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Abstract
Coherence scanning interferometry (CSI) is a common optical measurement method for measuring three-dimensional surface profiles. However, batwings and ghost steps are common obstacles in CSI. In this paper, we proposed a gap-matching algorithm with the improved complete ensemble empirical mode decomposition with adaptive noise (impCEEMDAN) to solve the above two obstacles without any priori knowledge of the surface geometry from the tested sample. A micro-component with 500 nm and 1200 nm step heights and a 10 µm standard step of were used as test samples to evaluate the accuracy of the proposed method. Simulations and experimental results show that this approach can effectively suppress the batwings effect and eliminate the ghost steps. Experiments also confirm that the approach has good accuracy and repeatability.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Coherence scanning interferometry (CSI) is a widely used technology in optical non-contact three-dimensional (3D) surface profile measurement with a high accuracy of nanometer order [1–3]. Scanning white-light interferometry uses a white-light source with a short coherence length and provides a narrowly localized interferogram to measure the surface profile without 2π-ambiguity [4]. Combining coherence and phase information on smooth surfaces can achieve the same high accuracy as phase-shifting interferometry [5–8]. However, sometimes phase peak does not fall within the zero order interference fringe of the coherence peak, and this causes artificial jumps that are termed ghost steps [4]. Moreover, Harasaki and Wyant have shown that coherence data is highly sensitive to diffraction, resulting in batwings effect at the edges of surface features such as steps and lines [9]. The batwings effect can give rise to systematic errors, causing extra non-linearity to the instrument transfer function (ITF) of a white light interferometer [10]. ITF represents the fraction of the amplitude of a sinusoidal measuring object as obtained by the measuring instrument, which is defined by P.de Groot et al. [11].
There have been many investigations on suppressing batwings effect and correcting fringe order errors. Roy et al. proposed the use of achromatic phase shifters on measuring geometric phases to minimize errors due to the batwings effect [12]. However, because this method needs a pair of switchable achromatic phase-shifters, the modification of the system light path becomes necessary. Ghim and Davies improved the dual-wavelength measurement [13] method and proposed a method to eliminate ghost steps based on typical value judgments [4]. Although this method does not require any improvement of the light source, it has strict requirements on the wavelength selection and the stability of the experimental environment. P.de Groot et al. analyzed the difference map between the coherence and the phase profiles to eliminate the ghost steps. They used a $3 \times 3$ pixel averaging filter and 3D surface fitting function to correct the wrong phase gaps [1]. This method demonstrates that nonlinear filtering of the profiles obtained from coherent peak detection can be employed to avoid phase jumps in the phase evaluation. Consequently, the nonlinear filtering seems an appropriate method to remove batwings [14]. However, it requires the selection of different 3D surface fitting functions according to different tested samples. A priori knowledge of surface geometry and the selection of filter template size will also affect the accuracy of measurement results.
In this paper, a gap-matching algorithm with the improved complete ensemble empirical mode decomposition with adaptive noise (impCEEMDAN) is proposed to suppress the batwings effect and eliminate ghost steps without any priori knowledge of the surface geometry from the tested sample. The impCEEMDAN, proposed by Colominas et al. [15], which is an improvement of the empirical mode decomposition (EMD) methods [16–17], can effectively reduce the amount of noise within the mode and can also remove the useless mode components to obtain less noisy and more physically decomposition signals. This algorithm divides the calculation into two main steps. Firstly, the interference signal is decomposed by impCEEMDAN to obtain a series of mode components. By comparing these components, one of them is selected as the decomposition signal. Gap-matching is then used to correct the fringe order errors obtained from coherence and phase information. Finally, 3D topography results can be obtained. The efficiency of the proposed method in solving the problems of batwings and ghost steps are verified by numerical simulations and experimental investigations.
2. Interference signals and impCEEMDAN
impCEEMDAN is an improvement to the EMD method, which can extract signals from data generated during nonlinear and non-stationary processes. It separates the oscillation modes of different scales in the signal by adding limited (non-integer) amplitude white noise. Finally, the original signal can be expressed as the sum of amplitude modulation and frequency modulation functions, called the “instrinic mode functions” (IMF). IMF can generate instantaneous frequencies, and the embedded structure can be clearly identified [15,17].
The main idea of the noise-assisted variation of EMD is to add some controlled noise to the signal to create new extreme values. In this way, in those parts where new extreme values are generated, the local mean is “forced” to stick to the original signal, while remaining unchanged in the rest of the signal (no generation of extreme values). That is, the algorithm is forced to focus on certain specific values of the scale energy space. Adding noise inevitably generates additional noise. In order to deal with the extra noise, impCEEMDAN uses each final mode to calculate the next mode. Each mode is calculated in the order in which the scheme is determined. The amount of noise present in the modes are reduced by defining the true mode as the difference between the current residual and its local mean.
We set I as the interference signal and the algorithm can be described as follows [15].
- (1) Calculate the local means of N realizations ${I^{(i )}} = I + {\beta _0}{E_1}({{w^{(i )}}} )$ to obtain the first residue using EMD: where, ${w^{(i )}}({i = 1,\ldots ,N} )$ is a realization of white Gaussian noise with zero mean and unit variance, and the magnitude of the additive noise $\beta > 0$. ${E_k}({\cdot} )$ is the operator for producing the $k$th mode obtained by EMD and $\left\langle \cdot \right\rangle $ is the function of averaging throughout the realizations. $M({\cdot} )$ is the operator which produces the local mean of the signal that is applied to. ${r_1}$ is the first layer signal residual.
- (2) At the first stage ($k = 1$) calculate the first mode: ${\widetilde d_1} = I - {r_1}$.
- (6) Go to step (4) for next k.
Constants ${\beta _k} = {\varepsilon _k}\textrm{std}({{r_k}} )$ are chosen to obtain a desired signal-noise ratio between the added noise and the residue to which the noise is added. ${\beta _k}$ is normalized based on standard deviation of noise to obtain noise realizations with smaller amplitudes for the late stages of the decomposition.
3. Gap-matching algorithm with impCEEMDAN and corresponding simulations
3.1. Gap-matching algorithm with impCEEMDAN
Due to the inconsistency of coherence and phase information, false height jumps will appear on the surface profile. Once the false jumps occur in the area where batwings appear, it will not be able to remove fringe order errors correctly. Therefore, solving batwings is the first step of the proposed algorithm. Since batwings effect is considered to be strongly nonlinear [14], we introduce impCEEMDAN to restore the modulation contrast of the interference signal. The flow chart of the proposed algorithm is shown in Fig. 1.
The algorithm can be described as follows:
Step 1: The interference signal is decomposed by impCEEMDAN to obtain a series of mode components. We need to choose one of these components as the decomposition signal. So we define two characteristic functions to filter the signal components:
where, ${F_1}$ is a matching function, and ${F_2}$ is a correlation function. ${I_k}$ is a signal of different mode components, I is the original signal, and n is the number of sampling points.For any signal component, if its ${F_1}$ is the minimum value of ${F_1}$ of all signal components and its ${F_2}$ is the minimum value of ${F_2}$ of all signal components, this component is considered to be the actual decomposition signal. It is worth noting that if all the signal components cannot satisfy these two conditions at the same time, then the component with the smallest ${F_1}$ among the signal components is selected as the decomposition signal.
Step 2: According to frequency domain analysis [18], for full-field imaging, the phase $\phi $ of each component pattern is mathematically extracted by the Fourier transform of the decomposition signal. A linear least squares fit to the phase data $({{\phi_j},{k_j}} )$ can provide a slope $\sigma \approx \textrm{d}\phi /\textrm{d}k$ and an intercept $A \approx {\phi _{k = 0}}$ for each pixel [1]. Fringe order ${m_p}$ of the phase data can be expressed as
It is worth noting that ${m_p}$ has uncertain fringe orders, which will cause false jumps on the surface profile. So coherence information is needed to eliminate the uncertain parts. Surface profile ${h_c}$ of the coherence information can be obtained from the slope $\sigma $[1]. $\sigma $ has a linear dispersion shift related to the wave number $k$. When k is the center wave number ${k_0}$, the shift will be eliminated. Therefore, fringe order ${m_c}$ of the coherence information can be expressed as
In order to eliminate the false jumps on the surface profile, a gap-matching algorithm is applied to eliminate inconsistencies between the fringe order ${m_c}$ and ${m_p}$. Defining that $\Delta m = {m_p} - {m_c}$ is the gap of the two fringe orders. If $\Delta m \ne 0$, errors will occur. The errors can be eliminated by adjusting the gap value. Therefore, the surface profile h obtained by the gap-matching algorithm can be described as
where ${h_p}$ is the surface profile obtained by the phase information.3.2. Simulations based on the Kirchhoff-modeling
In order to better prove that impCEEMDAN can effectively suppress the batwings effect, we use the Kirchhoff-modeling to simulate the CSI signals of one-dimensional rectangular grating structures with different heights. P. Lehmann and W. Xie et al. has conducted extensive researches on the Kirchhoff-modeling and explained the detailed modeling process [19,20]. Since our focus is on the effect of impCEEMDAN and the comparison of the filtering results, the process of modeling is not repeated here. We set the period of the rectangular grating to 6 µm (far from the lateral resolution limit), the numerical aperture to 0.55 and the center wavelength to 600 nm. Figure 2 shows the correlograms of the upper plateau and the upper edge of a rectangular grating with the height of 200 nm. The envelope peak positions of the signals in Figs. 2(a) and (b) are 68.02 and 72.52, respectively. The inconsistency of the envelope peak positions cause a false height jump at the edge of the grating. For Figs. 2(c) and (d), after impCEEMDAN (related parameters:$\varepsilon = 0.2,N = 50$), the envelope peak positions of the upper plateau and the upper edge are 67.96 and 68.17, respectively. The results show that after impCEEMDAN, only a small height jump occurs at the edge of the grating. Figure 3 shows the IMFs obtained by decomposing the signal of Fig. 2(b) by impCEEMDAN. It can be seen that the interference mode $\textrm{IM}{\textrm{F}_1}$ can be well separated from the background light and other modes.
Fig. 2. Correlograms of the upper plateau (blue solid lines) and the upper edge (red solid lines) of a rectangular grating with a height of 200 nm. (a) and (b) Original signal. (c) and (d) Signal filtered by impCEEMDAN.
Then, we simulated the CSI signals of the rectangular gratings with heights of 160 nm to 200 nm, respectively. By calculating the surface profiles of the rectangular gratings, the heights of the upper plateau and the upper edge of the rectangular gratings are obtained, as shown in Fig. 4(a). The average error between the upper plateau and upper edge before impCEEMDAN is 103.4 nm. However, after impCEEMDAN, the average error is 5.8 nm. Figure 4(b) shows the surface profiles of a rectangular grating with the height of 200 nm with and without impCEEMDAN. The height jumps of the rectangular edges are significantly reduced. The above simulation results show that impCEEMDAN can effectively separate the interference mode from the signal and minimize the influence of the batwings effect.
Fig. 4. The heights of the upper plateau and the upper edge of the rectangular gratings with heights of 160 nm to 200 nm. (a) The black solid line represents the heights of the upper edge without impCEEMDAN. The red solid line represents the heights of the upper edge with impCEEMDAN. The blue solid line represents the standard heights of the upper plateau. (b) The surface profiles of a rectangular grating with a height of 200 nm with and without impCEEMDAN.
4. Experimental results
A Mirau interference microscope [21] with an objective lens with magnification of $20 \times $ and numerical aperture of 0.4 is applied as a core component in our SWLI experimental setup. A piezoelectric transducer (Physik Instrumente, Germany) translator for moving the objective vertical scanning in nanometer range. The light source is a halogen lamp with a broad continuous spectrum. The test sample is placed on a stage, and the piezoelectric transducer is controlled by a computer to move it along the z direction with a specific interval. For each nanometer scale increment, the CCD record one fringe pattern and store it in the computer. The system and the interferogram are shown in Fig. 5.
4.1. Measurement of the step heights of 500 nm and 1200 nm
A sample with two step heights of 500 nm and 1200 nm is measured by our SWLI setup. Figure 6 shows the measurement results by gap-matching algorithm with and without the pre-process of impCEEMDAN. When the signal is not pre-decomposed by impCEEMDAN, the surface profile obtained by the gap-matching algorithm still has residual ghost steps, as shown in Fig. 6(d). Comparing Fig. 6(a) and Fig. 6(c), it is shown that the surface profile obtained by impCEEMDAN pre-decomposed can effectively suppress the batwings effect. So when combining the coherence and phase information, the gap-matching algorithm is used to obtain a more accurate surface profile without residual ghost steps, as shown in Fig. 6(e).
Fig. 6. Surface profiles calculated by using different algorithms, where (a), (b) and (d) are the results of gap-matching without impCEEMDAN, and (b), (c) and (e) are the results of gap-matching with impCEEMDAN.
More details of the gap-matching are shown in Fig. 7. Figures 7(a)-(c) and Figs. 7(d)-(f) show the gap-matching process with and without the pre-process of impCEEMDAN, respectively. Due to the batwings effect, the fringe order ${m_c}$ has ghost-like jumps at the edge, leading to a wrong gap-matching result, as shown in Figs. 7(b) and (c). However, the use of the gap-matching algorithm with impCEEMDAN avoids the above problem.
Fig. 7. Gap-matching process. (a) Fringe order ${m_p}$ and ${m_c}$ obtained by the phase and coherence information without impCEEMDAN, respectively. (b) Fringe order gap of (a). (c) Surface profile of (b). (d) Fringe order ${m_p}$ and ${m_c}$ obtained by the phase and coherence information with impCEEMDAN, respectively. (e) Fringe order gap of (d). (f) Surface profile of (e).
Figure 8 shows the result for the 3D topography measurements of the step sample using the gap-matching algorithm with impCEEMDAN (related parameters:$\varepsilon = 0.2,N = 50$).
Fig. 8. Measurement results of the surface with step heights of 500 nm and 1200 nm. (a) 3D topography results measured by the proposed algorithm. (b) Cross-sectional profile between points A and B in part (a).
4.2. Comparison of different algorithms
Figure 9 shows the measurement results of two different samples using three different algorithms. The samples used for measurement are the step shown in Sect. 4.1 and a Micro-Electro-Mechanical System (MEMS) block. Figures 9(a) and (d) use the algorithm proposed in the paper (related parameters:$\varepsilon = 0.2,N = 50$). Figures 9(b) and (e) use the algorithm proposed by Ref. [1]. Figures 9(c) and (f) use the algorithm proposed by Ref. [4]. As shown in Figs. 9(a)-(c), for the relatively simple surface structures, such as steps, these three algorithms can achieve satisfactory results. However, for the samples with relatively complex surface structures, although there are still error points in the measurement results measured by the proposed algorithm, compared with the other algorithms, the proposed algorithm still achieves better results.
Fig. 9. Measurement results using different algorithms for samples with step heights of 500 nm and 1200 nm and a MEMS block. (a) and (d) The proposed algorithm. (b) and (e) The algorithm proposed by Ref. [1]. (c) and (f) The algorithm proposed by Ref. [4]
4.3. Measurement of the 10 µm standard step height
A 10 µm standard step height (9.976${\pm} $0.028 µm) manufactured by VLSI Standards Inc. and calibrated by the National Institute of Standards and Technology (NIST) was measured. The measurement result is shown in Fig. 10. In order to show the accuracy of the proposed algorithm, we compared the height results measured by different algorithms. As shown in Table 1, the white light phase-shifting interferometry (WLPSI), fast Fourier transform (FFT) and proposed method determined the average step height value as 10.273 µm, 10.085 µm and 9.992 µm, respectively. The error of the step height measurement has been reduced from 0.297 µm to 0.016 µm. It can be seen that the measurement result of proposed method is the closest to the actual height of the step and the error of the measurement result is within the allowable error range (9.976${\pm} $0.028 µm).
Fig. 10. Measurement results of the surface with step heights of 10 µm. (a) 3D topography results measured by the proposed algorithm. (b) Cross-sectional profile between points A and B in part (a).
Table 1. Comparison of calculation results of different algorithms
Table 2 shows results of the 10 µm standard step height calibration calculated by the proposed algorithm. The standard deviation of 10 times is 0.011 µm. The results show that the algorithm is in good agreement with the specifications, which indicate that the instrument was well calibrated (related parameters:$\varepsilon = 0.3,N = 100$).
Table 2. Results of the 10 µm standard step height calibration
5. Discussion
When using impCEEMDAN for data processing, we need to set the amplitude of the added noise $\varepsilon $ and the number of ensemble members $N$. If the added noise amplitude is too small, it may not introduce the change of extrema that the EMD relies on. When the noise amplitude increases, the number of ensemble members should increase to reduce the effect of adding noise to the decomposition results. In most cases, $\varepsilon = 0.2$ and $N = 50$ can meet the measurement requirements. However, the setting values of the above parameters are not applicable in all cases. When the situations described in Table 3 occur, we should adjust the parameters according to the actual situations. In short, when the data is dominated by high-frequency signals, the noise amplitude may be smaller, and when the data is dominated by low-frequency signals, the noise amplitude may be increased.
Table 3. The parameters of impCEEMDAN and the scope of application (Parameters: the amplitude of the added noise $\varepsilon $ and the number of ensemble members $N$)
It should also be noted that when the measured surface structure is relatively complex, sometimes setting only one set of parameters does not achieve a good effect. The reason for the existence of error points is that for such relatively complex surface structure samples, it is not possible to use a single set of parameters for measurement, and the appropriate parameters should be selected according to different surface structures. However, if we do not know the surface structure of the item in advance, there may be more errors in the measurement results. Therefore, this requires us to further study the applicability of the algorithm and improve it so that it can adaptively adjust the parameters.
6. Conclusion
We have proposed a gap-matching algorithm with impCEEMDAN in scanning white-light interferometry. Simulations and experimental results demonstrate that, this approach can effectively suppress the batwings effect at the edges and fully eliminate the ghost steps in the measurement of the surface topography of the sample with the step height of less than the coherence length. Compared to conventional methods, this method can effectively improve the measurement accuracy and has good repeatability. The proposed algorithm provides a novel insight for solving the obstacles in the low-coherence interferometry and demonstrates a very strong potential for future application.
Funding
National Key Research and Development Program of China (2018YFF01013203).
Acknowledgments
The authors would like to thank Mr. Sijun Wang from Tiandy Technology Ltd. for the discussions on the algorithms and programming.
Disclosures
The authors declare no conflicts of interest.
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