1. Introduction

Coherence scanning interferometry (CSI) is a three-dimensional (3D) optical measuring technique enabling a height resolution down to the nanoscale. Generally, 3D optical microscopes are widely used in scientific and industrial applications, which require a fast and contactless measurement. Unfortunately, systematic deviations like the so called batwing effect [1–5] occur at certain surface topographies such as gratings or steep slopes. These discrepancies lead to significant reductions in the accuracy of CSI instruments. In order to increase the accuracy, the physical relations and dependencies of the systematic deviations have to be analyzed, for what analytical and numerical models are developed. Furthermore, reliable models can be applied in the measurement chain to reduce discrepancies in the evaluation software as already applied in scatterometry [6,7]. Therefore, models providing high accuracy and low computational as well as memory effort are needed.

In former studies analytical CSI models are developed based on Fourier optics [8,9], Kirchhoff’s diffraction theory [4,8,10,11] and on the theory of Richards and Wolf [4,5,10,12]. These models provide reasons for the occurrence of batwings, but have a limited validity with respect to small radii of curvature of the surface structure and multiple scattering effects. Furthermore, polarization and material dependencies are not predicted reliably. By taking into account sharp edges, multiple scattering, polarization as well as material effects, rigorous numerical models have to be applied.

In former studies various rigorous methods such as the rigorous coupled wave analysis (RCWA) [13–17], the boundary element method (BEM) [18–21], the finite difference time domain method (FDTD) [22–25] and the finite element method (FEM) [16,17,25] are adopted to conventional light microscopy and CSI. All of the named methods show different advantages with regard to computational effort, accuracy and simplicity of modeling with respect to various applications. So far, conical diffraction is applied to microscopic applications using RCWA [13] and FDTD [22–24]. With respect to CSI RCWA simulations are performed [14,15], which are restricted to periodic structures. Simulation models may show a lack of accuracy regarding rounded or rough surfaces, due to a sliced (RCWA) or rectangular (FDTD) discretization. Furthermore, RCWA is already applied in a model-based CSI system by de Groot et al. [26]. Besides the RCWA, FEM is often used in model-based scatterometry, due to high accuracy and flexibility [6,27,28]. It should be noted that the scattering at structures, which are invariant under translation in one direction, is correctly solved by considering conical diffraction at 2D structures and hence, a full 3D modeling of these structures is provided without large computational effort and memory requirement of discretizing 3D geometries. Throughout the whole paper, the modeling of conical illumination and diffraction with regard to 2D surface topographies is named 3D model. A comparison between 2D and 3D simulation models of the image formation as well as the scattering process is given in Table 1, where the model presented in this paper is marked by yellow backgroundcolor.

Table 1. Comparison between 2D and 3D simulation models. Advantages and disadvantages are marked by ’+’ and ’-’, respectively. The model presented in this study is highlighted by the yellow background.

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In this study we include conical diffraction in the 2D finite element model presented by Bischoff et al. [17] and Pahl et al. [25]. By reason of a triangular volume discretization, FEM is flexible with regard to the shape of the structure and can easily include spatially inhomogeneous material properties. Additionally, FEM is simply adaptable to versatile problems and, therefore, enables a comfortable and flexible investigation of various objects. An overview of FEM in electromagnetic scattering is given by Jin [29] and Volakis et al. [30]. Depending on the chosen boundary conditions, FEM can be applied to periodic [6,31–34] and non-periodic structures [35–39]. Though in this study we analyze periodic structures, we also clarify how to treat non-periodic structures. In case of periodic structures one period instead of the whole structure is modeled, due to Bloch’s theorem, reducing the computational demands.

The model is used to reproduce and investigate systematic deviations occurring at two types of surface topographies, rectangular and sinusoidal grating structures. Additionally, a chrome-on-glass grating is analyzed in order simulate the influence of different materials. Since rectangular gratings include edges and sinusoidal structures imply slopes, the main reasons for the systematic deviations mentioned above are taken into account. Furthermore, phase gratings provide high spatial frequencies and, therefore, are appropriate to study the transfer characteristics of measurement instruments. Industrial applications of grating structures are for example Fabry-Pérot filters [40] or through-silicon vias [41]. Thus, models predicting measurement errors occurring at these types of structures are of high interest.

In this contribution, the 3D transfer characteristics of the interferometers are included by a Fourier optics (FO) approach as investigated analytically by many researchers [9,42–45] and applied to FEM in [25], whereby in [25] a 2D approximation is applied. Two typical interferometer setups are considered, a Linnik interferometer and a Mirau interferometer [46,47].

In the first part of this paper, the 3D FEM scattering model of 2D structures is explained and the extension of the FO modeling to 3D is discussed. Further, the advantages with respect to the accuracy compared to the more economic 2D model are pointed out for a rectangular grating and a Linnik system applying different polarizations and illuminating wavelengths and, therefore, different height to wavelength ratios (HWRs). In the second part of the paper a sinusoidal and a chrome-on-glass grating are simulated and results are compared to measurement results using a Mirau interferometer. In case of the sinusoidal grating, a simulation including an atomic force microscope (AFM) measurement result as the scattering object is included to constitute a more realistic situation and to underline the industrial interest of our model. Systematic deviations of measurement results are reproduced by simulations in all cases.

2. Model

The model we present is split in two parts, the FEM near-field calculation, which simulates the light-surface interaction, and the image formation of the measurement instrument. As the modeling of the image formation is already discussed in detail for the 2D case by Pahl et al. [25], we focus on the differences and novelties of the 3D modeling at this point. In the first part of this section the FEM model is extended to conical illumination and detection as well as arbitrary materials, whereby the vector wave-equation is solved and modified for 2D geometries compared to the scalar Helmholtz equation, which is used in case of the 2D model. The second part contains the 3D image formation by the interference microscope and considers the reference mirror in the beam path in case of Mirau interferometers.

2.1 FEM near-field calculation

In this study we focus on periodic structures. Non-periodic structures can be treated as periodic structures by enlarging the period length or replacing the right and left boundaries with absorbing layers used for the top and bottom boundaries in the presented simulation setup. The modeling is based on the work of Bao et al. [32] as well as Zhou and Wu [34]. Figure 1(c) depicts the underlying geometry of the scattering problem. The main aspect of the model is to solve the vector equation

 

Fig. 1. (a) Schematic representation of a Linnik interferometer. The near-field scattered by the measurement object is simulated with FEM. The spatially incoherent Köhler illumination is sketched by the red lines, which are focused on the back focal plane of the objective, the imaging beam path is denoted by blue lines. (b) Schematic representation of a Mirau interference objective lens. As the reference mirror is included in the object arm O, the reference arm shown in (a), marked by R, is replaced by an absorber. (c) Geometry of the FEM setup with labels of the boundaries. To avoid reflections, the simulation area $\Omega =\Omega _1\cup \Omega _2$ is extended by absorbing layers called perfectly matched layer (PML) on the top and the bottom of the geometry.

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In order to solve the 3D problem for a 2D geometry, we have to take into account that the structure is invariant under a translation in one direction (here $y$). Thus, the electric field can be written as $\textbf {E}(x,y,z)=\textbf {E}(x,z)\exp (ik_{y,\textrm {i}}y)$ [48], where $k_{y,\textrm {i}}$ is the $y$-component of the incident wave vector

In order to obtain a boundary value problem, the model is extended by boundary conditions. As periodic structures are investigated, we apply quasi-periodic boundary conditions on the left and on the right hand side, marked as $\partial \Omega _{\textrm {l},\nu }$ and $\partial \Omega _{\textrm {r},\nu }$ with $\nu =1,2$ in Fig. 1(c). In this case, the electric field at the left boundary is related to the field at the right boundary by

To avoid reflections on the upper and the lower boundaries, the geometry is extended by absorbing layers known as perfectly matched layers (PMLs) [49]. In this case the PMLs are implemented as anisotropic material properties as described in [50]. The right and the left boundaries of the PMLs are also assumed to be quasi-periodic. Analogous to [32,34], the incident wave $\textbf {E}_{\textrm {i}}$ is included in the simulation using the relation $\textbf {E}=\textbf {E}_{\textrm {i}}+\textbf {E}_{\textrm {s}}$, whereas only the scattered wave $\textbf {E}_{\textrm {s}}=\textbf {E}-\textbf {E}_{\textrm {i}}$ is damped by the top PML. Therefore, the top and bottom boundary condition are chosen to be

The near- to far-field transform of the scattered electric field is computed along one line in the near-field as sketched in Fig. 1(c) by a red dotted line. There are different possibilities to compute the far-field as summarized in [51]. In case of periodic structures, the far-field can be described by an eigenvalue expansion similar to a Fourier series as explained by Jin [29]. Here, the Fourier coefficients are calculated by single-frequency discrete Fourier transforms (DFTs) along the near-field line. With regard to non-periodic structures, the far-field can be computed by other eigenvalue expansions in some cases or by using Greens integral theorem as explained elsewhere [29,51,52]. Thereby, the far-field can be computed along a closed surface around the scatterer in contrast to the line used in the periodic case, in which the scatterer is assumed to be infinitely extended. In this study the periodic eigenvalue expansion is used in order to compute the far-field, which is imaged by CSI as explained in the following section. The near-field line is placed $100$ nm above the upper plateau of the grating in order to avoid numerical noise, which may occur at the interface, and on the other hand to be close to the scattering surface. Moving the line in a height range between the upper plateau of the surface and $200$ nm above, does not change the results significantly.

2.2 3D simulation of the imaging process

Since the imaging process of a Linnik interferometer is discussed in detail for the 2D case in [25], at this point we mainly describe the extensions to this work, which are required to model 3D image formation, and Mirau objectives (see Fig. 1(b)).

2.2.1 Conical illumination

Considering an objective lens with circular aperture in the $xy$ plane leads to conical illumination. The wave vector $\textbf {k}_{\textrm {i}}$ of the incident wave after passing the objective lens (see Fig. 1(a)) is given by Eq. (2), whereby the 2D wave vector results for $\varphi =0$ and $\varphi =\pi$. The illumination geometry including the angles of incidence, the electric fields and the wave vectors is sketched in Fig. 2(a). The electric field $\tilde {\textbf {E}}_{\textrm {i}}$ above the objective is polarized in the $xy$ plane by the polarizer depicted by the orange line in Fig. 1(a), as the optical axis coincides with the $z$-axis, where $\textbf {e}_z$ is denoted as the unit vector along the $z$-axis, and the related wave vector $\tilde {\textbf {k}}_{\textrm {i}}$ above the objective is antiparallel to $\textbf {e}_z$ as depicted in Fig. 2(a). As the wave vector is deflected by the objective (see Eq. (2) and Fig. 2(a)), where $\theta _{\textrm {i}}$ and $\varphi$ are dependent on the point in the pupil plane, the electric field is still perpendicular. Thus, the electric field $\textbf {E}$ after deflection can be split in one component perpendicular to the plane of $\textbf {k}_{\textrm {i}}$ and $\textbf {e}_z$ (out-of-plane) and two in-plane components. The normalized direction $\textbf {e}_\perp$ of the out-of-plane component is given by

 

Fig. 2. (a) Schematic representation of conical illumination and the corresponding angles $\varphi$ and $\theta _{\textrm {i}}$ as described in the text. The optical axis corresponds to the $z$-axis. In order to keep the figure as clear as possible, the $x$- and $y$-axes are sketched on the left hand side of the figure, whereby the corresponding Fourier components $k_{x,y}$ are sketched around the optical axis. (b-e) Special cases of (a) for TM polarized incident light of $\varphi =\pi$ (b), $\varphi =3\pi /2$ (c) as well as for TE polarized incident light with $\varphi =\pi$ (d) and $\varphi =3\pi /2$ (e). (f) Sketch of the in- (blue) and out-of-plane (red) vectors used to determine the electric field components of the conical illumination. The unit vector along the optical ($z$-) axis is denoted by $\textbf {e}_z$.

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In order to underline the requirement of considering conical illumination, four special cases of Fig. 2(a) and Eq. (8) are depicted in Figs. 2(b)–2(e). Figure 2(b) shows the focusing of TM polarized light along the $x$-axis for a certain angle $\theta _{\textrm {i}}$ and $\varphi =\pi$. Obviously, the electric field is rotated by the objective lens as it is perpendicular to the wave vector $\textbf {k}_{\textrm {i}}$. Figure 2(d) shows the same situation except for the polarization. In this case the electric field oscillates in the direction perpendicular to the wave vector as well as the optical axis and, therefore, is not changed after being focused. Figures 2(c) and 2(e) depict the focusing along the $y$-axis for $\varphi =3\pi /2$. In this case the behavior of the electric field changes for TE and TM polarization as shown in the figures as the in- and out-of-plane amounts are transposed. In general, the in-plane component of the electric field is rotated as the wave vector is focused by the objective lens, whereas the out-of-plane component is consistent. The in- and out-of-plane directions are sketched in Fig. 2(f). The rotation of the electric field for arbitrary polarization and various angles of incidence is given by Eq. (8).

2.2.2 Simulation of the image formation

The axial-scan, which is induced by the piezo stage displayed in Fig. 1(a), can be considered in Fourier space [25] as well as in image space [17]. In this study the scan is recognized in Fourier space. For detailed description of the simulation of the vertical scan, we refer to [25], where the scan is explained extensively for the 2D case. Here we focus on the expansions of the 2D model to 3D.

Taking into account conical diffraction, the $y$-component $k_{y,\textrm {s}}=k_{y,\textrm {i}}$ of the scattered wave vector $\textbf {k}_{\textrm {s}}$ is considered in the equations by

The spatially extended light source fully illuminates the pupil planes of the objective lenses (see Fig. 1(a))) leading to spatially incoherent Köhler illumination. The total intensity

In the presented simulations the modeled measurement instruments are assumed to be ideal. Misalignments with regard to the focal position of the reference mirror, sometimes also referred to as defocus can be applied straightforward in the reference field similar to [54,55]. Misalignments and discrepancies between the two objectives of a Linnik interferometer (see Fig. 1(a)) lead to dispersion effects, which can be considered in the Fourier optics model as shown by Xie [10]. In general, Mirau objectives, which are described in the following section, show minor dispersion effects due to the common path configuration.

2.2.3 Mirau interferometer

Mirau type objectives are less sensitive to vibrations and dispersion effects. Furthermore, less space is needed, since just one objective is used and, thus, Mirau interferometers are commonly used in industrial and scientific applications.

Due to the integrated reference mirror, an obscuration effect must be considered in the signal modeling. Hence, the lower limit in the integral is changed to a minimum angle

3. Results

This section provides a comparison of simulation results calculated by the model explained in the previous section with measurement results obtained by a Linnik and two Mirau interferometers with regard to various surface topographies and material properties of the objects to be measured. Thus, the section is split in two parts. The first part includes simulation and measurement results obtained by a $100\times$ Linnik interferometer from a rectangular grating structure of the Simetrics RS-N resolution standard [56]. The advantages of the 3D modeling are pointed out by comparison with results calculated by the 2D approach [25] and measurement results for two different light sources. In the second part of this study a Rubert 543 sinusoidal structure [57] is simulated and measured with a $50\times$ Mirau interferometer. Further, simulation as well as measurement results of a United States Air Force (USAF) chrome-on-glass resolution target in order to prove the reliability of the simulation model with respect to different material properties is implied. In this case a $100\times$ Mirau interferometer is used. Thus, the reliability of our model is proved for rectangular gratings, a combination of different materials and surface slopes. Therefore, the main surface structures leading to large systematic deviations are considered.

For all simulation and measurement results the interference data are evaluated by an envelope evaluation algorithm based on Hilbert-transform [58] and more accurately by a lock-in phase evaluation technique [59] in order to reconstruct the surface topography. The evaluation algorithms are summarized and extensively explained by Tereschenko [60]. Throughout the whole paper, the envelope evaluation results are depicted in blue, whereas the phase evaluation results are marked by red lines. It should be noticed that the reconstructed surface structures are separated by an artificial offset.

For reference, all of the presented surface structures are measured by an atomic force microscope (AFM) [61–63], where the cone angle of the cantilever tip amounts approx. $8$ $^\circ$ [62]. In order to create a simulation setup as close to the measurement process as possible, the simulation of the sinusoidal grating is repeated including AFM measurement results in the FEM simulation. As shown in [63], the standard deviation of CSI measurement instruments is in the range of $1$ nm. Presented deviations between results observed by optical measurement techniques and AFM measurement results are in the range of $100$ nm and, therefore, can be identified as systematic deviations. These deviations are discussed and confirmed by simulation results in the following subsections.

3.1 RS-N grating structure investigated by a Linnik interferometer

In this subsection a $\textrm {NA}=0.9$, $100\times$ Linnik interferometer [4] is applied to detect the rectangular surface structure of a Simetrics RS-N grating (period length $L=6$ $\mathrm {\mu }$m, height $h=190$ nm) [56]. Measurements are performed with two light sources, a red LED and a royalblue (rb) LED. The spectra of the LEDs used for simulations are extracted from spectrometer measurement results presented by Xie et al. [4]. A set of $6$ sample points of the specra around the central wavelengths are taken for simulations. The material parameters used can be found in Table 2 in the Appendix. As the spectral bandwidths of the single color LEDs are relatively small and the relative electric permittivity $\varepsilon$ does not change significantly over this range, $\varepsilon$ is assumed to be constant for all wavelengths within one LED-spectrum and is defined by the permittivity occuring at $\lambda _0$. Discretizing the angles of incidence, $29$ values for $\theta _{\textrm {i}}$ and $57$ values for $\varphi$ are used. In order to reduce the computational effort, simulations including $\theta _{\textrm {i}}=0$ are skipped and calculations are only performed for a range of $0\leq \varphi \leq \pi /2$ exploiting symmetry properties of the scattering process. The given angle and wavelength discretization is the same in all simulations throughout the paper.In order to obtain the same central wavelength in interferograms (often referred to as effective wavelength $\lambda _{\textrm {eff}}$) for TE and TM polarized incident light, the 2D approximation assumes a perfect electric conductor (PEC) as the material of the measurement object as well as of the reference mirror (for further explanations see [25]). Using a PEC instead of realistic material coefficients in the extended model does not lead to significant changes in the results and, therefore, 3D modeling is identified as the main effect leading to differences between both models. Thus, only results considering the given material parameters are shown in this paper with regard to the 3D FEM model.

Figure 3 presents normalized interferograms of the 2D simulation model (Fig. 3(a)), the 3D simulation model (Fig. 3(b)) and a measured interferogram (Fig. 3(c)) obtained using TE polarized red illumination. Note that intensity offsets are eliminated from the interferograms. The bottom of the figure shows intensity cross sections of the interferograms along the $x$-axis at $z=z_{\textrm {l}}$ (marked in the interferograms by blue dashed lines). It should also be noticed, that the presented line plots are just shown to underline the behavior of the interferograms along the $x$-axis and do not relate to the reconstructed grating structures displayed in subsequent figures.

 

Fig. 3. Extract of interferograms simulated using the 2D approach (a), full 3D modeling (b) and measured by a Linnik interferometer (c). For illumination a TE polarized, red LED is assumed. The intensity along the blue dashed lines at $z_{\textrm {l}}$ for each interferogram is presented at the bottom of the colorplots.

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In all of the three interferograms the main intensity maximum as well as the related modulation by the grating structure are clearly visible. The phase progression between top and bottom plateaus is continuous in all cases. The frequencies of the three interference signals in $z$-direction and therefore the effective wavelengths are similar. In case of the 2D model, there are high frequency oscillations with regard to the grating structure beyond the focal plane. The lineplot at the bottom of the figure underlines these oscillations. In the 3D simulation result as well as in the measured interferogram these fluctuations are reduced, as can be seen in the lineplots in Fig. 3. The oscillations probably follow from the overweight of high angles of incidence occurring in the 2D case, due to the neglect of out-of-plane oblique incoming waves, which generally imply smaller angles of incidence with regard to the grating structure, as $k^2=k_{x,\textrm {s}}^2+k_{y,\textrm {i}}^2+k_{z,\textrm {s}}^2$. The 3D simulation model and the measured result show in general good agreement. Slight deviations arise in the amplitudes of the minor maxima. The amplitude of the measurement in the lineplot is marginally enhanced compared to the 3D simulation result. Regarding these lineplots, it should be noticed that the discretization in $z$-direction is given by the step size of $\Delta z=20$ nm. Therefore, the positions $z_{\textrm {l}}$ are not taken at exactly the same positions in the respective interferogram leading to deviations in the amplitudes. Nonetheless, a Linnik interferometer provides many degrees of freedom, which complicate the alignment. Additionally, two not totally identical objectives are installed and, thus, differences between the object and the reference arm appear inevitably. The mentioned effects and misadjustments lead to dispersion effects caused by different optical path lengths in glass, which result in non-uniform and asymmetric amplitudes of the interferogram intensity in Fig. 3(c). Further consequences of these dispersion effects are discussed later in this section. Simulated as well as measured interferograms (as exemplary shown in Fig. 3) are evaluated with respect to envelope and phase as explained in the beginning of this section. The resulting profiles are named envelope and phase profile, respectively. As the 2D model previously appeared to be less accurate for large height to wavelength ratios (HWRs) (cp. [25]), reconstructed grating structures are shown in Fig. 4 using a royalblue LED.

 

Fig. 4. Grating structures reconstructed by envelope (blue) and phase (red) evaluation using a royalblue LED as the illuminating light source. The first column shows the 2D approach (a, d, g), the second one the simulation results in consideration of a full 3D image formation (b, e, h) and the right column depicts measurement results obtained by the Linnik interferometer (c, f, i) for TE (a ,b, c), TM (d, e, f) and unpolarized (g, h, i) incident light, respectively. The wavelength used for the phase evaluation amounts to $\lambda _{\textrm {phase}}=560$ nm.

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In case of TE polarized light (Figs. 4(a)–4(c)) batwings occur and no batwings are observed using TM polarized illumination (Figs. 4(d)–4(f)) in the measured profiles as well as in profiles obtained by both simulation models. In the 2D model a PEC is applied in order to describe the surface material and thus the observed behavior can not be explained by material properties. Therefore, the appearance of the batwing effect probably follows from polarization dependent scattering behavior at the grating, which is further related to the HWR.

In case of the 2D model, a high frequency modulation at the grating structure can be observed for all polarizations (see Figs. 4(a), 4(d), 4(g)), which distorts the results, especially close to the edges compared to the measured results (see Figs. 4(c), 4(f), 4(i)). These effects are strongly reduced by applying the full 3D modeling and, thus, the agreement with measurement results is improved. In the 2D model larger angles of incidence are more significant compared to the modeling of full conical illumination as explained in the discussion of the interferograms above. As the deviation between measurement and 2D simulation significantly increases close to the edges, probably obscuration and multiple scattering effects as a consequence of large angles of incidence are the main reason for the high modulation. Additionally, interactions between TE and TM polarization are neglected in the 2D model. As profiles obtained using TM polarized light do not show overshoots, the modulation in case of TE and unpolarized illumination is probably further flattened in the 3D model, due to the TM component of the oblique incident waves.

Using unpolarized incident light (see Figs. 4(g), 4(h), 4(i)) an upward and downward batwing occurs in both simulations, which is smaller than the batwing caused by TE polarized light. In the measurement results the upward batwing is larger than the downward one, whereas the upward one is larger compared to the upward one in the simulation models and the downward one is reduced. It should be noticed, that the direction of the batwing is strongly affected by dispersion effects and misalignments of the interferometer. To underline this claim, two measurements of the same structure using unpolarized royalblue illumination are presented in Figs. 8(a) and 8(b) in Appendix. In the second measurement (Fig. 8(b)) the reference mirror and the grating structure are interchanged, and the depth scan is performed under the objective that was previously used to image the reference mirror. A comparison of both figures confirms the strong dependency of the direction of the batwings on dispersion effects provoked by slight misalignment and deviating optical components of the Linnik interferometer. By changing the objective, under which the depth scan is performed, the direction of the batwing is reversed as well. Further, small deviations between measurement and simulation results can be explained regarding an AFM measurement result of the RS-N standard (Appendix Fig. 8(c)), which reveals that the edges differ from the ideal rectangular structure. In sum, the results simulated by the 3D model (see Figs. 4(b), 4(e), 4(h)) are in better agreement to the measured profiles (see Figs. 4(c), 4(f), 4(i)) than the results obtained by the 2D model (see Figs. 4(a), 4(d), 4(g)). For further conviction compare Fig. 7 in Appendix, where the TE polarized case of red LED light corresponds to the evaluated interferograms of Fig. 3.

In order to reduce deviations inherent to the Linnik configuration occurring with regard to misalignments and dispersion, a Mirau based CSI is used in the following sections. Additionally, a AFM measured profile is used as an input of the simulation in order to avoid deviations according to non-ideal surface topographies in case of the sinusoidal surface structure, which includes roughness.

3.2 Sinusoidal structure analyzed by a Mirau interferometer

In order to investigate a sinusoidal surface structure of a Rubert 543 standard (period length $L=2.5$ $\mathrm {\mu }$m, peak-to-valley height $h=120$ nm) [57], a Mirau based CSI with a 50$\times$ Nikon objective ($\textrm {NA}=0.55$) [61] is used. The parameters used in the simulation are given in Table 2 in the Appendix. As no value of the radius of the reference mirror $r_{\textrm {ref}}$ is provided by the manufacturer, $r_{\textrm {ref}}=0.35$ mm is estimated experimentally.

Figure 5 displays simulation (Fig. 5(a)) and measurement (Fig. 5(b)) results of the sinusoidal topography. In Fig. 5(c) an AFM measurement result is applied as scattering geometry instead of an ideal sinusoidal structure, because the measurement object includes roughness (cp. [57]). Figure 9(a) in Appendix depicts an extract of the AFM measurement result, where the part used in the simulation is marked by red vertical lines. The profile section is chosen to be two periods long and assumed to be periodic. In general, the simulated profiles and the interferometric measurement result show a good agreement. In all cases the height of the envelope profile is enlarged by a factor of more than $3$ compared to the nominal height up to $h\approx 400$ nm. Due to the slope of the sinusoidal structure, the intensity of reflected light passing the objective is reduced as explained in [64]. In addition, the scattering at sinusoidal structures may lead to multiple reflections, which further influence the interferograms and therefore the measured profiles [65]. Probably these effects cause the increase of the amplitude obtained by envelope evaluation. Since the simulation confirms this observation, reasons can be studied in detail in future investigations.

 

Fig. 5. Simulation (a,c) and measurement (b) results of a Rubert 543 sinusoidal surface structure obtained by a Mirau interferometer using unpolarized red light illumination. The structure is assumed to be perfectly sinusoidal in (a), whereas in (c) an AFM measurement (cp. Figure 9(a)) result is the input profile of the FEM simulation. The envelope profile is displayed in blue, the phase profile in red, the unwrapped phase profile in green and the AFM measurement result in black. The wavelength used for the phase evaluation amounts to $\lambda _{\textrm {phase}}=680$ nm.

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The large amplitudes of the envelope profiles lead to $2\pi$ phase jumps in the phase evaluation in both, simulation and measurement, because the envelope result is used to determine the fringe order in the phase evaluation. The green lines in Fig. 5 show the unwrapped phase profiles of the simulation and the measurement result, respectively. The unwrapped phase of the simulation results show heights of $h\approx 95$ nm (Fig. 5(a)) and $h\approx 90$ nm (Fig. 5(c)), whereas the height of the unwrapped phase in measurement amounts to $h\approx 75$ nm. Hence, the amplitude of the phase evaluation is less compared to the nominal height of $120$ nm. For a better comparison, the structure measured by the AFM is plotted in Fig. 5 by black lines. This result is in agreement with observations presented by Lehmann et al. [12]. According to the shape of the real structure, the simulation result including the AFM result shows a slightly better agreement to the measured profile. As a Mirau interferometer is used, misalignment effects are significantly reduced compared to the Linnik setup, but nonetheless also Mirau interferometers inevitably suffer from dispersion phenomena which may be a reason for the systematic discrepancies between measurement and simulation [66].

In sum, the envelope evaluation demonstrates a larger surface height compared to the real structure in both, measurement and simulation. On the other hand, the amplitude of the unwrapped phase profile is less than the real amplitude in the simulation as well as in the measured result. The phase evaluation result exhibits differences in the surface height of approx. $15$ nm. Probably these deviations between measurement and simulation results follow from misalignments like defocus [55] and dispersion effects [66].

3.3 Investigation of an USAF resolution target using a Mirau interferometer

In order to verify the reliability of the simulation model with regard to different materials, a chrome-on-glass grating structure of a USAF resolution target (period length $L=6.96$ $\mathrm {\mu }$m, height $h\approx 55$ nm) is measured by a Mirau interferometer equipped with a $100\times$ Nikon objective lens ($\textrm {NA}=0.7$) [62].

Figure 6(a) displays the reconstructed surface structure obtained by the simulation assuming an ideal grating and Fig. 6(b) depicts the corresponding measurement result. In general, the results show good agreement even though the measured envelope profile is noisy. For comparison, Fig. 9(b) in the Appendix presents an AFM measurement result of the grating. The averaged grating height measured by the AFM is $h\approx 55$ nm. In both, simulation and interferometric measurement the envelope profiles are significantly higher compared to the AFM result. Furthermore, both envelope results show a downward overshoot and just a small ascending one. With regard to the AFM result, the surface structure is superimposed by roughness in reality. As this is not taken into account in the simulation, the differences between simulation and measurement appear evidently. Regarding the phase profile, the amplitudes of the structures are smaller compared to the AFM result. Additionally, the noise of the measured surface structure is reduced and the amplitude of the measured phase profile is further decreased by approx. $10$ nm compared to the simulation result. In consideration of the complex reflection coefficient of chrome, a phase shift occurs, which is not taken into account in the phase evaluation. Including the reflection coefficient a phase-shift $\Delta \phi$ caused by the additional phase of the complex reflectivity of approx. $\Delta \phi =18^\circ$ has to be considered. Therefore, the presented step height is reduced by $\Delta h=\lambda _{\textrm {phase}}/2\cdot \Delta \phi /360^\circ \approx 18$ nm, where $\lambda _{\textrm {phase}}=720$ nm is the effective wavelength used for phase analysis. Under these circumstances, the resulting step height fits well with the one pretended in the FEM simulation and hence to the amplitudes of the AFM profile.

 

Fig. 6. Simulation (a) and measurement (b) results of a USAF chrome-on-glass resolution target (period length $L=6.96$ $\mathrm {\mu }$m) obtained by a Mirau interferometer using unpolarized red illumination. The envelope profile is imaged in blue, the phase profile in red. The wavelength used for the phase evaluation amounts to $\lambda _{\textrm {phase}}=720$ nm. Subfigure (c) depicts the effective wavelengths $\lambda _{\textrm {eff}}$ of the simulated as well as the measured interference signals.

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As two different materials are used, Fig. 6(c) shows the central wavelengths of the interferograms (see e.g., Fig. 3) referred to as effective wavelength $\lambda _{\textrm {eff}}$ depending on the $x$-position on the grating for both, simulation (magenta) and measurement (black). The resulting effective wavelengths are linked to the grating structure. Slight differences probably follow from the estimated radius of the reference mirror as well as a not completely homogeneous pupil illumination, approximated in the simulation by the pupil function $P(\theta _{\textrm {i}})=\sqrt {\cos (\theta _{\textrm {i}})}$ in Eq. (11). Furthermore, $\lambda _{\textrm {eff}}$ is shorter at the upper plateaus of the grating (see Fig. 6(a) and Fig. 6(b)), compared to the effective wavelength at the lower ones. In view of the reflectivities of chrome and glass, the reflection coefficient of chrome decreases in relation to larger angles of incidence, whereby the reflectivity value of glass increases. Hence, the effective wavelengths change as expected, due to the different material properties of chrome and glass. In addition, the progress of the effective wavelength depending on the position on the grating demonstrates overshoots, which are probably caused by the edge diffraction at the steps between the plateaus, for both, simulation and measurement.

4. Conclusion

In this study we introduced a full 3D simulation model reproducing the image formation in CSI considering conical diffraction on 2D periodic surface structures. Therefore, an exact solution is provided, without the computational effort needed for discretizing 3D surface structures. Nevertheless, an extension to 3D surface structures would be straight forward. Significant improvements compared to simulations using a 2D model are presented for the example of a rectangular grating using red and royalblue LED illumination in a Linnik interferometer. In consideration of misalignments and dispersion effects in contrast to an ideal measurement instrument, the simulation model enables to predict measurement results reliably. Further demonstrations of the accuracy of the simulation model are presented by a sinusoidal grating profile as well as a chrome-on glass grating using a Mirau interferometer with two different objectives and thus the performance of the model with regard to edges, slopes and varying materials as the main sources of systematic deviations is demonstrated. For reference, AFM measurements are included for all of the studied surface structures. The discrepancies between the ideal surface structures used for the simulations and the real structures lead to deviations between measurement and simulation results. Here, AFM measurement data can be used as the input profiles of the simulations as shown by the example of a sinusoidal structure. Thereby, the profile fidelity of the simulation is improved, because no assumptions about the surface topography are made and, on the other hand, surface structures can be measured and applied in various simulations independently.

In the presented model FEM was used to calculate the near-field, due to its high accuracy and its versatile and flexible applicability. The main disadvantage of using FEM is a higher computational effort in many cases compared to the BEM or RCWA, for example. As the 3D modeling of the image formation in the microscope is independent of the rigorous field computation, the rigorous method could be exchanged with regard to the application and the corresponding demands. Furthermore, the presented model could be extended to describe the image formation of additional optical measurement instruments such as a confocal microscope or a laser focus sensor. Thus, the advantages of several measurement instruments could be investigated without conducting real measurements, which leads to a saving of time and costs.

In order to analyze the physical dependencies of systematic deviations in optical measurement techniques, parameter scans could be performed in order to examine the relations between these discrepancies with regard to various parameter setups. Additionally, the model could be expanded by the inclusion of misalignments and their effect on the measurement results could be investigated. As a consequence, the model could be applied to align and calibrate the measurement instrument. In order to investigate topographies, which vary in all three dimensions, the model can be extended to 3D structures without further effort. Nonetheless, as a full 3D geometry needs to be discretized, powerful computer with high memories are required. Thus, the investigation of 2D geometries is useful to test the reliability of simulation models, in particular because those structures occur in many industrial and scientific applications.

Finally, the presented simulation model could be included in the measurement chain in order to correct the occurring systematic errors directly. This kind of application is called model-based interferometry and part of modern approaches in measurement technology.

Appendix

A. Rectangular grating structure investigated by a Linnik interferometer

Figure 7 shows reconstructed rectangular grating structures measured by a Linnik interferometer (Figs. 7(c), 7(f), 7(i)) and the corresponding results obtained by a 2D simulation model (Figs. 7(a), 7(d), 7(g)) as well as a 3D simulation model (Figs. 7(b), 7(e), 7(h)) using red LED illumination. The profiles occurring by use of TE polarized light are related to the interferograms shown in Fig. 3. Figures 8(a) and 8(b) present grating profiles measured by the Linnik interferometer using unpolarized royalblue illumination. Figure 8(a) shows three periods of the same result given in Fig. 4(i). Compared to the profile displayed in Fig. 8(a), in Fig. 8(b) the reference mirror and the measurement object are interchanged, whereas the depth scan is still performed by moving the measurement object. Figure 8(c) demonstrates one period of an AFM measurement result of the same grating structure. The width of the edge is marked by black vertical lines and black arrows to underline that the structure is not perfectly vertical in reality.

 

Fig. 7. Rectangular grating structures reconstructed by envelope (blue) and phase (red) evaluation using a red LED light source. The first column shows the 2D approach (a, d, g), the second one the simulation results under consideration of conical illumination and diffraction (b, e, h) and the right column depicts measurement results obtained by the Linnik interferometer (c, f, i) for TE (a ,b, c), TM (d, e, f) and unpolarized (g, h, i) incident light, respectively. The wavelength used for the phase evaluation amounts to $\lambda _{\textrm {phase}}=780$ nm.

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Fig. 8. Reconstructed rectangular grating structures obtained by envelope (blue) and phase (red) evaluation using the Linnik interferometer with unpolarized royalblue illumination, where from (a) to (b) the measurement object and the reference mirror are interchanged still performing the depth scan by moving the measurement object. In (c) an AFM measurement result of one period of the RS-N grating using a cantilever tip of $<8$ $^\circ$ opening angle is displayed.

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B. AFM measurement results of the sinusoidal grating and the USAF target

Figure 9 presents extracts of AFM measurement results of the Rubert 543 sinusoidal grating (Fig. 9(a)) and the USAF target (Fig. 9(b)). The section of the sinusoidal structure used in the FEM simulation (see Fig. 5(c)) is marked by the red vertical lines.

 

Fig. 9. AFM measurement results of a Rubert 543 sinusoidal structure (a) and a USAF grating (b). The extract applied to the FEM simulation is marked by the vertical red lines.

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C. Parameters applied in the simulations

Table 2 represents the parameters used in the simulation model with regard to the different measurement instruments and objects.

Table 2. Parameter used for the simulations. Royalblue is abbreviated by (rb).

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Funding

Deutsche Forschungsgemeinschaft (LE 992/14-1).

Acknowledgments

The authors gratefully acknowledge the financial support of this research project GZ: LE 992/14-1 by the Deutsche Forschungsgemeinschaft (DFG). This project is a cooperation with the Institute of Process Measurement and Sensor Technology, TU Ilmenau, Germany.

Disclosures

The authors declare no conflicts of interest.

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