Design and Correction of optical Systems

Transcription

1 1 Design and Correction of optical Systems Part 13: Correction of aberrations 2 Summer term 2012 Herbert Gross

2 2 Overview 1. Basics Materials Components Paraxial optics Properties of optical systems Photometry Geometrical aberrations Wave optical aberrations Fourier optical image formation Performance criteria Performance criteria Correction of aberrations Correction of aberrations Optical system classification

3 3 Contents 13.1 Fundamentals of optimization 13.2 Optimization in optical design 13.3 Initial setups and correction methods 13.4 Sensitivity of systems 13.5 Miscellaneous

4 4 Optimization Topology of the merit function in 2 dimensions Iterative down climbing in the topology topology of meritfunction F start iteration path x 1 x 2

5 5 Nonlinear Optimization Mathematical description of the problem: n variable parameters m target values Jacobi system matrix of derivatives, Influence of a parameter change on the various target values, sensitivity function Scalar merit function Gradient vector of topology x r r r f (x) J i j f = x m r F( x) = w g j i= 1 F = x j i j i [ y f ( x) ] 2 i r Hesse matrix of 2nd derivatives H jk 2 = x F x j k

6 6 Nonlinear Optimization Linearized environment around working point Taylor expansion of the target function r f r f = 0 + J r x Quadratical approximation of the merit function Solution ny lineare Algebra system matrix A cases depending on the numbers of n / m Iterative numerical solution: Strategy: optimization of - direction of improvement step - size of improvement step A + = A 1 F ( ) T 1 A A A A ( AA ) T T r r x) = F( x ( 0 T 1 if if if m r ) + J x + > n m < n m 1 2 (under (over = r r x H x n determined) determined)

7 7 Optimization Linear algebra of optimization: 1. Exact solvable 2. Over determined : more parameter than targets solution not unique 3. Under determined : more tragets than parameter best approximation solution A + = A 1 T 1 ( A A) T T A ( AA ) A T 1 if if if m = n ( exact) m > n ( over determined) m < n ( under determined) parameter n parameter n parameter n targets m A x r f r targets A m f r targets m A x r x r f r

8 8 Calculation of Derivatives Derivative vector in merit function topology: Necessary for gradient-based methods g jk = r f j ( x) x k = x k f j r ( x) Numerical calculation by finite differences g jk = f right j x k f j Possibilities and accuracy f j (x k ) left f j-1 f j (x k ) f j right f j+1 forward central exact x k - x k x k x k x k x k + x k backward x k

9 9 Optimierung Effect of constraints x 1 path without constraint 0 local minimum path with constraint constraint x 1 < 0 global minimum initial point x 2

10 10 Boundary Conditions and Constraints Types of constraints 1. Equation, rigid coupling, pick up 2. One-sided limitation, inequality 3. Double-sided limitation, interval Numerical realizations : 1. Lagrange multiplier 2. Penalty function 3. Barriere function 4. Regular variable, soft-constraint F(x) F(x) penalty function P(x) p large barrier function B(x) p large F 0 (x) p small F 0 (x) p small x x x min permitted domain x max permitted domain

11 11 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

12 12 Local Optimization Algorithms Gauss-Newton method Normal equations r x = r ( ) T 1 T J J J f System 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 ( Jij Jij + λ Iij ) Jij f i = J T ij ( T ) 1 J J fi ij ij Conjugate gradient method Reduction of oscillations

13 Optimization Principle for 2 Degrees of Freedom 13 Aberration depends on two parameters Linearization of sensitivity, 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-linearity: 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

14 Optimization 14 Principle of searching the local minimum x 2 nearly ideal iteration path steepest descent topology of the merit function starting point method with compromise Gauss-Newton method quadratic approximation around the starting point x 1

15 Optimization Damping 15 Damping with factor l x j = 1 ( J ji Jij + λ I kk ) 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

16 Optimization: Convergence 16 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

17 Optimization and Starting Point 17 The initial starting point determines the final result p 2 Only the next located solution without hill-climbing is found D' A' C' B' A B p 1

18 Global Optimization: Simulated Annealing 18 Simulated Annealing: temporarily added term to overcome local minimum F merit function with additive term F(x)+ F esc F esc r ( x) = F ( F ( x) F ) 2 Optimization and adaptation of annealing parameters 0 e β r 0 conventional path F esc local minimum x loc global minimum x glo merit function F(x) x β = 1. 0 β = 0. 01

19 Optimization Merit Function in Optical Design 19 Goal of optimization: Find the system layout which meets the required performance targets according of the specification Formulation of performance criteria must be done for: - Apertur rays - Field points - Wavelengths - Optional several zoom or scan positions Selection of performance criteria depends on the application: - Ray aberrations - Spot diameter - Wavefornt description by Zernike coefficients, rms value - Strehl ratio, Point spread function - Contrast values for selected spatial frequencies - Uniformity of illumination Usual scenario: Number of requirements and targets quite larger than degrees od freedom, Therefore only solution with compromize possible

20 Optimization in Optical Design 20 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 ( ist soll j j ) Φ = g j f f j= 1, m 2 Problems: 1. linear dependence of variables 2. internal contradiction of requirements 3. initail value far off from final solution Types of constraints: 1. exact condition (hard requirements) 2. soft constraints: weighted target Finding initial system setup: 1. modification of similar known solution 2. Literature and patents 3. Intuition and experience

21 Parameter of Optical Systems 21 Characterization and description of the system delivers free variable parameters of the system: - Radii - Thickness of lenses, air distances - Tilt and decenter - Free diameter of components - Material parameter, refractive indices and dispersion - Aspherical coefficients - Parameter of diffractive components - Coefficients of gradient media General experience: - Radii as parameter very effective - Benefit of thickness and distances only weak - Material parameter can only be changes discrete

22 Constraints in Optical Systems 22 Constraints in the optimization of optical systems: 1. Discrete standsrdized radii (tools, metrology) 2. Total track 3. Discrete choice of glasses 4. Edge thickness of lenses (handling) 5. Center thickness of lenses(stability) 6. Coupling of distances (zoom systems, forced symmetry,...) 7. Focal length, magnification, workling distance 8. Image location, pupil location 9. Avoiding ghost images (no concentric surfaces) 10. Use of given components (vendor catalog, availability, costs)

23 Optimization in Optics 23 Typical in optics: Twisted valleys in the topology Selection of local minima LM 1 LM 2 LM 5 LM 4 LM 3

24 Optimization 24 Typical merit function of an achromate Three solutions, only two are useful r 2 Φ aperture reduced r 1 Φ good solution

25 Optimization of Achromatic Glasses 25 Perfect aplanatic line of corresponding glasses For one given flint a line indicates the usefull crown glasses and vice versa n n fixed flint glass line of minimal spherical aberration line of minimal spherical aberration ν fixed crown glass ν

26 Global Optimization 26 No unique solution reference design : F = solution 5 : F = solution 11 : F = Contraints not sufficient fixed: unwanted lens shapes solution 6 : F = solution 12 : F = Many local minima with nearly the same performance solution 1 : F = solution 7 : F = solution 13 : F = solution 2 : F = solution 8 : F = solution 14 : F = solution 3 : F = solution 9 : F = solution 15 : F = solution 4 : F = solution 10 : F = solution 16 : F =

27 Optimization 27 Illustration of not usefull results due to non-sufficient constraints negative edge thickness negative air distance lens thickness to large lens stability to small air space to small

28 Optimization: Discrete Materials 28 Special problem in glass optimization: finite area of definition with discrete parameters n, ν n Restricted permitted area as one possible contraint area of available glasses Model glass with continuous values of n, ν in a pre-phase of glass selection, freezing to the next adjacend glass area of permitted glasses in optimization ν

29 Correction Effectiveness 29 Effectiveness of correction features on aberration types Aberration Primary Aberration 5th Chromatic Spherical Aberration Coma Astigmatism Petzval Curvature Distortion 5th Order Spherical Axial Color Lateral Color Secondary Spectrum Spherochromatism Lens Parameters Lens Bending (a) (c) e (f) Power Splitting Power Combination a c f i j (k) Distances (e) k Stop Position Makes a good impact. Refractive Index (b) (d) (g) (h) Makes a smaller impact. Makes a negligible impact. Action Material Dispersion (i) (j) (l) Relative Partial Disp. GRIN Zero influence. Special Surfaces Cemented Surface b d g h i j l Aplanatic Surface Aspherical Surface Mirror Diffractive Surface Struc Symmetry Principle Field Lens Ref : H. Zügge

30 Correction Methods 30 Lens bending Lens splitting Power combinations (a) (b) (c) (d) (e) Distances (a) (b) Ref : H. Zügge

31 Optimization: Starting Point 31 Existing solution modified Literature and patent collections Principal layout 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 system Experience and genius

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

33 Initial Conditions 33 Valid for object in infinity: 1. Total refractive power 2. Correction of Seidel aberrations 2.1 Dichromatic correction of marginal ray axial achromatical 2.2 Dichromatic correction of chief ray achromatical lateral magnification 2.3 Field flattening Petzval 2.4 Distortion correction according to Berek 3. Tri-chromatical correction Secondary spectrum s 1 F' = = F' = ν F' = ν F' = n M M m= 1 M ω m= 1 M m= 1 M m= 1 0= ω m= 1 F' P = ν N m n= 1 ω ω ω M m m= 1 F' nm N 2 F' nm m n= 1 ν nm N n= 1 N pm n= 1 ω N pm n= 1 F' n nm nm F' nm F' ν nm nm N 2 Pnm F' nm m n= 1 ν nm

34 Strategy of Correction and Optimization 34 Usefull options for accelerating a stagnated optimization: split a lens increase refractive index of positive lenses lower refractive index of negative lenses make surface with large spherical surface contribution aspherical break cemented components use glasses with anomalous partial dispersion

35 Zero-Operations 35 Operationen with zero changes in first approximation: 1. Bending a lens. 2. Flipping a lens into reverse orientation. 3. Flipping a lens group into reverse order. 4. Adding a field lens near the image plane. 5. Inserting a powerless thin or thick meniscus lens. 6. Introducing a thin aspheric plate. 7. Making a surface aspheric with negligible expansion constants. 8. Moving the stop position. 9. Inserting a buried surface for color correction, which does not affect the main wavelength. 10. Removing a lens without refractive power. 11. Splitting an element into two lenses which are very close together but with the same total refractive power. 12. Replacing a thick lens by two thin lenses, which have the same power as the two refracting surfaces. 13. Cementing two lenses a very small distance apart and with nearly equal radii.

36 Methodological Approach of Shafer 36 The surfaces are bent in such a way that the chief ray works only as a concentric or an aplanatic ray for the main wavelength. This approach guarantees that there is no coma and no astigmatism in the system. The lenses are not necessarily thin components. The refractive indices are chosen properly to satisfy the Petzval formula The introduction of buried surfaces with equal refractive indices for both materials, but different dispersion numbers, is used to correct the chromatic errors in the axial and the lateral direction In the next step, the spherical aberration is tackled by the following points: - one surface is made aspherical - the surfaces are split to divide - the lenses are bent near the pupil location - high refractive indices are chosen - small air gaps between concentric surfaces are introduced - cemented surfaces, which are concentric with the chief ray are introduced - meniscus lenses, which are concentric to the chief ray are used

37 Methodological Approach of Shafer 37 advantages of this structural initial design : - The effect and the function of every surface in the system is understood - The influence of the surfaces on the different aberrations shows a good decoupling - If no solution is found, the status can be used as a starting design for numerical optimization stop chief ray C A A C C C C : concentric A : aplanatic

38 Sensitivity of a System 38 Sensitivity/relaxation: Average of weighted surface contributions of all aberrations Sp h Correctability: Average of all total aberration values Total refractive power k F = F + F 1 ω j j j= 2 Important weighting factor: ratio of marginal ray heights ω j = h j h 1 Kom a Ast CH L Inz- Wi

39 Sensitivity of a System 39 Quantitative measure for relaxation with normalization A k j j= 1 = ω A j j = 1 F F j = h j h 1 F j F Non-relaxed surfaces: 1. Large incidence angles 2. Large ray bending 3. Large surface contributions of aberrations 4. Significant occurence of higher aberration orders 5. Large sensitivity for centering Internal relaxation can not be easily recognized in the total performance Large sensitivities can be avoided by incorporating surface contribution of aberrations into merit function during optimization

40 Sensitivity of a System 40 Representation of wave Seidel coefficients [λ] Sph Koma Double Gauss 1.4/ Ast Petz Verz surfaces Sum Ref: H.Zügge

41 Microscopic Objective Lens 41 Incidence angles for chief and marginal ray marginal ray microscope objective lens Aperture dominant system Primary problem is to correct spherical aberration chief ray incidence angle

42 Photographic lens 42 Incidence angles for chief and marginal ray Photographic lens Field dominant system Primary goal is to control and correct field related aberrations: coma, astigmatism, field curvature, lateral color chief ray 60 incidence angle marginal ray

43 Principles of Glass Selection in Optimization 43 Design Rules for glass selection Different design goals: 1. Color correction: large dispersion difference desired 2. Field flattening: large index difference desired Ref : H. Zügge

44 Substitution of Standard Radii 44 Method: Insertion of nearest available radii Check of optimal combinations re-adjusting thicknessess In general system slightly decreased in performance Example : Photographic lens : Orange : original, black: new system

45 Substitution of Standard Radii 45 field nm nm nm 1. After optimization 0 rms After substitution field nm nm nm rms field nm nm nm After re-optimization 0 rms

46 Development of Microscopic Lens Step : Generation of high-na on axis a ) without meniscus lens numerical aperture : NA = b) 1 a-c meniscus lens NA = c ) 2 a-c meniscus lenses NA = d) 3 a-c meniscus lenses NA = e) 3 a-c meniscus lenses and 1 c-c meniscus lens NA = 0.589

47 Development of Microscopic Lens Step : Optimization on axis Correcting thickness Introducing free working distance Inserting finite field axis correction 1. version : 2. and 4. lens to thin, field 20 µm 2. version : free working distance to short, field 50 µm 3. version : field 100 µm

48 Development of Microscopic Lens Step : Improvements Additional degrees of freedom Material changes Avoiding strong bendings Optimization of lens-thickness Comparison of variants Fine-tuning

49 Development of Microscopic Lens 49 Overview of the steps of development 1. Correction on axis, quality good 2. Correction with field, compact, quality not sufficient 3. Correction with triplet, some surfaces obsolet, quality not sufficient 4. Correction with one doublet and larger air distances, quality not sufficient 5. Correction with two doublets, quality nearly good enough 6. Further optimization with glass choices, better field uniformity 7. with working distance and telecentricity

50 Summary of Important Topics 50 Non-least-square optimization works with local gradients There are several methods to accelerate the convergence In the usual damped-least-squares a damping factors controls the size and the direction of the improvement step Several types of constraints can be defined in the merit function A special problem in reality is the discrete nature of the available glasses Optimization in optics is highly non-linear Global methods are useful, but until now there is no break through There are characteristic steps for correction methods in optical design A complete and save cooking receipt is not known The choice of the initial setup determines the final result, therefore experience helps For geeting a manufacturing-friendly system, a relaxed design with small uniform aberrations surface contributions is helpful More complex designs are developed iterative in small controllable steps

51 Outlook 51 Next lecture: Part 14 Optical system classification 1. Date: Wednesday, Contents: 14.1 Overview and classification 14.2 Achromate 14.3 Collimator 14.4 Microscope optics 14.5 Photographic optics 14.6 Zoom lenses 14.7 Telescopes 14.8 Miscellaneous 14.9 Lithographic projection system