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Abstract
We present simulations and experimental validations for separating astigmatic figure error from misalignments in Nodal Aberration Theory (NAT) with a high-precision Cassegrain telescope. Both the primary mirror figure error and the secondary mirror misalignments induce binodal astigmatism for the telescope systems. The separation of these two aberration factors plays a crucial role in the telescope alignment process. In this study, the figure error of the aspheric primary mirror of the Cassegrain telescope induced by the mirror mounts was measured interferometrically utilizing a computer-generated hologram (CGH). According to the primary mirror figure error, the astigmatic node locations in the image plane were simulated using real raytracing. The center of the nodes was located on the field center, and the nodes were placed symmetrically with respect to the field center in the image plane. The telescope's alignment was performed using the simulation results, and the node locations were measured on a grid of field points interferometrically. Thereafter, secondary mirror misalignments around the coma-free pivot point were introduced into the optical system, and the node's center was shifted from the field center in the image plane as predicted by NAT. The simulations and interferometric field measurements were performed and compared on a grid of field points for the misaligned state in the presence of primary mirror figure error. The experimental results confirm the predictions from NAT. Statistical analysis was also performed to confirm the accuracy and stability of the measurements.
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1. Introduction
Nodal aberration theory (NAT) defines aberrations with the characteristics of misaligned optical systems and quantifies them. Roland V. Shack developed the vectorial form of the wave aberration expansion at the University of Arizona in 1976. The theory builds on H. H. Hopkins's wave aberration theory [1], and R. Buchroeder’s shifted aberration field centers concept [2]. The mathematical background of the theory was developed by K. P. Thompson and R. V. Shack [3,4]. Thompson further developed the mathematical background of the theory up to fifth-order aberrations [5–8]. Today, the well-established theoretical foundation of NAT plays a crucial role in optical imaging systems. Specifically, the theory offers alignment strategies for modern telescope systems [9] or describes aberrations of freeform optical systems [10–12]. The historical path from NAT to freeform was discussed in a recent report as part of a memorial conference session for Shack [13]. The root of NAT is based on a through-focus star plate image gathered by the 32’’ Ritchey-Chretien (RC) telescope located at Steward Observatory. This historically reproduced data was published for the first time by Thompson in 2005 [5]. Although no significant misalignment induced axial third-order coma appeared in this image, the axial astigmatism was substantial. It had two astigmatic nodes (zero astigmatisms) that led Shack to call it binodal astigmatism. These two astigmatic nodes were located in the image plane away from the field center, unlike binodal astigmatism led by a common state of misalignment. Axial astigmatism could not be explained with a sole misalignment. Thompson addressed this issue for the first time in his Ph.D. dissertation. The axial astigmatism was found to be generated by the improper mirror mounts of large telescopes, called mirror warpage, introducing field-constant astigmatism [4]. McLeod [14] and Sutherland [15] also pointed out mount-induced contribution to astigmatism in their studies. Schmid et al. studied mirror warpage, also called form error, in NAT and developed a mathematical background to address mount-induced axial astigmatism. Both types of binodal astigmatism generated by the primary mirror (set as the aperture stop) figure error and the secondary mirror misalignments were distinguished [16]. Fuerschbach et al. extended the concept of isolated figure error in NAT to define freeform surfaces, which was a significant step [17]. The experimental investigation of binodal astigmatism for separating the primary mirror figure error from misalignments for a telescope system in NAT has been challenging.
This study presents simulations and experimental investigations of binodal astigmatism induced by the primary mirror figure error and the secondary mirror misalignments for a high-precision Cassegrain telescope. The design of the Cassegrain telescope and experimental validation of misalignment induced binodal astigmatism in NAT were reported in earlier studies [18–20]. The separation and isolation of binodal astigmatism induced by these two aberration factors are critical for the alignments of the emerging telescopes. In this work, the mount-induced figure error on the telescope's primary mirror, which was set as an aperture stop, was measured individually and interferometrically using a customized computer-generated hologram (CGH). The simulations of the third-order astigmatism were performed according to the measured figure error to determine the astigmatic node locations on the image plane, and the telescope alignment was performed referring to the simulations. Thereafter, the secondary was purposedly misaligned to induce astigmatism, which was isolated through the coma-free pivot point of the secondary mirror [21]. Interferometric measurements were performed for a grid of field points for confirming the simulations from NAT.
The remainder of this paper is organized as follows. Section 2 provides the theoretical background of third-order astigmatism for the misaligned optical system in the presence of primary mirror figure error. Section 3 provides the Cassegrain telescope system, and the experimental setups for the primary mirror surface form measurement and the telescope alignment with the interferometric field measurements. Section 4 compares the simulations and experimental results before conclusions are drawn in Section 5.
2. Nodal property of third-order astigmatism in a misaligned Cassegrain in the presence of primary mirror figure error
The mathematical background for the nodal property of astigmatism in NAT is presented in the following, starting from the misaligned optical system, and then including the primary figure error. For third-order astigmatism, the vector form of the wave aberration expansion with respect to the medial surface in an optical system without symmetry (i.e., misaligned or intentionally perturbed) can be written as [5] (here, bold depicts a vector)
Fig. 1. Schematic description of the characteristic node locations for third-order astigmatism: (a) nominal aligned state, (b) binodal astigmatism induced by misalignment in which coma is corrected (coma-free pivot point), (c) binodal astigmatism induced solely by the primary mirror figure error, and (d) simultaneous contributions of misalignment and figure error.
The terms included in Eq. (1) with squared vectors are vector products [23–25] applied to aberration theory for the first time by Shack. They represent vector multiplication, which is not a commonly used operation, and discussed extensively by Thompson [5,26]. Equation (1) can be solved for locations where the astigmatic terms go to zero for H as [5]
Equation (6) presents the fundamental discovery of binodal astigmatism made by Shack [2]. The astigmatic aberration field contains two zeros, or nodes, in an optical system without symmetry. In a misaligned state, without figure error, the nodes are located as illustrated in Fig. 1(b). The a222 vector is directed from the optical axis ray to the midpoint of the ± ib222 vectors in the image plane. In a state of alignment but with figure error, the vector a222 is zero, and the nodes are symmetric with respect to the field center, as shown in Fig. 1(c). In the presence of primary mirror figure error and overall misalignment, the figure error contributes to the vectors of ± ib222 as seen in Fig. 1(d), and the node locations are shifted away from the field center not symmetrically with respect to the field center due to the presence of the vector a222. The mathematical background was developed by Schmid et al. [16] that the figure error is directly added into the existing misalignment induced astigmatic component according to where where $|{{C_{5,6}}} |$ and ${\boldsymbol{\xi }_{5,6}}$ are the values of magnitude and orientation of the astigmatic figure error in the wavefront of the primary mirror. C5 and C6 are terms of the Zernike polynomial coefficient values. The magnitude, $|{{C_{5,6}}} |,\; $ and the orientation, ${\boldsymbol{\xi }_{5,6}}$, of the astigmatism are calculated as The referring Fringe Zernike coefficients (ZFC) of the measured Zernike polynomial terms for third-order astigmatism is given as [27] where ϕ and $\rho $ are the azimuthal frequency and radial variable in the exit pupil, respectively.For the aligned telescope, in which a222 = 0, $\boldsymbol{B}_{222,MISALIGN}^2$ = 0, and $\boldsymbol{B}_{222,FIGURE}^2$ ≠ 0, considering Eq. (4), Eq. (1) reduces to
The astigmatism field response of a Cassegrain telescope for the nominal state, then with figure error or misalignments, and in the presence of both figure error and misalignments were simulated in full-field displays (FFDs) function of CODE V and shown in Fig. 2. Figure 2(a) shows the nominal state of the telescope by design. Figure 2(b) presents binodal astigmatism induced by primary figure error in which the astigmatic nodes are placed symmetrically about the field center. In Fig. 2(c), binodal astigmatism caused by the secondary misalignment is shown, and one of the astigmatic nodes is placed on the field center, while the second node is placed away from the field center. The astigmatic field response for the combination of figure error and misalignments is given in Fig. 2(d). The midpoint of the nodes is moved from the field center, and both nodes are found away from the field center. Thus, each astigmatic field response for figure error or misalignment, or both, has its own characteristics, separable by NAT.
Fig. 2. Simulated FFDs of Fringe Zernike astigmatism Z5/6 in a Cassegrain system (a) in the aligned state, (b) the primary mirror figure error with Z5 = 0.108 waves, and Z6 = 0.001 waves at 632.8 nm wavelength, (c) in the misaligned state with a 0.34 mm decenter (XDE) and 0.151° tilt (BDE) for the coma-free pivot point, (d) in the misaligned state plus primary mirror figure error. The dots (blue) depict the nodes’ locations.
3. Cassegrain telescope and experimental setups
3.1 Cassegrain telescope
The Cassegrain telescope consists of a parabolic concave primary mirror (M1) and a hyperbolic convex secondary mirror (M2), as shown in the optical design layout in Fig. 3. The radii of the primary and the secondary mirrors are −1700 mm and −300 mm, respectively. The primary mirror is a parabola, thus with a conic constant of −1, and the secondary mirror is a hyperbola with a conic constant of −1.737. The primary mirror is defined as the aperture stop. The f-number of the telescope is 12.7, and it is designed to have a diffraction-limited optical performance over ±0.11-degree full field of view (FOV). The overall specifications of the telescope system are given in Table 1.
Table 1. Specifications of the F/12.7 Cassegrain telescope
The Cassegrain telescope was conceived according to athermalized, high precision optomechanical requirements. Three bipod flexures hold a lightweight primary mirror from the side of the mirror with 120° radial symmetry. The bipods are bolted on a base plate mounted onto the mainframe of the telescope structure. Six metering rods mounted onto the mainframe hold the secondary mirror and its assembly. The telescope's secondary mirror is aligned with respect to the primary mirror utilizing a customized, piezo-actuated, five-axis flexure mechanism (X, Y, Z, and tip/tilt). This alignment mechanism is also used to introduce the secondary mirror misalignments to generate (on purpose) aberrations of misalignments in this work. The design, realization, and alignment of the Cassegrain system were reported comprehensively in a previous study [15].
3.2 Primary mirror figure error measurement
Surface figure error of the primary mirror was measured utilizing a customized computer-generated hologram (CGH) (Jenoptik AG, Jena, Germany) and a phase-shifting Fizeau interferometer (DynaFiz, Zygo Corp., Middlefield, CT, USA). The schematic layout of the measurement setup is shown in Fig. 4(a). The primary mirror was mounted in the telescope structure in the nominal vertical orientation. The interferometer was placed on a five-axis manual adjustment stage, and the CGH was placed on a three-point mount (one is magnetic) manual tip/tilt stage. The entire setup was placed on an optical table with vibration isolation stages. The wavefront error of the primary mirror was measured to be 0.19 waves PV (see Fig. 4(b)) and 0.024 RMS waves at 632.8 nm. As shown in Fig. 4(b), third-order astigmatism induced by the bipod mounts is the dominant aberration. For this wavefront measurement, the Fringe Zernike astigmatic terms Z5, and Z6, were calculated to be 0.108 waves and 0.001 waves at 632.8 nm, respectively. The orientation of the astigmatism is dominated by Z5, which is visually apparent in Fig. 4(b).
Fig. 4. Testing of the primary mirror: (a) Schematic surface form measurement setup of the primary mirror with a customized CGH and a phase-shifting Fizeau interferometer; (b) measurement data for the primary mirror wavefront error (λ= 632.8 nm).
3.3 Measurements at field points
The schematic design layout of the double-pass interferometric setup for the telescope alignment and the field measurements is shown in Fig. 5. The setup consists of the Cassegrain telescope, a 32’’ reference flat mirror with a motorized tip/tilt stage (QED Optics, Rochester, NY, USA), and a phase-shifting Fizeau interferometer that sits on a five-axis (X, Y, Z, and tip/tilt) customized manual stage. The interferometer with a transmission sphere (i.e., F/7.1) is confocal to the telescope in this configuration. The beam follows the path through the secondary mirror, the primary mirror, and the reference flat mirror, then reflected to generate an interference signal on the interferometer. The five-axis manual stage beneath the interferometer is used to move the interferometer focus through the field points on the image plane of the telescope for the field tests. The reference flat was brought into the closest permissible proximity of the telescope to minimize the optical path and on account of the turbulence effects. The experimental setup was put on an optical table (3.6 m in length and 1.5 m in width) with a vibration isolation stage with a cut-off frequency of 1.5 Hz (S2000A, Newport Corp., Irvine, CA), as shown in Fig. 6.
Fig. 5. Auto-collimation test setup in the double-pass interferometric measurement configuration and a grid of simulated field points on the image plane.
Fig. 6. Picture of the Cassegrain telescope placed in the auto-collimation configuration with the interferometer and the reference flat mirror on the optical table.
The field tests were performed according to the simulated field points described in Section 4. The focus of the interferometer is always perpendicular to the image plane of the telescope, as illustrated in Fig. 5. When the interferometer focus was moved from a field point to a new one on the image plane, the reference flat was adjusted to be perpendicular to the output light beam of the telescope employing the motorized tip/tilt stage.
4. Simulations and experimental results
4.1 Telescope alignment
In the absence of the primary figure error, the telescope is aligned utilizing a sensitivity matrix obtained from the optical design. The telescope alignment is achieved by decreasing the Zernike terms for the third-order aberrations [13]. This process is initially performed on-axis (i.e., Field 0 or 0°). Then, measurements from several field points are conducted to check the symmetry of the aberrations across the field around the field center to confirm the nominal state of the telescope and to make sure that the secondary mirror is not sitting on a coma-free pivot point [14].
In the presence of primary mirror figure error, the above strategy does not work for the telescope alignment since on-axis measurement does not give the minimum wavefront error as expected from a nominal design state. As predicted by NAT and summarized in Section 2, the minimum wavefront error is measured on the binodal positions in the image plane, not on the optical axis. Note that coma is first subtracted since third-order coma is not corrected for the Cassegrain system by design. For the aligned telescope, on-axis field measurements gave a wavefront error equal to the primary mirror figure error measured individually that was reported in Section 3.2. Therefore, a new strategy for the telescope alignment had to be developed in the presence of primary mirror figure error.
A new alignment strategy based on NAT was developed in the presence of primary mirror figure error. Initially, a simulation for the field dependency of Fringe Zernike astigmatism (Z5/6) for the measured primary mirror figure error was conducted in the form of full-field display (FFDs), as illustrated in Fig. 2(b). Binodal astigmatism is induced by the figure error in which two astigmatic nodes are located symmetrically around the field center. The astigmatic node locations in the image plane were calculated utilizing the FFDs and the analytical calculations in Table 2. According to the calculated astigmatic node locations, a 5 × 5 grid of field positions was determined and given in Table 3. This grid provides 25 samples that evenly span the intended field of view. Then, the real ray trace model of the Cassegrain telescope together with the Wavefront Map Analysis function in Zemax were used to generate interferograms for the associated field points. The Fringe Zernike terms of Z5 and Z6 were used to quantify astigmatism as
and the interferograms using analysis results were plotted for astigmatism, as shown in Fig. 8(a).Table 2. Coordinates of the astigmatic nodes as calculated analytically and experimentally.
Table 3. 5 × 5 grid of simulated field positions on the image plane.
The telescope was aligned employing the sensitivity matrix for decreasing the Zernike terms of third-order spherical aberration, coma, astigmatism. Then, the field tests were performed to find the astigmatic node locations in the image plane, which was used to calculate the a222 displacement vector induced by misalignments. The amounts of misalignments were calculated according to the equations given in Section 2 and then employed to align the secondary mirror. As a final step, the field tests were performed to confirm that the a222 displacement vector was zero, and the nodes were placed symmetrically concerning the field center.
The final interferometric field tests were conducted at the locations given in Table 3. The results are illustrated in Fig. 7. The field points were determined to measure the node locations and the midpoint of the astigmatic nodes (i.e., the field center since a222 is zero). A customized apparatus guided the focus of the interferometer on the focal plane, which was mounted onto the rear panel of the telescope. The field points were marked on it to minimize the positioning error of the field points induced by the manual stage beneath the interferometer. In addition, the uncorrected third-order coma aberration in the Cassegrain telescope was also used as a ruler for determining the field center and field positions since the coma changes linear with the field. The four field points located at the corners of the image plane were excluded from the simulations and tests since the primary mirror causes obscuration. The primary mirror inner hole diameter was not enough to pass the full light beam of the interferometer for the tests, and these fields were beyond the telescope design field of view (FOV) limit. Zernike polynomials were fit to the measured WFE, and the astigmatic terms Z5 and Z6 were extracted from the data or equivalently set to zero in the fitted dataset. The extracted measured interferograms for Fringe Zernike astigmatism Z5/6 illustrated in Fig. 8(b) were gathered by a customized MATLAB code that collects raw interferogram files, processes each field data, and results into a 5 × 5 grid. The interferograms show the magnitude and orientation of third-order astigmatism for the field points in the aligned state of the telescope for the presence of primary mirror figure error. The two astigmatic nodes were placed symmetrically concerning the field center in the image plane as predicted in the simulations by NAT. The locations of the astigmatic nodes were calculated from the experimental data utilizing a customized MATLAB code that finds the local minima of astigmatism in the image plane, as given in Table 2. The analytical and experimental node calculation results are in agreement within 0.05 wave RMS (@ λ= 632.8 nm), as seen in Fig. 11(a), and they are within one-hundredth of a degree (0.01). Hence, the results validate that the primary mirror figure error induces binodal astigmatism in NAT.
Fig. 7. Interferograms of the measurements for the aligned state of the telescope in the presence of primary mirror figure error (@ λ = 632.8 nm). The values on each data denote the RMS WFE for the field points.
Fig. 8. Fringe Zernike astigmatism (Z5/6) for the aligned state of the telescope in the presence of primary mirror figure error for the associated field points: (a) simulated interferograms, (b) processed measured interferograms (@ λ = 632.8 nm).
4.2 Misalignment and figure error
After performing the alignment of the Cassegrain telescope, simulations of the astigmatic field response for the secondary mirror misalignments for the presence of the primary mirror figure error were performed. For this purpose, the misalignments of XDE = 0.340 mm, and BDE= 0.151 degrees were introduced into the model using the real ray trace model of the optical design in CODE V. The secondary mirror was pivoted about a coma-free pivot point to adjust and isolate the astigmatic field response of the telescope. These specified amounts of misalignments hold the astigmatic nodes along the x-axis in the measurable FOV, where the a222 displacement vector induced by the misalignments is parallel to the x-axis. Thus, interferometric measurements on/nearby the nodes would be possible. A simulation of FFDs of Fringe Zernike astigmatism Z5/6 for the misaligned Cassegrain telescope in the presence of primary mirror figure error is shown in Fig. 2(d). The astigmatic nodes’ center was displaced from the field center by misalignment, and the symmetry of the nodes with respect to the field center was broken.
The FFDs simulations were performed for a 4 × 5 grid of field points, as shown in Table 4. The field positions were determined to measure the two-node locations and the midpoint between the nodes. The associated interferograms of Fringe Zernike astigmatism Z5/6 for the field points were generated for the simulation, as shown in Fig. 10(a). Consequently, the interferometric measurements were conducted according to the field points and three measurements were performed at each field point with 128 averages. The mean values of the RMS WFE for the associated interferograms are presented in Fig. 9. The interferograms of the Fringe Zernike astigmatism Z5/6 from these raw data were generated, as shown in Fig. 10(b). The locations of the astigmatic nodes were calculated analytically using NAT and experimentally, as shown in Table 5. The experimental results are consistent with the simulations as seen in Fig. 11(b), and they are within a three one-hundredths of a degree (0.03).
Fig. 9. Interferograms of the measurements for the misaligned state of the telescope in the presence of primary mirror figure error (@ λ = 632.8 nm). The values on each data show the RMS WFE for the field points. The measurements at the corners show slight obscuration of the primary mirror.
Fig. 10. Fringe Zernike astigmatism (Z5/6) for the misaligned state of the Cassegrain telescope in the presence of primary mirror figure error and for the associated field points: (a) simulated interferograms, (b) processed measured interferograms (@ λ = 632.8 nm).
Fig. 11. Simulations and experimental FFDs matching binodal astigmatism (Z5/6): (a) for the state of primary figure error (i.e., Z5 = 0.108 waves, Z6 = 0.001 waves), (b) for the state of combined primary figure error and a state of misalignment (i.e., XDE = 0.340 mm, BDE = 0.151 deg.).
Table 4. 4 × 5 grid of simulated field positions on the image plane.
Table 5. Coordinates of the astigmatic nodes as calculated analytically and experimentally.
When the measurements were analyzed, we saw that the axial interferometric measurement deviated by ∼0.02 waves from the primary mirror's expected field-constant astigmatic figure error. While small, the possible error sources were analyzed and classified, such as secondary mirror figure error, reference flat mirror figure error, residual alignment errors, and positioning of the interferometer focus on the focal plane. Residual alignment errors should not be critical since the amplitude of the a222 displacement vector is zero. The interferometric positioning error was also eliminated since the same guiding tool (i.e., the tool is mounted onto the baseplate and its end surface is located at the focal plane depicts the simulated field points on it) was used at the focal plane for the individual primary mirror measurement and the telescope’s axial field measurements. The astigmatic contribution of the secondary figure error was measured to be ∼0.006 waves RMS-WFE at 632.8 nm, and its contribution is insignificant. To see the contribution of the reference flat mirror, the real ray trace model of the telescope in a two-pass configuration with a reference flat mirror was simulated. The results showed that the astigmatic form error of the reference flat directly contributed to the field constant astigmatism, and we concluded that the deviation of the axial measurement (or measurements on the nodes) of the telescope from the primary figure error was caused by figure error of the reference flat mirror.
The influence of environmental effects such as vibration, temperature, or air turbulence on the stability and accuracy of the measurements were investigated through statistical analysis. For this purpose, multiple interferometric measurements (i.e., eight consecutive measurements) were conducted at the field point (i.e., −0.039°, −0.250° field point) in a misaligned state of the telescope. Figure 12(a) shows a line chart of the four parameters, Fringe Zernike astigmatism terms Z5, Z6, and pair Z5/6, and the RMS wavefront error are given for eight consecutive measurements. Figure 12(b) reports the deviations from the mean values for these four variables. The standard deviations for Z5, Z6, Z5/6, and RMS-WFE were calculated as 0.001, 0.002, 0.001and 0.001 waves at 632.8 nm, respectively.
Fig. 12. (a) Statistical analysis of the measurements at field point (−0.039°, −0.250°) in the misaligned state of the telescope, (b) the deviation from the mean values shows eight consecutive measurements data for that field point in four different variables (@ λ= 632.8 nm).
The influence of Zernike Fringe fit error analysis is considered for the annular aperture case of the obscured Cassegrain telescope, which has 20% secondary mirror obscuration in optical design. When the secondary mirror’s spider contribution to obscuration was taken into account, the overall obscuration was calculated to be 27%. A similar obscuration ratio was studied by Hou et al. [28], who showed that the Zernike Fringe fits are similar for both the annular and circular apertures when the obscuration is less than about 30%. Thus, the contribution of fit error is insignificant.
5. Conclusion
This work presented an experimental investigation of NAT for a high-precision Cassegrain telescope. The primary mirror figure error and the secondary misalignment both induced binodal astigmatism. In this work, we showed in an experiment that these two contributions to the astigmatic field response could be separated. Distinguishing these two aberration sources plays a crucial role in the telescope alignment of modern telescope systems. The primary mirror figure error induced by the mirror mount was measured utilizing a custom CGH. Simulations were performed, and measurements were made with the aligned telescope at twenty-five field points to validate binodal astigmatism induced by figure error. The astigmatic nodes were shown to be located symmetrically with respect to the field center as predicted by NAT. Thereafter, a secondary mirror misalignment was introduced into the telescope system. The simulations and interferometric measurements were performed for the misaligned state of the telescope in the presence of primary mirror figure error at twenty field points to validate the asymmetry of the astigmatic nodes from the field center. The secondary mirror was kept in a coma-free pivot position to isolate the astigmatic field response from the misalignment state. The experimental results were compared with the analytical calculations of NAT, and values agreed with each other within 0.06 waves RMS at 632.8 nm and a three one-hundredths of a degree (0.03). The analysis showed that the astigmatic figure error contribution of the reference flat mirror in a two-pass experimental setup was small. When measured, this error can be calibrated out. Statistical analysis was also conducted to demonstrate the accuracy and stability of the measurements. The analysis showed that the environment was not the limiting factor of uncertainty. As NAT predicts, the primary mirror figure error (set as the aperture stop) and the secondary mirror misalignment both induce binodal astigmatism, which NAT can help separate.
Funding
Fulbright Association (FY-2019-TR-PD-06); Robert E. Hopkins Center at the University of Rochester.
Acknowledgments
Özgür Karcı thanks the Fulbright Commission for their partial support and the University of Rochester for their hosting during his Fulbright fellowship and continued collaboration on experimentally validating Nodal Aberration Theory. Özgür Karcı thanks Synopsys Inc. for providing a free academic CODE V license.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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