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Abstract
Lateral resolving power is a key performance attribute of Fizeau interferometers, confocal microscopes, interference microscopes, and other instruments measuring surface form and texture. Within a well-defined scope of applicability, limited by surface slope, texture, and continuity, a linear response model provides a starting point for characterizing spatial resolution under ideal conditions. Presently, the instrument transfer function (ITF) is a standardized way to quantify linear response to surface height variations as a function of spatial frequency. In this paper, we build on the ITF idea and introduce terms, mathematical definitions, and appropriate physical units for applying a linear systems model to surface topography measurement. These new terms include topographical equivalents of the point-, line-, and edge-spread functions, as well as a complex-valued transfer function that extends the ITF concept to systems with spatial-frequency-dependent topography distortions. As an example, we consider the experimental determination of lateral resolving power of a coherence scanning interference microscope using a step-height surface feature to measure the ITF directly. The experiment illustrates the proposed mathematical definitions and provides a direct comparison to theoretical calculations performed using a scalar diffraction model.
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1. INTRODUCTION
When choosing an instrument configuration for a specific application in surface topography measurement, one of the most important considerations is the ability of the measuring instrument to resolve closely spaced surface features and measure their heights. Experienced users of optical instruments know that it is difficult to get accurate height measurements when the spacing of features approaches traditional resolution limits, such as the Rayleigh criterion, which depends on the wavelength and the numerical aperture of the illumination and imaging optics. This awareness leads to the common practice of specifying the resolving power of optical topography measuring instruments using classical optics parameters, even though the principles of dimensional metrology differ significantly from the formation of intensity images [1].
The temptation to use optical imaging principles and nomenclature when describing the instrument performance extends beyond the Rayleigh criterion to the modulation transfer function (MTF) for describing the spatial frequency response [2–4]. However, although the response for optical measurements of topography often resembles the imaging MTF, this is by no means always the case, and it is important to have distinctive terminology and definitions for topography measurement in place. After many years of discussion, the consensus in the international standards community and the literature is to define an instrument transfer function (ITF). While the MTF is based on image contrast, the ITF is based on surface heights [5–7]. Provided that the response is at least approximately linear, the ITF offers a more informative way to characterize the resolving power than a single number, such as the minimum spacing of neighboring surface features. Notably, the ITF is defined without reference to the measurement principle, and it is generally applicable to optical, tactile, or any other technique.
ITF characterization is now common for interferometers measuring optical components such as lenses, mirrors, and optical flats [8–12], and standardized performance parameters such as the lateral period limit are now defined in terms of the ITF [5,13]. The ITF also characterizes 3D optical microscopy, although with restrictions on surface slope, roughness, and continuity, under what is best described as ideal conditions [7]. Advanced physical-optics models have shown that a linear response requires a sufficiently high numerical aperture (NA) to capture all the significant diffracted light, and smooth object surfaces without multiple scattering [14,15]. If the surface has grooves or sharp steps, the height variations for these features cannot exceed an eighth of a wavelength within the optical resolution cell [16,17]. Nonetheless, ITF curves defined for idealized conditions provide a meaningful starting point for instrument specifications and determining basic metrological capability.
While the ITF is now a frequently used tool for characterizing topographical spatial resolution, at present, there is no consistent or standardized framework for a linear systems model of topography measurement. For optical methods of topography measurement, it is common to borrow terms such as the point spread function, which in imaging systems often relies on the coordinates expressed in wavelengths or normalized to focal length, and measurements of irradiance in the image plane that do not apply directly to surface topography analysis. This practice can lead to confusion and misinterpretation of the factors that influence the quality of topography measurements. The formation and interpretation of interference fringes for surface height, for example, can have significantly different response characteristics than optical imaging. What is missing is a linear systems model of topographical spatial resolution using consistent terms, definitions, mathematical derivations, and physical units specific to the dimensional metrology of surface form and texture.
In this paper, the ITF serves as a starting point to introduce terminology for a linear systems model adapted to the surface topography measurement. New definitions for relevant functions are directly linked via Fourier relationships to a complex-valued version of the ITF, without relying on image-related concepts such as pinhole apertures and irradiance distributions. The new definitions include physical units of length relevant to dimensional metrology with self-consistent magnitudes for transformed quantities such as surface height maps and their frequency-domain equivalents.
To illustrate the use of our proposed linear model of topography measurement, we reconsider the well-known step-height test for ITF evaluation, based on the measurement of a 40 nm tall sharp-edged surface feature [18–22]. This test is an ideal example in that it provides clear topographical equivalents to the optical imaging point-, line-, and edge-spread functions, leading to a complete representation of the topography measurement within the scope of applicability of a linear response model. Experimental results using an interference microscope are compared directly with predictions from scalar diffraction theory to illustrate the meaning, usefulness, and relevance of the new definitions for characterizing lateral resolution Finally, we consider some of the limitations and prospects of linear models for topography measurement, with suggestions for further work.
2. ITF CONCEPT AND DEFINITION
In linear systems theory, a transfer function quantifies the system’s output as a function of an input. Most often, transfer functions are defined for linear-shift invariant systems so that the system’s response to a linear input is also linear [23]. Transfer functions based on a frequency analysis quantify the amplitude response to sinusoidal basis functions representing the Fourier components of the input. For topography, the Fourier components of the ITF are equivalent to sinusoidal gratings parameterized by surface spatial frequency [21,24].
The definition of the ITF along a single surface coordinate is
where ${a_{{\rm{out}}}}$ and ${a_{{\rm{inp}}}}$ are the output and input amplitudes, respectively, for the spatial frequency $\nu $ of sinusoidal components of the topography along a specified profile direction [5]. Equations for surface profilecomponents consistent with this definition may be written as where ${h_{{\rm{out}}}}$ and ${h_{{\rm{inp}}}}$ are output and input values, respectively, in the $z$ direction for an $x,y,z$ coordinate system for which $x,y$ are lateral surface coordinates. The phase shifts ${\phi _{{\rm{out}}}}$ and ${\phi _{{\rm{in}}}}$ account for the lateral position along $x$ for the constituent profile component. The definition in Eq. (1) presents the ITF as a one-dimensional (1D) function of spatial frequency. In the more general case, the ITF is a function of two spatial frequencies $\nu ,\mu$, defined as the conjugate variables to the spatial coordinates $x,y,$ respectively.Figure 1 illustrates common isotropic ITF behavior for a surface-topography-measuring instrument using incoherent light, such as an interference microscope with a diffuse white light source [14]. Note that ${f_{{\rm{ITF}}}}$ is a unitless function with a value equal to one at zero spatial frequency [3]. Differences between input and output values at zero spatial frequency, rather than being included in the ITF, are characterized by a constant scale factor, defined as the amplification coefficient in ISO 25178-600 [5].
Fig. 1. Simulated ITF curve based on modeling of an interference objective with incoherent illumination. Here, the upper spatial frequency limit is 1.4 cycles/µm.
Figure 1 resembles the incoherent optical imaging MTF familiar to optical engineers. However, the ITF is a general concept applicable to a wide variety of instrument types, not just optical instruments. Common topography measurement techniques with ITF curves that differ from an imaging MTF include tactile methods [25], interference fringe contrast methods [26], focus variation microscopy [27,28], and laser point-scanning [29]. Because the ITF represents the complete instrument response, it is influenced by hardware, data acquisition, and software choices made in the instrument design.
3. MEASURING THE ITF
The experimental evaluation of the system’s response involves introducing a known input and recording the results. For evaluating the ITF, a pure sinusoidal input is desirable, but it is not easily realized over a wide range of spatial frequencies. One compromise is a frequency-chirped grating [30,31], shown in Fig. 2 as an illustrative simulation. Other examples of chirped gratings include those with amplitudes that decline with frequency so that the maximum slope for each frequency is a constant—a meaningful constraint for optical systems [32]. These types of material artifacts require certified calibration using reference metrology.
Fig. 2. Simulation of the measurement of chirped sinusoidal input topography and corresponding output for the ITF shown in Fig. 1.
Non-sinusoidal disturbances for linear systems analysis include the impulse, line, and step functions. A step function is a particularly useful disturbance. The method is often described as analogous to the slant-edge, knife-edge, or edge-spread function test for determining the MTF of optical imaging systems [33–35]. In topography, the corresponding object feature is a sharp transition in height between two otherwise flat surface areas [18–22]. Test samples for the step-height test are readily available, either as step standards for height calibration, or as edge features on a wide variety of example objects that are not necessarily designed for calibration. A step should be sufficiently high to provide data above the measurement noise level, but small enough, as noted in the Introduction, to avoid nonlinear behavior that can sometimes accompany discontinuous surface features [21,36]. Apart from the step itself, the object should be smooth and free of texture that could result in additional irrelevant surface structure in the topography map. A further requirement is that the edge should be sharp—the step transition should take place over a length scale that is much smaller than the topographical lateral resolution of the instrument. With these modest requirements, the blurred profile of the step, as shown in Fig. 3, provides information about the ITF without the need to calibrate the actual step-height value using independent reference metrology.
Fig. 3. Simulation of a topographical step-height feature, together with expected response, using the same ITF as for Fig. 2.
Figure 4 illustrates the principle of the step-height test. The Fourier transform of the step object—equivalent to the square root of the power spectral density—is compared with the frequency content of the measured topography, which is attenuated by the ITF [19,37]. As we shall see, practical data processing does not rely on this comparison of Fourier magnitudes; however, Fig. 4 illustrates that a simple step provides the information we need for a practical ITF determination. To further examine the step-height test for ITF, it is useful to develop terms and definitions for a linear model of surface topography measurement.
Fig. 4. Comparison of the 1D Fourier transform of the original sharp object step and the measurement shown in Fig. 3. Note the expected inverse frequency dependence for the Fourier magnitudes of the original sharp step.
4. INSTRUMENT RESPONSE TO TOPOGRAPHY: TERMS AND DEFINITIONS
A. Preliminaries—Fourier Transforms
Linear systems lend themselves naturally to Fourier frequency analysis. However, there is potential for inconsistency in the definition of Fourier transforms, depending on the context. For example, for surface topography parameters, it is common to normalize the forward Fourier transform by the evaluation length or area [38]. Conversely, software algorithms often have the exact opposite normalization and divide the discrete inverse Fourier transform by the number of samples in the frequency domain [39]. These differences can result in inconsistent physical units and ambiguous magnitudes for transformed quantities.
For clarity, we define the 1D Fourier transform of a spatial-domain function $f$ with a single variable $x$ at a frequency $\nu$ as
The inverse Fourier transform isFor a numerical calculation over a finite number $N$ of $x$ values, using the midpoint rule for numerical integration, the Fourier transform in Eq. (4) becomes
where $\Delta x$ is the sampling interval. The inverse Fourier transform is then where the frequency interval $\Delta \nu$ equals the inverse of the $x$-axis evaluation length $N \Delta x$:Note that the Fourier transform $F$ is a spectral density that scales with the inverse of the sampling interval $\Delta \nu$ in the frequency domain. As an example, the discrete Fourier transform of a real-valued sinusoid of amplitude ${a_{{o}}}$ and frequency $\nu$ has a magnitude of $a_{o}/2\Delta \nu$ at Fourier frequencies ${\pm}\nu$. The value of ${a_{{o}}}$ at the frequency $\nu$ can be calculated by multiplying the spectral density $F$ by the frequency interval $\Delta \nu$, equivalent to dividing by the evaluation length $N \Delta \nu$. In this paper, to avoid confusion, we apply consistently the symmetric definitions of Eqs. (5) and (7) without optional normalizations.
B. Topographical Transfer Function
In optical imaging, the MTF is the modulus of a complex optical transfer function (OTF) [3]. The OTF includes the effects of lateral image shifts resulting from imperfections in the imaging process as a phase in the complex valued OTF, particularly for asymmetric aberrations. In analogy with the OTF, we define a complex, Hermitian topographical equivalent of the OTF as the topographical transfer function (TTF):
The TTF fully characterizes linear instrument response, including lateral shifts of sinusoidal surface topography components as a function of spatial frequency, in the same way as the OTF does for imaging systems [7]. In principle, if the instrument response is linear, we can correct for imperfections and irregularities in the frequency domain using the function ${f_{{\rm{TTF}}}}$.
C. Topographical Point Spread Function
In optical imaging, the point spread function (PSF) is sometimes described as the normalized intensity distribution in the image of a point source [40]. For topography, one can imagine a narrow surface height feature, such as a spike or microsphere, that can serve as an analogous structure to a point source [41]. However, it is not necessary to imagine or to physically realize a point-like topography feature to define a function comparable to the imaging PSF for topography. Here we define the topographical point spread function (T-PSF) directly as the spatial domain equivalent of the TTF. This is consistent with the formal definition of the imaging OTF in the ISO 9334 standard as the Fourier transform of the imaging PSF [3].
From the observations above, we propose to define the T-PSF as a function ${P}$ such that
The inverse Fourier transform of both sides of Eq. (17) shows that which expresses the transfer function as a convolution in the spatial domain by the T-PSF. Because the TTF is a unitless ratio of input and output values, the T-PSF necessarily has units of inverse length squared, following the definition of the inverse Fourier transform for the spatial frequency variables $\nu ,\mu$.D. Topographical Line Spread Function
If we have an experimental determination of $P$, we can in principle calculate the complete TTF using a bivariate inverse Fourier transform. However, it is common to determine a cross-sectional representation of the TTF or its modulus of the ITF along a specific frequency coordinate, for example $\nu$, corresponding to the lateral coordinate $x$, as illustrated in Fig. 1. From the definition of the 2D Fourier transform in Eq. (9) and by setting $\mu = 0$ with implied units of spatial frequency, we have
The T-LSF is meaningful for predicting the measured topography of objects that vary with $x$ but have no $y$ dependence. It also provides a way to calculate the cross section of the TTF in one direction by its inverse Fourier transform. Conceptually, the T-LSF is the instrument response to a thin isolated line or wall on a flat surface. Although the T-LSF in principle can be measured using such an object, it is more common to use a step-height feature, as we now describe.
E. Topographical Edge Spread Function
An approach to the practical determination of the T-LSF is to define a topographical edge-spread function (T-ESF) as the convolution of the function $L$ with a unitless step function $s$ along the $x$ direction:
Consequently, a convenient calculation of the TTF is to Fourier transform the derivative of a measured topographical step-height profile—a data analysis similar to the knife-edge method for the optical MTF [42,43]. Note that Eq. (31) can also be derived by convolving a 2D step function with the T-PSF. Also note that we can measure the complete 2D TTF by evaluating the T-LSF along multiple directions on the surface [44].
The meaningful Fourier frequency range depends on the range of measured height values along $x$, that is, the field of view (FOV) in this direction. A larger FOV allows for calculating the TTF at lower spatial frequencies. With a sufficiently large FOV, there is no need to calibrate the physical height ${a_0}$ of the step, as the modulus ITF of the TTF by definition tends toward one as we approach a zero spatial frequency.
F. Summary of Terms and Definitions
Table 1 provides a summary of the topographical response functions. The table includes new definitions in the context of topography, such as the topographical point spread function (T-PSF) in analogy with the imaging point spread function (PSF). We present the imaging and topography versions of linear systems functions side by side. The unit column uses $\unicode{x00B5}{\rm m}$ as representative of length units applicable to topography-measurement-related terms.
Table 1. Comparison of Imaging and Topography System Response Functions, Including the Physical Units for Topography Measurements
5. EXPERIMENTAL DEMONSTRATION
A. Approach
To illustrate the use of the linear systems’ functions in Table 1, we performed an experimental determination of the TTF and its modulus—the ITF—using the step-height method. Figure 5 summarizes the sequence of events. For the experiment, each step in the calculation is compared with an independent prediction using a physical optics model for verification of the method and comparison with expectations using nominal configuration values for the instrument setup.
B. Measurement System and Procedure
The candidate instrument is a Zygo NewView 9000 coherence scanning interferometry (CSI) microscope. The operating principles of this instrument have been described in detail in several review articles [45–47]. CSI microscopes determine surface topography by interpretating the interference contrast and phase detected from a sequence of camera images during an axial scan of the interference objective. In our measurements, we use a measurement mode that relies on interference phase for the final topography map. This mode is known to have low noise for topography measurement and a linear response for smooth surfaces with small surface slopes and discontinuous step features of less than one-eight of the mean operating wavelength. The optical configuration is summarized in Table 2 for a ${{20}} \times$ Mirau interference objective and ${{2}} \times$ zoom. The nominal values in the table are taken from specifications for the objective and camera, and the measured mean spectral wavelength of the light source. The adjusted values are referred to later in this section, when the experimental and theoretical ITF curves are compared.
Table 2. Optical Configuration for the Experimental Work
A suitable sample part is a Zygo precision lateral calibration standard specimen, most often employed to calibrate the lateral $x,y$ scales. The specimen is an array of square raised features, 40 nm tall and arranged in a grid with a 200 µm pitch. For the present purpose, it provides edge features oriented in both $x$ and $y$ directions at multiple locations in the field of view. The required sharpness is verified by measuring the step at a significantly higher ${{100}} \times$ magnification. Prior to the measurement, the instrument is precisely focused using a sample grating and software to evaluate image sharpness.
C. Theory
To compare experimental results with expectations for each step in the Fig. 5 flowchart, we simulate results using a complete physical-optics model of the instrument, initially without assumptions regarding linearity. The model employs scalar diffraction theory in the Kirchoff approximation for the surface scattering, together with a Fourier optics analysis for the creation and interpretation of interference fringes at the camera via the imaging optics [48].
The simulation proceeds in reverse order to the Fig. 5 flowchart, with a calculation of the instrument response for a sequence of simulated single-frequency sinusoidal surface gratings. After verifying that the predicted instrument response is sufficiently linear, the resulting theoretical ITF curve is transformed to the T-LSF and T-ESF functions. Reference [14] provides details of the software implementation of this model for the calculation of spatial frequency response, together with methods for quantifying and verifying linearity.
D. Results
Figure 6 illustrates the appearance of the topography for one of the square features of a lateral calibration standard. An initial data analysis locates suitable edge features, from which profiles are extracted and averaged to produce the step-height curve shown in Fig. 7. The corresponding T-ESF is the measured step divided by the nominal step height value ${a_o}$. There are many strategies for performing edge finding and averaging for discretely sampled profiles, some of which are documented for the knife-edge test for the optical imaging MTF [33,49]. The derivative or gradient of Fig. 7 results in the T-LSF curves shown in Fig. 8. It is easier to detect some slight differences between experiment and theory in the T-LSF, with the experimental results slightly broader than theory using the nominal optical configuration in Table 2.
Fig. 6. Topographical image of a raised square topographical step. The upper image is from the top down, while the lower image is the cross-sectional profile showing height and width and two topographical steps on either side of the raised area.
Fig. 7. Experimental step height measurement for a ${{20}}\; \times$ Mirau interference objective, compared to theory for the nominal optical configuration defined in Table 2.
The difference between experiment and theory is more evident in the ITF curves in Fig. 9, where there is a clear difference in the mid-spatial frequencies around ${{0}}{\rm{.7}}\,\,\unicode{x00B5}{{\rm{m}}^{- 1}}$. This illustrates one of the useful characteristics of the ITF in that it clearly identifies differences between results. Repeated measurements reproduce the experimental data to a level approximated by the size of the markers in the graph, implying that the true resolving power of the instrument is slightly less than what one would expect from the nominal optical imaging parameters.
Fig. 9. ITF obtained from the Fourier transform of the T-LSF measurements in Fig. 9, compared to theory for the nominal optical configuration described in Table 2.
Observing the difference in Fig. 9, we can speculate on possible causes. To illustrate this, Fig. 10 shows a better match of experimental to theoretical ITF achieved by adjusting the imaging NA value and illumination fill factor. Both adjustments are plausible, given vignetting in the objective for the imaging NA and underfilling of the objective pupil in incoherent Köhler illumination. This finding implies that the topographical spatial resolution could be improved by adjusting the illumination to better fill the pupil plane, and by reviewing the optical design to make sure that the instrument makes best use of the available NA of the objective. However, it is equally probable that the loss of mid-spatial frequencies is attributable to factors unrelated to the optical imaging system, such as cross talk between camera pixels or the specific way in which the interference data are converted to surface heights. To determine the cause for this specific example requires other measurements that are beyond the scope of this paper. The key idea is that the experimental ITF evaluation provides valuable information regarding the true lateral resolving power of the instrument.
Fig. 10. Experimental ITF from Fig. 9, compared to theory for the adjusted optical configuration described in Table 2.
6. DISCUSSION AND CONCLUSIONS
Commonly observed effects of limited topographical spatial resolution include merging of neighboring surface height features, attenuated response to high spatial frequencies, and the smoothing of shallow, sharp-edged surface height transitions. To describe these effects within a linear systems model, we require terms, definitions, and physical units relevant to the dimensional metrology of surface form and texture. After describing in Section 2 the now standardized ITF curve for resolution in terms of surface spatial frequencies, we have introduced new terminology for impulse response and spread functions in Table 1 adapted to the measurement of surface topography. The mathematical development in Section 4 clarifies the meaning of each term, as well as the physical units. These new definitions are directly linked by Fourier relationships to the TTF, which is the complex-valued version of the ITF. The experimental measurement of the ITF using a step-height feature described in Sections 3 and 5 illustrates the usefulness of evaluating spatial resolution using these terms when the instrument response is at least approximately linear.
The present work leaves many important questions open. As a start, it is worth discussing whether the terminology introduced here, or other terms, should be standardized. Our view is that they should be, just as analogous terms in optical engineering have been standardized [2]. Terms related to topography measurement should include appropriate units of length, relevant to dimensional metrology of surface structure. The formal definition should be decided by organizations such as the International Standards Organization (ISO).
Another question is whether the step-height test is the best approach to characterizing the ITF. Almost certainly it is not, at least, not in a general way. Alternatives that may prove more effective depending on the instrument and the measurement range include topographical star patterns [50,51], periodic rectangular patterns [52], irregular roughness calibration geometries [53], and pseudo-random binary arrays [54,55]. Here, the determination of the ITF using a surface step has been useful to clarify the usage of the terms introduced in Table 1. It is also convenient in that test artifacts are readily available and may already be in the toolbox of the average instrument user. Furthermore, the test has been used in practice for more than two decades and is used for routine verification of ITF specifications [13]. However, the step-height test has some important weaknesses. Figure 1, for example, shows that the signal to noise for ITF evaluation decreases rapidly with spatial frequency. There is ongoing research into alternative methods with the goal of obtaining the best possible experimental determination of ITF for a given instrument type and range of application.
Given that ITF characterization assumes linearity, one might reasonably ask how we can quantify lateral resolution when the response is nonlinear, which is common. Mechanical stylus instruments, for example, have inherent morphological filtering—a nonlinear process—resulting from the finite radius of the stylus tip [25]. As noted in this paper, optical instruments can have narrow operating restrictions for linear behavior [16,56]. A more general characterization is the topographical spatial frequency response (T-SFR), which quantifies response to isolated sinusoidal surface structures at specific spatial frequencies and amplitudes, even if the measurement is nonlinear [7,14]. The ITF is then the limit case of the T-SFR when the response is linear within established boundaries. This terminology is also not standardized, but it has some foundation in ISO documents related to photography [4]. The straightforward Fourier relationships outlined in Table 1 do not apply to a nonlinear T-SFR. Hence the step-height test is also not applicable—the most appropriate nonlinear T-SFR evaluation is a sequence of isolated sinusoidal topography features or the equivalent.
Finally, perhaps the most interesting questions relate to the meaning and use of the ITF for determining metrological capability, traceability, and uncertainty. It can be argued that the ITF represents an error whenever its value differs from one. This implies that it should be corrected, as indeed it often is, especially when using instruments of different types to measure derived parameters such as the power spectral density [9,57–59]. A proper correction for complete surface topography, as opposed to a derived parameter, should use the TTF, which includes phase shifts, as well as the attenuation as a function of spatial frequency [7]. An alternative view is to regard the ITF simply as an instrument specification, which should be included in the definition of the measurand, defined as the scale-limited surface topography [60]. The contribution to uncertainty for the ITF is the uncertainty in its determination, not its nominal value.
Whether the intent is to compensate the ITF, include it in the definition of the measurand, or simply use it to optimize a measurement setup, knowledge of the ITF is essential. Here we have described one experimental approach, the well-known step-height test. The description includes defining terminology and linear systems functions interrelated with Fourier transforms, in the context of topography measurement. These definitions should prove useful for a variety of approaches both established and in development.
Useful next steps include establishing the contribution to the total measurement uncertainty of inaccuracies in determining the linear ITF, as well as the contribution from residual nonlinear response. Further useful research is to quantify the ITF for a wide variety of optical configurations and establish the scope of applicability of both experimental ITF methods more thoroughly, as well as software instrument models.
Funding
Zygo Corporation Internal Research and Development.
Acknowledgment
We would like to thank Dan Russano and Anthony DePasquale for valuable discussions and assistance in operating the Zygo NewView interference microscope.
Portions of this work were carried out in coordination with the EMPIR TracOptic project (20IND07), for which Zygo Corporation is an unfunded partner.
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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