## Abstract

We present a method to enhance the achievable lateral resolution of a multi-sensor scanning profile measurement method. The relationship between the profile measurement method considered and established shearing techniques is illustrated. Simulation and measurement results show that non-equidistant sensor spacing can improve the lateral resolution significantly.

©2010 Optical Society of America

## 1. Introduction

The measurement of a specimen’s topography is an essential part of the production process in many fields of industry. When only a single sensor is moved over the specimen under test - as in coordinate measuring machines - measurement results are distorted by height offsets of the scanning stage. When two sensors are used for the measurement process, the height errors of the scanning stage can be compensated for [1]. However, tilt errors of the scanning stage would still be present in the measurement results. A combination of three (or more) sensors offers the opportunity to compensate for the tilt of the scanning stage, in addition [2–4]. In [5] it was shown that small but systematic offset errors of the utilized sensors can accumulate to a parabolic topography error which can be orders of magnitudes larger than the systematic offset errors. The accumulation of these systematic sensor errors can be avoided, if additional information of the tilt of the measured sub-profile is provided [5,6]. The measurement technique considered in this paper uses an angle measurement device for additional tilt measurements of the scanning stage. This technique allows for a unique reconstruction (up to an unknown straight line) of the specimen’s profile. In addition, the height offsets of the scanning stage and the systematic sensor errors of the sensors are also estimated. This technique was named the Traceable Multi Sensor (TMS) technique.

The TMS-technique was originally developed for the analysis of measurement data of a compact surface interferometer with more than 100 pixels in one row. To apply the TMS-technique, a line sensor array of at least three distance sensors (e.g. arrays of chromatic sensors [7] or displacement interferometers [8]) is necessary. Depending on the sensor type, it is not always possible to place neighboring sensors arbitrarily close. Constructing an array out of several commercial chromatic sensors would lead to sensor spacing of typically a few millimeters, whereas the sensitive area of such a type of sensor is typically a few micrometers. As shown in section 3, the distance between neighboring points of the reconstructed profile cannot be shorter than half of the distance between neighboring sensors for equidistantly spaced sensors. Transferring the TMS-technique to other sensor types requires overcoming this limitation of the lateral resolution.

The relationship between the TMS-technique and common shearing techniques is illustrated in section 3. As main result we show in section 4 in terms of simulations that non-equidistant sensor spacing can improve the lateral resolution significantly. The simulation results are verified by measurements of a chirp specimen with sinusoidal waves of amplitude of 100 nm and wavelengths from 2.5 mm down to 19 µm (section 5).

## 2. Absolute profile measurement by a multi-sensor scanning technique

A diagram of the scanning profile measurement technique is shown in Fig. 1
. An array of (at least) three distance sensors is moved over the specimen while its position p_{i} is measured with a displacement interferometer. An autocollimator is employed to conduct tilt measurements of the sensor system at each stage position.

Imperfectness of the scanning stage leads to tilt (b_{i}) and offset (a_{i}) errors in each stage position (index i, i = 1,…, I) which cannot be ignored in case of high precision measurements. Each of the three distance sensors (index j, j = 1, 2, 3) has an unknown systematic offset error ε_{j}. The offset errors a_{i} and the systematic errors ε_{j} are unknown and have to be compensated by the reconstruction algorithm. To this end, a mathematical model of the measurement process is utilized. The profile is modeled at the equidistant reconstruction points x_{k} (k = 1,…, K) with chosen spacing d_{s}. Since measurements are also conducted at positions ${\tilde{x}}_{i,j}$ between the x_{k}, the topography height $f({\tilde{x}}_{i,j})$ at these positions is modeled as a linear combination of the $f({x}_{k})$ applying Lagrange interpolation [9]. With the introduced notation, the measurement m_{i,j} of the distance sensor j in the stage position i can be modeled as follows.

_{i}(offsets), b

_{i}(tilts), ε

_{j}(systematic sensor errors) and f(x

_{1}),..., f(x

_{K}) (topography) provided that the sensor positions s(j) and the measurement positions ${\tilde{x}}_{i,j}$ (calculated from the measured positions p

_{i}and the s(j)) are known. A system of linear equations is constructed to estimate the parameters f(x

_{k}), ε

_{j}, a

_{i}and b

_{i}. The measurement uncertainties of the m

_{i,j}and b

_{i}are used as weights for the least squares solution of the over-determined system of linear equations [10]. As shown in [5], a unique reconstruction (up to an unknown straight line) of the topography values f(x

_{k}) is possible only if the b

_{i}are measured in addition and added to the system of linear equations.

## 3. Lateral resolution for equidistant and non-equidistant sensor arrays

The reconstruction distance d_{s} is a parameter of the TMS-algorithm. It can be chosen independently of the scanning step d_{step} and the sensor distance σ subject to the constraint d_{s}>0.5σ for equidistant sensor spacing, provided the scanning step d_{step} is smaller than d_{s}. Otherwise, the system of linear equations has no unique solution. Due to the limitation of the reconstruction distance, wavelengths shorter than the sensor distance σ cannot be detected. The reason for the constraint d_{s}>0.5σ is illustrated in Fig. 2
.

Assuming a perfect scanning stage without any height errors and a sinusoidal topography with a wavelength λ corresponding to the distance σ of neighboring sensors (Fig. 2(a)), the measurement results displayed in Fig. 2(c) would be received. On the other hand, an ideal plane topography together with sinusoidal (λ = σ) scanning stage errors (Fig. 2(b)) would lead to the same measurement results. Since the TMS-algorithm estimates both topography and scanning stage errors from the measurands of the distance sensor array, there is no unique solution for the wavelengths λ = σ/N (N = 1,2,…).

To obtain a more mathematical description, the sensor array given in Fig. 1 is considered. In each stage position i the values measured by the three distance sensors are given by

_{i}of the scanning stage. For I stage positions, overall I height offsets a

_{i}have to be estimated. Equation (4.2) leads to overall I rows of the matrix of the system of linear equations. If the height offsets a

_{i}are not of interest, the system of linear equations given by (4.1) and (4.3) would be sufficient for the topography reconstruction. The terms ${\tilde{m}}_{i,1}$ and ${\tilde{m}}_{i,3}$ in Eq. (4.1) and (4.3) are already given as a convolution of the topography f(x) and the term $(\delta (x+{\sigma}_{1})-\delta (x))$ respectively $(\delta (x+{\sigma}_{2})-\delta (x))$ plus the systematic sensor errors and the tilt influence. Due to this convolution some spatial frequencies of the topography cannot be measured. This can be exemplified with the help of the transfer function in the frequency domain. The transfer function T(f

_{topo}) is obtained by the Fourier transformation of the term $(\delta (x+\sigma )-\delta (x))$.

*𝔉*denotes the Fourier transformation and f

_{topo}the spatial frequency of the topography. Figure 3 shows the absolute value of the transfer function (Eq. (5)) for two different sensor distances σ

_{i}.

Hence, a transfer function can be assigned to each sensor distance σ_{i} and is zero for the topography frequencies f_{topo} = Ν/σ_{i} (N = 1,2,…). For an equally spaced sensor array, all roots of the transfer functions are identical. Therefore, the wavelengths λ = σ/N (N = 1,2,…) cannot be reconstructed just as in lateral shearing interferometry [11–14]. This requires the reconstruction distance d_{s} to be larger than half of the first non-reconstructable wavelength (d_{s}>0.5σ). This is nearly the same problem which occurs in shearing measurement techniques. But note, that the TMS-technique allows for the estimation of the systematic errors ε_{j}, in addition. The topography reconstruction is performed by the TMS-algorithm in the spatial domain and not in the frequency domain like common shearing techniques. Improving the lateral resolution requires the compensation of the roots of the transfer-function. In the two dimensional case, this can be achieved e.g. by combining measurements with shears in a number of directions [12]. For profile measurements with the TMS-technique, the sensor distances σ_{i} have to be chosen in the same manner as the shears in one dimensional shearing techniques [13,14] to extend the lateral resolution. This means, that the sensor distances have to be relatively prime so that the first joint root of their transfer functions occurs at higher spatial frequencies. For this reason, the case shown in Fig. 3 is not an ideal choice for the sensor positions since the first joint root of the transfer function occurs at f_{topo} = 2/σ_{1}. In this case d_{s} has to be larger than 0.25σ_{1} improving the lateral resolution by only a factor of 2 compared to the case σ_{2} = σ_{1}. Note, that due to the continuous topography model introduced by the Lagrange interpolation in Eq. (1), the sensor distances can be chosen independently of the topography length in contrast to Fourier-based solutions of the shearing problem [13,14]. Furthermore, the scanning step d_{step} must only be chosen smaller than the reconstruction distance d_{s}.

## 4. Simulation results

Simulations have been carried out to show that non-equidistant sensor positions can be used for improving the lateral resolution of the TMS-method. There are three important parameters which have an impact on the reconstruction quality, namely the topography wavelength λ, the reconstruction distance d_{s} and the relative sensor spacing σ_{2}/σ_{1}. To give the reader a general impression on how different combinations of these three parameters influence the reconstruction quality, simulation results will be presented which show the reconstruction result in dependence of two parameters while the third one was fixed.

For the simulations a sensor consisting of three distance sensors was assumed. The distance between the first and the second sensor was set to σ_{1} = 100 µm, the second sensor distance σ_{2} was a parameter of the simulations. The sensor was moved with a scanning step of d_{step} = 20 µm over the specimen. Sinusoidal topographies with an amplitude of 100 nm and different wavelengths λ in the range of 50 µm – 210 µm were used. Typical measurement noise of the distance sensors (5 nm), autocollimator (0.5 arcsec (2.4 µrad)) and displacement interferometer (100 nm) as well as offset (5 µm) and tilt errors (2 arcsec (10 µrad)) of the scanning stage have been accounted for; the values in brackets indicate the standard deviations of the (normally distributed) random variables used to simulate the various error sources. The reconstruction quality has been assessed by the root mean square error (rms) after rotating and shifting the reconstructed topography relative to the desired topography so that the rms error is minimal.

First of all, we show in Fig. 4
the rms error against the reconstruction distance and the topography wavelength. Results for equidistant sensors (σ_{2} = σ_{1}, Fig. 4(a)) are shown on the left and for non-equidistant sensors (σ_{2} = 1.5σ_{1}, Fig. 4(b)) on the right.

For equally spaced sensors (Fig. 4(a)) and d_{s} = 0.5σ_{1}, the rms error is in the region of 100 nm and independent of the topography wavelength. A further reduction of d_{s} leads to a rise of the rms error up to 10^{11} m (not displayed in Fig. 4(a)). Topographies with a wavelength smaller than 2d_{s} are reconstructed as plane profiles resulting in an rms error of about $\frac{100\text{\hspace{0.17em}}nm}{\sqrt{2}}=70.2\text{\hspace{0.17em}}nm$. This is an effect of the TMS-algorithm and can be used to suppress aliasing effects [15]. Figure 4(b) shows the rms reconstruction errors for a sensor configuration with the transfer functions as displayed in Fig. 3. Since both transfer functions have a joint root at f_{topo} = 2/σ_{1}, d_{s} cannot be smaller than 0.25 σ_{1}.

Considering that the ratio of the sensor distances σ_{2}/σ_{1} is the fundamental parameter to reconstruct higher spatial frequencies f_{topo} = 1/λ, it is interesting to see how the rms error behaves in dependence on both of these parameters. This is shown in Fig. 5
and Fig. 6
for different settings of the reconstruction distance (d_{s}/σ_{1} = 1, 0.5, 0.3, 0.25). To ease the comparison with other sensor arrays, the spatial frequency is given in units of $1/{\sigma}_{1}$.

If the reconstruction distance is equal to the shorter sensor distance (d_{s} = σ_{1}, Fig. 5(a)), the rms reconstruction error is independent of the ratio of the sensor distances. Spatial frequencies above the Nyquist frequency f_{Nyq} = 1/(2d_{s}) (${f}_{topo}=0.5/{\sigma}_{1}$in Fig. 5(a) and ${f}_{topo}=1/{\sigma}_{1}$ in Fig. 5(b)) are not reconstructed due to the anti-aliasing property of the TMS-algorithm [15]. Equally spaced sensors require d_{s} to be larger than 0.5σ_{1}. Figure 5(b) shows the rms error at this limit. Even for spatial frequencies below ${f}_{topo}=1/{\sigma}_{1}$, the reconstruction of the topography fails for σ_{2} = σ_{1}. However, various sensor distance ratios exist which allow a reconstruction with d_{s} = 0.5σ_{1}. Figure 6(a) and Fig. 6(b) show that d_{s} can be set even smaller if the sensor distances are chosen appropriately.

In Fig. 6(b) the rms error is plotted for d_{s} = 0.25σ_{1}. The rms error increases for σ_{2}/σ_{1} = 1.5. The transfer functions for this ratio of sensor distances have a joint root at f_{topo} = 2/σ_{1} as shown in Fig. 3. Hence, d_{s} has to be larger than 0.25σ_{1}. The minimal rms value is increasing from 1.03 nm in Fig. 5(a) to 2.63 nm in Fig. 6(b). This is primarily an averaging effect, since the number of reconstruction points is quadrupled from Fig. 5(a) to Fig. 6(b).

The results shown in Fig. 5 and Fig. 6 have demonstrated that similar results can be expected for all wavelengths smaller than 2d_{s} and fixed values of d_{s} and σ_{2}/σ_{1}. For this reason it is helpful to plot the rms error against the ratio of sensor distances σ_{2}/σ_{1} and the reconstruction frequency f_{s} = 1/d_{s} (Fig. 7
).

For the simulation results shown in Fig. 7, the topography wavelength was set to 1 mm. Equally (σ_{2}/σ_{1} = 1) or semi-equally (σ_{2}/σ_{1} = 2) spaced sensors again lead to large reconstruction errors for ${f}_{s}\ge 2/{\sigma}_{1}$ (d_{s}<0.5σ_{1}). Moreover, a reconstruction frequency of ${f}_{s}=2/{\sigma}_{1}$ (d_{s} = 0.5σ_{1}) leads to an increased rms error for ratios of sensor distances close to σ_{2}/σ_{1} = 1. The peak at (σ_{2}/σ_{1} = 1.5, ${f}_{s}=4/{\sigma}_{1}$) represents the situation shown in Fig. 3, whereas the reconstruction distance d_{s} is set to half of the wavelength which cannot be reconstructed due to the joint root of the transfer functions. Figures 5-7 have shown, that not only single sensor distance ratios are critical, but that also ratios near the critical ones can lead to increasing reconstruction errors. This is caused by topography frequencies where the envelope of the transfer functions is close to zero and will be discussed in section 5. Nevertheless, non-equidistant sensor spacing can lead to an improved lateral resolution.

## 5. Measurement results

To verify the simulation results of the previous section, interferometric measurements on a chirp specimen have been carried out. Chirp specimens are used to characterize the lateral resolution of measurement setups [16,17] and are, therefore, ideal for testing the resolution enhancement by non-equidistant sensor spacing. The advanced fabrication technology-working group at PTB has manufactured the chirp specimen. The rotational symmetric topography was fabricated in a diamond turning process on a nickel layer on a copper specimen. Starting 16 mm from the center, several single sinusoidal wavelengths have been strung together. The largest wavelength was λ = 2.5 mm, then every following wavelength was reduced by 2% relative to the previous one down to λ = 1 mm. After this point the wavelengths were reduced by a factor of $2\cdot {10}^{-4}$down to λ = 19 μm to obtain a higher resolution. Figure 8 shows the wavelengths of the chirp specimen as a function of the distance to the center.

The interferometer used for the measurements has an aperture of 3 mm, a CCD with 165x165 pixels and a pixel distance of 19 µm. The compact interferometer has been moved over the specimen with a scanning step of d_{step} = 18.8 µm while its lateral position has been measured with a displacement interferometer and the tilt with an electronic autocollimator. In this configuration the TMS-technique is designed for profile measurement of moderately flat optical surfaces with peak to valley values up to 100 µm and length of several decimeters with a measurement uncertainty of about 10 nm.

In order to study the effect of the relative sensor spacing σ_{2}/σ_{1} only three pixels of one CCD row have been used. The three selected pixels establish a virtual sensor array of three distance sensors. The distance between the first two sensors was set to 36 pixels leading to σ_{1} = 684 µm. Different pixels were used as third distance sensor to simulate sensor arrays with different relative sensor spacing. The reconstruction distance was set to d_{s} = 30.4 µm. Note, that the reconstruction distance would be limited to d_{s}>342 µm for σ_{2} = σ_{1}. In order to compare the measurement results with a reference measurement, 110 pixels of the interferometer were used with the same reconstruction distance. Since the reconstruction distance d_{s} is larger than the sensor distance of the large sensor array (σ_{i} = 19 µm) by a factor of 1.6, all wavelengths larger than 2d_{s} can be reconstructed. The reconstructed topography obtained by this large sensor array is referred to as the high-resolution scan in the following.

The chirp specimen has a peak to valley of 1.2 µm due to the manufacturing process. This long-wave form deviation has been removed for clarity of presentation. To this end, a polynomial of degree 5 has been fitted to the high-resolution scan. This polynomial has been subtracted not only from the high-resolution scan, but also from all other profiles. Since Eq. (1) allows for a reconstruction only up to an arbitrary line, all other scans have additionally been aligned to the high-resolution scan by adding a polynomial of degree one.

The top of Fig. 9
shows a section of the right part of the reconstructed chirp specimen. The position is given as the distance to the center of the specimen to ease the comparison with Fig. 8. The blue curve shows the reconstruction result of the virtual three-point sensor with σ_{2}/σ_{1} = 1.028 and the high-resolution scan is drawn in orange in the background. On the left, the reconstructed part of the specimen has a local wavelength of λ = 680 µm, and on the right of λ = 170 µm. The reconstruction distance d_{s} = 30.4 µm would, in principle, allow the reconstruction of wavelengths down to λ = 2d_{s} = 60.8 µm, but for wavelengths below λ = 170 µm too many pixels of the used interferometer dropped out so that the construction of a virtual three-point sensor was not possible. The vertical green lines mark the positions where the chirp specimen has a local frequency corresponding to the roots of the transfer function of the smaller sensor distance σ_{1}. Note, that all wavelengths to the right of the first green line would not be measurable for σ_{2} = σ_{1}. The rms value of the difference between both topographies shown at the bottom of Fig. 9 is 7.8 nm. Obviously, the reconstruction contains some systematic errors due the reconstruction process.

A comparison of the transfer functions (top of Fig. 10
) and the spectrum of the difference between both reconstructed topographies (bottom of Fig. 10) demonstrates the influence of the transfer functions on the reconstruction result. The spectrum has peaks at the same frequencies where the envelope of both transfer functions is close to zero. Hence, a relative sensor spacing of σ_{2}/σ_{1} = 1.028 is not a good choice for the selected reconstruction distance.

A more appropriate choice of the relative sensor spacing would lead to an envelope of the transfer functions which has as few points as possible close to zero for frequencies smaller than the Nyquist frequency f_{Nyq} = 1/(2d_{s}). Figure 11
shows the reconstruction result for σ_{2}/σ_{1} = 1.583. At first glance, the difference between the high-resolution scan and the topography measured with the virtual three-point sensor (Fig. 11, bottom) reveals no systematic reconstruction errors. The rms value of the difference between both reconstructed topographies was reduced to 3.7 nm compared to 7.8 nm for σ_{2}/σ_{1} = 1.028.

In view of the spectrum (Fig. 12 , bottom) of the difference between both reconstructed topographies there are still some systematic reconstruction errors for the frequencies where the envelope of both transfer functions (Fig. 12, top) is close to zero. However, the largest amplitude in the spectrum is about 1 nm which is quite small compared to the noise of a single distance sensor (5 nm) of the three-point sensor.

For an equally spaced sensor array with σ_{2} = σ_{1} = 684 µm, the reconstruction distance would be limited to d_{s}>0.5σ_{1/2} = 342 µm. With non-equidistant sensor distances, the reconstruction distance was reduced by more than one magnitude down to d_{s} = 30.4 µm.

Finally, the proposed non-equidistant sensor arrangement will be compared with an equally spaced sensor arrangement. This requires the sensor distances ${\tilde{\sigma}}_{1}={\tilde{\sigma}}_{2}$ to be smaller than 2d_{s}. An equally spaced virtual three-point sensor was used for the results shown in Fig. 13
and Fig. 14
. Three neighboring pixels of the compact interferometer have been used for the reconstruction procedure. As a result, the sensor distances ${\tilde{\sigma}}_{1}={\tilde{\sigma}}_{2}=1/36\text{\hspace{0.17em}}{\sigma}_{1}=19\text{\hspace{0.17em} \mu m}$are now smaller than the reconstruction distance d_{s} = 30.4 µm. The measurement result for these sensor distances is corrupted by a long-wave error as shown on the top of Fig. 13.

The rms value of the difference topography shown at the bottom of Fig. 13 is 62.3 nm. Compared to the results shown in Fig. 11, this is an increase by more than one order of magnitude. The small sensor distances lead to a transfer function which is very small for low spatial frequencies (Fig. 14, top) compared to the transfer functions shown in Fig. 10 and Fig. 12. The transfer function has its maximum value at f_{topo} = 18/σ_{1}. The low values of the transfer function result in reconstruction errors for the lower frequency part of the specimen (Fig. 14, bottom) and are the reason for the significantly increased rms value. Hence, the combination of large non-equidistant sensor distances leads also to an improved topography reconstruction compared to an equally spaced sensor with smaller separation.

Non-equally spaced sensor design enables the transfer of the TMS-technique to other types of sensors. For instance, the TMS-technique will be used at the Nanometer Comparator at PTB as a straightness reference for measurements in the sub-nanometer range [18] with a non-equally spaced array of displacement interferometers. Moreover, the TMS-technique is not limited to an array of three distance sensors. A fourth or fifth sensor with additional sensor distances σ_{3} and σ_{4} could be used to improve the remaining systematic reconstruction errors shown in Fig. 12.

## 6. Conclusion

The limitation of the lateral resolution in absolute scanning profile measurement caused by equidistantly spaced sensors has been discussed. It was shown, that the proposed multi-sensor measurement technique can be primarily characterized as a shearing technique with additional estimation of systematic sensor errors. As a result, the transfer functions which can be associated to each sensor distance must not have joint roots for frequencies below the Nyquist frequency. Hence, as shown by simulations, the lateral resolution can be improved by non-equidistant positioning of the sensors. Interferometric measurements on a chirp specimen have shown that the lateral resolution can be improved by more than a factor of 10 compared to an equally spaced sensor design. Furthermore, it was shown that the topography reconstruction can be improved if the sensor distances are chosen in such a way that the envelope of the transfer functions is as large as possible for all frequencies below the Nyquist frequency.

## Acknowledgments

The authors acknowledge the financial support of the German Federal Ministry of Economics and Technology (Grant II D 5 - 14/06). The support of the Braunschweig International Graduate School of Metrology (IGSM) is also gratefully acknowledged.

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