1. Introduction

Freeform optics refer to surfaces without an axis of rotational symmetry, within or beyond the surface [1]. Freeform surfaces allow higher degrees of freedom in optical design independent of surface geometry, which enables improved performance and miniaturization of optical systems [2–5]. Applications that leverage the freeform advantage include wide field-of-view telescopes [6], hyperspectral imagers [7], spectrometers [8], and head-worn displays for augmented and virtual reality [9–13]. The widespread utilization of freeform surfaces is critically dependent on the availability of non-contact, rapid, part-independent, and preferably economic metrology tools with low uncertainty in form measurement, with a targeted uncertainty of less than one-tenth of the operating wavelength [14].

In this paper, we classify non-contact freeform metrology techniques into point-cloud, sub-aperture, and full-field methods based on the modality of acquisition of measurement data. Swept-source optical coherence tomography (SS-OCT) [15] and the more recent cascade-OCT [16] are demonstrated non-contact (i.e., optical) point-cloud metrology techniques for surface under test (SUTs) in the regime of 1-inch in size with uncertainty in the 60 nm range. OCT-based metrology is limited in the slope of (SUTs) that it can measure when operating in telecentric scanning mode. When scanning perpendicular to the part, while precise mechanical scanning is at play, the slope range may be extended at the expense of a higher uncertainty in measurements compared to telecentric scanning. Commercial point-cloud optical profilometers typically leverage advanced mechanical scanning together with scanning perpendicular to the SUT and can thus measure larger parts with extreme slopes but at the expense of higher uncertainty due to the added uncertainty in the scanning mechanism [17]. Also, the data acquisition time scales with the part's size, sag departure, and sampling density in all these point-cloud techniques, setting a fundamental limit to how fast one can measure. The measurement uncertainty for longer time-scale measurements may be influenced by environmental factors, such as thermal drift. The measurement uncertainty varies for different systems, but typically the uncertainty reaches about two to five times lower values than contact CMM that provide uncertainty of about 1 micron.

Sub-aperture techniques include methods that illuminate and measure overlapping segments on the SUT, and the measurements are combined to generate the full-aperture measurement. In sub-aperture stitching interferometry [18] the local fringe density is reduced by illuminating a small region of the SUT, circumventing the fringe density limitation of interferometry for measuring freeform optics [19]. Additional nulling elements can be used for sub-aperture interferometry to extend the system’s dynamic range [20,21]. Tilted wave interferometry [22], transverse translation diverse phase retrieval [23], and axial scanning using annular apertures [24] are other sub-aperture metrology techniques. Sub-aperture techniques often require high-precision kinematics, extensive calibration, and sophisticated stitching algorithms [25,26].

Full-aperture metrology enables single-shot data acquisition and eliminates mechanical scanning of the SUT. Structured light illumination [27] and phase measuring deflectometry (PMD) [28,29] are examples of full-field metrology techniques involving slope integration to reconstruct the shape of freeform surfaces. The ability to measure form with uncertainty in the subwavelength range remains under investigation. Recently, optical differentiation wavefront sensing (ODWS) has been demonstrated as an emerging slope measuring technique for freeform optics with dense sampling compared to the Shack-Hartmann method [30] and remains under investigation to optimize the trade-off between dynamic range and sensitivity. Finally, interferometric null test based on computer-generated holograms (CGH) is a well-established and prevalent freeform metrology technique for measuring optics with mm-level sag departures from a base surface with subwavelength uncertainty [31,32]. The challenge with CGHs is that they are part-specific and involve significant expense and lead time for fabrication.

To alleviate the part-specificity limitation of CGHs, adaptive nulls have been implemented using a deformable-mirror (DM) [33–35] or a spatial light modulator (SLM) either in transmission or in reflection [36–43]. DMs are limited by their actuator stroke amplitude, which restricts the maximum peak-valley (PV) wavefront they can generate. Most DMs generate up to 50 μm PV tilt, and the state-of-the-art is 80 μm PV. Thus, DM-based metrology systems often require additional custom nulling elements such as the Offner null or tilt of the SUT in order to measure larger departures [33].

Prior work on optical testing using SLMs investigated the measurement of mild optics such as a near-flat membrane mirror [36], sub-aperture measurement of a convex lens [37], and a progressive lens [38] - all of these works involve an SLM compensating less than one wave PV wavefronts at 632.8 nm. Twyman-Green interferometric configurations have been demonstrated using an SLM generating the reference wavefront for the testing of extremely mild (i.e., a few waves of PV departure) freeforms [39] and microelements [40]. More recently, Fizeau interferometric configurations using a transmissive SLM in the test beam have been implemented for nulling localized freeform errors on a flat mirror [41], or as a standalone null to measure a freeform mirror with a mild departure of about 20 µm from a base sphere [42]. In all of the prior works on SLM-based metrology, the SLMs were limited by large pixel size (e.g., 36 µm pixel SLM was used in [41,42]) and, consequently, a limited wavefront range compared to the current state-of-the-art in SLM. Modern reflective SLMs have a smaller pixel pitch (i.e., < 10 μm) together with a high pixel count (i.e., greater than 1080 pixels along each axis), enabling the metrology of freeform surfaces with more severe sag and slope departures. An example of the current state-of-the-art is the Holoeye GAEA-2 SLM with 2464 × 4164 pixels and a pitch of 3.74 µm.

The unique features of the SLM-based null test developed in this work are summarized in three main points: first, a high-definition, high phase modulation range, reflective SLM with 1152 × 1920 pixels is implemented in a Fizeau interferometric configuration for the first time, along with the use of phase wrapping to encode the maximum possible phase on the SLM. Using a reflective SLM in a Fizeau interferometer’s test beam necessitates a different architecture than the architecture for transmissive SLMs [41,42]. Second, the SLM is imaged to the SUT to ensure one-to-one mapping between the nulling wavefront on the SLM and the SUT. Finally, another unique feature is the implementation of precise SLM calibration at its operating point and optimization of the nulling wavefront generated by the SLM using a high-dynamic range wavefront sensor. All of the aforementioned system design and optimization strategies enabled the testing of a freeform mirror with a 65 mm clear aperture and 90.6 μm peak-valley sag departure from a sphere, which is both the largest aperture and the most severe freeform sag measured with an SLM-based system at this time.

A challenge common to all full-aperture metrology techniques is that the measurement is sensitive to the alignment of the SUT to the measurement system. Therefore, precise alignment and calibration techniques are critical for low-uncertainty measurements. In this paper, we detail the alignment and calibration processes that have been developed for the SLM-based null test, including novel fiducial-based strategies to align the SUT to the SLM. The alignment protocols are benchmarked by first testing a spherical mirror and a mild (15 degree) off-axis parabolic mirror with cross-validation on the same parts with conventional interferometry. Finally, the high repeatability of the freeform measurement with the SUT realigned each time further validates the robustness of the alignment process.

2. System overview and considerations for hardware implementation

The experimental setup in this paper builds on our prior work reporting on an SLM-based null test, where the architecture was first reported [43]. In this section, we further discuss key updates to the hardware. The wavelength of the source is 632.8 nm.

The new schematic of the system is shown in Fig. 1. First, a 2:1 beam expander (BE) was implemented using catalog NBK7 lenses of 200 mm and 400 mm focal lengths to supplement the internal 3x zoom of the interferometer and fill most of the detector. The lenses are slightly tilted to clear their back reflections from causing spurious fringes in the interferometer. Using a sensitivity analysis in optical design software, we find that tilting the lenses even slightly creates astigmatism, which is later compensated using the SLM, as discussed in Section 4.4.

 

Fig. 1. Schematic of the SLM-based null test.

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An SLM modulates the phase of linearly polarized light using birefringence. Polarization management will be discussed in Section 3. The SLM is tilted by a small angle (i.e., < 10 degrees) to clear the reflected beam from the incident beam. The SLM is imaged to the SUT with a singlet lens designed for the appropriate magnification between the SLM and the SUT. Imaging the SLM to the SUT is a unique feature of the null test architecture designed in this work, unlike CGH-based null tests or other SLM-based null test architectures in the literature. Imaging the SLM to the SUT ensures linear mapping between the nulling wavefront and the SUT to minimize measurement errors due to improper mapping, which has been discussed in CGH-based null tests [44]. The SUT is placed concentric to a fitted sphere from the focal plane of the imaging lens to minimize the residual wavefront departure to be compensated by the SLM. The SLM is modeled as a diffractive phase polynomial optimized in optical design software to null the nominal departure of the SUT from the fitted sphere. Here, a tilt carrier is added to the nulling wavefront to separate the diffraction orders since only the +1 order is used in the system, and all others are blocked at the focal plane of the singlet. Note that the SUT is aligned to the +1 order of the tilt carrier. As a result, the carrier is not visible in the final measurement.

Once the nulling phase and tilt carrier are generated in the simulation, they are encoded on the SLM. One approach is to encode an interferogram between the designed nulling wavefront and a reference wavefront on the SLM, which is known as interferogram-type CGH or ICGH [41,42,45]. The specific SLM used in this work allows 8-bit encoding, which results in 256 discrete phase levels between 0 and 2π phase. The ICGH is discretized over the allowed phase levels and a binary interferogram is encoded on the SLM. When the slope and PV of the nulling phase are large, the number of pixels within one period of the ICGH is limited, and encoding errors occur. Xue et al. show that by encoding an ICGH using at least 8 pixels per period (PPP), mild wavefronts (e.g., 18 λ PV Fringe Zernike Z5/Z6) can be generated with less than λ/30 RMS error. Generation of accurate wavefronts with ICGH avoids the need for calibration of the SLM since the fringe frequency alone determines the wavefront, and the amplitude reduction results only in diffraction efficiency reduction. Additionally, there is no need for a wavefront sensor to monitor the SLM wavefront, which significantly simplifies the hardware. However, the ICGH technique is severely limited in the wavefront PV and slope that can be accurately generated using the SLM, due to the need for greater than eight SLM PPP to minimize errors.

In this work, we wrap the phase between 0 to 4π, the limit of the SLM, to generate a kinoform that corresponds to more severe wavefronts compared to ICGH. The benefit of phase wrapping is that the limiting condition for the maximum wavefront slope is 2 PPP from the Nyquist criterion rather than 8 PPP in the case of ICGH, which allows the encoding of higher wavefront slopes. The combination of a high-definition SLM and phase wrapping is leveraged in this work to generate a 140 λ PV freeform wavefront, excluding the added tilt carrier of ∼ 700 λ PV. The relationship between the number of pixels on the SLM and the wavefront range is described in our previous publication [43]. This strategy enables the testing of almost six times larger freeform departure than previously reported in the literature.

As the wavefront encoded on the SLM becomes more severe in PV or slope, the number of SLM pixels sampling the kinoform reduces to the minimum of two PPP, and consequently, fewer number of phase levels are used to encode the phase compared to a mild wavefront. Thus, wavefront errors scale as a function of the severity of the wavefront written on the SLM with phase wrapping. The generation of accurate wavefronts by encoding the wrapped phase on the SLM relies on the precise calibration of the SLM at its operating point to generate a linear phase response. Even with a calibrated SLM, a wavefront sensor is needed to measure and optimize the SLM wavefront to account for encoding errors. As shown in Fig. 1, a high dynamic range Shack-Hartmann wavefront sensor (SHWS) is deployed in closed loop feedback to iteratively monitor and converge the SLM wavefront to the nominal. The final optimized nulling wavefront is then displayed on the SLM to null the nominal departure of the freeform SUT, and the residual interferogram is captured. Note that the SHWS, owing to its limited resolution, is only used for fine-optimization of the SLM wavefront over a small pupil diameter of 10.7 mm, while the SUT with several inches in diameter is measured interferometerically after the SLM nulls the sag departure. Since the interferometer has 1200 × 1200 pixels compared to 128 × 128 pixels of the SHWS, it provides about 90 times higher sampling compared to a SHWS for the surface figure measurement of the SUT.

Next, we discuss the trade-offs involved in selecting an SLM for the null test. The metrology capability of the system is dependent on the specifications of the SLM used, as discussed in [43]. In order to maximize the range of wavefronts generated by the SLM, the desired specifications for an SLM are a high pixel count (i.e., > 1080 pixels along each direction), a high phase modulation range to enable a large PV-wavefront, and a small pixel pitch (i.e., < 10 μm) to maximize the allowed wavefront slope as per the Nyquist criterion. An SLM typically creates 2π phase modulation but greater than 2π phase can be generated by phase wrapping. A high phase-retardation SLM with larger than 2π phase modulation can generate even larger PV wavefronts by wrapping at multiples of 2π. Thus, a larger PV wavefront can be obtained either by more pixel counts at 2π phase shift (e.g., Holoeye GAEA 2 SLM) or fewer pixels at 4π phase shift (e.g., Holoeye PLUTO SLM).

A key aspect of choosing an SLM is the phase noise level, which depends on whether the drive scheme of the SLM is analog or digital. In analog SLMs, the pixels are driven by digital to analog converters that generate the different voltage levels smoothly [46]. In digital SLMs, only two voltage levels are allowed, which necessitates a pulse-width modulation (PWM) scheme to generate different root-mean-square (RMS) voltage values between the low and high voltage levels [47]. Because of the flipping voltage levels in the PWM sequence, a significant phase flicker is created in digital SLMs, but not in analog SLMs [47,48]. Consequently, we selected a high-definition analog SLM from Meadowlark Optics with 1152 × 1920 pixels, 9.2 µm pitch, and 0.01 radian peak-valley (PV) phase flicker. To maximize the PV wavefront range of the SLM, we further had the SLM custom-made to generate 4π phase modulation at 633 nm, which yields a 729 µm PV tilt wavefront and a maximum slope of 3.94°. Table 1 lists the amounts of the various low order Fringe Zernike coefficients that the SLM can generate at the Nyquist limited slope of 3.94°.

Table 1. Wavefront range of the SLM in terms of Fringe Zernike coefficients

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3. In-situ SLM calibration and optimizing the operating point

In this Section, we detail polarization optimization and SLM calibration. Let us note that the SLM modulates the phase of the incident light through birefringence, i.e., the refractive indices of the liquid crystal layer of the SLM are different between the ordinary and extraordinary axes. With applied voltage to the pixels, the refractive index is varied along the extraordinary axis, creating phase modulation. Thus, the SLM imparts phase modulation only on the component of the incident light that is polarized parallel to the extraordinary axis, and the rest of the light is left unmodulated in phase. This unmodulated light creates spurious fringes or stray light in the interferometric system, and is minimized by feeding the SLM linearly polarized light aligned to its extraordinary axis. In our system, the output of the Fizeau interferometer, which is circularly polarized, is then first converted to linear polarization using a quarter-wave plate (QWP), followed by a half-wave plate (HWP) that is rotated to orient the axis of the polarized light so that it aligns to the extraordinary axis of the SLM. To find the orientation of the HWP that results in this polarization alignment, we display a binary grating on the SLM with one grey-level (GL) equal to 0 and the other GL equal to 64 (which corresponds to a π phase), and rotate the HWP until the intensity of the +1 order is maximized. Then the HWP is fixed, and the polarization alignment is complete.

The phase response of the SLM is non-linear by default and is calibrated in situ for the generation of a linear phase response to the input GL. The calibration of the SLM involves the generation of a look-up table (LUT), which is then applied to the SLM to generate a linear phase response. While several methods to calibrate SLMs are reported in the literature [49–52], the diffractive method is implemented here for in-situ calibration since it is quick and simple in hardware [53]. The working principle behind the diffractive method for SLM calibration relies on the modulation of the diffraction efficiency of the +1 order when the amplitude of a binary phase grating is modulated from 0 to integral multiples of 2π. The detailed theoretical background for the diffractive method for SLM calibration has been reported in literature [53–55] and is also detailed in Chaudhuri (2022) [59]. The schematic of the diffractive method of calibration is shown in Fig. 2(a). The polarized light from the interferometer is incident on the SLM, and binary phase gratings are displayed on the SLM consecutively. The binary (2-level) gratings have one GL fixed at 0, and the other GL varies from 0 to 255 to generate 256 total gratings. Examples of a few binary gratings generated for calibration are shown in Fig. 2(b). We generate the binary gratings at 16 PPP for the initial calibration here. The SLM is tilted to pass the +1 order of the binary gratings through the beam splitter and a long focal length lens. An iris at the focal plane of the lens selects only the +1 order and blocks the others. The intensity of the +1 order is measured using a calibrated power meter for each binary grating displayed on the SLM. Thus, we obtain 256 intensity readings, which form a sinusoidal curve versus the applied GLs, from the theory of the diffractive method.

 

Fig. 2. (a) Schematic of the in-situ diffractive calibration of the SLM, (b) Calibration greyscale maps showing a few binary gratings as examples, (c) Intensity measurements in the +1 order as a function of grey levels, and (d) Phase response curve of the calibrated SLM.

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The method for generating in-situ calibration of the SLM involves first, the measurement of the intensities for the 256 gratings without any calibration applied to the SLM. The measured intensities from the uncalibrated SLM are then fed into a proprietary application supplied by the SLM manufacturer, which generates a custom LUT to create a linear phase response. To validate this custom LUT, we apply it to the SLM and measure the intensities again for 256 gratings. From diffraction theory, the theoretical intensity maxima for the scattered light into the first diffraction order should occur at GLs 64 and 192 (corresponding to π and 3π phase, respectively), and the minima should occur at GL 0, 128, and 255 (corresponding to phase 0, 2π, and 4π). From Fig. 2(c), we observe that the intensity maxima and minima occur at the theoretical GLs, indicating effective calibration. Next, to extract the phase ($\emptyset $) from the intensity measurements, we normalize the intensities in the +1 order (${\textrm{I}_\textrm{1}}$) by the maxima (${\textrm{I}_{1,\; \textrm{max}}}$) and use the following relations [55]:

The phase modulation range obtained after calibration is 3.8π, as shown in Fig. 2(d), and the phase response is linear with an R² = 0.99 of the linear fit. One possible reason for the slight reduction from the 4π phase modulation range is the spatial non-uniformity of the liquid crystal layer in the SLM, which results in a slightly different phase response of one pixel from another. In this work, we have used a global calibration, i.e., the same calibration is applied to all SLM pixels. For this first-generation system, the global calibration works well since the phase response of the calibrated SLM is linear, and the slightly smaller phase modulation range than the theoretical 4π phase results in a small reduction of diffraction efficiency, which is not an issue in this system. However, in future work, the SLM calibration may be further enhanced through a pixel-wise or regional calibration of the SLM to account for spatial non-uniformity of the phase response of the SLM [49,55].

Since the SLM is operated at a non-zero angle of incidence (AOI) for the null test, it is important to study the impact of AOI on the SLM performance to select an optimal operating point. As AOI increases, it has been shown that the polarization of the light through the liquid crystal in the SLM becomes increasingly elliptical, especially at high GLs [56]. The ellipticity of polarization causes residual unmodulated light, which manifests as the higher intensity of the minima at GL 128. Let us define the extinction ratio (ER) as the ratio of the first intensity maxima to the minima at GL 128 of the intensity curve. We measured the intensity vs. GL for AOI from 0 to 10 degrees and plotted the calculated ER in Fig. 3(a). The ER decreases sharply from 169 at normal incidence to about 31 between 6 deg. and 8 deg., then to 19 at 10 deg. AOI. The lower the ER, the higher the unmodulated intensity in the +1 order, which causes ghost fringes in the interferometric null test. Thus, we limit the SLM tilt to 8 degrees in our system.

 

Fig. 3. (a) Extinction ratio for variation of AOI from 0 to 10 degrees and (b) Phase curves with the 16 PPP and with the recalibrated LUT generated at 6 PPP.

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The other factor affecting the SLM phase response is the grating frequency of the wrapped phase displayed on it. The initial calibration was carried out at 16 PPP grating frequency, and here we recalibrate the SLM at 6 PPP, which is close to the frequency of the tilt carriers designed for the null tests. To test the impact of the grating frequency of the calibration grating, we generate the phase response of the SLM by displaying 6 PPP binary gratings, with first the 16 PPP LUT applied to the SLM and again with the new 6 PPP LUT applied. The 6 PPP LUT results in a higher phase modulation range, as shown in Fig. 3(b).

4. Alignment protocol and system calibration

The SLM can be configured to display tailored phase functions that provide fiducials to facilitate the alignment of the SLM to the interferometer and the SUT to the SLM. While fiducials on a traditional CGH are multiplexed on the annulus of the pupil, the SLM can be reconfigured to display fiducials sequentially using the full aperture of the SLM, which maximizes the intensity of light that is directed into each fiducial. The complete process chain for testing an optic with the SLM-based null test is shown in the flowchart in Fig. 4(a). In this section, we detail the alignment and calibration procedures developed for the SLM-based null test with a catalog spherical mirror as the first test optic. The sphere has a radius of curvature of -150 mm, a 1-inch clear aperture, and is rated to have a surface quality <$\; \lambda /4$ PV.

 

Fig. 4. (a) Metrology process chain for the SLM-based null test, (b) SLM centering and focus alignment to the interferometer on-axis, with a displayed fiducial image on the SLM.

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4.1 Alignment of the SLM to the interferometer

The first step in the metrology process chain is the alignment of the SLM to the interferometer on-axis. Note that the active area of the SLM is rectangular, while the beam incident on it is circular, necessitating the centering of the SLM to the incident circular beam. Also, the interferometer camera needs to be focused on the SLM, which is conjugate to the test optic. Figure 4(b.1) shows the schematic of SLM alignment on-axis, with the lenses and waveplates slightly tilted to clear reflections from causing ghost fringes in the interferometer. First, the SLM is adjusted coarsely for rotation about the optical axis to bring the reflected light back to the interferometer. The SLM is then aligned to the interferometer through centering and focus adjustments first, and fine alignment of tip and tilt subsequently.

The centering and focus alignments are facilitated by displaying a fiducial image on the SLM, as shown in Fig. 4(b.2). The circle fiducial enables the lateral alignment of the SLM to the interferometer mask, and the two rectangles outside the circle fiducial contain tilt fringes that provide additional visual feedback for lateral centering of the SLM. The fiducial image is designed using two grey levels: 0 (in black) and a grey level of 192, which corresponds to 3π phase. The 3π phase added to the reflected beam by the SLM causes destructive interference, which shows up as dark regions on the intensity map observed interferometrically, as shown in Fig. 4(b.3). The set of pixels on the crosshairs and the circle that is black (0 phase shift) on the fiducial image act as a mirror with no added phase, and are observed as bright pixels on the interferometer intensity map, which are easily visible due to the dark background. The thickness of the fiducial artifacts on the SLM is 20 pixels thick, which creates an effective linewidth of 625.6 μm on the interferometer camera, and is clearly visible on the intensity map. First, the interferometer focus is adjusted until the crosshairs on the intensity map are visible sharply. Next, the SLM is laterally moved to align the circle fiducial to the interferometer’s mask. Once the SLM is centered and is in focus, the interferometer’s mask (shown in blue in Fig. 4(b.3)) and the focus setting remain fixed throughout the rest of the alignment steps in the metrology process chain. Next, the fiducial image is switched off, and the tip/tilt of the SLM is finely adjusted by minimizing Fringe Zernikes Z2 and Z3 interferometrically.

4.2 Alignment of the sphere to the SLM

Although the next element after the SLM in the system is the imaging singlet, we first align the sphere because fiducials can be projected onto the sphere to aid in alignment. The alignment of the sphere involves lateral alignment (centering), tip, tilt, and axial alignment. First, the sphere is centered by projecting line fiducials on the top, bottom, left, and right edges. Figure 5(a) shows the simulation layout to design a line fiducial for the top edge of the sphere, as shown in Fig. 5(a.1). The optimized phase for the line fiducial is composed of Y², and dominantly Y tilt, as shown in Fig. 5(a.2). Next, the axial location of the sphere is set by projecting a point fiducial consisting of power, designed to focus on the vertex of the sphere, as shown in Fig. 5(b). An iris is placed close to the vertex of the sphere to block all the orders other than the +1 order for which the fiducial is designed. The sphere is moved axially to minimize the Fringe Zernike Z4 in the interferometric measurement. In addition, the fiducial image with crosshairs is also displayed on the SLM along with the point fiducial to check for centering of the sphere with respect to the SLM. At this stage, the sphere is aligned to the SLM. Next, we finely align the transmission sphere to the optical axis, with the point-fiducial displayed as before, minimizing the residual tip and tilt. This step is similar to the alignment of the transmission sphere in a cat-eye configuration in conventional interferometry. The intensity map in Fig. 5(b.2) shows an astigmatic residual after aligning the SUT to the SLM. This astigmatism is from the tilt of the lenses of the 2:1 beam expander before the SLM. For now, we only focus on Fringe Zernike Z4 since the astigmatism is compensated at a later stage once all the elements are aligned in the test beam.

 

Fig. 5. (a) Design of the line fiducial for the top edge of the sphere, (b) Axial alignment of the sphere with a point-fiducial optimized and the corresponding interferometric intensity map, and (c) Intensity map after the sphere is aligned to the +1 order of tilt carrier, with the point fiducial and crosshair fiducials displayed on the SLM.

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Since a tilt carrier is added to the nulling wavefront for testing surfaces with the SLM-based metrology system, we now align the sphere to the +1 order of a tilt carrier to establish the alignment protocol for a generic SUT. A tilt carrier with 200 λ PV along the X and Y directions each was displayed in addition to the point fiducial. The SLM was physically tilted in the X and Y directions to compensate for the tilt carrier and bring the focus of the point fiducial to the iris at the vertex of the sphere. Note that each time the SLM is tilted, it needs to be centered as well, using the crosshair fiducial image. Thus, the SLM is centered with the crosshair fiducial image displayed in addition to the point fiducial and tilt carrier, and the intensity map with the sphere aligned to the +1 order of the tilt carrier is shown in Fig. 5(c). Again, we observe residual astigmatism in the intensity map as expected from the lens tilts in the beam expander.

4.3 Alignment of the imaging singlet between the SLM and the sphere

The alignment of the singlet lens is done in two parts – first, a flat mirror is placed at the nominal focal plane of the lens and is aligned to the SLM +1 order, and then the lens is inserted and aligned in a cat-eye configuration, with the flat untouched. To align the flat, we display a point fiducial on the SLM designed to focus at a point on the flat at its nominal location, and the tilt carrier is also added, as shown in Fig. 6(a.1). The flat is first coarsely aligned for tip and tilt by minimizing vignetting in the intensity map, and then the axial location is fixed by minimizing Fringe Zernike Z4, as shown in Fig. 6(a.2). Note that while the Z4 is -0.18 λ in this step, this is close enough to zero to enable final adjustment of the singlet lens to minimize Z4 after its insertion later. Next, only the tilt carrier is displayed, as shown in Fig. 6(b.1), and the flat is finely adjusted for tip and tilt by minimizing Z2 and Z3 interferometrically, as shown in Fig. 6(b.2). Next, the singlet lens is inserted with the flat untouched. Note that the double-pass wavefronts measured with the flat in place in Fig. 6(a),(b) are dominated by astigmatism, as expected from the lens tilts in the 2:1 beam expander. At this stage, we take note of the composition of Fringe Zernike Z5 and Z6 since they are used for feedback to align the singlet lens in the next step.

 

Fig. 6. Steps for insertion and alignment of the imaging singlet (a) Tip, tilt, and axial positioning of the flat at the nominal focal point of the singlet (not yet inserted) with a point fiducial and tilt carrier displayed on the SLM (b) The double-pass wavefront with only the tilt carrier displayed on the SLM before inserting the lens, and (c) Wavefront measurement with the lens inserted and aligned to the flat. All units of waves are at 632.8 nm.

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From simulations, we find that when the lens is inserted between the SLM and the flat, the astigmatism coefficients Fringe Zernike Z5 and Z6 remain the same as before inserting the lens. Thus, the lens is inserted with the tilt carrier displayed as shown in Fig. 6(c.1) and is aligned by monitoring the Fringe Zernikes Z5 and Z6, such that they are as close as possible to the values before inserting the lens as listed in Fig. 6(b.2), highlighted in green. This measurement with the lens aligned to the tilt carrier, shown in Fig. 6(c.2), accounts for the aberrations from all the elements in the system other than the SUT, which is compensated as discussed in the next subsection.

4.4 Measurement and compensation of system aberrations

Here with the sphere still blocked, the measured system aberrations in the cat-eye configuration are nulled iteratively by applying a compensation wavefront ${W_{comp}}\; $to the SLM, as shown in Fig. 7(a.1). The wavefront captured from the interferometer (Fig. 7(a.2)) has a different sampling (700 × 700) compared to the SLM (1132 × 1132). Thus, to generate the compensation wavefront ${W_{comp}}$ for the SLM from the interferometric measurement, we first fit Fringe Zernike polynomials to the interferometric measurement to generate a set of coefficients denoted as ${Z_{sys}}$, and then generate the compensation Zernike coefficients, ${Z_{comp}} = \frac{{{Z_{sys}}}}{2}$. ${W_{comp}}$ is then generated over a unit circle over the 1132 × 1132 grid for the SLM, using the coefficients ${Z_{comp}}$. After applying the compensation, the aberrations are nulled to 11.4 nm RMS, as shown in the double-pass interferometric measurement in Fig. 7(b). This residual is calibrated from the null test measurements, as discussed in the next section.

 

Fig. 7. (a) Flowchart for measurement and compensation of system aberrations by blocking the sphere and (b) Wavefront measurement after compensating the system aberrations. All units of waves are at 632.8 nm.

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5. Measurement of a sphere and off-axis conic and cross-validation with regular interferometry to benchmark system calibration

In this section, we report the null test measurements on the catalog sphere and an off-axis conic, with cross-validation of both measurements with a Zygo Verifire HD Fizeau interferometer.

5.1 Measurement of the sphere and cross-validation with conventional interferometry

With the system compensation and the tilt carrier displayed on the SLM as described in the previous section, the flat at the focus of the imaging lens is removed. After fine tip/tilt alignment of the sphere, the double-pass wavefront measurements are captured as ${\textrm{W}_{\textrm{test}}}$. The calibrated double-pass wavefront ${\textrm{W}_{\textrm{calibrated}\; }}$is then calculated as:

Figure 8(a) shows the mean and standard deviation maps for 15 consecutive measurements captured with the SLM-based system, with the piston, tip, tilt, and power terms removed, as is typical for interferometry. The mean and standard deviation maps from the Zygo Verifire system, over 15 measurements, are shown in Fig. 8(b). Note that the data drop-out in the 12 o’clock position on both measurements is due to a fiducial mark applied to the sphere to orient it similary between the two systems. The SLM-based null test measures the sphere as 0.16 μm PV and 22.14 nm RMS, where the Zygo measurement is 0.13 μm PV and 11.39 nm RMS. The close agreement between the measurements from the Zygo and the SLM-based system indicates reliable alignment and calibration processes. The standard deviation of the measurements in both systems is in the order of a few nm, which indicates stable measurements. One challenge of measuring such a near-perfect optic is that it is difficult to visually compare the signature of the surface measurements between the Zygo and the SLM-based systems, since the surface figure of the sphere is in the same order of magnitude as the measurement uncertainty of the SLM-based system.

 

Fig. 8. (a) Mean and standard deviation maps of 15 consecutive surface figure measurements of the sphere with the SLM-based system, and (b) Mean and standard deviation maps of 15 consecutive measurements with the Zygo Verifire system

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5.2 Measurement of an off-axis conic and cross-validation with conventional interferometry

To facilitate a more direct comparison of the measurement from the SLM-based system and the Zygo Verifire interferometer, a catalog off-axis parabolic (OAP) mirror with a 1-inch diameter, 15-degree off-axis angle, and effective focal length of 387.6 mm was measured in both systems. From optical design software, we find that the best-fit sphere radius of curvature (${R_s}$) that minimizes the residual peak-valley departure of the SUT when illuminated on-axis is 775.17 mm, which is close to twice the effective focal length of the OAP. The sag departure of the OAP from the best-fit sphere is a mild freeform, consisting of Fringe Zernike Z5 of 1.85 μm and Z8 of 0.13 μm as the dominant contributions, amounting to a PV freeform sag departure of 2 μm. The mild departure of the OAP from the fitted sphere enables direct measurement on-axis by placing it concentric to$\; {R_s}$ in both the SLM-based interferometric system and the Verifire system, instead of needing to tilt the OAP to null the departure. Here, by the term “on-axis”, we mean that the surface normal of the OAP at its vertex is parallel to the optical axis of the interferometer.

The OAP was measured in the SLM-based interferometric system, with a catalog NBK7 lens of focal length 317 mm, and the SLM tilted at 5 deg., as shown in Fig. 9(a). The lateral and axial alignment of the OAP follows the fiducial-based approach discussed before for the sphere. The clocking (XY plane orientation) of the OAP is adjusted using the sensitivity table from CODE V. The measured surface figure of the OAP with the SLM-based system is shown in Fig. 9(b), along with the measurement from the Zygo Verifire system in Fig. 9(c). The surface figure was measured to be 2.44 μm PV and 0.31 μm RMS from the SLM-based system, and the Verifire system measured 2.39 μm PV and 0.22 μm RMS. We note that the ratio of the PV to the RMS values in both measurements in close to 8:1, and PVr might be a relevant metric to consider in the furture instead of PV for a more robust metric with respect to noise [57]. Next, we generate a difference map between the measurments obtained from the two systems by using the 37 Fringe Zernike coefficients corresponding to the two measurments to generate the respective surface maps over the same unit circle and subtracting them. Figure 9(d) shows the difference map between the measurements made by the Verifire vs. the SLM system. The difference between the two measurements is 30 nm RMS, which is consistent with the target uncertainty for our metrology system.

 

Fig. 9. (a) Schematic for the on-axis measurement of the OAP with the SLM-based system, (b) Surface figure measurement by the SLM system, (c) Surface figure measurement by Zygo Verifire system, and (d) Difference map between the surface figure measurements from the Zygo Verifire system and the SLM system.

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6. Null test of a concave freeform optic

6.1 Null test design

The concave freeform SUT selected is the secondary mirror of a freeform telescope built, assembled, and tested in 2014 by Fuerschbach et al. [33] using a DM-based null test. The total clear aperture (CA) of the SUT is 74 mm; however, in the SLM-based system, we are able to test over 65 mm CA within the Nyquist limit of the SLM. The freeform mirror has a -378.84 mm radius of curvature, a total sag of 1.46 mm, and a maximum slope of 5.1 degrees evaluated over 65 mm. The radius of curvature of the best-fit sphere that minimizes the residual PV sag departure is the same as the base sphere radius for this optic. The freeform sag departure from the base sphere, evaluated over 65 mm CA, is 90.6 μm PV and is dominated by astigmatism, as shown in Fig. 10(a). With astigmatism removed, the coma and higher-order freeform composition can be visualized, which is a total of 9.8 μm PV, as shown in Fig. 10(b).

 

Fig. 10. (a) Total freeform sag departure of the part under test and (b) freeform departure with astigmatism removed.

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Even though the bulk of the departure of the freeform part is astigmatism which can be nulled by tilting the part, doing so introduces a source of uncertainty unless the SUT is tilted at a precise value as determined by Coddington’s equation. Thus, the architecture of the SLM-based test was specifically designed to avoid the need for tilting the SUT to minimize measurement uncertainty and the alignment complexity associated with tilting the SUT at a precise value. In the case of this freeform mirror, the SLM is optimized to null the entire freeform departure of 90.6 μm PV over the test aperture of 65 mm without tilting the part. The simulated null test layout is shown in Fig. 11(a), using a catalog NBK7 lens with a focal length of 60 mm, and SLM is tilted to 8 degrees to clear the reflected beam. The SUT is placed conjugate to the SLM, and the SLM is optimized to generate a null interferogram, and a tilt carrier is then added to the optimized phase function such that the diffraction orders are separated at the focal plane of the lens, as shown in Fig. 11(a). The nulling wavefront, excluding the tilt carrier, is composed of -63.12 λ of Z5, 2.65 λ of Z8, -2.37 λ of Z9, 0.44 λ of Z11, and -2.27 λ of Z12 as the dominant Fringe Zernike coefficients, with a 138.6 λ PV.

 

Fig. 11. (a) Null test simulation for the freeform SUT showing separation of diffraction orders at the focal plane of the lens (b) Simulated nulling phase function including the tilt carrier; the figure is scaled by 50 for visibility, (c) Simulated double-pass interferogram before optimization of the SLM; the figure scaled by 50 for visibility, (d) Null interferogram after SLM optimization, and (e) Residual wavefront after addition of a tilt carrier. All units of waves are at 632.8 nm.

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The wrapped phase function, including the tilt carrier, has a PV of 807 λ and a maximum slope of 3.62 deg., as shown in Fig. 11(b). Note that the phase in the picture is scaled 50 times for visibility. Before optimization of the SLM, the interferogram shows a PV of 289 λ, as shown in Fig. 11(c), scaled by 50 times for visibility. The SLM optimization creates a null interferogram, as shown in Fig. 11(d). Upon addition of the tilt carrier, the SLM is tilted to compensate for the tilt carrier and align the singlet and the freeform to the +1 order. This tilt creates a residual of 1.87 λ PV in simulation, as shown in Fig. 11(e). From the Scheimpflug principle, we know that in a tilted geometry, pupils do not image perfectly to one another [58]. The tilt of the SLM creates an elliptical projection of the wavefront pupil on the SUT, which results in a slight residual after adding the tilt carrier due to the mapping mismatch between the SLM and the SUT. This residual is calibrated from the null test measurements obtained experimentally.

6.2 Alignment of the freeform optic

First, the SUT is mounted and oriented in-plane using an inscribed line fiducial on the side of the SUT, indicating the upright direction. Next, we use line fiducials designed to be projected on the edges of the SUT to align it laterally to the SLM. The freeform SUT being fairly large, requires a lot of tilt in the line fiducial phase function to hit the edge, and the required tilt exceeds the Nyquist limited slope for the SLM. To circumvent this limitation, we designed a line fiducial optimized for the +2 order instead of the usual +1 order to hit the edge of the SUT. Using higher orders of the wrapped phase grating allows the amount of PV tilt to be scaled down by the order of the grating used. In this case, the +2 order uses half the tilt required for the +1 order to hit the edge of the SUT. The position of the different orders at the plane of the SUT is shown in Fig. 12(a) from the simulation in optical design software. The SUT is adjusted along the vertical direction such that the +2 order line fiducial hits the top of the SUT, as viewed by a white screen held at the edge of the SUT, as shown in Fig. 12(b). The next step is the axial positioning of the SUT, which is done by displaying a point fiducial as discussed before, and minimizing the Fringe Zernike Z4 term interferometrically. The tilt carrier is then displayed in addition to the point fiducial, and the SLM is tilted to align the +1 order to the SUT under test.

 

Fig. 12. (a) Simulated 2^nd order line fiducial designed to hit the top edge of the freeform SUT, (b) The aligned SUT showing the line fiducial hitting its top edge, and (c) Intensity map with the freeform SUT aligned to the tilt carrier.

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Once the freeform is aligned to the SLM, a flat is positioned at the nominal focal plane of the lens, using a point fiducial designed to focus on the flat. Since the flat is located very close to the SLM for this case, we find that the maximum local slope of the point fiducial exceeds the Nyquist limited slope on the SLM. To circumvent this slope limitation, we design the point fiducial over 40% of the SLM's active area, resulting in a smaller maximum slope due to a smaller pupil. The flat is then axially aligned, as before, by reducing the Fringe Zernike Z4 interferometrically to 0.03 λ. Once the flat is in place axially, the tilt carrier over the entire pupil of the SLM is displayed, and the tip/tilt of the flat is finely adjusted.

6.3 Nulling wavefront optimization with the SHWS

Next, we implement the SHWS feedback loop to optimize the nulling wavefront generated by the SLM. Note that the freeform is still blocked by the flat for this step, as shown in Fig. 13. A beam splitter is inserted and aligned between the SLM and the flat to direct the light to the SHWS arm. The SLM is imaged to the entrance pupil of the SHWS through a 1:1.25 beam expander. The flowchart outlining the methodology for iteratively optimizing the SLM wavefront to converge to the nominal is shown in Fig. 13.

 

Fig. 13. Schematic showing the SHWS wavefront monitoring arm and the flowchart to optimize the SLM wavefront.

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The lenses are aligned with only the tilt carrier displayed on the SLM using the SHWS measurement of the wavefront, and the alignment is guided by sensitivity analysis in CODE V. The datum wavefront, ${W_{ref}}$ is measured after the lenses are aligned to the tilt carrier of the SLM, and this measurement is calibrated out from subsequent wavefront measurements. Next, the nulling wavefront designed in simulation and shown in Fig. 14(a) is applied to the SLM and is measured by the SHWS, as ${W_o}$. The wavefronts ${W_{ref}}$ and ${W_o}\; $are fit to Fringe Zernike polynomials and are both generated over a unit circle over the same resolution as the SLM, to subtract them and generate the calibrated wavefront measurements ${W_{cal}}$. Fringe Zernike coefficients (${Z_{cal}}$) for ${W_{cal}}$ are used to generate the error Zernike coefficients, ${Z_e} = {Z_{cal}} - {Z_i}$, where ${Z_i}$ is the set of Fringe Zernikes for the nominal freeform wavefront. ${Z_e}$ is then subtracted from ${Z_i}$ to generate the corrected wavefront, and the corrected wavefront is applied to the SLM in the next iteration. The wavefront is again measured by the SHWS, and the process of error correction is repeated until the measured wavefront by the SHWS has converged to the nominal wavefront.

 

Fig. 14. (a) Nominal nulling wavefront displayed on the SLM (tilt carrier not shown) (b) top row: Iterative wavefront measurements with the datum wavefront calibrated out and bottom row: wavefront errors from the nominal. All units of waves are at 632.8 nm.

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The measured wavefronts with ${W_{ref}}$ calibrated out in each iteration are shown in Figs. 14(b.1), (b.3), (b.5), denoted as ${W_{cal,1}}$, ${W_{cal,2}}$, and ${W_{cal,3}},$ respectively. For each iteration, the wavefront error (${W_{err,1to3}}$) between the calibrated wavefront and the nominal wavefront applied to the SLM is shown in Figs. 14(b.2), (b.4), and (b.6). The wavefront error in the first iteration (Fig. 14(b.2)) is mainly Fringe Zernike Z6, with 9.97 λ PV. In subsequent iterations, we find the wavefront errors reduce and converge to 38 nm RMS as shown in Fig. 14(b.6). This small residual wavefront error is calibrated from the null test measurements of the part. The final optimized wavefront generated by the SLM is shown in Fig. 14(b.5), which is displayed on the SLM for the null test of the freeform, as discussed in Section 6.2.5.

6.4 Freeform null test result and cross-validation with other metrology

Once the wavefront generated by the SLM is optimized with the SHWS loop, the beam splitter is removed from the system, and the imaging singlet is inserted at its nominal location and is aligned with the tilt carrier displayed on the SLM. The system aberrations are captured in cat-eye configuration blocking the SUT with the flat and are compensated to 0.04 λ RMS.

Next, the flat is removed and the freeform SUT is illuminated with the optimized nulling wavefront shown in Fig. 14(b.5). The laboratory system for the SLM-based null test of the freeform SUT is shown in Fig. 15(a), and the surface figure measurement of the freeform mirror is shown in Fig. 15(b). This measurement has the residual error from the SHWS (Fig. 14(b.6)), and the residual from the tilt carrier (Fig. 11(e)) calibrated out. The surface figure measurement by the SLM-based system is 0.14 μm RMS and 1.07 μm PV, which is dominated by astigmatism. The same part measured with commercial point-cloud metrology (UltraSurf 5x) and with a deformable-mirror-based null test by Fuerschbach et al. in 2014 [12] also shows astigmatic residual on the part, as shown in Fig. 15(c) and Fig. 15(d), respectively. Note that the DM-based null test measurement is over the entire aperture of the part (i.e., 74 mm), whereas the SLM-based null test is performed over 65 mm, and the UltraSurf measurement is cropped over 65 mm to enable direct comparison with the SLM system. The UltraSurf and the SLM-based measurements agree within 40 nm RMS surface figure. While the astigmatic signature on the part is validated across all three metrology systems, it is observed that the orientation of astigmatism in the SLM-based null test is the dominant source of uncertainty when compared to the UltraSurf and the DM-based null test.

 

Fig. 15. (a) The SLM-based null test system in the laboratory, (b) Null test result with the SLM-based system, (c) Measurement of the same SUT with the UltraSurf 5x (d) Null test result from [33] obtained with a DM-based null test over the entire aperture of 74 mm. All three results are plotted on the same scale for direct comparison.

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6.5 Repeatability and sources of uncertainty

We report three sets of measurements on the freeform mirror, as shown in Figs. 16(a)-(c), with the freeform and imaging singlet between the freeform and the SLM re-inserted and re-aligned for each set. Note that the same phase function optimized with the SHWS is applied to the SLM for all three measurement sets. The mean of 15 measurements across the three sets shows high repeatability of the RMS surface figure with respect to the alignment of the freeform and the imaging lens. Since the misalignment of the part is not the dominant source of uncertainty as demonstrated in the repeatability analysis, next, we investigate the SHWS loop to determine the cause of the astigmatic uncertainty in the surface figure measurement of the freeform.

 

Fig. 16. Three sets of null test measurements with the freeform and the imaging singlet re-aligned.

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To study the repeatability of the SHWS optimization of the nulling wavefront, we evaluate the impact of the alignment of the lenses in the SH arm, which is performed by displaying only the tilt carrier on the SLM. The alignment of the lenses in the SHWS arm is challenging in practice, since the incident wavefront on the SLM is already astigmatic due to the tilted 2:1 beam expander before the SLM. The tilt of the lenses in the SH arm also creates astigmatism, which becomes a coupled alignment problem with the astigmatic incident beam. To guide the alignment of the lenses in the SH arm, we simulate the wavefront for the aligned case of the SH arm. This is done in two parts. First, we simulate the tilted lenses in the 2:1 beam expander arm in the configuration shown in Fig. 17(a), where the lens tilts are chosen manually to mimic the wavefront captured in the same configuration experimentally.

 

Fig. 17. (a) Simulation to reproduce the measurement of the aberrations from the tilted lenses in the 2:1 beam expander with associated Fringe Zernike terms for reference and (b) Simulation of the aligned case of the SH lenses, with the lens tilts of the 2:1 beam expander applied. All units of waves are at 632.8 nm.

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Once the lens tilts are finalized in this first step, another layout shown in Fig. 17(b) is created, where the 2:1 beam expander is tilted to the same amounts as the previous simulation, but the lenses in the SH arm are aligned. The Fringe Zernike coefficients Z5 to Z8 are shown below the two layouts. We observe that the Fringe Zernikes corresponding to the wavefront in the second layout is close to half of those obtained from the first, as expected because the second layout is single-pass, whereas the first layout is in double-pass configuration. Once we have the target Fringe Zernike coefficients for the aligned state of the SH arm, we can tailor the alignment of the lenses in the SH arm to target the Fringe Zernike coefficients obtained from the simulated values in Fig. 17(b), instead of monitoring only overall PV of the wavefront.

In Fig. 18, we show the null test results for the optimized nulling wavefront from two different alignment states of the SH arm. Figures 18(a)-(c) are the plots corresponding to the best-case SH arm alignment obtained experimentally, and Figs. 18(d)-(f) are the plots for another state of alignment. The Fringe Zernike coefficients corresponding to the datum wavefront in Fig. 18(a) are listed below the measured wavefront. It is observed that the Z5 coefficient for the datum is smaller than expected from simulation, and Z8 is larger than expected from simulation of the aligned state of the SH arm, compared with the values in Fig. 17(b). The freeform nulling wavefront from the null test simulation is applied to the SLM, and the wavefront measured by the SHWS with the datum measurement in Fig. 18(a) subtracted from it is shown in Fig. 18(b). The SLM wavefront is then iteratively optimized to converge to the nominal as described in Sec. 6.3, and the null test result with the optimized freeform wavefront applied to the SLM is presented in Fig. 18(c).

 

Fig. 18. Two sets of null test measurements obtained for two states of alignment of the SH arm; Set #1: (a) The datum wavefront obtained from the best case alignment of the SH arm, (b) Measured wavefront by the SHWS with the freeform wavefront displayed on the SLM and the datum in (a) calibrated, (c) Null test result with the optimized nulling wavefront applied to the SLM. Set #2: (d) Another datum wavefront from a different state of alignment of the SH arm, (e) Freeform wavefront measured by the SHWS with the datum in (d) subtracted, and (f) Null test result by applying the optimized nulling wavefront to the SLM. All units of waves are at 632.8 nm.

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Next, another state of alignment of the SH arm is captured in the datum wavefront in Fig. 18(d), with the Fringe Zernike composition listed below. While the PV wavefront seems even better than the previous datum wavefront in Fig. 18(a), a closer look reveals that the Fringe Zernike coefficients Z5 and Z6 are further from the simulated values compared to the previous state of alignment in Fig. 18(a). This indicates that the alignment of the SH arm is worse in this case, even though the overall PV wavefront is smaller than the previous datum wavefront. The freeform wavefront measured by the SHWS with the datum in Fig. 18(d) removed, is shown in Fig. 18(e). We find that even with the respective datums subtracted, the measured wavefronts in Fig. 18(b) and 18(e) differ in the Fringe Zernike compositions, especially in Fringe Zernike Z6. Thus, it is inferred that different amounts of aberrations are induced to the freeform wavefront propagating through the lenses in the SH arm, arising from different states of lens alignments in the SH arm. For the second alignment state, the iterative wavefront optimization is carried out with the SHWS to generate the optimized wavefront for the null test. The corresponding null test result is shown in Fig. 18(f). This result shows a more significant clocking error than the previous measurement in Fig. 18(c). Thus, the calibration of the datum wavefront does not take care of the induced aberrations introduced to the freeform wavefront propagating through the misaligned lenses, since induced aberrations are a function of the severity of the wavefront propagating through the system. The datum wavefront has zeros on the SLM, and the induced aberrations experienced by the severe freeform wavefront (140 λ PV) are more significant than the datum case.

While the misalignment of the lenses in the SH arm clearly contributes to measurement uncertainty, another limitation is the limited resolution (128 × 128 pixels) and the trade-off between sensitivity and dynamic range of the SHWS. This is a well-known trade-off of SHWS, which may become significant while measuring a large wavefront like the one for the freeform null test. Since the input to the SLM is predominantly Fringe Zernike Z5 for this part, it is likely that any loss of sensitivity will affect this component the most and will also result in an astigmatic error in the measurement. Emerging research on wavefront sensing techniques provides the path to incorporate a more advanced wavefront sensor to optimize the SLM, while still maintaining the high dynamic range as required to measure freeforms with ∼100 μm PV sag departures or even higher with next-generation SLMs [30]. Lastly, another source of uncertainty is the SLM pixelation and phase quantization, which is detailed in Chaudhuri (2022) [59]. For the freeform wavefront with a 138.6 μm PV, the impact of SLM pixelation and phase quantization results in a net encoding error of 41 nm RMS from the nominal wavefront, and the enoding error is minimized through the iterative optimization using the SHWS. Note that the encoding error due to SLM pixelation and phase quantization scales with the severity of the wavefront displayed on the SLM.

7. Conclusion

We reported on the hardware implementation with detailed calibration and alignment protocol of an SLM-based null test to measure concave freeform SUTs. Novel alignment strategies involving tailored fiducials projected on the part using the SLM are reported. We first benchmarked the alignment process chain by measuring simple parts such as a catalog sphere and a mild off-axis parabolic mirror, which were both cross-validated to the measurements with a commercial Fizeau interferometer within tens of nm RMS surface figure.

Finally, we reported the null test of a concave freeform optic with a freeform sag departure of 90.6 μm PV and a 65 mm clear aperture, which are both the largest test aperture and about six times more severe sag departure than those reported in the literature with SLM-based null tests. The combination of a high-definition SLM, phase wrapping, precise SLM calibration, and wavefront optimization using a high-dynamic range SHWS made it possible to test a significant freeform departure with an SLM-based null test for the first time. Even without physical fiducials on the freeform part, the measurement was highly repeatable with the freeform and imaging singlet re-aligned each time. Finally, sources of uncertainty were investigated. The dominant source of uncertainty is primarily the residual misalignment in the lenses in the SH arm, which was established through systematic measurements with different states of alignment of the SH arm. In future generations of the SLM-based null test system, the alignment of the SH arm may be improved through the implementation of motorized stages to generate a repeatable and precise state of alignment. In addition, induced aberrations may be modeled through the system to minimize uncertainty from misalignment of the SH arm. The alignment and calibration protocols developed in this work, as well as the identification of the sources of uncertainty, paves the way for the advancement of the SLM-based null test as a reconfigurable alternative to CGH testing of freeform optics.