Principles of Geometric Optics

Transcription

1 Principles o Geomeric Opics Joel A. Kubby Universiy o Caliornia a Sana Cruz. Inroducion...9. Relecion Reracion Paraxial Lens Equaion Tin Lens Equaion Magniicaion Aberraions Inroducion In is caper, we consider geomeric opics, wic is an approximaion o wave opics a can be used wen considering an opical sysem composed o elemens a are muc larger an e waveleng o lig going roug e sysem. Ten, we can ignore e wave naure o lig, aside rom is color, and assume a i will ravel in a sraig line, wic is oen called a ray. Aloug geomeric opics is only an approximaion o wave opics, i is ecnologically useul or e design and modeling o e adapive opical sysems a will be considered ere. I grealy simpliies e calculaions o a poin a allows inuiion o guide e design. Tis is no usually e case wen considering e Huygens inegral! To deermine e direcion in wic a lig ray will pass roug an opical sysem, we can apply Ferma s principle o leas ime or sores opical pa leng a was discussed in Caper. Ferma s principle is a e opical pa disance OPD beween poins A and B given by B OPD A,B n x d x (.) = is sorer an e opical pa leng o any oer curve a joins ese poins and lies in is cerain regular neigborood (Born and Wol, 006).. Relecion We consider e applicaion o Ferma s principle o wo simple opical suraces: a mirror a relecs lig and an inerace beween wo media a reracs lig. Wi ese wo simple opical suraces, we can undersand e mos imporan aspecs o geomerical opics or e design o opical sysems. In Figure., we sow a mirror surace were a lig ray saring rom poin A is releced o e mirror surace o reac poin B. Te quesion is wa pa will e lig ray ake? I we assume a e A 03 by Taylor & Francis Group, LLC 9

2 30 Principles A B d θ θ d x L x L FIGuRE. Relecion o e lig rom a mirror surace. Te lig ravels rom poin A o poin B by being releced o e mirror surace a a orizonal disance x rom poin A. Poin B is a orizonal disance L rom poin A. Te disance a e lig ravels rom poin A o e mirror is d and e disance a i ravels rom e mirror o poin B is d. To ind e angle o relecion, θ, given e angle o incidence, θ, we minimized e ravel ime along is pa according o e Ferma s principle o leas ime. orizonal disance beween poins A and B is L, en a wa poin x on e mirror will e lig beam be releced? We see a e lig will ake a pa a is a disance d rom poin A o e mirror and a pa a is a disance d rom e mirror o poin B. We assume a e lig is raveling in a vacuum a a speed c. Te ime required or e lig o ravel rom poin A o B along is pa is given by e ollowing equaion: d( x) d( L x) ( x) = + = ( c c c d ( x ) + d ( L x )) = ( ) c ( x L x ) To ind e minimum ime, we ake e derivaive o wi respec o x and se i equal o zero, wic gives d( x) = ( + x ) ( x) + + ( L x) ( L dx = x + x x L = x d d = sin( θ ) sin( θ ) L x sin( θ ) = sin( θ ) θ = θ. + L x ( x ) ) = 0 (.) We see a or e lig o ake e pa o leas ime o ge rom poin A o B by relecing o e mirror surace requires a e angle o incidence, θ, is equal o e angle o relecion, θ. Tis simple ormula allows us o calculae e pa lig will ake wen releced rom a mirror. As we sall see in Caper 9, Secion 3, i is ineresing o consider mirrors a are no la. In is case, e angle o incidence is equal o e angle o relecion, were e angles o incidence and relecion are deined by e local normal o e curved surace. 03 by Taylor & Francis Group, LLC

3 Principles o Geomeric Opics 3.3 Reracion We can also calculae e pa a ray o lig will ake wen i passes roug an inerace a separaes media wi wo dieren indices o reracion. Te speed o lig inside e media wi an index o reracion n is c/n, were c is e speed o lig in vacuum. Since n >, e speed o lig is always slower in media oer an vacuum. Given is speed o lig wiin e media, we can en calculae e ime a lig akes o ravel rom a poin A in e irs medium, wi index n, o a poin B in e second medium, wi index n, as sown in Figure.. Te ime required or e lig o ravel rom poin A o B along is pa is given by e ollowing equaion: ( n x ) = c d ( x) + n c d ( L x) = c ( n d ( x) + n d ( L x)) ( ) c n x = + + n + L x To ind e minimum ime, we ake e derivaive o wi respec o x and se i equal o zero, wic gives: d( x) x c n x = ( + ) ( d x )+ n + ( L x ) ) ( ) ( L x) = 0 = n x + x n n x n L = x d d = n sin( θ ) n sin( θ ) n sin( θ ) = n sin( θ ) L x + L x ( (.3) We recognize is as Snell s law rom Caper. A d θ n x θ d n L x B FIGuRE. Reracion o e lig roug an inerace. Te lig rom poin A is inciden a an angle θ on an inerace beween wo media. Te upper media as an index o reracion n and e lower media as an index o reracion n. Te cange in e index bends e pa o e lig a an angle θ, causing i o pass roug poin B. 03 by Taylor & Francis Group, LLC

4 3 Principles.4 Paraxial Lens Equaion I is also useul o consider reracion rom a curved surace, suc as e circular arc sown in Figure.3. Wen e arc separaes wo regions o dieren indices o reracion, n and n, a ray o lig will be reraced a e inerace according o Snell s law. I we coose e correc sape or e inerace, we can cause lig a originaes rom a poin O along e axis o e arc o pass roug a poin I a is also along e axis. Tis is e geomery o a lens and i enables an objec a one poin in an opical sysem o be imaged a a dieren poin in e sysem. To ind e equaion a describes e acion o e lens, we consider e ac a a lig ray raveling rom posiion O o posiion I mus ake e same amoun o ime regardless o e pa. Tis means a e ime aken or lig o ravel rom poin O o P and en rom poin P o I mus equal e amoun o ime aken or lig o ravel direcly rom poin O o I. O course, e spaial disance OP + PI is greaer an e disance OI, bu i n is smaller an n, en lig will ravel slower inside e media wi index n and us will ake e same amoun o ime along e direc pa OI as i would along e pa OP + PI. Tis is because more o e pa o lig is inside e media wi iger index. To esimae ese ravel imes, i is useul o make an approximaion or lig rays a are close o e opical axis (Feynman 966). Tese are called paraxial rays. Consider e rig riangle wi sides o leng s > d > sown in Figure.4. From e Pyagorean eorem, we know a s = + d = s d = ( s d)( s + d) To simpliy is, we can approximae a s d, so a s + d s and subsiue Δ = (s d) o obain = ( s d)( s + d) s (.4) s P O S R V Q C S I n n FIGuRE.3 Reracion a a curved inerace. Te lig raveling rom poin O (objec) o poin P is reraced a e inerace beween wo media wi indices o reracion n and n. Te reraced lig inersecs e opical axes a poin I (image). Te curved surace as a radius o curvaure R. Δ s d FIGuRE.4 Te paraxial approximaion o ind e dierence in opical pa dierence or wo rays a are close o e opical axis. 03 by Taylor & Francis Group, LLC

5 Principles o Geomeric Opics 33 We can use is paraxial approximaion o ind e ravel imes or lig along e dieren pas sown in Figure.3. Te ravel imes along e pas rom poin O o P and rom poin P o I are given by OP n c OP n = PI = c PI Te ravel imes along e pa rom poin O o I are given by n c OV n c VQ n c QC n = = = = c CI OV VQ QC CI Ten, e oal ravel imes along e pas OPI and OI are as ollows: OPI OI n c OP n = + c PIs n c OV n = + VQ + QC + CI c We can en use e paraxial equaion o simpliy is equaion: OP = OQ + = OV + VQ + = OV + VQ + s PI = QI + = QC + CI + = QC + CI + s Te dierence in e ravel imes along e wo roues would be OPI n c OP OV n OI = ( )+ PI VQ QC CI c n n = VQ + VQ s + s For e ravel imes along e wo roues o be equal, we need n VQ n VQ + s = c s n n + = VQ c s n n c s c c n n VQ + = n s s n 03 by Taylor & Francis Group, LLC

6 34 Principles We can apply e paraxial approximaion (Equaion.4) o ind e leng VQ: R = QC + 3 = QC + R R QC = VQ = R Subsiuing or VQ en gives n s n n n + = s R Consider wa appens wen e disance s becomes very large, a is, or an objec a a very long disance. In e limi a s, we ave n s ( n n ) nr = s = R n n = ' Here, e lig comes o a ocus a a se disance ino e media wi index n. Tis is deined as e ocal poin wiin medium. Since e lig is coming rom ininiy, e lig rays would be parallel and e waverons would be plane waves. I s, we ave n s ( n n ) nr = s = R n n = Here, e lig comes o a ocus a a se disance ino e media wi index n. Tis is e ocal poin wiin medium..5 Tin Lens Equaion In mos opical sysems, we would like o ave e lig pass rom one poin s o anoer poin s, bo o wic are in air raer an inside some maerial suc as glass. Tis is accomplised using a in lens a as wo curved suraces, as sown in Figure.5. Since a lens is usually used in air, we ave se n =. Wen describing e objec and image disances, and e radii o curvaure o e wo suraces o e lens, e ollowing convenions or e signs o e disances and radii o curvaure are used (Feynman 966): O P S S R V Q V C I n FIGuRE.5 Reracion a wo curved ineraces o orm a biconvex lens. 03 by Taylor & Francis Group, LLC

7 Principles o Geomeric Opics 35. Te objec disance s is posiive i e poin O is o e le o e lens and is negaive i e poin is o e rig o e lens.. Te image disance s is posiive i e poin I is o e rig o e lens and is negaive i e poin is o e le o e lens. 3. Te radius o curvaure o e lens is posiive i e cener o e radius o curvaure is o e rig o e lens and is negaive i e cener is o e le o e lens. For e example sown in Figure.5, bo s and s are posiive. Te radius o curvaure on e leand side o e lens is posiive and a on e rig-and side o e lens is negaive. Since is lens as wo convex suraces, i is called a biconvex lens. To solve e lens equaion, we again calculae e ravel imes or wo dieren pas direc rom poin O o I and rom poin O o P and rom P o I and equae em. Te ravel ime along e sraig pa beween O and I is given by e sum o e ravel imes along segmens OV, VQ, QV, and V I: c OV n c VQ n c QV = = ' = ' ' = c V ' I OV VQ QV V I Ten e oal ravel imes along pas OPI and OI are OPI OI c OP = + c PI c OV V I n = ( + ' )+ c VQ + QV ' We can en use e paraxial approximaion (Equaion.4) o simpliy is equaion: OP = OQ + = OV + VQ + = OV + VQ + s PI = QI + = QV ' + V ' I + = QV ' + V ' I + s Te dierence in e ravel imes along e wo dieren roues would be OPI c OP PI OV V I n OI = ( + ' ) c VQ + QV ' = + s n VQ + QV ' s c I e magniude o e radius o curvaure on bo sides o e lens is e same, en we ave VQ = QV ' = R 03 by Taylor & Francis Group, LLC

8 36 Principles Te dierence in e ravel ime can en be wrien as OPI n OI = + s s R For e ravel imes along e wo roues o be equal, we need n + s s = R s + s = ( ) n R + = n s s R. (.5) In e limi a s, we ave = n s R R s = n = In e limi a s, we ave s R = n = Tereore, e ocal lengs would be e same on eier side o e lens, and e lig a comes in rom ininiy is broug o a ocus a a disance rom e lens. I e radii o curvaure R and R on eier side o e lens are no equal, and using e convenion a e radius o curvaure is posiive i e cener o e radius o curvaure is o e rig o e lens (R ) and is negaive i e cener is o e le o e lens (R ), en we would ave and VQ R QV = ' = R + = ( n ) s s R R In e limi a s, we ave = ( n ) s R R RR s = =. n R R 03 by Taylor & Francis Group, LLC

9 Principles o Geomeric Opics 37 Subsiuing or e ocal leng in Equaion.5, we ave + = (.6) s s Tis is called e lensmaker s equaion, wic provides a relaionsip beween e ocal leng and e disances o e objec and e image. I bo e objec and e image disances s and s, respecively, are equal, en or s = s = s: + = = s = s s s In general, e objec and e image may no be poins, as sown in Figure.5, bu raer exended objecs, as sown in Figure.6. Here e objec is sown as an arrow a exends above e opical axis. We can ind ou were e image is posiioned by considering wo principal lig rays, one rom e ip o e objec a is parallel o e opical axis and one rom e ip o e objec a passes roug e cener o e lens. Tis is called ray racing. Te lig ray rom e ip o e objec a is parallel o e opical axis is equivalen o a lig ray rom ininiy, and ereore, aer passing roug e lens, i will inersec e opical axis on e rig-and side o e lens a e ocal poin. Te lig ray a passes roug e cener o e lens as a symmeric opical pa roug e in lens, and ereore i can be drawn as a sraig line. Tis ray will be reraced by a cerain amoun in raveling rom air, wi n = n =, ino e glass wi n = n, wi e delecion being given by Snell s law, as described in Equaion.3. Tis lig ray will be reraced a second ime wen i exis e lens, and ereore i will coninue along e same direcion aer passing roug e lens. For a in lens, we can draw a ray as a sraig line. Were ese wo rays inersec on e rig-and side o e lens, an image will be ormed. Since is image is on e opposie side o e lens rom e objec, i is called a real image. I a screen were o be placed a is locaion, an image o e objec would be seen, aloug e image is invered (i.e., upside down), and e arrow in e image may be dieren in leng rom e arrow sown as e objec. We will discuss is in Secion.6, wen we consider e magniicaion o e lens. S S Objec Real image Figure.6 Ray racing o ind e objec and e image or a in biconvex lens. Te objec is locaed a a disance S o e le o e lens, and e image is locaed a a disance S o e rig o e lens. Te posiion o e image can be ound by inding e inersecion o wo lig rays emanaing rom e ip o e objec. Tree.principal lig rays are sown. One passes roug e ocal poin on e le-and side o e lens, wile anoer passes roug e ocal poin on e rig-and side o e lens. Tese lig rays appear o come in rom ininiy. A ird lig ray passes roug e cener o e lens. Since e lens is considered o be a in lens, is lig ray is drawn as a sraig line rom e ip o e objec o e ip o e image. (Credi: Wiki.) 03 by Taylor & Francis Group, LLC

10 38 Principles Virual image Objec S S Figure.7 A virual image ormed rom an objec a S a is closer an one ocal leng o e surace o e lens. In is case, a real image is no ormed o e rig o e lens, bu raer a virual image is ormed o e le o e lens. Te lig rays on e rig-and side o e lens appear o be emanaing rom is virual image. (Credi: Wiki.) S Objec Virual image S Figure.8 Biconcave lens composed o wo concave lens suraces. (Credi: Wiki.) We noe ere is a ird ray a can be raced, as sown in Figure.6. Tis ray passes roug e ocal poin on e le-and side o e lens. From Equaion.6, we know a a lig ray coming rom ininiy oward e lens rom e rig-and side would pass roug e ocal poin on e le-and side o e lens, and ereore we can draw is lig ray parallel o e opical axis on e image side o e lens. We see a is lig ray also inersecs e real image were e irs wo lig rays inersec. Since we only need o ind e posiion o e real image a e inersecion o any o ese lines, any wo lig rays are suicien or inding e objec locaion and size. Consider wa appens wen e objec disance S is closer o e lens an e ocal leng. Tis siuaion is sown in Figure.7. We can draw e lig ray rom e ip o e objec a is parallel o e opical axis. I inersecs e opical axis a a disance o e rig o e lens. We can also draw e lig ray a goes roug e cener o e lens, bu we see a ese wo lig rays do no inersec a any poin on e rig-and side o e lens. I we exend ese wo lines o e le o e objec, as sown by e dased lines in Figure.7, we see a ey inersec a a disance S o e le o e lens. From convenion above, S as a negaive value since i is o e le o e lens. Te lig rays rom e objec appear as oug ey were coming rom a virual image a a disance S o e le o e lens. Tis image is called a virual image since i would no be visible i a screen were o be locaed a a poin. In addiion o aving convex suraces, a lens can also be ground o ave concave suraces, as sown in Figure.8. A lens wi wo convex suraces is called a biconcave lens. Using ray racing rom an objec 03 by Taylor & Francis Group, LLC

11 Principles o Geomeric Opics 39 locaed a a disance S o e le o e lens, wic is urer an one ocal leng o e le o e lens, will orm a virual image on e same side o e lens a S, wic is closer an e ocal leng. I is also possible o ave a lens wi one convex surace and one concave surace. Tis orm o a lens is called a meniscus lens. I is also possible o ave a lens wi one side planar and e oer side eier convex or concave. Tese are called plano-convex and plano-concave lenses, respecively..6 Magniicaion Again rom Figure.6, we can deermine e raio o eigs o e objec and e image. We ave redrawn is igure wi some similar riangles, were e eig o e rig riangle a includes e objec is y, and e eig o e rig riangle a includes e image is y (Figure.9). From e similar rig riangles on e le-and side o e lens, we can ind e magniicaion M: y y M y S = = y S We can also use similar riangles on e rig-and side o e lens o ind a y y M y S = S = y Since y is below e opical axis, i is a negaive quaniy by convenion, we say a e magniicaion is negaive, and i resuls in an invered image. Te virual image in Figure.8 is posiive, and ereore, in is case, e magniicaion is posiive and e virual image is no invered. Combining ese wo equaions, we ind M y = y S S S S S S = S S ( )( ) = + = SS = S + S = S + S SS S + S = + = S S We recover Equaion.6, e lensmaker s ormula. S S y Objec Real image y Figure.9 Magniicaion o a lens. Te eig o e objec is y and a o e image is y. (Credi: Wiki.) 03 by Taylor & Francis Group, LLC

12 40 Principles.7 Aberraions Te resricion o e in lens analysis o monocromaic paraxial rays leads o opical aberraions or real lenses. For an exended lens a admis lig rays away rom e axis, e paraxial approximaion used in Equaion.4 is no longer valid. In general, e opical surace obained rom e principle o leas ime is no longer a sperical surace, bu raer a iger-order surace an a spere. Noneeless, a sperical surace is muc easier o abricae by grinding and polising an an asperical surace, and ereore, lenses wi sperical suraces are oen used. As e lig rays become urer removed rom e opical axis, ey no longer come o a ocus a one poin, bu raer ocus a dieren poins depending on eir disance rom e axis. Te resuling aberraion is known as sperical aberraion, since i resuls rom e sperical surace o e lens. An example is sown in Figure.0. Te lig rays a e edge o e lens come o a ocus closer o e lens an ose rom e paraxial region near e opical axis. In addiion o e sperical sape o e lens, sperical aberraion can arise rom index mismaces beween e lens and e sample. Wen e lig ravels beween regions o dieren indices o reracion, e lig rays can be ben a e inerace according o Snell s law, as described in Equaion.3. Tis aberraion can be e dominan aberraion wen imaging deeply ino a specimen since i increases wi dep ino e sample. Anoer aberraion is caused by e waveleng dependence o e index o reracion n. Cromaic aberraion arises wen muliwaveleng lig is reraced a an inerace beween wo regions wi dieren indices o reracion. Te dieren colors o lig can be reraced by diering amouns, as sown in Figure.. A amiliar example is e decomposiion o wie lig by a prism or waer drople, wic orms e amiliar colored rainbow wen sunlig is decomposed ino is specral componens. Te cause or e specral decomposiion is e waveleng dependence o e index o reracion. Te Sperical aberraion Figure.0 Sperical aberraion. Te lig rays a are arer rom e opical axis come o a ocus a dieren disances rom e lens. (Credi: Wiki.) Cromaic aberraion Figure. Cromaic aberraion. Dieren colors o lig are reraced by dieren amouns wen passing roug regions wi dieren indices o reracion. Tis is because e index o reracion depends on e waveleng o lig. (Credi: Wiki.) 03 by Taylor & Francis Group, LLC

13 Principles o Geomeric Opics 4 Crown Flin Acromaic double FIGuRE. Acromaic double lens o compensae or cromaic aberraion due o e waveleng dependence o e indices o reracion. Here one lens is biconvex and is made o crown glass wi one index o reracion, and e oer lens is plano convex and is made o lin glass wi a dieren index o reracion. Te lens pair ends o compensae or e dispersion o muliwaveleng lig. waveleng dependence o e index o reracion is called dispersion, since i causes lig o be dispersed ino is specral componens. A soluion o overcome cromaic aberraion is o use wo dieren lenses wi wo dieren sapes (e.g., convex and concave radii o curvaure) made wi opical maerials wi dieren dispersion caracerisics (e.g., crown and lin glasses). An example is sown in Figure.. In addiion, lenses can ave oer opical aberraions suc as coma, asigmaism, ield curvaure, and disorions (barrel and pincusion). Misalignmen beween opical elemens can cause urer aberraion as discussed in Capers 4 and 9. Wile e primary goal o adapive opics in biological imaging is o overcome sample-induced aberraions, as discussed in Caper 3, e adapive opical sysem can also overcome opical sysem induced aberraions due o aberraions in e opical componens and misalignmens beween componens. In some siuaions, sysem aberraions can dominae specimeninduced aberraions. Here, adapive opics can be used o relax sysem olerances o bring down e cos o ig-perormance opical sysems or o improve e perormance o lower-perormance opical sysems. Reerences Feynman, R. P., R. B. Leigon, and M. Sands, Te Feynman Lecures on Pysics, Addison-Wesley, Reading, MA, 966. Born M. and E. Wol, Principles o Opics, 7 ed., Cambidge: Cambridge Universiy Press, by Taylor & Francis Group, LLC