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Scanning Wavefront Sensor for Measurement of Highly Divergent Wavefronts
8th IFAC Symposium on Mechatronic Systems 8th IFAC Symposium on Mechatronic Systems 8th IFACAustria, Symposium Systems Vienna, Sept. on 4-6,Mechatronic 2019 8th IFAC Symposium on Mechatronic Systems Vienna, Available online at www.sciencedirect.com Vienna, Austria, Austria, Sept. Sept. 4-6, 4-6, 2019 2019 8th IFACAustria, Symposium Systems Vienna, Sept. on 4-6,Mechatronic 2019 Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-15 (2019) 25–30
Scanning Wavefront Sensor for Scanning Wavefront Sensor for Scanning Wavefront Sensor for Measurement of Highly Divergent Scanning Wavefront Sensor for Measurement of Highly Divergent Measurement of Highly Divergent Wavefronts Measurement of Highly Divergent Wavefronts Wavefronts Martin E.Wavefronts Fuerst ∗∗∗ Georg Schitter ∗∗∗
Martin E. Fuerst Georg Schitter Martin Martin E. E. Fuerst Fuerst ∗ Georg Georg Schitter Schitter ∗ ∗ ∗ ∗ Martin E. Fuerst Georg Schitter Doppler Laboratory for Precision Engineering for ∗ ∗ Christian Christian Doppler Laboratory for Precision Engineering for Christian Doppler Laboratory for Precision Engineering for ∗ Automated In-Line Metrology, Automation and Control Institute, Christian Doppler Laboratory for Precision Engineering for Automated In-Line Metrology, Automation and Control Institute, ∗ Automated In-Line Metrology, Automation and Control Institute, Christian Doppler Laboratory for Precision Engineering for Vienna University of Technology (e-mail: [email protected]) Automated In-Line Metrology, Automation and Control Institute, Vienna University of Technology (e-mail: [email protected]) Vienna University of [email protected]) Automated In-Line Metrology, (e-mail: Automation and Control Institute, Vienna University of Technology Technology (e-mail: [email protected]) Vienna University of Technology (e-mail: [email protected]) Abstract: This paper deals with the characterization of freeform optics by means of wavefront Abstract: This paper deals deals with with the the characterization characterization of of freeform freeform optics optics by by means means of of aaa wavefront wavefront Abstract: This paper sensor. Freeform optics show an increasing demand since they have the potential to improve Abstract: This paper deals with the characterization of freeform optics by means of a wavefront sensor. Freeform optics show an increasing demand since they have the potential to improve sensor. Freeform optics show an increasing demand since they have the potential to improve Abstract: This paper deals with the characterization of freeform optics by means of a wavefront optical system performance while reducing size, weight and complexity. To directly measure sensor. Freeform optics show an increasing demand since they have the potential to improve optical system system performance performance while while reducing reducing size, size, weight weight and and complexity. complexity. To To directly directly measure measure optical sensor. Freeform optics showwhile anoptical increasing since they have the potential to measure improve the performance of freeform part, a novel scanning wavefront sensor approach is optical system performance reducing size, weight and complexity. To directly the performance performance of aa a freeform freeform optical part,demand novel scanning wavefront sensor approach is the of optical part, aaameasuring novel scanning wavefront sensor approach is optical system performance while reducing size, weight and complexity. To directly measure proposed which is faster than a coordinate machine and more flexible than an the performance of a freeform optical part, novel scanning wavefront sensor approach is proposed which is faster than a coordinate measuring machine and more flexible than an proposed which is than a coordinate machine and more flexible than an the performance a freeform ameasuring novel scanning sensor approach is interferometer. challenges in the limited dynamic range of aa Shack-Hartmann sensor proposed whichThe isof faster faster than lie aoptical coordinate measuring machine more flexible than an interferometer. The challenges lie in the thepart, limited dynamic rangewavefront ofand Shack-Hartmann sensor interferometer. The challenges lie in limited dynamic range of a Shack-Hartmann sensor proposed which is faster than a coordinate measuring machine and more flexible than an concerning wavefront slope and curvature. Both limits can be overcome by repositioning the interferometer. The challenges lie in the limited dynamic range of a Shack-Hartmann sensor concerning wavefront wavefront slope slope and and curvature. Both Both limits can can be be overcome by by repositioning repositioning the the concerning interferometer. The challenges in dynamic range ofsetup a Shack-Hartmann sensor sensor and orienting it tangentially to the wavefront. An automatic is developed and first concerning wavefront andliecurvature. curvature. Both limits limits be overcome overcome repositioning the sensor and and orienting orienting itslope tangentially to the the limited wavefront. An can automatic setup isbydeveloped developed and first sensor it tangentially to the wavefront. An automatic setup is and first concerning wavefront and curvature. Both limits can be overcome repositioning the measurements are demonstrated on a highly divergent wavefront. It is shown that the dynamic sensor and orienting itslope tangentially the wavefront. An automatic isbydeveloped and first measurements are demonstrated demonstrated on to a highly highly divergent wavefront. Itsetup is shown shown that the dynamic dynamic measurements are on a divergent wavefront. It is that the sensor and orienting it tangentially to the wavefront. An automatic setup is developed and first range of the sensor is increased by scanning the sensor along the wavefront and orienting the measurements are demonstrated on a highly divergent wavefront. It is shown that the dynamic range of of the the sensor is is increased increased by by scanning scanning the the sensor along along the wavefront wavefront and and orienting orienting the the range measurements are to demonstrated on in a highly divergent wavefront. isare shown that the sensor wavefront each position. Measurements taken over aa dynamic range of range oftangential the sensor sensor isthe increased by scanning the sensor sensor along the theItwavefront orienting the sensor tangential to the wavefront in each position. position. Measurements are takenand over range of sensor tangential to the wavefront in each Measurements are taken over a range of ◦ ◦ range of the sensor is increased by scanning the sensor along the wavefront and orienting the ±15 , which is an improvement by a factor 5 compared to the typical dynamic range of ±3 for sensor tangential to the wavefront in each position. Measurements are taken over a range of ◦ ◦ ◦ , which is an improvement by a factor 5 compared to the typical dynamic range of ±3◦ for ±15 is improvement by compared to dynamic ±3 ±15 ◦ , which sensor tangential the wavefront each 55position. Measurements taken range over aof of aa static Shack-Hartmann sensor. , which is an an to improvement by a ainfactor factor compared to the the typical typicalare dynamic range ofrange ±3◦ for for ±15 static Shack-Hartmann sensor. ◦ a static Shack-Hartmann sensor. , which is an improvement by a factor 5 compared to the typical dynamic range of ±3◦ for ±15 a static Shack-Hartmann sensor. 2019, IFAC (International Federation of Automaticwavefront Control) Hosting by Elsevier Ltd. All system, rights reserved. a©static Shack-Hartmann sensor. Keywords: Freeform optics, optical metrology, measurement, scanning Keywords: Freeform optics, optical metrology, wavefront measurement, scanning system, Keywords: Freeform optics, optical metrology, wavefront measurement, scanning system, Shack-Hartmann sensor, surface metrology Keywords: Freeform optics, optical metrology, wavefront measurement, scanning system, Shack-Hartmann sensor, surface metrology Shack-Hartmann sensor, surface metrology Keywords: Freeform optics, optical metrology, wavefront measurement, scanning system, Shack-Hartmann sensor, surface metrology Shack-Hartmann sensor, surface metrology 1. INTRODUCTION ferometric 1. INTRODUCTION INTRODUCTION ferometric techniques techniques offer offer sub-nm sub-nm resolution resolution and and are are thus thus 1. ferometric techniques offer sub-nm resolution are aa well known solution for fast measurement of flats and 1. INTRODUCTION ferometric techniques offer sub-nm resolution and and are thus thus well known solution for fast measurement of flats and a well known solution for fast measurement of flats and 1. INTRODUCTION Freeform optics have a high potential for small packaging ferometric techniques offer sub-nm resolution and are thus spheres. Aspheric surfaces can also be measured with a well known solution for fast measurement of flats and Freeform optics have a high potential for small packaging spheres. Aspheric surfaces can also be measured with Freeform optics have a high potential for small packaging spheres. Aspheric surfaces can also be measured with size and high optical performance in light distribution and a well known solution for fast measurement of flats and interferometric techniques through the use of null-optics Freeform optics have a high potential for small packaging spheres. Aspheric surfaces can also be measured with size and and high high optical optical performance performance in in light distribution distribution and and interferometric techniques through the use of null-optics size interferometric techniques through the use of null-optics Freeform optics have performance a high potential for small packaging light detection applications. Assefa et al. (2018) demonAspheric surfaces can also be measured with (refractive or diffractive). However, this constitutes a huge size and high optical in light light distribution and spheres. interferometric techniques through the use of null-optics light detection applications. Assefa et al. (2018) demon(refractive or diffractive). However, this constitutes a huge light detection applications. Assefa et al. (2018) demonor However, constitutes a size and high optical performance in light and (refractive strate the potential of freeform optics energy interferometric techniques through this the tool, use of null-optics loss of universality in the measurement as a new surlight detection applications. Assefa et for al.distribution (2018) efficient demon(refractive or diffractive). diffractive). However, this constitutes a huge huge strate the potential of freeform optics for energy efficient loss of universality in the measurement tool, as a new surstrate the potential of freeform optics for energy efficient loss of universality in the measurement tool, as a new surlight detection applications. Assefa et al. (2018) demonillumination with LEDs, while Reimers et al. (2017) were (refractive or diffractive). However, this constitutes a huge face geometry requires a new null-lens [Kim et al. (2004)]. strate the potential of freeform optics for energy efficient loss of universality in the measurement tool, as a new surillumination with LEDs, while Reimers et al. (2017) were face geometry requires a new null-lens [Kim et al. (2004)]. illumination with LEDs, while Reimers et al. (2017) were face geometry requires a new null-lens [Kim et al. (2004)]. strate the potential of freeform optics for energy efficient able to design a freeform spectrometer that is five times loss of universality in the measurement tool, as a new surThe fabrication of an application-specific null-lens is costly illumination with LEDs, while Reimers et al. (2017) were face geometry requires a new null-lens [Kim et al. (2004)]. able to to design design aa freeform freeform spectrometer spectrometer that that is is five five times times The fabrication of an application-specific null-lens is costly able The fabrication of an application-specific null-lens is costly illumination with LEDs, while Reimers et al.designs. were face more compact than comparable conventional Curgeometry requires a new null-lens [Kim et al. (2004)]. able to design a freeform spectrometer that is(2017) five times and also introduces additional uncertainties (alignment, The fabrication of an application-specific null-lens is costly more compact than comparable conventional designs. Curand also introduces additional uncertainties (alignment, more compact than comparable conventional designs. Curalso additional uncertainties (alignment, able to designinvestigates a freeform spectrometer that is five optics times rent research the potential of freeform The fabrication of anresidual application-specific null-lens is costly fabrication quality, aberrations) to the measuremore compact than comparable conventional designs. Cur- and and also introduces introduces additional uncertainties (alignment, rent research research investigates the potential of freeform optics fabrication quality, residual aberrations) to the measurerent investigates the potential of freeform optics fabrication quality, residual aberrations) to the measuremore compact than comparable conventional designs. Curin the micro scale, e.g. aspheric contact lenses [Li and and also introduces additional uncertainties (alignment, ment procedure. Also, interferometric methods are very rent research investigates the potential of freeform optics fabrication quality, residual aberrations) to the measurein the the micro micro scale, scale, e.g. e.g. aspheric aspheric contact contact lenses lenses [Li [Li and and ment procedure. Also, interferometric methods are very in ment procedure. Also, interferometric methods are very rent research investigates the potential of freeform optics Fang (2019)] and the macro scale, e.g. telescope mirrors fabrication quality, residual aberrations) to the measuresensitive to vibrations and thus difficult to use in industrial in the micro scale, e.g. aspheric contact lenses [Li and ment procedure. Also, interferometric methods are very Fang (2019)] (2019)] and and the the macro macro scale, scale, e.g. e.g. telescope telescope mirrors mirrors sensitive to vibrations and thus difficult to use in industrial Fang sensitive to vibrations and thus difficult to use in industrial in the micro scale, e.g. aspheric contact lenses [Li and [Graves et al. (2019)]. In the first case, medical benefits are ment procedure. Also, interferometric methods are very Fang (2019)] and the macro scale, e.g. telescope mirrors environments. sensitive to vibrations and thus difficult to use in industrial [Graves et al. (2019)]. In the first case, medical benefits are environments. [Graves et al. (2019)]. In the first case, medical benefits are environments. Fang (2019)] and the macro scale, e.g. telescope mirrors expected, while in the second case, astronomical images sensitive to vibrations and thus difficult to use in industrial [Graves et al. (2019)]. In the first case, medical benefits are environments. expected, while while in in the the second case, case, astronomical astronomical images images Coordinate measuring machines (CMMs) are very flexexpected, [Graves et while al. (2019)]. themay firstcase, case, medicalthrough benefits are environments. of unprecedented be achieved the Coordinate measuring machines (CMMs) are very flexexpected, in quality theInsecond second astronomical images Coordinate measuring machines (CMMs) are very flexof unprecedented quality may be achieved through the ible and can be equipped with tactile or optical Coordinate measuring machines (CMMs) are very nonflexof unprecedented quality may be achieved through the expected, while in the second case, astronomical images reduction of aberrations. ible and can be equipped with tactile or optical nonof unprecedented quality may be achieved through the ible and can be equipped with tactile or optical nonreduction of of aberrations. aberrations. Coordinate measuring machines (CMMs) are very flexcontact probes. For optical (possibly coated) parts, nonible and can be equipped with tactile or optical reduction of unprecedented quality may be achieved through the contact probes. For optical (possibly coated) parts, nonreduction of aberrations. contact probes. For optical (possibly coated) parts, nonThe high-precision manufacturing of freeform parts relies ible and can be equipped with tactile or optical probes are favoured as they do not damage the contact probes. For optical (possibly coated) parts, nonThe high-precision high-precision manufacturing of of freeform freeform parts parts relies relies contact reduction of aberrations. probes are favoured as they do not damage the The manufacturing contact probes are favoured as they do not damage the on high-precision metrology [Fang et al. (2013)]. In gencontact probes. For optical (possibly coated) parts, nonsurface under test. Henselmans et al. (2011) developed The high-precision manufacturing of freeform parts relies probes are favoured as they do not damage the on high-precision high-precision metrology metrology [Fang [Fang et et al. al. (2013)]. (2013)]. In In gengen- surface under test. Henselmans et al. (2011) developed on surface under test. Henselmans et al. (2011) developed The high-precision manufacturing of freeform parts relies eral, there is a large number of measurement methods for contact probes are favoured as they do not damage the on high-precision metrology [Fang et al. (2013)]. In gena non-contact measurement machine for freeform optics surface under test. Henselmans et al. (2011) developed eral, there is a large number of measurement methods for aa non-contact measurement machine for freeform optics eral, there is a large number of measurement methods for non-contact measurement machine for freeform optics on high-precision metrology [Fang et al. (2013)]. In gennon-optical freeform surfaces such as turbine blades, ausurface under test. Henselmans et al. (2011) developed which resembles a giant CD-player and utilizes a probe eral, there is a large number of measurement methods for a non-contact measurement machine for freeform optics non-optical freeform freeform surfaces surfaces such such as as turbine turbine blades, blades, auau- which resembles aa giant CD-player and utilizes a probe non-optical resembles and a probe eral, thereparts isfreeform a large number ofsuch measurement methods for which tomotive or airplane fuselage: tactile and optical coabased non-contact measurement machine for utilizes freeform on the differential confocal method. Compared to non-optical surfaces as turbine blades, auwhich resembles a giant giant CD-player CD-player and utilizes a optics probe tomotive parts or airplane fuselage: tactile and optical cobased on the differential confocal method. Compared to tomotive parts or airplane fuselage: tactile and optical cobased on the differential confocal method. Compared to non-optical freeform surfaces such as turbine blades, auordinate measuring machines, interferometric and deflecwhich resembles a giant CD-player and utilizes a probe interferometric methods, which are typically “one-shot” tomotive parts or airplane fuselage: tactile and optical cobased on the differential confocal method. Compared to ordinate measuring measuring machines, machines, interferometric interferometric and and deflecdeflec- interferometric methods, which are typically “one-shot” ordinate interferometric methods, which are typically “one-shot” tomotive parts or airplane fuselage: tactile and optical cotometric techniques, profilometry and laser trackers [Savio based on the differential confocal method. Compared measurements, CMMs are relatively slow as they have to ordinate measuring machines, interferometric and deflecinterferometric methods, which are typically “one-shot” tometric techniques, profilometry and laser trackers [Savio measurements, CMMs are relatively slow as they have to tometric techniques, profilometry and laser trackers [Savio measurements, CMMs are relatively slow as have ordinate measuring machines, interferometric and deflecet al. (2007)]. However, only few of these are methods, arepoint. typically “one-shot” acquire surface geometries point by For rotationally tometric techniques, profilometry and laser techniques trackers [Savio measurements, CMMs arewhich relatively slow as they they have to to et al. al. (2007)]. (2007)]. However, only few few of these these techniques are interferometric acquire surface geometries point by point. For rotationally et However, only of techniques are acquire surface geometries point by point. For rotationally tometric techniques, profilometry and laser trackers [Savio suitable for freeform optics because optical parts typically measurements, CMMs are relatively slow as they have to symmetric parts, this can be mitigated by scanning a single et al. (2007)]. However, only few of these techniques are acquire surface geometries point by point. For rotationally suitable for for freeform optics optics because because optical parts parts typically symmetric parts, this can be mitigated by scanning a single suitable −5ofoptical symmetric parts, this can be mitigated by scanning a single et al. (2007)]. However, only few these techniques are acquire require very fine tolerances (< 10 are typically medium surface geometries point by point. For rotationally −5 line; for freeform surfaces, the whole geometry has to be suitable for freeform freeform optics because optical parts typically typically symmetric parts, this can be mitigated by scanning a single −5 ), require very fine tolerances (< 10 ), are typically medium line; for for freeform freeform surfaces, surfaces, the the whole whole geometry geometry has has to to be be require very fine tolerances (< 10 ), are typically medium line; suitable for freeform optics because parts typically sized (range of mm to m) and are [Savio symmetric parts, this can be mitigated bytimes scanning ),optical aretransparent typically medium scanned, resulting in long measurement (minutes to require very fine tolerances (< 10−5often line; for freeform surfaces, the whole geometry hasa single to be sized (range of mm to m) and are often transparent [Savio scanned, resulting in long measurement times (minutes to −5 sized (range of mm to m) and are often transparent [Savio scanned, resulting in long measurement times (minutes to require very fine tolerances (< 10 ), are typically medium et al. (2007)]. line; for freeform surfaces, the whole geometry has to be sized (range of mm to m) and are often transparent [Savio hours). Another recent example of an optical, scanning scanned, resulting in long measurement times (minutes to et al. (2007)]. hours). Another recent example of an optical, scanning et al. (2007)]. hours). Another recent example of an optical, scanning sized (range of mm to m) and are often transparent [Savio scanned, resulting in long measurement times (minutes to et al. (2007)]. method for freeform surface metrology is reported [Zapico hours). Another recent example of an optical, scanning method for freeform surface metrology is reported [Zapico The main methods to be considered are coordinate method for freeform surface metrology reported [Zapico The two main methods to be considered are coordinate et al.two (2007)]. hours). Another recent example of anis optical, scanning method for freeform surface metrology is reported [Zapico The two main methods to be considered are coordinate measuring machines and InterThe two main methods tointerferometric be consideredmethods: are coordinate measuring machines and interferometric methods: Intermethod for freeform surface metrology is reported [Zapico measuring machines and InterThe two main methods be consideredmethods: are coordinate measuring machines andtointerferometric interferometric methods: Intermeasuring machines and interferometric 2405-8963 © © 2019 2019, IFAC (International Federation methods: of AutomaticInterControl) Copyright IFAC 98 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 98 Copyright © under 2019 IFAC IFAC 98 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 98 10.1016/j.ifacol.2019.11.644 Copyright © 2019 IFAC 98
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et al. (2018)], using a conoscopic holography sensor for onmachine-measurement of CNC-machined freeform parts. Fang et al. (2013) point out a second way of freeform part metrology: Instead of measuring the freeform part surface, the freeform part performance can be measured. This can be achieved by measuring the optical wavefront that is transmitted or reflected by an optical system, for instance with a Shack-Hartmann sensor. A ShackHartmann sensor is a camera-based optical wavefront sensor that allows to measure a region of the wavefront reflected or transmitted by an optical system [Platt and Shack (2001)]. It consists of a lenslet array that segments the incoming wavefront onto an imaging chip, where the positions of the focussed spots are detected. From this spot pattern, the wavefront shape can be reconstructed, as long as the curvature of the incoming wavefront is not large enough to create ambiguities [Rockt¨ aschel and Tiziani (2002)]. Shack-Hartmann sensors are small, robust to vibrations and fast, so they are well suited for application in industrial environments [Thier et al. (2013)]. Their capability to characterize lenses that are bigger than the sensor aperture was recently demonstrated [Burada et al. (2017), Pant et al. (2015)]. This was done by manual, linear repositioning of the sensor and subsequent stitching of the acquired wavefront subaperture images.
Fig. 1. The working principle of a Shack-Hartmann wavefront sensor. a) Each lenslet forms a focussed spot on the image sensor. The deviations of the spots from their reference positions can be identified. b) An angle α between wavefront and sensor causes the spots to leave their subapertures. c) High wavefront curvature leads to ambiguities in spot assignment.
The contribution of this paper is to propose an automatic setup that allows for arbitrary repositioning and reorientation of a Shack-Hartmann sensor. As discussed in section 2, this allows the measurement of arbitrary wavefront shapes and is especially useful for highly divergent wavefronts because the wavefront curvature automatically decreases with an increased measurement distance. In combination with a reliable stitching algorithm, a measurement of the complete, global, wavefront exiting a freeform optic allows characterization of the part performance. Compared to an interferometric approach, this technique does not need any null-optics and is less sensitive to environmental vibrations. Compared to a coordinate measuring machine, the Scanning Shack-Hartmann Sensor (SSHS) approach offers higher speed and avoids damaging the part (compared to a tactile CMM). In section 3, an automatic setup is constructed and wavefront measurements on a highlydivergent optic are taken. The device under test for this demonstration is not a freeform optic, but its numerical aperture (NA) exceeds the dynamic range of a static Shack-Hartmann sensor and thus serves to demonstrate the measurement principle. In section 4, it is demonstrated that the dynamic range of the Shack-Hartmann sensor is increased by rotational repositioning. Further it is shown that aberrations can be visualized by numerically subtracting the dominant low-order aberrations tip, tilt and defocus from individual subaperture recordings. Finally, section 5 concludes the paper and briefly outlines future work.
in Figure 1 a). From the recorded spot pattern, local slopes can be calculated and then used to reconstruct the wavefront shape incident on the sensor [Platt and Shack (2001)]. Shack-Hartmann sensors offer a high measurement speed, are relatively robust to vibrations and have subwavelength resolution [Thier et al. (2013)]. The Shack-Hartmann sensor’s dynamic range is limited by two wavefront characteristics: wavefront slope and wavefront curvature [Rockt¨ aschel and Tiziani (2002)]. Wavefront slope describes the average angle between wavefront and sensor aperture. If this angle is too high, the lenslet array does not form spot images on the sensor chip (see Figure 1 b)). Spot deviation ∆x can be related to incidence angle α and lenslet focal length with the relation ∆x = fL · tan(α) (1) (adapted from [Chernyshov et al. (2005)]). To ensure that spots are registered within their respective subapertures, this limits the maximum allowed incidence angle to PL αmax = (2) 2 · fL (with lenslet pitch PL as in Fig. 1 a)). For typical ShackHartmann sensor parameters (Pl = 100 − 200µm, fl = 3 − 10mm), αmax is only a few degrees (1-5◦ ). Wavefront curvature describes the local variation of wavefront slope. If the curvature is too high, lenslets may form spots in the same position on the image chip, leading to ambiguities (see Figure 1 c)).
2. SCANNING SHACK-HARTMANN SENSOR A Shack-Hartmann sensor (Figure 1) consists of a microlens array and an image sensor. Each microlens produces a focused spot on the image sensor, the position of the spot being dependent on the local slope of the wavefront incident on the microlens. This is illustrated
Figure 2 shows a schematic of a transmissive freeform lens. Instead of measuring the surface geometry of the lens, it is proposed to measure the resulting wavefront. This constitutes a measurement of part performance instead of 99
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rot. stage and SHS
lower curvature incoming wavefront
high curvature region
27
Shack-Hartmann sensor
fiber coupler
z-stage
high-NA lens
sensor aperture d
freeform optics
scan trajectory
focal distance
x-stage
measuring distance rmeas
Fig. 2. The proposed measurement strategy for freeform optics utilizing a scanning wavefront sensor. The incoming, collimated light is shaped by the optics and the resulting wavefront is observed. part surface, thus evading the insecurities typically related to measuring surfaces in order to predict performance. The Shack-Hartmann sensor is limited by its finite aperture size and its dynamic range considering maximum wavefront curvature. Both limitations could be overcome by the use of additional optical components to collimate and resize the wavefront. However, the added optical components have to be very specific and add additional aberrations and alignment problems [Fuerst et al. (2018)]. With a scanning setup, wavefront slope and curvature can be dealt with by increasing the measurement distance and repositioning the Shack-Hartmann sensor along the expected wavefront shape. Considering a spherical wave, the incidence angle α (comp. Fig. 1 b)) between Shack-Hartmann sensor and wavefront is related to the measurement distance rmeas (comp. Fig. 2))as rmeas =
d 2 · sin α
(3)
with d being the diameter of the sensor aperture. It follows, that the angle α drastically decreases when the ratio rmeas d is increased: α = arcsin
d 2 · rmeas
Fig. 3. Schematic view of the setup used to investigate the proposed measurement strategy for freeform optics by testing it on a high-NA optic. A laser diode is coupled to an optical fiber and the light exiting the fiber at a fiber coupler (FC) is used to illuminate a highNA lens. The resulting, highly divergent wavefront is measured with the scanning Shack-Hartmann sensor.
(4)
For typically acceptable incidence angles of below 5◦ , a measurement distance of at least 5 times the sensor aperture is required. To keep the incidence angle below 3◦ , the sensor is placed 10 times its aperture diameter from the spherical wave origin. Instead of optical insecurities, this adds mechanical insecurities that need to be evaluated for the system. Once these are known, the same measurement system can be used for a large range of optical parts without the need for any null-optics or reference shapes. To fully scan the wavefront produced by a freeform lens, a 5-degrees-of-freedom positioning system is required. Position data and wavefront data is required to reconstruct the global wavefront. In case of rotational symmetry of the part (e.g. aspherical lenses), 3 degrees of freedom are sufficient as in that case, a scan along a single line crossing optical axis is sufficient. Therefore, a first experimental 3DOF-system was constructed to investigate a rotationally symmetric lens, as described in the following section. 100
Fig. 4. The setup used for first investigations of the proposed measurement strategy. A 635 nm laser diode was coupled to a single-mode fiber to provide a monochromatic near-point light source for the highNA lens. The Shack-Hartmann sensor is mounted on two motorized linear stages and one motorized rotary stage. 3. EXPERIMENTAL SETUP To investigate the measurement capabilities of the 3degrees-of-freedom scanning Shack-Hartmann setup, a rotationally symmetric, highly divergent wavefront is investigated. Such highly divergent wavefronts occur for instance in high-resolution microscopy or confocal displacement sensors. The lateral resolution of these applications is dictated by the numerical aperture (NA) of the lens or lens stack which is proportional to the divergence angle of the focussed beam. Since high numerical apertures are needed for small measuring spots, the resulting wavefront is highly curved and thus exceeds the dynamic range of a single, static Shack-Hartmann sensor. With the Scanning Shack-Hartmann Sensor (SSHS), these limitations can be overcome.
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3.1 Experiments With the constructed 3-degrees-of-freedom setup, the wavefront generated by a high-NA lens is investigated. A 635 nm laser diode was coupled to a single-mode optical fiber. The other end of the fiber was attached to a fiber coupler without collimating optics, thus serving as a monochromatic near-point light source for the high-NA lens. A circular trajectory was chosen to scan the wavefront as that shape is close to the expected (ideal) shape of the investigated wavefront. Figure 5 illustrates the reasons for choosing a circular scanning trajectory and orienting the sensor tangentially to the wavefront: To scan a divergent wavefront, the sensor has to be moved from the optical axis of the optic under test. This position can be characterized by an angle α. For a linear scan trajectory, using a single, linear stage, the incidence angle β between wavefront and sensor aperture equals α, typically exceeding the dynamic range of a Shack-Hartmann sensor for angles of approximately 3 − 5◦ . Addition of a rotational stage on which the sensor is mounted removes this limitation, as the stage angle θ can be adjusted to equal α, keeping β close to zero. In this case, however, the measurement distance would increase with α. To keep the measurement distance (distance between wave origin and sensor) constant, a second linear stage (Z-direction in Fig. 5) is added and a circular scan trajectory is chosen. The radius of the scanning trajectory was chosen to be large enough that the wavefront curvature incident on the sensor is small enough to not exceed the dynamic range of the Shack-Hartmann sensor (compare Equation 4) and the spacing of the measurement positions was chosen so that the subapertures overlap. The Shack-Hartmann sensor was moved to 11 discrete positions spaced 3◦ apart in a distance of 54 mm to the focal point of the high-NA lens (the “origin” of a spherical wave). At each position, the Shack-Hartmann sensor is automatically pointed towards the expected position of the focal point and an image is recorded. For the presented experiments, this was done in a purely feed-forward controlled way and with coarse, 101
Fig. 5. Comparison of a linear and a circular scan trajectory. For sensor positions off the optical axis, an angle β between sensor and wavefront occurs. This can be negated by rotating the sensor by an angle θ. Note that α = β = θ.
2 1
100 50 0 -50
y [mm]
Figure 3 depicts the selected use case and the scanning setup. The setup consists of two motorized linear stages and one motorized rotary stage, meaning the ShackHartmann sensor can be positioned with 3 degrees of freedom (moving along a 2D-trajectory with arbitrary orientation). As this setup does not provide enough degrees of freedom to scan a fully freeform wavefront without any symmetries, a highly divergent, rotationally symmetric, wavefront was chosen as a test case. The linear stages (VT-80, PI Physik Instrumente, Braunschweig, Germany) provide a travel range of 200 mm and 50 mm, respectively, while their internal encoders have a resolution of 500 nm. The rotary stage (ELL-8M model from Thorlabs, USA) can be rotated by 360◦ , and offers an encoder resolution of 1.4 mdeg. A Shack-Hartmann sensor (model HR-2 from Optocraft, Erlangen, Germany) is used for the wavefront measurements. It has a rectangular aperture of 11.2 × 7 mm over which 85 × 53 lenslets with 130 µm pitch are distributed and its measurement rate is up to 18 Hz. Figure 4 shows the setup as assembled in the laboratory.
wavefront [ ]
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0 dPV
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0 x [mm]
1
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Fig. 6. An exemplary wavefront image from the measurement position -15◦ . The image is dominated by the low-order aberrations tip, tilt and defocus. The difference between the highest and the lowest point (peak-to-valley) is P V = 225λ = 142 µm and the distance between these two values is dP V = 4.044 mm. manual alignment. As dicussed in the results section, the information collected by such a coarse run can be used to automatically improve the scanning trajectory for another measurement. From the spot patterns, local wavefront gradients (slopes) are calculated and a 2D-polynomial function is fitted to match the observations (least-square-fit). Next, the wavefront shape is decomposed into Zernike base polynomials and their respective coefficients are calculated. Zernike polynomials are widely used in optics since they directly relate to basic optical aberrations such as tip, tilt, defocus, coma or astigmatism [Dai (2008)]. This is done to analyze the recorded wavefront images and also allows to numerically compensate for low-order aberrations. 4. RESULTS AND DISCUSSION Figure 6 shows a single measured subaperture on the described trajectory. The imaged wavefront subaperture resembles part of a sphere, which is expected when measuring on a divergent wavefront. From the detected peak-tovalley variation of P V = 225λ = 142 µm and the measured
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Low order Zernike modes 100
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Fig. 7. A map of residual wavefront errors, when the loworder aberrations tip, tilt and defocus are subtracted from the recorded wavefront shown in Figure 6. All measurements were taken for a single wavelength (635 nm).
Fig. 8. Recorded low order Zernike coefficients via measurement position. The defocus- and tilt- (Y) coefficients are approximately constant while the tip (x) coefficient varies linearly. This can be explained by positioning errors.
distance from peak to valley dP V = 4.044 mm, the radius R of the observed sphere can be calculated: d2 + P V 2 = 57.66mm (5) R = PV 2 · PV
Figure 8 compares the calculated low-order Zernike coefficients for each individual measurement position. These low order Zernike polynomials describe the wavefront tilt (X and Y) and defocus (deviation from the plane wave), as observed in the wavefront region incident on the sensor. Since a circular trajectory was chosen for the experiments and a divergent wavefront is under test, a constant defocus component is expected and can be observed (fig. 8, top line). The constant tilt (Y) coefficient can be explained by the scanning wavefront sensor and the optic under test not being set up at the exact same height from the optical table. This means that the optical axis of the high-NA lens is not in the same plane as the circular trajectory which is also visible in Figure 6. The linearly varying tilt (X) coefficient results from a mismatch of the center of the scanning trajectory and the focal point along the optical axis. As a next step, this could be automatically compensated by re-calculating the measurement positions and repeating the scanning measurement. The presented low order aberrations result from geometrical mismatches and misalignments and can be identified as such. To evaluate the optical quality of the system under test, higher order aberrations need to be analyzed. This can be done by correcting each individual subaperture as shown exemplarily in Figure 7 and interpreting this set of 102
Higher-order Zernike modes
0.1
Zernike coeff. [ ]
This is in agreement with the chosen measurement distance. Zernike analysis shows that the image is dominated by tip, tilt and defocus components. Defocus is responsible for the sphericity of the image while tip and tilt cause the visible shift of the center of the image. By subtracting these low-order aberration terms, the aberrations superimposed on the general sphere shape can visualized. This is shown in Figure 7, where only the residual wavefront error is visible. Peak-to-valley, this residual error is 0.2 λ for the presented subaperture, indicating a good quality lens.
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Astigmatism 0° Astigmatism 45° Koma x Koma y Spherical aberration
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sensor position [°] Fig. 9. The higher-order Zernike coefficients for astigmatism, coma and spherical aberration vary with the measurement position, but are relatively small in magnitude. images. Figure 9 presents the single-subaperture higherorder aberration coefficients over measurement position. It can be seen that these higher-order aberrations are consistently much smaller in magnitude than the low order aberrations but do increase towards the ends of the measurement range. However, since the coefficients are calculated for individual subapertures, they are not representative for the complete (global) wavefront (e.g. a coma aberration in a single subaperture does not indicate a coma aberration of the global wavefront). A more meaningful metric is the residual wavefront error, which is the peakto-valley number of wavefront error when the low order aberrations are subtracted. This residual error is below 0.7 λ for all measured positions, providing a first estimate of the optical system’s quality. For more valid measurements, the wavefront information of all subapertures needs to be combined with each other and the positioning data, which is part of ongoing work.
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In summary, it is shown that the dynamic range limitations of the Shack-Hartmann sensor can be overcome by repositioning and tangential orientation of the Shack-Hartmann sensor to acquire wavefront images of subapertures that can be further processed to obtain the global wavefront information. 5. CONCLUSION The Scanning Shack-Hartmann approach offers a versatile, flexible way to measure freeform optics with reasonable measurement accuracy and measurement times. A first automatic setup is proposed and demonstrated to show the feasibility of the concept by taking measurements on a highly divergent wavefront. It is shown that the dynamic range of the Shack-Hartmann sensor can be increased by repositioning the sensor with respect to the wavefront. Future investigation is focussed on the tradeoffs between measurement speed, positioning accuracy, and measurement resolution. Next steps will be automatic correction of misalignments and evaluation of the global wavefront shape. For this purpose, stitching algorithms [Chen et al. (2017)], that incorporate both the measurement data and the position data, have to be developed. ACKNOWLEDGEMENTS The financial support by the Christian Doppler Research Association, the Austrian Federal Ministry for Digital and Economic Affairs, and the National Foundation for Research, Technology and Development, as well as MICROEPSILON MESSTECHNIK GmbH & Co. KG and ATENSOR Engineering and Technology Systems GmbH is gratefully acknowledged. REFERENCES Assefa, B.G., Pekkarinen, M., Saastamoinen, T., Biskop, J., Kuittinen, M., Turunen, J., and Saarinen, J. (2018). Design and characterization of 3D-printed freeform lenses for random illuminations. Light-Emitting Diodes: Materials, Devices, and Applications for Solid State Lighting XXII, (February), 53. doi:10.1117/12.2288223. Burada, D.R., Pant, K.K., Bichra, M., Khan, G.S., Sinzinger, S., and Shakher, C. (2017). Experimental investigations on characterization of freeform wavefront using Shack–Hartmann sensor. Optical Engineering, 56(08), 1. doi:10.1117/1.OE.56.8.084107. Chen, S., Xue, S., Wang, G., and Tian, Y. (2017). Subaperture stitching algorithms: A comparison. Optics Communications, 390(January), 61–71. doi: 10.1016/j.optcom.2016.12.067. Chernyshov, A., Sterr, U., Riehle, F., Helmcke, J., and Pfund, J. (2005). Calibration of a Shack-Hartmann sensor for absolute measurements of wavefronts. Applied Optics, 44(30), 6419–6425. doi:10.1364/AO.44.006419. Dai, G.M. (2008). Wavefront optics for vision correction Vol. 179. Bellingham: SPIE press. Fang, F.Z., Zhang, X.D., Weckenmann, A., Zhang, G.X., and Evans, C. (2013). Manufacturing and measurement of freeform optics. CIRP Annals - Manufacturing Technology, 62(2), 823–846. doi:10.1016/j.cirp.2013.05.003. 103
Fuerst, M., Unger, S., Ito, S., and Schitter, G. (2018). Wavefront measurement based feedback control for automatic alignment of a high-NA optical system. Journal of Physics: Conference Series, 1065, 32001. doi: 10.1088/1742-6596/1065/3/032001. Graves, L.R., Smith, G.A., Apai, D., and Kim, D.W. (2019). Precision Optics Manufacturing and Control for Next - Generation Large Telescopes. Nanomanufacturing and Metrology, (0123456789). doi:10.1007/s41871-019-00038-2. URL https://doi.org/10.1007/s41871-019-00038-2. Henselmans, R., Cacace, L.A., Kramer, G.F.Y., Rosielle, P.C.J.N., and Steinbuch, M. (2011). The NANOMEFOS non-contact measurement machine for freeform optics. Precision Engineering, 35(4), 607–624. doi: 10.1016/j.precisioneng.2011.04.004. Kim, B.C., Saiag, T., Wang, Q., Soons, J., Polvani, R.S., and Griesmann, U. (2004). The Geometry Measuring Machine ( GEMM ) Project at NIST. Proceedings of the 2004 ASPE Winter Top. Meeting, North Carolina, USA, 3–6. Li, Q. and Fang, F. (2019). Advances and challenges of soft contact lens design for myopia control. Applied Optics, 58(7), 1639. doi:10.1364/AO.58.001639. Pant, K.K., Burada, D.R., Bichra, M., Singh, M.P., Ghosh, A., Khan, G.S., Sinzinger, S., and Shakher, C. (2015). Subaperture stitching for measurement of freeform wavefront. Applied Optics, 54(34), 10022. doi: 10.1364/AO.54.010022. Platt, B.C. and Shack, R. (2001). History and Principles of Shack-Hartmann Wavefront Sensing. Journal of Refractive Surgery, 17(5), S573–S577. doi:10.3928/1081597x-20010901-13. Reimers, J., Bauer, A., Thompson, K.P., and Rolland, J.P. (2017). Freeform spectrometer enabling increased compactness. Light: Science & Applications, 6(7), e17026. doi:10.1038/lsa.2017.26. Rockt¨aschel, M. and Tiziani, H.J. (2002). Limitations of the Shack-Hartmann sensor for testing optical aspherics. Optics and Laser Technology, 34(8), 631–637. doi: 10.1016/S0030-3992(02)00069-5. Savio, E., De Chiffre, L., and Schmitt, R. (2007). Metrology of freeform shaped parts. CIRP Annals - Manufacturing Technology, 56(2), 810–835. doi: 10.1016/j.cirp.2007.10.008. Thier, M., Paris, R., Thurner, T., and Schitter, G. (2013). Low-latency shack-hartmann wavefront sensor based on an industrial smart camera. IEEE Transactions on Instrumentation and Measurement, 62(5), 1241–1249. doi:10.1109/TIM.2012.2223333. Zapico, P., Pati˜ no, H., Vali˜ no, G., Fern´andez, P., and Rico, J.C. (2018). CNC centralized control for digitizing freeform surfaces by means of a conoscopic holography sensor integrated in a machining centre. Precision Engineering, 55(November 2018), 474–483. doi: 10.1016/j.precisioneng.2018.11.001.
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IFAC PapersOnLine 52-15 (2019) 25–30
Scanning Wavefront Sensor for Scanning Wavefront Sensor for Scanning Wavefront Sensor for Measurement of Highly Divergent Scanning Wavefront Sensor for Measurement of Highly Divergent Measurement of Highly Divergent Wavefronts Measurement of Highly Divergent Wavefronts Wavefronts Martin E.Wavefronts Fuerst ∗∗∗ Georg Schitter ∗∗∗
Martin E. Fuerst Georg Schitter Martin Martin E. E. Fuerst Fuerst ∗ Georg Georg Schitter Schitter ∗ ∗ ∗ ∗ Martin E. Fuerst Georg Schitter Doppler Laboratory for Precision Engineering for ∗ ∗ Christian Christian Doppler Laboratory for Precision Engineering for Christian Doppler Laboratory for Precision Engineering for ∗ Automated In-Line Metrology, Automation and Control Institute, Christian Doppler Laboratory for Precision Engineering for Automated In-Line Metrology, Automation and Control Institute, ∗ Automated In-Line Metrology, Automation and Control Institute, Christian Doppler Laboratory for Precision Engineering for Vienna University of Technology (e-mail: [email protected]) Automated In-Line Metrology, Automation and Control Institute, Vienna University of Technology (e-mail: [email protected]) Vienna University of [email protected]) Automated In-Line Metrology, (e-mail: Automation and Control Institute, Vienna University of Technology Technology (e-mail: [email protected]) Vienna University of Technology (e-mail: [email protected]) Abstract: This paper deals with the characterization of freeform optics by means of wavefront Abstract: This paper deals deals with with the the characterization characterization of of freeform freeform optics optics by by means means of of aaa wavefront wavefront Abstract: This paper sensor. Freeform optics show an increasing demand since they have the potential to improve Abstract: This paper deals with the characterization of freeform optics by means of a wavefront sensor. Freeform optics show an increasing demand since they have the potential to improve sensor. Freeform optics show an increasing demand since they have the potential to improve Abstract: This paper deals with the characterization of freeform optics by means of a wavefront optical system performance while reducing size, weight and complexity. To directly measure sensor. Freeform optics show an increasing demand since they have the potential to improve optical system system performance performance while while reducing reducing size, size, weight weight and and complexity. complexity. To To directly directly measure measure optical sensor. Freeform optics showwhile anoptical increasing since they have the potential to measure improve the performance of freeform part, a novel scanning wavefront sensor approach is optical system performance reducing size, weight and complexity. To directly the performance performance of aa a freeform freeform optical part,demand novel scanning wavefront sensor approach is the of optical part, aaameasuring novel scanning wavefront sensor approach is optical system performance while reducing size, weight and complexity. To directly measure proposed which is faster than a coordinate machine and more flexible than an the performance of a freeform optical part, novel scanning wavefront sensor approach is proposed which is faster than a coordinate measuring machine and more flexible than an proposed which is than a coordinate machine and more flexible than an the performance a freeform ameasuring novel scanning sensor approach is interferometer. challenges in the limited dynamic range of aa Shack-Hartmann sensor proposed whichThe isof faster faster than lie aoptical coordinate measuring machine more flexible than an interferometer. The challenges lie in the thepart, limited dynamic rangewavefront ofand Shack-Hartmann sensor interferometer. The challenges lie in limited dynamic range of a Shack-Hartmann sensor proposed which is faster than a coordinate measuring machine and more flexible than an concerning wavefront slope and curvature. Both limits can be overcome by repositioning the interferometer. The challenges lie in the limited dynamic range of a Shack-Hartmann sensor concerning wavefront wavefront slope slope and and curvature. Both Both limits can can be be overcome by by repositioning repositioning the the concerning interferometer. The challenges in dynamic range ofsetup a Shack-Hartmann sensor sensor and orienting it tangentially to the wavefront. An automatic is developed and first concerning wavefront andliecurvature. curvature. Both limits limits be overcome overcome repositioning the sensor and and orienting orienting itslope tangentially to the the limited wavefront. An can automatic setup isbydeveloped developed and first sensor it tangentially to the wavefront. An automatic setup is and first concerning wavefront and curvature. Both limits can be overcome repositioning the measurements are demonstrated on a highly divergent wavefront. It is shown that the dynamic sensor and orienting itslope tangentially the wavefront. An automatic isbydeveloped and first measurements are demonstrated demonstrated on to a highly highly divergent wavefront. Itsetup is shown shown that the dynamic dynamic measurements are on a divergent wavefront. It is that the sensor and orienting it tangentially to the wavefront. An automatic setup is developed and first range of the sensor is increased by scanning the sensor along the wavefront and orienting the measurements are demonstrated on a highly divergent wavefront. It is shown that the dynamic range of of the the sensor is is increased increased by by scanning scanning the the sensor along along the wavefront wavefront and and orienting orienting the the range measurements are to demonstrated on in a highly divergent wavefront. isare shown that the sensor wavefront each position. Measurements taken over aa dynamic range of range oftangential the sensor sensor isthe increased by scanning the sensor sensor along the theItwavefront orienting the sensor tangential to the wavefront in each position. position. Measurements are takenand over range of sensor tangential to the wavefront in each Measurements are taken over a range of ◦ ◦ range of the sensor is increased by scanning the sensor along the wavefront and orienting the ±15 , which is an improvement by a factor 5 compared to the typical dynamic range of ±3 for sensor tangential to the wavefront in each position. Measurements are taken over a range of ◦ ◦ ◦ , which is an improvement by a factor 5 compared to the typical dynamic range of ±3◦ for ±15 is improvement by compared to dynamic ±3 ±15 ◦ , which sensor tangential the wavefront each 55position. Measurements taken range over aof of aa static Shack-Hartmann sensor. , which is an an to improvement by a ainfactor factor compared to the the typical typicalare dynamic range ofrange ±3◦ for for ±15 static Shack-Hartmann sensor. ◦ a static Shack-Hartmann sensor. , which is an improvement by a factor 5 compared to the typical dynamic range of ±3◦ for ±15 a static Shack-Hartmann sensor. 2019, IFAC (International Federation of Automaticwavefront Control) Hosting by Elsevier Ltd. All system, rights reserved. a©static Shack-Hartmann sensor. Keywords: Freeform optics, optical metrology, measurement, scanning Keywords: Freeform optics, optical metrology, wavefront measurement, scanning system, Keywords: Freeform optics, optical metrology, wavefront measurement, scanning system, Shack-Hartmann sensor, surface metrology Keywords: Freeform optics, optical metrology, wavefront measurement, scanning system, Shack-Hartmann sensor, surface metrology Shack-Hartmann sensor, surface metrology Keywords: Freeform optics, optical metrology, wavefront measurement, scanning system, Shack-Hartmann sensor, surface metrology Shack-Hartmann sensor, surface metrology 1. INTRODUCTION ferometric 1. INTRODUCTION INTRODUCTION ferometric techniques techniques offer offer sub-nm sub-nm resolution resolution and and are are thus thus 1. ferometric techniques offer sub-nm resolution are aa well known solution for fast measurement of flats and 1. INTRODUCTION ferometric techniques offer sub-nm resolution and and are thus thus well known solution for fast measurement of flats and a well known solution for fast measurement of flats and 1. INTRODUCTION Freeform optics have a high potential for small packaging ferometric techniques offer sub-nm resolution and are thus spheres. Aspheric surfaces can also be measured with a well known solution for fast measurement of flats and Freeform optics have a high potential for small packaging spheres. Aspheric surfaces can also be measured with Freeform optics have a high potential for small packaging spheres. Aspheric surfaces can also be measured with size and high optical performance in light distribution and a well known solution for fast measurement of flats and interferometric techniques through the use of null-optics Freeform optics have a high potential for small packaging spheres. Aspheric surfaces can also be measured with size and and high high optical optical performance performance in in light distribution distribution and and interferometric techniques through the use of null-optics size interferometric techniques through the use of null-optics Freeform optics have performance a high potential for small packaging light detection applications. Assefa et al. (2018) demonAspheric surfaces can also be measured with (refractive or diffractive). However, this constitutes a huge size and high optical in light light distribution and spheres. interferometric techniques through the use of null-optics light detection applications. Assefa et al. (2018) demon(refractive or diffractive). However, this constitutes a huge light detection applications. Assefa et al. (2018) demonor However, constitutes a size and high optical performance in light and (refractive strate the potential of freeform optics energy interferometric techniques through this the tool, use of null-optics loss of universality in the measurement as a new surlight detection applications. Assefa et for al.distribution (2018) efficient demon(refractive or diffractive). diffractive). However, this constitutes a huge huge strate the potential of freeform optics for energy efficient loss of universality in the measurement tool, as a new surstrate the potential of freeform optics for energy efficient loss of universality in the measurement tool, as a new surlight detection applications. Assefa et al. (2018) demonillumination with LEDs, while Reimers et al. (2017) were (refractive or diffractive). However, this constitutes a huge face geometry requires a new null-lens [Kim et al. (2004)]. strate the potential of freeform optics for energy efficient loss of universality in the measurement tool, as a new surillumination with LEDs, while Reimers et al. (2017) were face geometry requires a new null-lens [Kim et al. (2004)]. illumination with LEDs, while Reimers et al. (2017) were face geometry requires a new null-lens [Kim et al. (2004)]. strate the potential of freeform optics for energy efficient able to design a freeform spectrometer that is five times loss of universality in the measurement tool, as a new surThe fabrication of an application-specific null-lens is costly illumination with LEDs, while Reimers et al. (2017) were face geometry requires a new null-lens [Kim et al. (2004)]. able to to design design aa freeform freeform spectrometer spectrometer that that is is five five times times The fabrication of an application-specific null-lens is costly able The fabrication of an application-specific null-lens is costly illumination with LEDs, while Reimers et al.designs. were face more compact than comparable conventional Curgeometry requires a new null-lens [Kim et al. (2004)]. able to design a freeform spectrometer that is(2017) five times and also introduces additional uncertainties (alignment, The fabrication of an application-specific null-lens is costly more compact than comparable conventional designs. Curand also introduces additional uncertainties (alignment, more compact than comparable conventional designs. Curalso additional uncertainties (alignment, able to designinvestigates a freeform spectrometer that is five optics times rent research the potential of freeform The fabrication of anresidual application-specific null-lens is costly fabrication quality, aberrations) to the measuremore compact than comparable conventional designs. Cur- and and also introduces introduces additional uncertainties (alignment, rent research research investigates the potential of freeform optics fabrication quality, residual aberrations) to the measurerent investigates the potential of freeform optics fabrication quality, residual aberrations) to the measuremore compact than comparable conventional designs. Curin the micro scale, e.g. aspheric contact lenses [Li and and also introduces additional uncertainties (alignment, ment procedure. Also, interferometric methods are very rent research investigates the potential of freeform optics fabrication quality, residual aberrations) to the measurein the the micro micro scale, scale, e.g. e.g. aspheric aspheric contact contact lenses lenses [Li [Li and and ment procedure. Also, interferometric methods are very in ment procedure. Also, interferometric methods are very rent research investigates the potential of freeform optics Fang (2019)] and the macro scale, e.g. telescope mirrors fabrication quality, residual aberrations) to the measuresensitive to vibrations and thus difficult to use in industrial in the micro scale, e.g. aspheric contact lenses [Li and ment procedure. Also, interferometric methods are very Fang (2019)] (2019)] and and the the macro macro scale, scale, e.g. e.g. telescope telescope mirrors mirrors sensitive to vibrations and thus difficult to use in industrial Fang sensitive to vibrations and thus difficult to use in industrial in the micro scale, e.g. aspheric contact lenses [Li and [Graves et al. (2019)]. In the first case, medical benefits are ment procedure. Also, interferometric methods are very Fang (2019)] and the macro scale, e.g. telescope mirrors environments. sensitive to vibrations and thus difficult to use in industrial [Graves et al. (2019)]. In the first case, medical benefits are environments. [Graves et al. (2019)]. In the first case, medical benefits are environments. Fang (2019)] and the macro scale, e.g. telescope mirrors expected, while in the second case, astronomical images sensitive to vibrations and thus difficult to use in industrial [Graves et al. (2019)]. In the first case, medical benefits are environments. expected, while while in in the the second case, case, astronomical astronomical images images Coordinate measuring machines (CMMs) are very flexexpected, [Graves et while al. (2019)]. themay firstcase, case, medicalthrough benefits are environments. of unprecedented be achieved the Coordinate measuring machines (CMMs) are very flexexpected, in quality theInsecond second astronomical images Coordinate measuring machines (CMMs) are very flexof unprecedented quality may be achieved through the ible and can be equipped with tactile or optical Coordinate measuring machines (CMMs) are very nonflexof unprecedented quality may be achieved through the expected, while in the second case, astronomical images reduction of aberrations. ible and can be equipped with tactile or optical nonof unprecedented quality may be achieved through the ible and can be equipped with tactile or optical nonreduction of of aberrations. aberrations. Coordinate measuring machines (CMMs) are very flexcontact probes. For optical (possibly coated) parts, nonible and can be equipped with tactile or optical reduction of unprecedented quality may be achieved through the contact probes. For optical (possibly coated) parts, nonreduction of aberrations. contact probes. For optical (possibly coated) parts, nonThe high-precision manufacturing of freeform parts relies ible and can be equipped with tactile or optical probes are favoured as they do not damage the contact probes. For optical (possibly coated) parts, nonThe high-precision high-precision manufacturing of of freeform freeform parts parts relies relies contact reduction of aberrations. probes are favoured as they do not damage the The manufacturing contact probes are favoured as they do not damage the on high-precision metrology [Fang et al. (2013)]. In gencontact probes. For optical (possibly coated) parts, nonsurface under test. Henselmans et al. (2011) developed The high-precision manufacturing of freeform parts relies probes are favoured as they do not damage the on high-precision high-precision metrology metrology [Fang [Fang et et al. al. (2013)]. (2013)]. In In gengen- surface under test. Henselmans et al. (2011) developed on surface under test. Henselmans et al. (2011) developed The high-precision manufacturing of freeform parts relies eral, there is a large number of measurement methods for contact probes are favoured as they do not damage the on high-precision metrology [Fang et al. (2013)]. In gena non-contact measurement machine for freeform optics surface under test. Henselmans et al. (2011) developed eral, there is a large number of measurement methods for aa non-contact measurement machine for freeform optics eral, there is a large number of measurement methods for non-contact measurement machine for freeform optics on high-precision metrology [Fang et al. (2013)]. In gennon-optical freeform surfaces such as turbine blades, ausurface under test. Henselmans et al. (2011) developed which resembles a giant CD-player and utilizes a probe eral, there is a large number of measurement methods for a non-contact measurement machine for freeform optics non-optical freeform freeform surfaces surfaces such such as as turbine turbine blades, blades, auau- which resembles aa giant CD-player and utilizes a probe non-optical resembles and a probe eral, thereparts isfreeform a large number ofsuch measurement methods for which tomotive or airplane fuselage: tactile and optical coabased non-contact measurement machine for utilizes freeform on the differential confocal method. Compared to non-optical surfaces as turbine blades, auwhich resembles a giant giant CD-player CD-player and utilizes a optics probe tomotive parts or airplane fuselage: tactile and optical cobased on the differential confocal method. Compared to tomotive parts or airplane fuselage: tactile and optical cobased on the differential confocal method. Compared to non-optical freeform surfaces such as turbine blades, auordinate measuring machines, interferometric and deflecwhich resembles a giant CD-player and utilizes a probe interferometric methods, which are typically “one-shot” tomotive parts or airplane fuselage: tactile and optical cobased on the differential confocal method. Compared to ordinate measuring measuring machines, machines, interferometric interferometric and and deflecdeflec- interferometric methods, which are typically “one-shot” ordinate interferometric methods, which are typically “one-shot” tomotive parts or airplane fuselage: tactile and optical cotometric techniques, profilometry and laser trackers [Savio based on the differential confocal method. Compared measurements, CMMs are relatively slow as they have to ordinate measuring machines, interferometric and deflecinterferometric methods, which are typically “one-shot” tometric techniques, profilometry and laser trackers [Savio measurements, CMMs are relatively slow as they have to tometric techniques, profilometry and laser trackers [Savio measurements, CMMs are relatively slow as have ordinate measuring machines, interferometric and deflecet al. (2007)]. However, only few of these are methods, arepoint. typically “one-shot” acquire surface geometries point by For rotationally tometric techniques, profilometry and laser techniques trackers [Savio measurements, CMMs arewhich relatively slow as they they have to to et al. al. (2007)]. (2007)]. However, only few few of these these techniques are interferometric acquire surface geometries point by point. For rotationally et However, only of techniques are acquire surface geometries point by point. For rotationally tometric techniques, profilometry and laser trackers [Savio suitable for freeform optics because optical parts typically measurements, CMMs are relatively slow as they have to symmetric parts, this can be mitigated by scanning a single et al. (2007)]. However, only few of these techniques are acquire surface geometries point by point. For rotationally suitable for for freeform optics optics because because optical parts parts typically symmetric parts, this can be mitigated by scanning a single suitable −5ofoptical symmetric parts, this can be mitigated by scanning a single et al. (2007)]. However, only few these techniques are acquire require very fine tolerances (< 10 are typically medium surface geometries point by point. For rotationally −5 line; for freeform surfaces, the whole geometry has to be suitable for freeform freeform optics because optical parts typically typically symmetric parts, this can be mitigated by scanning a single −5 ), require very fine tolerances (< 10 ), are typically medium line; for for freeform freeform surfaces, surfaces, the the whole whole geometry geometry has has to to be be require very fine tolerances (< 10 ), are typically medium line; suitable for freeform optics because parts typically sized (range of mm to m) and are [Savio symmetric parts, this can be mitigated bytimes scanning ),optical aretransparent typically medium scanned, resulting in long measurement (minutes to require very fine tolerances (< 10−5often line; for freeform surfaces, the whole geometry hasa single to be sized (range of mm to m) and are often transparent [Savio scanned, resulting in long measurement times (minutes to −5 sized (range of mm to m) and are often transparent [Savio scanned, resulting in long measurement times (minutes to require very fine tolerances (< 10 ), are typically medium et al. (2007)]. line; for freeform surfaces, the whole geometry has to be sized (range of mm to m) and are often transparent [Savio hours). Another recent example of an optical, scanning scanned, resulting in long measurement times (minutes to et al. (2007)]. hours). Another recent example of an optical, scanning et al. (2007)]. hours). Another recent example of an optical, scanning sized (range of mm to m) and are often transparent [Savio scanned, resulting in long measurement times (minutes to et al. (2007)]. method for freeform surface metrology is reported [Zapico hours). Another recent example of an optical, scanning method for freeform surface metrology is reported [Zapico The main methods to be considered are coordinate method for freeform surface metrology reported [Zapico The two main methods to be considered are coordinate et al.two (2007)]. hours). Another recent example of anis optical, scanning method for freeform surface metrology is reported [Zapico The two main methods to be considered are coordinate measuring machines and InterThe two main methods tointerferometric be consideredmethods: are coordinate measuring machines and interferometric methods: Intermethod for freeform surface metrology is reported [Zapico measuring machines and InterThe two main methods be consideredmethods: are coordinate measuring machines andtointerferometric interferometric methods: Intermeasuring machines and interferometric 2405-8963 © © 2019 2019, IFAC (International Federation methods: of AutomaticInterControl) Copyright IFAC 98 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 98 Copyright © under 2019 IFAC IFAC 98 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 98 10.1016/j.ifacol.2019.11.644 Copyright © 2019 IFAC 98
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et al. (2018)], using a conoscopic holography sensor for onmachine-measurement of CNC-machined freeform parts. Fang et al. (2013) point out a second way of freeform part metrology: Instead of measuring the freeform part surface, the freeform part performance can be measured. This can be achieved by measuring the optical wavefront that is transmitted or reflected by an optical system, for instance with a Shack-Hartmann sensor. A ShackHartmann sensor is a camera-based optical wavefront sensor that allows to measure a region of the wavefront reflected or transmitted by an optical system [Platt and Shack (2001)]. It consists of a lenslet array that segments the incoming wavefront onto an imaging chip, where the positions of the focussed spots are detected. From this spot pattern, the wavefront shape can be reconstructed, as long as the curvature of the incoming wavefront is not large enough to create ambiguities [Rockt¨ aschel and Tiziani (2002)]. Shack-Hartmann sensors are small, robust to vibrations and fast, so they are well suited for application in industrial environments [Thier et al. (2013)]. Their capability to characterize lenses that are bigger than the sensor aperture was recently demonstrated [Burada et al. (2017), Pant et al. (2015)]. This was done by manual, linear repositioning of the sensor and subsequent stitching of the acquired wavefront subaperture images.
Fig. 1. The working principle of a Shack-Hartmann wavefront sensor. a) Each lenslet forms a focussed spot on the image sensor. The deviations of the spots from their reference positions can be identified. b) An angle α between wavefront and sensor causes the spots to leave their subapertures. c) High wavefront curvature leads to ambiguities in spot assignment.
The contribution of this paper is to propose an automatic setup that allows for arbitrary repositioning and reorientation of a Shack-Hartmann sensor. As discussed in section 2, this allows the measurement of arbitrary wavefront shapes and is especially useful for highly divergent wavefronts because the wavefront curvature automatically decreases with an increased measurement distance. In combination with a reliable stitching algorithm, a measurement of the complete, global, wavefront exiting a freeform optic allows characterization of the part performance. Compared to an interferometric approach, this technique does not need any null-optics and is less sensitive to environmental vibrations. Compared to a coordinate measuring machine, the Scanning Shack-Hartmann Sensor (SSHS) approach offers higher speed and avoids damaging the part (compared to a tactile CMM). In section 3, an automatic setup is constructed and wavefront measurements on a highlydivergent optic are taken. The device under test for this demonstration is not a freeform optic, but its numerical aperture (NA) exceeds the dynamic range of a static Shack-Hartmann sensor and thus serves to demonstrate the measurement principle. In section 4, it is demonstrated that the dynamic range of the Shack-Hartmann sensor is increased by rotational repositioning. Further it is shown that aberrations can be visualized by numerically subtracting the dominant low-order aberrations tip, tilt and defocus from individual subaperture recordings. Finally, section 5 concludes the paper and briefly outlines future work.
in Figure 1 a). From the recorded spot pattern, local slopes can be calculated and then used to reconstruct the wavefront shape incident on the sensor [Platt and Shack (2001)]. Shack-Hartmann sensors offer a high measurement speed, are relatively robust to vibrations and have subwavelength resolution [Thier et al. (2013)]. The Shack-Hartmann sensor’s dynamic range is limited by two wavefront characteristics: wavefront slope and wavefront curvature [Rockt¨ aschel and Tiziani (2002)]. Wavefront slope describes the average angle between wavefront and sensor aperture. If this angle is too high, the lenslet array does not form spot images on the sensor chip (see Figure 1 b)). Spot deviation ∆x can be related to incidence angle α and lenslet focal length with the relation ∆x = fL · tan(α) (1) (adapted from [Chernyshov et al. (2005)]). To ensure that spots are registered within their respective subapertures, this limits the maximum allowed incidence angle to PL αmax = (2) 2 · fL (with lenslet pitch PL as in Fig. 1 a)). For typical ShackHartmann sensor parameters (Pl = 100 − 200µm, fl = 3 − 10mm), αmax is only a few degrees (1-5◦ ). Wavefront curvature describes the local variation of wavefront slope. If the curvature is too high, lenslets may form spots in the same position on the image chip, leading to ambiguities (see Figure 1 c)).
2. SCANNING SHACK-HARTMANN SENSOR A Shack-Hartmann sensor (Figure 1) consists of a microlens array and an image sensor. Each microlens produces a focused spot on the image sensor, the position of the spot being dependent on the local slope of the wavefront incident on the microlens. This is illustrated
Figure 2 shows a schematic of a transmissive freeform lens. Instead of measuring the surface geometry of the lens, it is proposed to measure the resulting wavefront. This constitutes a measurement of part performance instead of 99
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rot. stage and SHS
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Fig. 2. The proposed measurement strategy for freeform optics utilizing a scanning wavefront sensor. The incoming, collimated light is shaped by the optics and the resulting wavefront is observed. part surface, thus evading the insecurities typically related to measuring surfaces in order to predict performance. The Shack-Hartmann sensor is limited by its finite aperture size and its dynamic range considering maximum wavefront curvature. Both limitations could be overcome by the use of additional optical components to collimate and resize the wavefront. However, the added optical components have to be very specific and add additional aberrations and alignment problems [Fuerst et al. (2018)]. With a scanning setup, wavefront slope and curvature can be dealt with by increasing the measurement distance and repositioning the Shack-Hartmann sensor along the expected wavefront shape. Considering a spherical wave, the incidence angle α (comp. Fig. 1 b)) between Shack-Hartmann sensor and wavefront is related to the measurement distance rmeas (comp. Fig. 2))as rmeas =
d 2 · sin α
(3)
with d being the diameter of the sensor aperture. It follows, that the angle α drastically decreases when the ratio rmeas d is increased: α = arcsin
d 2 · rmeas
Fig. 3. Schematic view of the setup used to investigate the proposed measurement strategy for freeform optics by testing it on a high-NA optic. A laser diode is coupled to an optical fiber and the light exiting the fiber at a fiber coupler (FC) is used to illuminate a highNA lens. The resulting, highly divergent wavefront is measured with the scanning Shack-Hartmann sensor.
(4)
For typically acceptable incidence angles of below 5◦ , a measurement distance of at least 5 times the sensor aperture is required. To keep the incidence angle below 3◦ , the sensor is placed 10 times its aperture diameter from the spherical wave origin. Instead of optical insecurities, this adds mechanical insecurities that need to be evaluated for the system. Once these are known, the same measurement system can be used for a large range of optical parts without the need for any null-optics or reference shapes. To fully scan the wavefront produced by a freeform lens, a 5-degrees-of-freedom positioning system is required. Position data and wavefront data is required to reconstruct the global wavefront. In case of rotational symmetry of the part (e.g. aspherical lenses), 3 degrees of freedom are sufficient as in that case, a scan along a single line crossing optical axis is sufficient. Therefore, a first experimental 3DOF-system was constructed to investigate a rotationally symmetric lens, as described in the following section. 100
Fig. 4. The setup used for first investigations of the proposed measurement strategy. A 635 nm laser diode was coupled to a single-mode fiber to provide a monochromatic near-point light source for the highNA lens. The Shack-Hartmann sensor is mounted on two motorized linear stages and one motorized rotary stage. 3. EXPERIMENTAL SETUP To investigate the measurement capabilities of the 3degrees-of-freedom scanning Shack-Hartmann setup, a rotationally symmetric, highly divergent wavefront is investigated. Such highly divergent wavefronts occur for instance in high-resolution microscopy or confocal displacement sensors. The lateral resolution of these applications is dictated by the numerical aperture (NA) of the lens or lens stack which is proportional to the divergence angle of the focussed beam. Since high numerical apertures are needed for small measuring spots, the resulting wavefront is highly curved and thus exceeds the dynamic range of a single, static Shack-Hartmann sensor. With the Scanning Shack-Hartmann Sensor (SSHS), these limitations can be overcome.
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3.1 Experiments With the constructed 3-degrees-of-freedom setup, the wavefront generated by a high-NA lens is investigated. A 635 nm laser diode was coupled to a single-mode optical fiber. The other end of the fiber was attached to a fiber coupler without collimating optics, thus serving as a monochromatic near-point light source for the high-NA lens. A circular trajectory was chosen to scan the wavefront as that shape is close to the expected (ideal) shape of the investigated wavefront. Figure 5 illustrates the reasons for choosing a circular scanning trajectory and orienting the sensor tangentially to the wavefront: To scan a divergent wavefront, the sensor has to be moved from the optical axis of the optic under test. This position can be characterized by an angle α. For a linear scan trajectory, using a single, linear stage, the incidence angle β between wavefront and sensor aperture equals α, typically exceeding the dynamic range of a Shack-Hartmann sensor for angles of approximately 3 − 5◦ . Addition of a rotational stage on which the sensor is mounted removes this limitation, as the stage angle θ can be adjusted to equal α, keeping β close to zero. In this case, however, the measurement distance would increase with α. To keep the measurement distance (distance between wave origin and sensor) constant, a second linear stage (Z-direction in Fig. 5) is added and a circular scan trajectory is chosen. The radius of the scanning trajectory was chosen to be large enough that the wavefront curvature incident on the sensor is small enough to not exceed the dynamic range of the Shack-Hartmann sensor (compare Equation 4) and the spacing of the measurement positions was chosen so that the subapertures overlap. The Shack-Hartmann sensor was moved to 11 discrete positions spaced 3◦ apart in a distance of 54 mm to the focal point of the high-NA lens (the “origin” of a spherical wave). At each position, the Shack-Hartmann sensor is automatically pointed towards the expected position of the focal point and an image is recorded. For the presented experiments, this was done in a purely feed-forward controlled way and with coarse, 101
Fig. 5. Comparison of a linear and a circular scan trajectory. For sensor positions off the optical axis, an angle β between sensor and wavefront occurs. This can be negated by rotating the sensor by an angle θ. Note that α = β = θ.
2 1
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Figure 3 depicts the selected use case and the scanning setup. The setup consists of two motorized linear stages and one motorized rotary stage, meaning the ShackHartmann sensor can be positioned with 3 degrees of freedom (moving along a 2D-trajectory with arbitrary orientation). As this setup does not provide enough degrees of freedom to scan a fully freeform wavefront without any symmetries, a highly divergent, rotationally symmetric, wavefront was chosen as a test case. The linear stages (VT-80, PI Physik Instrumente, Braunschweig, Germany) provide a travel range of 200 mm and 50 mm, respectively, while their internal encoders have a resolution of 500 nm. The rotary stage (ELL-8M model from Thorlabs, USA) can be rotated by 360◦ , and offers an encoder resolution of 1.4 mdeg. A Shack-Hartmann sensor (model HR-2 from Optocraft, Erlangen, Germany) is used for the wavefront measurements. It has a rectangular aperture of 11.2 × 7 mm over which 85 × 53 lenslets with 130 µm pitch are distributed and its measurement rate is up to 18 Hz. Figure 4 shows the setup as assembled in the laboratory.
wavefront [ ]
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0 dPV
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0 x [mm]
1
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Fig. 6. An exemplary wavefront image from the measurement position -15◦ . The image is dominated by the low-order aberrations tip, tilt and defocus. The difference between the highest and the lowest point (peak-to-valley) is P V = 225λ = 142 µm and the distance between these two values is dP V = 4.044 mm. manual alignment. As dicussed in the results section, the information collected by such a coarse run can be used to automatically improve the scanning trajectory for another measurement. From the spot patterns, local wavefront gradients (slopes) are calculated and a 2D-polynomial function is fitted to match the observations (least-square-fit). Next, the wavefront shape is decomposed into Zernike base polynomials and their respective coefficients are calculated. Zernike polynomials are widely used in optics since they directly relate to basic optical aberrations such as tip, tilt, defocus, coma or astigmatism [Dai (2008)]. This is done to analyze the recorded wavefront images and also allows to numerically compensate for low-order aberrations. 4. RESULTS AND DISCUSSION Figure 6 shows a single measured subaperture on the described trajectory. The imaged wavefront subaperture resembles part of a sphere, which is expected when measuring on a divergent wavefront. From the detected peak-tovalley variation of P V = 225λ = 142 µm and the measured
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Low order Zernike modes 100
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Fig. 7. A map of residual wavefront errors, when the loworder aberrations tip, tilt and defocus are subtracted from the recorded wavefront shown in Figure 6. All measurements were taken for a single wavelength (635 nm).
Fig. 8. Recorded low order Zernike coefficients via measurement position. The defocus- and tilt- (Y) coefficients are approximately constant while the tip (x) coefficient varies linearly. This can be explained by positioning errors.
distance from peak to valley dP V = 4.044 mm, the radius R of the observed sphere can be calculated: d2 + P V 2 = 57.66mm (5) R = PV 2 · PV
Figure 8 compares the calculated low-order Zernike coefficients for each individual measurement position. These low order Zernike polynomials describe the wavefront tilt (X and Y) and defocus (deviation from the plane wave), as observed in the wavefront region incident on the sensor. Since a circular trajectory was chosen for the experiments and a divergent wavefront is under test, a constant defocus component is expected and can be observed (fig. 8, top line). The constant tilt (Y) coefficient can be explained by the scanning wavefront sensor and the optic under test not being set up at the exact same height from the optical table. This means that the optical axis of the high-NA lens is not in the same plane as the circular trajectory which is also visible in Figure 6. The linearly varying tilt (X) coefficient results from a mismatch of the center of the scanning trajectory and the focal point along the optical axis. As a next step, this could be automatically compensated by re-calculating the measurement positions and repeating the scanning measurement. The presented low order aberrations result from geometrical mismatches and misalignments and can be identified as such. To evaluate the optical quality of the system under test, higher order aberrations need to be analyzed. This can be done by correcting each individual subaperture as shown exemplarily in Figure 7 and interpreting this set of 102
Higher-order Zernike modes
0.1
Zernike coeff. [ ]
This is in agreement with the chosen measurement distance. Zernike analysis shows that the image is dominated by tip, tilt and defocus components. Defocus is responsible for the sphericity of the image while tip and tilt cause the visible shift of the center of the image. By subtracting these low-order aberration terms, the aberrations superimposed on the general sphere shape can visualized. This is shown in Figure 7, where only the residual wavefront error is visible. Peak-to-valley, this residual error is 0.2 λ for the presented subaperture, indicating a good quality lens.
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sensor position [°] Fig. 9. The higher-order Zernike coefficients for astigmatism, coma and spherical aberration vary with the measurement position, but are relatively small in magnitude. images. Figure 9 presents the single-subaperture higherorder aberration coefficients over measurement position. It can be seen that these higher-order aberrations are consistently much smaller in magnitude than the low order aberrations but do increase towards the ends of the measurement range. However, since the coefficients are calculated for individual subapertures, they are not representative for the complete (global) wavefront (e.g. a coma aberration in a single subaperture does not indicate a coma aberration of the global wavefront). A more meaningful metric is the residual wavefront error, which is the peakto-valley number of wavefront error when the low order aberrations are subtracted. This residual error is below 0.7 λ for all measured positions, providing a first estimate of the optical system’s quality. For more valid measurements, the wavefront information of all subapertures needs to be combined with each other and the positioning data, which is part of ongoing work.
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In summary, it is shown that the dynamic range limitations of the Shack-Hartmann sensor can be overcome by repositioning and tangential orientation of the Shack-Hartmann sensor to acquire wavefront images of subapertures that can be further processed to obtain the global wavefront information. 5. CONCLUSION The Scanning Shack-Hartmann approach offers a versatile, flexible way to measure freeform optics with reasonable measurement accuracy and measurement times. A first automatic setup is proposed and demonstrated to show the feasibility of the concept by taking measurements on a highly divergent wavefront. It is shown that the dynamic range of the Shack-Hartmann sensor can be increased by repositioning the sensor with respect to the wavefront. Future investigation is focussed on the tradeoffs between measurement speed, positioning accuracy, and measurement resolution. Next steps will be automatic correction of misalignments and evaluation of the global wavefront shape. For this purpose, stitching algorithms [Chen et al. (2017)], that incorporate both the measurement data and the position data, have to be developed. ACKNOWLEDGEMENTS The financial support by the Christian Doppler Research Association, the Austrian Federal Ministry for Digital and Economic Affairs, and the National Foundation for Research, Technology and Development, as well as MICROEPSILON MESSTECHNIK GmbH & Co. KG and ATENSOR Engineering and Technology Systems GmbH is gratefully acknowledged. REFERENCES Assefa, B.G., Pekkarinen, M., Saastamoinen, T., Biskop, J., Kuittinen, M., Turunen, J., and Saarinen, J. (2018). Design and characterization of 3D-printed freeform lenses for random illuminations. Light-Emitting Diodes: Materials, Devices, and Applications for Solid State Lighting XXII, (February), 53. doi:10.1117/12.2288223. Burada, D.R., Pant, K.K., Bichra, M., Khan, G.S., Sinzinger, S., and Shakher, C. (2017). Experimental investigations on characterization of freeform wavefront using Shack–Hartmann sensor. Optical Engineering, 56(08), 1. doi:10.1117/1.OE.56.8.084107. Chen, S., Xue, S., Wang, G., and Tian, Y. (2017). Subaperture stitching algorithms: A comparison. Optics Communications, 390(January), 61–71. doi: 10.1016/j.optcom.2016.12.067. Chernyshov, A., Sterr, U., Riehle, F., Helmcke, J., and Pfund, J. (2005). Calibration of a Shack-Hartmann sensor for absolute measurements of wavefronts. Applied Optics, 44(30), 6419–6425. doi:10.1364/AO.44.006419. Dai, G.M. (2008). Wavefront optics for vision correction Vol. 179. Bellingham: SPIE press. Fang, F.Z., Zhang, X.D., Weckenmann, A., Zhang, G.X., and Evans, C. (2013). Manufacturing and measurement of freeform optics. CIRP Annals - Manufacturing Technology, 62(2), 823–846. doi:10.1016/j.cirp.2013.05.003. 103
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