MIT 2.71/2.71009/21/05 wk3-b-1Lenses and Imaging (Part II)• Review from Part I• Thin lens– matrix ray-tracing– imaging condition– real and virtual images• Thick lens• Generalized optical systems:– principal planes– effective, front and back focal lengths• Reflective opticsMIT 2.71/2.71009/21/05 wk3-b-2Refraction at spherical surfaceoptical axisoff-axis rayRefraction at positive spherical surface:()11111xRnnnnnxx⎥⎦⎤⎢⎣⎡′−′−′=′=′αα1α1α′1x1x′x: ray heightα: ray directionnn’R: radiusof curvaturePowerPowerMIT 2.71/2.71009/21/05 wk3-b-3The power of surfaces• Positive power bends rays “inwards” (converging bundle)• Negative power bends rays “outwards” (diverging bundle)Simple sphericalrefractor (positive)1nR>0R<0Simple sphericalrefractor (negative)1n–()()()0 1leftright>++−=−≡RnRnnP()()()0 1leftright<−+−=−≡RnRnnP+MIT 2.71/2.71009/21/05 wk3-b-4The power of surfaces• Positive power bends rays “inwards” (converging bundle)• Negative power bends rays “outwards” (diverging bundle)+n1R<0Simple sphericalrefractor (negative)–()()()0 1leftright>−−−=−≡RnRnnP()()()0 1leftright<+−−=−≡RnRnnPR>0n1Simple sphericalrefractor (positive)MIT 2.71/2.71009/21/05 wk3-b-5(Paraxial) Ray-tracing in generalinxoutxinαoutαninnoutOPTICALSYSTEMFor an arbitrary ray entering with• lateral departure displacement xinwrt the optical axis• angle of departure αinwrt the optical axis• in a departure space of refractive index nindetermine the ray’s• lateral arrival displacement xoutwrt the optical axis• angle of arrival αoutwrt the optical axis• in an arrival space of refractive index noutupon exiting the optical systemMIT 2.71/2.71009/21/05 wk3-b-6Matrix ray-tracing in generalinxoutxinαoutαninnoutOPTICALSYSTEM⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛ininin22211211outoutoutxnMMMMxnαα⎟⎟⎠⎞⎜⎜⎝⎛=22211211MMMMMis the “system matrix,” found as the product, in reverse order, of the matricesof the elements (free space, spherical surfaces)comprising the systemMIT 2.71/2.71009/21/05 wk3-b-7Matrix ray-tracinginxoutxinαoutαninnoutOPTICALSYSTEM⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛ininin22211211outoutoutxnMMMMxnαα•Power• Imaging condition• Lateral magnification• Angular magnification12M021=M01121=MM02121=MMMIT 2.71/2.71009/21/05 wk3-b-8Thin lens in airRR′−)(n=1n=1⎟⎟⎠⎞⎜⎜⎝⎛ininxα⎟⎟⎠⎞⎜⎜⎝⎛outoutxα⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛ininoutout????xxααObjectiveObjective: specify input-output relationshipnMIT 2.71/2.71009/21/05 wk3-b-9Thin lens in airRR′−)(++Model: refraction from firstfirst (positive) surface+ refraction from secondsecond (negative) surfaceIgnoreIgnore space in-between (thin lens approx.)n=1n=1nnMIT 2.71/2.71009/21/05 wk3-b-10Thin lens in airRR′−)(++⎟⎟⎠⎞⎜⎜⎝⎛ininxα⎟⎟⎠⎞⎜⎜⎝⎛11xnα⎟⎟⎠⎞⎜⎜⎝⎛11xnα⎟⎟⎠⎞⎜⎜⎝⎛outoutxα⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−−=⎟⎟⎠⎞⎜⎜⎝⎛inin111011xRnxnαα⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛′−−=⎟⎟⎠⎞⎜⎜⎝⎛11outout1011xnRnxααnnn=1n=1MIT 2.71/2.71009/21/05 wk3-b-11Thin lens in airRR′−)(n=1n=1⎟⎟⎠⎞⎜⎜⎝⎛ininxα⎟⎟⎠⎞⎜⎜⎝⎛outoutxα()⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛′−−−=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−−⎟⎟⎠⎞⎜⎜⎝⎛′−−=⎟⎟⎠⎞⎜⎜⎝⎛ininoutoutininoutout10 11 1 110 1 110 1 1xRRnxxRnRnxααααnMIT 2.71/2.71009/21/05 wk3-b-12Thin lens in airRR′−)(n=1n=1⎟⎟⎠⎞⎜⎜⎝⎛ininxα⎟⎟⎠⎞⎜⎜⎝⎛outoutxα()⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛′−−−=⎟⎟⎠⎞⎜⎜⎝⎛ininoutout10 11 1 1xRRnxααnRnP1−=RnP′−=′1PowerPower of thefirst surfacePowerPower of thesecond surfaceMIT 2.71/2.71009/21/05 wk3-b-13Thin lens in airPP′n=1n=1⎟⎟⎠⎞⎜⎜⎝⎛ininxα⎟⎟⎠⎞⎜⎜⎝⎛outoutxα⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛ininlensthin outout10 1xPxαα()⎟⎠⎞⎜⎝⎛′−−=′−+−=′+=RRnRnRnPPP11111lensthin Lens-maker’sformulanMIT 2.71/2.71009/21/05 wk3-b-14Thin lens in airPP′n=1n=1⎟⎟⎠⎞⎜⎜⎝⎛ininxα⎟⎟⎠⎞⎜⎜⎝⎛outoutxα⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛ininlensthin outout10 1xPxααninoutinlensthin inout xxxP=−=ααRay bending is proportionalproportionalto the distanceto the distance from the axisMIT 2.71/2.71009/21/05 wk3-b-15The power of lenses• Positive power bends rays “inwards” (converging bundle)• Negative power bends rays “outwards” (diverging bundle)+ ++ NPlano-convexlensBi-convexlensPlano-concavelensBi-concavelens– – –N+Simple sphericalrefractor (positive)1nR>0R>0R<0nn11 11nn11 11R<0R>0R<0R>0R<0Simple sphericalrefractor (negative)1n–MIT 2.71/2.71009/21/05 wk3-b-16Positive thin lens in airRay bending is proportionalproportionalto the distanceto the distance from the axisobject at ∞MIT 2.71/2.71009/21/05 wk3-b-17Positive thin lens in airthin lens as a “black box”f0 ,inin=αxinlensthin outinout ,xPxx−==αlensthin outin1 Pfxf =⇒−=αFocal pointFocal point = image of an object at ∞Focal lengthFocal length = distance between lens & focal point0lensthin >PRealReal imageMIT 2.71/2.71009/21/05 wk3-b-18Negative thin lens in airRay bending is proportionalproportionalto the distanceto the distance from the axisobject at ∞MIT 2.71/2.71009/21/05 wk3-b-19Negative thin lens in airVirtualVirtual imagelensthin 1Pf =fstill applies, now with0 0lensthin <⇒⇒<fP(to the “left”“left”)object at ∞MIT 2.71/2.71009/21/05 wk3-b-20Imaging condition: ray-tracing2ndFPobjectimage• Image point is located at the common intersection of all rays whichemanate from the corresponding object point• The two rays passing through the two focal points and the chief raycan be ray-traced directlyn=1n=1chief ray1stFPMIT 2.71/2.71009/21/05 wk3-b-21Imaging condition: ray-tracingF’• (ABF)~(FLN) and (F’CD)~(MLF’) are pairs of similar trianglesABFCDMLN(CD)(LN) (ML)(AB) C)F((CD))F(L(LM) (FL)(LN)(AF)(AB)==′=′=xffx’2fxx =′)F(L (FL) C)F( (AF)′=′MIT 2.71/2.71009/21/05 wk3-b-222ndFPobjectimage•