Content uploaded by Efren Santamaría Juárez
Author content
All content in this area was uploaded by Efren Santamaría Juárez on Nov 29, 2022
Content may be subject to copyright.
Testing concave optical surface by Shack–Hartmann test
using microholes
Alberto Jaramillo Nunez ,aEfren Santamaria Juarez ,a,b,*
Javier Muñoz-López,aand Alejandro Cornejo Rodrigueza
aInstituto Nacional de Astrofísica, Óptica y Electrónica, Department of Optics, Puebla, México
bUniversidad Nacional Autónoma de México, Instituto de Ciencias Aplicadas y Tecnología,
Coyoacán, México
Abstract. We present experimental results for testing concave optical surfaces by the Shack–
Hartmann (S–H) test, where a set of microholes is used instead of the array of lenses employed in
the original (S–H) plate. © 2022 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10
.1117/1.OE.61.10.104104]
Keywords: instrumentation; measurement; metrology.
Paper 20220632G received Jun. 16, 2022; accepted for publication Oct. 4, 2022; published
online Oct. 27, 2022.
1 Introduction
After several attempts to produce a Shack–Hartman (S–H) plate without a lens array with a short
focal length distance, we propose to produce with a microlithographic technique with satisfac-
tory results. This proposition was reported for the first time in OSA-OPTICA meeting by
Núñez et al.,1considering the principle of the obscure camera to produce images.2This paper
is organized into three sections. In Sec. 2, we present the basic principles of the (S–H) method,3
seeking a simple understanding of the classical Hartman4,5and Ronchi6,7methods. Sec. 3covers
our experience in producing the S–H plate without a small lens array, and we refer to previous
studies based on the same principle to leave open holes in the S–H plate. In Sec. 4, we describe
our experimental setup and results obtained by testing a concave parabolic surface using an
incoherent light source. Section 5concludes this paper.
2 Relations Between the Hartmann, Ronchi, and Shack–Hartmann
Methods
For a comprehensive understanding of the uniqueness and interesting method of the S–H plate3
used for testing optical surfaces, in the following paragraphs, we describe its origin based on an
analysis of the Hartmann and Ronchi techniques.
Granados-Agustín et al.8explained in detail the difference between the Hartmann and Ronchi
tests: the Hartmann screen must cover the exit pupil of the optical element under test either as a
single surface or as a complete optical system, and the Hartmanngram obtained is registered in a
plane close to the center of curvature (cc) in the case of testing a single optical surface, or near the
focal plane of the lens system for the entire optical system, as shown in Fig. 1(a). In contrast,
in the Ronchi test, the screen with the ruling is in one plane located close to the cc for a single-
surface system, or near the focal plane of the lens in the optical system being tested, and the
Ronchigram is observed over the surface and exit pupil, as shown in Fig. 1(b). Another important
difference between the two methods is that in the Hartmanngram bidimensional, information is
retrieved, and from the Ronchigram, only unidimensional information is recuperated.
However, even considering the uncovered differences between the experimental arrange-
ments for the Hartmann and Rochi methods, Cordero-Davila et al.9proved that the same math-
ematical analysis can be applied to both testing procedures.
*Address all correspondence to Efren Santamaria Juarez, efren.santamaria@icat.unam.mx
0091-3286/2022/$28.00 © 2022 SPIE
Optical Engineering 104104-1 October 2022 •Vol. 61(10)
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 29 Nov 2022
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
Hence, an important step in combining these two methods was the development of Platt and
Shack,3currently known as the S–H testing technique. In Fig. 2, a small plate with the structure
of the Hartmann plate with holes containing small lenses, with a short focal distance, is located in
a plane close to the cc of the optical surface under test, leading to a short distance of the plate of
the Shack–Hartmanngram image with the spots produced by the microlens array.
3 Shack–Hartmann Plate with Microholes
In this study, we propose a system in which we remove the lenses used at each position of the
S–H plate, leaving the hole open under the condition that the holes must have diameters
<0.5 mm. From our perspective, based on a study by Cordero-Davila et al.,10 they presented
a system in which two Ronchi gratings were located perpendicular to each other; these gratings
have a period of 1.8 lines∕mm producing a square hole array. Instead of an array of square
apertures, we substituted it with an array of microholes with the same period.
Owing to the technical problems of manufacturing the S–H plate using short focal lens arrays
and their particular application in different studies, an alternative modification to eliminate the
use of small lenses was developed for astronomical telescopes by Chanan et al.11 They used a set
Fig. 1 Experimental schemes for the Hartmann and Ronchi test (a) Hartmann and (b) Ronchi.
cc is the center curvature, Fis a point light source, and F0its image.
Fig. 2 A typical image obtained by Hartmann test and Ronchi test: (a) Hartmanngram and
(b) Ronchigram.
Nunez et al.: Testing concave optical surface by Shack–Hartmann test using microholes
Optical Engineering 104104-2 October 2022 •Vol. 61(10)
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 29 Nov 2022
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
of microprisms instead of lenses and later left the microholes open in the S–H plate.12 Previously,
Polo et al.13 applied the same principle of using open microholes in an S–H plate to test an
EUV lithography optical system. Notably, in the previous research, they considered the Fresnel
diffraction concepts of the images produced by the S–H plate method. Other interesting papers
related with the subject of this paper are Refs. 14 and 15.
The main idea behind our work for using microholes in the S–H plate was to consider the
principle of a well-known obscure camera.2However, we also considered Fresnel’s diffraction
theory16,17 for a small pinhole illuminated by spherical wavefronts. Special focus is on the
described studies reported by Born et al.,17 Sec. 8.8, where there is a conclusion about the resem-
blance with the known Fraunhofer diffraction pattern but with the main principal intensity
maxima with a squared tubular shape.
An important aspect to consider in the classical Hartmann test is the size and distance
between holes of the Hartmann screen plate. The size must avoid producing diffraction images,
and the distances between holes must avoid the overlapping of images in the Hartmanngram
(see Ref. 18). Therefore, we consider the aforementioned requirements to improve the S–H plate.
In the next section, the experimental results are described.
3.1 Experimental Results
To find the best images produced for testing an optical concave surface with the S–H plate with
microholes, a careful analysis was performed on the diameter of the holes, revealing the close
dependence between the coherence of the light source employed, the type of the optical surface
under test, and the distance between the S–H patterns with respect to the S–H plate, as shown
in Fig. 3.
Figure 2shows the diagram of the experimental setup for testing a concave parabolic surface
with a diameter of 100 mm and radius of curvature of 1000 mm, and Fig. 4shows a picture of the
experimental setup. In our first attempt, the S–H plate was manufactured using a 3D machine;
unfortunately, the geometry of the microholes created in the plate was of low quality. Therefore,
the next S–H plate was produced by a microlithographic technique and contained microholes
with diameters of 100 μm, with a pitch of 400 μm, and the full size was a square plate of side
8 mm [see Figs. 5(a) and 5(b)]. The light source used was an LED with a pinhole in front, 0.3 mm
in diameter, to produce a point light source (PS), which in turn produced spherical wavefronts
incident on the optical surface under test. The reflected wavefront returned to the image ðPS 0Þof
Fig. 3 Setup for the Shack–Hartmann method. PS is the point source, PS 0is the PS image, and
cc is the center of curvature.
Nunez et al.: Testing concave optical surface by Shack–Hartmann test using microholes
Optical Engineering 104104-3 October 2022 •Vol. 61(10)
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 29 Nov 2022
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
the point light source, and the S–H plate could be located in front or behind the image of the light
source. The PS and its image ðPS 0Þwere aligned with the paraxial cc of the concave surface.
Figures 6(a) and 6(c) show the Shack–Hartmanngrams observed at different positions of
the S–H plate. The S–H plate was located before (inside) and after (outside) the cc at a distance
of 10 mm.
Figure 6(b) shows close to the cc of the optical surface.
An interesting and important point to mention about the close relations between the three
testing methods, Hartmann, Ronchi, and S–H, is the fact that after careful analysis in Fig. 7,at
the edges of the pattern, either along the vertical or horizontal axes, a small shear can be noticed,
similar to the frequently observed effect with the Ronchi gratings with a small period. In our
experiment, a pitch of 400 μmof the S–H plate produced light diffraction effects.
4 Comparison of Results with Different Methods
We compare the results shown in Fig. 6, obtained by the proposed S–H plate with microholes,
against those obtained using the classical Hartmann test Fig. 8, and the spotfield pattern coming
Fig. 4 Experimental setup for S–H test, PS is the point light source.
Fig. 5 (a) View of lithographic S–H screen and (b) S–H screen view in a microscope.
Fig. 6 Shack–Hartmanngram located (a) inside of cc, (b) close to cc, and (c) outside the cc.
Nunez et al.: Testing concave optical surface by Shack–Hartmann test using microholes
Optical Engineering 104104-4 October 2022 •Vol. 61(10)
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 29 Nov 2022
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
from S–H test performed with a commercial wavefront sensor Thorlabs-WFS150C (Fig. 9).
Both patterns of Fig. 9were obtained with S–H wavefront sensor Thorlabs-WFS150C with
a rectangular array of lenses of 146 μmin diameter and a lenslet pitch of 150 μm. Finally, the
fringe patterns are obtained through the Ronchi test (Fig. 10).
It is important to clarify that the pictures of Fig. 8with the clasical Hartmann screen it is
located in front of the surfaces with holes 6-mm in diameter [Figs. 8(a) and 8(b)] and holes 3-mm
in diameter [Figs. 8(c) and 8(d)]. In both cases, the separation between holes at the Hartmann
screen measures 10 mm from center to center. The size of the Hartmann screen is 10-mm in
diameter to cover the diameter of the mirror under test.
It is observed that the results presented in Fig. 6obtained with the S–H plate proposed are in
good agreement with those obtained with the classic Hartmann test (Fig. 8) and with the spotfield
obtained with the wavefront sensor Thorlabs-WFS150C (Fig. 9). In addition, the same geometry
is observed along the fringes in the Ronchi pattern (Fig. 10).
In the first two methods, classical Hartmann and spotfield pattern S–H test, two-dimensional
information from the surface is obtained. On the other hand, the Ronchi test gives us one-dimen-
sional information from the surface.7In all mentioned test methods, it is possible to recover the
wavefront by calculating the slopes of the rays perpendicular to the wavefront. All the experi-
ments used a point source of white LED light.
Fig. 8 Classical Hartman test (a), (c) inside of cc and (b), (d) outside the cc at 10 mm.
Fig. 7 Shack–Hartmanngram located (a) inside and (b) outside the cc.
Nunez et al.: Testing concave optical surface by Shack–Hartmann test using microholes
Optical Engineering 104104-5 October 2022 •Vol. 61(10)
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 29 Nov 2022
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
5 Conclusions
The testing of a concave surface described in this paper is an example of the use of small pinholes
instead of lenses in the S–H plates, which is an easier and faster alternative for testing optical
surfaces, as well as an entire optical system. The proposed S–H plate can be constructed using
different techniques, depending on the application; one example was already mentioned with a
3D equipment or in our case, using microlithography equipment. The size of the pinholes and the
pitch between them are parameters that can be modified. An important parameter in the exper-
imental work is the distance between the S–H plate and the image plane for registering the
Shack–Hartmanngrams, mainly owing to the Fresnel diffraction phenomena.
Acknowledgments
The authors would like to thank Dr. Ismael Cosme Bolaños and the laboratory’s technicians at
the INAOE Electronics Department for producing the S–H plates. The authors were also grateful
to Guadalupe Flores Serrano for the editorial job. Dr. Eduardo Tepichín Rodríguez and to the
Image Sciences and Vision Physics group for using his S–H filter with the wavefront sensor
Thorlab WFS150C. When this study was conducted, Dr. Efrén Santamaría Juárez was in the
INAOE Optics Department. We would like to thank Editage for English language editing.
The authors declare that they have no competing financial interests or personal relationships
that could influence the work reported in this paper.
References
1. A. J. Núñez, E. S. Juárez, and A. C. Rodríguez, “Shack–Hartmann test using micro-holes for
testing a concave optical surface,”in Front. Opt./ Laser Sci., p. JW6A.7, Optica Publishing
Group (2020).
Fig. 10 Ronchi pattern (a) inside of cc and (b) outside the cc; a Ronchi ruling with 20 lines/cm was
used.
Fig. 9 Spotfield pattern (a) inside to cc and (b) outside the cc; with a commercial Shack–Hartmann
with lenses Thorlabs WFS150C.
Nunez et al.: Testing concave optical surface by Shack–Hartmann test using microholes
Optical Engineering 104104-6 October 2022 •Vol. 61(10)
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 29 Nov 2022
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
2. E. Hecht, Optics, Ch. 3, 4th ed., Addison-Wesley (2002).
3. B. C. Platt and R. V. Shack, “Lenticular Hartmann screen,”Opt. Sci. Cent. Newsl. 5,15–16
(1971).
4. J. Hartmann, “Bemerkungen uber den Bau und die Justirung von Spektrograpen,”
Zt. Instrumentenkd. 20,48–57 (1900).
5. J. Hartmann, “Objektuvuntersuchungen,”Zt. Instrumentenkd. 24, 1 (1904).
6. V. Ronchi, “Le Frange di Combinazioni Nello Studio delle Superficie e dei Sistemi Ottici,”
Riv. Ottica Mecc. Precis. 2,9–35 (1923).
7. D. Malacara, Optical Shop Testing, Wiley Series in Pure and Applied Optics, Wiley (2007).
8. F. S. Granados-Agustín et al., “Analysis of the common characteristics of the Hartmann,
Ronchi, and Shack–Hartmann tests,”Optik 125(2), 667–670 (2014).
9. A. Cordero-Davila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Ronchi and Hartmann
tests with the same mathematical theory,”Appl. Opt. 31, 2370–2376 (1992).
10. A. Cordero-Dávila et al., “Ronchi test with a square grid,”Appl. Opt. 37, 672–675 (1998).
11. G. A. Chanan, “Design of the Keck observatory alignment camera,”Proc. SPIE 1036,
59–71 (1989).
12. G. Chanan et al., “Fresnel phasing of segmented mirror telescopes,”Appl. Opt. 50, 6283–
6293 (2011).
13. A. Polo et al., “Wavefront measurement for EUV lithography system through Hartmann
sensor,”Proc. SPIE 7971, 808–814 (2011).
14. C. Pannetier and F. Hénault, “Shack–Hartmann versus reverse Hartmann wavefront sensors:
experimental results,”Opt. Lett. 45, 1746–1749 (2020).
15. F. Hénault, “Fresnel diffraction analysis of Ronchi and reverse Hartmann tests,”J. Opt. Soc.
Am. A 35, 1717–1729 (2018).
16. E. Hecht, Optics, Ch. 10, 4th ed., Addison-Wesley (2002).
17. M. Born et al., Principles of Optics: Electromagnetic Theory of Propagation, Interference
and Diffraction of Light, Ch. 8, 7th ed., Cambridge University Press (1999).
18. J. E. A. Landgrave and J. R. Moya, “Effect of a small centering error of the Hartmann screen
on the computed wave-front aberration,”Appl. Opt. 25, 533–536 (1986).
Alberto Jaramillo Nunez received his BSc degree in physics in 1990 from the Benemerita
Universidad Autonoma de Puebla (BUAP) and his MSc degree in optics in 1992 from the
Instituto Nacional de Astrofísica, Optica y Electrónica (INAOE), and his PhD in sciences in
1996 from the INAOE. His major research interests are interferometry and optical
instrumentation.
Efren Santamaria Juarez received his BS degree in applied physics from the Meritorious
Autonomous University of Puebla in 2016 and his MS and PhD degrees in optics sciences from
the National Institute of Astrophysics, Optics, and Electronics during 2015 to 2019. He is a
postdoc at the Institute of Applied Sciences and Technology of the National Autonomous
University of Mexico. His major research interests are optical instrumentation, testing, and
metrology.
Javier Muñoz-López received his BSc degree in physics from the BUAP in 1992, his MSc
degree in optics from the INAOE in 1995, and his PhD in optics also from INAOE in 1999.
His research interests include scattering and optical instrumentation.
Alejandro Cornejo Rodriguez received his BS degree in physics from the UNAM in 1964, his
MS degree in optics from the University of Rochester, Rochester, New York, USA, in 1968, and
his PhD from Tokyo Institute of Technology, Tokyo, Japan, in 1982. He isan emeritus researcher
of the INAOE, Puebla, Mexico. His research interests include optical design, optical testing, and
instrumentation. He is a member of SPIE.
Nunez et al.: Testing concave optical surface by Shack–Hartmann test using microholes
Optical Engineering 104104-7 October 2022 •Vol. 61(10)
Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 29 Nov 2022
Terms of Use: https://www.spiedigitallibrary.org/terms-of-use