www.iap.uni-jena.de

Advanced Lens Design

Lecture 2: Optimization I

2013-10-22

Herbert Gross

Winter term 2013

2

Preliminary Schedule

1 15.10. Introduction Paraxial optics, ideal lenses, optical systems, raytrace, Zemax handling

2 22.10. Optimization I Basic principles, paraxial layout, thin lenses, transition to thick lenses, scaling, Delano diagram, bending

3 29.10. Optimization II merit function requirements, effectiveness of variables

4 05.11. Optimization III complex formulations, solves, hard and soft constraints

5 12.11. Structural modifications zero operands, lens splitting, aspherization, cementing, lens addition, lens removal

6 19.11. Aberrations and performance Geometrical aberrations, wave aberrations, PSF, OTF, sine condition, aplanatism, isoplanatism

7 26.11. Aspheres and freeforms

spherical correction with aspheres, Forbes approach, distortion correction, freeform surfaces, optimal location of aspheres, several aspheres

8 03.12. Field flattening thick meniscus, plus-minus pairs, field lenses

9 10.12. Chromatical correction

Achromatization, apochromatic correction, dialyt, Schupman principle, axial versus transversal, glass selection rules, burried surfaces

10 17.12. Special topics symmetry, sensitivity, anamorphotic lenses

11 07.01. Higher order aberrations high NA systems, broken achromates, Merte surfaces, AC meniscus lenses

12 14.01. Advanced optimization strategies

local optimization, control of iteration, global approaches, growing requirements, AC-approach of Shafer

13 21.01. Mirror systems special aspects, bending of ray paths, catadioptric systems

14 28.01. Diffractive elements color correction, straylight suppression, third order aberrations

15 04.02. Tolerancing and adjustment tolerances, procedure, adjustment, compensators

1. Delano diagram

2. Nonlinear optimization

3. Optimization in optical design

4. Initial system selection

5. Thick lenses and bending

3

Contents

Delano Diagram

Special representation of ray bundles in

optical systems:

marginal ray height

vs.

chief ray height

Delano digram gives useful insight into

system layout

Every z-position in the system corresponds

to a point on the line of the diagram

Interpretation needs experience

CRyy

lens

y

field lens collimatormarginal ray

chief ray

y

y

y

lens at

pupil

position

field lens

in the focal

plane

collimator

lens

MRyy

5

Delano Diagram

Delano’s

skew ray

Image

d2 d1

yC yM

(yC,yM) yM

a a

b

c c

d d

yC Delano ray (blue)=

Chief ray (red) in x +

Marginal ray (green) in y

Delano Diagram =

Delano ray projected

into the xy-Plane

Substitution

x -->

y = Pupil coordinate

= yc Field coordinate

Stop

Lens

a b

c

d

y (or yM)

y

y

y

Ref.: M. Schwab / M. Geiser

chief ray

marginal ray

Delano diagram:

projection

along z

b

x

y

marginal

ray

chief

ray

skew ray

y

y

y

y

diagram

image

object

Pupil locations:

intersection points with y-axis

Field planes/object/image:

intersectioin points with y-bar axis

Construction of focal points by

parallel lines to initial and final line

through origin

y

y

object

plane

lens

image

plane

stop and

entrance pupil

exit pupil

y

y

object

space

image

space

front focal

point Frear focal

point F'

Delano Diagram

Delano Diagram

Influence of lenses:

diagram line bended

Location of principal planes

y

y

strong positive

refractive power

weak positive

refractive power

weak negative

refractive power

y

y

object space image space

principal

plane

yP

yP

Location of principal planes in the Delano diagram

Triplet Effect of stop shift

Delano Diagram

y

y

object

plane

lens L1

lens L2

lens L3

image

plane

stop shift

y

y

object

spaceimage

space

principal plane

yP

yP

Vignetting :

ray heigth from axis

Marginal and chief ray considered

Line parallel to -45° maximum diameter

yya

Delano Diagram

object

pupil

chief ray

marginal ray

coma ray

yyy + y

y

y

maximum height at

lens 2

system polygon line

lens 1

lens 2

lens 3

D/2

Delano Diagram

Microscopic system

y

y

eyepiece

microscope

objective tube lens

object

image at infinity

aperture

stop

intermediate

image

exit pupil

telecentric

Delanos y-ybar diagram

Simple implementation in Zemax

11

Delano Diagram in Zemax

Example:

- Lithographic projection lens

- the bulges can be seen by characteristic arcs

- telecentricity: vertical lines

- diameter variation

- pupil location

12

Delano Diagram in Zemax

telecentric

image

telecentric

object

pupil

largest beam

diameter: surface 19

Dmax/2

1

23456

78

910

11

12

13

1415

16171819

2021

22

2324

2526

27

28

29

3031

32333435

3637

3839

40

41

42

430

smallest

beam

diameter:

surface 25

yMR

yCR

negative

lenses

positive

lenses

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

13

Complex topology of the merit function:

1. many local minima

2. function not differentiable

3. function not smooth

4. value of global minimum not known

14

Optimization – Merit Function

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

15

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

16

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

17

Gauss-Newton method

Normal equations

System matrix

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

Conjugate gradient method

Reduction of oscillations

Local Optimization Algorithms

fJJJxTT

1

i

T

ijijij

T

ijj fJIJJx 12

i

T

ijij

T

ijj fJJJx 1

TTJJJA

1

18

Control function

Gradient method with steepest descent

Changing directions:

zig-zac-path with poor convergence

Optimal damping of step size

Steepest Descent Method

)(xFg

x x

x1

x2 F1

F2

F3

F4

s1

s2

s3

s4interative

improvement

steps

levels merit

function

gg

ggT

TT

JJ

xxxgff TTT 200

0 xg

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

20

Optimization Damping

Damping with factor l

Damping defines the orientation

and the size of the improvement

step

kjkkkijjij fJIJJx 1

x1

x2 F1

F2

F3

F4

improvement

steps

merit function

levels

x2x1

Ref: C. Menke

21

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

22

Adaptation of direction and length of

steps:

rate of convergence

Gradient method:

slow due to zig-zag

Optimization: Convergence

0 10 20 30 40 50 60-12

-10

-8

-6

-4

-2

0

2

Log F

iteration

steepest

descent

conjugate

gradient

Davidon-

Fletcher-

Powell

23

Optimization and Starting Point

The initial starting point

determines the final result

Only the next located solution

without hill-climbing is found

24

x2

A

A'

B'

B

C'

D'

x1

attraction

to A'

attraction

to B'

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,

25

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

26

System development flow chart

Development in Optics

requirements

no

fix specification

define merit function

define constraints

search start system

requirements

reachable ?

rough optimizationstructural changes

requirements reduced

better inital system

yes

improved optimization

convergence ?nominor changes of goals

and system

yes

fine tuning

norm radii

tolerancing

mechanical housing

adjustment....

end

1. definition

phase

2. initial

design

4. refined

optimization

5. finishing

calculations

3. orientation phase

Existing solution modified

Literature and patent collections

Principal layout with ideal lenses

successive insertion of thin lenses and equivalent thick lenses with correction control

Approach of Shafer

AC-surfaces, monochromatic, buried surfaces, aspherics

Expert system

Experience and genius

Optimization: Starting Point

object imageintermediate

imagepupil

f1

f2 f

3f4

f5

28

Decomposition of ABCD-Matrix

2x2 ABCD-matrix of a system in air: 3 arbitrary parameters

Every arbitrary ABCD-setup can be decomposed into a simple system

Decomposition in 3 elementary partitions is alway possible

Case 1: C # 0 one lens, 2 transitions

System data

MA B

C D

L

f

L

1

0 1

1 01

1

1

0 1

1 2

LD

C1

1

LA

C2

1

fC

1

Output

xo

Input

xi

Lens

f

L2

L1

Decomposition of ABCD-Matrix

Case 2: B # 0 two lenses, one transition

System data:

MA B

C D f

L

f

1 01

11

0 1

1 01

12 1

fB

A1

1

L B

fB

D2

1

OutputInput

Lens 1

f1

L

Lens 2

f2

Pre-Calculations

Zero-order properties of the system:

- focal length

- magnification

- pupil size and location

- size/length of the system, image location

Pre-Calculation of the system structure, which is independent of lens bendings,

Analytical 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

31

Initial Conditions

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

1s

N

n

nm

M

m

m FF11

''

N

n nm

nmM

m

m

FF

11

2 ''

N

n nm

nmM

m

pmm

FF

11

''

N

n nm

nmnmM

m

m

FPPF

11

2 ''

N

n nm

nmM

m n

F

n

F

11

''

N

n

nm

M

m

pm F11

'0

32

Introduction of a finite lens thickness from an ideal setup

Goal: angles of chief ray and marginal ray not changed

First principal plane at location of ideal lens

2nd principal plane shifted by z

Change of radii to get the same

ray path

33

Introduction of Thick Lenses

P P'

z

N N'

r2r1

z

s'PsP

t

P=P'b) ideal lens

a) real thick lens

21

1,

1'

r

tf

n

ns

r

tf

n

ns PP

)1()()1(

'

21

21

ntrrn

trrnt

ssz PP

1

')0(

11

)0(

'

'1

1

j

Pjj

j

Pjj

s

srr

s

srr

Different shapes of singlet lenses:

1. bi-, symmetric

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

bi-convex lens

P'P

plane-convex lens

P'P

positive

meniscus lens

P P'

bi-concave lens

P'P

plane-concave

lens

P P'

negative

meniscus lens

Lens shape

Bending of a Lens

Bending: change of shape for

invariant focal length

Parameter of bending

Principal planes are moving

Incidence angles and most aberrations are changing

12

21

RR

RRX

X = +1

X > +1

X = 0

X = -1

meniscus lensX < -1

biconvex lens

biconcave lens

planconvex lens

planconcave lens

planconvex lens

planconcave lens

meniscus lens

Ray path at a lens of constant focal length and different bending

Quantitative parameter of description X:

The ray angle inside the lens changes

The ray incidence angles at the surfaces changes strongly

The principal planes move

For invariant location of P, P‘ the position of the lens moves

P P'

F'

X = -4 X = -2 X = +2X = 0 X = +4

Lens bending und shift of principal plane

12

21

RR

RRX

Magnification parameter M:

defines ray path through the lens

Special cases:

1. M = 0 : symmetrical 4f-imaging setup

2. M = -1: object in front focal plane

3. M = +1: object in infinity

The parameter M strongly influences the aberrations

1'

21

2

1

1

'

'

s

f

s

f

m

m

UU

UUM

Magnification Parameter

M=0

M=-1

M<-1

M=+1

M>+1

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:

usually no complete correction possible

Spherical Aberration: Lens Bending

object

plane

image

plane

principal

plane

diameter

bending

X

Correction of spherical aberration:

Splitting of lenses

Distribution of ray bending on several

surfaces:

- smaller incidence angles reduces the

effect of nonlinearity

- decreasing of contributions at every

surface, but same sign

Last example (e): one surface with

compensating effect

Correcting Spherical Aberration: Lens Splitting

Transverse aberration

5 mm

5 mm

5 mm

(a)

(b)

(c)

(d)

Improvement

(a)à(b) : 1/4

(c)à(d) : 1/4

(b)à(c) : 1/2

Improvement

Improvement

(e)

0.005 mm

(d)à(e) : 1/75

Improvement

5 mm

Ref : H. Zügge

39

Correcting Spherical Aberration: Cementing

0.25 mm0.25 mm

(d)

(a)

(c)

(b)

1.0 mm 0.25 mm

Crown

in front

Filnt

in front

Ref : H. Zügge

Correcting spherical aberration by cemented doublet:

Strong bended inner surface compensates

Solid state setups reduces problems of centering sensitivity

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 ray

40