1. Introduction

Aspherical surfaces are key components of modern optical imaging systems [1, 2]. Since such surfaces are described not only by a radius but by several other parameters, their nominal form has a high degree of freedom [3]. For this reason, aspheres are used to compensate for several different imaging errors at the same time. In this way, smaller and lighter imaging systems with better optical properties can be built [4]. However, due to the high degree of freedom of such surfaces, producing and measuring them is challenging.

In interferometric measurements, asphericity leads to effects that make evaluating the fringes of an ordinary interferometer challenging or even impossible. These effects include sub-sampling due to an excessive fringe density, and retrace and vignetting effects.

To accommodate these challenges, the tilted-wave interferometer (TWI) has been proposed [5–7]. This measurement system combines interferometric measurements taken using a special measurement setup with ray tracing, perturbation methods and mathematical evaluation procedures. Figure 1 shows the schematic setup of the TWI. The setup is based on a Twyman-Green interferometer [7] and differs in two essential aspects from a standard Twyman-Green interferometer: First, an additional beam stop in the Fourier plane of the imaging optics limits the fringe density at the detector [5]. Second, instead of a single-point light source, a two-dimensional point light source array is used for the illumination of the specimen. This arrangement generates differently tilted wavefronts behind the collimator that illuminate the specimen under test [5, 7]. As a result, different sources, depending on the local slope of the topography to be measured, provide measurement data to the image detector (CCD). In order to reconstruct the form of the topography from this measurement data, a specific evaluation procedure is needed. The main idea of the evaluation principle is that a deviation of the topography to be measured T leads to a characteristic change in the optical path length differences (OPDs) L measured at the CCD between the reference arm and the (constant) test arm of the interferometer [5]. The OPDs of selected source-pixel pairs are measured with the measurement setup described above and then compared to OPDs calculated for the known design of the topography using a model of the interferometer setup. The difference between the measured and the calculated OPDs (L and L₀) is then explained by the deviation of topography ΔT from its design topography T₀ [5, 6]. The deviation of the topography from its design ΔT may be parametrized in terms of Zernike polynomial functions Z_j [5,6],

 

Fig. 1 Setup of the TWI with some example rays traced through the system.

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A model of the interferometer is used to calculate the OPDs L₀ by means of ray tracing methods. A linear approach is assumed between the change in the OPDs and the change in the topography close to the solution, i.e.:

In Eq. (2)L denotes the measured OPDs, L₀ marks the OPDs calculated for the design topography, and c is a vector containing the Zernike coefficents c_j. The entries of the Jacobian matrix J₀ contain the partial derivatives of the OPDs L_i (one element of vector L) with respect to a Zernike coefficient $c_j:J_{0_{ij}}=\frac{\partial L_i}{\partial c_j}$. These partial derivatives can be calculated directly from the ray tracing data, as has been shown recently [8].

The TWI reconstruction problem is solved by applying least squares to estimate the topography parameters. The linear model is an approximation only; depending on the actual non-linearity of the relation between the design deviation and the change of the OPDs, the method is applied iteratively [5,6].

Since simulations are part of the data evaluation, the design model of the interferometer used for these simulations needs to be adapted to the real measurement setup in a so-called “calibration procedure” [10]. This calibration procedure is based on a black-box model, i.e. instead of the correction of the positions and parameters of all elements, their influence on the resulting wavefronts, arriving or leaving the test space, is corrected [10]. Throughout this paper, a perfect calibration of the interferometer model is assumed.

Due to the measurement principle of the TWI (cf. [5, 7]), the OPDs are calculated separately from several sub-interferograms at delimited areas (patches) at the CCD image sensor. For each patch k, the measured OPDs L_meas are only known up to an unknown offset b_k, i.e. $L_i=L_{{\text{meas}}_{i_k}}+b_k$; this has to be taken into account when solving the inverse problem of the TWI reconstruction. The ambiguity due to the unknown (patch) offsets has not been mentioned explicitly in the TWI literature [5–7, 10] because it is inherent to all single-frequency interferometry measurements. Nevertheless, this ambiguity is an important issue in order to provide absolute measurements of aspheres and freeform surfaces.

In this paper, the accuracy of simple approaches currently used to manage the unknown offsets is investigated. The results of the paper show that simple strategies can result in a significantly diminished accuracy of the absolute form reconstruction. In order to achieve a highly accurate form reconstruction, additional information is needed. In this paper, the type of additional information as well as its required accuracy are explored in virtual experiments. For this purpose, the TWI evaluation procedure is adapted to enable the processing of the additional information. The results of this paper can be regarded as a basis for absolute TWI measurements, while the knowledge gained about the required accuracy is important for choosing a suitable experimental realization.

The paper is organized as follows: In Section 2, the problem resulting from the unknown patch offsets is demonstrated, and simple approaches to accommodating this problem are presented and analyzed by means of virtual experiments. The investigations show that simple methods, such as referencing the OPDs to the OPD of each patch or ignoring the piston parameter, result in diminished reconstruction accuracy. A way to evaluate additional information is presented and investigated by means of virtual experiments in Section 3. Finally, some conclusions are drawn.

2. Description of the problem

A measured OPD $L_{{\text{meas}}_{i_k}}$ can be expressed as

2.1. Simple approaches to deal with the unknown patch offsets

One simple way to manage the unknown patch offsets b_k is to reference each OPD to the OPD of a chosen source-pixel pair within the same patch, and to take this referencing into account for the determination of the Jacobian matrix J₀ as well (see Fig. 2(b)). However, such a procedure does not solve the ambiguity of the reconstruction problem, since the absolute patch offsets are still unknown and the absolute positioning of the topography to be measured cannot be determined in this way. The reason for this is that a position error along the optical axis corresponds to the piston parameter of the topography, which cannot usually be distinguished from the topography parameters of defocus and spherical aberration.

If the columns of the Jacobian matrix J₀ were orthogonal, one remedy would be to ignore the piston parameter of the topography, as it only corresponds to an offset of the topography and is not part of its actual form. However, the columns of the Jacobian matrix are usually non-orthogonal; such a model reduction then leads to systematic errors for an imperfectly aligned topography. The contribution of the existing unknown piston (offset) to the data is then distributed over the non-orthogonal estimation parameters (defocus and spherical aberration) (see Fig. 2(c)). Thus, in order to overcome the ambiguity created by the unknown patch offset changes, appropriate additional information has to be included.

 

Fig. 2 Example of a comparison of the reconstruction results for the reconstruction of simulated data: (a) shows the deviation of the topography from its design. The difference between the reconstructed and the real topography is shown in (b) for referencing each OPD to a chosen OPD of each patch, in (c) for referencing the OPDs to a chosen OPD of each patch and not estimating the piston parameter, and in (d) with additional absolute data and assuming that the data are affected only by small measurement errors which can be modelled by white Gaussian noise (standard deviation 5 nm). (e) shows the Zernike coefficients τ_j corresponding to the residual topography T_rec − T of (b), and (f) shows the Zernike coefficients τ_j corresponding to the residual topography T_rec − T shown in (c). The topography reconstructions were realized by using a Zernike polynomial function of 153 Zernike coefficients.

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2.2. Reconstruction examples

The effects of the different reconstruction methods of Section 2.1 are demonstrated with the help of virtual experiments in a reconstruction example of an aspherical topography whose design parameters are shown in Table 1 (design 1).

Table 1. Parameters of the design of the two aspheres used in the examples.

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A deviation of the topography T from its design T₀ was simulated by adding randomly selected Zernike polynomial functions to the design of the topography. The Zernike coefficients were taken from a normal distribution with a mean of 0 nm and a coefficient-dependent standard deviation. The coefficient-dependent standard deviations were chosen in such a way that the amplitude drops exponentially from the chosen maximum of 1 µm the higher the azimutal Zernike index and the radial Zernike index were. In this way, the simulated deviations correspond to typical topography deviations (see e.g. [2]). Furthermore, the standard deviation of the Zernike coefficient which corresponds to the defocus was set as a value typical of this parameter: c₅ = 1 µm. The first Zernike coefficient (piston) corresponds to an offset and, therefore, to a position error along the optical axis. Thus, the piston coefficient was set to zero and a position error along the optical axis was simulated instead (normal distribution with mean 0 nm, standard deviation 1 µm). The resulting standard deviations used to draw Zernike coefficients from a normal distribution in order to simulate a deviation from the design topography are shown in Fig. 3. The RMS (root-mean-square) deviation of the topography T from its design in this example amounted to 659 nm (see Fig. 2(a)). For these examples, the deviation of the topography from its design was reconstructed using the first 153 Zernike polynomials. The number of Zernike polynomials can be adapted, depending on the measurement setup and the expected deviation of the topography from its design, e.g. to account for higher frequency deviations due to the manufacturing process of the aspheres. Furthermore, 13 350 source-pixel combinations were used for the reconstruction. Angular and lateral position errors of the topography were not taken into account.

 

Fig. 3 Standard deviations used to draw random Zernike coefficients that represent the deviation of the topography from its design from a normal distribution.

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Figure 2(b) shows the difference between the reconstructed and the real topography for referencing the OPDs of each patch to a chosen OPD of the patch. Zernike polynomial functions were fitted to this difference. The corresponding Zernike coefficients τ_j are shown in Fig. 2(e). As expected, the residual is dominated by the Zernike parameters of defocus (j = 5, magnitude of 10⁻⁷ m) and spherical aberration (j = 13, magnitude of 10⁻⁸ m). Figure 2(c) shows the difference between the reconstructed and the real topography of the same simulated data when the OPDs are referenced to a chosen OPD of each patch and the piston parameter is ignored. The corresponding Zernike coefficients τ_j of the residual topography T_rec − T are shown in Fig. 2(f). For the chosen example, this method leads to a small improvement of the result. Nevertheless, since the piston parameter is not equal to zero and therefore contributes to the data, its contribution is distributed to the parameters which are non-orthogonal to the piston (defocus, spherical aberration). Therefore, a method such as this will not provide a result of the highest possible accuracy. Figure 2(d) will be discussed in Chapter 3.

3. Remedy: Including additional absolute data

In order to achieve highly accurate absolute reconstruction results, at least one absolute patch offset b_k must be known. In this section, we propose a method to include and evaluate such absolute data. The proposed approach accompanies additional OPD data from an additional patch. Therefore, we also propose an appropriate common processing of the ordinary and additional data. For this purpose, we propose to process the measured OPDs L_meas with $L_{{\text{meas}}_{i_k}}=L_i-b_k+{\epsilon }_i$ and to include the estimation of the unknown patch offsets of each patch in such a way that the vector of unknown parameters is extended to include the unknown patch offsets b_k. In addition, we propose a measurement setup that allows the measurement of one absolute OPD. It is important that the additional measurement be chosen in such a way that the measured absolute OPD relates to the specimen’s surface. A method that merely measures the absolute position of the topography’s mount is not sufficient, since the position of the real topography relative to the mount, as well as the unknown fabrication deviations of the topography, lead to further deviations of the topography’s position.

A simulation study is used to investigate the influence of the accuracy of the additional absolute measurement on the reconstruction result. For the simulation of the proposed additional measurement, we assume that, for each pixel, an absolute OPD is measurable. We divide the measurements for the additional patch into an absolute offset value $b_{L_{\text{add}}}$ and a measurement part which contains the information of the form ΔL_add. This notation facilitates the investigation of the influence of errors in the absolute measurement, as shown in Section 3.3.

The vector containing the data for the reconstruction of the topography ΔL = L_meas − L₀ is then extended to include the additional data ΔL_add and its absolutely measured patch offset $b_{L_{\text{add}}}$:

At this point, the Jacobian matrix needed for the calculation of the topography parameters has the following structure:

The index max marks the number of rays of the main measurement, the index max_add assigns the number of rays of the additional measurement and the index a tags the number of Zernike polynomial functions that are used to reconstruct the topography’s form.

To make it possible to incorporate different previously known measurement uncertainties, we propose to solve the inverse problem by applying weighted least squares (see below). The covariance matrix of the input parameters (measurements) has the following form:

${\sigma }_{\mathrm{\Delta }{L}_1}=\dots ={\sigma }_{\mathrm{\Delta }{L}_{{\max }_{\text{add}}}}=\sigma$ corresponds to the uncertainty of the measured OPDs and ${\sigma }_{b_{L_{\text{add}}}}$ corresponds to the uncertainty of the absolute patch offset change of the additional measurement $\mathrm{\Delta }{b}_{L_{\text{add}}}$. Using this Jacobian matrix, the inverse problem of the TWI reconstruction is solved in a least-squares method [11] by minimizing

3.1. Proposed measurement setup

For the acquisition of the additional data, only the central light source of the two-dimensional light source array is activated. Furthermore, only the source-pixel combinations of a single patch produced by the central source are used for the reconstruction procedure. In order to use the existing TWI design and to be able to compare the results to former results, the reference arm is deactivated during the acquisition of the additional measurement. Instead, a semitransparent mirror is placed behind the collimator and used as a reference. The position of the semitransparent mirror is assumed to be calibrated together with the model of the TWI, thus introducing an absolute reference to the OPDs. A sketch of this setup is shown in Fig. 4. At the CCD image sensor, the rays reflected at the semitransparent mirror interfere with the rays that are reflected at the topography to be measured. Thus, the OPDs between these rays are measured. The setup can be used to measure absolute OPDs by applying wavelength shifting [12], multi-wavelength interferometry [13] or white light interferometry [14] for the additional measurement. If methods containing several wavelengths are applied, the use of a Fizeau objective would be preferred in order to prevent effects of dispersion. A TWI setup containing a Fizeau objective was recently invented at the University of Stuttgart [15] and an experimental realization is currently under development.

 

Fig. 4 Schematic sketch of the proposed measurement setup of the TWI for including an additional absolute measurement.

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3.2. Reconstruction using additional data

By way of example, Fig. 2(d) shows the difference between the reconstructed and the real topography from Section 2 using the proposed additional measurement for simulated data by assuming an error of the additional absolute measurement dominated by white Gaussian noise (standard deviation 5 nm). The results show that, for the assumed circumstances, the RMS-reconstruction error can be reduced to the sub-nanometer range if additional absolute data of the proposed form is included in the evaluation.

3.3. Influence of the size of the errors of the absolute measurement data on the accuracy of the reconstruction result

The deviation of the reconstructed and the real topography shown in Fig. 2(d) is based on the assumption that the additional absolute measurement data is affected only by small measurement errors which can be modelled by white Gaussian noise. In this section, the influence of the the size of the measurement error of the additionally measured patch offset on the accuracy of the reconstruction result is investigated. The goal of this investigation is to identify the limits of the acceptable measurement error in order to achieve a significant improvement of the reconstruction result by including additional absolute data. To this end, a normally distributed measurement error (mean 0 nm) with different standard deviations was added to the measured patch offset of the additional measurement. All remaining measurement values were corrupted by adding white Gaussian noise (mean 0 nm, standard deviation 5 nm) to the simulated data.

Due to the above-mentioned subdivision of the additional data into one part explaining the form of the topography and one part belonging to the absolute offset (patch offset), correlations between the data can be neglected. Therefore, only the diagonal entries of the matrix V_in deviate from zero, as shown in Eq. (6).

The virtual experiments were performed for two different design topographies with the parameters shown in Table 1. The deviations of the topographies from their designs in each realization were simulated in the above-mentioned way (see Section 3.2), so that different random topography deviations were investigated. In addition to the measurement errors caused by noise (standard deviation 5 nm), an additional error was added to the measurement of the additional patch offset. The standard deviation of this additional error was varied between 0 nm and 1 mm. For the reconstruction with different sizes of added errors, the same topography deviation was used. Altogether, 50 different topography deviations were realized and reconstructed assuming different standard deviations of the measurement errors of the additional data and applying different reconstruction methods. After the reconstruction, the RMS reconstruction error (RMSE) and the PV reconstruction error (PV=peak to valley) of each reconstruction result were calculated. These evaluation parameters of the 50 topography realizations were averaged for each chosen standard deviation of the added errors of the additional patch offset measurement and then plotted as a function of the chosen standard deviation. The results are shown in Fig. 5. Figures 5(a) and 5(b) show the mean RMS reconstruction error as a function of the standard deviation of the errors of the additional absolute patch offset measurement for the topography with design 1 and with design 2, respectively. The mean RMS reconstruction error increases in both cases, starting at a sub-nanometer value (additional measurement error 0 nm) with increasing measurement error of the additional patch offset measurement. Without including any additional data, and by simply referencing the OPDs to the chosen OPD of a patch, the mean RMS reconstruction error amounts to 42 nm (design 1) and 23 nm (design 2), respectively, and is plotted in both diagrams for better orientation as a dotted red line. As expected, the investigation shows that too large a measurement error of the additional absolute patch offset measurement leads to the same results one would achieve by simply referencing the OPDs of each patch to a chosen OPD without including any additional data. Figures 5(c) and 5(d) show the mean PV reconstruction error of 50 realizations as a function of the standard deviation of the additional absolute patch offset measurement for the topography with design 1 and with design 2, respectively. The mean PV reconstruction error shows the same dependence on the standard deviation of the error of the additional absolute patch offset measurement as the RMS reconstruction error. If the standard deviation of the error of the absolute patch offset measurement amounts to 0 nm, the mean PV reconstruction error has a magnitude in the single nanometer range and increases with increasing measurement error until a value of 155 nm (for design 1) and of 79 nm (for design 2) is reached, respectively. These maximum values correspond to the mean PV reconstruction errors that occur for the simple method of referencing the OPDs to a chosen OPD without including any additional measurement information. The results show that the measurement error of the additional absolute patch offset measurement should be smaller than 500 nm to achieve a significant improvement (factor 2 to 4) of the reconstruction result by including additional data.

 

Fig. 5 In (a) and (b) the mean RMS reconstruction errors in dependence on the chosen standard deviation of the added errors of the additional absolute patch offset measurement are shown for a topography with design 1 and for a topography with design 2, respectively (blue points). Additionally, in (c) and (d) the mean PV reconstruction errors in dependence on the chosen standard deviation of the added errors of the additional absolute patch offset measurement are shown for a topography with design 1 and for a topography with design 2, respectively (blue points). The dotted red line marks in each diagram the mean RMS- or PV-reconstruction error if no additional data were included and the OPDs were referenced to a chosen OPD of a patch. The dashed green line assigns the mean RMS- or PV-reconstruction error if no additional data were included, the OPDs were referenced, and additionally the piston parameter was not estimated.

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Each diagram also shows a dashed green line. This line depicts the corresponding mean reconstruction error if the OPDs of each patch are simply referenced to a chosen OPD while the piston parameter was not estimated. This case was investigated because, for all estimated parameters with the exception of the piston parameter, a low uncertainty would result if the columns of the Jacobian matrix were orthogonal to the column of the piston parameter. Yet the diagrams show that the improvement of the reconstruction accuracy, which can be achieved by applying this method, depends on the basic form of the topography. The reconstruction result may be determined to be superior or inferior to the result obtained by applying the simple method of referencing the OPDs of each patch to a chosen OPD.

In order to show that this method (referencing OPDs and ignoring the piston parameter) is only reasonable if the specimen is perfectly aligned along the optical axis (and therefore the piston does not contribute to the data), the RMS reconstruction error of all 50 topography realizations was plotted as a function of the position error along the optical axis for the different reconstruction methods. The results are shown in Fig. 6. The diagrams reveal that the only method which systematically depends on the position error along the optical axis is the method which references the OPDs of each patch to a chosen OPD while ignoring the piston parameter. This method can only achieve good results for near-perfect positioning of the topography, e.g. better than 100 nm. For some specimens, a positioning with this degree of accuracy might be achieved by starting the positioning at the cat’s eye position and applying an additional distance interferometer. Nevertheless, for certain specimens, this solution does not work (e.g. specimens with a central aperture, or those where the cat’s eye cannot be reached mechanically). For such specimens, the axial positioning might be even worse than 1 µm. Therefore, the inclusion of an additional absolute measurement and the evaluation of its absolutely known patch offset becomes even more important. For both design topographies in Fig. 6, a position error of 2 µm along the optical axis leads to an RMS reconstruction error of approximately 80 nm for the reconstruction method that ignores the piston parameter.

 

Fig. 6 RMS reconstruction error as a function of the position error of the topography along the optical axis for a topography with design 1 (a) and a topography with design 2 (b) for different reconstruction methods: Referencing the OPDs of each patch to a chosen OPD, with including additional absolute patch offset data having a measurement error of 100 nm, and referencing the OPDs of each patch to a chosen OPD as well as ignoring the piston parameter.

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This systematic behavior results from the non-orthogonality of the Jacobian-matrix concerning its first column (piston). A position error along the optical axis cannot usually be distinguished from defocus and spherical aberration. Thus, the contribution of the existing offset (piston) is distributed to the other parameters if the piston parameter is ignored. Furthermore, the diagrams show that all other reconstruction methods are not dependent on the position error along the optical axis (for position errors of a few mircometers); the diagrams also underline the significant improvement of the reconstruction accuracy if additional absolute data with low measurement errors is included.

4. Conclusions

The TWI is a special interferometer for the highly accurate interferometric measurement of aspheres and freeform surfaces. The evaluation of interferometric phase maps merely allows relative phase maps to be determined [9]; the measurement of absolute OPDs is not possible. Due to the measurement principle of the TWI, relative OPDs are determined separately from many sub-interferograms at delimited areas (patches) at the CCD image sensor. Therefore, the OPDs of different patches each have an unknown offset. Simple approaches to managing the unknown patch offsets, such as referencing the OPDs of each patch to a chosen OPD, can diminish the final reconstruction accuracy significantly.

We have demonstrated that additional information is needed for a highly accurate reconstruction of the topography’s form. Furthermore, we have proposed an improved evaluation method which enables the processing of additional absolute measurement information, and we have investigated the required accuracy of the additional data. The investigations of this paper show that additional absolute OPD data of one patch is sufficient to allow an accurate estimation of the spherical parameters of the topography. Including a suitable additional measurement thus makes it possible to measure the absolute form using the TWI.

The simulation study revealed that an accuracy of the additional, absolute measurement of 500 nm already reduces the reconstruction error (RMSE) in the examples from 42 nm to 9 nm (topography design 1) and from 23 nm to 11 nm (topography design 2), respectively.

This knowledge about the required accuracy is needed for choosing an appropriate experimental setup which provides the additional absolute measurement data.

Furthermore, we have shown that the method, which simply references the OPDs of each patch to a chosen OPD and ignores the piston parameter, might be a solution if the position error of the topography along the optical axis is small (i.e. smaller than ≈ 100 nm), but otherwise leads to a systematic error. For some specimens, a positioning with this degree of accuracy might be achieved by starting at the cat’s eye position and applying an additional distance interferometer.