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