1. Introduction

Many optical surfaces are either approximately plane or spherically shaped, and several optical techniques are available for their accurate form measurement [1]. Typically, interferometers (e.g. Fizeau type) are based on a matched wavefront which is either a plane or spherical wave. For most applications these wavefronts can be generated with satisfying quality, and high accurate form measurement is achieved provided that the deviation of the surface from an ideal plane or sphere is small.

In order to reduce the number of components of optical systems, surfaces with greater deviations from a plane or a sphere are increasingly used in industry. When testing such specimens by an interferometer utilizing a spherical or planar reference wave, the calculation of the surface becomes inaccurate or even impossible if the fringe density is too high compared to the pixel distance of the interferometer. An alternative then is to use computer generated holograms (CGHs) to create a wavefront adjusted to the particular optical surface. While this technique can in principle be applied to test arbitrary surfaces, the design of an appropriate CGH is involved and needs to be done anew for every specimen under test. Furthermore, the wavefront generated by the CGH is not perfectly known and this leads to systematic deviations of the reconstructed surface. There are several techniques in classical interferometry to compensate these systematic errors [2, 3], and some of these methods can also be used with CGHs [4–7]. Nonetheless, the lateral resolution that can be achieved is for both, classical interferometry and CGHs, limited by that of the CCD of the interferometer.

Sub-aperture interferometry (see, e.g., [8–11]) can be used to reach higher lateral resolutions and to test complex surfaces by combining the measurements of several small sub-surfaces of the specimen using a small interferometer. Since the effective pixel distance (CCD pixel distance traced through the optical system onto the surface) of a small interferometer is small, larger deviations of the surface under test from a plane can be dealt with in this way. Usually sub-aperture measurements are combined using stitching techniques. However, this can lead to an accumulation of systematic interferometer errors, and for large surfaces the resulting surface error can be one order of magnitude larger than the systematic errors of the sub-surface [11], or even more if the interferometer is very small compared to the specimen. In principle, a parabolic systematic interferometer error is indistinguishable from a parabolic topography distribution when only sub-topography measurements are taken into account [12]. Hence, when using only sub-surface information the parabolic part of the topography remains unknown [10].

For many applications a profile measurement of the specimen is sufficient (e.g. synchrotron optics). Recently, the Traceable Multi Sensor method (TMS) [12, 13] has been proposed which reconstructs surface profiles by a particular combination of sub-profile measurements. By using additional tilt measurements of the scanning stage as well as a particular design of experiment, this method allows both, scanning stage errors and systematic errors of the sub-profile measurements, to be eliminated. That is, TMS is an absolute measurement technique, which allows the parabolic part of the topography to be determined exactly. In addition, the systematic sensor errors are obtained. Hence, while sharing the flexibility of sub-aperture stitching procedures, this method does not suffer from the accumulation of systematic errors. TMS focuses on profile measurements of nearly flat specimens which could not be measured with one large interferometer due to the limited dynamic range and limited lateral resolution of a large interferometer. However, TMS requires the sub-profile measurements to be carried out at exactly prescribed lateral positions which is often difficult to realize in practice. In this paper we extend TMS to overcome this limitation. We propose the use of additional measurements of the lateral position of the scanning stage and, using these measurements, we extend the mathematical reconstruction procedure underlying TMS. As a result, the experimental constraints of TMS can be relaxed which significantly eases the application of this technique. Furthermore, we investigate the resolution of the novel procedure as well its accuracy by accounting for the most dominant error influences. Using realistic simulation scenarios we show that an accuracy in the nanometer range can be achieved and that the lateral resolution is not limited by the algorithm.

2. TMS

Figure 1 schematically shows a TMS set-up. The interferometer is scanned along the surface and measures at each position a sub-profile. Each pixel j(j = 0, …, M−1) of the interferometer is treated as an independent distance sensor suffering from an individual systematic error ε_j due to an imperfect reference surface of the interferometer. An autocollimator is used for the additional measurement of the tilt angle b_i of the scanning stage at each measurement position.

 

Fig. 1. Sketch of TMS. The thick red horizontal line represents the interferometer and the blue vertical lines represent the systematic sensor errors ε_j.

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The position of pixel j in the interferometer is denoted by s (j) with s(0) = 0. The lateral coordinate of the first pixel (j = 0) of the interferometer in the measurement position i is named p_i, the distance between neighboring measurement positions is called d_step and the effective pixel distance d_pix1. In each measurement position, the scanning stage introduces a height offset a_i and a tilt angle b_i. For small tilt angles we can use the approximation tan(b_i)≈b_i. Therefore, in position p_i the distance m_i,j between the j-th pixel and the surface can be modeled by

When applying TMS [12] to the set-up in Fig. 1, the scanning step d_step, the shortest distance d_pix1 between neighboring pixels and the distance d_s between adjacent reconstructed surface points are all the same. As a consequence, all pixels measure the surface f(x) at the same positions x_k(k = 0, ..,N −1) (except for the boundaries), and Eq. (1) reduces to

where the implicit dependence of x_k on (i, j) has been suppressed. In Eq. (2) the surface enters only at the discrete locations x_k(k = 0, ..,N −1), and the reconstruction of the surface heights f(x_k) from the measurements is a discrete task. Note that Eq. (2) is linear in the unknowns f(x_k), ε_j, a_i and b_i provided that the positions s(j) are known. As shown in [12], the surface heights f(x_k) can be reconstructed up to an unknown straight line if the tilt angle b_i is measured in addition. Note that the tilt angle may be measured with a systematic offset without affecting the reconstruction result. Since no knowledge about the systematic sensor errors ε_j and the height offset a_i is required for this reconstruction, there is no need to calibrate the systematic errors of the interferometer reference or the scanning stage prior to the measurement.

3. Extended TMS

Application of TMS assumes that for each measurement position of the interferometer all its pixels are placed exactly at the lateral positions x_k. However, positioning errors or distortion of the interferometer lead to lateral measurement locations which differ from the x_k which then results in reconstruction errors for TMS [14]. Furthermore, it is desirable to extend the TMS algorithm to also cope with the case where d _pix1, d_s and d_step are no longer identical. Exact positioning of the interferometer is more difficult to realize than the determination of its actual position. For this, we propose to extend the TMS measurement set-up by an additional measurement of the lateral position of the interferometer, which can be done with the help of a displacement interferometer, cf. Fig. 1 for a schematic and Fig. 2 for a prototype set-up. The proposed set-up now allows for more flexibility in the scanning steps with the actual positions of the scanning stage being accounted for. The reconstruction algorithm underlying TMS is based on the fact that the surface is measured at the discrete lateral positions x_k repeatedly by many different interferometer pixels. This is utilized to account for the additional unknowns in Eq. (2), i.e. offsets a_i and systematic errors ε_j. Since for the extended TMS the surface may be measured at many more positions, several measurements by different interferometer pixels at the same surface position may no longer be available. As a consequence, immediate application of the algorithm underlying TMS is not possible. In contrast to common stitching techniques, interpolation of the measured values in every sub-profile at the grid points x_k as a preprocessing to the TMS algorithm is not useful because both, topography heights and interferometer errors would be affected by such an interpolation. Now, the idea is to still model the surface at a set of positions x_k, but to use an interpolation scheme to account for the measurements made at locations different from the x_k. In this way, the TMS data analysis scheme can still be applied and the surface be reconstructed at the positions x_k. Effectively, a continuous reconstruction of the surface is being carried out in this way, and the three values d _pix1, d_step and d_s need no longer be identical.

 

Fig. 2. Photo of the new extended TMS measurement set-up (under construction) with an autocollimator (left red box), a compact interferometer (middle green box) and an additional displacement measurement interferometer (right blue box).

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A continuous model of the surface is derived from the discrete surface heights f(x_k) by applying Lagrange interpolation [15]. The surface height f(x̃) at position x̃(x̃ ≠ x_k; k = 0, ..,N −1) is given as a linear combination of the surface values f(x_k) in the neighborhood of x̃, and Eq. 2 now reads:

where

In Eq. (3) o is the chosen degree of the interpolation polynomial, c _N the vector with all interpolation coefficients and f _N the vector with the desired surface values. The coefficients c_k(x̃) can be calculated independent of the f_N. When a measurement position matches one of the x_k’s, all the coefficients c_k (x̃) would be zero apart from one and Eq. (3) reduces to Eq. (2).

In general, Eq. (3) differs from Eq. (2), but Eq. (3) is still linear with respect to the unknowns f(x_k), ε_j, a_i and b_i. Therefore, the least-squares estimation procedure described in [12] can be applied as well, the only difference being the construction of the design matrix according to Eq. (3) instead of Eq. (2). The positions x̃ can be calculated from the measured positions p_i and the known pixel positions s(j).

Since now the x_k are chosen only for the purpose of analysis (and not for the design of the experiment), we have the freedom to determine them in some optimal manner. To this end (and prior to the analysis), we choose the x_k equidistantly and place this grid in an optimal way: we shift the whole set of x_k such that the accumulated squared distances between every actual measurement position x̃ and its particular nearest neighbor from the set of x_k’s is minimum.

The o+1 points used for the interpolation are chosen in such a manner that $\frac{o+1}{2}$ points are on the left hand side and $\frac{o+1}{2}$ on the right hand side of x̃, and hence o has to be an odd number. A transfer function can be associated with the interpolation scheme [16] and this transfer function depends on the chosen degree of the interpolation polynomial. Figure 3 shows that the higher the degree of the interpolation polynomial, the more rectangular the shape of the transfer function. Ideally, a rectangular transfer function is sought which correctly interpolates all continuous functions containing only spatial frequencies below the Nyquist frequency. The drawback of a high interpolation degree is increasing computing time and memory requirements due to a decrease in the sparsity of the design matrix implied by Eq. (3). In any case, a too high degree is not required since other error influences such as errors of the lateral position measurements then become the dominant error source, cf. section 4.

The symmetric interpolation scheme of Eq. (3) cannot be applied in the boundary of the surface, and an appropriate treatment is required there. For this, the surface may be assumed to be periodic (f (x) = f (x +Nd_s)), axial symmetric (f(x ₀−Δx) = f (x ₀ +Δx), f(x _N−1−Δx) = f(x _N−1−Δx)), point symmetric (f (x ₀−Δx) = 2f(x ₀) − f(x ₀ +Δx), f(x _N−1+Δx) = 2f(x _N−1)− f(x _N−1 −Δx)) or surrounded by zeros. To avoid such assumptions, it is also possible to use an asymmetric interpolation scheme or to reduce the degree of interpolation near the margins down to a nearest neighbor interpolation. From our experience we found that the assumption of point symmetry appears to perform best and subsequent results are based on this assumption. To account for the boundary conditions, Eq. (3) has to be modified according to

 

Fig. 3. Transfer functions for the polynomial interpolation for different degrees of the polynomials; f_nyq denotes the Nyquist frequency.

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The total number of points used beyond the surface margins depends on the degree of interpolation. Therefore the matrix A in Eq. (5) is a N ×(N +o) matrix with a N ×N identity matrix inside. Vector c _N+o represents now also the interpolation coefficients for the external surface points. The matrices L and R are designed in such a way, that the desired boundary conditions are fulfilled.

The surface can be uniquely reconstructed only up to an arbitrary straight line. Unlike the suggestion made in [12], we found it more advantageous (in terms of the condition number of the resulting design matrix, computing time and required memory) to account for this by setting f ₀ = f _N−1 = 0. Note finally, that one of the systematic sensor errors still has to be set to zero, e.g. ε ₀ = 0 as explained in [12]. Otherwise a constant offset between surface and sensor system would not be distinguishable from an additive constant for all systematic sensor errors.

4. Results

In order to quantitatively assess the performance and benefit of the extended TMS we have conducted simulations taking account for the most relevant error sources. We first identified the most important error sources. To this end, we conducted simulations with the parameters of table 1 (M = 100, d _pix1 = 18.9μm) by successively setting the value of each error parameter to zero. As a result, the lateral positioning error characterized by σ _pos1 (for TMS), the lateral position measurement error σ _pos2 (for extended TMS) and the assumed pixel distance d _pix2 (cf. section 4.3) for the reconstruction process turned out to be the most important error sources.

Table 1. Summary of simulation parameters

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4.1. Simulation scenario and quality assessment

The simulated measurement accounts for more parameters than those considered in Eqs. (1)-(3). Figure 4 gives an overview of the simulation scenario. Tilt b_i and offset a_i of the interferometer lead to lateral displacements in real measurements and this has been considered in the simulation. The interferometer was assumed to be approximately 5mm above the surface. The point of rotation was set to the first interferometer pixel. The measurement directions of the interferometer pixels are assumed to be parallel (according to a planar wavefront). These measurement directions change with a tilt of the interferometer and lead to lateral displacements on the surface. These displacements were considered when simulating the measurements by calculating the geometric distances between every pixel and the surface under test in the direction of the tilted interferometer. Note that for the analysis these lateral displacements were not accounted for, cf. Eqs. (1)-(3). The interferometer is not capable to measure the complete geometric distance, and hence from every sub-profile the mean value was subtracted. Finally, the systematic interferometer errors were added to these sub-profiles. Autocollimator and displacement measurements suffer from adjustment errors. To account for this, a constant offset angle bsys was added to the autocollimator measurements b̃_i and a systematic adjustment error p_sys was considered for the measurements of the displacement interferometer p̃_i. Table 1 gives a summary of all parameters used for the simulations. Due to computing time, for some simulations the number of pixels was reduced to 10 and the pixel distance set to 189μm instead of 18.9μm. Furthermore, the maximum lateral dimension of the measured profile was limited to 80mm. Since TMS does not suffer from accumulation of errors, the maximum length of the profile is not limited by the algorithm.

 

Fig. 4. Sketch of simulation scenario. The simulated measurements p̃_i and b_i account for systematic (p_sys, b_sys) and random measurement errors. N(μ,σ ²) represents a random value drawn from a Gaussian distribution with μ mean and variance σ ².

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Assessment of reconstructed surfaces was done in terms of the root mean square error. Since the surface is invariant upon a rotation and a shift, the results were assessed after the surface has been rotated and shifted such that the root mean square error is minimum.

As test surface, an extremely steep surface (Fig. 5, left) with a Peak to Valley (PV) of 15μm was considered to explore the reconstruction accuracy of the algorithm. In addition, sinusoidal functions (Fig. 5, right) were utilized to explore the resolution which can be achieved.

 

Fig. 5. Simulated surfaces used for the investigations of the extended TMS algorithm. On the left a steep surface with a PV of 15μm, on the right a sinusoidal surface with a PV of 200nm.

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4.2. Accuracy and resolution

To explore the accuracy that can be reached, the test surface from the left hand side of Fig.5 has been utilized. Figure 6 shows resulting rms errors for the TMS and for the extended TMS algorithm in dependence on the size of the positioning error. From table 1 we used here the particular settings M = 10, d _pix1 = d _pix2 = 189μm, σ _pos2 = 250nm. The size σ _pos1 of the positioning error is given in units of the reconstruction distance d_s which eases comparison with other set-ups. For low positioning errors the normal and extended TMS algorithm lead to nearly identical results. When σ _pos1 is larger than $\frac{1}{128}{d}_s$ (≈ 1.5μm here), the results of TMS become worse, while the extended TMS is not affected at all. Hence, Fig. 6 demonstrates that the extended TMS algorithm does not suffer from positioning errors of the scanning stage (provided that the actual measurement locations have been measured sufficiently accurate).

In order to study the lateral resolution which can be achieved we used a sinusoidal surface with a PV of 200nm and different wavelengths (Fig. 5, right). The better the resolution the smaller the wavelength of a sinusoidal test function which can still be reconstructed. Figure 7 shows the resulting average rms errors of 10 realization for TMS and extended TMS (simulation parameters: M = 10, d _pix1 = d _pix2 = 189μm, σ _pos2 = 250nm). The number of realizations was chosen that the standard deviation for every parameter set is about a tenth of the mean value. While TMS suffers from positioning errors for high spatial frequencies, the extended TMS procedure is almost independent of this error source. Hence, the extended TMS method has an improved resolution as compared to TMS when positioning errors emerge, provided that their position is determined accurately. Without any measurement errors the transfer function of the entire algorithm would be identical to the transfer function of the applied interpolation (cf. Fig. 3). Realistic simulation scenarios (M = 100, d _pix1 = d _pix2 = 18.9μm, σ _pos1 = 5μm) have shown, that the lateral positioning measurement error is the most important error source regarding lateral resolution (Fig. 8). For high spatial frequencies and large lateral position measurement errors the reconstructed surface is nearly averaged to zero and the rms error converges to $\frac{100\mathrm{nm}}{\sqrt{2}}=71\mathrm{nm}$. When the size σ _pos2 of the lateral position measurement error exceeds a tenth of the distance ds between neighboring reconstruction points, the surface reconstruction for short wavelength is limited by this error rather than by the degree of interpolation. When σ _pos1 is larger than half of the reconstruction distance, several peaks occur for the reconstruction error. These peaks correspond to peaks of the condition number of the design matrix of the system of linear equations. Avoiding this instability demands a lateral position measurement error smaller than half of the reconstruction distance.

 

Fig. 6. Comparison of TMS and extended TMS in dependence on the positioning error.

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Fig. 7. Reconstruction error for different positioning errors and spatial surface wavelength for TMS and extended TMS.

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Fig. 8. Influence of lateral measurement error (σ _pos2) for high spatial frequencies for extended TMS. The maximum allowed degree of interpolation was o = 41.

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4.3. Optimization and limitations

The described optimal choice of evaluation grid (section 3) is important to reduce interpolation errors. To analyze the effect of this pre-optimization, the reconstruction points x_k were shifted by Δ_pos relative to their optimal position as displayed in Fig. 9. As expected, the rms error increases with increasing Δ_pos, cf. Fig. 10; the minimal rms error is about 2.9nm (simulation parameters: M = 10, d _pix1 = d _pix2 = 189μm, σ _pos1 = 5μm, σ _pos2 = 250nm). When the grid is shifted only by ±0.5ds, the rms error increases by 1nm. For higher shifts the increase of the rms error is caused by effects at the surface margins.

 

Fig. 9. Locations of interferometer measurements and choice of evaluation grid. Red vertical lines represent the measurement positions of the interferometer pixels, and the x_k the chosen evaluation grid. The lower part of the figure shows the evaluation grid obtained from the pre-optimization procedure.

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For the design matrix of the system of linear equations according to Eqs. (1)-(3) the effective pixel distance d _pix1, which is the distance of CCD pixels traced through the optical system of the interferometer, has to be known. Errors of this parameter lead to a scaling of the reconstructed surface perpendicular to the scanning direction; Fig. 11 shows results of extended TMS in dependence on the relative error of the pixel distance d _pix1 as introduced in table 1. In contrast to Fig. 10, Fig. 11 shows not the rms errors, but the reconstruction error at each position of the reconstructed surface for one typical simulation. When the assumed pixel distance is too large, the reconstructed surface is scaled down in y-direction. If an error below 10nm is requested for the topography displayed on the left hand side of Fig. 5, a relative error for the effective pixel distance below 10–3 is required.

 

Fig. 10. Influence of the choice of the evaluation grid for extended TMS for the surface on the left hand side of Fig. 5.

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Fig. 11. Influence of the relative pixel distance error on the reconstruction result of extended TMS.

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Near the surface margins, a special treatment is required as explained in section 3. To study the influence of such a treatment we ran simulations without noise. When the degree of interpolation is reduced, the reconstruction error increases close to the margins (cf. Fig. 12). Reduction of the interpolation degree is connected with a loss of lateral resolution near the surface margins, caused by inferior transfer functions. The reconstruction error was for point symmetric extension less than for all other boundary treatments. Note that for point symmetric extension the profile and its first derivative are continuous at the boundaries. Furthermore the degree of interpolation needs not be reduced. However, the point symmetric extension may introduce an oscillation ringing close to the margins. This ringing depends on how the surface is running into the margins, and for some surfaces it is better to slightly reduce the degree of interpolation when approaching the margins. The used interpolation scheme is only a local approximation of the surface, but as a result of the least squares optimization all surface values are linked with each other. This is why improperly chosen surface extensions may lead to global reconstruction errors. In case of an asymmetric interpolation scheme the reconstruction error is in the μm range and not displayed here. The remaining peaks of the reconstruction error at the positions 10mm, 20mm and 70mm in Fig. 11 are caused by the linear approximation in Eq. 1. The tilt of the sensor leads to a displacement in the lateral position which is not considered in Eq. 1. Especially at positions with large surface slope this modeling error leads to deviations for the reconstructed surface.

 

Fig. 12. Reconstruction errors for different surface extensions.

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For the simulations the lateral resolution of the interferometer was limited solely by the effective pixel distance. The resolution of real interferometers is, however, limited by illumination, the optical system as well as the phase retrieval algorithm and it can be characterized by the Height Transfer Function [17]. In this paper pixels were treated as point sensors and the simulation scenario was one-dimensional. That is, two dimensional averaging of pixels and movements transverse to the scanning direction were not considered here.

The three values d _pix1 (pixel distance), d _step (scanning step) and d_s (reconstruction distance) need not be identical. However, there may still exist some constraints. We found that the reconstruction distance should not be smaller than half of the distance between two neighboring pixels. Furthermore, the scanning step has to be smaller than the reconstruction distance. In case of an equidistant sensor array this leads to the conditions

Future work is addressed to investigate a possible relaxation of these constraints.

5. Conclusion

A scanning procedure for the optical profile measurement of large, moderately flat surfaces has been proposed. The procedure extends a recently introduced scanning method which allows for the elimination of scanning stage errors and systematic interferometer errors. While that procedure requires the measurements to be carried out at prescribed lateral locations, the novel procedure overcomes this limitation which is often difficult to realize in practice. The lateral resolution of the reconstructed topography is only limited by the resolution of the interferometer, not by the algorithm. Results of a realistic simulation scenario show that highly accurate profile measurements with high lateral resolution can be achieved, and these results encourage the employment of the proposed procedure for the profile measurement of large and nearly flat surfaces with a dynamic range which is too large for full aperture measurements.