1. Introduction

A variety of measurement technologies used to characterize surface roughness, topography or film thickness have been reported [1–3]. The most common techniques for precision areal profilometry are atomic force microscopy (AFM) [4–6], confocal laser scanning microscopy (CLSM) [7] and low-coherence interferometry (LCI) [8]. It was shown before that AFM is capable of achieving axial resolutions of down to 1 Å for a measurement range of 2.5 µm [9]. Consequently, the scanning of larger areas like 2000 x 2000 µm² with resolutions of about 60 nm can take up to 30 minutes [10,11]. In CLSM, a surface is usually scanned in the axial and lateral direction where areas of the same size can be measured in about 15 minutes with an axial resolution of 5 - 40 nm [12,13]. Faster approaches utilize multi-probe setups in order to parallelize data acquisition but are strongly limited in the lateral resolution [14,15]. A commonly known LCI technique is optical-coherence tomography (OCT) where typical axial resolutions between 1 - 10 µm are achieved for a lateral measurement range of several 100 µm [16–18]. Recent works have shown that axial resolutions in the sub-nm range are possible with a fiber-based common path configuration for single point measurements [19]. The necessity of mechanical scanning of samples or parts of a setup introduces issues regarding the accuracy and repeatability of results [8,19]. In order to avoid this, different full-field approaches were developed recently. Based on a hyperspectral imager and a frequency comb, Zhu et al. have shown a method to image height profiles in an axial range of 250 µm with an error of about 100 nm [20]. The lateral measurement range was limited by the combs spectral width to only 20 µm. A more advanced method substitutes the frequency comb for a pinhole array in order to decode one areal dimension on a hyperspectral imager [21]. This allowed for an acquisition of 2500 independent probing points and increased the measurement range, light efficiency as well as the tilt angle acceptance. One drawback is that the method is only able to make use about 50 % of the detector size to image µm²-sized samples which decreases its lateral resolution. The axial measurement range was about 825 µm where a resolution of 6 nm was achieved.

The purpose of this work is to present an alternative, dispersion-encoded low-coherence interferometer (DE-LCI) setup to capture full-field surface information within milliseconds with high resolution and on comparatively large areas. In this context, the setup will provide a tunable measurement range and resolution, a high-dynamic-range in terms of high axial resolution and a large axial measurement range at the same time. In particular, the decoupling of axial as well as the lateral measurement range from the axial resolution is discussed.

2. Experimental approach

For the purpose of measuring the surface topography of technical samples with high spatial resolution, a modified low-coherence interferometry approach was used [22]. The setup consisted primarily of a broadband light source, an interferometric core and an imaging spectrometer (Fig. 1). The distinguishing feature of the interferometric core was an element with known dispersion. In this work, the dispersive element (DE) used was a plate of N-BK7 glass ($t_{DE}$ = 2 mm). The light of a broadband laser-driven plasma light source ($\Delta \lambda$ = 170 - 2100 nm, EQ-99X, Energetiq Technology, Inc., USA) was split 50:50 using a cube beamsplitter. In the reference arm, the light was transmitted through the dispersive element before and after the reflection on a plane mirror. In the second arm, the sample with a specific height profile in the x-y plane denoted by $z_n(x_n,y_n)$ was positioned to reflect the incoming light. After the recombination of the light from both arms, the imaging configuration L1 was used to guide it towards an imaging spectrometer for spatially-resolved spectral decomposition. The imaging spectrometer was built using a CMOS camera (2048 x 2048 px², acA2040-90umNIR, Basler AG, Germany) and a slit of 10 µm width. In consequence, a detectable spectral range of $\Delta \lambda$ = 333 nm ($\lambda$ = 447 - 780 nm) with a spectral resolution of $r_{spec}$ = 0.2 nm was achieved, where the typical integration time was between 50 - 200 ms, depending on the sample reflectivity.

 

Fig. 1. Principle profilometry setup with WLS - white light source, BS - beamsplitter, DE - dispersive element (having the thickness $t_{DE}$ and the refractive index $n^{DE}(\lambda )$), REF - reference mirror, SMP - sample profile including the points $z_1(x_1,y_1)$ and $z_2(x_2,y_2)$ which are imaged with a given magnification M (typically M = 1.3 or 4) by the L1 - imaging configuration, relayed by a FM - folding mirror onto the slit of the IMSPEC - imaging spectrometer as magnified points $z_1'(x_1,y_1)$ and $z_2'(x_2,y_2)$.

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In order to qualify the interferometric method, measurements using two different height standards were performed. For the evaluation of heights in the nm-range, a silicon height standard with a 100 nm nominal step height (Simetrics VS, Simetrics GmbH, Germany) was examined with an imaging magnification of M = 4 over a length of $x$ = 450 µm which resulted in a lateral resolution of about 5 µm. A second height standard (EN14-3, Physikalisch-Technische Bundesanstalt (PTB), Germany) was used to measure nominal height steps of 1, 5 and 20 µm over a length of $x$ = 1.5 mm utilizing an imaging magnification of M = 1.3. Three-dimensional information was gathered by moving the imaging configuration L1 and therefore stacking height profiles along the $y$-direction.

2.1 Data analysis

In its general form, the signal at the imaging spectrometer can be described using [23],

 

Fig. 2. Plot of the simulated signal for single points on a profile with a) $z_1(x_1,y_1)$ and b) $z_2(x_2,y_2)$ where the corresponding equalization wavelengths are $\lambda _{eq1}$ and $\lambda _{eq2}$.

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By using an imaging spectrometer the detection of individual spectra for every point along a line in the (x,y)-plane of the sample became possible. For this purpose, the collimated light of the system was imaged on a slit. This slit was afterwards imaged onto a grating spectrometer where the grating was chosen according to the desired spectral resolution (300 grooves/mm, 500 nm reflective blaze grating, GR25-0305, Thorlabs, USA). Consequently, every height change on the sample’s surface (e.g. $z_1$, $z_2$ and $z_n$) along the x-dimension changed the path length $\delta (x)$. This led to different phases for every position on the sample in the $x$-dimension with corresponding equalization wavelengths (e.g. $\lambda _{eq1}$, $\lambda _{eq2}$ and $\lambda _{eqn}$) (Fig. 3). The determination of the equalization wavelength can be performed analytically by calculating the first derivative of phase with respect to the wavelength and solving for $\lambda _{eq}$

 

Fig. 3. Plot of simulated phase signals which were calculated based on Eq. (2) with the equalization wavelengths $\lambda _{eq1}$ and $\lambda _{eq2}$ which can be determined analytically by calculating the derivative of the phase with respect to the wavelength and solving for $\lambda _{eq}$, Eq. (3). Correspondingly, two different heights $z_1(x_1,y_1)$ and $z_2(x_2,y_2)$ can be calculated according to Eq. (4).

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Here $n^{DE}_{g}(\lambda _{eq})$ is the group refractive index of the DE material and the factor of $\frac {1}{2}$ considers that the light passes the DE with the thickness $t_{DE}$ twice in a Michelson configuration. The axial measurement range can be calculated from this equation as the range of differences $\Delta z$, determined by the range of equalization wavelengths detectable with the specific setup. While spectral information and therefore the height of a sample was contained in one camera axis, the other axis stored the spatial information of the surface profile distribution along the x-dimension. According to this approach, the surface profile along a line was gathered in a scan-free, single-shot approach. This minimized distortive influences and guaranteed a fast data acquisition.

In order to acquire information on the areal surface profile of a sample, the imaging lens L1 (see Fig. 1) was actuated by a translation stage. In this way, single-shot spectral images and hence line profiles were gathered continuously in the y-direction over a given range in one lateral dimension which typically was 250 µm. As the imaging lens was placed after both interferometric arms and the sample was not moved between measurements, negative influences such as small shifts were eliminated.

2.2 High-dynamic-range design

In contrast to conventional LCI approaches, the proposed setup relies on a broadband spectral operation in order to maximize the usable measurement range $\Delta z$ in the axial dimension, Eq. (4). Dependent on the material used as DE and the spectral bandwidth, the thickness can be used to proportionally tune $\Delta z$. This enables a modification of the measurement range and resolution on demand by changing either the thickness of the DE or its material. During all presented experiments, the same DE made of N-BK7 with $t_{DE}$ = 2mm was used. If other materials were to be chosen, the Sellmeier-like slope of the wavelength-dependent refractive index could be utilized to manipulate the axial measurement range. The spectral range of $\Delta \lambda$ = 333 nm and spectral resolution of $r_{spec}$ = 0.2 nm allowed the calculation of the system resolution $r_{sys}$ based on the equalization wavelength as the only measure for axial height changes,

A more in-depth analysis of a narrowed region-of-interest close to the equalization wavelength shows that not only $\lambda _{eq}$, but also the spectral intensity, determined by the phase at this point, is $\delta$- and therefore height-dependent (Fig. 4(a)). The amplitude of the signal at $\lambda _{eq}$ is sensitive to small height changes. This behavior makes it possible to use a fit of the signal amplitude in order to increase the precision of the height measurement independently from the measurement range where the variation of $\delta$ was within $r_{sys}$. Due to the ability to search for the minimum of the error sum of squares (SSE), it was possible to find the best fit for the amplitude close to $\lambda _{eq}$ (Fig. 4(b)). The best fit data enables the determination of the height at a point on the sample with higher resolution than the determination of the equalization wavelength alone. The limitation of the resolution was determined by the systems noise behavior. It is visible that the measured spectrum has some intensity noise, which was considered as the main noise component during the evaluation of the resolution.

 

Fig. 4. Visualization of the fitting routine with a) plot of a measured intensity signal having a narrowed region-of-interest at $\lambda _{eq}$ with a selection of fit curves which were modeled with Eq. (1) and (2) where the OPD is separated by $\Delta \delta =$ 5 nm for each iteration and b) corresponding error sum of squares (SSE) for the different $\Delta \delta$ and a interpolating curve plotted in black where the arrows indicate a magnified plot of the area in close proximity of the minimum SSE while a detailed description of the fitting routine can be found in the appendix and Fig. 8(b).

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In a separate measurement, the setup was used in the same way as in profilometry experiments. In contrast the arms of the interferometer were adjusted so that no interference was visible. Spectra of the light source passing through the complete setup were taken for every point along the spatial dimension. The average of 10 full spectral measurements was calculated and the relative mean intensity noise was computed. The distribution of the intensity noise along the spatial dimension was checked by taking the histogram where a subset of 400 spectra, one at every 5^th pixel, in the spatial domain was used for analysis. A normal distribution was found. On this basis, a mean value of $\Delta I$ = -21.25 $\pm$ 0.3 dB was determined. In order to estimate the resolution resulting from from the noise of the system, Eq. (2) was solved for the path length difference $\delta$

3. Results

3.1 Height step characterization

The quality and characteristics of the optical setup were evaluated by analyzing different height steps. For this purpose, a silicon-based step standard (VS 0.10, Simetrics GmbH, Germany) was examined on a length of 450 µm in 10 repetitive measurements (Fig. 5). The results of this examination proved the capability to resolve nm-sized height steps with a measured height of $h$ = 101.8 $\pm$ 0.1 nm which is in good agreement to the nominal value of 100 $\pm$ 7 nm quoted by the manufacturer. The corresponding RMS error with regard to the nominal value was 1.1 nm. The Ra roughness of the silicon surface could be measured with 0.8 nm which scales with a factor of about 8.7 to Rt = 7.0 nm [25]. This is within the range of 6 - 10 nm quoted by the manufacturer [26]. The recorded and measured profiles show bat-wing effects on the sharp edges [27]. The measurement error increases in these regions due to diffraction effects and deflections. It was visible that deviations of up to 20 nm could occur, marked with blue ellipses in Figs. 5(b) and (c). These deviations were attributed to diffraction effects visible as additional intensity modulations in the spectral interference raw data (Fig. 5(b)). For calibration purposes, the oscillations of the diffraction can be modeled as Fourier filtering by the aperture of the capturing optical system [27]. In relation to the simulated raw data (Fig. 5(a)), it was visible that not only diffraction occurs, but also other distortions. In case of a flat, properly aligned sample, the spatial distribution of the maxima and minima is parallel to the $x$-axis of the plot (see Fig. 5 (a)). It can be seen in the actual measured data, that this was not the case (Fig. 5 (b)). This was the result of a slight tilt of the sample (about 0.11 ${\raise0.7ex\hbox{${nm}$} \!\mathord{\left/ {\vphantom {{nm} {\mu m}}}\right.}\!\lower0.7ex\hbox{${\mu m}$}}$) with regard to the sample arm. It was corrected during post-processing of the final measured profiles assuming a linear tilt.

 

Fig. 5. Results of the measurement of a 100 $\pm$ 7 nm nominal height standard with a) simulated spectral interference signal over a spectral range of 333 nm and lateral dimension of 450 µm with the equalization wavelength $\lambda _{eq}$ marked and b) corresponding measured spectral interferences data with visible intensity modulations due to diffraction (marked with blue ellipses) and c) calculated mean height profile $z(x)$ from the raw data with diffraction-induced batwing effects on the sharp edges (marked with blue ellipses).

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Three-dimensional information can be obtained by translating the imaging configuration L1 (see Fig. 1) in order to image different parts of the sample on the slit (Fig. 6(a)). This method has the advantage that neither the sample nor the reference mirror have to be moved, which excludes possible translational deviations in the interferometric signal. By stacking height profiles along the $y$-direction, the three-dimensional surface of a precision-turned height standard (EN14-3, PTB, Germany) could be acquired. The standard provided grooves of defined depths with nominal steps of 1, 5 and 20 µm which were subject to a series of measurements. In total, an area of 1.5 x 0.25 mm² was covered. The recorded data included measured steps of 971.26 $\pm$ 0.31, 4951.40 $\pm$ 0.28 and 19924.00 $\pm$ 0.36 nm (Fig. 6(b)). As a matter of comparison, the same profile was evaluated using a tactile profilometer. The interferometric results showed an overall averaged RMS error of 26.9 nm with regard to the tactile measurements. According to the calibration certificate of the standard, the initial calibration was performed on a comparable tactile profilometer. The quoted uncertainty for the nominal height steps is $\pm$ 33 nm. Furthermore, the high axial resolution leads to the ability to capture roughness data in the nm-range on all height steps, see inset in Fig. 6 (b). A value of Rq = 26.7 nm was calculated. It is visible that some edge effects and noise occurred on the slopes of the steps. In the current optical design, utilizing an imaging system with a NA of 0.06, a large lateral measurement range could be covered while data on the slopes with an 70° angle could not be gathered reliably. This is due to the comparatively low lateral resolution of 5 µm. Depending on the application, the setup can be optimized to increase the sensitivity on these parts of the sample. The measurement of this standard was performed in order to demonstrate the three-dimensional acquisition capabilities of the setup as well as the combination of the axial measurement range in the µm-regime together with the ability to resolve nm-sized features such as the roughness of the standards surface. The results of this sample also highlights the capability of the approach to decouple the axial resolution from the lateral measurement range as nm features can be detected while measuring over a range of 1.5 mm. This distinguishes the approach clearly from other techniques such as confocal microscopy.

 

Fig. 6. Plot of the a) averaged three-dimensional surface of a precision turned groove standard (Gaussian filter applied to reduce edge effects for display purposes) with marked depth levels where the red arrow indicates b) averaged line profile at y = 100 µm where recorded depths of 971.26 $\pm$ 0.31, 4951.40 $\pm$ 0.28 and 19924.00 $\pm$ 0.36 nm could be measured with an mean RMS error of 26.9 nm with respect to a measurement on a tactile profilometer while having the ability to capture roughness information which is shown in the inset where a value of Rq = 26.7 nm was calculated.

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3.2 Resolution characterization

The error of the system can be analyzed utilizing two measures where one is the repeatability defined by the standard deviation $\sigma _z(x)$ of multiple profiles $z_i(x)$ gathered in a short time frame. The second is the resolution calculated as the standard deviation $\Delta z_{min}$ of a feature such as the height $h_i$ (Fig. 7(a)). In order to analyze the repeatability, the structure presented in Fig. 5(c) was measured N = 10 times in a row without any other delay than the acquisition and data transfer time. The analysis of the standard deviation allows to conclude on the repeatability,

 

Fig. 7. a) Depiction of the calculation of repeatability as the standard deviation $\sigma _z(x)$ of multiple profiles $z_i(x)$ according to Eq. (10) as well as the resolution $\Delta z_{min}$ as the standard deviation of the feature height $h_i$ according to Eq. (11) and b) Plot of the spatially resolved repeatability of the nm-sized height standard of Fig. 5 where the impact of diffraction is visible at $x =$ 150 - 210 µm and $x =$ 300 - 350 µm with an inset to visualize the magnitude of $\sigma _z(x)$ between $x =$ 50 - 150 µm.

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While the repeatability was a measure for temporal fluctuations occur from one measurement to the other, the ability to resolve structures along the spatial domain (here denoted as the $x$-coordinate) is independent from these fluctuations. A measure for the resolution can be found in the standard deviation $\Delta z_{min}$ of features sizes such as the height of structures $h_i$ as it can be assumed that in-between short time frames such as the acquisition time of single frames, the sample does not change,

The standard deviation of the feature size represents a cumulative measure for the resolution which includes influences of the optical setup, the electronics, the calibration routines and the data processing alike. During the data analysis of the presented results, it became obvious that part of the difference between the calculated minimal resolution of 0.029 nm, see Eq. (8), and the minimal measured resolution of 0.1 nm is introduced by data processing routines. As the recorded profiles usually were tilted by a minor degree, an appropriate tilt correction was performed based on the linear fit of every captured surface profile. Although the tilt correction was optimized, a minor influence on the standard deviation cannot be excluded.

In contrast to the silicon standard, the µm-sized height standard has a significantly higher roughness which leads to scattering. In consequence, the measurements on this sample were affected by noise which led to a number of outlier data points. The implemented post-processing routines accounted for these outliers and corrected them. The outlier correction was performed by the detection of rising edges using data of the first derivative of the profile with respect to the $x$-coordinate in combination with the correction of the difference between the outlier and the mean value of five previous data points. In the analysis of the step heights and its standard deviation it could be detected that the outlier correction scheme influences the profile on a sub-nm level. As outliers occur on different spatial positions for consecutive measurements, a higher standard deviation was measured as opposed to measurements were less outliers occurred like on the silicon height standard.

On the basis of the calculated resolution, the dynamic range of the setup was calculated with the measurement range of $\Delta z$ = 79.91 µm and $\Delta z_{min}$ ranging from 0.1 - 0.36 nm as DR = 2.22 - 7.99×10⁵. Compared to the latest findings of other areal profilometer approaches such as of Reichold et al. [21], the achieved dynamic range is about 1.6 - 5.8 times higher.

4. Conclusion

Within this work, an alternative approach to high resolution, high-dynamic-range surface profilometry was presented. The approach utilized a low-coherence interferometer in conjunction with a defined dispersive element in order to control the axial resolution as well as the measurement range. By the application of an imaging spectrometer for signal detection, it was possible to gather surface profiles along a line of up to 1.5 mm length without the need for mechanical scanning. Specifically, a decoupling of the axial resolution from the lateral measurement range was accomplished where nm-features were captured on mm-long profiles. Full areal surface information could be acquired with the same approach. It was shown that a minimal axial resolution of 0.1 nm is achievable. Experiments on a height standard were performed and showed an RMS error of 1.1 nm regarding the nominal height of 100 nm. The dynamic range of the developed setup was demonstrated by the measurement of a PTB-calibrated, µm-sized, turned-groove standard where nominal depths of 1, 5 and 20 µm could be analyzed simultaneously. The mean RMS error of these measurements was 26.9 nm with respect to a reference measurement on a tactile profilometer. In conjunction with a total possible axial measurement range of 79.91 µm, a maximal dynamic range of DR = 7.99×10⁵ could be achieved, which is nearly a 6-fold increase compared to other current approaches known from literature [21]. This allows both measurements of large axial heights on large lateral ranges with high resolution while simultaneously gathering data of nm and sub-nm features which can be used for e.g. roughness evaluation. Scattering of sample surfaces was identified as a reason for increased error due to the outlier correction mechanism applied in post-processing. Future work will concentrate on this algorithm in order to reduce the dependency of the resolution from the surface properties of the sample. Furthermore, additional work is planned to optimize the analysis algorithms in order to reduce processing times from currently 2 seconds per profile.

The authors are aware that an evaluation of interferometric data is possible in the depth space, too, as typically done in OCT [16]. Different dispersion in the interferometer arms can be compensated by multiplying the spectral data with an appropriate phase function [28]. After resampling to wavenumber and Fourier transformation this leads to a depth signal with three contributions: first the desired peak in depth space, second the broadened mirror term and third the broadened DC term. The extremum in phase within the spectrum, associated with the equalization length (see Fig. 5), leads to an overlap of the three terms, influencing both, the peak position and phase. For this reason, a simple evaluation as used in [19] will induce large errors. Nevertheless, considering the three terms is possible as shown in full range OCT [29], but requires a more elaborate compensation for the desired accuracy. Because this kind of evaluation can increase the measurement range considerably, this approach will be presented in a future publication.

In essence, the dispersion-encoded profilometer approach has proven to be flexible in measurement range as well as highly resolving in the axial dimension with a measurement range of nearly 80 µm and a resolution of 0.1 nm. With these characteristics, the approach is potentially useful for process or production accompanying tasks as well as for scientific purposes.

Appendix

The data analysis of the captured spectra within this work was based on the determination of the equalization wavelength $\lambda _{eq}$. While Eq. (3) describes an analytical way of calculating $\lambda _{eq}$ from the phase data of a spectrum, the method is problematic in its implementation for measured data. In order to analyze the phase signal, a $\cos ^{-1}$-operation has to be performed which results in data which is wrapped in the range of $[-\pi ,\pi ]$. In order to avoid phase wrapping, the data processing established within this work relied on a short-time Fourier transform (STFT) to estimate $\lambda _{eq}$. In STFT a window of fixed width is discretely moved along the measured signal while a Fourier transform is performed for every position. The resulting spectrogram allows the determination of the equalization wavelength (Fig. 8(a)). The estimated equalization wavelength was used in conjunction with Eq. (4)

(12)$$z\left(\lambda_{e q}\right)=\frac{\delta_{e q}}{2}=\left[n_{g}^{D E}\left(\lambda_{e q}\right)-1\right] \cdot \frac{t_{D E}}{2}$$

(13)$$\delta_{e q}=\left[n_{g}^{D E}\left(\lambda_{e q}\right)-1\right] \cdot t_{D E}$$

where the value for the group refractive index of the dispersive element $n_g^{DE}(\lambda _{eq})$ as well as its thickness $t_{DE}$ are used to calculate the initial estimate for the path length difference $\delta _{eq}$. These preliminary calculations of $\lambda _{eq}$ and $\delta _{eq}$ were the basis for the subsequent fitting routine.

 

Fig. 8. a) STFT spectrogram with the detected equalization wavelength $\lambda _{eq}$ used as a basis for fitting and b) visualization of the simplified two-step fitting process based on SSE determination for simulated data sets in two ranges ($\Delta \delta _1$ and $\Delta \delta _2$) for the path length difference $\delta$.

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The fitting routine developed was a two-step process. Using a range of $\Delta \delta _1$ = $\pm$ 1 µm having a step size of 2 nm centered around the calculated $\delta _{eq}$, a set of simulated spectra based on Eq. (1) and (2) was calculated. The determination of the SSE of these calculated spectra with respect to the measured spectrum enabled the estimation of a more precise value for the path length difference $\delta _1$ at the minimum of the SSE curve (Fig. 8(b)). The calculated $\delta _1$ was used in a second iteration of the routine to calculate another set of spectra with a finer spacing in $\Delta \delta _2$ = $\pm$ 140 nm with steps of 0.02 nm. Comparable to the first processing iteration, the SSE of the calculated spectra with respect to the measured spectrum was evaluated. The minimum SSE indicates the path length difference $\delta _2$ which can be used to calculate the height at a point of the sample (Fig. 8 (b) and Fig. 4 (b)).

The described method was chosen in contrast to other established fitting algorithms to ensure the convergence to the global minimum instead of a local minimum which can be the case due to the oscillating nature of the data. The iterative fitting approach bears further potential for optimizations regarding the processing time which will be in focus of future work.

Funding

Sächsische Aufbaubank (100316880); Sächsisches Staatsministerium für Wissenschaft und Kunst (4-7544.10/7/3).

Acknowledgments

The authors would like to thank the group members of the WHZ Optical Technologies- as well as the Fraunhofer AZOM-group for their fruitful discussions and preparatory lab work.

Disclosures

The authors declare no conflicts of interest.

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