Lens Design I. Lecture 9: OptimizationI Herbert Gross. Summer term

Transcription

1 Lens Design I Lecture 9: OptimizationI Herbert Gross Summer term

2 2 Preliminary Schedule Basics Properties of optical systrems I Properties of optical systrems II Properties of optical systrems III Advanced handling I Aberrations I Aberrations II Wave aberrations, Zernike polynomials Introduction, Zemax interface, menues, file handling, preferences, Editors, updates, windows, coordinates, System description, 3D geometry, aperture, field, wavelength Diameters, stop and pupil, vignetting, Layouts, Materials, Glass catalogs, Raytrace, Ray fans and sampling, Footprints Types of surfaces, cardinal elements, lens properties, Imaging, magnification, paraxial approximation and modelling, telecentricity, infinity object distance and afocal image, local/global coordinates Component reversal, system insertion, scaling of systems, aspheres, gratings and diffractive surfaces, gradient media, solves Add fold mirror, scale system, slider, multiconfiguration, universal plot, diameter types, lens catalogs Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams, Aberration expansions, Primary aberrations Aberrations III Point spread function, Optical transfer function Optimization I Principles of nonlinear optimization, Optimization in optical design, Global optimization methods, Solves and pickups, variables, Sensitivity of variables in optical systems Optimization II Systematic methods and optimization process, Starting points, Optimization in Zemax Imaging Fundamentals of Fourier optics, Physical optical image formation, Imaging in Zemax Correction I Correction II Symmetry principle, Lens bending, Correcting spherical aberration, Coma, stop position, Astigmatism, Field flattening, Chromatical correction, Retrofocus and telephoto setup, Design method Field lenses, Stop position influence, Aspheres and higher orders, Principles of glass selection, Sensitivity of a system correction

3 3 Contents 1. Principles of nonlinear optimization 2. Optimization in optical design 3. Global optimization methods 4. Optimization in Zemax

4 4 Basic Idea of 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 f (x) J i j f x m F( x) w g j i1 F x j i j i y f ( x) 2 i Hesse matrix of 2nd derivatives H jk 2 F x x j k

6 Optimization Principle for 2 Degrees of Freedom 6 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

7 7 Nonlinear Optimization Linearized environment around working point Taylor expansion of the target function f f 0 J x Quadratical approximation of the merit function Solution by 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 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)

8 8 Calculation of Derivatives Derivative vector in merit function topology: Necessary for gradient-based methods g jk f j ( x) x k x k f j ( 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 Effect of Constraints on Optimization 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 Optimization Minimum Search 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

13 13 Optimization and Starting Point 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

14 14 Optimization Merit Function in Optical Design 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

15 15 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 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

16 16 Parameter of Optical Systems 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

17 17 Constraints in Optical Systems Constraints in the optimization of optical systems: 1. Discrete standardized 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)

18 18 Lack of Constraints in Optimization 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

19 19 Optimization in Optics Typical in optics: Twisted valleys in the topology Selection of local minima LM 1 LM 2 LM 5 LM 4 LM 3

20 20 Optimization Landscape of an Achromate Typical merit function of an achromate Three solutions, only two are useful r 2 aperture reduced r 1 good solution

21 21 Optimization: Discrete Materials Special problem in glass optimization: finite area of definition with discrete parameters n, n n Restricted permitted area as one possible contraint Model glass with continuous values of n, n in a pre-phase of glass selection, freezing to the next adjacend glass area of permitted glasses in optimization area of available glasses n

22 22 Global Optimization: Escape method of Isshiki Simulated Annealing: temporarily added term to overcome local minimum F merit function with additive term F(x)+F esc F esc ( x) F F ( x) F 2 Optimization and adaptation of annealing parameters 0 e 0 conventional path F esc local minimum x loc global minimum x glo merit function F(x) x = 1. 0 = = 0. 5

23 23 Global Optimization 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 =

24 Saddel Point Method Saddel points in the merit function topology Systematic search of adjacend local minima is possible Exploration of the complete network of local minima via saddelpoints M 2 S M 1 F o

25 Saddel Point Method Example Double Gauss lens of system network with saddelpoints

26 Merit Function in Zemax Default merit function 1. Criterion 2. Ray sampling (high NA, aspheres,...) 3. Boundary values on thickness of center and edge for glass / air 4. Special options Add individual operands Editor: settings, weight, target actual value relative contribution to sum of squares Several wavelengths, field points, aperture points, configurations: many requirements Sorted result: merit function listing 26

27 Merit function in Zemax If the number of field points, wavelengths or configurations is changed: the merit function must be updated explicitly Help function in Zemax: many operands 27

28 Merit Function in Zemax Classical definition of the merit function in Zemax: Special merit function options: individual operands can be composed: - sum, diff, prod, divi,... of lines, which have a zero weight itself - mathematical functions sin, sqrt, max... - less than, larger than (one-sided intervals as targets) Negative weights: requirement is considered as a Lagrange multiplier and is fulfilled exact Optimization operands with derivatives: building a system insensitive for small changes (wide tolerances) Further possibilities for user-defined operands: construction with macro language (ZPLM) General outline: - use sinple operands in a rough optimization phase - use more complex, application-related operands in the final fine-tuning phase 28

29 Variables in Zemax Defining variables: indicated by V in lens data editor toggle: CNTR z or right mouse click Auxiliary command: remove all variables, all radii variable, all distances variable If the initial value of a variable is quite bad and a ray failure occurs, the optimization can not run and the merit function not be computed 29

30 Variable Glass in Zemax Modell glass: characterized by index, Abbe number and relative dispersion Individual choice of variables Glass moves in Glass map Restriction of useful area in glass map is desirable (RGLA = regular glass area) Re-substitution of real glass: next neighbor in n-n-diagram Choice of allowed glass catalogs can be controlled in General-menu Other possibility to reset real glasses: direct substitution 30

31 Optimization Methods Available in Zemax General optimization methods - local - global Easy-one-dimensional optimizations - focus - adjustment - slider, for visual control Special aspects: - solves - aspheres - glass substitutes 31

32 Methods Available in Zemax Classical local derivative: - DLS optimization (Marquardt) - orthogonal descent Hammer: - Algorithm not known - Useful after convergence - needs long time - must be explicitely stopped Global: - global search, followed by local optimization - Save of best systems - must be explicitely stopped 32

33 Conventional DLS-Optimization in Zemax Optimization window: Choice of number of steps / cycles Automatic update of all windows possible for every cycle (run time slows down) After run: change of merit function is seen Changes only in higher decimals: stagnation Window must be closed (exit) explictly 33