Desig ad Correctio of Optical Systems Lecture : Materials ad compoets 07-04-4 Herbert Gross Summer term 07 www.iap.ui-jea.de
Prelimiary Schedule - DCS 07 07.04. Basics.04. Materials ad Compoets 3 8.04. Paraxial Optics 4 05.05. Optical Systems 5.05. Geometrical Aberratios 6 9.05. Wave Aberratios 7 6.05. PSF ad Trasfer fuctio 8 0.06. Further Performace Criteria 9 09.06. Optimizatio ad Correctio 0 6.06. Correctio Priciples I 3.06. Correctio Priciples II 30.06. Optical System Classificatio Law of refractio, Fresel formulas, optical system model, raytrace, calculatio approaches Dispersio, aormal dispersio, glass map, liquids ad plastics, leses, mirrors, aspheres, diffractive elemets Paraxial approximatio, basic otatios, imagig equatio, multi-compoet systems, matrix calculatio, Lagrage ivariat, phase space visualizatio Pupil, ray sets ad samplig, aperture ad vigettig, telecetricity, symmetry, photometry Logitudial ad trasverse aberratios, spot diagram, polyomial expasio, primary aberratios, chromatical aberratios, Seidels surface cotributios Fermat priciple ad Eikoal, wave aberratios, expasio ad higher orders, Zerike polyomials, measuremet of system quality Diffractio, poit spread fuctio, PSF with aberratios, optical trasfer fuctio, Fourier imagig model Rayleigh ad Marechal criteria, Strehl defiitio, -poit resolutio, MTF-based criteria, further optios Priciples of optimizatio, iitial setups, costraits, sesitivity, optimizatio of optical systems, global approaches Symmetry, les bedig, les splittig, special optios for spherical aberratio, astigmatism, coma ad distortio, aspheres Field flatteig ad Petzval theorem, chromatical correctio, achromate, apochromate, sesitivity aalysis, diffractive elemets Overview, photographic leses, microscopic objectives, lithographic systems, eyepieces, sca systems, telescopes, edoscopes 3 07.07. Special System Examples Zoom systems, cofocal systems
3 Cotets. Dispersio. Glass map 3. Materials 4. Leses ad compoets 5. Aspheres 6. Diffractive elemets
4 Importat Test Wavelegths i [m] Name Color Elemet 48.3 UV Hg 80.4 UV Hg 96.778 UV Hg 3.5663 UV Hg 334.478 UV Hg 365.046 i UV Hg 404.656 h violett Hg 435.8343 g blau Hg 479.994 F' blau Cd 486.37 F blau H 546.0740 e grü Hg 587.568 d gelb He 589.938 D gelb Na 63.8 HeNe-Laser 643.8469 C' rot Cd 656.75 C rot H 706.588 r rot He 85. s IR Cä 03.98 t IR Hg 060.0 Nd:YAG-Laser
5 Chromatical Evaluatio of Optical Systems Chromatical performace evaluatio of optical systems: Usage of oe mai (cetral) wavelegth ad two secodary waveleghts Mai wavelegth st secodary wavelegth d secodary wavelegth e 546.07 gree F' 480.0 bue C' 643. 8 d 587.56 yellow F 486. blue C 656. 3 Additioal defiitio of wvalegths at the boudaries of the used spectral rage, e.g. - oe further wavelegth ear to the UV edge (g, i) - oe further wavelegth ear to the IR-edge (s,t) red red
Atomic model for the refractive idex: oscillator approach of atomic field iteractio Sellmeier dispersio formula: correspodig fuctio Special case of coupled resoaces: example quartz, degeerated oscillators Atomic Model of Dispersio j j j j j j i r i c f m c Ne i 0 0-0 0 0 0 3 4 5 6 7 log [mm] visible 0.4 0.7 (UV) (UV) 3 (IR) 4 (IR) vis () j j j C B A 4 0 j j j o C B B A 6
7 Dispersio Dispersio: Refractive idex chages with wavelegth Normale dispersio: larger idex for shorter wavelegths, Ray bedig of blue rays stoger tha red Notice: d d 0 ormal aomal ormal Diffractio dispersio is aomalous with d/d > 0 () The differet sig allows for chromatic correctio i diffractive elemets. i () o
8 Dispersio formulas Schott formula empirical Sellmeier Based o oscillator model 4 6 a a a a a a o ( ) A B C 3 4 5 8 Bausch-Lomb empirical Herzberger Based o oscillator model 4 D E ( ) A B C F ( o) a a3 ) ao a ( o o mit 0.68 mm o o Hartma Based o oscillator model ( ) a o a a 3 a4 a 5
9 Dispersio ad Abbe umber Descriptio of dispersio: Abbe umber Visual rage of wavelegths: typically d,f,c or e,f,c used e e F ' C' F ' C' refractive idex.8.75.7 Typical rage of glasses e = 0...00.65 SF flit Two fudametal types of glass: Crow glasses: small, large, dispersio low Flit glasses: large, small, dispersio high.6.55.5.45 BK7 crow 0.5 0.75.0.5.5.75.0
0 Dispersio ad Partial Dispersio Glasses o ormal lie: global slope proportioal to local (blue/red) slope.050 idex Abbe average slope local slope at blue/red dispersio.990.930 P blue SF66.870 P red.80.750 P blue LAFN7.690 Pred.630.570.50 P blue.450 0.4 0.43 0.46 0.49 0.5 0.55 0.58 0.6 0.64 0.67 0.7 P red BK7 [mm]
Curvatures c j of the radii of a les Focal power at the ceter wavelegth e for a thi les Differece i focal powers for outer wavelegths F', C' with the Abbe umber Focal legth at the ceter wavelegth Differece of the focal legths for outer wavelegths Achromatizatio coditio for two thi leses close together Abbe Number ad Achromatizatio, r c r c c c c F e e e ) ( ) )( ( e e e e C F C F C F F c c F F F ) ( ) ( ' ' ' ' ' ' c F f e e e ) ( e e e F C C F F C C F f c c f f f ' ' ' ' ' ' ' ' ) ( ) )( ( ' ' C F e e 0 f f F F F
Glass Diagram Usual represetatio of glasses: diagram of refractive idex vs dispersio () Left to right: Icreasig dispersio decreasig Abbe umber
3 Glass Glass blocks Striae i glass Ref: P. Hartma / Schott
4 Glass Diagram: Chages of Numbers 300 50 Number of glass types i the Schott catalog 73 5 ECO split cosolidatio 00 50 00 50 0 0 early days 886 44 0 8 67 systematic developmet 96 54 03 0 Split ito gree ad Pb glasses reductio due to cost 0 04 0 86 8 5 0 850 870 890 90 930 950 970 990 00 Year Ref: P. Hartma / Schott
5 Developmet of the Glas Map First ad curret Schott glass map 00 / 886 00 886
6 Plastic Materials Plastics i the - - diagram.00.95 refractive idex.90.85.80.75.70.65 aorgaic glasses.60.55.50.45 CR39 o o COC CMMA oo PMMA o MR8 o SCMA DPSC o o PS PC o SAN oo SMA SMMA FMS o plastics.40 90 80 70 60 50 40 30 0 0 Abbe umber
7 Relative Partial Dispersio Relative partial dispersio : Chage of dispersio slope with Differet curvature of dispersio curve.54.53 Defiitio of local slope for selected wavelegths relative to secodary colors P F ' C'.5.5.5 i - g g - F F - e F - C C - s C - t () Special -selectios for characteristic rages of the visible spectrum.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 400 500 600 700 800 900 000 00 g : 435 m UV edge e : 546 m d : 588 m mai color F' : 480 m C' : 644 m F : 486 m C : 656 m. secodary color. secodary color s : 85 m IR edge t : 04 m IR edge
8 Partial Dispersio ad Normal Lie The relative partial dispersio chages approximately liear with the dispersio for glasses P b, a, d, P 0.6 Nearly all glasses are located o the ormal lie i a P--diagram P gf The slope of the ormal lie depeds o the selectio of wavelegths 0.55 Glasses apart from the ormal lie shows aomalous partial dispersio P 0.5 P Cs P a d b P 0.45 80 60 40 0 these material are importat for chromatical correctio of higher order
9 Relative Partial Dispersio Log crow ad short flit as special realizatios of large P Log crow Short flit Crow Flit Ref.:H. Zuegge
0 Aomalous Partial Dispersio There are some special glasses with a large deviatio from the ormal lie Of special iterest: log crows ad short flits P g,f lie of ormal dispersio SF N-SF57 KZFSN4 FK5 FK5 PSK53A ZKN7 LAK8 LASFN30 P g,f heavy flits with character of log crows flit log crows log crow short flit short flits crow ormal lie
Plastics Material Idex at 546 m Abbe umb er Max Temp Therm expa 0-6 K - Scatt er i % Tras. 3mm, PMMA - Polymethyl-Methacrylat.4980 57 90 65 9.9 PC - Polycarboat - Makrolo, Lexa.59037 30 0 69 4 87.0 CR39 - DEGBAC - Gießharz.50 57.8 00 0.3 PS - Polystyrol.590 30.8 80 70 3 89.06 DPSC - Dipheyl-sulfidcarboat.6 6.0 CMMA.50 56.0 Styrol, SAN.566 34.7 95 65 4 90.09 SMA.585 3.3 SMMA.568 33.5 FMS.508 34.0 SCMA.535 4.5 COC.533 56.0 MR8.60 4.0 Desity g/cm 3
Properties of Plastic Materials. Stress iduced birefrigece durig processig. Geeratio of local ihomogeieties of the refractive idex i die castig 3. Water itake (swellig) : chage of shape (up to 4%) ad decrease i the refractive idex 4. Electro-static charge 5. Agig due to cold formig, polymerizatio, opalescece, yellowig 6. Strog thermal variatio of the refractive idex 7. Limitig temperature (above the trasitio temperature the material is destroyed) 00... 0 C 8. For a icreased abrasive hardess ad for the prevetio from chargig ad swellig,special coatigs may have to be applied. 9. Durig the coolig process sigificat chages occur i the volume caused by shrikig. There are two differet types of plastics a. thermosets, shrikig 0.4%...0.7% b. thermoplasts, shrikig 4%...4%
3 Usage of Plastics i Optical Systems Most attractive use of plastics: Cosumer optics - beefit of light weight - critical cost - high umber of pieces Advatages for special compoets due to maufacturig techique: - complex surface shapes, arrays, aspheres - for ijectio mouldig cost of complex shape oly for master piece Typical products with plastics compoets: - Eye glasses - bioculars - photographic leses - pic-up objective leses - illumiatio systems
4 Plastics vs Glass Materials Compariso plastics with glasses property uit rage plastics rage glass refractive idex.49...6.44...95 dispersio 5...57 0...90 uiformity of the refractive idex 0-3...0-4 0-4...0-6 temperature depedece of the refractive idex 0-6*K - -00...-60-0...+0 Vickers hardess N/mm 0...90 3000...7000 thermal expasio 0-6*grd - 70...00 5...0 thermal coductivity Wm -grd- 0.5...0.3 0.5...4 iteral trasmissio i the gree rage 0.97...0.993 0.999 stress - optical coefficiet 0 - Pa - 40 3 stress- birefrigece 5*0-5...0-3 0 desity g/cm 3.05...3.3...6. water itake % 0...0.8 0
5 UV- ad IR - Materials material refractive idex -rage (mm) UV IR P MgF.389 0. - 9.0 ZS.5 0.4-4.5.83 0.7 CaF, calcium fluoride.4 0. -.5 47.53 0.397 ZSe.44 0.5 -.0 34.0 0.9 MgO.69...737 0.8-9.5 53.4 CdTe.70 0.9-3.0 diamod.3757 0.5...3.7, 6.0... () 387 0.469 germaium 4.003.0...5 70.4 0.556 silico 3.433...5 40.6 0.53 BaF.474 0.8... 8.7 0.65 SiO, quartz.544 0.5...4.0 () 69.9 0.653 Al O 3, sapphire.769 0.7...5.5 () 6.66 0.650
6 Les Compoets Variability of geometry
7 Optical Compoets Complexity:. Low. Medium 3. high
8 Les shape Differet shapes of siglet leses:. bi-, symmetric. plae covex / cocave, oe surface plae 3. Meiscus, both surface radii with the same sig Covex: bedig outside Cocave: hollow surface Pricipal plaes P, P : outside for mesicus shaped leses P P' P P' P P' P P' P P' P P' bi-covex les plae-covex les positive meiscus les bi-cocave les plae-cocave les egative meiscus les
9 Cardial Elemets of a Les Focal poits:. icomig ray parallel to the axis itersects the axis i F. ray through F is leaves the les parallel to the axis The focal legths are refereced o the pricipal plaes F frot focal plae f P P' f ' F' back focal plae pricipal plaes s BFL Nodal poits: Ray through N goes through N ad preserves the directio odal plaes N N' u' u
30 Cardial Elemets of a Les Pricipal plae P: icomig ray hits itersectio poit with P is trasferred with the same height h to P Q h Q' P P' pricipal plaes Special case of icidet ray parallel to the axis: pricipal plae P : locatio of apparet ray bedig P P' pricipal plaes
3 Mai properties of a les Mai otatios ad properties of a les: - radii of curvature r, r curvatures c sig: r > 0 : ceter of curvature is located o the right side - thickess d alog the axis - diameter D - idex of refractio of les material Focal legth (paraxial) Optical power Back focal legth itersectio legth, measured from the vertex poit c r c r yf ' f, f ' ta u F s f f s ' f ' F ' ' P' y ta u'
3 Notatios of a les P pricipal poit S vertex of the surface F focal poit O s f itersectio poit of a ray with axis focal legth PF y u F S P P' N N' S' u' F' r radius of surface curvature y' O' d thickess SS s f f' s' refrative idex f BFL s P s' P' f' BFL a d a'
33 Bedig of a Les Bedig: chage of shape for ivariat focal legth Parameter of bedig X R R R R X < - X = - X = 0 meiscus les placovex les placocave les bicovex les bicocave les X = + placovex les placocave les X > + meiscus les
34 Les bedig ud shift of pricipal plae Ray path at a les of costat focal legth ad differet bedig Quatitative parameter of descriptio X: The ray agle iside the les chages X R R R R The ray icidece agles at the surfaces chages strogly The pricipal plaes move For ivariat locatio of P, P the positio of the les moves P P' F' X = -4 X = - X = 0 X = + X = +4
35 Magificatio Parameter Magificatio parameter M: defies ray path through the les M<- M U ' U U ' U m m f s f s' M=- Special cases:. M = 0 : symmetrical 4f-imagig setup. M = -: object i frot focal plae 3. M = +: object i ifiity M=0 The parameter M strogly iflueces the aberratios M=+ M>+
36 Aspheres - Geometry Referece: deviatio from sphere Deviatio z alog axis Better coditios: ormal deviatio r s y z(y) deviatio z y height y tagete z(y) deviatio z alog axis z height y sphere perpedicular deviatio r s aspherical shape spherical surface z aspherical cotour
37 Coic Sectios Explicite surface equatio, resolved to z Parameters: curvature c = / R coic parameter Ifluece of o the surface shape cx y c x z y Parameter Surface shape = - paraboloid < - hyperboloid = 0 sphere > 0 oblate ellipsoid (disc) 0 > > - prolate ellipsoid (cigar ) Relatios with axis legths a,b of coic sectios a b c b a b c a c
38 Simple Asphere Parabolic Mirror Equatio Radius of curvature i vertex: R s Perfect imagig o axis for object at ifiity Strog coma aberratio for fiite field agles Applicatios:. Astroomical telescopes. Collector i illumiatio systems z y R s axis w = 0 field w = field w = 4
39 Simple Asphere Elliptical Mirror Equatio Radius of curvature r i vertex, curvature c eccetricity Two differet shapes: oblate / prolate Perfect imagig o axis for fiite object ad image loactio Differet magificatios depedig o used part of the mirror Applicatios: Illumiatio systems s z cy ( ) y c s' F F'
40 Geeral Aspherical Surface Coic surface as basic shape Additioal correctio of the sag by a Taylor expasio Oly eve powers: o kik at r=0 z( x, y) cx y c x y k max k a k x y k Mostly rotatioal symmetric shape cosidered z( r) Problems with this represetatio:. added cotributios ot orthogoal, bad performace durig optimizatio. o-ormalized represetatio, coefficiets deped o absolute size of the diameter (very small high order coefficiets for large diameters) 3. Oscillatory bahavior, large residual slope error ca occur 4. i optics slope ad ot sag is relevat 5. the coefficiets ca ot be measured/are hard to cotrol, toleracig is critical ad comlicated 6. the added sag is alog z, more importat is a correctio perpedicular to the surface for strog aspheres c r c r k max k a k r k
4 Aspherical Expasio Order Improvemet by higher orders Geeratio of high gradiets y(r) 00 6. order 50 D rms [mm] 0 3 0 4. order 8. order. order 0. order -50 0 0-00 0 0. 0.4 0.6 0.8 r 0 0 0-4 6 8 0 4 order k max
4 Deviatio of Light Mechaisms of light deviatio ad ray bedig Refractio Reflectio Diffractio accordig to the gratig equatio si ' si ' ' g si si m o Scatterig ( o-determiistic) refractio reflectio diffractio scatterig les mirror gratig scatter plate
43 Diffractive Elemets Origial les height profile h(x) Wrappig of the les profile: h red (x) Reductio o maximal height h Digitalizatio of the reduced profile: h q (x) h z ray refracted at Fresel les ray refracted at smooth asphere 3 h h h(x) : cotiuous profile h red (x) : wrapped wrappig h q (x) : quatized reduced profile profile h
44 Realizatios of Diffractive Elemets DOE's blazed DOE's quatized DOE's biary gratig gratig of idex example: HOE surface cotour multi phase level phase gratig amplitude gratig
45 Diffractio Orders Usually all diffractio orders are obtaied simultaeously Blazed structure: suppressio of perturbig orders Uwated orders: false light, cotrast ad efficiecy reduced diffractive structure m+3 m+ diffractio orders m+ m m- m- m-3 desired order
46 Diffractig Surfaces Surface with gratig structure: ew ray directio follows the gratig equatio Local approximatio i the case of space-varyig gratig width s' s ' mg gˆ e ' d Raytrace oly ito oe desired diffractio order Notatios: g : uit vector perpedicular to grooves d : local gratig width m : diffractio order e : uit ormal vector of surface s e g p p grooves s Applicatios: - diffractive elemets - lie gratigs d - holographic compoets
47 Diffractive Optics: Local micro-structured surface Locatio of ray bedig : macroscopic carrier surface Directio of ray bedig : local gratig micro-structure local gratig g(x,y) thi layer bedig agle m-th order les macroscopic surface curvature
Gratig Equatio Itesity of gratig diffractio patter (scalar approximatio g >> ) Product of slit-diffractio ad iterferece fuctio Maxima of patter: coicidece of peaks of both fuctios: gratig equatio 0.9 0.8 I N g ug si ug Nug si ug N si g si si m o 0.7 Agle spread of a order decreases with growig umber od periods N 0.6 0.5 Oblique phase gradiet: - relative shift of both fuctios - selectio of peaks/order - basic priciple of blazig 0.4 0.3 0. 0. 0-3 - - 0 3 u = si
Blaze Gratig Blaze gratig (echelette): - facets with fiite slope - additioal phase shifts the slit diffractio fuctio - all orders but oe suppressed Blaze coditio is oly valid for - oe wavelegth - oe icidece agle 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0 slit diffractio workig order suppressed orders m B - m B - m B m B + m B +
50 Trasitio Refractive - Diffractive Phase of refractig blaze gratig: covolutio of prism trasmissio with periodic iterferece fuctio T( x) T ( r / d ) prism ( x) T ( diff ) periodic Notatio: L total width g legth of period blaze agle m idex of periods q order of phase jump with height h Complete trasmissio:. oe period with liear phase. total width 3. periodicity of cell L q = q = q = 4 q = 8 g qh T( x) rect x g e i rect x L x m mg
5 Trasitio Refractive - Diffractive Higher umber of periods Icreasig order q Widths of orders decrease Limitig case: oe prism, slit diffractio prism / iterferece 0.5 0-0 0 0 40 60 80 00 0.5 total L / g = 8 0.5 0-0 0 0 40 60 80 00 0.5 L / g = 4 0 0 0.5-0 0 0 40 60 80 00-0 0 0 40 60 80 00 0.5 L / g = 0 0-0 0 0 40 60 80 00-0 0 0 40 60 80 00
Achromatic Hybrid Les Les with diffractive structured surface: hybrid les Refractive les: dispersio with Abbe umber = 5...90 refractive les blue gree red Diffractive les: equivalet Abbe umber d d 3.453 F Combiatio of refractive ad diffractive surfaces: achromatic correctio for compesated dispersio C diffractive les R D red gree blue Usually remais a residual high secodary spectrum Broadbad color correctio is possible but complicated hybrid les blue gree red