Optical Performance Factors

Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.11 2 v www.cvimellesgriot.com 1 v Fundamental Optics Fundamental Fundamental Optics

wavelength l Refraction of light at a dielectric boundary 1 2 n n Melles Griot applications engineers are able to provide a Melles Griot applications engineers are able to material 1 index material 2 index APPLICATION NOTE APPLICATION Technical Assistance Technical is Detailed performance analysis of an optical system software. accomplished by using computerized ray-tracing CVI If you systems. ray-tracing analysis of simple catalog-component of your optical need assistance in determining the performance your particular system, or in selecting optimum components for application, please contact your nearest CVI Melles Griot office. Figure 1.14 Figure , (1.20) is the angle of refraction, and both angles 2 v 2 sinv 2 n alignment, impact the performance of an optical system. alignment, impact the = 1 is the angle of incidence, is the angle of incidence, 1 v sinv 1 n are measured from the surface normal as shown in figure 1.14. After paraxial formulas have been used to select values for component focal to select values formulas have been used After paraxial As in any is to select actual lenses. diameter(s), the final step length(s) and selection process involves a number of tradeoffs engineering problem, this where ABERRATIONS determine the precise performance of a lens system, we can trace the To at each optical interface law path of light rays through it, using Snell’s called ray tracing, process, to determine the subsequent ray direction. This When this process is completed, is usually accomplished on a computer. it is typically found that not all the rays pass through the points or posi- deviations from ideal imaging These tions predicted by paraxial theory. are called lens aberrations. at the interface between direction of a light ray after refraction The index of refraction is given isotropic media of differing two homogeneous, law: by Snell’s DIFFRACTION poses nature, its wave Diffraction, a natural property of light arising from Diffraction is always a fundamental limitation on any optical system. if the system has significant present, although its effects may be masked When an optical system is essentially free from aberrations, aberrations. and it is referred to as its performance is limited solely by diffraction, diffraction limited. the focal length(s) and In calculating diffraction, we simply need to know factors such as aperture diameter(s); we do not consider other lens-related shape or index of refraction. and aberrations decrease Since diffraction increases with increasing f-number, determining optimum system performance with increasing f-number, of these factors often involves finding a point where the combination has a minimum effect. Performance Factors Performance cost, weight, and environmental factors. including performance, real optical systems is limited by several factors, performance of The magnitude of these and light diffraction. The including lens aberrations with relative ease. effects can be calculated such as lens manufacturing tolerances and Numerous other factors, component considered explicitly in the following discussion, Although these are not that if calculations indicate that a lens system in mind it should be kept in practice it may fall short only just meets the desired performance criteria, In critical applications, of this performance as a result of other factors. performance is it is generally better to select a lens whose calculated significantly better than needed. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.11 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:28PMPage1.12

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics v by developingamethodofcalculatingaberrationsresultingfromthe excessively tediousandtimeconsuming.Seidel*addressedthisissue real lenssurfaces. Beforetheadventofelectroniccomputers, thiswas As alreadystated,exactraytracingistheonlyrigorousway toanalyze the paraxialprediction. theory, aberrationsarereallyameasureofhowtheimagediffersfrom would formitsimageatthepointandtosizeindicatedbyparaxial rations. Becauseaperfectopticalsystem(onewithoutanyaberrations) real performancebecausesinv marginal rays),paraxialtheoryyieldsincreasinglylargedeviationsfrom high f-numberlenses).Withmorehighlycurvedsurfaces(andparticularly The assumptionthatsinv this approximationusingtheparaxialformulas. expansions areused.Designofanyopticalsystemgenerallystartswith called first-orderorparaxialtheorybecauseonlythefirsttermsofsine 1.12 von* Ludwig Seidel,1857. with theirarguments(i.e., replacesinv The firstapproximationwecanmake istoreplaceallthesinefunctions functions inSnell’s lawcanbeexpandedinaninfiniteTaylor series: The firststepindevelopingtheseroughguidelinesistorealizethatthesine better startingpointforanyfurthercomputeroptimization. in theinitialstagesofsystemspecification,butcanalsohelpachievea method forquicklyestimatinglensperformance. This notonlysavestime ing easiertouseandarereadilyavailable, itisstillquiteusefultohavea Even thoughtoolsforthepreciseanalysisofanopticalsystemarebecom- widely used. powerful ray-tracingsoftware, Seidel’s formulaforsphericalaberrationisstill good descriptionofopticalsystemimagequality. Infact, evenintheeraof system ofclassifyingthem,whichmakes analysismuchsimpler, givesa In actualpractice, aberrationsoccurincombinationsratherthan alone. This terms inthesineexpansions. aberrations withoutactuallytracinglargenumbersofraysusingallthe eral color. Seideldevelopedmethodstoapproximateeachofthese tion. Inpolychromaticlighttherearealsochromaticaberrationandlat- are sphericalaberration,astigmatism,fieldcurvature, coma,anddistor- system intoseveraldifferentclassifications. Inmonochromaticlightthey To simplifythesecalculations, Seidelputtheaberrationsofanoptical called Seidelaberrations. 1 3 /3! term.The resultantthird-orderlensaberrationsaretherefore i ... ! / ! / ! / ! / sin v v v v vvv 111 Fundamental Optics −+−+− + − + =− 3 3579 = 1 5 v is reasonablyvalid forv ≠ v. These deviationsareknownasaber- 1 7 1 with 1 9 v 1 itself andsoon).This is

close tozero(i.e., (1.21) spherical aberration(TSA).These quantitiesarerelatedby which theseraysintercepttheparaxialfocalplaneiscalledtransverse rays) iscalledlongitudinalsphericalaberration(LSA).The heightat rays)andtheraysthatgothroughedgeoflens(marginal (paraxial between theinterceptofraysthatarenearlyonopticalaxis it focuses(crossestheopticalaxis).The distancealongtheopticalaxis ther fromtheopticalaxisrayenterslens, thenearertolens shows thesituationmoretypicallyencounteredinsinglelenses. The far- collimated light.AllrayspassthroughthefocalpointF Figure 1.15illustrateshowanaberration-freelensfocusesincoming SPHERICAL ABERRATION Figure 1.15 for agiventaskarepresentedlaterinthischapter. However, thethird- present. Parameters forchoosingthebestlensshapeandorientation conjugate Spherical aberrationisdependentonlensshape, orientation,and TSA = LSA# longitudinal sphericalaberration ratio, aswellontheindexofrefractionmaterials tan(u Spherical aberrationofaplano-convexlens ″ ). aberration-free lens u ″ transverse sphericalaberration paraxial focalplane Fundamental Optics LSA www.cvimellesgriot.com ″ . The lowerfigure F F TSA ″ ″ (1.22) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.13 www.cvimellesgriot.com Fundamental Optics Fundamental Fundamental Optics sagittal image (focal line) paraxial meridional, rays. Rays not in this plane are referred to as skew Rays meridional, rays. focal plane ASTIGMATISM the natural asym- is focused by a spherical lens, When an off-axis object system appears to have two different The metry leads to astigmatism. focal lengths. the plane containing both optical axis and object As shown in figure 1.16, Rays that lie in this plane are called plane. point is called the tangential tangential, or ray goes from the object point through the or principal, chief, The rays. the plane perpendicular to the lens system. The center of the aperture of or the principal ray is called the sagittal tangential plane that contains radial plane. figure illustrates that tangential rays from the object come to a focus closer The When the image is evaluated to the lens than do rays in the sagittal plane. we see a line in the sagittal direction. A line in at the tangential conjugate, Between these conjugate. the tangential direction is formed at the sagittal Astigmatism the image is either an elliptical or a circular blur. conjugates, is defined as the separation of these conjugates. amount of astigmatism in a lens depends on lens shape only when The with the lens itself. there is an aperture in the system that is not in contact although in many cases (In all optical systems there is an aperture or stop, strongly Astigmatism it is simply the clear aperture of the lens element itself.) depends on the conjugate ratio. (focal line) v tangential image ≠ (1.23) sagittal plane principal ray optical system . f

3 f/# . 0 067 = tangential plane

Astigmatism represented by sectional views Astigmatism represented

al axis al c opti produce spherical or cylindrical surfaces. The manufacture of The surfaces. produce spherical or cylindrical object point spot size due to spherical aberration spot size due to spherical Figure 1.16 Figure Theoretically, the simplest way to eliminate or reduce spherical aberration simplest way the Theoretically, an (i.e., radius of curvature with a varying the lens surface(s) is to make to exactly compensate for the fact that sin v aspheric surface) designed with high surface most lenses however, practice, In at larger angles. by grinding and polishing techniques that accuracy are manufactured naturally chromatic, spherical aberration of a plano-convex lens used aberration of a monochromatic, spherical order, by ratio can be estimated at infinite conjugate of complex, and it is difficult to produce a lens aspheric surfaces is more aberration completely. sufficient surface accuracy to eliminate spherical these aberrations can be virtually eliminated, for a chosen Fortunately, by combining the effects of two or more spherical (or set of conditions, cylindrical) surfaces. spherical aberration, In general, simple positive lenses have undercorrected spherical aberration. By and negative lenses usually have overcorrected glass with a negative lens combining a positive lens made from low-index it is possible to produce a combination in which made from high-index glass, powers do not. The the spherical aberrations cancel but the focusing series such as the LAO simplest examples of this are cemented doublets, properly used. which produce minimal spherical aberration when 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.13 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:28PMPage1.14

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.14 Figure 1.17 some extentbycombiningpositiveandnegativelenselements. lenses haveoutward curvingfields. Field curvature canthusbecorrectedto Positive lenselementsusuallyhaveinward curvingfields, andnegative the blurfromfieldcurvature to avalue of0.25itsoriginalsize. height. Therefore, byreducingthefieldangleone-half, itispossibletoreduce Field curvature varies withthesquareoffieldangle orthesquareofimage correspond tothetangentialandsagittalconjugates. problem iscompoundedbecausetwoseparateastigmaticfocalsurfaces field curvature (seefigure1.19). Inthepresenceofastigmatism,this to imagebetteroncurvedsurfacesthanflatplanes. This effectiscalled Even intheabsenceofastigmatism,thereisatendencyopticalsystems FIELD CURVATURE marginal rays. placing anaperture, orstop, inanopticalsystemtoeliminatethemore surfaces. Alternatively, asharperimagemaybeproducedbyjudiciously As withsphericalaberration,correctioncanbeachievedbyusingmultiple still exhibitcomaoffaxis. Seefigure1.18. corrected andthelensbringsallraystoasharpfocusonaxis, alensmay appears asacharacteristiccomet-like flare. Evenifsphericalaberrationis object points. Anoff-axisobjectpointisnotasharpimagepoint,butit a comaticcircle. This causesblurringintheimageplane(surface)ofoff-axis in figure1.17,eachconcentriczoneofalensformsring-shapedimagecalled of magnification.This givesrisetoanaberrationknownascoma.Asshown In sphericallenses, differentpartsofthelenssurfaceexhibitdegrees COMA Fundamental Optics Imaging anoff-axis pointsource byalenswithpositivetransversecoma S 1 1 ′ 1 ′ 1 S P, O 1′ 1 33 points onlens 42 2 3′ 4′ 2′ Figure 1.18 Figure 1.19 1′ 1 1 1′ 0 2′ 4′ 3′ 4 spherical fosurfae Positive transversecoma Field curvature positive transversecoma 4 corresponding points on 4′ 60∞ 1 3′ 3 1 focal plane ′ 2′ S 2 Fundamental Optics www.cvimellesgriot.com Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 1.15 ) 2 2 3 y y y y y — chromatic aberration longitudinal www.cvimellesgriot.com )( 2 2 3 red focal point v v v Fundamental Optics Fundamental —— —— — Fundamental Optics blue focal point )( red light ray 3 2 2 fv fv f f f — — (f Aperture Angle Field Image Height blue light ray . From Snell’s law (see equation 1.20), it can be law (see Snell’s . From Longitudinal chromatic aberration Longitudinal chromatic white light ray Material Properties Lateral Spherical Astigmatism Longitudinal Spherical Coma Curvature Field Distortion Chromatic Aberration seen that light rays of different wavelengths or colors will be refracted wavelengths seen that light rays of different 1.21 shows the index is not a constant. Figure at different angles since collimated light is incident on a positive the result when polychromatic index of refraction is higher for shorter wave- lens element. Because the closer to the lens than the longer wavelengths. these are focused lengths, the axial distance from Longitudinal chromatic aberration is defined as of spherical aberration, the nearest to the farthest focal point. As in the case of chromatic aberration. positive and negative elements have opposite signs aberration to form Once again, by combining elements of nearly opposite corrected. It is neces- a doublet, chromatic aberration can be partially so that characteristics, sary to use two glasses with different dispersion negative element can balance the aberration of the stronger, the weaker positive element. Figure 1.21 Figure of Aberrations with Aperture, Variations Field Angle, and Image Height CHROMATIC ABERRATION CHROMATIC described are purely a function of the shape aberrations previously The light. they can be observed with monochromatic and of the lens surfaces, are used to trans- arise when these optics however, Other aberrations, index of refraction of a The wavelengths. form light containing multiple Known as dispersion, this is discussed material is a function of wavelength. in BARREL DISTORTION DISTORTION PINCUSHION Pincushion and barrel distortion Pincushion and barrel Furthermore, the amount of distortion usually increases with the amount Furthermore, OBJECT Figure 1.20 Figure DISTORTION but may also be distorted. may have curvature image field not only The point may be formed at a location on this surface image of an off-axis The distor- This by the simple paraxial equations. other than that predicted to (where rays from an off-axis point fail tion is different from coma plane). Distortion means that even if a perfect meet perfectly in the image formed, its location on the image plane is not off-axis point image is correct. effect of this can be seen as two different The increasing image height. barrel (see figure 1.20). Distortion does kinds of distortion: pincushion and it simply means that the image shape does not lower system resolution; to the shape of the object. Distortion is a sepa- not correspond exactly predicted location ration of the actual image point from the paraxially or as an absolute value on the image plane and can be expressed either as a percentage of the paraxial image height. has opposite types of It should be apparent that a lens or lens system This or backward. distortion depending on whether it is used forward a photograph, and then used means that if a lens were used to make in the final screen in reverse to project it, there would be no distortion systems at 1:1 magnification perfectly symmetrical optical Also, image. have no distortion or coma. 1ch_FundamentalOptics_Final_a.qxd 6/15/2009 2:28 PM Page 1.15 2:28 PM 6/15/2009 1ch_FundamentalOptics_Final_a.qxd 1ch_FundamentalOptics_Final_a.qxd 6/15/20092:29PMPage1.16

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics 1.16 Figure 1.22 is inadequate. Inthesecases, exactraytracingisabsolutelyessential. having largeaperturesoraangularfieldofview, third-ordertheory to quantifyaberrations. However, inhighlycorrectedsystemsorthose For manyopticalsystems, thethird-ordertermisallthatmaybeneeded stop location. magnification dependsoncolor. Lateralcolorisverydependentonsystem is whyraysintercepttheimageplaneatdifferentheights. Statedsimply, wavelength, bluelightisrefractedmorestronglythanredlight,which positive lensandaseparateaperture. Becauseofthechangeinindexwith Figure 1.22showsthechiefrayofanopticalsystemconsistingasimple Lateral coloristhedifferenceinimageheightbetweenblueandredrays. LATERAL COLOR aperture Fundamental Optics Lateral Color blue lightray red lightray focal plane lateral color estimate thespotsizeofadoublet,tablesin Although thereisnosimpleformulathatcanbeusedto purely monochromaticlight. for focusingcollimatedlightorcollimatingpointsources, evenin chromatic aberration,theyareoftensuperiortosimplelenses Because achromaticdoubletscorrectforsphericalaswell Simple Lenses Achromatic Doublets Are Superiorto catalog achromaticdoublets. sample values thatcanbeusedtoestimatetheperformanceof APPLICATION NOTE Fundamental Optics www.cvimellesgriot.com Spot Size Spot give