Advanced Lens Design

Transcription

1 Advanced Lens Design Lecture 2: Optimization I Herbert Gross Winter term

2 2 Preliminar Schedule Introduction Paraxial optics, ideal lenses, optical sstems, ratrace, Zemax handling Optimization I Basic principles, paraxial laout, thin lenses, transition to thick lenses, scaling, Delano diagram, bending Optimization II merit function requirements, effectiveness of variables Optimization III complex formulations, solves, hard and soft constraints Structural modifications zero operands, lens splitting, aspherization, cementing, lens addition, lens removal Aberrations and performance Geometrical aberrations, wave aberrations, PSF, OTF, sine condition, aplanatism, isoplanatism Aspheres and freeforms spherical correction with aspheres, Forbes approach, distortion correction, freeform surfaces, optimal location of aspheres, several aspheres Field flattening thick meniscus, plus-minus pairs, field lenses Chromatical correction Achromatization, apochromatic correction, dialt, Schupman principle, axial versus transversal, glass selection rules, burried surfaces Special topics smmetr, sensitivit, anamorphotic lenses Higher order aberrations high NA sstems, broken achromates, Merte surfaces, AC meniscus lenses Advanced optimization local optimization, control of iteration, global approaches, strategies growing requirements, AC-approach of Shafer Mirror sstems special aspects, bending of ra paths, catadioptric sstems Diffractive elements color correction, stralight suppression, third order aberrations Tolerancing and adjustment tolerances, procedure, adjustment, compensators

3 3 Contents 1. Delano diagram 2. Nonlinear optimization 3. Optimization in optical design 4. Initial sstem selection 5. Thick lenses and bending

4 Delano Diagram Special representation of ra bundles in optical sstems: marginal ra height MR vs. chief ra height CR Delano digram gives useful insight into sstem laout Ever z-position in the sstem corresponds to a point on the line of the diagram Interpretation needs experience lens at pupil position field lens in the focal plane collimator lens marginal ra lens field lens collimator chief ra

5 5 Delano Diagram Delano ra (blue)= Chief ra (red) in x + Marginal ra (green) in Delano Diagram = Delano ra projected into the x-plane M a c b d a C Delano s skew ra marginal ra chief ra Lens b x M ( C, M ) c C Image d Substitution x --> Stop d 1 d 2 = Pupil coordinate = c Field coordinate a (or M ) b diagram Delano diagram: projection along z c d skew ra chief ra image object marginal ra Ref.: M. Schwab / M. Geiser

6 Delano Diagram Pupil locations: intersection points with -axis exit pupil Field planes/object/image: intersectioin points with -bar axis stop and entrance pupil lens object plane image plane Construction of focal points b parallel lines to initial and final line through origin front focal point F image space object space rear focal point F'

7 Delano Diagram Influence of lenses: diagram line bended weak negative refractive power weak positive refractive power strong positive refractive power Location of principal planes principal plane P object space image space P

8 Delano Diagram Location of principal planes in the Delano diagram P principal plane object space image space P Triplet Effect of stop shift lens L1 lens L2 lens L3 stop shift object plane image plane

9 Delano Diagram Vignetting : ra heigth from axis a Marginal and chief ra considered sstem polgon line lens 1 lens 2 maximum height at lens 2 Line parallel to -45 maximum diameter lens 3 D/2 object coma ra marginal ra pupil + chief ra

10 Delano Diagram Microscopic sstem microscope objective aperture stop tube lens telecentric object intermediate image image at infinit exit pupil eepiece

11 Delano Diagram in Zemax Delanos -bar diagram Simple implementation in Zemax 11

12 Delano Diagram in Zemax Example: - Lithographic projection lens - the bulges can be seen b characteristic arcs - telecentricit: vertical lines - diameter variation - pupil location MR D max /2 pupil smallest beam diameter: surface 25 largest beam diameter: surface positive lenses negative lenses telecentric image telecentric object 0 CR 12

13 13 Basic Idea of Optimization Topolog of the merit function in 2 dimensions Iterative down climbing in the topolog topolog of meritfunction F start iteration path x 1 x 2

14 14 Optimization Merit Function Complex topolog of the merit function: 1. man local minima 2. function not differentiable 3. function not smooth 4. value of global minimum not known

15 15 Nonlinear Optimization Mathematical description of the problem: n variable parameters m target values Jacobi sstem matrix of derivatives, Influence of a parameter change on the various target values, sensitivit function Scalar merit function Gradient vector of topolog x f (x) J i j f x m F( x) w g j i1 F x j i j i f ( x) 2 i Hesse matrix of 2nd derivatives H jk 2 F x x j k

16 Optimization Principle for 2 Degrees of Freedom 16 Aberration depends on two parameters Linearization of sensitivit, Jacobian matrix Independent variation of parameters Vectorial nature of changes: Size and direction of change f 2 Vectorial decomposition of an ideal step of improvement, linear interpolation Due to non-linearit: change of Jacobian matrix, next iteration gives better result 0 B x 2 =0.1 x 1 =0.035 target point x 2 =0.07 A initial point x 1 =0.1 C 0 f 2

17 17 Nonlinear Optimization Linearized environment around working point Talor expansion of the target function f f 0 J x Quadratical approximation of the merit function Solution b lineare Algebra sstem matrix A cases depending on the numbers of n / m Iterative numerical solution: Strateg: optimization of - direction of improvement step - size of improvement step A F 1 A T 1 A A A T T A AA x) F( x ( 0 T 1 if if if m ) J x n m n m 1 2 (under (over x H x n determined) determined)

18 18 Local Optimization Algorithms Gauss-Newton method Normal equations x T 1 T J J J f Sstem matrix A T 1 T J J J Damped least squares method (DLS) Daming reduces step size, better convergence without oscillations ACM method according to E.Glatzel Special algorithm with reduced error vector x x j j T 2 1 T J ij Jij Iij Jij f i J T ij T 1 Jij Jij fi Conjugate gradient method Reduction of oscillations

19 Steepest Descent Method Control function f T 0 f 0 2g T x x T x x 2 levels merit function F 3 F1 F 2 Gradient method with steepest descent g x 0 g x x F(x) Changing directions: zig-zac-path with poor convergence s 1 s 2 s 3 s 4 interative improvement steps F 4 Optimal damping of step size T T g J J g T g g x 1

20 20 Optimization Minimum Search Principle of searching the local minimum x 2 nearl ideal iteration path steepest descent topolog of the merit function starting point method with compromise Gauss-Newton method quadratic approximation around the starting point x 1

21 21 Optimization Damping Damping with factor l x j 1 J ji Jij Ikk J jk fk Damping defines the orientation and the size of the improvement step x 2 F 1 merit function levels F 3 F 2 F 4 improvement steps x1 x 2 x 1 Ref: C. Menke

22 22 Optimization Algorithms in Optical Design Local working optimization algorithms nonlinear optimization methods methods without derivatives derivative based methods simplex conjugate directions single merit function no single merit function least squares descent methods adaptive optimization nonlinear inequalities undamped damped steepest descents variable metric line search additive damping orthonorm alization conjugate gradient Davidon Fletcher multiplicative damping second derivative

23 23 Optimization: Convergence Adaptation of direction and length of steps: rate of convergence Gradient method: slow due to zig-zag Log F steepest descent -4-6 conjugate gradient Davidon- Fletcher- Powell iteration

24 24 Optimization and Starting Point The initial starting point determines the final result Onl the next located solution without hill-climbing is found x 2 attraction to A' D' attraction to B' A' C' B' A B x 1

25 25 Optimization in Optical Design Merit function: Weighted sum of deviations from target values Formulation of target values: 1. fixed numbers 2. one-sided interval (e.g. maximum value) 3. interval g f ist j j f j1, m soll j 2 Problems: 1. linear dependence of variables 2. internal contradiction of requirements 3. initail value far off from final solution Tpes of constraints: 1. exact condition (hard requirements) 2. soft constraints: weighted target Finding initial sstem setup: 1. modification of similar known solution 2. Literature and patents 3. Intuition and experience

26 26 Sstem Design Phases 1. Paraxial laout: - specification data, magnification, aperture, pupil position, image location - distribution of refractive powers - locations of components - sstem size diameter / length - mechanical constraints - choice of materials for correcting color and field curvature 2. Correction/consideration of Seidel primar aberrations of 3rd order for ideal thin lenses, fixation of number of lenses 3. Insertion of finite thickness of components with remaining ra directions 4. Check of higher order aberrations 5. Final correction, fine tuning of compromise 6. Tolerancing, manufactabilit, cost, sensitivit, adjustment concepts

27 Development in Optics Sstem development flow chart 1. definition phase requirements fix specification define merit function define constraints 2. initial design search start sstem 3. orientation phase rough optimization requirements reachable? no structural changes requirements reduced better inital sstem es improved optimization minor changes of goals and sstem no convergence? 4. refined optimization es fine tuning norm radii tolerancing mechanical housing adjustment finishing calculations end

28 28 Optimization: Starting Point Existing solution modified Literature and patent collections Principal laout with ideal lenses successive insertion of thin lenses and equivalent thick lenses with correction control object pupil intermediate image image f 1 f 2 f 3 f 4 f 5 Approach of Shafer AC-surfaces, monochromatic, buried surfaces, aspherics Expert sstem Experience and genius

29 Decomposition of ABCD-Matrix 2x2 ABCD-matrix of a sstem in air: 3 arbitrar parameters Ever arbitrar ABCD-setup can be decomposed into a simple sstem Decomposition in 3 elementar partitions is alwa possible Case 1: C # 0 one lens, 2 transitions Sstem data M A B L L C D f L f L 2 1 A 1 C 1 C D 1 C Input x i Lens f Output x o L 2 L 1

30 Decomposition of ABCD-Matrix Case 2: B # 0 two lenses, one transition M A B L 1 0 C D f f 2 1 Sstem data: f 1 B A 1 L B f 2 B D 1 Input f 1 Lens 1 Lens 2 f 2 Output L

31 31 Pre-Calculations Zero-order properties of the sstem: - focal length - magnification - pupil size and location - size/length of the sstem, image location Pre-Calculation of the sstem structure, which is independent of lens bendings, Analtical conditions for these 3rd order corrections - field flattening - achromatism, apochromatism - distortion-correction - anastigmatism - aplanatism - isoplanatism Lens bending with 3rd order lens contributions - spherical aberration - coma - astigmatism

32 32 Initial Conditions Valid for object in infinit: 1. Total refractive power 2. Correction of Seidel aberrations 2.1 Dichromatic correction of marginal ra axial achromatical 2.2 Dichromatic correction of chief ra achromatical lateral magnification 2.3 Field flattening Petzval 2.4 Distortion correction according to Berek 3. Tri-chromatical correction Secondar spectrum s 1 F' F' F' F' n M M m1 M m1 M m1 M m1 0 m1 F' P N m n1 M m m1 F' nm N 2 F' nm m n1 nm N n1 N pm n1 N pm n1 F' n nm nm F' nm F' nm nm N 2 PnmF' nm m n1 nm

33 33 Introduction of Thick Lenses Introduction of a finite lens thickness from an ideal setup Goal: angles of chief ra and marginal ra not changed First principal plane at location of ideal lens 2nd principal plane shifted b z s' P n 1 n f t r 1, s P n 1 n f t r 2 a) real thick lens r 1 P P' r 2 z s P s' P t( n 1) n ( r 1 r1 r2 t r ) t ( n 1) 2 N z N' Change of radii to get the same ra path b) ideal lens s P P=P' t s' P z r r r j j1 (0) j r s 1 s (0) j1 P s' 1 s' j j P' 1

34 Lens shape Different shapes of singlet lenses: 1. bi-, smmetric 2. plane convex / concave, one surface plane 3. Meniscus, both surface radii with the same sign Convex: bending outside Concave: hollow surface Principal planes P, P : outside for mesicus shaped lenses P P' P P' P P' P P' P P' P P' bi-convex lens plane-convex lens positive meniscus lens bi-concave lens plane-concave lens negative meniscus lens

35 Bending of a Lens Bending: change of shape for invariant focal length Parameter of bending X R R 1 2 R 2 R 1 X < -1 X = -1 meniscus lens planconvex lens planconcave lens Principal planes are moving Incidence angles and most aberrations are changing X = 0 biconvex lens biconcave lens X = +1 planconvex lens planconcave lens X > +1 meniscus lens

36 Lens bending und shift of principal plane Ra path at a lens of constant focal length and different bending Quantitative parameter of description X: The ra angle inside the lens changes X R R 1 2 R 2 R 1 The ra incidence angles at the surfaces changes strongl The principal planes move For invariant location of P, P the position of the lens moves P P' F' X = -4 X = -2 X = 0 X = +2 X = +4

37 Magnification Parameter Magnification parameter M: defines ra path through the lens M<-1 M U ' U U ' U 1 m 1 m 2 f s 1 2 f s' 1 M=-1 Special cases: 1. M = 0 : smmetrical 4f-imaging setup 2. M = -1: object in front focal plane 3. M = +1: object in infinit M=0 The parameter M strongl influences the aberrations M=+1 M>+1

38 Spherical Aberration: Lens Bending Spherical aberration and focal spot diameter as a function of the lens bending (for n=1.5) Optimal bending for incidence averaged incidence angles Minimum larger than zero: usuall no complete correction possible object plane principal plane image plane diameter bending X

39 39 Correcting Spherical Aberration: Lens Splitting Transverse aberration Correction of spherical aberration: Splitting of lenses (a) 5 mm Distribution of ra bending on several surfaces: - smaller incidence angles reduces the effect of nonlinearit - decreasing of contributions at ever surface, but same sign Last example (e): one surface with compensating effect (b) (c) 5 mm 5 mm Improvement (a)à(b) : 1/4 Improvement (b)à(c) : 1/2 5 mm (d) Improvement (c)à(d) : 1/ mm (e) Improvement (d)à(e) : 1/75 Ref : H. Zügge

40 40 Correcting Spherical Aberration: Cementing Correcting spherical aberration b cemented doublet: Strong bended inner surface compensates Solid state setups reduces problems of centering sensitivit In total 4 possible configurations: 1. Flint in front / crown in front 2. bi-convex outer surfaces / meniscus shape Residual zone error, spherical aberration corrected for outer marginal ra 1.0 mm 0.25 mm Crown in front (a) (b) Filnt in front (c) 0.25 mm (d) 0.25 mm Ref : H. Zügge