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Metrology and Sensing
Lecture 7: Wavefront sensors
2017-11-30
Herbert Gross
Winter term 2017
2
Preliminary Schedule
No Date Subject Detailed Content
1 19.10. Introduction Introduction, optical measurements, shape measurements, errors,
definition of the meter, sampling theorem
2 26.10. Wave optics Basics, polarization, wave aberrations, PSF, OTF
3 02.11. Sensors Introduction, basic properties, CCDs, filtering, noise
4 09.11. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry
5 16.11. Interferometry I Introduction, interference, types of interferometers, miscellaneous
6 23.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods
7 30.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods
8 07.12. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight,
Scheimpflug setup
9 14.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology
10 21.12. Holography Introduction, holographic interferometry, applications, miscellaneous
11 11.01. Measurement of basic
system properties Bssic properties, knife edge, slit scan, MTF measurement
12 18.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy
13 25.01. Metrology of aspheres
and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms
14 01.02. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous
15 08.02. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method
Hartmann Shack Wavefront Sensor
detector
xDmeas
u
hf
array
spot
offset
Darray
Dsub
refractive
index n
wave-
front
Lenslet array divides the wavefront into subapertures
Every lenslet generates a simgle spot in the focal plane
The averaged local tilt produces a transverse offset of the spot center
Integration of the derivative matrix delivers the wave front W(x,y)
5
Typical setup for component testing
Lenslet array
Hartmann Shack Wavefront Sensor
6
test surface
telescope for adjust-
ment of the diameter
detector
collimator
lenslet
array
fiber
illumination
beam-
splitter
Spot Pattern of a HS - WFS
Aberrations produce a distorted spot pattern
Calibration of the setup for intrinsic residual errors
Problem: correspondence of the spots to the subapertures
7
Array Signal
array of phase spot pattern cross section of the spot pattern
original discretized and quantized with noise
Lenslet array
ideal signal
Real signal:
1. discretization
2. quantization
3. noise
8
Dynamic range: ratio of spot diameter to size of sub-aperture
Averaging of wavefront slope inside sub-aperture
HS - WFS : Size of Sub-Apertur
I(x)
xp
lens focal length f
Dsub
Dspot
/2
point
spread
function
Dspot
/2incoming wave
front W(xp)
9
Setup of a Hartmann-Shack - WFS
pupil
plane
straylight
stop
Dpup D
arr
sensor
farr
array relay lenstelescopefocal
plane
mrel
1. Simple setup
2. With telecope
and relay lens
10
test surface
telescope for adjust-
ment of the diameter
detector
collimator
lenslet
array
fiber
illumination
beam-
splitter
General Setup of a HS - WFS
Generalized setup:
adaptation of diameter
Wavefront is scaled
Relay lens
Adaptation of sensor
size
f2
f1
f1
f2
pupil
plane
sensor
plane
Dpup D
sens
Wpv
W'pv
pup
senspupsens
DDWW ,
focal
plane
x'
farr
array relay lens
x
sensor
11
Real Measurement of a HS-WFS
Problem in practice: exact determination of the spot centroid:
- noise
- discretization
- quantization
- broadening by partial coherence
- broadening by local curvature
- error by centroid affecting coma
- error by partly illuminated pixels
original discretized and quantized with noise
13
Parametrization of a HS-WFS
Layout parametrization:
Fresnel number
Fill factor
Spot size
Accuracy:
f [mm]
Nsub
0 10 20 30 40 50 600
10
20
30
40
50
60
70
80
90
100
Dspot
= 0.01
Dspot
= 0.05
Dspot
= 0.10
Dspot
= 0.20
Dspot
= 0.40
sensitivity
spatial resolution
accuracy
dynamic
range
f
D
Nf
DN sub
sub
measF
44
2
22
2
array
meas
D
D
meas
sub
F
sub
sub
spotD
Nf
N
D
D
fD
2
2
2
Fsub
spot
ND
D
2
1
relm
f
Pkmin
2max2
f
Dh sub
14
Parametrization of a HS-WFS
Wavefront detectability
Dynamic range
f [mm]
Nsub
0 10 20 30 40 50 600
10
20
30
40
50
60
70
80
90
100
minimum
spot size
maximum
spot size
minimum
spatial
resolution
possible
parameter area
dynamic
range limit
rel
sub
mkP
DhR
2min
max
subarray NDW minmin
40002
1
2min
max
f
rel
sub
rel
spotsubN
mPk
hDh
mPk
DDR
15
Die Wellenfläche wird über die Subapertur gemittelt
Fresnelzahl der Arraylinse, Spotgröße / Subaperturdurchmesser
Auflösung, mit Pixelgröße p und Subaperturzahl N
Zuordnung der Spots zu den Subaperturen, Dynamikbereich
Spotablage subpixelgenau messen
Probleme mit teilausgeleuchteten Subaperturen
Keine Probleme mit spektraler Breite, Kohärenz und Polarisation
x
W
n
fx
fN sub
F
4
2
f
mess
NNp
W
4min
Problems with a Hartmann-Shack - Sensor
16
Tilted sensor plane
Rotated sensor in the azimuth
Scattering of focal lengths of the lenslets
Average of slope inside the subaperture area
Errors in the wavefront reconstruction algorithms
Coma of lenses
Wrong focal length due to dispersion for different wavelength
Sensor plane not exactly matched with focal plane
Partly illuminated lenslets
Electronical noise
Zernike errors due to bad known normalization radius / edge of pupil
Geometrical distortions of the array
Truncation of spot by the corresponding subaperture / cross talk
Discrete finite number of pixels
Quantization of signal on the detector
Errors in the HS - Wavefrontsensor
17
Theoretical largest curvature: R = f
Real size of point spread function:
Larger curvature: cross talk generates errors
Dynamic Range due to Local Curvature
R
f
Dsub
R
f
Dsub
Dspot
a) b)
FN
fR
2
11
min
18
HS-WFS : Discretization Errors
Signal errors due to finite pixel size discretization of the point spread function
N: number of pixels per sub-aperture
N = 4
N = 6
N = 8
N = 12
N = 16
xc
Dspot
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 1 2 3 4 5 6
19
Averaging of the wavefront over the finite size of the subaperture
Number of subapertures (linear) : ns
index of the Zernikes: n
Error decreases with growing ns
Error larger for higher order Zernikes n
Averaging Error by Subapertures
n
c/c
1 2 3 4 5 6 7 8 9
Wrms Wpv
n
c in
1 2 3 4 5 6 7 8 9
0
0.02
0.04
0.06
ns = 32
ns = 64
ns = 100
ns = 144
ns = 200
0
0.05
0.1
0.15
0.2
0.25
ns = 32
ns = 64
ns = 100
ns = 144
ns = 200
nn1 2 3 4 5 6 7 8 9
0
0.005
0.01
0.015
0.02
0.025
0.03
ns = 32
ns = 64
ns = 100
ns = 144
ns = 200
1 2 3 4 5 6 7 8 90
0.05
0.1
0.15
0.2
0.25
0.3ns = 32
ns = 64
ns = 100
ns = 144
ns = 200
20
Larger errors of Zernike polynomials (radial order n) for small number of
sub-apertures (ns)
Larger gradients of high order Zernikes suffers strongly from averaging
PV value less sensitive as rms value
Larger ns more accurate and stable
Averaging Error by Subapertures
n
c/c
1 2 3 4 5 6 7 8 9
Wrms Wpv
n
c in
1 2 3 4 5 6 7 8 9
0
0.02
0.04
0.06
ns = 32
ns = 64
ns = 100
ns = 144
ns = 200
0
0.05
0.1
0.15
0.2
0.25
ns = 32
ns = 64
ns = 100
ns = 144
ns = 200
nn1 2 3 4 5 6 7 8 9
0
0.005
0.01
0.015
0.02
0.025
0.03
ns = 32
ns = 64
ns = 100
ns = 144
ns = 200
1 2 3 4 5 6 7 8 90
0.05
0.1
0.15
0.2
0.25
0.3ns = 32
ns = 64
ns = 100
ns = 144
ns = 200
21
Determination of wavefront of a microscopic lens
Number ns of subapertures (linear):
16 , 32 , 64 , 100
Calculated:
1. gradient of wavefront
2. reconstructed wavefront
3. errors Zernikes
Errors due to averaging and shifted center
of the subaperture
16
32
64
100
5*r(AP)
dW / dr
W in
16
32
64
100
real
5*r(AP)
cj in
16
32
64
100
Nr j
Averaging of Subapertures: Example
22
Relative size of the spot in a HS WFS:
determined by Fresnel number
Small NF: large PSF, crosstalk of neighbouring apertures
Larger error of centroid calculation for subapertures at the edge
Fresnel Number and Crosstalk
Fsubsub
spot
ND
f
D
D
2
122
I(x)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
1.2
1.4
x
Nf = 1
Nf = 2
Nf = 5
Nf = 20
xs/Dsub
0 5 10 15 20 25 300
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
NF
all subapertures
only 1
neighbouring
subaperture
23
HS-WFS : Partly Illuminated Sub-Apertures
Partly illuminated sub-aperture:
change of centroid and error of signal
Wrong signal for constant phase
plateaus
subapertures
illumination
12 3
45
6
partly
illuminated
focal
plane
f
z
detector
centroidpartial illuminated
lens
xo
x
o
o
24
HS-WFS : Partly Illuminated Sub-Apertures
Example
Change of point spread function due to partly illumination
25
Comparison HS-Sensor - Interferometer
Feature HS-WFS
PSI-interf.
1 Complexity of the setup + 2 Cost of the equipment, additional high-quality components + 3 Spatial resolution +
4 Robust measurement under environmental conditions + 5 Noise due to image processing and pixelated sensor +
6 Perturbation by coherent scattering and straylight + 7 Algorithm for reconstructing the wavefront +
8 Test systems with central obscuration +
9 Speed of measurement + 10 Absolute accuracy +
11 Measurement possible independently of wavelength, polarization and coherence
+
26
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'
27
Schematic drawing of
transverse aberrations
Distance of planes limited: overlap of spots
Coherent coupling of sub-aperture fields, interference induces errors of centroid
Hartmann Method
Hartmann
screenfocal
plane
first measuring
planesecond measuring
plane
y'
D
s1'
sy'
s2'
ideal
real with
aberrations
28
Possible geometry of the pinholes:
- number of pinholes,
- size of holes
- distance / geometry
Parameters determine the accuracy
Hartmann Method: Pinhole Array Geometry
c) hexagonalb) cartesiana) polar d) Albrecht grid
120 points 3000 positions for a 3.6° azimuthal scan
29
Hartmann Method Properties
z-positions critical for large spots diameters
No dependence on spectral range and polarization
Coherence is critical, interference for overlapping pinhole images
Apodization not critical
Averaging gives stable data evaluation
30
Separated spots in case of diffraction
s
obj
s
gesamt
sd
ddf
Dddd
d
dd
44.2' 2121
1
2)(
Hartmann Method
Hartmann
diaphragm
ideal
image
plane
2. camera
planeobject
plane
geometrical
projected size
of pinhole
finite site
source field
of view
broadening
due to
diffraction
spot
intensity
pinhole
defocus and
diffraction
convolution
with field of
view
total
d1
f
Dobj
d's
DAiry
/2
D'Feld
/2
ds
h
d2
32
Apodized beam: centroid rays pass through the perfect image point
A cetroid error is eliminated
Reconstruction of the transverse aberrations delivers the wave aberration
x
dxxR
yxW0
'1
),(
Hartmann Method in Case of Apodization
Hartmann
diaphragm
I(y)
image
plane
1. measuring
plane
y
marginal
ray
centroid
ray
transverse
ray aberration
intensity
2. measuring
planepinhole lens under
test
33
Hartmann Sensor
Small power transmission
Problem: diffraction spreading of light pencils
z
z = 0 z = f/4 z = fz = f/2z = - f z = - f/2 z = - f/4
image
plane
Hartmann
screen
transmission T
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
Nsub
Dsub
= 0.5 mm
Dsub
= 0.4 mm
Dsub
= 0.3 mm
Dsub
= 0.2 mm
Dsub
= 0.1 mm
34
Pyramidal Wavefront Sensor
Pyramidal components splits the
wavefront
Signal evaluation analog to 4-quadrant
detector
pupil
pyramid
prism
pupil images
in 4
quadrants
relay lens for
pupil imaging
astigmatism
x-tilt
focussed
beam profile
at pyramid
1
43
2corresponding
points
4321
4321
EEEE
EEEEsc
36
Moving a knife edge perpendicular through
the beam cross section
Relationship between power
transmission and intensity:
Abel transform for circular
symmetry
Example: geometrical spot with spherical aberration
dr
drrrIxP
x
22
)(2)(
z
beforecaustic
zone rays belownear paraxial
focus
Knife Edge Method
y
x
beam
x
knife edge
movement
37
Eyepiece with strong zonal pupil aberration
Illumination for decentered pupil :
dark zones due to vignetting Kidney beam effect
Zonal Aberration
eyepiecelens and
pupil of
the eye
retina
caustic of the pupil
image enlarged
instrument
pupil
39
Method very similar to moving knife edge
Integration of slit length must be inverted:
- inverse Radon transform
- corresponds to tomographic methods
Slit-Scan-Method
y
x
beam x
moving slit
d
41
Generalized concept:
filtering the wave
Realizations:
1. Foucualt knife edge
2. slit
3. Toepler schlieren method
4. Ronchi test
5. wire test
6. Lyot test (/4 wire)
General Filter Techniques
42
Knife edge filter for defocussing
Changing intensity distribution as
a function of the filter position
General Filter Techniques
43
Measurement of surfaces by fringe deformation
Grating creates reference: fringe of 1st order after Ronchi grating
Evaluation of the lateral aberrations of the wavefront by
Explanation geometrical or wave-optical
Ronchi Method
lens under test Ronchi-
grating
Relay
optic
0. order
-1. order
+1. order
R
y
y
W
R
x
x
W
pp
,
44
Setup
Problem: superposition of perturbing higher orders
Ronchi Method
xp
yp
y
x
Ronchi-
grating
R
L xy
g
surface
under
test
45
Setup:
First slit selects one cross section
Lens images the slit
45° rotated toroidal lens inverts space coordinate/angle
Phase Space Analyzer
d
toroidal lens
fT
y
x
1
1
y
x
4
4
y
x
3
3
a
z
lens fL
slit
2s
receiver
plane
y
x
2
2
b
slit
image
Cylindrical lens under 45°:
Focal lengths fx = +fT and fy = -fT
Rotation by 45°
Angle deviation u is transformed into spatial v offset
Imaging of the slit: angle aberration u gives transverse aberration y
Cylindricalal lens under 45°
diverging
-fT
y
u
v
x
focussing
+ fT
toroidal lens
fT
x
y
u
v
a) b)
Phase Space Analyzer in Case of Sph Aberration z = 50
100
150
200
250
300
350
400
Image of a slit for a system with
spherical aberration
Image distance z is variied
Cubic angle aberration more clearly
seen in defocussed planes
Not trivial: adjustment of proper slit width
y y'
slit
toroidal
lenslens
x'
slit
toroidal
lenslensx
a
db
image
plane
a) cross
section
perpendicular
to slit
b) cross
section
parallel to slit
Imaging of the slit
Special case
Slit coordinates:
angle v creates deviation in x'
perpendicular to the slit
Special case:
thin slit and toroidal lens under 45°
Defocussing c4: slit rotates
fba
111
Lfd
vfbf
fx
f
bx L
T
L
L
1'
ufbf
fy
f
by L
T
L
L
1'
xf
bx
L
1' ufb
f
fy L
T
L
'
2
4
2 41'
APLT
L
r
cx
f
b
f
fy
Phase Space Analyzer
Application:
Measurement of microscopic lenses
Three colored points in intermediate
image plane
Measurement of:
- axial color
- defocussing
- spherical aberration
Phase Space Analyser
source
collectorcondenser
microscopic
test lens
mirror
objective
observation
pinholesslit
toroidal lens
azimuth 45°
filter