Design and Correction of Optical Systems

Transcription

1 Design and Correction of Optical Sstems Lecture 4: Further topics Herbert Gross Summer term 5

2 Preliminar Schedule 5.4. Basics.4. Materials and Components Paraial Optics Optical Sstems Geometrical Aberrations 6.5. Wave Aberrations PSF and Transfer function Further Performance Criteria 9.6. Optimization and Correction 7.6. Correction Principles I 4.6. Correction Principles II.7. Optical Sstem Classification Law of refraction, Fresnel formulas, optical sstem model, ratrace, calculation approaches Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors, aspheres, diffractive elements Paraial approimation, basic notations, imaging equation, multi-component sstems, matri calculation, Lagrange invariant, phase space visualization Pupil, ra sets and sampling, aperture and vignetting, telecentricit, smmetr, photometr Longitudinal and transverse aberrations, spot diagram, polnomial epansion, primar aberrations, chromatical aberrations, Seidels surface contributions Fermat principle and Eikonal, wave aberrations, epansion and higher orders, Zernike polnomials, measurement of sstem qualit Diffraction, point spread function, PSF with aberrations, optical transfer function, Fourier imaging model Raleigh and Marechal criteria, Strehl definition, -point resolution, MTF-based criteria, further options Principles of optimization, initial setups, constraints, sensitivit, optimization of optical sstems, global approaches Smmetr, lens bending, lens splitting, special options for spherical aberration, astigmatism, coma and distortion, aspheres Field flattening and Petzval theorem, chromatical correction, achromate, apochromate, sensitivit analsis, diffractive elements Overview, photographic lenses, microscopic objectives, lithographic sstems, eepieces, scan sstems, telescopes, endoscopes Special Sstem Eamples Zoom sstems, confocal sstems Further Topics New sstem developments, modern aberration theor,...

3 3 Contents. Aberration theor. Phase retrieval 3. Etended Zernike approach 4. Freeform surfaces 5. Etended depth of focus 6. Digital processing 7....the end

4 4 Strehl Ratio and PSF-Peak Height for Aberrations Differences between Strehl and I peak, if the profile is structured.8 defocussing c 4.8 astigmatism c c 4 [l].5.5 c 5 [l] spherical aberration c 9 coma c 8.8 Strehl peak c 9 [l].5.5 c 8 [l]

5 Variation of Performance with Field Position PSF as a function of the field height = = % = 4% = 6 % = 8 % = % Interpolation is critical Orientation of coma in the field Small field area with approimatel shiftinvariant PSF: isoplanatic patch

6 Variation of Performance with Field Position Strehl ratio:. Photo objective lens: strong dependence. Microscope objective lens: weak dependence D S photo objective D S micro objective nm 468 nm 587 nm nm 468 nm 587 nm field field

7 Variation of Performance with Field Position Variation of the wave aberration with field location Zernike coefficients can be interpolated as a function of field position Useful approach for space-variant deconvolution c j in l 4 3 c 4 defocus c 5 astigmatism c 8 coma c 9 spherical c trefoil c astigmatism 5. order c 5 coma 5. order c 6 spherical 5. order W rms.5.4 polchromatic.3.. diffraction limit nm 546 nm 643 nm field relative field position

8 Aplanatic and Perfect Imaging Perfect imaging on ais due to conic section - not aplanatic: linear grows of coma with field size D spot m] 5 w in Aplanatic: - Perfect stigmatic imaging on ais, spherical corrected - linear coma vanishes: good correction off-ais but near to ais - quadratic grows of spot size due to astigmatism - aplanatic and perfect marginal ra quite different ideal lens ideal ras real ras real lens sin u ideal' =.77 sin u real' = D spot m] w in

9 Perfect imaging as special cases: Eamples 9 Perfect Imaging n s z r F F z R ),, ( R z n z n env r d n r n cosh ) ( F s s' F' aplanatic concentric d F z aplanatic ellipsoid hperboloid Mawell lens Mikhaelian lens

10 Order (wave aberration) Spatial pupil resolution Non-centered Freeforms Field dependence Comment Options for Surface Contributions 4th order, Seidel 4 N N N Y well known 6th order, Shack/Thompson 6 low Y N Y components circular smmetric Wavefront, Hopkins/Welford all Y Y Y N one ra onl Aldis all Y N N N one ra onl Aldis generalized all Y Y Y N one ra onl Wavefront all Y Y Y N onl numerical, huge information Zernike high ~ Y Y N problem induced aberrations

11 Kingslakes Diagram Practical problem in analsis of classical spot diagrams: relation between deviations and pupil location is lost Idea of Kingslake: transverse aberrations of spot points drawn in pupil intersection points D and D at ever point in the pupil sampling grid In principle corresponding to wave gradient in eit pupil Problems:. proper representation of quite different scales. distorted grid in case of induced aberrations 3. no field dependence seen p p J. Palmer, Lens aberration data, Monographs on Apllied Optics, A. Hilger, 97

12 Kingslakes Diagram Etended Etension of Kingslakes representation for surface contributions.8 M M M 3 Problem: compaction of high compleit, limited clearness.6.4. M asphere M freeform M 3 freeform

13 Aldis Theorem Aldis theorem: surface contribution of transverse aberration of all orders Calculation b tracing two ras:. paraial marginal ra. finite ra H: Lagrange invariant H nu k k A: Paraial refraction invariant Transverse aberrations object A j n j i j n D nu s' k D nu s' k k k zk zk j ( h jc j u j k j k j Ajz Ajz j j ) Ds Ds j j zj zj A j j D sj sj s' zjszj Aj j H Ds j s s' s image j P arbitrar finite ra paraial marginal ra u u' ' surfaces P' D',D'

14 Aldis Theorem Advantage of Aldis theorem: contain all orders Larger differences for surfaces/cases with higher order contributions Usuall, the reference is the paraial ra, therefore distortion is taken into account A known formulation is available for aspherical surfaces in centered sstems A specialized equation must be used for the case of image in infinit More general 3D geometries are not supported More general formulations are possible (Brewer) Disadvantage of Aldis theorem: onl for one ra

15 Aldis Theorem Δ' 3 Eample Achromate - Seidel and Aldis contributions at evere surface and in summar Surface - r p Differences to Seidel terms due to higher order at cemented surface for larger pupil radii Δ' Surface Aldis higher orders Seidel r p F/ Achromat, f = Transverse spherical aberration? Δ' Surface 3 - r p -.5 Δ' Sum Surfaces to 3 - r p Ref: H. Zügge -

16 Phase retrieval Given object I object () Known illumination, usuall incoherent Known measured image intensities I image (',z) for several z-values To be calculated: transfer function of the sstem / pupil function Relationship: convolution mask lens pupil tube lens measuring planes illumination I object () I pupil ( p ) = A pupil ( p )ep [ iw( p ) ] I image (',z)

17 Phase Retrieval Principle of phase retrieval for metrolog of optical sstems Measurement of intensit caustic z-stack Reconstruction of the phase in the eit pupil image plane eit pupil optical sstem z z wave front W(,) 3D- intensit image stack I(,,z)

18 Phase Space Interpretation Known measurement of intensit in defocussed planes: - Several rotated planes in phase space - Information in and near the spatial domain Calculation of distribution in the Fourier plane Wave equation is valid Principle : Tomograph pupil plane desired distribution p minimal defocus plane phase space angle measuring planes image plane ' maimal defocus plane

19 Conventional usage of Zernike coefficients: - description of wave front in pupil - determines the PSF intensit in the reference plane More general approach due to Braat (5) according to an old idea of Zernike (93) - epansion of the intensit I(,,z) in the image domain in all dimensions - lateral epansion into Zernikes - aial Talor epansion with coefficients c nm : classical Zernike coefficients This gives an analtical representation of the volume distribution of intensit Etended Zernike Approach l j l q j p l l l j l l j m j l m b p nm lj ) ( m n q m n p p j l j l m nm lj l l iz m n m nm r l r J b iz e m i c i r r J z r E, ) ( cos 4 ) (,, 9

20 Etended Zernike Approach Eample: - intensit z-stack for coma - calculation with diffraction integral / etended Zernike approach - nearl perfect result without differences diffraction integral c coma =.5 z = etended Nijboer- Zernike

21 Publications of Freeform Sstems In the TOP-5 journals Ref.: W. Ulrich

22 Crossed Clindrical Lenses Eample of two aspherical clindrical lenses with different focal lengths Due to difference, the numerical apertures are different The wavefront shows the deviations in the 45 directions The spot diagram has etreme small diameters onl along the aes -z-section NA =.4 -z-section NA =.57 5 l

23 Schiefspiegler-Telescopes Pseudo-3D-laouts: eccentric part of aismmetric sstem common ais Remaining smmetr plane.5 mirror M image mirror M 3 mirror M.5 used eccentric subaperture field points of figure M M 3 M

24 HMD Projection Sstem Special anatomic requirements Aspects:. Ee movement. Pupil size 3. Ee relief 4. Field size 5. See-through / look-around 6. Brightness 7. Weight and size 8. Stereoscopic vision 9. Free-forme surfaces and DOE spectacles ee ais ee ball free space for HMD iris ear D retina L

25 HMD Projection Lens 5 l coma, l W rms, l 8 Refractive 3D-sstem Free-formed prism 4 4 One coma nodal point Two -4 astigmatism nodal -4 points -6-6 image free formed surface binodal points 6 total internal reflection 4 ee pupil - free formed surface -4-6 field angle

26 TMA Telescopes Three mirrors without central obscuration with corrected astigmatism (TMA = three mirror ananstigmates) At least one mirror conve Incidence angles properl adjusted: proper matching of the conditions M3 f tan Rcos i f sag in sum over all three imaging mirrors Best case (cheap): all spherical real: at least one aspherical mirror for spherical correction R cos i M (stop) M

27 Asmmetric Double-TMA Design 3. Draft design of a double TMA corresponding to the ideas of the literature (Zema: Double TMA spectrometer v8) Basis: TMA telescopes (6 mirrors) - plane grating - currentl freeform surfaces used - ecellent performance - spectral resolution appro..3 nm - performance on ais and for slit length 6 mm - size mm - VIS range.4 -. m entrance slit grating eit slit

28 8 Freeform Sstems: Applications General purpose: - freeform surfaces are useful for compact sstems with small size - due to high performance requirements in imaging sstems and limited technological accurac most of the applications are in illumination sstems - mirror sstems are developed first in astronomical sstems with complicated smmetr-free geometr to avoid central obscuration HMD Head mounted device with etreme size constraints HUD Head up displa, onl few surfaces allowed Schiefspiegler - astronomical sstems without central obscuration - EUV mirror sstems for net generation lithograph sstems Illumination sstems Various applications, smooth and segmented

29 Etended polnomials in,: Zernike epansion Etended Forbes asphere Epansion in other orthogonal polnomial sstems: Legendre, Chebchev,... Fourier epansion Epansion into non-orthogonal local shifted Gaussian functions (RBF) Cubic spline, locall in patch j,k defined as polnomials of order 3 9 Freeform Sstems: Equations of Description,, ) ( ) ( ), ( m n m n m n a c c c c z ), ( ) ( ) ( ), ( j j j Z c c c c c z ) sin( ) cos( ), ( a r Q m b m a a r a r Q a r c a r a r r c cr z m n m n m n m n m n n n n k m j m n jkmn k j a z 3 3, ), ( m n w w nm n n e a c c c c z, ) ( ) ( ), ( m n ik ik nm m n e B c c c c z, Re ) ( ) ( ), (

30 3 Freeform Surface Representations Representation support area Orthogonalit Basis geometr Boundar Zema Remark Talor polnomials global no cartesian arbitrar es ma 3 terms Zernike polnomials global spatial polar circular es Legendre polnomials D global spatial cartesian rectangular no General Forbes D global slope polar circle no ma 36 terms biconic basic shape Fourier description global spatial cartesian rectangular no onl azimuthal Radial shifted bases local, with no arbitrar arbitrar no overlap Splines (cubic) local spatial arbitrar arbitrar no Wavelets local spatial arbitrar arbitrar no appropriate for midfrequenc errors

31 Depth of Focus Depth of focus depends on numerical aperture. Large aperture:. Small aperture: small depth of focus large depth of focus Ref: O. Bimber

32 Depth of Focus Schematic drawing of the principal ra path in case of etended depth of focus Where is the energ going? What are the constraints and limitations? conventional ra path beam with etended depth of focus Dz Dz

33 EDF and MTF-Profile Sstem with EDF can be defined as an enlarged constant aial OTF-distribution. With clear pupil function With manipulated pupil function Reduced contrast Preserved resolution Enlarged constant OTF-distribution Pupil manipulation Aial MTF-distribution: l W Aial MTF-distribution: l W Ref: S. Förster

34 EDF with Comple Toraldo Mask I(r,z) I() I(z) I(z) depth for 8%: 3 RE

35 EDF with Chirped Ring Pupil Absorption filter with profil Corresponds to a linear sequence of single psf-peaks m n P( r) ( ) cos nr m n p pupil transmission P(r) Transmission quite small T ( m ) Etrem good edf performance p ' image plane f arra of single distributions ' z z

36 EDF with Chirped Ring Pupil Eample with m = 8 peaks P(r p ) r p Intensität

37 Fourier Filtering Objective tube lens image Object Pupil with phase mask digital image I image (') transfer function Computer Image Digital optics with pupil phase mask Primar image blurred Digital reconstruction with the help of image digital restored the sstem transfer function

38 Cubic Phase Plate (CCP) Phase Mask with cubic polnomial shape cubic phase PSF MTF P( ) e i 3 für sonst Effect of mask: - depth of focus enlarged - Psf broadened, but nearl constant - Deconvolution possible Problems : - variable psf over field size - noise increased - finite chief ra angle - broadband spectrum in VIS - Imageartefacts -section z -section

39 Cubic Phase Mask : PSF and OTF Conventional imaging Sstem with cubic phase mask focus defocussed focus defocussed

40 Etended Depth of Focus: Microscopic Imaging Conventional microscopic image Image with phase mask with / without deconvolution Ref: E. Dowski

41 Deconvolution: Influence of Regularization = = -5 = -3 = - = =

42 Aliasing and Moire in Digital Image Processing Ref: M. Seesselberg

43 Image qualit with Real Objects a) object b) good image c) defocussed d) aial chromatic aberration e) lateral chromatic aberration f) spherochromatism g) chromatical astigmatism

44 Real Image with Different Chromatical Aberrations original object good image color astigmatism l 6% lateral color aial color 4 l

45 Time is Over

46 Wh Aberration Theor? Understanding optical sstems is onl possible with aberration theor Correction of sstems is efficient with detailed analsis of aberrations and the methods to prevent or compensate them after a proper classification Especiall the decomposition of the total aberrations into the surface contributions helps for analzing and improving sstems Allows qualified performance assessment But:. the classical aberration theor is restricted to the geometrical picture. the classical aberrations theor mostl assumes circular smmetr 3. complete general geometries are complicate to implement, the single numbers becomes matrices and are hard to interprete 4. the digital image processing approaches of toda reduce the necessit of perfectl corrected analogue sstems 4. the application to real human image perception is still complicated

47 Feedback nothing clear? to much stuff? to complicated? Ref: D. Shafer

48 Optics Man msterious things and new notations Ref: T. Kaiser

49 Thank ou for attending the lecture