Design and Correction of Optical Systems

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1 Desig ad Correctio of Optical Sstems Lecture 3: Paraial optics Herbert Gross Summer term 207

2 2 Prelimiar Schedule - DCS Basics Materials ad Compoets Paraial Optics Optical Sstems Geometrical Aberratios Wave Aberratios PSF ad Trasfer fuctio Further Performace Criteria Optimizatio ad Correctio Correctio Priciples I Correctio Priciples II Optical Sstem Classificatio Law of refractio, Fresel formulas, optical sstem model, ratrace, calculatio approaches Dispersio, aormal dispersio, glass map, liquids ad plastics, leses, mirrors, aspheres, diffractive elemets Paraial approimatio, basic otatios, imagig equatio, multi-compoet sstems, matri calculatio, Lagrage ivariat, phase space visualizatio Pupil, ra sets ad samplig, aperture ad vigettig, telecetricit, smmetr, photometr Logitudial ad trasverse aberratios, spot diagram, polomial epasio, primar aberratios, chromatical aberratios, Seidels surface cotributios Fermat priciple ad Eikoal, wave aberratios, epasio ad higher orders, Zerike polomials, measuremet of sstem qualit Diffractio, poit spread fuctio, PSF with aberratios, optical trasfer fuctio, Fourier imagig model Raleigh ad Marechal criteria, Strehl defiitio, 2-poit resolutio, MTF-based criteria, further optios Priciples of optimizatio, iitial setups, costraits, sesitivit, optimizatio of optical sstems, global approaches Smmetr, les bedig, les splittig, special optios for spherical aberratio, astigmatism, coma ad distortio, aspheres Field flatteig ad Petzval theorem, chromatical correctio, achromate, apochromate, sesitivit aalsis, diffractive elemets Overview, photographic leses, microscopic obectives, lithographic sstems, eepieces, sca sstems, telescopes, edoscopes Special Sstem Eamples Zoom sstems, cofocal sstems

3 3 Cotets. Paraial approimatio 2. Ideal surfaces ad leses 3. Imagig equatio 4. Matri formalism 5. Lagrage ivariat 6. Phase space cosideratios 7. Delao diagram

4 4 Paraial Approimatio Paraialit is give for small agles relative to the optical ais for all ras Large umerical aperture agle u violates the paraialit, spherical aberratio occurs Large field agles w violates the paraialit, coma, astigmatism, distortio, field curvature occurs

5 5 Paraial approimatio Paraial approimatio: Law of refractio for fiite agles I, I si-epasio Small icidece agles allows for a liearizatio of the law of refractio Small agles of ras at ever surface si I si I i i i i si i i i i All optical imagig coditios become liear (Gaussia optics), calculatio with ABCD matri calculus is possible No aberratios occur i optical sstems There are o trucatio effects due to trasverse fiite sized compoets Serves as a referece for ideal sstem coditios Is the fudamet for ma sstem properties (focal legth, pricipal plae, magificatio,...) The sag of optical surfaces (differece i z betwee verte plae ad real surface itersectio poit) ca be eglected All waves are plae of spherical (parabolic) i 2 The phase factor of spherical waves is quadratic R E( ) E 0 e

6 6 Paraial approimatio Talor epasio of the si-fuctio Defiitio of allowed error 0-4 Deviatio of the various approimatios: - liear: 5 - cubic: 24-5th order: 542 si() eact si() liear cubic 5th order = 5 = 24 = 52 deviatio 0-4 [ ]

7 7 Paraial Approimatio Law of refractio si I si I Epasioi of the sie-fuctio: si 3 3! ! Liearized approimatio of the law of refractio: I ----> i i i Relative error of the approimatio i i I I si i arcsi i- I) / I =.9 =.7 = i

8 8 Geeralized Paraialit Pitfalls i the classical defiitio of paraialit:. Cetral obscurartio ad rig-shaped pupil: - Paraial margial ra of o relevace - Referece o cetroid ra M F 2. Geeral 3D sstem without straight ais: Cetral ra as referece, calculated fiite parabasal ras i the eighborhood of the real chief ra Distortio iformatio is lost M circular smmetric asphere Geeral quasi-parabasal ras: - macroscopic astigmatism - aberratios referece defiitio more complicated - separated view o cheif ra / margial ra M 2 pupil freeform M 3 freeform image

9 9 Optical imagig Optical Image formatio: All ra emergig from oe obect poit meet i the perfect image poit Regio ear ais: gaussia imagig ideal, paraial Image field size: Chief ra field poit O 2 chief ra pupil stop Aperture/size of light coe: margial ra defied b pupil stop obect ais margial ra optical sstem O O image O 2

10 Sigle surface imagig equatio Thi les i air focal legth Thi les i air with oe plae surface, focal legth Thi smmetrical bi-les Thick les i air focal legth f r s s 2 r r f r f 2 r f r r d r r f Formulas for surface ad les imagig 0

11 Sigle Surface Sigle surface betwee two media Radius r, refractive idices, Imagig coditio, paraial s s r f Abbe ivariat alterative represetatio of the imagig equatio Q s r s r s obect s arbitrar ra verte S C s image r ra through ceter of curvature C pricipal plae surface

12 2 Imagig b a Les Imagig with a les Locatio of the image: les equatio s s f Size of the image: Magificatio m s s obect sstem les image -s f +s

13 3 Imagig b a Les Rages of imagig Locatio of the image for a sigle les sstem s < f image virtual magified image image F Obekt s F Chage of obect loactio s = f F Image could be:. real / virtual 2. elarged/reduced 3. i fiite/ifiite distace image at ifiit 2f > s > f image real magified obect F obect F s F image s s = 2f image real : obect F F image s obect s > 2f F image image real reduced F s

14 4 Imagig equatio s Imagig b a les i air: les makers formula s s f real obect real image 4f 2f virtual image real image Magificatio m s s - 4f -2f 2f 4f s Real imagig: s < 0, s > 0 Itersectio legths s, s measured with respective to the pricipal plaes P, P real obect virtual image -2f virtual obect virtual image - 4f

15 Trasfer Legth / Total Track L Distace obect-image: (trasfer legth) L = s + s L f 2 m m Two solutio for a give L with differet magificatios m 6 L mi = 4f m ma = L / f - 2 m L 2 f 2 L f 2 L f 5 4 No real imagig for L < 4f 3 magified 2 4f-imagig reduced L / f

16 6 Magificatio Lateral magificatio for fiite imagig Scalig of image size m f ta u f ta u pricipal plaes obect focal poit focal poit F P P F z f f z image s s

17 7 Agle Magificatio Afocal sstems with obect/image i ifiit Defiitio with field agle w agular magificatio ta w ta w h h w h w h Relatio with fiite-distace magificatio m f f

18 8 Newto Formula Imagig equatio accordig to Newto: distaces z, z measured relative to the focal poits z z f f focal poit F P P focal poit F image obect -z -f f z -s s pricipal plaes

19 9 Graphical Image Costructio after Listig Graphical image costructio accordig to Listig b 3 special ras: 3 F. First parallel through ais, through focal poit i image space F F 2 2. First through focal poit F, the parallel to optical ais P P 3. Through odal poits, leaves the les with the same agle Procedure work for positive ad egative leses For egative leses the F / F sequece is reversed 3 F 2 F P P

20 20 Geeral Graphical Ra Costructio First ra parallel to arbitrar ra through focal poit, becomes parallel to optical ais Arbitrar ra: - costat height i pricipal plaes S S - meets the first ra i the back focal plae, desired ra is S Q arbitrar ra S S desired output ra Q F parallel ra through F F f P P

21 Two leses with distace d Focal legth distace of ier focal poits e Sequece of thi leses close together Sequece of surfaces with relative ra heights h, paraial Magificatio F F d F F F 2 2 e f f d f f f f f k F k F k k k k k r h h F k k k s s s s s s m 2 2 Multi-Surface Sstems 2

22 22 Pricipal Plaes P P L L 2 F Sstem of two separated thi leses Variatio of the back pricipal plae as a fuctio of the distributio of refractive power plate plate

23 23 Scheimpflug Imagig Imagig with tilted obect plae If pricipal plae, obect ad image plae meet i a commo poit: Scheimpflug coditio, sharp imagig possible Scheimpflug equatio s s ta ta ta ta tilted obect h pricipal plae sstem optical ais s h s tilted image

24 24 Scheimpflug Sstem Geeral :. Image plae is tilted 2. Magificatio is aamorphic Eample : Scheimpflug-Imagig m m m 2 o m o si si s ta s ta obect plae les image plae s s

25 25 Matri Formulatio of Paraial Optics Liear relatio of ra trasport Simple case: free space propagatio ra u Advatages of matri calculus:. simple calculatio of compoet combiatios 2. Automatic correct sigs of properties 3. Eas to implemet u B z Geeral case: paraial segmet with matri ABCD-matri : A u C B M D u u u ra A B C D u z

26 Matri Calculus Paraial ratrace trasfer Matri formulatio Matri formalism for fiite agles Paraial ratrace refractio Iserted Matri formulatio U d U i i i U U U U i i U U U d U 0 U U 0 u D C B A u ta ta 26

27 27 Matri Formulatio of Paraial Optics Liear trasfer of spatio coordiate ad agle u ABu u CDu Matri represetatio A u C B M D u u Lateral magificatio for u=0 A / m Agle magificatio of cougated plaes Refractive power for u=0 D u / u C u / Compositio of sstems MM k M k... M 2 M Determiat, ol 3 variables detm ADBC

28 Sstem iversio Trasitio over distace L Thi les with focal legth f Dielectric plae iterface Afocal telescope A C B D M 0 L M 0 f M 0 0 M 0 L M Matri Formulatio of Paraial Optics 28

29 29 Matri Formulatio of Paraial Optics Calculatio of itersectio legth Magificatios:. lateral 2. agle 3. aial, depth A s B s C s D AD BC m C s D AD BC C s D A C s ds ds AD BC C s D 2 Pricipal plaes Focal poits a H a F AD BC C A C D A a H C D a F C

30 Decompositio of ABCD-Matri 22 ABCD-matri of a sstem i air: 3 arbitrar parameters Ever arbitrar ABCD-setup ca be decomposed ito a simple sstem Decompositio i 3 elemetar partitios is alwa possible Case : C # 0 oe les, 2 trasitios Sstem data M A B L L C D 0 f L f L 2 A C C D C Iput i Les f Output o L 2 L

31 Decompositio of ABCD-Matri Case 2: B # 0 two leses, oe trasitio M A B L 0 C D f 0 0 f 2 Sstem data: f B A L B f 2 B D Iput f Les Les 2 f 2 Output L

32 32 Helmholtz-Lagrage Ivariat Product of field size ad umercial aperture is ivariat i a paraial sstem Derivatio at a sigle refractig surface:. Commo height h: 2. Triagles 3. Refractio: 4. Elimiatio of s, s,w,w L u u h su s u w s, w w w s The ivariace correspods to:. Eerg coservatio 2. Liouville theorem 3. Ivariat phase space volume (area) 4. Costat trasfer of iformatio margial ra u h u w w chief ra s surface s

33 33 Helmholtz-Lagrage Ivariat Product of field size ad umercial aperture is ivariat i a paraial sstem L u u The ivariat L describes to the phase space volume (area) The ivariace correspods to. Eerg coservatio 2. Liouville theorem 3. Costat trasfer of iformatio margial ra obect u chief ra sstem ad stop u image

34 34 Helmholtz-Lagrage Ivariat Basic formulatio of the Lagrage ivariat: Uses image heigth, ol valid i field plaes Geeral epressio:. Triagle SPB 2. Triagle ABO w CR s CR EP w s s EP MR arbitrar z Q S s Ep pupil p s chief ra margial ra image CR B A P O CR z 3. Triagle SQA 4. Gives u s MR L u 5. Fial result for arbitrar z: CR L w s MR MR w ( z) u s s w u w s CR EP ( z) MR EP

35 35 Helmholtz-Lagrage Ivariat Simple eample: - A microscope is a 4f-sstem with obective les (f ob = 3 mm) ad tube les (f ob = 80 mm) - the umerical aperture is NA = 0.9 ad the itermediate Image size D = 2 ima = 25 mm - magificatio - image sided aperture - pupil size - obect field m u ima f TL / f 60 u ob ob / m 0.05 Dpup fob NA2. 7 mm ob ima uima / uob 0. 42mm obective les focal legth 3 mm NA = 0.9 tube les: focal legth 80 mm image diameter 25 mm u ob ob pupil u ima ima

36 36 Helmholtz-Lagrage Ivariat Geometrical optic: Etedue, light gatherig capacit Paraial optic: ivariat of Lagrage / Helmholtz Geeral case: 2D L Geo L D field siu 2 u u Ivariace correspods to coservatio of eerg space Iterpretatio i phase space: costat area, ol shape is chaged at the trasfer through a optical sstem aperture small u u 2 u 3 medium aperture 2 2 large aperture 3 3 agle u

37 37 Phase Space Direct phase space represetatio of ratrace: spatial coordiate vs agle 2 2 space domai 2 2 z phase space u u u 2 u u 2 u

38 38 Phase Space z u I u I

39 39 Phase Space Direct phase space represetatio of ratrace: spatial coordiate vs agle les les z gri 4 free les trasfer free 3 trasfer 3 les 2 5 free trasfer 2 free les 2 trasfer 6 u

40 40 Phase Space Gri les with aberratios i phase space: - cotiuous beded curves - aberratios see as oliear agle or spatial deviatios z u u u

41 4 Ucertait Relatio i Optics. Slit diffractio Diffractio agle iverse to slit width D D D D 2. Gaussia beam Costat product of waist size w o ad divergece agle o w 0 0 o w o z

42 42 Phase Space i Optics Agle u is limited Tpical shapes: Ra : poit (delta fuctio) Coheret plae wave: horizothal lie Eteded source : area Isotropic poit source: vertical lie Gaussia beam: elliptical area with miimal size u LED poit source spherical wave ra gaussia beam plae wave (laser) u Rage of small etedues: modes, discrete structure Rage of large etedues: quasi cotiuum quasi cotiuum discrete mode poits

43 43 Eample Phase Space of a Arra Simplified pictures to the chages of the phase space desit. Etedue is elarged, but o complete fillig. ) before arra u 2) separatio of subapertures u 3) les effect of subapertures u 4) i focal plae u 5) i 2f-plae u 6) far awa u

44 44 Aalog Optics - Mechaics Mechaics Optics spatial variable Impuls variable p=mv agle u, directio cosie p, spatial frequec s, k equatio of motio spatial domai Lagrage Eikoal equatio of motio phase space Hamilto Wiger trasport potetial mass m ide of refractio miimal priciple Hamilto priciple Fermat priciple Liouville theorem costat umber of particles costat eerg sstem fuctio impuls respose poit spread fuctio wave equatio Schrödiger Helmholtz propagatio variable time t space z ucertait h/2p l/2p cotiuum approimatio classical mechaics geometrical optics eact descriptio quatum mechaics wave optics

45 Delao Diagram Special represetatio of ra budles i optical sstems: margial ra height MR vs. chief ra height CR Delao digram gives useful isight ito sstem laout Ever z-positio i the sstem correspods to a poit o the lie of the diagram Iterpretatio eeds eperiece les at pupil positio field les i the focal plae collimator les margial ra les field les collimator chief ra

46 46 Delao Diagram Delao ra (blue)= Chief ra (red) i + Margial ra (gree) i Delao Diagram = Delao ra proected ito the -Plae M a c b d a C Delao s skew ra margial ra chief ra Les b M ( C, M ) c C Image d Substitutio --> Stop d d 2 = Pupil coordiate = c Field coordiate a (or M ) b diagram Delao diagram: proectio alog z c d skew ra chief ra image obect margial ra Ref.: M. Schwab / M. Geiser

47 Delao Diagram Pupil locatios: itersectio poits with -ais eit pupil Field plaes/obect/image: itersectioi poits with -bar ais stop ad etrace pupil les obect plae image plae Costructio of focal poits b parallel lies to iitial ad fial lie through origi frot focal poit F image space obect space rear focal poit F

48 Delao Diagram Ifluece of leses: diagram lie beded weak egative refractive power weak positive refractive power strog positive refractive power Locatio of pricipal plaes pricipal plae P obect space image space P

49 Delao Diagram Afocal Kepler-tpe telescope les obective itermediate focal poit les 2 eepiece Effect of a field les les obective itermediate focal poit field les les 2 eepiece

50 Delao Diagram Microscopic sstem microscope obective aperture stop tube les telecetric obect itermediate image image at ifiit eit pupil eepiece

51 5 Delao Diagram Cougated poit are located o a straight lie through the origi cougate lie cougate lie with m = Distace of a sstem poit from origi gives the sstems half diameter cougate poits pricipal poit obect space image space curve of the sstem les les 2 maimum height of the coma ra at les 2 les 3 D/2

52 Delao Diagram Locatio of pricipal plaes i the Delao diagram P pricipal plae obect space image space P Triplet Effect of stop shift les L les L2 les L3 stop shift obect plae image plae

53 Delao Diagram Vigettig : ra heigth from ais a Margial ad chief ra cosidered sstem polgo lie les les 2 maimum height at les 2 Lie parallel to -45 maimum diameter les 3 D/2 obect coma ra margial ra pupil + chief ra

Design and Correction of Optical Systems

Design and Correction of Optical Systems Deig ad Correctio of Optical Stem Lecture 3: Paraial optic 06-04-0 Herbert Gro Summer term 06 www.iap.ui-ea.de Prelimiar Schedule 06.04. Baic 3.04. Material ad Compoet 3 0.04. Paraial Optic 4 7.04. Optical