Design and Correction of Optical Systems

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1 Deig ad Correctio of Optical Stem Lecture 3: Paraial optic Herbert Gro Summer term 06

2 Prelimiar Schedule Baic Material ad Compoet Paraial Optic Optical Stem Geometrical Aberratio Wave Aberratio PSF ad Trafer fuctio Further Performace Criteria Optimizatio ad Correctio Correctio Priciple I Correctio Priciple II.06. Optical Stem Claificatio Law of refractio, Freel formula, optical tem model, ratrace, calculatio approache Diperio, aormal diperio, gla map, liquid ad platic, lee, mirror, aphere, diffractive elemet Paraial approimatio, baic otatio, imagig equatio, multi-compoet tem, matri calculatio, Lagrage ivariat, phae pace viualizatio Pupil, ra et ad amplig, aperture ad vigettig, telecetricit, mmetr, photometr Logitudial ad travere aberratio, pot diagram, polomial epaio, primar aberratio, chromatical aberratio, Seidel urface cotributio Fermat priciple ad Eikoal, wave aberratio, epaio ad higher order, Zerike polomial, meauremet of tem qualit Diffractio, poit pread fuctio, PSF with aberratio, optical trafer fuctio, Fourier imagig model Lie of ight, apodizatio, edge ad lie, pupil aberratio, ie coditio, iduced aberratio, vectorial aberratio Fourier imagig, cautic Priciple of optimizatio, iitial etup, cotrait, eitivit, optimizatio of optical tem, global approache Smmetr, le bedig, le plittig, pecial optio for pherical aberratio, atigmatim, coma ad ditortio, aphere Field flatteig ad Petzval theorem, chromatical correctio, achromate, apochromate, eitivit aali, diffractive elemet Overview, photographic lee, microcopic obective, lithographic tem, eepiece, ca tem, telecope, edocope Special Stem Eample Zoom tem, cofocal tem Further Topic New tem developmet, moder aberratio theor,...

3 3 Cotet. Paraial approimatio. Ideal urface ad lee 3. Imagig equatio 4. Matri formalim 5. Lagrage ivariat 6. Phae pace coideratio

4 Modellig of Optical Stem Pricipal purpoe of calculatio: Imagig model with level of refiemet Stem, data of the tructure (radii, ditace, idice,...) Aali imagig aberratio theorie Sthei le deig Fuctio, data of propertie, qualit performace (pot diameter, MTF, Strehl ratio,...) Paraial model (focal legth, magificatio, aperture,..) liear approimatio Aaltical approimatio ad claificatio (aberratio,..) Talor epaio Geometrical optic (travere aberratio, wave aberratio, ditortio,...) with diffractio approimatio --> 0 Wave optic (poit pread fuctio, OTF,...) Ref: W. Richter

the paraialit, pherical aberratio occur Large field agle w

5 Paraial Approimatio Paraialit i give for mall agle relative to the optical ai for all ra Large umerical aperture agle u violate the paraialit, pherical aberratio occur Large field agle w violate the paraialit, coma, atigmatim, ditortio, field curvature occur

6 Paraial approimatio Paraial approimatio: Small agle of ra at ever urface Small icidece agle allow for a liearizatio of the law of refractio All optical imagig coditio become liear (Gauia optic), calculatio with ABCD matri calculu i poible No aberratio occur i optical tem There are o trucatio effect due to travere fiite ized compoet Serve a a referece for ideal tem coditio I the fudamet for ma tem propertie (focal legth, pricipal plae, magificatio,...) The ag of optical urface (differece i z betwee verte plae ad real urface iterectio poit) ca be eglected All wave are plae of pherical (parabolic) The phae factor of pherical wave i quadratic E( ) E i i 0 e i R

7 7 Paraial approimatio Talor epaio of the i-fuctio Defiitio of allowed error 0-4 Deviatio of the variou approimatio: - liear: 5 - cubic: 4-5th order: 54 i() eact i() liear cubic 5th order = 5 = 4 = 5 deviatio 0-4 [ ]

8 Paraial Approimatio Law of refractio i I i I Epaioi of the ie-fuctio: i 3 3! ! Liearized approimatio of the law of refractio: I ----> i i i Relative error of the approimatio i i I I i i arci i- I) / I =.9 =.7 = i

9 Optical imagig Optical Image formatio: All ra emergig from oe obect poit meet i the perfect image poit Regio ear ai: gauia imagig ideal, paraial Image field ize: Chief ra field poit O chief ra pupil top Aperture/ize of light coe: margial ra defied b pupil top obect ai margial ra optical tem O O image O

10 Sigle urface imagig equatio Thi le i air focal legth Thi le i air with oe plae urface, focal legth Thi mmetrical bi-le Thick le i air focal legth f r r r f r f r f r r d r r f Formula for urface ad le imagig

11 Sigle Surface Sigle urface betwee two media Radiu r, refractive idice, Imagig coditio, paraial r f Abbe ivariat alterative repreetatio of the imagig equatio Q r r obect arbitrar ra verte S C image r ra through ceter of curvature C pricipal plae urface

12 Imagig b a Le Imagig with a le Locatio of the image: le equatio f Size of the image: Magificatio m obect tem le image - f +

13 Imagig b a Le Rage of imagig Locatio of the image for a igle le tem < f image virtual magified image image F Obekt F Chage of obect loactio = f F Image could be:. real / virtual. elarged/reduced 3. i fiite/ifiite ditace image at ifiit f > > f image real magified obect F obect F F image = f image real : obect F F image obect > f F image image real reduced F

14 Imagig equatio Imagig b a le i air: le maker formula f real obect real image 4f f virtual image real image Magificatio m - 4f -f f 4f Real imagig: < 0, > 0 Iterectio legth, meaured with repective to the pricipal plae P, P real obect virtual image -f virtual obect virtual image - 4f

15 Magificatio Lateral magificatio for fiite imagig Scalig of image ize m f ta u f ta u pricipal plae obect focal poit focal poit F P P F z f f z image

16 Agle Magificatio Afocal tem 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-ditace magificatio m f f

17 Newto Formula Imagig equatio accordig to Newto: ditace z, z meaured relative to the focal poit z z f f focal poit F P P focal poit F image obect -z -f f z - pricipal plae

18 Graphical Image Cotructio after Litig Graphical image cotructio accordig to Litig b 3 pecial ra: 3 F. Firt parallel through ai, through focal poit i image pace F F. Firt through focal poit F, the parallel to optical ai P P 3. Through odal poit, leave the le with the ame agle Procedure work for poitive ad egative lee For egative lee the F / F equece i revered 3 F F P P

19 Geeral Graphical Ra Cotructio Firt ra parallel to arbitrar ra through focal poit, become parallel to optical ai Arbitrar ra: - cotat height i pricipal plae S S - meet the firt ra i the back focal plae, deired ra i S Q arbitrar ra S S deired output ra Q F parallel ra through F F f P P

20 Two lee with ditace d Focal legth ditace of ier focal poit e Sequece of thi lee cloe together Sequece of urface with relative ra height h, paraial Magificatio F F d F F F 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 m Multi-Surface Stem

21 Pricipal Plae P P L L F Stem of two eparated thi lee Variatio of the back pricipal plae a a fuctio of the ditributio of refractive power plate plate

22 Scheimpflug Imagig Imagig with tilted obect plae If pricipal plae, obect ad image plae meet i a commo poit: Scheimpflug coditio, harp imagig poible Scheimpflug equatio ta ta ta ta tilted obect h pricipal plae tem optical ai h tilted image

23 Scheimpflug Stem Geeral :. Image plae i tilted. Magificatio i aamorphic Eample : Scheimpflug-Imagig m m m o m o i i ta ta obect plae le image plae

24 Matri Formulatio of Paraial Optic Liear relatio of ra traport Simple cae: free pace propagatio ra u Advatage of matri calculu:. imple calculatio of compoet combiatio. Automatic correct ig of propertie 3. Ea to implemet u B z Geeral cae: paraial egmet with matri ABCD-matri : A u C B M D u u u ra A B C D u z

25 Matri Calculu Paraial ratrace trafer Matri formulatio Matri formalim for fiite agle Paraial ratrace refractio Ierted 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 Matri Formulatio of Paraial Optic Liear trafer of patio coordiate ad agle u ABu u CDu Matri repreetatio A u C B M D u u Lateral magificatio for u=0 A / m Agle magificatio of cougated plae Refractive power for u=0 D u / u C u / Compoitio of tem MM k M k... M M Determiat, ol 3 variable detm ADBC

27 Stem iverio Traitio over ditace L Thi le with focal legth f Dielectric plae iterface Afocal telecope A C B D M 0 L M 0 f M 0 0 M 0 L M Matri Formulatio of Paraial Optic

28 Matri Formulatio of Paraial Optic Calculatio of iterectio legth Magificatio:. lateral. agle 3. aial, depth A B C D AD BC m C D AD BC C D A C d d AD BC C D Pricipal plae Focal poit a H a F AD BC C A C D A a H C D a F C

29 Helmholtz-Lagrage Ivariat Product of field ize ad umercial aperture i ivariat i a paraial tem Derivatio at a igle refractig urface:. Commo height h:. Triagle 3. Refractio: 4. Elimiatio of,,w,w L u u h u u w, w w w The ivariace correpod to:. Eerg coervatio. Liouville theorem 3. Ivariat phae pace volume (area) 4. Cotat trafer of iformatio margial ra u h u w w chief ra urface

30 Helmholtz-Lagrage Ivariat Product of field ize ad umercial aperture i ivariat i a paraial tem L u u The ivariat L decribe to the phae pace volume (area) The ivariace correpod to. Eerg coervatio. Liouville theorem 3. Cotat trafer of iformatio margial ra obect u chief ra tem ad top u image

31 Helmholtz-Lagrage Ivariat Baic formulatio of the Lagrage ivariat: Ue image heigth, ol valid i field plae Geeral epreio:. Triagle SPB. Triagle ABO w CR CR EP w EP MR arbitrar z Q S Ep pupil p chief ra margial ra image CR B A P O CR z 3. Triagle SQA 4. Give u MR L u 5. Fial reult for arbitrar z: CR L w MR MR w ( z) u w u w CR EP ( z) MR EP

32 Helmholtz-Lagrage Ivariat Simple eample: - A microcope i a 4f-tem with obective le (f ob = 3 mm) ad tube le (f ob = 80 mm) - the umerical aperture i NA = 0.9 ad the itermediate Image ize D = ima = 5 mm - magificatio - image ided aperture - pupil ize - obect field m u ima f TL / f 60 u ob ob / m 0.05 Dpup fob NA. 7 mm ob ima uima / uob 0. 4mm obective le focal legth 3 mm NA = 0.9 tube le: focal legth 80 mm image diameter 5 mm u ob ob pupil u ima ima

33 Helmholtz-Lagrage Ivariat Geometrical optic: Etedue, light gatherig capacit Paraial optic: ivariat of Lagrage / Helmholtz Geeral cae: D L Geo L D field iu u u Ivariace correpod to coervatio of eerg pace Iterpretatio i phae pace: cotat area, ol hape i chaged at the trafer through a optical tem aperture mall u u u 3 medium aperture large aperture 3 3 agle u

34 Phae Space Direct phae pace repreetatio of ratrace: patial coordiate v agle pace domai z phae pace u u u u u u

35 Phae Space z u I u I

36 Phae Space Direct phae pace repreetatio of ratrace: patial coordiate v agle le le z gri 4 free le trafer free 3 trafer 3 le 5 free trafer free le trafer 6 u

37 Phae Space Gri le with aberratio i phae pace: - cotiuou beded curve - aberratio ee a oliear agle or patial deviatio z u u u

38 Ucertait Relatio i Optic. Slit diffractio Diffractio agle ivere to lit width D D D D. Gauia beam Cotat product of wait ize w o ad divergece agle o w 0 0 o w o z

Design and Correction of Optical Systems

Design and Correction of Optical Systems Desig ad Correctio of Optical Sstems Lecture 3: Paraial optics 207-04-28 Herbert Gross Summer term 207 www.iap.ui-ea.de 2 Prelimiar Schedule - DCS 207 07.04. Basics 2 2.04. Materials ad Compoets 3 28.04.