Shack-Hartmann sensor based on a cylindrical microlens array.

Abstract

We present a Shack-Hartmann wavefront sensor (SHWS) based on a cylindrical microlens array as a device for measuring highly aberrated wavefronts. Instead of the typical spot pattern created by a conventional SHWS, two orthogonal line patterns are detected on a CCD and are superimposed. A processing algorithm uses the continuity of the focal line to extend the dynamic range of measurement by localizing the line, even if it leaves the CCD area confined by the corresponding microcylinder. The measurement of a wavefront from a progressive addition lens with an 80 lambda peak-to-valley value reveals the capabilities of the sensor.

Full text

Page 1

Shack–Hartmann sensor based on a cylindrical
microlens array
M. Ares, S. Royo, and J. Caum
Center for Sensor, Instrumentation and System Development, Technical University of Catalonia (CD6-UPC),
Rambla Sant Nebridi 10, 08222 Terrassa, Spain
Received November 3, 2006; revised December 21, 2006; accepted December 24, 2006;
posted January 8, 2007 (Doc. ID 76759); published March 5, 2007
We present a Shack–Hartmann wavefront sensor (SHWS) based on a cylindrical microlens array as a device
for measuring highly aberrated wavefronts. Instead of the typical spot pattern created by a conventional
SHWS, two orthogonal line patterns are detected on a CCD and are superimposed. A processing algorithm
uses the continuity of the focal line to extend the dynamic range of measurement by localizing the line, even
if it leaves the CCD area confined by the corresponding microcylinder. The measurement of a wavefront from
a progressive addition lens with an 80? peak-to-valley value reveals the capabilities of the sensor. © 2007
Optical Society of America
OCIS codes: 120.2650, 120.3930, 120.3940, 120.5050.
The Shack–Hartmann wavefront sensor (SHWS) is a
well-known setup for measuring wavefront shapes.1,2
In a single shot, the wavefront is spatially sampled
using a microlens array that creates a spot pattern
on a photodetector (typically a CCD camera) placed
at the focal distance of the microlenses. The displace-
ment of the spots created by the wavefront to be mea-
sured from the spots of a reference wavefront (usu-
ally a plane beam) allows for the measurement of the
average wavefront slopes across each microlens aper-
ture. The original wavefront shape is usually ob-
tained by fitting them to a polynomial basis.3,4
Although typical resolutions of the SHWS are not
as high as those obtained in interferometry, the
SHWS has a much larger dynamic range and a larger
tolerance to non-vibration-isolated environments.
The trade-off between resolution and dynamic range
must be managed in the design of the sensor by
choosing the f number of the lenslet that matches the
requirements of the specific application.5To measure
the wavefront correctly, each of the detected spots
must be assigned unequivocally to the microlens that
refracted it. In the classical SHWS, samples are re-
quired to lie in the photodetector area corresponding
to a given microlens, so the dynamic range becomes
quite limited. A great deal of effort has been devoted
to developing schemes for extending the dynamic
range of the sensor once the f number is fixed.6,7
To deal with highly aberrated wavefronts with
abrupt shape changes, we propose a SHWS based on
a cylindrical microlens array. Thus, we replace the
conventional array of spherical microlenses with an
array of microcylinders that focuses the wavefront to
be measured onto the CCD in the form of focal lines,
instead of focal spots. This means that spots in a
given focal line are now continuously connected, al-
lowing us to perform a correct assignment of all the
data, even in cases when the wavefront has locally
important slope changes. The two-dimensional infor-
mation on slope that is present on a spot pattern is
obtained by superimposing the two line patterns with
the microcylinders oriented in orthogonal directions.
The expansion of dynamic range using a cylindrical
microlens array is illustrated in Fig. 1. A typical spot
and vertical line patterns from a complex-shaped
wavefront passing through a conventional SHWS
and through the sensor that we propose have been
drawn as an example. The part of the wavefront that
passes through the central column of the array of
spherical microlenses (numbered as 1) or through the
central microcylinder (1) is properly localized in both
cases. However, regarding areas 3 and 5, it may be
seen how in the conventional SHWS the spots tagged
with capital letters leave their original subapertures
in area 3 and merge with those belonging to area 5.
This implies an uncertainty in the assignment of
spots a5 and A5, b5 and B?, and g5 and G5, and, con-
sequently, a loss of these data for wavefront recon-
struction. The use of an array of cylindrical micro-
lenses solves this problem as all the wavefront
samples refracted by each microcylinder are con-
nected in the same focal line and could be easily
tracked. Thus, samples within line numbers 5 and 3
are assigned to microcylinders 5 and 3, respectively.
Fig. 1.
patterns of a complex wavefront with (a) a conventional
SHWS with spherical microlenses that has an uncertainty
on the localization of spots a5 and A5, b5 and B?, and g5
and G5; and (b) our sensor based on an array of microcyl-
inders from which all the data are unequivocally localized.
(Color online) Example to illustrate the detected
April 1, 2007 / Vol. 32, No. 7 / OPTICS LETTERS
769
0146-9592/07/070769-3/$15.00© 2007 Optical Society of America

Page 2

The experimental setup developed to test the de-
scribed measurement principle is depicted in Fig. 2.
A 635 nm laser diode coupled with a monomode opti-
cal fiber acts as a point source that is collimated us-
ing a diffraction-limited achromat. The collimated
beam crosses the optical object of interest, and the
transmitted wavefront is conjugated with a cylindri-
cal microlens array through an afocal 4:1 beam-
reducer system. Microcylinders of f=10.9 mm focal
length and d=500 ?m width sample the wavefront,
creating a focal line pattern recorded by a mono-
chrome CCD detector of pixel dimensions p_x=p_y
=4.65 ?m. The array is placed onto a high-precision
rotary stage, which allows the vertical and horizontal
line patterns to be recorded. Once the two orthogonal
line patterns are captured, a processing algorithm is
applied, which localizes the lines, even if they leave
their corresponding microcylinder areas. First, in
each line pattern, a segmentation procedure sepa-
rates the connected active pixels from the back-
ground. Next, each of the resulting lines is tagged
with ascending X and Y values starting from the cen-
tral cylindrical microlens. Once each of the lines is
tagged, both line patterns are superimposed and the
areas with the active pixels of the intersection are ob-
tained. These areas are equivalent to the spots of a
conventional SHWS. The centroids ??x
these active areas are calculated using a classical
center-of-mass computation. The wavefront slopes in
the x and y directions (using the displacement of the
aberrated centroids from the corresponding ones of
the reference wavefront) are then calculated follow-
ing the usual SHWS geometrical approximation:
?X,Y?,?y
?X,Y?? of
u?X,Y?=
?W
?x
=
M??x,ABERR
?X,Y?
− ?x,REF
f
?X,Y??p _ x
,
v?X,Y?=
?W
?y
=
M??y,ABERR
?X,Y?
− ?y,REF
f
?X,Y??p _ y
,
?1?
where M=1
From these data, the wavefront is easily recon-
structed in terms of the circular Zernike polynomial
decomposition.8
An important issue to be considered is the calibra-
tion of the proposed sensor. In a first approximation,
there are two main sensor parameters for which a
difference between the real and nominal values may
affect the accuracy of the measurement: the CCD
pixel size and the spacing between the rear principal
plane of the lenslet array and the CCD chip (focal
length of the microlenses).9We calibrated the sensor
using the spherical wavefront created by the fiber-
optic point light source of our system. The fiber tip
4is the magnification of the afocal system.
was placed 535 mm in front of the object plane of the
4:1 beam reducer. By simply removing the collimat-
ing lens, the divergent spherical wavefront with a
well-known 535 mm radius of curvature (ROC) must
be sensed. The plane wave created when the large
achromat is working as a collimator is used as the
reference beam. This calibration has two main ad-
vantages: (a) the same collimated wavefront used as
the reference in the calibration will be used when
measuring any phase object, and (b) a high-quality
spherical wavefront has been used for calibration
without any mechanical changes to the experimental
setup, avoiding the introduction of potential addi-
tional errors.
By means of a linear fit of the displacement of the
reference and the aberrated centroids for the x ?y? po-
sitions, the curvature, C, of the measured spherical
wavefront multiplied by the ?p_x/f?−1??p_y/f?−1? fac-
tor was obtained. Taking into consideration that the
reference curvature of the spherical wavefront is C
=?ROC·M2?−1=?535·1/42?−1mm−1because of the 4:1
afocal reducer, we got p_x/f and p_y/f calibrated val-
ues of 4.685?10−4and 4.687?10−4. The spherical
wavefront measurement was also used to compute
the sensitivity (or vertical resolution) of the sensor,
defined as the smallest amount of wavefront aberra-
tion that the sensor can reliably measure. Ten succes-
sive measurements were taken. From these, the rms
difference between the measured u and v slopes and
the average slopes was calculated. A rms value of
0.017 mrad was obtained. From that slope sensitivity,
calculating the sensitivity of the sensor in terms of
the wavefront is straightforward using the relation
proposed by Neal et al.10:
rms wavefront error= ?rms slope error?d?N, ?2?
where N is the number of samples and d is the width
of the microlenses. This results in a sensitivity of
?/25 at the wavelength used ??=635 nm?.
To show the capabilities of the system, we mea-
sured a commercial progressive addition lens (PAL)
that had nominal null far vision power and +2D
power addition. A 20 mm diameter area was in-
spected, covering the power progression corridor and
part of the lateral blurred zones where considerable
astigmatism is present. The intersection between the
two orthogonal line patterns recorded by the CCD is
shown in Fig. 3(a). The deflection of the focal lines vi-
sually shows the power progression of the PAL from
the plane far vision zone to the highest-power near-
vision zone. As expected, in the regions where the
wavefront is more aberrated, the width of the focal
line increases from the diffraction-limited size and is
also displaced outside the corresponding microcylin-
der area on the CCD array. From the 74 u and v pairs
measured, we performed the wavefront reconstruc-
tion in terms of the Zernike basis. Figure 3(b) shows
how the wavefront is almost flat in the far vision area
and increases its curvature along the corridor follow-
ing the increase in lens power. The distribution of as-
tigmatic aberrations in the lateral areas of the verti-
cal corridor is also clearly shown. The measured
Fig. 2.
sion with the cylindrical microlens array sensor.
Scheme of the setup to test an object by transmis-
770
OPTICS LETTERS / Vol. 32, No. 7 / April 1, 2007

Page 3

peak-to-valley (PV) of more than 80? reveals the
huge dynamic range of the sensor.
Summarizing, we have developed a SHWS sensor
based on a cylindrical microlens array. The use of cy-
lindrical microlenses combined with the image pro-
cessing algorithm presented extends the dynamic
range of the conventional SHWS sensor, allowing for
the measurement of highly aberrated wavefronts. To
improve measurement accuracy, the sensor was cali-
brated using a simple method. A sensor sensitivity of
?/25 at ?=635 nm was also obtained. Finally, a wave-
front transmitted by a commercial PAL with more
than 80? PV height was tested with the sensor to
demonstrate its complex-shaped wavefront measure-
ment performance.
The authors thank the Spanish Ministry of Educa-
tion and Science for the AP2003-3140 grant received,
and for the project DPI2005-00828, which has par-
tially funded this research. M. Ares’s e-mail address
is miguel.ares@oo.upc.edu.
References
1. G. Y. Yoon, T. Jitsuno, M. Nakatsuka, and S. Nakai,
Appl. Opt. 35, 188 (1996).
2. T. M. Jeong, M. Menon, and G. Yoon, Appl. Opt. 44,
4523 (2005).
3. J. Primot, G. Rousset, and J. C. Fontanella, J. Opt. Soc.
Am. A 7, 1598 (1990).
4. L. Seifert, H. J. Tiziani, and W. Osten, Opt. Commun.
245, 255 (2005).
5. R. R. Rammage, D. R. Neal, and R. J. Copland, Proc.
SPIE 4779, 161 (2002).
6. N. Lindlein and J. Pfund, Opt. Eng. 41, 529 (2002).
7. J. Lee, R. V. Shack, and M. R. Descour, Appl. Opt. 44,
4838 (2005).
8. D. Malacara and S. L. DeVore, Optical Shop Testing
(Wiley, 1992), p. 461.
9. A. Chernyshov, U. Sterr, F. Riehle, J. Helmcke, and J.
Pfund, Appl. Opt. 44, 6419 (2005).
10. D. R. Neal, D. J. Armstrong, and W. T. Turner, Proc.
SPIE 2993, 211 (1997).
Fig. 3.
intersection of the line patterns in the x and y directions
recorded by the CCD, (b) reconstruction of the wavefront
transmitted by the PAL.
(Color online) PAL measured with the sensor: (a)
April 1, 2007 / Vol. 32, No. 7 / OPTICS LETTERS
771