Comparison of Fourier optics-based methods for modeling coherence scanning interferometry

Abstract. Coherence scanning interferometry (CSI) is a widely used optical method for surface topography measurement of industrial and biomedical surfaces. The operation of CSI can be modeled using approximate physics-based approaches with minimal computational effort. A critical aspect of CSI modeling is defining the transfer function for the imaging properties of the instrument to predict the interference fringes from which topography information is extracted. Approximate methods, for example, elementary Fourier optics, universal Fourier optics, and foil models, use scalar diffraction theory and the imaging properties of the optical system to model CSI surface topography measurement. In this work, the simulated topographies of different surfaces, including various sinusoids, two posts, and a step height, calculated using the three example methods are compared. The presented results illustrate the agreement between the three example models.


Introduction
Interference microscopy, particularly coherence scanning interferometry (CSI), 1 is a popular optical technique for high-precision surface topography measurement. 2,35][6][7] The development of these models addresses the practical need for a better understanding of the instrument characteristics and performance specifications, optimization of instrument configurations for good practice, and uncertainty estimation using virtual instruments.
1][12] A number of practical approximate models have been developed with known limitations in their validity ranges, 11,13 including the neglect of near-field and polarization effects, multiple scattering, and surface films. 14These approximate models serve a useful purpose constrained by the fundamental limits of scalar diffraction and linear imaging theory. 15,16sing approximate models, the formation of interference fringes can be considered as a linear filtering operation characterized by a transfer function (TF) in the spatial frequency domain.Linear systems theory has been extensively applied to 2D optical imaging. 10The linear systems theory approach to interferometric imaging allows in many cases for the compensation of measurement errors by the application of an inverse filter. 17Furthermore, approximate models are easier to implement than more rigorous solvers of Maxwell's equations and are computationally efficient and can provide insight into fundamental sources of measurement error related to light scattering and imaging. 18lementary Fourier optics (EFO), 19,20 universal Fourier optics (UFO), 21,22 and the foil model 16,23 benefit from scalar approximation methods that consider the imaging properties of the optical system.These models assume that local surface curvatures are small enough to comply with Kirchhoff's approximation; 11 however, each method uses a different approach to model the surface and the TF.EFO models the surface as a phase object together with classical Fourier optics methods and a 2D partially coherent optical TF.EFO methods, along with a 2D representation of the propagating light field, have been used to model an interference microscope 19,24 and to predict the linear instrument TF and residual nonlinear measurement errors for optical measurements of surface topography. 25The UFO method also uses the phase object approximation and a 2D TF, where the 2D TF equals the horizontal cross-section of a 3D TF. 21In the foil model, the surface is defined as a 3D thin foil-like object, and the 3D TF maps this surface to the interference fringes. 16The foil model has been used in various surface topography measurement applications including signal modeling, 16,26 calibration and adjustment of the 3D TF 27 and lens aberration compensation 28 in a CSI instrument.Applications of the foil model are not limited to interference microscopy but can also be extended to 3D image formation in focus variation microscopy. 29hile the mathematical derivations in the literature for the EFO, UFO, and foil models differ from each other, we shall show here that they predict the same measurement results within their respective validity regimes.To demonstrate the comparability of the EFO, UFO, and foil models, we perform numerical calculations based on simulated object profiles that include sinusoids, step heights, and closely space rectangular surface features, for different instrument configurations, such as numerical aperture (NA) and light source spectrum.Some primary results have been previously presented. 30These results demonstrate the consistency of these approximate methods based on similar scattering and imaging theories and improve confidence in approximate methods as a foundation for the development of virtual CSI instruments.

EFO, UFO, and Foil Models
The EFO, UFO, and foil models are well-established approximate models.5,31 In all three models, imaging of the surface topography is described as a linear filtering process characterized by a TF.In the following sections, we briefly describe how the TF, object, and image are simulated in each model.

EFO
In the EFO model, the contribution of surface topography in interference microscopy modeling is approximated by introducing a phase shift proportional to the surface heights z ¼ h o ðxÞ to the object light field (i.e., the light field immediately after reflection).Assuming uniform monochromatic illumination and surface reflectivity, the 2D object field is approximated as 19 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 4 ; 1 5 1 U o ðxÞ ¼ exp½−i2πKh o ðxÞ; (1) where K ¼ 2∕λΩ is the interference fringe frequency, λ is the wavelength of the incident light, and Ω is the obliquity factor that approximates the effect of the illumination geometry by integrating over all incident angles.This approximation is a significant simplification compared to pupil integration methods, 32 including 3D TF models that calculate the contribution of each incident wave vector within the pupil plane independently. 33This simplification enables a classical 2D Fourier optics analysis, at the expense of disregarding focus effects on surfaces with large height variations.The image field is obtained by applying a filtering operation in the spatial frequency domain using a 2D partially coherent TF (PCTF) 10 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 7 ; 7 0 0 where k x is the projection of the scattered wave vector in the pupil plane; Ũs ðk x Þ and Ũo ðk x Þ are the Fourier transforms or plane wave spectra of the image and object fields, respectively; and Õðk x Þ is the PCTF.As an example, for an interference microscope with Köhler illumination and a filled illumination pupil of the same size as the imaging pupil, the PCTF is similar in form to the modulation TF for conventional imaging E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 7 ; 6 1 3 where k max ¼ 2A N ∕λ and A N is the value of the NA.The scattered field in the image plane U s ðxÞ is given by the inverse Fourier transform of Ũs ðk x Þ.For a broadband source, Eqs. ( 1)-( 3) are repeated to give U s ðx; KÞ.Although it is possible to simulate the interference fringes in the EFO model (inverse Fourier transform of U s ðx; KÞ along z-axis), surface topography can be calculated directly from the image field. 19,20The limits of applicability for EFO modeling are reported elsewhere. 20

UFO
In the UFO model, the optical field U o ðx; yÞ on a surface h o ðx; yÞ immediately after reflection is given as where K z is the component of K along the z-axis, and 31 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 7 ; 4 1 3 In Eq. ( 5), k s and k i are the scattered and incident wave vectors characterized by scattered and incident angles θ s and θ i in relation to the z-axis, respectively, and k 0 ¼ 1∕λ is the wavenumber.Unlike the EFO model, the effect of multiple illumination incident angles and orientations is included in the UFO method, to account for focus effects.
The interference intensity between the object and the reference field in the K-space results from frequency domain filtering of the Fourier representation of the object field Ũo ðKÞ as 34 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 7 ; 3 0 1 ĨðKÞ ∼ Ũ0 ðKÞ HðKÞ; ( where HðKÞ is the 3D optical TF of the imaging system.An analytical form for the 3D TF follows from the 3D correlation of the spherical caps corresponding to the incident and scattered wave vectors. 35,36It has been shown that the shape of the 3D TF of a diffraction-limited interference microscope with uniform monochromatic pupil illumination depends on the surface under investigation. 31,35For piecewise continuous surfaces, the normalized 3D TF for monochromatic light of wavenumber k 0 results in 34 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 1 1 7 ; 2 0 3 where K z;min , K z;0 , and K z;max are given as To consider polychromatic light, the individual monochromatic 3D TFs are superimposed after weighting to account for the spectral distribution.

Foil Model
Consider a monochromatic plane wave U i ðrÞ ¼ expð2πik i :rÞ propagated with a 3D wave vector k i illuminating a 3D scattering object with a surface height function of h o ðx; yÞ.Using the integral theorem of Helmholtz and Kirchhoff, the scattered field can be expressed as a surface integral. 13Applying Kirchhoff's boundary conditions 11 and the free-space Green's function into the integral, the scattered far-field can be written as 16 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 9 ; 1 1 4 ; 5 3 1 where K ¼ k s −k i , k s is the scattering wave vector, k 0 ¼ 1∕λ is the wavenumber, and R is the amplitude reflection coefficient.The term Rδ½z − h o ðx; yÞ is proportional to what is referred to as the "foil model" of the surface. 16Note that the Fourier transform of the foil representation of a surface, expressed by a 1D Dirac delta function follows the surface height with respect to the z-coordinate, resulting in the phase object representation of the electric field shown in Eq. ( 4) as follows: Therefore, the phase object presented in the UFO model and the surface foil described in the foil model both offer an equivalent approach to modeling the object.
In interference microscopy, the scattered field over the surface is obtained by a 3D surface TF (STF) of a microscope objective with a finite NA and a pupil apodization function of PðkÞ.The STF with regards to the incident wave vector k i is a truncated spherical shell expressed as 16,23 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 1 1 4 ; 3 2 5 For an ideal aplanatic case, . Using the definition of the STF and the foil model of the surface, Eq. ( 9) can be re-written as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 2 ; 1 1 4 ; 2 6 2 where FðKÞ is the 3D Fourier transform of the foil model of the surface.Using Eq. ( 12) and considering all possible incident wave vectors, the Fourier transform of the interference term between the incident and scattered field is given as 23 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 3 ; 1 1 4 ; 1 8 6 This equation represents the product of the 3D Fourier transform of the foil model of the surface and the optical 3D TF according to the foil model considering all possible incident and scattered wavevectors.The interference fringes can be obtained by applying an inverse Fourier transform to Eq. ( 13).
In this section, the simulated TFs and measurement results for the EFO, UFO, and foil models are compared for measurements that use interference phase to determine surface profiles.In all simulations, the light source is assumed to have a Gaussian wavenumber spectrum with a mean wavelength of 0.57 μm similar to a common CSI instrument.The interference microscope is configured with Köhler illumination and the objective aperture is fully filled (the illumination pupil is equal to the observation pupil).We also assume that the objective pupil function is consistent with an aplanatic imaging system 33 satisfying Abbe's sine condition.

Comparison of the Simulated Profiles
The detailed specification of the nominal profiles and the optics, including NA, objective, and full-width at half maximum (FWHM) wavelength bandwidth of the light source, considered for simulation are shown in Table 1.For each test, all models share the same lateral resolution, which is embedded within the simulated TF and determined by the NA specific to that particular test.
As a part of CSI modeling, the EFO, UFO, and foil models can simulate the interference signal.Figure 1 shows an example of the simulated CSI signal obtained by the (a) EFO, (b) UFO, and (c) foil models for the S1 test in Table 1.In interference microscopy, the surface topography can be obtained using an appropriate surface reconstruction method, e.g., envelope detection, 37 frequency domain analysis (FDA), 38 and the correlogram correlation method. 39The FDAenvelope method provides a first estimation of the surface height corresponding to the location of the coherence envelope.This can be achieved by fitting a linear model to the Fourier component phases in the spatial frequency domain.Using FDA-phase, the height value is calculated by interpolating the linear fit at the spatial frequency for which the Fourier magnitude is greatest.In the context of the UFO model, it has been shown that the lateral resolution for interference microscopy can be enhanced by selecting specific Fourier components, rather than using linear phase fitting. 34However, in this paper, the primary goal is the comparison of different scattering and imaging models, rather than the comparison of reconstruction algorithms.Hence, the interference signal data generated by the EFO, UFO, and foil models are analyzed using the same FDA-phase algorithm.Simulated measurement results obtained by the EFO, UFO, and foil models using the configurations shown in Table 1 are illustrated in Fig. 2. In Fig. 2, (a)-(i) show the nominal and reconstructed profiles obtained by the EFO, UFO, and foil models along the x-axis while (a')-(i') illustrate the difference between the reconstructed and nominal profiles (that is, the predicted height measurement error) for each modeling method.Finally, the relative signal strength of the interference fringe data for the UFO, EFO and foil models are shown in (a")-(i").The relative signal strength plot shows the amplitude of the interference signal, normalized to the highest signal in the dataset.The fringe signal data for the S1 test in Fig. 2(b") clearly shows that at steep slopes, the signal level (interference fringe contrast) is lower than at the peak and valley positions.Due to the overlapping of the data, the individual curves cannot be distinguished in most of the subplots of Fig. 2. The inset of Fig. 2(h) illustrates the differences between the reconstructed profiles, in this case, considered negligibly small.In Fig. 2(a"), at the edges of the step where the signal data are low, the data points corresponding to the relative signal value below the minimum modulation threshold are removed from the reconstructed profile.The so-called batwing effect, 40 which appears when discontinuous surfaces with sharp edges (step heights smaller than the coherence length) are measured with CSI, can be seen in Fig. 2(a').However, since the simulated data are analyzed using an FDA-phase algorithm, the batwing effect is not significant. 41n the S2 to S4 tests (sinusoidal profiles with the same maximum slope angles and minimum curvatures), increasing the NA causes the height error to decrease in all methods.The DS test with the height range of AE5.15 μm, demonstrates that the EFO, UFO and foil models are not restricted to small surface heights (e.g., ≤λ∕4).In the TP1 and TP2 tests, post separations are chosen to be close to the Sparrow resolution limit 42 (0.95 and 0.43 μm, respectively).Figures 2(h), 2(h'), 2(i), and 2(i') illustrate that higher NA results in higher lateral resolution and amplitude of the simulated profile in all three models.Additionally, smoothing of peaks in TP1 and TP2 tests is anticipated in a linear process, attributed to the presence of out of pass band diffraction orders.
The comparison of the different simulated profiles obtained by EFO, UFO, and foil models shows that there is good agreement between these three approaches.The root-mean-square (RMS) of the difference between the simulated profiles obtained by each two models is within the sub-nanometer range.This confirms that they are based on common physical assumptions, even though they use different approaches to model the surface and TF.

Comparison of the TFs
In addition to the simulation and comparison of the various profiles for the three models illustrated in Fig. 2, a second comparison directly compares the TFs of the EFO, UFO, and foil models.Figure 3 shows the x; z-plane cross-sectional view of the simulated 3D [(a) to (d)] foil and [(a') to (d')] UFO TFs that are obtained using the analytical and numerical models respectively, and [(a") to (d")] 2D EFO TF along the x-axis.In Fig. 2, the behavior of the TF corresponding to each model is observed for four different NAs of [(a) to (a")] 0.8, [(b) to (b")] 0.55, [(c) to (c")] 0.3, and [(d) to (d")] 0.15.It should be noted that all TFs are simulated considering a Gaussian light source with a mean wavelength of 0.57 μm, an FWHM bandwidth of 0.08 μm and Fig. 1 The interference fringe pattern simulated by (a) EFO, (b) UFO, and (c) foil corresponding to the S1 test in Table 1.
Fig. 2 (a)-(i) Nominal and simulated profiles obtained by the EFO, UFO, and foil models along the x -axis.The specification of the samples and optics used at each row are provided in Table 1.(a')-(i') Difference between the simulated and nominal profiles for EFO, UFO, and foil models.(a")-(i") relative signal strength of the fringe data in the EFO, UFO, and foil models.
a fully filled objective aperture that obeys Abbe's sine condition.Comparing the rows of Fig. 3, it is evident that increasing the NA causes the TF to broaden along the x-axis.Figure 3 also shows that, despite the small differences between the 3D TFs of the UFO and foil models around the side lobes, they are in general agreement since the simulated profiles obtained by these models are almost identical, as shown in Fig. 2. Apart from the numerical calculation of the 3D TF in the foil model and the analytical result according to Eqs. ( 7) and ( 8) used in the UFO model, the TFs differ by the factor in square brackets in Eq. ( 12) used in the foil model.This so-called inclination factor is derived from the Kirchhoff approximation 11 assuming plane wave illumination.In the UFO model, the inclination factor has been adapted to a microscope with Köhler illumination and is considered in the 3D TF according to Eq. ( 7).Nevertheless, the 3D TFs show slight deviations for larger values of K x and K y , especially for high NAs (≥0.7). Figure 4 illustrates the subtraction of the magnitude of the normalized 3D TF in the foil model from that in the UFO model for the NAs of (a) 0.8 and (b) 0.3.The average of the RMS of the difference between the magnitude of the normalized TFs of the UFO and foil models over the range of provided NAs is of the order of 10 −3 .The 2D TF of the EFO model corresponds to the integration of the 3D TFs along the z-axis.

Implications for CSI
All the models predict that there is the potential for errors attributable to the imaging of interference fringes through the optical system, even under ideal conditions with diffraction-limited optics.As has been well established in the literature, for the linear instrument response in interferometry, all possible diffracted light needs to be captured by the objective lens, necessitating a sufficiently large NA for the optical instrument.This ensures the inclusion of the higher diffraction orders from steep slopes on smooth, continuous surfaces while limiting surface discontinuities (abrupt changes within a resolution cell) to less than a certain height (i.e., ≤λ∕4). 43redictions of residual nonlinear behavior for phase-based measurements can be calculated with any one of the three models compared in this paper, as a function of surface slope, surface height range, and optical configuration.This capability is one of the principal practical benefits of theoretical modeling, providing a way to optimize measurement configurations for the best results.

When to Use a 3D Model in Place of a 2D Model
As we have seen, all models should yield the same results for topography measurements based on the interference phase, assuming negligible polarization effects.This raises the question of where the models differ, and specifically, when a 2D model such as the EFO approach is no longer adequate for predicting measurement results and a 3D model is necessary.
The key difference between 2D and 3D approaches becomes apparent if we calculate surface profile using the shape of the interference fringes rather than the mean phase at the plane that intersects the object surface.The FDA-envelope analysis method is one such approach to topography measurement that is influenced by the shape of the interference pattern, particularly at high NA, when focus effects become significant.The disparity between the reconstructed profiles using 2D and 3D TFs at high NA is illustrated in the example presented in Fig. 5, which shows the predicted measurement error for tests S1, S4, and S5.The EFO model uses the obliquity factor that is equivalent to averaging the frequency projections along the vertical axis, enabling the application of a linear 2D TF.While Fig. 2 indicates that the FDA-phase provides nearly identical results for all models, this simplification lacks details about the variation in possible K z values as a function of K x , as presented in the 3D TFs in the foil and UFO models.Although this approximation is acceptable at low NA values, it proves inadequate at high NAs.For high NAs (above 0.2), the obliquity factor approximation fails to accurately represent the shape of the interference signal, and consequently, the reconstructed profile.Figure 5 illustrates the height error obtained by the foil, UFO, and foil models using the same FDA-envelope analysis.As shown in Fig. 5, the 3D foil and UFO models yield nearly identical results.However, the 2D EFO model underestimates the measurement error as the NA increases.Importantly, this discrepancy does not improve in the limit of small surface heights for coherence-based measurements-the EFO model is not up to the task of correctly predicting the shape of the interference fringes at high NA, even for nominally flat surfaces. 20

Conclusion
Due to the wide range of CSI applications, the development of physics-based models to predict interference signals and analyze measurement results is of great interest.Despite the limitations of the approximate models, they can provide a powerful means for CSI modeling using basic scalar diffraction and linear imaging theory.The EFO, UFO, and foil models are approximate models based on scalar diffraction theory.These models benefit from the linear nature of their imaging theories so that the transfer characteristic of a CSI instrument can be defined by a linear filtering operation.
In the foil model, the 3D object is defined as a thin foil-like model, and the 3D STF is calculated by numerical integration.The EFO method simplifies the surface topography to a phase object at a constant equivalent wavelength and uses an analytical form for the 2D partially coherent optical TF to map the object field to the image field in the spatial frequency domain.In a similar manner to the EFO model, the UFO approach treats the object as a phase object, but preserves the effects of multiple illumination incident angles, and relies on an analytical 3D optical TF to calculate the interference signal.
In this paper, we demonstrate the degree of agreement for these three approximate scaler diffraction and imaging models using software simulations.The RMS of the difference between the simulated profiles obtained by each two models is within the sub-nanometer range.The cross-sectional view of the UFO and foil 3D TFs in the xz-plane and the 2D TF of EFO along the x-axis are in good agreement, so the average of the RMS of the difference between the magnitude of the normalized TFs of the UFO and foil models over the provided range of NAs is of the order of 10 −3 .The EFO, UFO, and foil models applied to various 2D profiles, including sinusoids, step, and rectangular surface features, for different instrument configurations illustrate the applicability of these methods for piecewise-continuous, relatively smooth surfaces.
In future work, we intend to compare the simulated profiles obtained by these models with a more comprehensive range of profiles, including various slope angles and curvatures within their validity range.Furthermore, to verify the measurement results, we will compare the results of these theoretical predictions with the experimental results.

Fig. 4
Fig. 4 Difference between the magnitude of the normalized TFs in the foil and UFO models for the NA of (a) 0.8 and (b) 0.3.

Fig. 5
Fig. 5 Difference between the reconstructed and nominal profiles for EFO, UFO, and foil models associated with (a) S1, (b) S4, and (c) S5 tests.Reconstructed profiles are obtained using the same FDA-envelope analysis for all models.

Table 1
Summary of simulated samples and optics.
The mean wavelength equals 0.57 μm for all simulations.