Lens Design II - uni-jena.deDesign+II... · Circular homogeneous illuminated aperture: intensity...
Transcript of Lens Design II - uni-jena.deDesign+II... · Circular homogeneous illuminated aperture: intensity...
-
1
www.iap.uni-jena.de
Lens Design II
Lecture 1: Aberrations and optimization
2019-10-16
Herbert Gross
Winter term 2019
-
2 2
Preliminary Schedule Lens Design II 2018
1 16.10. Aberrations and optimization Repetition
2 23.10. Structural modificationsZero operands, lens splitting, lens addition, lens removal, material selection
3 30.10. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres
4 06.11. FreeformsFreeform surfaces, general aspects, surface description, quality assessment, initial systems
5 13.11. Field flatteningAstigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses
6 20.11. Chromatical correction IAchromatization, axial versus transversal, glass selection rules, burried surfaces
7 27.11. Chromatical correction IISecondary spectrum, apochromatic correction, aplanatic achromates, spherochromatism
8 04.12. Special correction topics I Symmetry, wide field systems, stop position, vignetting
9 11.12. Special correction topics IITelecentricity, monocentric systems, anamorphotic lenses, Scheimpflug systems
10 18.18. Higher order aberrations High NA systems, broken achromates, induced aberrations
11 08.01. Further topics Sensitivity, scan systems, eyepieces
12 15.01. Mirror systems special aspects, double passes, catadioptric systems
13 22.01. Zoom systems Mechanical compensation, optical compensation
14 29.01. Diffractive elementsColor correction, ray equivalent model, straylight, third order aberrations, manufacturing
15 05.02.
-
3
1. Geometrical aberrations
2. Wave aberrations and Zernikes
3. Point spread function
4. Modulation transfer function
5. Principles of optimization
Contents
-
4
Optical Image Formation
optical
system
object
plane
image
plane
transverse
aberrations
longitudinal
aberrations
wave
aberrations
Perfect optical image:
All rays coming from one object point intersect in one image point
Real system with aberrations:
1. tranverse aberrations in the image plane
2. longitudinal aberrations from the image plane
3. wave aberrations in the exit pupil
4
-
5
x, y object coordinates, especially object height
x', y' image coordinates, especially image height
xp,yp coordinates of entrance pupil
x'p, y'p coordinates of exit pupil
s object distance form 1st surface
s' image distance form last surface
p entrance pupil distance from 1st surface
p' exit pupil distance from last surface
Dx' sagittal transverse aberration
Dy' meridional transverse aberration
5
Notations for an Optical System
x
y
y
s
p
s'
p'
xP
yp
y'
x'
y'
x'P
y'p
object
plane
entrance
pupil
exit
pupil
image
plane
zDx'
Dy'
system
surfaces
xP
yp
x'P
y'p
P'P'0
P
-
6
Polynomial Expansion of Aberrations
Representation of 2-dimensional Taylor series vs field y and aperture r
Selection rules: checkerboard filling of the matrix
Constant sum of exponents according to the order
Field y
Spherical
y0 y 1 y 2 y3 y 4 y 5
Distortion
r
0
y
y3
primary
y5
secondary
r
1
r 1
Defocus
Aper-
ture
r
r
2
r2y Coma primary
r 3
r 3 Spherical
primary
r
4
r
5
r 5 Spherical
secondary
DistortionDistortionTilt
Coma Astigmatism
Image
location
Primary
aberrations /
Seidel
Astig./Curvat.
cos
cos
cos2
cos
Secondary
aberrations
cos
r 3y2
cos2
Coma
secondary
r4y cos
r2y
3 cos
3
r2y
3 cos
r1 y
4
r1 y
4 cos
2
r 3y2
r12 y
r12 y
6
-
7
Pupil Sampling
Ray plots
Spot
diagrams
sagittal ray fan
tangential ray fan
yp
Dy
tangential aberration
xp
Dy
xp
Dx
sagittal aberration
whole pupil area
7
-
8
Typical chart of aberration representation
Reference: at paraxial focus
Spherical Aberration
Wave aberration
tangential sagittal
22
Primary spherical aberration at paraxial focus
Transverse ray aberration yD xD yD
Pupil: y-section x-section x-section
0.01 mm 0.01 mm 0.01 mm
Geometrical spot through
focus
0.0
2 m
m
Modulation Transfer Function MTF
MTF at paraxial focus MTF through focus for 100 cycles per mm
1
0
0.5
0 100 200cyc/mm z/mm
1
0
0.5
-0.02 0.020.0
z/mm-0.02 0.020.0-0.01 0.01
Ref: H. Zügge
8
-
9
Surface Contributions: Example
Seidel aberrations:
representation as sum of
surface contributions possible
Gives information on correction
of a system
Example: photographic lens
1
23 4
5
6 8 9
10
7
Retrofocus F/2.8
Field: w=37°
SI
Spherical Aberration
SII
Coma
-200
0
200
-1000
0
1000
-2000
0
2000
-1000
0
1000
-100
0
150
-400
0
600
-6000
0
6000
SIII
Astigmatism
SIVPetzval field curvature
SVDistortion
CIAxial color
CIILateral color
Surface 1 2 3 4 5 6 7 8 9 10 Sum
9
-
10
Wave Aberration in Optical Systems
Definition of optical path length in an optical system:
Reference sphere around the ideal object point through the center of the pupil
Chief ray serves as reference
Difference of OPL : optical path difference OPD
Practical calculation: discrete sampling of the pupil area,
real wave surface represented as matrix
Exit plane
ExP
Image plane
Ip
Entrance pupil
EnP
Object plane
Op
chief
ray
w'
reference
sphere
wave
front
W
y yp y'p y'
z
chief
ray
wave
aberration
optical
systemupper
coma ray
lower coma
ray
image
point
object
point
10
-
11
Pupil Sampling
y'p
x'p
yp
xp x'
y'
z
yo
xo
object plane
point
entrance pupil
equidistant grid
exit pupil
transferred grid
image plane
spot diagramoptical
system
All rays start in one point in the object plane
The entrance pupil is sampled equidistant
In the exit pupil, the transferred grid
may be distorted
In the image plane a spreaded spot
diagram is generated
11
-
12
Zernike Polynomials
+ 6
+ 7
- 8
m = + 8
0 5 8764321n =
cos
sin
+ 5
+ 4
+ 3
+ 2
+ 1
0
- 1
- 2
- 3
- 4
- 5
- 6
- 7
Expansion of wave aberration surface
into elementary functions / shapes
Zernike functions are defined in circular
coordinates r,
Ordering of the Zernike polynomials by
indices:
n : radial
m : azimuthal, sin/cos
Mathematically orthonormal function
on unit circle for a constant weighting
function
Direct relation to primary aberration types
n
n
nm
m
nnm rZcrW ),(),(
01
0)(cos
0)(sin
)(),(
mfor
mform
mform
rRrZ mnm
n
12
-
13
Airy distribution:
Gray scale picture
Zeros non-equidistant
Logarithmic scale
Encircled energy
Perfect Lateral Point Spread Function: Airy
DAiry
r / rAiry
Ecirc
(r)
0
1
2 3 4 5
1.831 2.655 3.477
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2. ring 2.79%
3. ring 1.48%
1. ring 7.26%
peak 83.8%
r
z
I(r,z)
13
-
14
0
2
12,0 Iv
vJvI
0
2
4/
4/sin0, I
u
uuI
-25 -20 -15 -10 -5 0 5 10 15 20 250,0
0,2
0,4
0,6
0,8
1,0
vertical
lateral
inte
nsity
u / v
Circular homogeneous illuminated aperture: intensity distribution
transversal: Airy
scale:
axial: sinc
scale
Resolution transversal better
than axial: Dx < Dz
Ref: M. Kempe
Scaled coordinates according to Wolf :
axial : u = 2 p z n / NA2
transversal : v = 2 p x / NA
Perfect Point Spread Function
NADAiry
22.1
2NA
nRE
r
z
lateral
aperture
cone
axial
image plane
optical
axis
14
-
15
Spherical aberration Astigmatism Coma
c = 0.2
c = 0.3
c = 0.7
c = 0.5
c = 1.0
Psf with Aberrations
Zernike coefficients c in
Spherical aberration,
Circular symmetry
Astigmatism,
Split of two azimuths
Coma,
Asymmetric
15
-
16
),(ˆ),( yxIFvvH PSFyxOTF
*
2
( ) ( )2 2
( )
( )
x xp p p
OTF x
p p
f v f vP x P x dx
H v
P x dx
Optical Transfer Function: Definition
Normalized optical transfer function (OTF) in frequency space:
Fourier transform of the Psf- intensity
Absolute value of OTF: modulation transfer function MTF
Gives the contrast at a special spatial frequency of a
sine grating
OTF: Autocorrelation of shifted pupil function, Duffieux-integral
Interpretation: interference of 0th and 1st diffraction of
the light in the pupil
x
y
x
y
L
L
x
y
o
o
x'
y'
p
p
light
source
condenser
conjugate to object pupil
object
objective
pupil
direct
light
at object diffracted
light in 1st order
16
-
17
Optical Transfer Function of a Perfect System
Aberration free circular pupil:
Reference frequency
Maximum cut-off frequency:
Analytical representation
Separation of the complex OTF function into:
- absolute value: modulation transfer MTF
- phase value: phase transfer function PTF
'sinu
f
avo
'sin222 0max
un
f
navv
2
000 21
22arccos
2)(
v
v
v
v
v
vvHMTF
p
),(),(),( yxPTF
vvHi
yxMTFyxOTF evvHvvH
/ max0
0
1
0.5 1
0.5
gMTF
17
-
18
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
18
-
19
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
19
-
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
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 jist
j
soll
j m
2
1,
20
-
21
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
21
-
22
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
22
-
23
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. Zügge23