imaging and aberration theory - iap.uni-jena.deand+aberration+theory... · l o n g i t u d i n al...
TRANSCRIPT
www.iap.uni-jena.de
Imaging and Aberration Theory
Lecture 5: Aberrations representations
2018-11-16
Herbert Gross
Winter term 2018
2
Schedule - Imaging and aberration theory 2018
1 19.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 26.10.Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 02.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 09.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas
5 16.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 23.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders
7 30.11. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 07.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 14.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 21.12.Sine condition, aplanatism and isoplanatism
Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics
11 11.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations
12 18.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement
13 25.01. Point spread function ideal psf, psf with aberrations, Strehl ratio
14 01.02. Transfer function transfer function, resolution and contrast
15 08.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability
1. Stop shift formulas
2. Lens aberration contributions
3. Pupil aberrations
4. Representation of geometrical aberrations
5. Representation of wave aberrations
6. Miscellaneous
7. Aberration measurements
3
Contents
Refracting Surface VII: Eikonal
Eikonal of 4th order
Coefficients
1. Spherical aberration
2. Astigmatism
3. Field curvature
4. Distortion
5. Coma
pppppp
ppppA
yyxxyxp
pssCyyxxyx
p
pssDyyxxyx
p
pssP
yyxxp
pssAyx
p
sSyx
p
psKL
22
4
322
4
322
4
22
2
4
22222
4
4222
4
4)4(
2
)(
2
)(
4
)(
2
)(
88
)(
''
11)'(
''
11)(
''
11)'(
''
11)(
''
11)'(
''
11)'(
''
11)'(
'
1
'
1''
11)'(
3
2
3
3
2
3
2
3
2
2
2
23
snnsQQ
R
bnnC
snnsQQQ
snnsQ
R
bnnD
snnsQQQ
snnsQQ
R
bnnP
snnsQ
R
bnnA
snnsQ
R
bnnS
sRssn
sRsns
R
bnnK
p
pp
ppp
pp
pp
p
pp
4
Stop Shift Formulas
If the stop is moved, the chief ray takes a modified way through the system
Approach: expansion of the surface coefficient formulas for small changes in the
pupil position p, p‘
The stop shift formulas shows the change of the Seidel coefficients due to this
effect
Also possible:
set of formulas for object or image shift
applicable for curved objects
imageplane
ray
y'chief ray 1
chief ray 2
ExP 1ExP 2
p'1
rpmax
z
p'2
5
Stop Shift Formulas
Stop shift formulas excplicite with the help of the
moving parameter
sph
coma
ast
curv
dist
Mix of aberration types due to stop shift: induced aberrations
Examples:
1. spherical aberration induces coma
2. coma induces astigmatism
,
h
hhE oldnew
II SS
IIIII SESS
IIIIIIIII SESESS 2
IVIV SS
IIIIVIIIVV SESESSESS 3233
old
position
new
position
old
chief raynew
chief ray
oldhnewh
6
Lens Contributions of Seidel
In 3rd order (Seidel) :
Additive contributions of thin lenses (equal w) to the total aberration value
(stop at lens position)
Spherical aberration
X: lens bending
M: position parameter
Coma
Astigmatism
Field curvature
Distortion
2
22
23
3 2
)1(
2
)1(2
1
2
1)1(32
1M
n
nnM
n
nX
n
n
n
n
fnnSlens
MnX
n
n
fnsClens )12(
1
1
'4
12
2'2
1
sfAlens
2'4
1
snf
nPlens
0lensD
7
Lens Contributions of Seidel
Spherical aberration
Special impact on correction:
1. Special quadratic dependence on
bending X
Minimum at
2. No correction for small n and M
3. Correction for large
n: infrared materials
M: virtual imaging
Limiting value
2
22
23
3 2
)1(
2
)1(2
1
2
1)1(32
1M
n
nnM
n
nX
n
n
n
n
fnnSlens
sphW
X
M = 6M = - 6
M = - 3M = 0
M = 3
n = 1.5
X
n
nMsph min
2 1
2
2
M
n n
ns
0
2
2
2
1
8
Photographic lens
Incidence angles for chief and
marginal ray
Field dominant system
Quasi symmetry can be seen at the
surface contributions of field aberrations
Symmetry disturbed for spherical
aberration
incidence angle
chief
ray
Photographic lens
60
40
20
0
20
40
60
marginal
ray
1
1 2 3 4 5 6 7 8 9 10 11 12 13
2 3
4
7
6 8
1112 13
sum
spherical
coma
astigmatism
curvature
distortion
axial
chromatic
lateral
chromatic
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.2
0
0.2
-0.4-0.2
0
0.20.4
-0.02
0
0.02
1 2 3 4 5 6 7 8 9 10 11 12 13
-0.01
0
0.01
9
Microscopic Objective Lens
Incidence angles for chief and
marginal ray
Aperture dominant system
marginal ray
chief ray
incidence angle
0 5 10 15 20 2560
40
20
0
20
40
60
microscope objective lens
10
Microscopic Lens
Large distance system
Problems with large diameters
Aplanatic front group
Not corrected for curvature and
distortion
Astigmatic contributions of
cemented surfaces corrected by
rear group
Sign of lateral chromatic
aberration in front group
spherical
coma
astigmatism
curvature
distortion
axial
chromatic
lateral
chromatic
10 151 sum
3174
810
1316
20 23
29
-1
0
1
-0.02
0
0.02
-2
0
2
-2
0
2
-4-2024
-0.05
0
0.05
-1
0
1
5 20 25
11
Lithographic Lens
Large effect of mirror on field curvature
Typical bulge structure shows the correction
of field curvature according to the Petzval
theorem
-0.2
0
0.2
10 20 30 40 6050
1. bulge 2. bulge 3. bulge1. waist 2. waist
-1
0
1
concave
mirror1. intermediate
image2. intermediate
image
12
Primary Aberration Spot Shapes
Simplified set of Seidel formulas:
field point only in y‘ considered
Spherical aberration S:
circle
Coma:
shifted circle
Astigmatism:
focal line
Field curvature:
circle
Distortion:
shifted point
PpPppp
PpPppp
ryPryCrSx
yDryPAryCrSy
sin'''2sin'''sin'''
''cos'')''2()2cos2('''cos'''
223
3223
6222
33
''''
sin''',cos'''
p
pppp
rSyx
rSxrSy
422222
22
''''''2''
2sin'''' ,)2cos2(''''
pp
PpPp
ryCryCyx
ryCxryCy
0',cos'''2' 2 xryAy Pp
24222
22
'''''
sin'''',cos''''
p
PpPp
ryPyx
ryPxryPy
0',''' 3 xyDy
13
Primary Aberration Spot Shapes
Schematically:
14
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
15
Longitudinal aberrations s
Transverse aberrations y
Representation of Geometrical Aberrations
Gaussian image
plane
ray
longitudinal
aberration
s'
optical axis
system
U'reference
point
reference
plane
reference ray
(real or ideal chief ray)
transverse
aberrationy'
optical axis
system
ray
U'
Gaussian
image
plane
reference ray
longitudinal aberration
projected on the axis
l'
optical axis
system
ray
l'o
logitudinal aberration
along the reference ray
16
Representation of Geometrical Aberrations
ideal reference ray angular aberrationU'
optical axis
system
real ray
x
z
s' < 0
W > 0
reference sphere
paraxial ray
real ray
wavefront
R
C
y'
Gaussian
reference
plane
U'
Angle aberrations u
Wave aberrations W
p
pp
pp y
yxW
y
R
u
yy
y
Rs
),(
'sin
'''
2
17
Angle Aberrations
Angle aberrations for a ray bundle:
deviation of every ray from common direction of the collimated ray bundle
Representation as a conventional spot diagram
Quantitative spreading of the collimated
bundle in mrad / °
perfect
collimated
real angle
spectrum
real beamu
z
Typical representation:
Longitudinal aberration as function of aperture
(pupil coordinate)
If correction at the edge: maximum residuum
at the zone
Typical: largest gradients at the edge
Correcting aspheres or high NA of
higher order:
oscillatory behavior
2/1
Aperture Dependence of Longitudinal Aberration
-0.04 0 0.04 0.08 0.12 0.16 0.2
rp
s'
l = 644 nm
l = 546 nm
l = 480 nm
rp
s'-3.00E-003-2.40E-003-1.80E-003-1.20E-003-6.00E-0040.00E+0006.00E-0041.20E-0031.80E-0032.40E-003
19
spherical aberration4 colors
coma in zone4 colors
coma in full field4 colors
image shells/astigmatism4 colors
distortionmain color
chromatical differencein magnification
secondarychromatic
Longitudinal Aberration Chart
20
Pupil Sampling
Ray plots
Spot
diagrams
sagittal ray fan
tangential ray fan
yp
y
tangential aberration
xp
y
xp
x
sagittal aberration
whole pupil area
Tangential / sagittal / skew rays
View along optical axis
22
Ray Selection Planes
x
ytangential
meridional
rays
x
ysagittal
rays
x
yskew
rays
Transverse Aberrations
Typical low order polynomial contributions for:
defocus, coma, spherical aberration, lateral color
This allows a quick classification of real curves
linear:
defocus
quadratic:
coma
cubic:
spherical
offset:
lateral color
pprSy cos''' 3
)2cos2('''' 2
PpryCy
pprKy cos'''
23
Combinations of basic shapes
24
Interpretation of Transverse Aberrations
defocus+spherical+coma
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
defocus+comaspherical+coma+distortion
spherical 3rd+spherical 5th
spherical+defocus
defocus+spherical 5th
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
yp
y'
yp
y'
yp
y'
yp
y'
yp
y'
yp
y'
Transverse Aberrations
y x
xp
yp
1
-1
1-1
5 m5 m
l= 486 nm
l= 588 nm
l= 656 nm
Dy
Dx
tangential sagittal
Classical aberration curves
Strong relation to spot diagram
Usually only linear sampling along the x-, y-axis
no information in the quadrant of the aperture
25
Best resolution:
- bright central peak in spot
- tangent at transverse aberration
curve
Best contrast:
- mean straight line over complete
pupil of transverse aberration curve
- smallest maximal deviation
Different criteria give slightly different
best image planes
Best Image plane
0
z
Wrms
0
z
Ds
u
y
best matching
in the
centre
minimal difference
ymaxy
min
ymax
26
Transverse Aberrations
Characteristic chart for the representation of transverse aberrations
axis field zone full field
meridional
deviation
sagittal
deviation
yp
y
x
yy
x
yp yp
xp xp
wavelengths:
365 nm
480 nm
546 nm
644 nm
27
Pupil Aberration
rp
'yp/yp
1%
2%
Characteristic chart for the representation of pupil aberration
Distortion of the pupil grid from the entrance to the exit pupil
Pupil aberration can be interpreted as the spherical aberration of the chief ray for the
pupil imaging
yp
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
xp
ideal
real
28
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
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
Variation of Chromatical Aberrations
Representation of the image location
as a function of the wavelength
axial chromatical shift
Representation of the chromatical
magnification difference with field
height
lateral chromatical aberration
l
-100 -80 -60 -40 -20 0 20 40 60 80 100
500 nm
s' [m]400 nm
600 nm
y' [m]
field height y/ymax
-15 -12 -9 -6 -3 0 3 6 9 12 15
Airy1
31
Spot Diagram
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
32
Spot Diagram
Variation of field and color
Scaling of size:
1. Airy diameter (small circle)
2. 2nd moment circle (larger circle, scales with wavelength)
3. surrounding rectangle 486 nm 546 nm 656 nm
axis
fieldzone
fullfield
33
Gaussian Moment of Spot
Spot pattern with transverse aberrations xj and yj
1. centroid
2. 2nd order moment
3. diameter
Generalized:
Rays with weighting factor gj:
corresponds to apodization
Worst case estimation:
size of surrounding rectangle Dx=2xmax, Dy = 2ymax
xN
xS jj
1
yN
yS jj
1
M rN
x x y yG j S j Sj
22 21
GMD 2
M rN
g x x y yG
G
j j S j S
j
22 21
34
Practical problem in analysis of classical
spot diagrams:
relation between deviations and pupil location
is lost
Idea of Kingslake:
transverse aberrations of spot points drawn
in pupil intersection points
x and y at every surface in the pupil
sampling grid
r at all surfaces in the pupil sampling grid
Problems:
1. proper representation of quite different
scales
2. distorted grid in case of induced aberrations
35
Kingslakes Diagram
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
xp
yp
J. Palmer, Lens aberration data, Monographs on Apllied Optics, A. Hilger, 1971
Extension of Kingslakes
representation for surface
contributions
Problem:
compaction of high
complexity,
limited clearness
36
Kingslakes Diagram Extended
x
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
y
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
M3
M1
M2
Aberrations of a Single Lens
Single plane-convex lens,
BK7, f = 100 mm, l = 500 nm
Spot as a function of field position
Coma shape rotates according to circular symmetry
Decrease of performance with the distance to the axis
Example HMD without symmetry
x
y
y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
eye
pupil
image
total
internal
reflection
free formed
surface
free formed
surface
field angle 14°
37
negativ spherical aberration
intrafocal: compact broadened spot with bright edge
extrafocal: ring structure
positiv spherical aberration
intrafocal: ring structurewith bright center
extrafocal: ring structure with bright outer ring
coma
bending of caustic
shifted center of gravity
Caustic of Spherical Aberration and Coma
Ref: W. Singer
spherical negativ
spherical positiv
coma
z
38
Wave Aberration
Definition of the peak valley value
exit
aperture
phase front
reference
sphere
wave
aberration
pv-value
of wave
aberration
image
plane
39
Wave Aberrations
Classification of wave aberrations for one image point:
Zernike polynomials
Mean root square of wave front error
Normalization: size of pupil area
Worst case / peak-valley wave front error
Generalized for apodized pupils (non-uniform illumination)
dydxAExP
ppppmeanpp
ExP
rms dydxyxWyxWA
WWW222 ,,
1
pppppv yxWyxWW ,,max minmax
pppp
w
meanppppExPw
ExP
rms dydxyxWyxWyxIA
W2)(
)(,,,
1
40
Relation :
wave / geometrical aberration
Primary Aberrations
Type
ypxp
ypxp
ypxp
ypxp
ypxp
4
1 rcW
cos3
2 yrcW
222
3 cosrycW
22
4 rycW
cos3
5 rycW
sin' 3
1 rcx
cos' 3
1 rcy
2sin' 2
2 yrcx
0'x
cos' 2
3 rycy
sin' 2
4 rycx
cos' 2
4 rycy
0'x
3
5' ycy
Spherical
aberration
Symmetry to
Periodicity
axis
constant
point
1 period
Wave aberration Geometrical spot
Coma
Astigmatism
Field
curvature
(sagittal)
Distortion
Symmetry to
Periodicity
Symmetry to
Periodicity
Symmetry to
Periodicity
Symmetry to
Periodicity
one plane
1 period
two planes
2 period
one plane
1 period
axis
constant
point
1 period
one straight line
constant
one straight line
2 periods
two straight lines
1 period
)2cos2('2
2 yrcy
Ref: H. Zügge
41
Typical Variation of Wave Aberrations
Microscopic objective lens:
Changes of rms value of wave aberration with
wavelength
Common practice:
1. diffraction limited on axis for main
part of the spectrum
2. Requirements relaxed in the outer
field region
3. Requirement relaxed at the blue
edge of the spectrum
Representation of the wave aberration
with field position
Achroplan 40x0.65
on axis
field zone
field edge
diffraction limit
l [m]0.480 0.6440.562
0
Wrms [l]
0.12
0.06
0.18
0.24
0.30
Wrms [l]
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
diffraction limit
field
w [°]0 2.8 5.6 8.4 11.2 14
l= 480 nm
l= 546 nm
l= 644 nm
42
Typical Variation of Wave Aberrations
Representation of the wave aberration
for defocussing at several field points
- decrease of performance with field
height
- field curvature
Wavefront over the pupil as surface
Wprms [l]
-0.4 -0.2 0 0.2 0.40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
axis
zone
full field
axis field zone full field+1 l
-1 l
0
43
Typical Variation of Wave Aberrations
Representation of the wave aberration
as a function of field and wavelength
for a microscopic lens
Analysis:
1. diffraction limited correction near to
axis for medium wavelength range
2. no flattening
3. blue edge more critical than red edge
0.06
0.09
0.12
0.1
5
0.15
0.1
8
0.180.21
0.24
0.0
714
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64
y'
l
[m]
diffraction
limit
44
Contributions of the lower Zernike coefficients per surface,
In logarithmic scale not additive
(Fringe convention)
Error in additivity due to
numerical reasons for
astigmatism
Effect of induced aberrations
and grid distortion in the
range of l / 20 in this case
45
Zernike Coefficients per Surface
3
Log cjs
index j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250
0.02
0.04
0.06
0.08
0.1
0.12
0.14
cj [l]
index j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25-3
-2
-1
0
1
2M1 M1 M3
tilt def ast com sph
Example system: plane symmetric TMA system
nearly diffraction limited correction for a small field
of view
M1: off axis asphere, M2, M3: freeforms
F-number 1.8, field -1°...+1°
46
TMA System
x = -1° x = +1°
y = +1°
x = 0°
y = -1°
y = 0°
field
angles x/y
M1
circular
symmetric
asphere
M2
pupil
freeform M3
freeform
image
M2
freeform
M3
freeform
M1
asphere
Surface contributions of every mirror with parabasal reference
pupil rescaling neglected
Dominating astigmatism
Sum of wave aberration not additive,
difference due to induced aberrations
47
Wavefront Contribution of every Surface
sum of surface contributions
M1 M2 M3
Low order Zernikes as a function of the field position
Completly different distributions,
Complete characterization gives a huge amount of detailed information.
Also analytical solution for lower orders provided in the literature
48
Zernikes as Function of the Field
R. Gray, C. Dunn, K. Thompson, J. Rolland, Opt. Expres 20(2012) p. 16436, An analytic expression for the field
dependence of Zernike polynomilas in rotational symmetric optical systems
astigmatismcomaspherical aberration
49
PSD Ranges
Typical impact of spatial frequency
ranges on PSF
Low frequencies:
loss of resolution
classical Zernike range
High frequencies:
Loss of contrast
statistical
Large angle scattering
Mif spatial frequencies:
complicated, often structured
fals light distributions
log A2
Four
low spatial
frequency
figure errormid
frequency
range micro roughness
1/l
oscillation of the
polishing machine,
turning ripple
10/D1/D 50/D
larger deviations in K-
correlation approach
ideal
PSF
loss of
resolution
loss of
contrast
large
angle
scattering
special
effects
often
regular
Spatial Frequency of Wavefront Aberrations
The spatial frequency determines the effect of the wave front aberration
Characteristic ranges, scaled on the diameter of the pupil:
- figure error : Zernike
causes resolution loss
- midfrequency range
- high frequency : roughness
causes contrast loss
log I(r)
0 5 10 15 20 2510
-6
10-5
10-4
10-3
10-2
10-1
100
r/rairy
g/D = 0.5
g/D = 4
g/D = 2
g/D = 1
g/D = 0.25
g/D = 0.07g/D = 0.1
50
Typical setup for component testing
Lenslet array
Hartmann Shack Wavefront Sensor
subaperture
point spread
function
2-dimensional
lenslet array
a) b)
test surface
telescope for adjust-
ment of the diameter
detector
collimator
lenslet
array
fiber
illumination
beam-
splitter
Spot Pattern of a HS - WFS
a) spherical aberration b) coma c) trefoil aberration
Aberrations produce a distorted spot pattern
Calibration of the setup for intrinsic residual errors
Problem: correspondence of the spots to the subapertures
21
1121 )''(''
yy
yssss y
Hartmann Method
Similar to Hastmann Shack Method with simple hole mask and two measuring
planes
Measurement of spot center position as geometrical transverse aberrations
Problems: broadening by diffraction
Hartmann
screen with
hole mask
test optic
y
focal
plane
first measuring
plane
second measuring
plane
reference
plane
s'1
y
2
s'
s'
y1
y2
u'
Hartmann Method
Real pinhole pattern with signal
Problems with cross talk and threshold
L
threshold
peak
minimum
c)
D
d a
b
a) b)
Testing with Twyman-Green Interferometer
Short common path,
sensible setup
Two different operation
modes for reflection or
transmission
Always factor of 2 between
detected wave and
component under test
detector
objective
lens
beam
splitter 1. mode:
lens tested in transmission
auxiliary mirror for auto-
collimation
2. mode:
surface tested in reflection
auxiliary lens to generate
convergent beam
reference mirror
collimated
laser beam
stop
Interferograms of Primary Aberrations
Spherical aberration 1 l
-1 -0.5 0 +0.5 +1
Defocussing in l
Astigmatism 1 l
Coma 1 l
56
Interferogram - Definition of Boundary
Critical definition of the interferogram boundary and the Zernike normalization
radius in reality
57