www.iap.uni-jena.de

Lens Design I

Lecture 9: Optimization I

2017-06-08

Herbert Gross

Summer term 2017

2

Preliminary Schedule - Lens Design I 2017

1 06.04. Basics Introduction, Zemax interface, menues, file handling, preferences, Editors, updates,

windows, coordinates, System description, 3D geometry, aperture, field, wavelength

2 13.04. Properties of optical systems I Diameters, stop and pupil, vignetting, Layouts, Materials, Glass catalogs, Raytrace,

Ray fans and sampling, Footprints

3 20.04. Properties of optical systems II

Types of surfaces, cardinal elements, lens properties, Imaging, magnification,

paraxial approximation and modelling, telecentricity, infinity object distance and

afocal image, local/global coordinates

4 27.04. Properties of optical systems III

Component reversal, system insertion, scaling of systems, aspheres, gratings and

diffractive surfaces, gradient media, solves

5 04.05. Advanced handling I Miscellaneous, fold mirror, universal plot, slider, multiconfiguration, lens catalogs

6 11.05. Aberrations I Representation of geometrical aberrations, Spot diagram, Transverse aberration

diagrams, Aberration expansions, Primary aberrations

7 18.05. Aberrations II Wave aberrations, Zernike polynomials, measurement of quality

8 01.06. Aberrations III Point spread function, Optical transfer function

9 08.06. Optimization I Principles of nonlinear optimization, Optimization in optical design, general process,

optimization in Zemax

10 15.06. Optimization II Initial systems, special issues, sensitivity of variables in optical systems, global

optimization methods

11 22.06. Advanced handling II System merging, ray aiming, moving stop, double pass, IO of data, stock lens

matching

12 29.06. Correction I Symmetry principle, lens bending, correcting spherical aberration, coma,

astigmatism, field curvature, chromatical correction

13 06.07. Correction II Field lenses, stop position influence, retrofocus and telephoto setup, aspheres and

higher orders, freeform systems, miscellaneous

1. Principles of nonlinear optimization

2. Optimization in optical design

3. General process

4. Optimization in Zemax

3

Contents

Basic Idea of Optimization

iteration

path

topology of

meritfunction F

x1

x2

start

Topology of the merit function in 2 dimensions

Iterative down climbing in the topology

4

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

Hesse matrix of 2nd derivatives

Nonlinear Optimization

x

)(xf

j

iji

x

fJ

21

)()(

m

i

ii xfywxF

j

jx

Fg

kj

kjxx

FH

2

5

Optimization Principle for 2 Degrees of Freedom

Aberration depends on two parameters

Linearization of sensitivity, Jacobian matrix

Independent variation of parameters

Vectorial nature of changes:

Size and direction of change

Vectorial decomposition of an ideal

step of improvement,

linear interpolation

Due to non-linearity:

change of Jacobian matrix,

next iteration gives better result

f2

0

0

C

B

A

f2

initial

point

x1

=0.1

x1

=0.035

x2

=0.07

x2

=0.1

target point

6

Linearized environment around working point

Taylor expansion of the target function

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

Nonlinear Optimization

xJff 0

xHxxJxFxF

2

1)()( 0

)determinedover(

)determinedunder(1

1

1

nmifAAA

nmifAAA

nmifA

ATT

TT

7

Derivative vector in merit function topology:

Necessary for gradient-based methods

Numerical calculation by finite differences

Possibilities and accuracy

Calculation of Derivatives

)()(

xfx

xfg jx

k

j

jk k

k

j

right

j

jkx

ffg

fj(x

k)

xk

exact

forward

backward

central

xk

xk x

k

xk-x

k xk+x

k

fj-1

fj

fj+1

fj(x

k)

right

left

8

Effect of constraints

Effect of Constraints on Optimization

initial

point

global

minimum

local

minimum

x2

x1

constraint

x1

< 0

0

path without

constraint

path with

constraint

9

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

Boundary Conditions and Constraints

x

F(x)

permitted domain

F0(x) p

small

penalty

function

P(x)

p large

xmaxx

min

x

F(x)

permitted domain

F0(x)

barrier

function

B(x)

p large

p small

10

Local working optimization algorithms

Optimization Algorithms in Optical Design

methods without

derivatives

simplexconjugate

directions

derivative based

methods

nonlinear optimization methods

single merit

function

steepest

descents

descent

methods

variable

metric

Davidon

Fletcher

conjugate

gradient

no single merit

function

adaptive

optimization

nonlinear

inequalities

least squares

undamped

line searchadditive

damping

damped

multiplicative

damping

orthonorm

alization

second

derivative

11

Principle of searching the local minimum

Optimization Minimum Search

x2

x1

topology of the

merit function

Gauss-Newton

method

method with

compromise

steepest

descent

nearly ideal iteration path

quadratic

approximation

around the starting

point

starting

point

12

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

Optimization Merit Function in Optical Design

13

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

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

Optimization in Optical Design

g f fj j

ist

j

soll

j m

2

1,

14

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

Parameter of Optical Systems

15

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)

Constraints in Optical Systems

16

Illustration of not usefull results due to

non-sufficient constraints

Lack of Constraints in Optimization

negative edge

thickness

negative air

distance

lens thickness to largelens stability

to small

air space

to small

17

Typical in optics:

Twisted valleys in the topology

Selection of local minima

Optimization in Optics

LM1 LM2

LM3

LM4

LM5

18

Typical merit function of an achromate

Three solutions, only two are useful

Optimization Landscape of an Achromate

aperture

reduced

good

solution

1

2

3

r1

r2

19

System Design Phases

1. 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

20

Strategy of Correction and Optimization

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

21

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

Merit Function in Zemax

22

Default merit function

Merit Function in Zemax

23

If the number of field points, wavelengths or configurations is changed:

the merit function must be updated explicitly

Help function in Zemax: many operands

24

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

Merit Function in Zemax

25

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

Variables in Zemax

26

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

Variable Glass in Zemax

27

General optimization methods

- local

- global

Easy-one-dimensional optimizations

- focus

- adjustment

- slider, for visual control

Special aspects:

- solves

- aspheres

- glass substitutes

Optimization Methods Available in Zemax

28

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

Methods Available in Zemax

29

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

Conventional DLS-Optimization in Zemax

30