThe main reason,and this involves a veryfrom this design,while keeping the performance,obscure point about aberration theory,is that forbut another design was also shown where theconventional lenses the image aberrations and thepupil aberrations are inter-related in a way thatthe effect of the diffractive surfaces by combiningglasses with a very low index of refraction with