Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field
The application of a Shack-Hartmann sensor with holographic lenslet array to the measurements of wavefront aberrations in a speckle field is offered. The main feature of the method is that the tested wave front can be compared with an arbitrary wavefront preliminary recorded in the holographic me...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2008
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| Zitieren: | Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field / D.V. Podanchuk, V.N. Kurashov, V.P. Dan'ko, M.M. Kotov, N.S. Sutyagina // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 1. — С. 29-33. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1186712017-05-31T03:09:30Z Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field Podanchuk, D.V. Kurashov, V.N. Dan’ko, V.P. Kotov, M.M. Sutyagina, N.S. The application of a Shack-Hartmann sensor with holographic lenslet array to the measurements of wavefront aberrations in a speckle field is offered. The main feature of the method is that the tested wave front can be compared with an arbitrary wavefront preliminary recorded in the holographic memory of the array. An iterative algorithm of the sensor work for measuring the variable speckled wavefronts is offered. The experimental results of measurements of the curvature of a spherical speckled wave are presented. The possibility to use the method proposed in the analysis of deformations of a rough surface is shown. 2008 Article Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field / D.V. Podanchuk, V.N. Kurashov, V.P. Dan'ko, M.M. Kotov, N.S. Sutyagina // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 1. — С. 29-33. — Бібліогр.: 7 назв. — англ. 1560-8034 PACS 42.15.Fr, 42.30.Ms, 42.30.Rx, 42.40.Eq, 42.40.My http://dspace.nbuv.gov.ua/handle/123456789/118671 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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English |
| description |
The application of a Shack-Hartmann sensor with holographic lenslet array to
the measurements of wavefront aberrations in a speckle field is offered. The main feature
of the method is that the tested wave front can be compared with an arbitrary wavefront
preliminary recorded in the holographic memory of the array. An iterative algorithm of
the sensor work for measuring the variable speckled wavefronts is offered. The
experimental results of measurements of the curvature of a spherical speckled wave are
presented. The possibility to use the method proposed in the analysis of deformations of
a rough surface is shown. |
| format |
Article |
| author |
Podanchuk, D.V. Kurashov, V.N. Dan’ko, V.P. Kotov, M.M. Sutyagina, N.S. |
| spellingShingle |
Podanchuk, D.V. Kurashov, V.N. Dan’ko, V.P. Kotov, M.M. Sutyagina, N.S. Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Podanchuk, D.V. Kurashov, V.N. Dan’ko, V.P. Kotov, M.M. Sutyagina, N.S. |
| author_sort |
Podanchuk, D.V. |
| title |
Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field |
| title_short |
Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field |
| title_full |
Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field |
| title_fullStr |
Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field |
| title_full_unstemmed |
Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field |
| title_sort |
shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118671 |
| citation_txt |
Shack-hartmann wavefront sensor with holographic lenslet array for the aberration measurements in a speckle field / D.V. Podanchuk, V.N. Kurashov, V.P. Dan'ko, M.M. Kotov, N.S. Sutyagina // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 1. — С. 29-33. — Бібліогр.: 7 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T14:25:32Z |
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2025-07-08T14:25:32Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 29-33.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
29
PACS 42.15.Fr, 42.30.Ms, 42.30.Rx, 42.40.Eq, 42.40.My
Shack-hartmann wavefront sensor with holographic lenslet array
for the aberration measurements in a speckle field
D.V. Podanchuk, V.N. Kurashov, V.P. Dan’ko, M.M. Kotov, N.S. Sutyagina
Optical Processing Laboratory, Faculty of Radiophysics,
Taras Shevchenko Kyiv National University, 64, Volodymyrska str., 01033 Kyiv, Ukraine
Abstract. The application of a Shack-Hartmann sensor with holographic lenslet array to
the measurements of wavefront aberrations in a speckle field is offered. The main feature
of the method is that the tested wave front can be compared with an arbitrary wavefront
preliminary recorded in the holographic memory of the array. An iterative algorithm of
the sensor work for measuring the variable speckled wavefronts is offered. The
experimental results of measurements of the curvature of a spherical speckled wave are
presented. The possibility to use the method proposed in the analysis of deformations of
a rough surface is shown.
Keywords: sensor, speckle, holographic lenslet array, aberration measurement.
Manuscript received 29.01.08; accepted for publication 07.02.08; published online 31.03.08.
1. Introduction
The occurrence of phase and amplitude fluctuations in a
tested beam, which appear as a result of the laser beam
scattering by the randomly inhomogeneous medium,
complicates or makes it impossible to carry out
measurements by the wavefront sensors. Such a problem
arises for aberration measurements of human eye using
the laser illumination or, for example, for the precision
monitoring of deformations of a rough surface by a
wavefront sensor, etc. The laser illumination scattered
by the surface gets to the wavefront sensor. Because of
the microstructure of the eye retina or the rough surface,
the speckle field considerably reducing the accuracy of
the aberration determination arises. In this case, the
random phase and amplitude modulation of the speckle
field should be considered to be an obstacle when
determining a large-phase distortion of the investigated
wavefront. There are several ways to reduce this
phenomenon [1, 2]. Most common are the methods of
time averaging of speckles and the use of illuminations
with different wavelengths. Each of these methods has
its advantages and peculiarities but doesn’t solve the
problem of the aberration measurement accuracy
increase in corpore. In this paper, the iterative algorithm
of the work of a Shack-Hartmann wavefront sensor with
holographic lenslet array is proposed for the wavefront
aberration measurement in the presence of a speckle
field [3].
2. Principles of the method
The Shack-Hartmann wavefront sensor belongs to the
group of sensors that measure the local slopes of a
wavefront and determine the phase of an optical wave by
the discrete set of its first derivatives [4]. The inves-
tigated wavefront is directed to the lenslet array that
forms a hartmannogram in the focal plane, whose image
is recorded by a CCD-photodetector. The displacements
of hartmannogram spots in relation to the corresponding
lenslet optical axes are proportional to the wavefront
slope averaged by each subaperture. The centers of spots
are considered to be the centroids:
;
,
,
∑
∑
=
ji
ij
ji
ijij
I
Ix
x ;
,
,
∑
∑
=
ji
ij
ji
ijij
I
Iy
y (1)
where Iij is the intensity in the corresponding pixel; xij, yij
are its coordinates; the summation being made over all
pixels from the subaperture that belongs to the current
lenslet. The wavefront shape reconstruction from the
experimentally determined local slopes is made by the
modal method, in which the wavefront phase is
represented as the expansion in basis functions, the so-
called modes. Generally, the basis of Zernike
polynomials is used, since they describe standard
aberrations [5].
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 29-33.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
30
Actually, the usual Shack-Hartmann sensor with a
refractive lenslet array estimates the tested wavefront
shape from its deviation from a plane wave. The
abilities of modern holography allow creating a more
universal wavefront sensor with comprehensive
facilities [6, 7]. The basic difference of its work from
the classic analog is the ability to compare the tested
wavefront with an arbitrarily shaped wavefront which
is recorded in the holographic memory of a lenslet
array. Thus, an iterative algorithm of the sensor work
on this basis will be developed for investigating the
time-varying wave fronts distorted by speckles. At the
first stage, the initial state of a speckled wavefront is
recorded into the memory of a holographic lenslet
array. The sensor will register the tested wavefront
deflection from the initial stage. When the limit of the
measurement range defined by the wavefront distortion
degree is reached, the next holographic lenslet array is
recorded, and so on. The whole dynamics of changing
the tested wavefront shape can be defined by the
sequential summation of the local slopes determined
for every subaperture on different stages of the iterative
process. Hereby, the value of local slope that can be
measured for n iterations is determined as
,
,
1
1
∑
∑
=
=
∆
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
Φ∂
∆
=⎟
⎠
⎞
⎜
⎝
⎛
∂
Φ∂
n
m
m
n
n
m
m
n
f
y
y
f
x
x
(2)
where Φ is the phase of the investigated wavefront, ∆xm
and ∆ym are the maximal displacements of a focused spot
that can be detected by the sensor for the m-th iteration,
and f is the focal length of the lenslet array. Thus, the
iterative algorithm of recording the holographic lenslet
array and measuring the aberrations with the
compensation of distortions for sequential stages of a
speckled wavefront is realized. It’s obvious that the
extension of the range of speckled wavefront measure-
ments depends on the number of iterations.
3. Experimental setup
For the experimental verification of the iterative
method of the holographic Shack-Hartmann wavefront
sensor work, the laboratory setup was engineered; its
optical scheme being shown on Fig. 1. A laser beam (λ
= 0.63 µm) is divided into two beams by beam splitter
BS1. One beam (the higher beam on the scheme) forms
the reference wave for the holographic lenslet array
recording. This beam passes through refractive lenslet
array LA and, by means of beam delivery system L1–
L2 and mirrors M1 and M5, is directed to the
holographic plate at an angle of about 10°. The dia-
meter of the lenslet array lenses is 0.4 mm, and the
focal length is 24 mm.
The beam which passed through BS1 forms the
object and the tested waves for the recording and
reconstruction of holograms. The beam that is reflected
from M2 and M4 mirrors forms the reference plane
wave. The tested spherical wavefront is formed by the
optical system of L3 and L4 lenses. The curvature of this
beam can be changed by the micro-objective L3 shifting
along the optical axis. On the outlet of the system,
diffusive plate DP (ground glass) is placed. Statistical
parameters of the speckle field are determined by a
rough surface shape. The object wave also can be
formed after the reflection from steel disc SD with rough
surface that is placed instead of M4 mirror. With the L5–
L6 lenses system, the surfaces of DP and reference
mirror M4 are transposed to the recording plane of
holograms. During the hologram recording, one of the
beams – the tested one or the beam reflected from M2
mirror – is used as the object beam. The hologram
recording is realized on holographic plates PFG-01.
During the measurements, the holographic lenslet array
is illuminated by the tested wave, so the reconstructed
wave forms the hartmannogram on the surface of a
photodetector CCD1 (768×576, the pixel size is 10 µm).
CCD2
CCD1
Col.1
M3
M1
M5
L5
L6
L1
L2
LA
to PC
HLA
∆
M4
DP
L3
Col.2
Col.3 L4
M2
SD
BS 3
BS1
BS2
Fig. 1. Optical setup of a holographic Shack-Hartmann wavefront sensor: Col1-Col3 are collimators; LA/HLA is the
refractive/holographic lenslet array; BS1–BS3 are beam-splitters; M1–M5 are mirrors; L1–L6 are lenses; DP is the ground
glass; SD is the rough-surface steel disc; CCD1 and CCD2 are photodetectors.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 29-33.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
31
To determine the wavefront local slopes, the array of
14×14 hartmannogram spots is used; it corresponds to
the area of about 2 mm in diameter on the object plane.
The tested wavefront curvature radius determination is
realized by the value of the defocusing coefficient in the
Zernike polynomial expansion of the wave front.
The veracity of the wavefront reconstruction by a
wavefront sensor with holographic lenslet array can be
reduced by the inaccuracy of the holograms positioning
in the exposure place. This leads to the appearance of an
additional aberration in the reconstructed wave front. To
estimate these errors, the following experiments are
performed. The tested wavefront shape on the stage of
the HLA reconstruction is set to be the same, as it was
during the hologram recording. Then the hologram is
extracted from a holder and then mounted on that place
again. During the data processing, the hartmannogram
formed by a plane wave with the help of the refractive
lenslet array is used as the reference, and the object
hartmannogram is formed by the reference beam with
the help of the holographic lenslet array. If the hologram
is mounted precisely, the hartmannograms are identical,
and, as a result of the reconstruction, we will obtain the
plane wave. But in practice, there is some difference in
hartmannograms that is revealed as the presence of some
aberrations in the reconstructed wavefront. The
aberration rate was quantitatively evaluated as the
wavefront phase standard deviation from the plane
wavefront phase throughout the tested aperture. The
accuracy of the hologram positioning into the exposition
place was visually controlled by the maximum fringe
width of an interference image, which appears as a result
of the interference between the wave reconstructed by
the hologram and the object wave from the refractive
array. In this case, the average standard deviation of the
reconstructed wavefront phase was ~ 0.05 λ. This posi-
tioning method for holograms is used in the further
work.
4. Results and discussion
At the first stage of the investigations, the influence of
the speckle field on the results of wavefront
measurements and reconstruction with a Shack-
Hartmann sensor is estimated. Thereto, the holographic
lenslet array is recorded with a plane reference wave
formed by mirror M4. Thus, the diffractive analog of
refractive LA is obtained. A clean or speckled tested
wave front with the curvature that varied by the shifting
of microobjective L3 is inputed to the sensor. The
subimage of the speckle field and the fragment of
corresponding noised hartmannogram of the plane wave
front are depicted in Fig. 2a,b. As a diffuser, the one-side
ground glass plate is used. Its surface is polished by a
diamond paste with grains from 7 to 10 µm; thus, the
correlation radius of the speckle field in the hologram
plane is rcor = 15 µm. Figure 2b shows that the focused
spots on the hartmannogram are splitted and noised.
Therefore, the centroid estimation error increases and the
wavefront reconstruction error is revealed as the
appearance of an additional aberration. The first ten
coefficients of the Zernike polynomial expansion (up to
the 3-rd order) for the reconstructed wave front are
depicted in Fig. 2c (marks □ and ■ represent the plane
wavefront, ◊ and ♦ represent the spherical wave). In case
of a clean wave front (white marks), its shape is
generally determined by defocusing aberration
(coefficient C3). But when the speckles appear (black
marks), the defocus influence is reduced, because the
rest expansion coefficients increase; thus, higher order
aberrations “appear” in the reconstructed wave front,
even when the plane wave is analyzed.
At the second stage, the application of the Shack-
Hartmann sensor for spherical wavefront measurements
in a speckle field is examined. The hologram of the
refractive lenslet array with a plane reference beam
reflected from M4 mirror is preliminarily registered.
Then the obtained holographic lenslet array is
reconstructed with the tested spherical wave. The
dependence of the wavefront curvature on the micro-
objective L3 shifting for the clean nonspecled beam is
shown in Fig. 3 (solid line a). To make these measure-
ments, the ground glass is replaced by a transparent plate
with the same thickness. Then the corresponding
dependence is measured with a diffuser (marks b in
Fig. 3). One can see that the curvature values for the
speckled wave front obtained with the usual technique
are significantly different from those for a clean beam
without speckles.
a b
98 1 54
-1
0
1
2
3
4
0 2 3 6 7
Ck , λ
k
50
∆, mm:
0
c
Fig. 2. a) Image of a speckle field with the correlation radius
rcor = 15 µm, b) corresponding noised harmannogram, c)
Zernike coefficients up to the 3-rd order for clean (white
marks) and speckled (black marks) wave fronts.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 29-33.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
32
a b
Fig. 4. Hartmannogram of the wave front reflected from the
rough surface obtained with refractive (a) and holographic (b)
LA.
a
100
1000
10000
0 20 40 60 80
Cu
rv
.
ra
di
us
, m
m
Defl. at the center, µm
HLA1 HLA2 HLA3
S-H sensor
interferometry
b
Fig. 5. a) Interferogram and a phase map of the wave front
reflected from the rough surface with a 30-µm strain; b)
measurements of the curvature radius of the sample shape.
Fig. 3. Wavefront curvature measurements: a) in the absence of
speckles; b) in the speckle field with a holographic copy of the
usual lenslet array; c) in the speckle field with a holographic
lenslet array and the iterative recording method; d) the error
appearance because of overrunning the iteration range.
In Fig. 3 (marks c on the diagram), we also show
results of the application of the iterative holographic
lenslet array recording method to speckled wave front
measurements (correlation radius rcor = 15 µm). Three
iterations were made when changing the wavefront
curvature from 0.9 m–1 to +1.6 m–1: the first holographic
array was recorded for a wavefront curvature of 0.93 m–1,
the second – for the plane wave, and the third for a
curvature of +0.86 m–1. One can see that the usage of the
proposed method considerably reduces the curvature
measurement error down to several per cent. In our case,
the iterative recording methods allow obtaining the
reliable results when the wavefront curvature increases
by 0.9 m–1 in comparison with the wavefront recorded in
the holographic memory of the array. When the wave-
front curvature overruns the measurement range of the
holographic lenslet array, the rapid increase in the error
is shown (dashed line d on the diagram).
The iterative recording algorithm was also applied
to surface deformation measurements by means of the
reflected beam analysis. Metallic disc SD with a
diameter of 20 mm clamped around the periphery is used
as a sample (see Fig. 1). During the measurements, it can
be deformed with a flexure at the center controlled by
the micrometer screw. Since the disc surface is rough,
the speckles appear, when the probe beam is scattered
after reflection. In this case, the application of a usual
sensor is impossible because of the high noise level on
the hartmannogram (Fig. 4a). But if the reflected
speckled beam is used as the reference for the recording
of the holographic lenslet array, then a noiseless
hartmannogram with well-focused spots is formed in the
focal plane of the lenslet array, while the hologram is
being reconstructed (Fig. 4b). During the deformations
of a sample, these spots are shifted proportionally to the
local slopes of the surface, as it is in the absence of a
speckle field; thus, the determination of a surface shape
deviation from the initial shape of the sample becomes
possible. We sequentially recorded three HLAs with
different states of the sample: with zero strain, and with
deflections at the center of 30 and 50 µm.
The subsidiary control of the introduced surface
deformation is realized by real-time holographic
interferometric methods. For this, the hologram of an
unstrained tested object is recorded; refractive lenslet
array LA is removed from the beam to form the
reference plane wave. The reconstruction of the
hologram with the reference beam leads to the
generation of a fringe pattern corresponding to the
interference between waves reflected from the tested
surface before and after the deformation. The fringe
pattern is observed with a camera CCD2 focused on the
plane of a hologram.
The interferogram obtained by the holographic
interferometry method for a deflection of 30 µm at the
center and superimposed with the phase map of the
reconstructed wavefront is shown in Fig. 5a. One can see
-1.0
-0.5
0.0
0.5
1.0
1.5
1/R, m-1
-60 -40 -20 0 20
∆, mm
(a)
(b)
(c)
(d)
HLA1HLA2HLA3
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 29-33.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
33
that they are identical. However, the interferogram
analysis is complicated because of the appearance of a
speckle field. The greater the deformation, the greater
the error of the interferogram processing. For example,
when the deformation at the center is above 80 µm, the
rings are almost indistinguishable against the speckled
background. The sample’s deformation was quan-
titatively evaluated by the surface curvature radius.
Experimental values of the curvature radius are
computed from the reconstructed Zernike coefficients
and from the interference fringe radius. The results are
given in Fig. 5b. The data obtained for different methods
well coincide. The parts of the experimental curve
obtained for different HLAs are uninterrupted, and it is
more even in comparison with the curve for inter-
ferometric measurements.
5. Conclusion
The discussed iterative method of the holographic lenslet
array recording advances the Shack-Hartmann wavefront
sensor features. The possibility of its use for the
measurement of aberrations in a speckle field by the
holographic compensation of distortions for sequential
states of the speckled wavefront is shown. The usage of
a reversible material for the holographic lenslet array
recording is expected to allow the realization of the
Shack-Hartmann sensor, which easily accommodates to
the dynamic changes of the wavefront. Such a sensor can
find applications in optical metrology, e.g. for the real-
time testing of a rough surface deformation.
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