Subaperture stitching for measurement of freeform wavefront

Abstract

A method based on subaperture stitching for measurement of a freeform wavefront is proposed and applied to wavefronts calculated from the slope data acquired using a scanning Shack Hartmann sensor (SHS). The entire wavefront is divided into a number of subapertures with overlapping zones. Each subaperture is measured using the SHS, which is scanned over the entire wavefront. The slope values and thus the phase values of separately measured subapertures cannot be connected directly due to various misalignment errors during the scanning process. The errors lying in the vertical plane, i.e., piston, tilt, and power, are minimized by fitting them in the overlapping zone. The radial and rotational misalignment errors are minimized during registration in the global frame by using active numerical alignment before the stitching process. A mathematical model for a stitching algorithm is developed. Simulation studies are presented based on the mathematical model. The proposed mathematical model is experimentally verified on freeform surfaces of a cubic phase profile.

Full text

Subaperture stitching for measurement of
freeform wavefront
KAMAL K. PANT,1DALI R. BURADA,2MOHAMED BICHRA,3MAHENDRA P. S INGH,1AMITAVA GHOSH,1
GUFRAN S. KHAN,2,*STEFAN SINZINGER,3AND CHANDRA SHAKHER2
1Instruments Research and Development Establishment, Dehradun 248008, India
2Indian Institute of Technology, Delhi, New Delhi 110016, India
3Fachgebiet Technische Optik, Technische Universität Ilmenau, Ilmenau 98684, Germany
*Corresponding author: gufranskhan@iddc.iitd.ac.in
Received 28 August 2015; revised 26 October 2015; accepted 2 November 2015; posted 3 November 2015 (Doc. ID 248165);
published 25 November 2015
A method based on subaperture stitching for measurement of a freeform wavefront is proposed and applied to
wavefronts calculated from the slope data acquired using a scanning Shack Hartmann sensor (SHS). The entire
wavefront is divided into a number of subapertures with overlapping zones. Each subaperture is measured using
the SHS, which is scanned over the entire wavefront. The slope values and thus the phase values of separately
measured subapertures cannot be connected directly due to various misalignment errors during the scanning
process. The errors lying in the vertical plane, i.e., piston, tilt, and power, are minimized by fitting them in
the overlapping zone. The radial and rotational misalignment errors are minimized during registration in the
global frame by using active numerical alignment before the stitching process. A mathematical model for a stitch-
ing algorithm is developed. Simulation studies are presented based on the mathematical model. The proposed
mathematical model is experimentally verified on freeform surfaces of a cubic phase profile. © 2015 Optical
Society of America
OCIS codes: (120.3940) Metrology; (220.4840) Testing; (120.3930) Metrological instrumentation.
http://dx.doi.org/10.1364/AO.54.010022
1. INTRODUCTION
Freeform surfaces are nonrotationally symmetric surfaces having
advantages over rotationally symmetric surfaces in nonimaging
and imaging applications, such as a compact projection system,
head-up displays, lithography, computational imaging, space
optics, and optical microsystems. In the illumination industry
freeform surfaces are used for controlling the desired illumination
distribution on the given plane [1]. The use of freeform surfaces
in the optical system provides more degrees of freedom to the
optical designer for better control of aberrations and allows one
to develop compact and light-weight systems [2,3]. Along with
the advantages, there are numerous challenges in the fabrication
and characterization of these surfaces. CNC-based manufactur-
ing technologies are used to fabricate freeform surfaces [4,5]. The
measured form error on the fabricated surface is limited by the
availability of metrology tools [6–8]. Coordinate measuring
machines can be utilized to measure complex surfaces but are
limited in spatial resolution and accuracy [9]. Profilometric mea-
surement is more accurate but limited by scanning probe-
induced errors and provides only 2D profiles. Interferometry
with null optics or computer-generated holograms has been used
to characterize aspheric and freeform surfaces [10–13]. These
auxiliary elements are customized, surface specific, and expensive
to manufacture. Non-null-based interferometric systems devel-
oped by QED technology, USA, have combined the stitching
technique with variable optical null elements to test less complex
surfaces, e.g., mild aspheric surfaces, and rely on high-precision
mechanical stages [14]. Bothe et al. [15] utilized a fringe deflec-
tometry technique mainly for industrial applications for 3D
shape measurement. Li et al. [16] reported a wet cell base inter-
ferometric technique for a miniature freeform surface fabricated
by the microinjection molding method.
Stitching techniques are primarily used to test large optics
and have been reported by Otsubu et al. [17] in which the phase
of small subapertures are measured interferometrically and
stitched together while correcting misalignment errors based on
the overlapping area. Burge et al. [18]haveextendedthis
approach to measure large off-axis aspheric surfaces using null-
based interferometry. The interferometric methods are limited
to rotationally symmetric aspheric surfaces. The wavefront mea-
surement capability of the Shack Hartman sensor (SHS) has
brought its applications in the aspheric shape measurement
[19]. Floriot et al. [20] have reported a method for surface met-
rology using a stitching Shack Hartmann profilometric head,
10022 Vol. 54, No. 34 / December 1 2015 / Applied Optics Research Article
1559-128X/15/3410022-07$15/0$15.00 © 2015 Optical Society of America

which is limited to wavefront shapes of low complexity and rota-
tionally symmetric profiles like aspheres. Xu et al. [21] proposed
a method for testing a rotationally symmetric aspheric surface
with several annular subapertures measured using a SHS. The
stitching algorithms applied in the above-mentioned contribu-
tions are limited to flat, spherical, and rotationally symmetric
aspheric surfaces. These algorithms cannot be directly applied to
freeform surfaces as such.
In this paper, it is proposed to use a scanning SHS to
measure freeform wavefronts using a subaperture stitching
approach. A mathematical model for subaperture stitching for
a freeform wavefront is developed. The stitching algorithm has
been verified by performing simulation studies on a freeform
wavefront. The experiments have been performed using scan-
ning SHS for the freeform wavefront measurement. To the best
of our knowledge, the proposed method for the measurement
of a freeform wavefront is reported for the first time with the
advantages of being compact, low cost, less vibration sensitive,
and non-null in nature as compared to null-based interferomet-
ric approaches.
2. MEASUREMENT SCHEME
To measure a freeform wavefront using a scanning SHS, the
full wavefront is divided into subapertures with overlapping
zones. The SHS is scanning the entire wavefront in the Xand
Ydirections. During the scanning process various misalign-
ment errors may appear and limit the stitching accuracy. To
connect the slope values of separately measured subapertures
in the overlapping zone, the misalignment errors are minimized
by fitting them with respect to the reference subaperture.
Misalignment errors lying in the vertical planes, i.e., piston, tip,
tilt, and power, are corrected using least-square optimization
based on the fixed overlapping correspondence between the
adjacent subapertures.
The lateral misalignment includes radial shift and rotation
of subapertures with respect to their nominal position and these
errors cannot be corrected using the least-square fit due to
inconsistency in overlapping zones. We utilize the active align-
ment of the central subaperture and the minimum two extreme
subapertures (alignment subapertures) lying in the Xand Y
directions to minimize the lateral error in an iterative manner
with respect to their geometrical profile before taking the final
measurement. The flow diagram of the measurement scheme is
given in Fig. 1.
Once these subapertures are properly aligned, it is assumed
that the rest of the subapertures are also aligned. All the sub-
apertures are measured and stitched together to represent the
full wavefront. The details of the stitching algorithm, where
various misalignment errors lying in the vertical planes are cor-
rected, are given in Section 2.A, while the scheme for the lateral
misalignment correction is given in Section 2.B.
A. Vertical Alignment of Subapertures Using
Least-Square Optimization
For any two adjacent subapertures iand jrepresented by the
solid and the dashed rectangular boundaries, respectively, in
Fig. 2, any arbitrary point Plying in the overlapping area needs
to be represented by the same slope value SPx; yas shown by
the dashed lines. Due to misalignment errors, the jth subaper-
ture is shifted from its nominal position and therefore point P
will also shift to a new position P1with a different slope value
Sp1x;yrepresented in the global coordinate frame. The ith
subaperture is assumed as a reference subaperture.
The slope relationship between the ith subaperture and the
jth subaperture in the overlapping area is
Sjx;ySix; yX
L
m1
ξmχmx;y;(1)
where Six;yhSxi
Syi iand Sjx;yhSxj
Syjiare the slope
values in the xand ydirections. χmx; yhχxm
χymiare the
Fig. 1. Flow diagram of the measurement scheme.
Fig. 2. Illustration of misalignment in the overlapping zones
between adjacent subapertures.
Research Article Vol. 54, No. 34 / December 1 2015 / Applied Optics 10023

corresponding misalignment functions for m1;2;3…Land
ξmare the misalignment coefficients which need to be calcu-
lated and fitted to the misaligned subapertures before the stitch-
ing process. For the misalignment errors lying in the vertical
plane, stitching is realized by calculating and compensating
misalignment coefficients corresponding to piston, tip, tilt,
and power between the adjacent subapertures. To calculate
these misalignment coefficients, a least-square fitting method
is used where the sum of the squared difference of the slope
distribution in the overlapping area is minimized by defining
an objective function as
X
i;j
i∩jSjx;y−Six;y−X
L
m1
ξmχmx;y2
minF:(2)
Equation (2) can be generalized for Nnumber of subaper-
tures as
X
N
j1X
i≠j
i∩jSjx;y−Six; yX
L
m1
ξjm χjmx; y2
minF:(3)
The partial derivative of Eq. (3) with respect to each misalign-
ment coefficient transforms to a vector/matrix relationship as
UVW: (4)
If there are npoints in the overlapping area, Uis a vector of
dimension n; 1:
Un;1X
N−1
j1X
i∩j
ΔSnX 1;ΔSSjx;y−Six;y:(5)
Vis a matrix of dimensions n; Las
Vn; LPN−1
j1P
i∩j
χ1nX 1χ2nX 1…χLnX 1nX 1;i≠j
0; ij:(6)
Wis a vector of dimensions L; 1as
WL; 1X
N−1
j1X
i∩j
ξmnX 1:(7)
To calculate the vector W, whose elements are the various
fitting coefficients, Eq. (4) can be written as
WV−1U: (8)
Each subaperture is now corrected by fitting the various mis-
alignment coefficients calculated using Eq. (8) and inserting
them in Eq. (1).
After aligning the subapertures, the slope data of the indi-
vidual subaperture is stitched in the Xand Ydirections as
SXPN
i1SiX
Pn;SYPN
i1SiY
Pn:(9)
Here, SXand SYare the full slope values of the stitched
aperture in the Xand Yaxes.
The wavefront can be calculated by integrating the slope
value as
Φx;yZSXdxSYdy:(10)
Φx;yrepresents the fully stitched wavefront.
B. Lateral Alignment of Subapertures Using Active
Numerical Control
Due to the misalignment of the scanning stages, subapertures
are radially shifted and rotated from their nominal positions
in the (X;Y) plane, which causes the inconsistencies in the
overlapping region. In this case, the above-described stitching
method cannot be directly applied. An active alignment of sub-
apertures is required during their registration to the global
frame. The misaligned subaperture is related to their nominal
position using the following relationship:
Sjx;yRzXSmjx;yT: (11)
Here, Sjx;yis the slope value of the subapertures at their
nominal positions, Smjx; yis the slope value of the misaligned
subapertures, Rzis the rotational matrix in the zdirection, and
Tis the translation vector in the xand ydirections. Any lateral
misalignments that include rotational and translation of suba-
pertures is minimized iteratively by minimizing Rzand Tby
using active alignment and simultaneously updating the new
position of Smjx; y. The updated slope value of the subaper-
ture is correlated to its theoretical nominal position and any
residual slope error is minimized by aligning the subapertures.
During the alignment process, the descending direction of Rz
and Tis being ensured by utilizing a set of library which ar-
chives the residual slope error of the central and the alignment
subapertures for different values of Rzand Tbased on Eq. (11).
The library of the residual slope errors is created beforehand for
the specific freeform surface.
Softwareis developed to perform the active alignment and the
subsequent stitching of the subapertures based on the math-
ematical model explained in Section 2.A.Thedimensionsof
the model frame and subaperture registration are automatically
done based on the input parameters, such as the size of the sub-
aperture, the overlapping area, and the number of subapertures.
3. NUMERICAL VERIFICATION
The wavefront stitching procedure is tested numerically as well as
experimentally on the example of a cubic wavefront generated by
a phase plate with a physical dimension of 14 mm ×14 mm.
The profile is described by a seventh-order polynomial with
maximum sag of 1.119 mm, as shown in Fig. 3(a). The maxi-
mum slopes of the freeform surface in the xand ydirections are
11.88 mrad and 14.09 mrad, respectively. To perform the sim-
ulation studies, the evolution of the freeform wavefront during
propagation from a plane wave illuminating the ideal cubic phase
plate is simulated numerically using a ray-tracing model [22]. In
the practical situation, there is a physical separation between the
freeform surface and the lenslet array of the SHS. To incorporate
this limitation in the simulations, the transmitted wavefront is
measured at the distance of 18 mm from the cubic surface.
The peak to valley (PV) of the wavefront is 579.1 waves, as
shown in Fig. 3(b).
The lateral extent of the freeform wavefront is 10.4 mm×
10.4 mm, which is larger than the detector size of a typical
10024 Vol. 54, No. 34 / December 1 2015 / Applied Optics Research Article

SHS. Therefore, nine subapertures have been simulated using
the ray-trace method, as shown in Fig. 4(a). The size of an indi-
vidual subaperture is 4mm ×4mm and an overlapping area
of 20% is maintained. Individual subapertures are stitched
together using the developed stitching software. The bottom
left subaperture of Fig. 4(b) is assumed as a reference subaper-
ture with respect to which the other subapertures are aligned
and stitched. Figure 4(c) shows the stitched wavefront map.
The PV value is 579.1 waves, which matches to the ideal full
wavefront, as shown in Fig. 3(b).
In the practical situations, the scanning stage carrying the
SHS will introduce misalignment errors during the measure-
ment of subapertures with respect to the reference subaperture.
To analyze the effect of these errors, a simulation study is per-
formed by incorporating various misalignment errors in the
simulated subapertures lying in the vertical and lateral planes.
A. Effects of Vertical Misalignment
To analyze the effects of vertical misalignment errors, a random
tilt of magnitude up to 0.05 deg and defocus of 10 μm has been
induced in each subaperture except the reference subaperture.
The stitched map has the wavefront profile of 579.06 waves
PV, as shown in Fig. 5(a). The stitched wavefront is compared
with the ideal wavefront [as shown in Fig. 3(b)] and the residual
wavefront error is 0.095 waves PV, as shown in Fig. 5(b). This
shows that the stitching software is able to cope up with any
vertical misalignment of subapertures with respect to the refer-
ence subaperture.
B. Effects of Lateral Misalignment
To analyze the effects of lateral misalignments, radial shift and
rotations, we have given a lateral error of 10 μm in both the X
and Ydirections and a rotation of 0.1 deg in each subaperture
except the reference subaperture. All the subapertures in the
presence of these misalignment errors are stitched together.
The final stitched wavefront map has a PV of 586.9 waves.
The stitched wavefront is compared with the ideal wavefront
[as shown in Fig. 3(b)] and a relatively large residual wavefront
error of PV 13.4 waves is found, as shown in Fig. 6(a).
Fig. 3. (a) Cubic phase plate (freeform surface), (b) propagated
wavefront (PV 579.1 waves) at 18 mm distance.
Fig. 4. (a) Ray-trace method to simulate subapertures, (b) nine sub-
apertures of size 4×4mm, (c) stitched wavefront (PV 579.1 waves).
Research Article Vol. 54, No. 34 / December 1 2015 / Applied Optics 10025

To minimize the residual wavefront error resulting from
the lateral misalignment, the subapertures are actively aligned
before the measurement, as described in Section 2.B. Iterative
optimization of the radial misalignment errors within the limit
of 5 μm and rotation of 0.05 deg resulted in the stitched wave-
front with a PV of 578.98 waves. The residual wavefront error
map is shown in Fig. 6(b) with a PV of 0.56 waves. The above
simulation studies show that lateral misalignment during the
subaperture measurement limits the stitching accuracy and
can be minimized by actively aligning the subapertures before
the measurements. In an experiment, high-precision computer-
controlled scanning stages are used to perform these operations
easily and in an automated fashion.
4. EXPERIMENTAL RESULTS
The schematic of the optical layout for the scanning SHS experi-
ment is shown in Fig. 7(a). It consists of a laser source, a spatial
filtering assembly, a collimating optics, the freeform surface
under test, and a SHS coupled with the postprocessing comput-
ing system. The collimated beam is transmitted through the free-
form surface. The same specimen for which the simulation
studies are presented has been used for the experiment. It has
been fabricated on PMMA by ultraprecision machining at the
Institute for Micro and Nanotechnologies at the Technische
Universität Ilmenau Germany [23]. The SHS (OPTOCRAFT
GmbH make) is placed as close to the specimen as possible to
minimize the propagation error. The dynamic range of the SHS
Fig. 5. Effects of vertical misalignment: (a) Stitched wavefront (PV
579.06 waves), (b) residual wavefront error (PV 0.095 waves).
Fig. 6. Effects of lateral misalignment: (a) Residual wavefront (PV
13.4 waves) before active alignment, (b) residual wavefront error (PV
0.56 waves) after active alignment.
Fig. 7. (a) Schematic of the experimental setup, (b) the mounted
test specimen and the SHS.
10026 Vol. 54, No. 34 / December 1 2015 / Applied Optics Research Article

offers the maximum slope measurement as 18.86 mrad. The
maximum slope of the freeformsurface is 14.09 mrad, and hence
can be measured by the employed SHS.
The distance between the specimen vertex and the lenslet
array is 18 mm. The freeform surface is mounted on six degrees
of freedom mount and the SHS on five degrees of freedom
mount with computer-controlled translation stages of the
positioning accuracy of 1 μm [Fig. 7(b)]. The SHS has been
moved in a raster fashion and multiple subapertures of area
4mm×4mmof the test specimen are measured with an over-
lapping area of 20% of the subaperture. The measured suba-
pertures are shown in Fig. 8(a). All nine subapertures are
stitched together. The experimentally stitched wavefront map
has a PV of 584 waves, as shown in Fig. 8(b).
The wavefront has been fitted with a higher polynomial to
remove the noise. The fitted wavefront (PV 582.3 waves) is
shown in Fig. 8(c).The experimentally stitched wavefront is
compared point to point with the ideal freeform wavefront
as shown in Fig. 3(b). The residual wavefront has a PV of
15.16 waves, as shown in Fig. 9.
The residual error comprises the fabrication error in the part
propagated at the detection plane, stitching errors, and any
material inhomogeneity. Numerical analysis on the same speci-
men presented in Section 3shows that the stitching errors are of
the order of submicrometer order. Hence the major part of the
error is the fabrication error propagated at the detection plane.
5. CONCLUSIONS
In this paper, we have presented for the first time a metrology
scheme for the measurement of a freeform wavefront by suba-
perture stitching techniques using a scanning SHS. The proposed
scheme utilizes active numerical alignment to minimize lateral
misalignment, which is more challenging to correct. Vertical mis-
alignment errors are further corrected using a least-square fitting
method. A mathematical model for subaperture stitching is de-
veloped. The stitching algorithm has been verified by performing
simulation as well as experimental studies on a freeform wave-
front. The proposed method is an efficient and cost-effective ap-
proach, e.g., for freeform metrology in nonimaging applications.
By increasing the accuracy of the misalignment correction during
the stitching, it can be applied for precision freeform optics mea-
surement. Hardware can be realized on the above-described
approach with the advantage of being compact, vibration insen-
sitive, and cost effective, and can be used as an in situ metrology
tool on polishing machines for profiling freeform surfaces.
Funding. Department of Science and Technology (DST),
India; German Academic Exchange Service (DAAD), Germany;
Indo-German DST-DAAD Project Based Personnel Exchange
Programme 2013-2015.
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