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Design and Correction of Optical
Systems
Lecture 8: Further performance criteria
2019-06-03
Herbert Gross
Summer term 2019
2
Preliminary Schedule - DCS 2019
1 08.04. BasicsLaw of refraction, Fresnel formulas, optical system model, raytrace, calculation
approaches
2 15.04. Materials and ComponentsDispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,
aspheres, diffractive elements
3 29.04. Paraxial OpticsParaxial approximation, basic notations, imaging equation, multi-component
systems, matrix calculation, Lagrange invariant, phase space visualization
4 06.05. Optical SystemsPupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,
photometry
5 13.05. Geometrical AberrationsLongitudinal and transverse aberrations, spot diagram, polynomial expansion,
primary aberrations, chromatical aberrations, Seidels surface contributions
6 20.05. Wave AberrationsFermat principle and Eikonal, wave aberrations, expansion and higher orders,
Zernike polynomials, measurement of system quality
7 27.05. PSF and Transfer functionDiffraction, point spread function, PSF with aberrations, optical transfer function,
Fourier imaging model
8 03.06. Further Performance CriteriaRayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based
criteria, further options
9 17.06. Optimization and CorrectionPrinciples of optimization, initial setups, constraints, sensitivity, optimization of
optical systems, global approaches
10 24.06. Correction Principles ISymmetry, lens bending, lens splitting, special options for spherical aberration,
astigmatism, coma and distortion, aspheres
11 01.07. Correction Principles IIField flattening and Petzval theorem, chromatical correction, achromate,
apochromate, sensitivity analysis, diffractive elements
12 08.07. Optical System ClassificationOverview, photographic lenses, microscopic objectives, lithographic systems,
eyepieces, scan systems, telescopes, endoscopes
1. PSF: line of sight and apodization
2. Edges and lines
3. Pupil aberrations
4. Sine condition
5. Induced aberrations
6. Vectorial aberrations
7. Fourier imaging formation
8. Caustics
Contents
3
4
PVW
Rayleigh Criterion
The Rayleigh criterion
gives individual maximum aberrations
coefficients,
depends on the form of the wave
Examples:aberration type coefficient
defocus Seidel 25.020 a
defocus Zernike 125.020 c
spherical aberration
Seidel 25.040 a
spherical aberration
Zernike 167.040 c
astigmatism Seidel 25.022 a
astigmatism Zernike 125.022 c
coma Seidel 125.031 a
coma Zernike 125.031 c
4
a) optimal constructive interference
b) reduced constructive interference
due to phase aberrations
c) reduced effect of phase error
by apodization and lower
energetic weighting
d) start of destructive interference
for 90° or /4 phase aberration
begin of negative z-component
Rayleigh criterion:
1. maximum of wave aberration: Wpv < /4
2. beginning of destructive interference of partial waves
3. limit for being diffraction limited (definition)
4. as a PV-criterion rather conservative: maximum value only in 1 point of the pupil
5. different limiting values for aberration shapes and definitions (Seidel, Zernike,...)
Marechal criterion:
1. Rayleigh crierion corresponds to Wrms < /14 in case of defocus
2. generalization of Wrms < /14 for all shapes of wave fronts
3. corresponds to Strehl ratio Ds > 0.80 (in case of defocus)
4. more useful as PV-criterion of Rayleigh
Criteria of Rayleigh and Marechal
14856.13192
Rayleigh
rmsW
5
PV and Wrms-Values
PV and Wrms values for
different definitions and
shapes of the aberrated
wavefront
Due to mixing of lower
orders in the definition
of the Zernikes, the Wrms
usually is smaller in
comparison to the
corresponding Seidel
definition
6
In the case of defocus, the Rayleigh and the Marechal criterion delivers
a Strehl ratio of
The criterion DS > 80 % therefore also corresponds to a diffraction limit
This value is generalized for all aberration types
8.08106.08
2
SD
Strehl Ratio Criterion
aberration type coefficient Marechal
approximated Strehl
exact Strehl
defocus Seidel 25.020 a 7944.0 8106.08
2
defocus Zernike 125.020 c 0.7944 0.8106
spherical aberration
Seidel 25.040 a 0.7807 0.8003
spherical aberration
Zernike 167.040 c 0.7807 0.8003
astigmatism Seidel 25.022 a 0.8458 0.8572
astigmatism Zernike 125.022 c 0.8972 0.9021
coma Seidel 125.031 a 0.9229 0.9260
coma Zernike 125.031 c 0.9229 0.9260
7
8
Performance Criteria Overview
Applications
Geometrical
model
Diffraction
model
Longitudinal
aberrations
Transverse
aberration curves
Spot diagrams
Wave aberrations
AdvantagesQuantitative
numbers
scaling on Rayleigh unit
scaling on Airy diameter
rms
pv
scaling on Airy diameters
rms,pv
Rayleigh/Marechal
Zernike decomposition
RepresentationsLimitations
Problems
astigmatism
axial chromatical
field curvature
information on sensor
positioning
not useful in the field
not defined for afocal
no information on skew rays
camera lenses
simple direct analysis
possible
allows Seidel surface
decomposition
1 curve per field point
to be re-defined for afocal
no information on skew rays
any
illustrative
describes resolution
calculation fast
analysis complicated
any
direct measurable
scaling on wavelength
all orders in Zernikes
only one field and wavelength
normalization radius of Zernikes
Zernikes only circular pupil
Point spread
function
Modulation transfer
function
Strehl ratio
scaling on Airy diameter
Hopkins number
microscopy
astronomy
diffraction limited
direct relation to resolution
easy white light formulation
computational problems for large
aberrations
hard to correct directly
camera lenses
lithography
projection lenses
direct analysis possible
easy white light formulation
computational problems for large
aberrations
analysis complicated
hard to correct directly
Centroid Ray
Deviation of centroid ray from chief ray
First possibility:
- asymmetrical apodization
- coincidence in image plane
Second possibility:
- coma phase aberration
- coincidence in pupil
image
planepupil
chief ray
apodization
I(yp)
yp
coma
centroid ray for
coma
centroid ray for
apodization
9
Chief ray:
centroid of geometrical pupil area
Centroid ray:
centroid of energy
Due to wave equation the centroid propagates along a straight line:
Line of sight
Wave aberrations of odd order in the azimuthal term influence the centroid
- tilt and coma-like aberrations of any order
- centroid has an offset against the peak of intensity
- simple calculation possible:
dydxzyxIy
PdydxzyxI
dydxzyxIyzy Cen
c ),,(1
),,(
),,()()(
,...5,3,1
1
)( )1(22
)(n
n
ExP
CR
c cnA
zzy
Line of Sight
dydxy
Adydx
dydxyzy CR
c
1)()(
10
Pupil with apodization
e.g. non-homogeneous asymmetrical
illumination
Line of Sight for Apodization
exit pupil with
apodization
image
plane
centroid line
chief
ray
intensity
reference
sphere
yp
xp
peak
centroid
pupil
y
z
centroid
image
11
Psf with Coma
Defocus: centroid moves on a straight
line (line of sight)
Peak of intensity moves on a curve
(bananicity)
c8 = 0.3
c8 = 0.5
c15 = 0.5
acoma = 1.7
x I(z)
zz
-8 -6 -4 -2 0 2 4 6 8-0.1
-0.05
0
0.05
0.1
0.15
y
z
solid lines : peak
dashed lines : centroid
c8 = 0.3
c8 = 0.5
c15
= 0.5
acoma
= 1.7
12
Point Spread Function with Apodization
w
I(w)
1
0.8
0.6
0.4
0.2
00 1 2 3-2 -1
Airy
Bessel
Gauss
FWHM
w
E(w)
1
0.8
0.6
0.4
0.2
03 41 2
Airy
Bessel
Gauss
E95%
Apodisation of the pupil:
1. Homogeneous
2. Gaussian
3. Bessel
Psf in focus:
different convergence to zero forlarger radii
Encircled energy:
same behavior
Complicated:Definition of compactness of thecentral peak:
1. FWHM: Airy more compact as GaussBessel more compact as Airy
2. Energy 95%: Gauss more compact as AiryBessel extremly worse
13
14
Focus Spot Size of a Lens
Changing the NA/aperture D of a focussing lens:
- small values of D:
diffraction dominates, Airy formula
- large D:
geometrical aberrations dominate
Total aberrations:
superposition of both effects
Dspot
NA / D
Log Dspot
NA / D
diffraction
Airy
diffraction
Airy
geometrical
aberration
geometrical
aberration
total
total
Focal Diameter Determination
15
1 2 3 4 5 6
Pupil shape
definiton of focus
Gauss radius w
no trunc..
Super gauss w,m=6
no trunc.
Gauss 2a=3w
with trunc.
Gauss a=w with
trunc.
Circle homoge-
neous D=2a
Linear / homog. slit width
2a
1. Zero point of intensity
#
#
#
1.426
1.220 ( Airy )
1.00
I=0.5 ( FWHM )
0.375
0.513
0.644
0.564
0.519
0.443
I=0.1353 (1/e2 )
0.637
=2/
0.831
1.059
0.914
0.822
0.697
I=0.01
0.966
1.122
1.491
1.238
1.092
( Peak )
0.908
I=0.001
1.183
2.109
1.695
2.925
1.174
( Peak )
0.969
E=0.86466
0.637
=2/
0.818
1.008
0.890
1.378
0.658
E=0.95
0.779
1.040
1.201
1.104
3.915
1.989
Focal diameter:
Basic scaling factor
Additional factor b
Factor differs in size by
one magnitude
NAD foc
b
wbeamradius
truncationno
w
f
aradiusat
truncationwith
a
f
NA
Small aperture:
Diffraction limited
Spot size corresponds
to Airy diameter
Spot size depends on
wavelength
Large aperture:
Diffraction neglectible
Aberration limited
Geometrical effects not
wavelength dependent
But: small influence of
dispersion
Log Dfoc
Log sinu
f=1000 , 500 , 200 , 100 , 50 , 20 , 10 mm
= 10 m
= 1 m
550 nm
1
0
-1
-2-2-3 -1 0
Focussing by a Lens: Diffraction and Aberration
16
Comparison Geometrical Spot – Wave-Optical Psf
aberrations
spot
diameter
DAiry
exact
wave-optic
geometric-optic
approximated
diffraction limited,
failure of the
geometrical model
Fourier transform
ill conditioned
Large aberrations:
Waveoptical calculation shows bad conditioning
Wave aberrations small: diffraction limited,
geometrical spot too small and
wrong
Approximation for the
intermediate range:
22
GeoAirySpot DDD
17
Normalized axial intensity
for uniform pupil amplitude
Decrease of intensity onto 80%:
Scaling measure: Rayleigh length
- geometrical optical definition
depth of focus: 1RE
- Gaussian beams: similar formula
22
'
'sin' NA
n
unRu
Depth of Focus: Diffraction Consideration
2
0
sin)(
u
uIuI
2' o
un
R
udiff Run
z
2
1
sin493.0
2
12
focal
plane
beam
caustic
z
depth of focus
0.8
1
I(z)
z-Ru/2 0
r
intensity
at r = 0
+Ru/2
18
Diffraction at an edge in Fresnel
approximation
Intensity distribution,
Fresnel integrals C(x) and S(x)
scaled argument
Intensity:
- at the geometrical shadow edge: 0.25
- shadow region: smooth profile
- bright region: oscillations
22
)(2
1)(
2
1
2
1)( tStCtI
FNxz
xz
kt 2
2
Fresnel Edge Diffraction
t-4 -2 0 2 4 6
0
0.5
1
1.5
I(t)
19
ESF with defocussing ESF with spherical aberration
x
IESF
(x)
-10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
W20
= 0.0
W20
= 0.1
W20
= 0.2
W20
= 0.3
W20
= 0.4
W20
= 0.5
W20
= 0.7
x
IESF
(x)
-8 -6 -4 -2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
W40
= 0.0
W40
= 0.1
W40
= 0.2
W40
= 0.3
W40
= 0.4
W40
= 0.5
W40
= 0.7
Incoherent Edge Spread Function
20
Line image: integral over point sptread function
LSF: line spread function
Realization: narrow slit
convolution of slit width
But with deconvolution, the PSF can be reconstructed
dyyxIxI PSFLSF ),()(
Integration
intens
ity
x
Line spread function
PSF
dyyxIxI PSFLSF ),()(
Line Image
21
Line image:
Fourier transform of pupil in one dimension
Line spreadfunction with aberrations
Here: defocussing
pppp
pp
xxR
i
pp
iLSF
dydxyxP
dydxeyxP
xI
pi
2
22
,
,
)(
x
ILSF
(x)
-10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
W20
= 0.0
W20
= 0.1
W20
= 0.2
W20
= 0.3
W20
= 0.4
W20
= 0.5
W20
= 0.7
Line Spread Function
22
ESF, PSF and ESF-Gradient
Typical behavior of intensity of an edge image for residual aberrations
The width of the distribution roughly corresponds to the diameter of the PSF
Derivative of the edge spread function:
edge position at peak
location
y'0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.1 -0.05 0 0.05 0.1 0.15
derivation of
edge spread
function
edge spread
function
point spread
function
23
Spherical aberration of the chief ray / pupil imaging
Exit pupil location depends on the field height
Pupil Aberrations
yobject
sP
chief rays
pupil position
pupil
location
24
Pupil Aberration
Interlinked imaging of field and pupil
Distortion of object imaging corresponds to spherical aberration of the pupil
imaging
Corrected spherical pupil aberration: tangent condition
O O’
stop and
entrance pupil
optical system
exit pupil
objectimage
Object imaging Pupil imaging
Blue rays
Red rays
Marginal rays
Marginal raysChief rays
Chief rays
.tan
'tanconst
w
w
25
Eyepiece with pupil aberration
Illumination for decentered pupil :
dark zones due to vignetting
Pupil Aberration
eyepiecelens and
pupil of
the eye
retina
caustic of the pupil
image enlarged
instrument
pupil
26
Sine Condition
Lagrange invariante for paraxial angles U, U‘
sin-condition:
extension for finite aperture angle u
Corresponds to energy conservation in the system
Constant magnification for alle aperture zones
Pupil shape for finite aperture is a sphere
Definition of violation of the sine condition:
OSC (offense against sine condition)
OSC = 0 means correction of sagittal coma (aplanatic system)
'sin'
sin
'' un
un
Un
nUm
'sin''sin UynUny
'sin''sin uynuny
y y'
equal magnification
in every zone of the
aperture cone
n'
z
UU'
y'
y
n
27
Optical path difference for two object points between object and image space
28
Sine Condition
P
P'
Po P'o
dy
dy sindy'
'
dy' sin'
'
optical
system
If for example a small field area and a widespread ray bundle is considered, a perfect
imaging is possible
The eikonal with the expression
can be written for dL=0 as
In the special case of an angle 90°we get with cos()=sin(u) the Abbe sine condition
with the lateral magnification
29
Abbe Sine Condition
P
Qs
u dr'
P'
Q's'
u'dr
rdsnrdsnL
'''d
'cos'cos
'cos''cos
'''
b
nn
drndrn
rdsnrdsn
rd
rdm
'
'sin'
sin
un
unm
Tangential and Sagittal Coma
2 terms of tangential transverse aberration:
- Sagittal coma depends on xp, describes the asymmetry
- Tangential coma depends on yp, corresponds to spherical aberration under skew conditions
larger by a factor of 3
Only asymmetry removed with sine condition: sagittal coma vanishes
exit
pupiltangential
coma rays
sagittal
coma rays
image
plane
y'
x'
yp
xp
chief ray
coma
spot0°
90°
45°
wavefront
with coma
ys'
2 2' 3 ' 'p p s ty R x y y y
yt'
30
Decomposition of coma:
1. part symmetrical around
chief ray: skew spherical
aberration
2. asymmetrical part:
tangential coma
Skew spherical aberration:
- higher order aberration
- caustic symmetric around
chief ray
Skew Spherical aberration
upper
coma ray
chief
ray
lower
coma ray exit
pupil
y'p
ideal image
location
S
sagittal image
point
tangential
image point
T
upper
coma ray
chief
ray
lower
coma ray
exit
pupil
y'p
ideal image
plane
common
intersection
point
2
lowcomupcom
tangcoma
yyy
2
lowcomupcom
skewsph
yyy
31
Transfer of Energy in Optical Systems
Conservation of energy
Invariant local differential flux
Assumption: no absorption
Delivers the sine condition
'22 PdPd
ddudAuuLPd cossin2
T 1
y
dA dA's's
EnP ExP
n n'
F'F
y'
u u'
'sin''sin uynuyn
32
Sine condition not fulfilled:
- nonlinear scaling from entrance to exit pupil
- spatial filtering on warped grid, nonlinear sampling of spatial frequencies
- pupil size changes
- apodization due to distortion
- wave aberration could be calculated wrong
- quantitative mesaure of offence against the sine condition (OSC):
distortion of exit pupil grid
Sine Condition
xo xp
sphere distorted exit
pupil surface
object
plane
exit
pupil
optical
system
sx
u
xp
x'o
u'
x'p
x'p
image
plane
entrance
pupil
grid
distortion
1sin
unf
xD
ap
p
33
Photometric effect of pupil distortion:
illumination changes at pupil boundary
Effect induces apodization
Sign of distortion determines the effect:
outer zone of pupil brighter / darker
Additional effect: absolute diameter of
pupil changes
OSC and Apodization
focused +20 m +50 m-20 m-50 m
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-0.05 - barrel
+0.05 - pincushion
no
distortion
rp
intensity
34
General Aplanatic Surface
General approach
of Fermat principle:
aplanatic surface
Cartesian oval,
4th order
Special case OPD = 0:
Solution is spherical aplanatic surface
35
P
S
P'
oval
surface
r
ss'
z
nsnszsrnszrn '')'(')( 2222
2
2
2
2222222
22222222
2222
''
01'/'/'/21'/
'/2'/2'/
0)'/(')(
nn
snr
nn
snz
nnrnnnnzsnnz
nzsnsnnzrzsszrnn
zsnnrnszrn
nsns ''
Isoplanatism Condition of Staeble-Lihotzky
Sagittal coma aberration:
from the geometry of the figure and Lagrange invariant
Condition of Staeble-Lihotzky
Problems:
- no quantitative measure
- only tangential rays are considered
- integral criterion
m
un
un
m
sSss
p
p'sin'
sin''''
m
ssS
sS
un
un
m
yy
sphp
p
s'''
''
'sin'
sin''
exit pupil
real
tangential
image plane
ideal
gaussian
image plane
Q' P'
Q'chief ray
optical axis
marginal ray
s
Q't
projection of
sagittal coma ray y'
s'
S'
s'
last
surface
sp'
u'
ys'
ys'
P't
36
Piecewise Isoplanatism
Invariance of PSF: to be defined
Possible options:
1. relative change of Strehl
2. correlation of PSF's
Examples for microscopic lenses
with and without flattening
correction
In medium field size:
small isoplanatic patches
On axis:
large isoplanatic area
Criteria not useful at the
edge for low performance
Plane MO 100x1.25
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Strehl correlation
Strehl correlation
no plane MO
40x0.85normalized
field position
SystemMO plane 100x1.25
isoplanatic patch size in m
MO not plane 40x0.85isoplanatic patch size in
m
Strehl 1%Psf
correlation 0.5%
Strehl 1%Psf
correlation 0.5%
on axis 70 72 81 100
half field 3.8 3.8 27 3.1
field zone 2.5 2.5 29 39
full field 45 3.8 117 62
37
exit pupil
realtangential
image plane
idealgaussian
image plane
Q' P'
Q'chief ray
optical axis
marginal ray
s
Q'P'
t t
y's
y't
sagittalcoma ray
yo
s'
S's'
Offence Against the Sine Condition
Conradys OSC (offense against sine condition):
- measurement of deviation of sagittal coma
- quantitative validation of the sine condition
Only sagittal coma considered
in case of OSC=0 the Staeble-Lihotzky-
condition is automatically fulfilled
OSC allows for the definition of surface
contribution
''
''
'sin'
sin1
'
''
p
p
t
stOSC
ss
sS
unm
un
y
yy
OSCtppcoma yrryW )0,,(
ExP
yp
z
CR
Q'1
Q's
Q'
P'1
P'
ideal
y's
y'y's
y't
k kkk
CR
kkkkOSC
unh
inQQ
u
w
'''
)'(
sin
sin )(
1
1
'sin'
sin3'
un
unmyyt
38
Overview on conditions for aberrations and aplanatism-isoplanatism
39
Overview Aplanatism-Isoplanatism
Nr Sine
cond.
Iso-planat cond.
Isoplanatism
condition
Spherical
aberration
Sagittal
coma
Tangential
coma
Imaging system
1 # # # # # general
2a # OSC=0, Conrady # 0 # isoplanatic-I
2b # Staeble-Lihotzky / Berek
# 0 0 isoplanatic-II
3a 0 0 axial aplanatic
3b 0 (skew) 0 0 off-axis aplanatic
Tangential coma
Isoplanatism
Staeble-
Lihotzky
Sagittal coma
Spherical aberration
Isoplanatism
Conrady
OSC
sine condition
off-axis
Aplanatism
0
0 00 0
0
Skew Spherical aberration
sine condition
axial
Aplanatism
0
0 0
Special idea of Seidel to consider the 3rd order as a perturbation of the paraxial ray
Independent changes/contributions of every surface aberration to the final transverse
aberration
Therefore special reference on paraxial fundamental properties
40
Seidel Approach
P P'0
initial path
paraxial ray
perturbation at
1st surface
y
y'0
y'(1)
1 2 3 4
y'(2)
y'(3)
y'(4)
y'perturbation at
2st surfaceperturbation at
3rd surface
perturbation at
4th surface
Aberration expansion: perturbation theory
Linear independent contributions only in lowest correction order:
Surface contributions of Seidel additive
Higher order aberrations (5th order,...): nonlinear superposition
- 3rd oder generates different ray heights and angles at next surfaces
- induces aberration of 5th order
- together with intrinsic surface contribution: complete error
Separation of intrinsic and induced aberrations: refraction at every surface in the system
Induced Aberrations
PP'0
initial path
paraxial ray
intrinsic
perturbation at
1st surface
y
1 2 3
y'
intrinsic
perturbation at
2nd surface
induced perturbation at 2nd
surface due to changed ray height
change of ray height due to the
aberration of the 1st surface
P'
41
42
Induced Aberrations
Example spherical aberration
Induced effects for higher orders
3rd order
spherical
7th order
spherical
5th order
spherical
S(3rd)
intr
intrinsic induced
S(5th)
intr
S(7th)
intr
S(5th)
indu
S(7th)
indu
9th order
spherical S(9th)
intr S(9th)
intr
+2
+4
+6
Example Gabor telescope
- a lens pre-corrects a spherical mirror to obtain vanishinh spherical aberration
- due to the strong ray deviation at the plate, the ray heights at the mirror changes
significantly
- as a result, the mirror has induced
chromatical aberration, also the
intrinsic part is zero by definition
Surface contributions and chromatic
difference (Aldi, all orders)
Induced Aberrations
1 2 3-1.5
-1
-0.5
0
0.5
1
1.5
= 400 nm
1 2 3-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
= 700 nmmirror
contribution
to color
surfaces surfaces
difference
heigth
difference
with
wavelength= 400 nm
= 700 nm
43
Wave aberration field
indices
Normalized field vector: H
normalized pupil vector: rp
angle between H and rp:
Expansion according to the invariants
for circular symmetric components
Vectorial Aberrations
x
yrp
s
p
s'
p'
xP
yp
x'
y'
x'P
y'p
object
plane
entrance
pupil
exit
pupil
image
plane
z
system
surfaces
P'
P
H
nmj
n
pp
m
p
j
klmp rrrHHHWrHW,,
,
mnlmjk 2,2
y
Hrp
field1
1
pupil
cos,, 22 ppppp rHrHrrrHHH
44
Wave aberration field
until the 6th order
Analogue:
transverse aberrations
with
Vectorial Aberrations
ord j m n Term Name
0 0 0 0 000W uniform Piston
2
1 0 0 HHW
200 quadratic piston
0 1 0 prHW111
magnification
0 0 1 pp rrW020 focus
4
0 0 2 2040 pp rrW
spherical aberration
0 1 1 ppp rHrrW131
coma
0 2 0 2222 prHW
astigmatism
1 0 1 pp rrHHW220
field curvature
1 1 0 prHHHW311
distortion
2 0 0 2400 HHW
quartic piston
6
1 0 2 2240 pp rrHHW
oblique spherical aberration
1 1 1 ppp rHrrHHW331
coma
1 2 0 2422 prHHHW
astigmatism
2 0 1 pp rrHHW
2
420 field curvature
2 1 0 prHHHW
2
511 distortion
3 0 0 3600 HHW
piston
0 0 3 3060 pp rrW
spherical aberration
0 1 2 ppp rHrrW
2
151
0 2 1 2242 ppp rHrrW
0 3 0 3333 prHW
Wn
RH
pr'
'
45
Wave aberration
with shift vector
In 3rd order:
1. spherical
2. coma
3. astigmatism
4. defocus
5. distortion
Systems with Non-Axisymmetric Geometry
q nmj
n
pp
m
pq
j
qqklmp rrrHHHWrHW,,
000,
jjoj HH
p
q
q q
qqqqq
q q
qqqq
q
q
p
q q
q q
qqqq
q q
p
q
q
q
q
ppp
q
q
q
pp
q
qp
rWHW
HWHHWHHW
r
WW
HWWHWW
rWHWHW
rrrWHW
rrWrHW
2
,3110,311
0
2
,31100,3110
2
0,311
2
2
,222,220
0,222,22
2
0,222,220
22
,2220,222
2
0,220
,1310,131
2
,040
2
2
2
1
2
12
2
1
2
1
2
1
,
46
47
Vectorial Aberration Contributions
Idea of nodal points:
image points of the tilted component axes
Every component has its individual axis, the aberrations are symmetric around this axis
(circular symmetric sub-system)
The axis are bended towards the image plane
Every circular symmetric component therefore has an individual aberration center j in
the image plane
The interaction and overlay of the various centers are complicated y
x
lens 2
lens 3
aberration
contribution
lens 3
aberration
contribution
lens 2
3
2
lens 1
bended axis
rays
aberration
contribution
lens 1
1
Superposition of aberrations in the image plane
Vector sum of A-contributions
- center locations influences by surface location/orientation
- weighting/strength Wklm defined by surface shape
Sum of contributions:
influences by both quantities
Vectorial Aberration Contributions
48
j
x
y
1Wklm,1
A1
2
Wklm,2
A2
3
Wklm,3
A3
(0,0)
total
Wklm
H arbitrary field
point (x,y)
HA2
HA3
HA1
Expanded and rearranged 3rd order expressions:
- aberrations fields
- nodal lines/points for vanishing aberration
Example coma:
abbreviation: nodal point location
one nodal point with
vanishing coma
Nodal Theory
ppp
q
q
q
o
q
qcoma rrrW
W
HWW
,131
,131
,131
)(
131
,131
,131
,131
131 c
q
j
q
q
W
W
W
W
a
pppo
c
coma rrraHWW 131
)(
131
zero
coma
green zero
coma
blue
zero
coma
total
49
Generalized Nodal Locations
50
Wave aberration terms of can have nodal lines
in case of more complicates surfaces
Example 1:
linear coma node line for coma together
with skew spherical aberration
Example 2:
higher order coma nodal line
Ref: T. Yang / X. Zhang
HMD Projection Lens
eye
pupil
image
total
internal
reflection
free formed
surface
free formed
surface
field angle 14°
y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
binodal
points
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
astigmatism, 0 ... 1.25 coma, 0 ... 0.34 Wrms
, 0.17 ... 0.58
Refractive 3D-system
Free-formed prism
One coma nodal point
Two astigmatism nodal
points
51
Fourier Optics – Point Spread Function
Optical system with magnification m
Pupil function P,
Pupil coordinates xp,yp
PSF is Fourier transform
of the pupil function
(scaled coordinates)
pp
myyymxxxz
ik
pppsf dydxeyxPNyxyxgpp ''
,)',',,(
pppsf yxPFNyxg ,ˆ),(
object
planeimage
plane
source
point
point
image
distribution
52
Fourier Theory of Incoherent Image Formation
objectintensity image
intensity
single
psf
object
planeimage
plane
Transfer of an extended
object distribution I(x,y)
In the case of shift invariance
(isoplanasy):
incoherent convolution
Intensities are additive
dydxyxIyyxxgyxI psfinc
),()','()','(2
),(*),()','( yxIyxIyxI objpsfimage
dydxyxIyyxxgyxI psfinc
),(),',,'()','(2
53
Fourier Theory of Incoherent Image Formation
object
intensity
I(x,y)
squared PSF,
intensity-
response
Ipsf
(xp,y
p)
image
intensity
I'(x',y')
convolution
result
object
intensity
spectrum
I(vx,v
y)
optical
transfer
function
HOTF
(vx,v
y)
image
intensity
spectrum
I'(vx',v
y')
produkt
result
Fourier
transform
Fourier
transform
Fourier
transform
54
Testchart Visual Acuity
Snellen test chart
55
Visual Acuity
Recognition of simple geometrical
shapes :
1. Landolt ring with gap
2. Letter 'E'
Blur of image on retina with distance
a)
5a
a
a
3a
3a
a
3a
b)
distance 6.096 m
block
letter
E 8.9
mm
image
height
25 meye
original blur : a/2 blur : a blur : 2a blur : 3a blur : 4a
3a
56
USAF Test Target
0 1
10
2
3
4
5
6
6
5
4
3
2
1
6
5
4
3
2
2 31
2
3
2
4
5
6
57
Siemens Star Test Plate
58
Real Image with Different Chromatical Aberrations
original object good image color astigmatism 2
6% lateral color axial color 4
59
Early investigations on caustics:
Leonardo da Vinci 1508
Caustics at mirrors and lenses
Caustics
envelope
caustic curve
envelope
caustic curve
envelope
caustic curve
with cusp
Ref: J. Nye, Natural focusing
60
More general:
caustic occurs at every wavefront with concave
shape as locus of local curvature
Physically:
- crossing of rays indicates a caustic
- interference with diffraction ripple and ringing
is seen
Caustics
unique wave
front
rays
no unique ray
direction
amplitude variation due to
interference
Ref: J. Nye,
Natural focusingRef: W. Singer
61
Caustic: envelope of rays
Locus of local curvature
Calculation:
caustic:
ray direction:
rays:
L distance PC
variation of point on wavefront:
solution condition for linear system:
equation of caustic
Caustics
wave
front
rays
caustic
curve
P1
P2
C12
zyx ssss
cccc zyxr
sLrrc
zc
yc
xc
sLz
sLyy
sLxx
0
0
0
Lsyy
sLx
x
sL
Lsyy
sLx
x
sLy
Lsyy
sLx
x
sLx
zzz
y
yy
xxx
ddd
dddd
dddd
0
1
1
1
yx
y
yy
xxx
ss
sy
sL
x
sL
sy
sL
x
sL
62
Special case of one dimension x-z
Example: spherical aberration for focussing through
plane interface
Ray direction
Variation
Geometry and law
of refraction
Approximation of
small x:
caustic curve
Caustics
x
Wsx
0
01
Lsxx
sL
Lsxx
sL
zz
xx
dd
dd
22 xq
xn
a
xn
x
Wsx
refracting
surface
caustic
x
x
q
a
n
z
sx
2
22
2
)1(1q
xn
n
a
x
s
sL
x
z
3/23/12 )1(2
3cc xqn
nn
qz
63
