Imaging and Aberration Theory

Transcription

1 Imagig ad Aberratio Theory Lectre 10: Sie coditio, alaatim ad ilaatim Herbert Gro Witer term 015

2 Prelimiary time chedle Paraxial imagig araxial otic, fdametal law of geometrical imagig, comod ytem Pil, Forier otic, il defiitio, baic Forier relatiohi, hae ace, aalogy otic ad Hamiltoia coordiate mechaic, Hamiltoia coordiate Eikoal Fermat ricile, tatioary hae, Eikoal, relatio ray-wave, geometrical aroximatio, ihomogeeo media Aberratio exaio igle rface, geeral Taylor exaio, rereetatio, vario order, to hift formla Rereetatio of aberratio differet tye of rereetatio, field of alicatio, limitatio ad itfall, mearemet of aberratio Sherical aberratio heomeology, h-free rface, kew herical, correctio of h, aherical rface, higher order Ditortio ad coma heomeology, relatio to ie coditio, alaatic ytem, effect of to oitio, vario toic, correctio otio Atigmatim ad crvatre heomeology, Coddigto eqatio, Petzval law, correctio otio Chromatical aberratio Dierio, axial chromatical aberratio, travere chromatical aberratio, herochromatim, ecodary oectrm 10 Sie coditio, alaatim ad Sie coditio, iolaatim, relatio to coma ad hift ivariace, il iolaatim aberratio, Herchel coditio, relatio to Forier otic Wave aberratio defiitio, vario exaio form, roagatio of wave aberratio Zerike olyomial ecial exaio for circlar ymmetry, roblem, calclatio, otimal balacig, iflece of ormalizatio, mearemet PSF ideal f, f with aberratio, Strehl ratio Trafer fctio Trafer fctio, reoltio ad cotrat Additioal toic Vectorial aberratio, geeralized rface cotribtio, Aldi theorem, itriic ad idced aberratio, revertability

3 3 Cotet 1. Pil aberratio. Sie coditio 3. Iolaatim 4. Herchel coditio 5. Relatio to Forier otic ad hae ace

4 4 Sie Coditio Lagrage ivariate for araxial agle U, U i-coditio: exteio for fiite aertre agle Correod to eergy coervatio i the ytem Cotat magificatio for alle aertre zoe Pil hae for fiite aertre i a here Defiitio of violatio of the ie coditio: OSC (offee agait ie coditio) OSC = 0 mea correctio of agittal coma (alaatic ytem) y iu yiu y i yi U i m U i y y y U U y z

5 5 Otical Sie Coditio Coditio for fiite agle m iu iu Coditio for obect at ifiity Coditio for afocal ytem I the formlatio m y iu y iu the agittal magificatio i ed x y h f H 1 h1 H k h k U x U h iu y x y U U x y y z y

6 6 Abbe Sie Coditio If for examle a mall field area ad a wideread ray bdle i coidered, a erfect imagig i oible The eikoal with the exreio dl dr dr ca be writte for dl=0 a dr dr dr coq dr coq coq coq I the ecial cae of a agle 90 we get with co(q)=i() the Abbe ie coditio i m i with the lateral magificatio dr m dr Q dr q q Q dr P P

7 7 Derivatio of the Sie Coditio rface P y i Q i margial ray agittal image lae ideal image lae P o S w C w M o P o R chief ray y M M P From geometry P 0 CQ, M o QC i i( i) i i i R R R R M y y R R M M o M CM o PPo P C o Refractio Diviio ad btittio i ii i M R i i i i R i i i i M R y R y yi y i

8 8 Vectorial Sie Coditio Geeral vectorial ie coditio: atial freqecie / directio coie are liear related from etrace to exit il xi yi m x m y xo yo c x c y obect lae image lae o y0 y o q 0 z o yi q i i z i ytem y i Geeralizatio ca be alied for aamorhic ytem

9 9 Trafer of Eergy i Otical Sytem Coervatio of eergy Ivariat local differetial flx Amtio: o abortio Deliver the ie coditio d d P d P P L i co da d d T 1 yi y i y y F F da da EP ExP

10 10 Pil Shere Sie coditio flfilled: liear calig from etrace to exit il Pil rface mt be erical The il height cale with the ie of the agle obect y o etrace il here exit il here image y h EP =R EP i() h ExP =R ExP i() R EP R ExP obect y o herical il rface eqiditat h =R i() agle ot eqiditat

11 11 Pil Ditortio Sie coditio flfilled: liear calig from etrace to exit il Offece agait the ie coditio (OSC): Exit il grid i ditorted Coeqece: 1. Photometric effect cae aodizatio. Wave aberratio cold be calclated wrog 3. Satial filterig o wared grid D f xa i 1 exit il x o obect ditorted etrace il rface here otical ytem x ditorted exit il grid x x

12 1 Pil Ditortio Afocal ytem: x w x w Lagrage ivariat (claical Lagrage ivariat for il imagig) w Magificatio m w Sie coditio x m x Flfillmet of the ie coditio: liear calig of etrace to exit il etrace il x otical ytem exit il x field agle w w x z x

13 13 Sie Coditio i Microcoic Obective Le Tyical high-na ytem Virtal il located iide obect lae il exit il rear to Tyical grid ditortio chief ray 1 x

14 14 OSC ad Aodizatio Photometric effect of il ditortio: iteity 1.8 illmiatio chage at il bodary Effect idce aodizatio barrel Sig of ditortio determie the effect: oter zoe of il brighter / darker Additioal effect: abolte diameter of ichio o ditortio il chage r -50 mm -0 mm foced +0 mm +50 mm

15 15 Pil Aberratio Sherical aberratio of the chief ray / il imagig Exit il locatio deed o the field height chief ray y obect P il oitio il locatio

16 16 Pil Aberratio Iterliked imagig of field ad il Ditortio of obect imagig correod to herical aberratio of the il imagig Corrected herical il aberratio: taget coditio otical ytem ta w cot. ta w O O obect image exit il to ad etrace il Ble ray Red ray Obect imagig Margial ray Chief ray Pil imagig Chief ray Margial ray

17 17 Pil Aberratio Eyeiece with il aberratio itrmet il eyeiece le ad il of the eye retia Illmiatio for decetered il : dark zoe de to vigettig catic of the il image elarged

18 Wavefrot ad Sot for Coma Coma Seidel travere aberratio Wavefrot for coma y S r co q C y r ( co q ) ( A P ) y r co q D y 3 3 P P x S r i q C y r i q P y r iq 3 P W r coq x y y 3 3 P W with x r i q, y r co q y x Relatiohi W y W x, y R x R Here W x R R x y Rr Rr iq coq i q x W y R R x 3y Rr co y q

19 Wavefrot ad Sot for Coma Schematic geometry: Notice the dobled revoltio i the image lae de to combied effect of azimthal rotatio ad tilt of wavefrot 45 exit il 0 y x tagetial coma ray agittal coma ray y 0 coma ot x chief ray 90 image lae wavefrot with coma 45 ray roectio agittal coma ray roectio chief ray wavefrot with coma tagetial coma ray image lae

20 Tagetial ad Sagittal Coma term of tagetial travere aberratio: - Sagittal coma deed o x, decribe the aymmetry - Tagetial coma deed o y, correod to herical aberratio der kew coditio larger by a factor of 3 Oly aymmetry removed with ie coditio: agittal coma vaihe exit il 0 y tagetial coma ray y coma ot 45 x agittal coma ray x y y t chief ray image lae y R x y y y 3 t 90 wavefrot with coma

21 1 Sie Coditio ad Coma Liear coma (all order) Travere aberratio Sagittal coma y = 0, x = a R y c, ag y cma m m R W c,max (a,0, y ) a Wc ( x, y, y) cm y y x y y il c R W y rface R c m R y chief ray m c agittal ray m m x (m 1) y x y er coma ray tagetial coma T S m1 agittal coma S Sie coditio flfilled: liear agittal coma vaihe axiliary axi er coma ray C ceter of crvatre lower coma ray atigmatic differece betwee coma ray If i additio herical aberratio i corrected (alaatic): alo tagetial coma vaihe chief ray lower coma ray agittal ray S T chief ray ot axiliary axi

22 Skew Sherical aberratio Decomoitio of coma: 1. art ymmetrical arod chief ray: kew herical aberratio y kewh y com y lowcom er coma ray y agittal image oit S T chief ray tagetial image oit. aymmetrical art: tagetial coma y tagcoma y com y lowcom lower coma ray exit il ideal image locatio y Skew herical aberratio: - higher order aberratio - catic ymmetric arod chief ray er coma ray chief ray commo iterectio oit exit il lower coma ray ideal image lae

23 3 Alaatic ad Perfect Imagig Perfect imagig o axi de to coic ectio - ot alaatic: liear growth of coma with field ize 100 D ot mm] w i Alaatic: - Perfect tigmatic imagig o axi, herical corrected - liear coma vaihe: good correctio off-axi bt ear to axi - qadratic grow of ot ize de to atigmatim - alaatic ad erfect margial ray qite differet ideal le ideal ray real ray real le i ideal = i real = D ot mm] w i

24 Sherical Corrected Srface Seidel cotribtio of herical aberratio with Relt Vaihig cotribtio: 1. firt bracket: vertex ray. ecod bracket: cocetric 3. bracket: alaatic rface Dicio with the Delao formla. cocetric correod to i = i 3. alaatic coditio correod to i = 4 Q S h h 1 R Q 1 1 R h h S h R k k k SPH SPH U i i i i h U U i i i i i 1 1 1

25 5 Alaatic Srface with Vaihig Sherical Aberratio Alaatic rface: zero herical aberratio: 1. Ray throgh vertex. cocetric 3. Alaatic Coditio for alaatic rface: r Virtal image locatio Alicatio: 1. Microcoic obective le. Iterferometer obective le 0 d hyerboloid oblate ellioid oblate ellioid rolate ellioid + ower erie + ower erie + ower erie + ower erie vertex here cocetric here alaatic S

26 6 Alaatic Lee Alaatic lee Combiatio of oe cocetric ad oe alaatic rface: zero cotribtio of the whole le to herical aberratio Not efl: 1. alaatic-alaatic. cocetric-cocetric beded lae arallel late, early vaihig effect o ray A-A : arallel offet A-C : covergece ehaced C-A : covergece redced C-C : o effect

27 Geeral Alaatic Srface Geeral aroach of Fermat ricile: alaatic rface Carteia oval, 4th order Secial cae OPD = 0: Soltio i herical alaatic rface 7 P S P oval rface r z z r z r ) ( ) ( 0 1 / / / 1 / / / / 0 ) / ( ) ( r z r z z z z r z z r z r z r

28 8 Iolaatim Geeral defiitio of iolaatim: - Ivariace of erformace for mall lateral hift of the field oitio - herical aberratio ot ecearily corrected Ual imle cae: ear to axi Coeqece: - vaihig liear growig coma - catic ymmetrical arod chief ray

29 9 Iolaatim Coditio of Staeble-Lihotzky Sagittal coma aberratio: from the geometry of the figre ad Lagrage ivariat Coditio of Staeble-Lihotzky Problem: - o qatitative meare - oly tagetial ray are coidered - itegral criterio y y i S m i S S m i m i h m P t Q t y margial ray chief ray roectio of agittal coma ray Q y y otical axi exit il lat rface S Q real tagetial image lae P ideal gaia image lae

30 30 Iolaatim from Wave Aberratio Lateral hift of obect oit W dw dy i Chage i image dy P chief ray dy dy R d R d dy dy mdy R R R R Chage of wave aberratio R d dw dy i mdy i R Iolaatim: chage i eqal dw = dw R d i m i R 1 i R d 1 0 m i R

31 Iolaatim Berek coditio of roortioality Berek coicidece coditio Iolaatim i cae of defocig: ca oly be flfilled i oe lae or for telecetricity 31 m S m S i i. i i cot m S z z z z 1 co 1 1 co 1 d

32 3 Piecewie Iolaatim Ivariace of PSF: to be defied Poible otio: 1. relative chage of Strehl. correlatio of PSF Examle for microcoic lee with ad withot flatteig correctio I medim field ize: mall iolaatic atche O axi: large iolaatic area Criteria ot efl at the edge for low erformace Sytem MO lae 100x1.5 iolaatic atch ize i mm Strehl 1% Pf correlatio 0.5% Strehl Plae MO 100x1.5 correlatio MO ot lae 40x0.85 iolaatic atch ize i mm Strehl 1% Pf correlatio 0.5% o axi half field field zoe fll field o lae MO 40x0.85 Strehl correlatio ormalized field oitio

33 33 Offece Agait the Sie Coditio Corady OSC (offee agait ie coditio): - mearemet of deviatio of agittal coma - qatitative validatio of the ie coditio y P 1 CR Q 1 OSC y t y i S 1 y m i t Oly agittal coma coidered i cae of OSC=0 the Staeble-Lihotzkycoditio i atomatically flfilled ExP y t Q Q y y y P ideal z W coma ( y, r,0) r y t OSC OSC allow for the defiitio of rface cotribtio OSC y t i 3y m i i w1 i 1 k ( Qk Q k ) i h k k ( CR) k k k margial ray otical axi chief ray agittal coma ray S Q t Q y t Q y real tagetial image lae P t y o P ideal gaia image lae exit il

34 34 OSC Coma ad iolaatim are trogly coected exit il y x tagetial coma ray y coma ot agittal coma ray x chief ray image lae Vectorial OSC: liear calig of atial freqecie: ertrbatio of the liearity v al 1 m v v v v al 1 al x, y W

35 35 Geeral Ivariat of Welford Rotatio arod axi for mall agle calclatio of chage i wave aberratio Welford coditio (calar trile rodct) dw d, D, e d, D, e All other coditio ca be obtaied a ecial cae: 1. ie coditio. off axi iolaatim 3. Herrchel coditio 4. Smith co-ivariat axi of rotatio Srface d d ray D q D q

36 36 Co-Coditio of Smith From Eikoal theory: Geeral coditio of Smith: Ivariace of the calar rodct d e d e dr coq dr coq dr P Q q P dr Q q Secial cae: P o axi, q = 90 : Secial cae: P o axi, q = 0 : Abbe ie coditio, ivariat travere magificatio Herchel coditio, ivariat axial magificatio

37 37 Herchel Coditio Herchel coditio: Ivariace of the deth magificatio z i z i I ricile ot comatible with the ie coditio Therefore a erfect imagig of a volme i imoible P dz Q P dz Q

38 38 Overview Alaatim-Iolaatim Overview o coditio for aberratio ad alaatim-iolaatim Nr Sie cod. Iolaat cod. Iolaatim coditio Sherical aberratio Sagittal coma Tagetial coma Imagig ytem 1 # # # # # geeral a # OSC=0, Corady # 0 # iolaatic-i b # Staeble-Lihotzky / Berek # 0 0 iolaatic-ii 3a 0 0 axial alaatic 3b 0 (kew) 0 0 off-axi alaatic Iolaatim Corady OSC Iolaatim Staeble- Lihotzky ie coditio off-axi Alaatim ie coditio axial Alaatim Tagetial coma 0 0 Sagittal coma Sherical aberratio 0 Skew Sherical aberratio 0 0

39 39 Overview Overview o ivariat ad coditio geeral ivariace (Welford) oly tralatio oly travere tralatio co-law (Smith) chage obect oitio dy off axi iolaatim ecial o axi y=0 axi iolaatim (Staeble-Lihotzki / Berek) ecial for h=0 ie coditio (Abbe) tralatio alog z off axi z-ivariace (Herrchel ecial o axi x=y=0 o axi z-ivariace (claical Herrchel)

40 40 Phae Sace: 90 -Rotatio Traitio il-image lae: 90 rotatio i hae ace Plae Forier ivere Margial ray: ace coordiate x ---> agle q Chief ray: agle q ---> ace coordiate x Forier lae il image locatio margial ray x q x q chief ray f

41 41 Nearfield - Farfield f-et: Forier-cogated lae Agle ad atial freqecy are eqivalet q v obect ace atial domai coordiate x x il Forier domai agle/freqecy x Agle- ad atial coordiate are iterchaged: x ---> q q ---> x Correod to earfield <---> farfield Relatiohi: x q, x f q f f f

42 4 Helmholtz-Lagrage Ivariat Prodct of field ize y ad mercial aertre i ivariat i a araxial ytem L y y The ivariat L decribe to the hae ace volme (area) The ivariace correod to 1. Eergy coervatio. Lioville theorem 3. Cotat trafer of iformatio margial ray obect y y chief ray ytem ad to image

43 43 Helmholtz-Lagrage Ivariat Baic formlatio of the Lagrage ivariat: Ue image heigth, oly valid i field lae Geeral exreio: 1. Triagle SPB. Triagle ABO w y CR y CR ExP w ExP y MR arbitrary z y Q S Ex il y chief ray margial ray image y CR B A P y O y CR z 3. Triagle SQA 4. Give y MR L y 5. Fial relt for arbitrary z: CR L w y y MR MR w ( z) y y w w CR ExP ( z) MR ExP

44 44 Photometry i Phae Sace Radiatio traort i otical ytem Phae ace area chage it hae Fiite chief ray agle: arallelogram geometry i i iw y y le to y y y y y U w U

45 Aberratio i Phae Sace 45 Agle diviatio de to aberratio i the il il x Icreaed atial extetio i the foc regio foc x x

46 46 Ray Catic Secial cae of vaihig determiate of Jacobia matrix: ray catic Siglare oltio of the wave eqatio Two ray directio i oe oit Secial characterizatio with More- ad Malov idex catic x c x

47 47 Ucertaity Relatio i Otic 1. Slit diffractio q Diffractio agle ivere to lit q width D q D D D. Gaia beam x Cotat rodct of wait ize w o ad divergece agle q o w 0 q0 q o w o z

48 48 Light Sorce i Phae Sace Agle i limited Tyical hae: Ray : oit (delta fctio) Coheret lae wave: horizothal lie Exteded orce : area Iotroic oit orce: vertical lie Gaia beam: ellitical area with miimal ize LED oit orce herical wave ray gaia beam lae wave (laer) x qai cotim Rage of mall etede: mode, dicrete trctre Rage of large etede: qai cotim dicrete mode oit x