Design and Correction of Optical Systems
Lecture 11: Correction principles II 2014-06-18 Herbert Gross
Summer term 2014 www.iap.uni-jena.de
2 Preliminary Schedule
Law of refraction, Fresnel formulas, optical system model, raytrace, calculation 1 09.04. Basics approaches Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors, 2 16.04. Materials and Components aspheres, diffractive elements Paraxial approximation, basic notations, imaging equation, multi-component 3 23.04. Paraxial Optics systems, matrix calculation, Lagrange invariant, phase space visualization Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry, 4 30.04. Optical Systems photometry Longitudinal and transverse aberrations, spot diagram, polynomial expansion, 5 07.05. Geometrical Aberrations primary aberrations, chromatical aberrations, Seidels surface contributions Fermat principle and Eikonal, wave aberrations, expansion and higher orders, 6 14.05. Wave Aberrations Zernike polynomials, measurement of system quality Diffraction, point spread function, PSF with aberrations, optical transfer function, 7 21.05. PSF and Transfer function Fourier imaging model Rayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based 8 28.05. Further Performance Criteria criteria, further options Principles of optimization, initial setups, constraints, sensitivity, optimization of 9 04.06. Optimization and Correction optical systems, global approaches Symmetry, lens bending, lens splitting, special options for spherical aberration, 10 11.06. Correction Principles I astigmatism, coma and distortion, aspheres Field flattening and Petzval theorem, chromatical correction, achromate, 11 18.06. Correction Principles II apochromate, sensitivity analysis, diffractive elements Overview, photographic lenses, microscopic objectives, lithographic systems, 12 25.06. Optical System Classification eyepieces, scan systems, telescopes, endoscopes
13 02.07. Special System Examples Zoom systems, confocal systems
14 09.07. Further Topics New system developments, modern aberration theory,... 3 Contents
1. Field flattening and Petzval theorem 2. Field lenses 3. Chromatical correction 4. Achromate and apochromate 5. Diffractive elements 6. Miscellaneous 4 Petzval Theorem for Field Curvature
. Petzval theorem for field curvature: 1 nk 'nk 1. formulation for surfaces nm ' Rptz k nk nk 'rk
2. formulation for thin lenses (in air) 1 1 Rptz j n j f j . Important:
- no dependence on bending
- no dependence on stop location
. Natural behavior: image curved object towards system plane
. Problem: collecting systems with f > 0: If only positive lenses: R always negative ptz R
ideal optical system real image . Typical scaling for single lens: image plane R shell ptz n 1.6 f 5 Petzval Theorem for Field Curvature
. Goal: vanishing Petzval curvature 1 1 Rptz j n j f j
1 h 1 and positive total refractive power j f j h1 f for multi-component systems
. Solution: General principle for correction of curvature of image field: 1. Positive lenses with: - high refractive index - large marginal ray heights - gives large contribution to power and low weighting in Petzval sum 2. Negative lenses with: - low refractive index - samll marginal ray heights - gives small negative contribution to power and high weighting in Petzval sum 6 Flattening Meniscus Lenses
. Possible lenses / lens groups for correcting field curvature . Interesting candidates: thick mensiscus shaped lenses r 2 2 1 nk 'nk 1 n 1 d d Rptz k nk nk 'rk n f n r1r2 r 1
1. Hoeghs mensicus: identical radii 2 (n 1) d - Petzval sum zero F' nr 2 - remaining positive refractive power
2. Concentric meniscus, r r d - Petzval sum negative 2 1 - weak negative focal length 1 (n 1)d (n 1)d F' - refractive power for thickness d: R n r r d ptz 1 1 nr1(r1 d)
3. Thick meniscus without refractive power n 1 1 (n 1)2 d Relation between radii 0 r2 r1 d n Rptz nr1 nr1 d (n 1) 7
Correcting Petzval Curvature
r r . Triplet group with + - + 1 2 r 3
d/2 collimated
n 2 n1
. Effect of distance and 1/Rpet [1/mm] refractive indices 10-1
SF66 / FK3 / SF66
10-2
BK7
-3 10 r1 50 70 100 [mm]
From : H. Zügge 8 Field Curvature
1 F . Correction of Petzval field curvature in lithographic lens j for flat wafer R j n j
h j . Positive lenses: Green hj large F Fj j h1 . Negative lenses : Blue hj small
. Correction principle: certain number of bulges
Microscope Objective Lens
a) . Possible setups for flattening single meniscus the field lense . Goal: b) - reduction of Petzval sum two meniscus - keeping astigmatism corrected lenses
c) symmetrical triplet
d) achromatized meniscus lens
e) two meniscus lenses achromatized
f) modIfied achromatized triplet solution
Field Flatness
one waist two waists
. Principle of multi-bulges to reduce Petzval sum
1 1 n' r k n f p k k . Seidel contributions show principle
Petz Petz
1. bulge 1. waist 2. bulge 2. waist 3. bulge
0.2
0
-0.2
10 20 30 40 50 60
Size Reduction by Aspheres
1. bulge 2. bulge 3. bulge 10 8 . Sensitivity of refractive lenses 6 spherical for aspheres 4 2 0 1.5
1 coma 0.5
0 0.5 0.4 0.3 astigmatism 0.2 0.1 0 0.4
0.3
0.2 distortion 0.1
0 5 10 15 20 25 30 35 40 45 50 55 sum
pupil
aspherical 12 Flattening Field Lens
Effect of a field lens for flattening the image surface
1. Without field lens 2. With field lens
curved image surface image plane
image flat shell image
field lens 13 Field Lenses
. Field lens: in or near image planes . Influences only the chief ray: pupil shifted . Critical: conjugation to image plane, surface errors sharply seen
marginal ray
chief ray
field lens in intermediate original lens L image plane 2 shifted pupil pupil lens L1 14 Field Lens im Endoscope
Ref : H. Zügge 15 Chromatical Aberrations
. Various cases of chromatical aberration correction
a) axial and lateral color corrected b) axial color corrected
F FC C MR CR
FC FC
c) lateral color corrected d) no color corrected
F C C F
F C F C 16 Axial Colour: Achromate
(a) (b) (c) . Compensation of axial colour by appropriate glass choice
. Chromatical variation of the spherical aberrations: spherochromatism (Gaussian aberration)
. Therefore perfect axial color correction BK7 BK7 F2 BK7 N-SSK8F2 (on axis) are often not feasable n = 1.5168 n = 1.5168 1.6200 n = 1.5168 1.6177 = 64.17 = 64.17 36.37 = 64.17 49.83 F= 1 F = 2.31 -1.31 F = 4.47 -3.47
r p r p r p 1 1 1
486 nm 588 nm 656 nm
z z z -2.5 0 -0.20 0 0.20 -100 0 100
Ref : H. Zügge
Achromate
. Achromate: - Axial colour correction by cementing two Crown in different glasses front - Bending: correction of spherical aberration at the full aperture - Aplanatic coma correction possible be clever choice of materials
. Four possible solutions: - Crown in front, two different bendings - Flint in front, two different bendings Flint in front . Typical: - Correction for object in infinity - spherical correction at center wavelength with zone - diffraction limited for NA < 0.1 - only very small field corrected
solution 1 solution 2
Achromate: Realization Versions
. Advantage of cementing: solid state setup is stable at sensitive middle surface with large curvature
. Disadvantage: a) flint in front loss of one degree of freedom
. Different possible realization forms in practice
edge contact cemented
b) crown in front
edge contact cemented contact on axis broken, Gaussian setup
Achromate : Basic Formulas
. Idea: 1. Two thin lenses close together with different materials 2. Total power F F1 F2
F F 3. Achromatic correction condition 1 2 0 1 2 . Individual power values 1 1 F1 F F F 2 1 2 1 1 1 . Properties: 2 1. One positive and one negative lens necessary 2. Two different sequences of plus (crown) / minus (flint) 3. Large -difference relaxes the bendings 4. Achromatic correction indipendent from bending 5. Bending corrects spherical aberration at the margin 6. Aplanatic coma correction for special glass choices 7. Further optimization of materials reduces the spherical zonal aberration
Achromate: Correction
sMR' . Cemented achromate: 6 degrees of freedom: aplanatic 3 radii, 2 indices, ratio 1/2 case
. Correction of spherical aberration: diverging cemented surface with R positive spherical contribution 1
for nneg > npos case with 2 solutions . Choice of glass: possible goals case with 1. aplanatic coma correction one solution 2. minimization of spherochromatism case without solution, and coma correction 3. minimization of secondary spectrum only spherical minimum
. Bending has no impact on chromatical correction: is used to correct spherical aberration at the edge
. Three solution regions for bending 1. no spherical correction 2. two equivalent solutions 3. one aplanatic solution, very stable
Achomatic solutions in the Glass Diagram
flint crown negative lens positive lens
Achromat
Achromate
. Achromate
. Longitudinal aberration . Transverse aberration . Spot diagram 486 nm 587 nm 656 nm y'
= 486 nm axis
rp 1
= 656 nm sinu' = 587 nm 1.4° 486 nm 587 nm 656 nm
2° s' [mm] 0 0.1 0.2
Coma Correction: Achromate
Image height: y’ = 0 mm y’ = 2 mm Achromat . Bending of an achromate Pupil section: meridional meridional sagittal bending Transverse y' y' y' - optimal choice: small residual spherical Aberration: 0.05 mm 0.05 mm 0.05 mm aberration (a) - remaining coma for finite field size . Splitting achromate: additional degree of freedom: - better total correction possible (b) - high sensitivity of thin air space . Aplanatic glass choice: vanishing coma (c) . Cases: a) simple achromate, Achromat, splitting sph corrected, with coma b) simple achromate, (d) coma corrected by bending, with sph Achromat, aplanatic glass choice c) other glass choice: sph better, coma reversed d) splitted achromate: all corrected (e) e) aplanatic glass choice: all corrected Wave length:
Ref : H. Zügge
Achromate
. Residual aberrations of an achromate . Clearly seen: 1. Distortion 2. Chromatical magnification 3. Astigmatism
25 Spherochromatism: Achromate
. Residual spherochromatism of an achromate . Representation as function of apeture or wavelength
longitudinal aberration defocus variation rp 1 656
pupil height :
rp = 1 r = 0.707 587 p rp = 0.4 rp = 0
486 nm 587 nm 656 nm 486 z z 0 0.04 0.08 0.12 -0.04 0 0.04 0.08 26 Axial Colour: Achromate and Apochromate
. Effect of different materials . Axial chromatical aberration changes with wavelength . Different levels of correction: apochromate achromate 1.No correction: lens, one zero crossing point 644 2.Achromatic correction: C' - coincidence of outer colors - remaining error for center residual error apochromate wavelength - two zero crossing points 3. Apochromatic correction: lens - coincidence of at least three e 546 colors residual error achromate - small residual aberrations singlet - at least 3 zero crossing points - special choice of glass types F' s' with anomalous partial 480 dispertion necessery -100 0 100 27 Axial Colour: Apochromate
Anormal partial dispersion and normal line
Pg,F 0.6500
N-SF10 SF15 N-SF1 N-SF57 SF10 N-SF15 N-SF6 SF66 N-LAF7 N-SF64 N-SF4 N-SF8 N-SF5 SF14 SFL57 N-SF19 N-LLF1 N-LAF36 N-LASF40 SF57 SF11 N-SF56 0.6125 N-BAF52 N-BAF51 BASF51 GG375G34 N-BASF2 SF6G05 N-BAF3 SF6 N-LF5 F5 N-F2 K5G20 N-BAK4 N-BAF10 SF56A F2G12 N-SK2 N-SK18 N-LAF3 N-BAF4 SF4 N-LASF35 SK10G10 N-BALF5 SF1 N-LLF6 SF8G07 BAK1G12 N-SSK8 N-LASF46 SK4G13 N-BALF4 SF5 LASFN9 N-SK15 SSK5G06 SF2 N-SSK5 N-LASF45 N-LASF36 SF5G10 N-SK4 LAFN7 N-K5 F2 N-KZFS12 0.5750 SK51 N-BASF64 N-KF9 LF5G15 N-BAK2 LF5 K7 KZFSN5 N-LASF43 N-SK11 N-LASF31 N-SK5 N-LAF2 LLF1 N-PSK53 N-LASF41 N-PSK57 N-LAF33 N-KZFS11 N-PK51 KZFSN4 N-PK52 N-PSK58 BK7G25 N-FK51 KZFS4G20 0.5375 N-PSK3 N-LASF30 N-LASF44 N-KZFS4 N-LAF32 N-FK5 N-LAF21 N-BK10 N-LAK7 N-LAK33 N-LAF35 N-LAK12 N-LAF28 N-LAK21 N-LAK8 N-LAF34 N-SK16 N-LAK10 normal N-ZK7 LAK9G15 N-LAK9 N-SSK2 line N-PSK3 N-SK14 K10 LAKN13 N-LAK22 N-LAK34 BK7G18 SK5G06 N-BAK1 LAKL12 N-KZFS2 N-BK7 N-LAK14 N-SK10 0.5000 90 80 70 60 50 40 30 20 28 Relative Partial Dispersion
. Preferred glass selection for apochromates
N-SF1 N-SF6 N-SF57 N-SF66 P-SF68 P-SF67
N-FK51A N-PK52A N-PK51 N-KZFS12 N-LAF21 N-KZFS4 N-LAF35 N-LAF33 N-LAK10 N-LASF41 N-KZFS2 N-LAF37 29 Axial Colour : Apochromate
P gF . Choice of at least one special glass
0,62 N-FS6 . Correction of secondary spectrum: (2) anomalous partial dispersion 0,60
. At least one glass should deviate 0,58 significantly form the normal glass line (1)+(2) T 0,56 N-KZFS11 (3) 656nm 0,54 (1) N-FK51 588nm 90 80 70 60 50 40 30 20
486nm
z 436nm z -0.2mm -0.2mm 0 1mm 30
Correction of Spherochromatism: Splitted Achromate
. Broken-contact achromate: different ray heights allow fof y red
correcting spherochromatism blue
rp
SK11 SF12
486 nm 587 nm 656 nm
z in [mm] 0 0.2 0.4 31 Axial Color Correction with Schupman Lens
wavelength . Non-compact system in mm 0.656 . Generalized achromatic condition Schupman lens with marginal ray hieghts yj achromate y2 y2 1 F 2 F 0 v 1 v 2 1 2 0.571 . Use of a long distance and
negative F2 for correction . Only possible for virtual imaging
0.486 s -50 mm 0 50 mm
first lens second lens positive positive
intermediate virtual image image 32 Two-Lens Apochromate
. Special glasses and very strong bending allows for apochromatic correction
. Large remaining spherical zonal aberration
. Zero-crossing points not well distributed over wavelength spectrum
656nm
588nm
486nm
z 436nm -2mm 5m 0 z 33 Buried Surface
. Cemented surface with perfect refrcative index match . No impact on monochromatic aberrations . Only influence on chromatical aberrations . Especially 3-fold cemented components are advantages . Can serve as a starting setup for chromatical correction with fulfilled monochromatic correction . Special glass combinations with nearly perfect parameters
Nr Glas nd nd d d 1 SK16 1.62031 0.00001 60.28 22.32 F9 1.62030 37.96 d d d 2 SK5 1.58905 0.00003 61.23 20.26 1 2 3 LF2 1.58908 40.97 3 SSK2 1.62218 0.00004 53.13 17.06 F13 1.62222 36.07 4 SK7 1.60720 0.00002 59.47 10.23 BaF5 1.60718 49.24
34 Principles of Glass Selection in Optimization
. Design Rules for glass selection
. Different design goals: 1. Color correction: index n large dispersion positive lens field flattening difference desired Petzval curvature 2. Field flattening: large index difference + + desired
negative lens color correction + -
availability of glasses - - dispersion
Ref : H. Zügge
Achromatic Hybrid Lens
. Lens with diffractive structured blue surface: hybrid lens green refractive red lens . Refractive lens: dispersion with Abbe number = 25...90
. Diffractive lens: equivalent Abbe
number red d 3.453 green d diffractive F C blue lens . Combination of refractive and diffractive surfaces: achromatic correction for compensated dispersion R D . Usually remains a residual high blue green secondary spectrum hybrid red lens . Broadband color correction is possible but complicated
Diffractive Optics: Dispersion
. Dispersion by grating diffraction: e e 3.330 Abbe number F ' C' Relative partial dispersion P g F ' 0.2695 g,F ' F ' C' Consequence :
Large secondary spectrum PgF'
0.8 . -P-diagram
SF59 SF10 0.6 F4 BK7 KzFS1 normal line
0.4
DOE
0.2 e -20 0 20 40 60 80
Diffractive Optics: Singlet Solutions
y d y' 50 mm p . Combination of DOE all 2. order and aspherical carrier
yp
s' -0.5 mm 0 50 mm y d y' p diffractive surface, phase aspherical
yp
s' -0.5 mm 0 y a d y' 50 mm p refractive surface, aspherical yp
s' -0.5 mm 0 y' 50 mm y diffractive d,a p surface, carrier aspherical
yp
s' -0.5 mm 0
Hybrid - Microscope Objective Lens
diffractive . Data : element - = 193 nm - NA = 0.65 - b = 50
- sfree = 7.8 mm
. Properties: - short total track - extreme large free working distance - few lenses
Parameter of Excentricity
. Relative position of stop inside system . Quantitative measure: h h Parameter of excentricity c CR MR (h absolut values) hCR hMR . Special cases: c = 1 image plane c = -1 pupil plane c = 0 same effective distance from image and pupil
pupil image
marginal ray hMR (ima) hCR hCR
chief ray
surface no j
40 Parameter of Excentricity
50 . Example: 40 30 excentricity for all surfaces marginal ray height 20 pupil . Change: 10 c = +1 ...-1 ...+1 image 0 object chief ray height -10 surface 0 5 10 15 20 25 30 35 index 1
0.5
excentricity
0
-0.5
-1 surface index 0 5 10 15 20 25 30 35
9 11 13 15 17 1 2 3 4 19 5 7 22 23 26 28 30 32 34
object stop image 41 Influence of Stop Position on Performance
. Ray path of chief ray depends on stop position
stop positions spot 42 Effect of Stop Position
stop . Example photographic lens
. Small axial shift of stop changes tranverse aberrations
. In particular coma is strongly influenced
Ref: H.Zügge 43 Microscopic Objective Lens Initial System
a) negative 1. achromate
2. b) reversed positive achromate
3. c) symmetric to c)
positive achromate
4. d) non-infinite positive achromate
5. e) AC-meniscus lenses