Imaging and Aberration Theory

Transcription

1 Imagig ad Aberratio Theory Lecture 9: Chromatical aberratio 07-- Herbert Gro Witer term 07

2 Prelimiary time chedule 6.0. Paraxial imagig paraxial optic, fudametal law of geometrical imagig, compoud ytem Pupil, ourier optic, pupil defiitio, baic ourier relatiohip, phae pace, aalogy optic ad 3.0. Hamiltoia coordiate mechaic, Hamiltoia coordiate Eikoal ermat priciple, tatioary phae, Eikoal, relatio ray-wave, geometrical approximatio, ihomogeeou media Aberratio expaio igle urface, geeral Taylor expaio, repreetatio, variou order, top hift formula Repreetatio of aberratio differet type of repreetatio, field of applicatio, limitatio ad pitfall, meauremet of aberratio Spherical aberratio pheomeology, ph-free urface, kew pherical, correctio of ph, apherical urface, higher order Ditortio ad coma pheomeology, relatio to ie coditio, aplaatic ytem, effect of top poitio, variou topic, correctio optio Atigmatim ad curvature pheomeology, Coddigto equatio, Petzval law, correctio optio 9.. Chromatical aberratio Diperio, axial chromatical aberratio, travere chromatical aberratio, pherochromatim, ecodary poectrum 0 Sie coditio, aplaatim ad Sie coditio, ioplaatim, relatio to coma ad hift ivariace, pupil 8.. ioplaatim aberratio, Herchel coditio, relatio to ourier optic Wave aberratio defiitio, variou expaio form, propagatio of wave aberratio 5.0. Zerike polyomial pecial expaio for circular ymmetry, problem, calculatio, optimal balacig, ifluece of ormalizatio, meauremet 3.0. Poit pread fuctio ideal pf, pf with aberratio, Strehl ratio Trafer fuctio trafer fuctio, reolutio ad cotrat Additioal topic Vectorial aberratio, geeralized urface cotributio, Aldi theorem, itriic ad iduced aberratio, revertability

3 3 Cotet. Material diperio. Partial diperio 3. Aomalou partial diperio 4. Axial chromatical error 5. Achromatic 6. Apochromate 7. Spherochromatim 8. Chromatical variatio of magificatio 9. Example

4 4 Diperio ad Abbe umber Decriptio of diperio: refractive idex.8 viible Abbe umber C.75 D large Viual rage of wavelegth: typically d,,c or e,,c ued.7.65 S lit e e C.6.55 Typical rage of glae e = Two fudametal type of gla: Crow glae: mall, large, diperio low lit glae: large, mall, diperio high.5.45 D mall BK7 crow [mm]

5 Curvature c of the radii of a le ocal power at the ceter wavelegth e for a thi le Differece i focal power for outer wavelegth, C with the Abbe umber ocal legth at the ceter wavelegth Differece of the focal legth for outer wavelegth Achromatizatio coditio for two thi lee cloe together Abbe Number ad Achromatizatio, r c r c c c c e e e D ) ( ) )( ( e e e e C C C c c D D D ) ( ) ( c f e e e D ) ( e e e C C C C f c c f f f D D D ) ( ) )( ( C e e 0 D f f 5

6 6 Gla Diagram Uual repreetatio of glae: diagram of refractive idex v diperio () Left to right: Icreaig diperio decreaig Abbe umber

7 7 Diperio Material with differet diperio value: - Differet lope ad curvature of the diperio curve - Stroger chage of idex over wavelegth for large diperio - Iverio of idex equece at the boudarie of the pectrum poible refractive idex flit mall lope large.65 crow large lope mall.65 SK8A.6 VIS

8 Atomic model for the refractive idex: ocillator approach of atomic field iteractio Sellmeier diperio formula: correpodig fuctio Special cae of coupled reoace: example quartz, degeerated ocillator Atomic Model of Diperio i r i c f m c Ne i log [mm] viible (UV) (UV) 3 (IR) 4 (IR) vi () C B A 4 0 o C B B A 8

9 9 Diperio formula Schott formula empirical Sellmeier Baed o ocillator model 4 6 a a a a a a o ( ) A B C Bauch-Lomb empirical Herzberger Baed o ocillator model 4 D E ( ) A B C ( o) a a3 ) ao a ( o o mit 0.68 mm o o Hartma Baed o ocillator model ( ) a o a a 3 a4 a 5

10 0 Relative partial diperio Relative partial diperio : Chage of diperio lope with Differet curvature of diperio curve Defiitio of local lope for elected wavelegth relative to ecodary color P C i - g g - - e - C C - C - t () Special -electio for characteritic rage of the viible pectrum.49 = 656 / 04 m far IR = 656 / 85 m ear IR = 486 / 546 m blue edge of VIS = 435 / 486 m ear UV = 365 / 435 m far UV.48 i : 365 m UV edge g : 435 m UV edge e : 546 m d : 588 m mai color : 480 m C : 644 m : 486 m C : 656 m. ecodary color. ecodary color : 85 m IR edge t : 04 m IR edge

11 Partial Diperio ad Normal Lie The relative partial diperio chage approximately liear with the diperio for glae P b, a, d, P 0.6 Nearly all glae are located o the ormal lie i a P--diagram P g The lope of the ormal lie deped o the electio of wavelegth 0.55 Glae apart from the ormal lie how aomalou partial diperio DP 0.5 P C P a d b DP thee material are importat for chromatical correctio of higher order

12 Partial Diperio Aormal partial diperio ad ormal lie P g, N-K5 N-PK5 ormal lie GG375G34 N-BA5 N-BA3 N-BA5 K5G0 N-BAK4 N-BA0 N-SK N-SK8 N-LA3 SK0G0 N-BAL5 N-LL6 BAKG N-SSK8 SK4G3 N-BAL4 N-SK5 SSK5G06 N-SK4 N-SSK5 N-K5 SK5 N-BAK N-K9 K7 N-SK N-PK5 N-SK5 N-PSK53 N-PSK57 N-PSK58 BK7G5 N-PSK3 N-K5 N-BK0 BK7G8 N-BK7 N-LAK7 N-LAK N-ZK7 N-SK6 N-PSK3 N-SK4 SK5G06 N-LL N-LA N-BA4 BAS5 5 N-BAK N-LAK4 S5 N-LA7 N-S64 N-S8 N-S5 N-S9 N-LAS40 N-L5 N-BAS N- G N-LAK33 N-LAK N-LAK9 N-LAK LAKL N-SK0 N-S N-S0 N-S5 S0 N-LAK8 S S5 N-LAS45 N-LAS36 LAN7 N-KZS N-BAS64 L5G5 L5 KZSN5 N-LAS43 N-LAS3 N-LA LL N-LAS4 N-LA33 N-KZS KZSN4 KZS4G0 N-LAS30 N-LAS44 N-KZS4 N-LA N-LA3 LAK9G5 K0 N-LAK34 N-KZS N-S4 N-S6 S4 N-S57 N-LA35 N-LA8 N-LA34 N-LAK0 N-SSK LAKN3 S66 SL57 S57 S S N-S56 S6G05 S6 S56A S4 N-LAS35 S8G07 N-LAS46 LASN9 S5G0

13 3 Aomalou Partial Diperio There are ome pecial glae with a large deviatio from the ormal lie Of pecial iteret: log crow ad hort flit P g, lie of ormal diperio S N-S57 KZSN4 K5 K5 PSK53A ZKN7 LAK8 LASN30 P g, heavy flit with character of log crow flit log crow log crow hort flit hort flit crow ormal lie

14 4 Aomalou Partial Diperio Normal glae: Partial diperio chage liear with Abbe umber Defiitio of P deped o elected wavelegth Normal lie defied by ad K7 P P P P P C, t C,, e g, i, g d d d d d Deviatio from liear behavior: aomalou partial diperio DP P a d b DP P g D d The value of DP deped o the wavelegth electio Typical DP coidered at the red ad the blue ed of the viible pectrum ormal lie D P g real curve Large deviatio value DP are eceary for apochromatic chromatical correctio d

15 5 Aomalou Partial Diperio Arrow i the gla map: idicatio of the deviatio from the ormal lie P h Vertical compoet: at the red horizotal: at the blue ed of the pectrum P a d b DP ormal lie Gla D d DP h arrow of deviatio DP tc d gla locatio DP h blue ide red ide d

16 6 Chromatical Aberratio Axial chromatical aberratio: - diperio of margial ray - differet image locatio Travere chromatical aberratio: - diperio of chief ray - differet image ize obect ideal image ideal image margial ray chief ray margial ray axial chromatical aberratio ExP ExP chief ray travere chromatical aberratio

17 7 Overview o Chromatical Aberratio. Primary/t order chromatical aberratio: - axial chromatical aberratio error of the margial ray by diperio - travere chromatical aberratio error of the chief ray by diperio. Higher order chromatical aberratio: - ecodary pectrum reidual axial error, if oly elected wavelegth are coicidig - pherochromatim chromatical variatio of the pherical aberratio, oberved i a achromate - chromatical variatio of all moochromatic aberratio e.g. atigmatim, coma, pupil locatio,...

18 8 Chromatical Aberratio Variou cae of chromatical aberratio correctio a) axial ad lateral color corrected b) axial color corrected MR CR C C C C c) lateral color corrected d) o color corrected C C C C

19 9 Axial Chromatical Aberratio Axial chromatical aberratio: Higher refractive idex i the blue reult i a horter iterectio legth for a igle le The colored image are defocued alog the axi Defiitio of the error: chage i image locatio / iterectio legth Correctio eed everal glae with differet diperio Sigle le: ormal diperio blue iterectio legth i horter tha red P Notatio: white. CHL = chromatical logitudial. AXCL = axial chromatic D CHL C e blue C gree red

20 0 Axial Chromatical Aberratio Logitudial chromatical aberratio for a igle le Bet image plae chage with wavelegth bet image plae = 648 m = 546 m = 480 m defocu z Ref : H. Zügge

21 Secodary Spectrum P Simple achromatizatio / firt order correctio: - two glae with differet diperio - equal iterectio legth for outer wavelegth (blue, red C) white C ecodary pectrum Reidual deviatio for middle wavelegth (gree e): ecodary pectrum ( ) P, C P D SSP C f ( ), C 644 C e achromate blue red gree D 546 e iglet reidual error achromate e C 480 D

22 Achromate: Baic ormula Idea:. Two thi lee cloe together with differet material. Total power 3. Achromatic correctio coditio 0 Idividual power value Propertie:. Oe poitive ad oe egative le eceary. Two differet equece of plu (crow) / miu (flit) 3. Large -differece relaxe the bedig 4. Achromatic correctio idipedet from bedig 5. Bedig correct pherical aberratio at the margi 6. Aplaatic coma correctio for pecial gla choice 7. urther optimizatio of material reduce the pherical zoal aberratio

23 3 Achromate Compeatio of axial colour by appropriate gla choice (a) (b) Chromatical variatio of the pherical aberratio: pherochromatim (Gauia aberratio) Therefore perfect axial color correctio (o axi) are ofte ot feaable BK7 =.568 = 64.7 = BK7 = = = = = = r p r p 486 m 588 m 656 m Dz Dz -00 Ref : H. Zügge

24 4 Achromate Achromate Logitudial aberratio Travere aberratio Spot diagram Dy 486 m 587 m 656 m = 486 m axi r p = 587 m = 656 m iu m 587 m 656 m D [mm]

25 Achromate: Correctio Cemeted achromate: 6 degree of freedom: 3 radii, idice, ratio / D MR aplaatic cae Correctio of pherical aberratio: divergig cemeted urface with poitive pherical cotributio for eg > po Choice of gla: poible goal. aplaatic coma correctio. miimizatio of pherochromatim 3. miimizatio of ecodary pectrum Bedig ha o impact o chromatical correctio: i ued to correct pherical aberratio at the edge Three olutio regio for bedig. o pherical correctio. two equivalet olutio 3. oe aplaatic olutio, very table cae without olutio, oly pherical miimum R cae with olutio cae with oe olutio ad coma correctio

26 6 Bedig of a Achromate - Aplaatic Cae D too mall: o pherical correctio Aplaatic cae: ame zero poit of bedig for coma ad pherical aberratio oly oe olutio for pherical Large D: Two olutio for bedig with corrected pherical correctio o coicidece with coma zero poit aberratio 0 aplaatic lie of crow pherical aberratio crow o pherical correctio due to mall D aplaatic cae coma flit alterative value for the crow gla coma corrected o pherical correctio due to mall D X aplaatic cae 3 pherical aberratio crow crow / / / / bedig for correctio X zero

27 7 Aplaatic Achromate Appropriate gla combiatio for aplaatic correctio Cae of NA = 0. with Rayleigh rage R u = mm Compario of reidual aberratio lit Crow Vedor crow coma z 8 zoal pherical [mm] ecodary pectrum [mm] S66 EL4 Hoya S5 BAL5 Ohara S6 N-BAL5 Schott S57 KZ Schott S58 S-TIL Ohara S ADC Hoya S AD Hoya

28 8 Achromatic Solutio i the Gla Diagram large -differece give relaxed bedig crow poitive le flit egative le Achromat

29 9 Achromate Correctio axial color require a larger -differece i the gla map: if the differece D become maller, the axial focal power are icreaig Correctio of the pherical aberratio require a igificat maller i the poitive crow le e) N-LAK33 LAK9 d) K0 good olutio a) D à 0 TI3 b) D < 0 D mall TIN5 c) D very mall S

30 or oe give flit a lie idicate the uefull crow glae ad vice vera Perfect aplaatic lie of correpodig glae (corrected for coma) Coditio: Optimizatio of Achromatic Glae fixed flit gla lie of miimal pherical aberratio fixed crow gla lie of miimal pherical aberratio r 30

31 3 Achromate Reidual aberratio of a achromate Clearly ee:. Ditortio. Chromatical magificatio 3. Atigmatim

32 Surface ad Le cotributio of Axial Color Coiderig the Abbe ivariat Derivative after the wavelegth Summig over all urface of a ytem with the margial ray height ratio ad the propagatio of the ratio Surface ummatio for axial chromatical aberratio with the urface cotributio coefficiet h h d d r d d d d r d d r r Q Q D D D N N N CHL r r D N CHL N N N N N N CHL K Q CHL Q K 3

33 33 Geeral Achromatizatio Cotributio of a thi le to the axial chromatical aberratio Axial chromatical aberratio of a ytem of thi lee K D CHL le CHL N f Coditio of achromatizatio of a ytem of lee 0 Special cae of lee cloe together 0 Coditio of apochromatic (polychromatic) correctio with the partial relative diperio P 0

34 Dialyt approach: Achromatizatio with two lee at fiite ditace Scalig parameter k: With fiite margial ray height ocal legth coditio Achromatizatio ocal legth of the lee Le ditace a f t k a f b k f f 0 b b b a a a f y f y k f f a b a b a b k k f f f k k d a b ) ( Dialyt-Achromat 34

35 35 Dialyt Achromat Uage of oly oe gla material with achromatic correctio: dialyt achromate No real imagig poible Parameter: Setup kf f a f b k f ( k ) k le a k t k f le b image plae y a y b t f a

36 36 Axial Color Correctio with Schupma Le Special layout of dialyte approach accordig to Schupma Mirror guaratee real imagig f = -00 mm mirror f = 300 mm real image

37 37 Axial Colour : Apochromate Choice of at leat oe pecial gla P g Correctio of ecodary pectrum: aomalou partial diperio 0,6 0,60 N-S6 () At leat oe gla hould deviate igificatly form the ormal gla lie 0,58 0,56 ()+() T N-KZS (3) 656m 588m 0,54 () 90 N-K m -0.mm Dz -0.mm 436m 0 mm Dz

38 ocal power coditio Achromatic coditio Secodary pectrum Curvature of lee Parameter E The 3 material are ot allowed to be o the ormal lie The triagle of the 3 poit hould be large: mall c give relaxed deig P P P r r c 3,, a a c b c a a P P E f c 3,, b b a c c a b P P E f c 3,, c c b a c a c P P E f c [ ] b a c a c b c b a c a P P P P P P E Apochromate 38

39 39 Relative Partial Diperio Preferred gla electio for apochromate N-S N-S6 N-S57 N-S66 P-S68 P-S67 N-K5A N-PK5A N-PK5 N-KZS N-KZS4 N-LA33 N-LAS4 N-LA37 N-LA N-LA35 N-LAK0 N-KZS

40 40 Axial Colour: Achromate ad Apochromate Effect of differet material Axial chromatical aberratio chage with wavelegth Differet level of correctio:.no correctio: le, oe zero croig poit.achromatic correctio: - coicidece of outer color - remaiig error for ceter wavelegth - two zero croig poit 3. Apochromatic correctio: - coicidece of at leat three color - mall reidual aberratio - at leat 3 zero croig poit - pecial choice of gla type with aomalou partial dipertio eceery apochromate iglet C reidual error apochromate e reidual error achromate achromate D le

41 4 Spherochromatim Spherochromatim: variatio of pherical aberratio with wavelegth, Alterative otatio: Gauia chromatical error Idividual curve of pherical aberratio with color Covetioal achromate: - coicidig image locatio for red (C ) ad blue ( ) o axi (paraxial) - differece ad ecodary pectrum for gree (e) - but differet iterectio legth for fiite aperture ray r p Better balacig with half pherochromatim o axi 480 m 644 m aperture m 480 m 546 m 644 m D i R U 0 D ec 0. mm D 0. mm D chl D tot

42 Spherochromatim: Correctio by plitted achromate Split of cemeted urface: reduced zoal reidual aberratio poible a) Claical achromate Larger ditace of air gap: reduced pherochromatim Correctio priciple: Differet ray height at ecod le ad differet depedecie o ray height: ocu Spherical aberratio ~ ~ 4 b) Splitted achromate zoe mall Dy red blue c) Splitted achromate with large air gap pherochromatim mall Ref: D. Oche 4

43 43 New Achromate Covetioal achromate: trog bedig of image hell, typical R ptz.3 f Petzval hell mea image hell y Special electio of glae:. achromatizatio. Petzval flatteig Reidual field curvature: Combied coditio R ptz But uually o pherical correctio poible 0 0 f f R P elected crow gla perfect image plae lie of olutio for flit gla

44 44 New Achromate Thi coditio correpod to the requiremet to fid two glae o oe traight lie through the origi i the gla map D 0 Abbe umber PSK5 LAK33 K5 LL LAS40 S idex Example: K5 / PSK5: D = S / N-LAS40: D = LL / LAK33: D = The olutio i well kow a imple photographic le (ladcape le) LL LAK33 top Origi

45 45 Priciple of Gla Selectio i Optimizatio Deig rule for gla electio Differet deig goal:. Color correctio: idex large diperio differece deired poitive le field flatteig Petzval curvature. ield flatteig: large idex differece + + deired egative le color correctio + - availability of glae - - diperio Ref : H. Zügge

46 46 Burried Surface Nearly equal refractive idice Differece i Abbe umber ot larger tha 30.9 Gla Gla D D KZN N-PK KZN PSK N-LL Ultra KZSN L N-PSK SK SK N-SSK SK PSK SSK SSK4A LAKL N-PSK N-SK SK SK N-SK N-S4 N-LAK SL4 N-LAK SL56 LAN S LA D 30 D 30 D

47 47 Burried Surface Cemeted compoet with plae outer urface or ceter wavelegth oly plae parallel plate, ot ee i collimated light Curved cemeeted urface: - diperio for outer pectral weavelegth - color correctio without diturbig the mai wavelegth Example gree udeflected a) iglet b) color corrected iglet corrected Dz

48 48 Lateral Color Aberratio Diperio of the chief ray deviatio i the le Effect reemble the diperio of a prim i the upper part of the le I the image plae, the differece i the colored ray agle caue chage i the ray height The lateral color aberratio correpod to a chage of magificatio with the wavelegth diperio prim effect y Dy CHV chief ray z top image plae

49 49 Chromatic Variatio of Magificatio Lateral chromatical aberratio: Higher refractive idex i the blue reult i a troger ray bedig of the chief ray for a igle le The colored image have differet ize, the magificatio i wavelegth depedet Defiitio of the error: chage i image height/magificatio Correctio eed everal glae with differet diperio The aberratio trogly deped o the top poitio Dy Dy CHV CHV y y y y y e C C top red Dy CHV blue referece image plae

50 Surface ad Le cotributio of Lateral Color If the imagig of the etrace to the exit pupil uffer from axial chromatical aberratio, thi deliver a error of the exit pupil locatio ad alo of the chief ray agle: cheomatical lateral aberratio Travere chromatical aberratio of a le ytem Surface cotributio coefficiet of lateral color Correpodig le ummatio formula p p p p CHV Q H D y y H CHV CHV D p p p p CHV Q y y D p p p CHV y y 50

51 5 Lateral Color Correctio: Priciple of Symmetry Perfect ymmetrical ytem: magificatio m = - Stop i cetre of ymmetry Symmetrical cotributio of wave aberratio are doubled (pherical) Aymmetrical cotributio of wave aberratio vaihe W(-x) = -W(x) Eay correctio of: coma, ditortio, chromatical chage of magificatio frot part rear part 3

52 5 Chromatic Variatio of Magificatio Repreetatio of CHV:. Spot diagram. Magificatio m() 3. Travere aberratio: offet of chief ray referece chromatical magificatio differece pot diagram Y field height CHV 0.08 travere aberratio curve Dy axi field tagetial Dy field agital Dx Dy y p y p x p

53 Lateral color: Stop hift theorem Lateral color (LAC) for two differet top poitio a ad b (with top ize that defie the ame margial ray) relate to logitudial color (LOC) like thi: LAC a LAC b Dq LOC Where Dq i the top hift parameter which i the ame for every urface Dq If there i logitudial color i the ytem, there will be a top poitio for which lateral color vaihe If there i o logitudial color, lateral color i idepedet of the top poitio h a p h h b p 3 3 Ref: D. Oche Spot diagram for maximal field at differet top poitio 53

54 A M I fo : e c a f r u S oitarugifoc latot laixa fo oitarugifoc y a R m µ m m ) g e d ( : : A M I tops : J B O y r i A 3. m µ e r a 0 0 : : A M I : e c a f r u S e l a c S G E O S M R i e l d m m : A M I tops : J B O A M I : e c a f r u S tuoyal Lateral color: Stop hift theorem Example ytem with four N-BK7 lee corrected for e-lie Goal: Correct alo for C ad lie Spot diagram for maximal field with Airy dik : J B O : A M I m m ) g e d ( fo oitarugifoc tops. id the top poitio for which lateral color vaihe t i U y r i A. m µ e r a : u i d a R m µ 3 : S M R i e l d u i d a r : e l a c S G E O r a b 0 0 e c e r e f e R : y a R. Eure the ytem ha logitudial color top f e i h C margaid XMZ.tfihpotSleipieB top latot laixa :htgel mm Correct logitudial color there ad move top back oitarugifoc XMZ.tfihpotSleipieB fo r a b u i d a r t i U e c e r e f e R : u i d a R ) g e d ( f e i h C margaid xmz.tfihpotsleipieb top :htgel N-S6 tuoyal mm Ref: D. Oche xmz.tfihpotsleipieb margaid 54

55 55 Chromatic Variatio of Magificatio Impreio of CHV i real image Typical colored frige blue/red at edge viible Color equece deped o ig of CHV origial without lateral chromatic aberratio 0.5 % lateral chromatic aberratio % lateral chromatic aberratio

56 56 Chromatical Differece i Magificatio Color rig are hardly ee due to colored image Lateral hift of colored pf poitio Ref: J. Kaltebach

57 57 Axial Chromatical Aberratio Special effect ear black-white edge boarder mageta blue boarder Ref: J. Kaltebach