成像光学基础

成像光學基礎GaussianOpticsandParaxialOptics„Conventionalgeometricalopticsandopticalsystemdesigndealwithrefractionorreflectionatinterfacesbetweentwomedia,whichisgovernedbySnell’sLawnsinθ=n’sinθ’,whichisaverynon-linearrelationship.„ParaxialOptics:forraysclosetoopticalaxis,orwhenθissmall,sin(θ)≅θ,Snell’sLawbecomesnθ=n’θ’.„GaussianOptics:nθ=n’θ’relationshipisextendedtoallregionsofaopticalsystem.„BothparaxialandGaussianopticsarefirst-orderoptics.CardinalPointsofanGaussianOpticalSystemfrontfocalpointfrontprinciplepointRearprinciplepointrearfocalpointEFLEFLBFLFFLfrontandrearnodalpointsInairnodalpointscoincidewithprinciplepointsGaussianImageFormationDistancesfromfocalplanes:m=η’/η=-f/x=-x’/f’x=-f/mx’=-f’mxx’=ff’Distancesfromprincipleplanes:s=f+x=(1–1/m)fs’=f’+x’=(1–m)f’s’/s=-m(f’/f)f/s+f’/s’=1m=-f/(s-f)=-(s’-f’)/f’PP’ff’’xx’ss’FF’ηη’η’objectplanefrontfocalplanefrontprincipleplanerearprincipleplanerearfocalplaneimageplaneηηη’SignificantPointsandQuantities(1)f’,k’：焦距及光焦度(2)P,P’:主平面(3)δ,δ’:主平面距，向左為負，向右為正(4)hP,hP:邊緣光及主光線之高度(5)u,u’:邊緣光在物像空間之角度(6)u,u’:主光在物像空間之角度(7)η,η’:物高與像高(8)l,l’:物距與像距，向左為負，向右為正Pδδ'η'ηMarginalrayChiefrayhPuuu’u'P'f’,Kll’hPUsefulRelationships„放大率M=像高除物高„物像距T=像距-物距已知'f,M求Tll,',已知TllM,',,求'f已知Tllf,',,'求M')11(fMl−=lMMf−=1'''flfM+=')1('fMl−=Mlf−=1'''''flfM−=')12(fMMT−−=TMMMf12'2−+−=01)2'(2=+−+MfTMPδδ'η'ηMarginalrayChiefrayhPuuu’u'P'f’,Kll’hPFocalLengthandMagnification„照相機:焦距愈長，放大率愈大，視場愈小„投影機:焦距愈長，投影倍率愈小照相機投影機MarginalandChiefRayChiefray:-originatesfromtheedgeofobjectfield-travelstothecenterofaperturestop-thentotheedgeofimagefieldMarginalray:-originatesfromthecenterofobjectfield-travelstotheedgeofaperturestop-thentothecenterofimagefieldmarginalmarginalchiefchieflens(stop)apertureanglefieldangleopticalaxisobjectimageSagittalandTangentialPlanes„子午面(Tangential)：通過光軸和物高之單一平面„弧矢面(Sagittal)：通過主光線且垂直子午面之平面組„通常只分析子午及弧矢面之成像特性RingTangentialSagittalApertureandFieldStopInanopticalsystem,theelement,beittherimofalensoraseparatediaphragm,whichdeterminetheamountoflightreachingtheimageisknownastheaperturestop.Inanopticalsystem,theelementlimitingthesizeorangularbreadthoftheobjectthatcanbeimagedbythesystemiscalledfieldstop.AperturestopFieldstopStop,ChiefRayandMarginalRay„光欄(光圈,stop)控制入光量大小控制像差„邊緣光(MarginalRay)通過光欄邊緣定義孔徑角„主光線(ChiefRay)通過光欄中心定義視場角,定義軸外光束之參考線EntranceandExitPupil„物像空間無法看到光圈„入瞳(Entrancepupil):光欄對物空間之成像„出瞳(Exitpupil):光欄對像空間之成像Vignetting•可以降低大角度的像差,使MTF提昇,但會降低周邊亮度•眼睛可接受之vignetting為30-40%•背投系統最好是沒有漸暈F-number„定義D:entrancepupildiameter„照度與F/#平方反比„投影機鏡頭LCD:F/1.7-2.5DLP:F/2.0-2.4θsin21/#≈=DffGeometricalAberrations„DeparturesofarealopticalsystemfromtheidealizedconditionofGaussianopticsareknownasaberrations.„Snell’sLaw:nsinθ=n’sinθ’„Departuresdueton=n(λ)arecalledchromaticaberrations.¾Chromaticaberrationsexisteveninparaxialregion„Departuresduetonon-linearityofsinθtermarecalledmonochromaticaberrationssin(θ)=θ-+-+……θ3θ5θ73!5!7!thirdorderaberrationfifthorderaberration7thorderaberrationGeometricalAberrations„像差:光線或光束距離理想位置之偏差„研究像差的分類及行為模式„垂軸像差:光線在成像面截點和理想像高之差(dx’,dy’)„軸向像差:光線對交點距理想點之軸向距離„色差:不同波長成像差異TypesofAberrations„Chromaticaberrations¾Axialchromaticaberration(axialcolor)¾Lateralchromatic(lateralcolor)„Mono-chromaticaberration¾Spherical¾Coma¾Astigmatism¾Fieldcurvature¾DistortionChromaticAberrations(1)„Axial(longitudinal)chromaticaberration(軸向色差)¾Focallengthchangeswithcolor¾Whenbluefocusistotheleftofredfocus,axialcolorissaidtobepositive.¾Forconvexlens,redfocusisfarthestandistotheright;forconcavelensitisfarthestandtotheleftΔlFC’=lF’-lC’ΔlFC’circleofleastconfusionPositiveaxialcolornegativeaxialcolorLateralChromaticAberrationƒLateralchromaticaberration,orlateralcolor(軸向色差)¾Lateralcolorisachangeinmagnificationduetodifferencesinfocallengthwithcolor¾Redlighthaslongerfocallengththantheblue,resultinginalargerredimage¾LateralcolorisafunctionoffieldangleΔyFCSphericalAberration(球差)•Sphericalaberrationisanaxialaberration.Itcanbedefinedasvariationoffocuswithaperture.•Foragivensimplelens,sphericalaberrationgenerallyincreaseswithapertureheight,i.e.,itcanbereducedbystoppingdowntheaperture.•Foragivenapertureandfocallength,theamountofsphericalaberrationinasimplelensisafunctionofobjectpositionandtheshape,orbendingofthelens.•Sphericalaberrationcanalsobecontrolledbysplittingthelens•SphericalaberrationreducesthecontrastofanimageEffectoflensbendingonsphericalaberrationParaxialfocusSphericalAberrationSPDandRayFanPlotUnder-correctedSphericalAber.atbestfocusUnder-correctedSphericalAber.atparaxialfocuscoma•Comacanbedefinedasthevariationofmagnificationwithapertureheight•ComacanbecontrolledbyshiftingtheaperturestoporselectivelyaddinglensesComa(慧差)•Anastigmaticimageresultswhenlightinoneplane(yz)isfocuseddifferentlyinanotherplane(xz)•Astigmatismisessentiallyacylindricaldepartureofthewavefrontfromitsideasphericalshape•AstigmatismcanbecontrolledbyselectivelylocatingandbendingthelensesAstigmatism(像散)WhereDoesAstigmatismComeFrom?•Whenaconeoflightentersalenssurfaceobliquely,itextendsovermoresurfaceinthe“y”directionthanin“x”direction•Thiswillintroducemorepowerinthe“y”dir