Adaptive modal decomposition based overlapping-peaks extraction for thickness measurement in chromatic confocal microscopy

Jiafu Li; Yanlong Zhao; Hua Du; Xiaoping Zhu; Kai Wang; Mo Zhao

doi:10.1364/OE.410177

1. Introduction

The spectral confocal technology, including chromatic confocal microscopy and chromatic confocal probe, has been widely used in biomedical science, material science, and micro or macro-scale surface metrology [1–3]. Among them, chromatic sensors have the characteristic of no mechanical axial scanning. To achieve its depth discrimination capability, this method need to calibrate the relationship between wavelength and displacement, and then extract the peak wavelength of sample surface from the reflected light signal, which can be used to calculate the corresponding displacement or height by dispersion relation [4,5].

Meanwhile, chromatic probe also is a common approach to measure the thickness of transparent materials, whose measurement error can be less than 0.5µm [6,7]. In chromatic confocal systems (CCS), the spectral axial response signal (sARS) obtained by spectrometer contains two sub reflection intensity distributions, which correspond to the upper and lower surfaces of the transparent materials, respectively [8,9]. Therefore, the accurate and reliable peaks extraction of sARS has significant influences on the thickness measurement uncertainty.

At present, most researches focus on extracting peaks from the independent sub-reflection intensity distribution for displacement or a thicker transparent sample [10]. These methods can be categorized as linear and nonlinear [11]. The centroid algorithm is a classical linear method but would yield height errors due to the pitch-acquisition errors or slicing uniformity [12]. Nonlinear fitting algorithms (NFAs) use different mathematical models to describe the shape of reflected light intensity [13–16], such as Gaussian, sinc2, parabolic fitting and intensity point spread function [17]. Besides, while the transparent film is sufficiently thin, the sub reflection intensity distributions are no longer independent of each other, and the objective function of the above methods should be modified. As the intersection or superposition of two sub reflection intensity distributions may not have been taken into account, the existing algorithms could generate large peak extraction error.

Thus, in this paper, we described a new algorithm to extract the overlapping peak wavelengths using modal decomposition approach, whose objective is to separate the sub reflection intensity distributions from sARS before peak calculation. The rest of the paper is organized as follows. In Section 2, the principle of thickness measurement and the problem of peaks overlapping for ultrathin film are revisited. Section 3 introduces the principle of proposed modal decomposition algorithm (MDA) and its iterative solution process. Subsequently, some simulation and experiments on performance evaluation are presented in Sections 4 and 5, and the results are also compared with the two other existing algorithms, such as Gaussian fitting algorithm (GFA), and sinc2 fitting algorithms (SFA). Finally, some conclusions are made in Section 6.

2. Peak overlapping problem in ultra-thin film thickness measurement

Figure 1 shows the thickness measuring principle for a transparent plate based on CCS. The thickness d represents the distance between the upper and lower surfaces of the film. The white light source emitted from the Y-type coupling fiber can form various monochromatic lights with continuous focus after passing through the dispersion lens. Thus, two wavelengths λ₁ and λ₂ will be focused on the two surfaces of the measured sample, respectively. The dispersion relation between thickness and wavelengths can be expressed as [18,19]

(1)$$d = \frac{{r \times (S({\lambda _2}) - S({\lambda _1}))}}{{({z_0} + S({\lambda _2}) - S({\lambda _0})) \times \tan [\arcsin \frac{{{n_1} \times \sin (\arctan (r/({z_0} + S({\lambda _2}) - S({\lambda _0}))))}}{{{n_2}}}]}}.$$

Where r is the radius of the dispersive lens, its value has been determined at the design stage, S denotes the characterized displacement wavelength relationship [20], n₁ is the refractive index of air, n₂ is the refractive index of transparent sample, its value can be obtained by the refractive index measuring instrument, z₀ could be chosen as an arbitrary sampled point in the dispersion field, λ₀ is the corresponding wavelength. Therefore, these parameters can be taken as the prior information in the measurement process.

Fig. 1. Schematic diagram of thickness measurement for transparent specimens consisting of two surfaces with refractive index n₂.

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On the premise of sensor structure parameters and film materials, the key to calculate the thickness is to extract the corresponding wavelength λ_i (i=1, 2) accurately from the sARS I(λ) of sample surface detected by the photo detector. Based on the reversibility principle of optical path, the intensity I(λ) is constructed with two sub-intensity distributions, that is, upper surface sub-distribution s₁, and lower surfaces sub-distribution s₂. The signal level will be highest for the wavelength corresponding to the measuring surface. When the thickness is large enough, the peak spacing of reflected lights could be three times larger than their full-width-at-half-maximum (FWHM). As shown in Fig. 1, it is very clear and easy to distinguish the borderline between the two sub-distributions, and the position of peak is not influenced by other light intensity distributions. So GFA and SFA for single peak can be directly used to extract the wavelength λ_i from the corresponding distribution s_i, whose objective functions can be written as

(2)$$\begin{array}{cc} {\textrm{single GFA}:}&{\left|{{s_i}(\lambda ) - {a_i} \times \textrm{exp} [ - \frac{{{{(\lambda - {\lambda_i})}^2}}}{{2 \times \sigma_i^2}}]} \right|\to \min ,}\\ {\textrm{single SFA :}}&{\left|{{s_i}(\lambda ) - {{[\frac{{\sin [(\lambda - {\lambda_i})/2]}}{{(\lambda - {\lambda_i})/2}}]}^2}} \right|\to \min .} \end{array}$$

Otherwise, Fig. 2 shows the normalized simulated sARS of ultra-thin film. There will be no definite boundary between the two sub-distributions as the reflected light intensities corresponding to two surfaces of the sample exist crossover. Then, the traditional single peak algorithm need be changed for extracting multiple peaks simultaneously, whose objective functions can be re-represented as

(3)$$\begin{array}{cc} {\textrm{GFA :}}&{\left|{I(\lambda ) - \sum\limits_{i = 1}^2 {[{a_i} \times \textrm{exp} [ - \frac{{{{(\lambda - {\lambda_i})}^2}}}{{2 \times \sigma_i^2}}]]} } \right|\to \min ,}\\ {\textrm{SFA :}}&{\left|{I(\lambda ) - \sum\limits_{i = 1}^2 {{{[\frac{{\sin [(\lambda - {\lambda_i})/2]}}{{(\lambda - {\lambda_i})/2}}]}^2}} } \right|\to \min .} \end{array}$$

Fig. 2. Spectral axial response signal with overlap phenomenon.

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Among the above methods, the two sub-distributions were considered as a whole, and the results λ_i tend to be equal to the wavelength corresponding to local maximum. But the overlapping of sub-modes will change the correspondence between the theoretical peak λ_i’ and the wavelength of distribution-maximum, which causes the calculation error of the existing methods. Then, the key to improve the calculation accuracy may be how to separate the sub-modes from the sARS I(λ).

3. Peak extraction algorithm based on modal iterative decomposition

A peak extraction method of multi-wavelength light intensity based on modal decomposition is proposed. The reflected light intensity of each measuring surface is regarded as a probability distribution function with limited width, that is, the sampling sARS I(λ) related to the transparent specimen is composed of two probability distribution modes s_i(λ) (i=1, 2) and a residual sequence. The extraction of peak value needs to meet the requirement of minimizing the between-class variance function f(λ₁, λ₂), which can be expressed as [21,22]

(4)$$\min f(\lambda {}_1,{\lambda _2}) = \min \sum\limits_{i = 1}^2 {\int {{s_i}(\lambda ) \times {{(\lambda - {\lambda _i})}^2}} \textrm{d}\lambda } , {\lambda _{\min }} \le \lambda \le {\lambda _{\max }}. $$

where λ_min and λ_max are the minimum and maximum detection wavelength of spectrometer, respectively. The constraint conditions of objective function can be written as

(5)$$\sum\limits_{i = 1}^2 {s_i^{}(\lambda ) = I(\lambda ),{\lambda _{\min }} \le \lambda \le {\lambda _{\max }}}. $$

It can be seen from the above formula that the solution of this function is a process of obtaining two or more sub modes s_i(λ) and their corresponding centers λ_i, which is consistent with the idea of transforming a complex global solution problem into several local sub function solutions by the multiplier alternating direction method, that is, the peak wavelength can be extracted by this method.

Based on the basic ideas of alternate direction method of multipliers, by introducing a penalty term and Lagrange multiplier γ(λ), the constrained variational problem of Eq. (4) can be solved by the unconstrained augmented Lagrangian function L, which can be expressed as

(6)$$L(\{{s_i^{}} \},\{{\lambda_i^{}} \},\gamma ) = \int {\left\{ {\alpha \times s_i^{}(\lambda ) \times {{(\lambda - {\lambda_i})}^2} + {{\left|{I(\lambda ) - \sum\limits_{i = 1}^2 {s_i^{}(\lambda )} } \right|}^2} + \left\langle {\gamma (\lambda ),I(\lambda ) - \sum\limits_{i = 1}^2 {s_i^{}(\lambda )} } \right\rangle } \right\}} \textrm{d}\lambda,$$

where α denotes the bandwidth parameter of the reflected light intensity.

This transformation can avoid the hypothesis of strictly convex of the original function, which ensures the robustness of the convergence process. The added penalty term can guarantee the differentiability of the Lagrangian function and improve the convergence characteristics of the algorithm. In the iterative process, Eq. (6) can be transformed into the following equivalent minimization problem

(7)$${s_i}^{n + 1}(\lambda ) = \textrm{arg min}\left( {\int {\left\{ {\alpha \times s_i^n(\lambda ) \times {{(\lambda - \lambda_i^n)}^2} + {{\left|{I(\lambda ) - \sum\limits_{j = 1,j \ne i}^2 {s_j^n(\lambda )} + {\gamma_i}^n(\lambda )} \right|}^2}} \right\}} \textrm{d}\lambda } \right),$$

where n is the number of updates.

The solution of this quadratic optimization problem is readily found by letting the ﬁrst variation vanish for the reflected light intensity

(8)$${s_i}^{n + 1}(\lambda ) = \frac{{I(\lambda ) - \sum\limits_{j = 1,j \ne i}^2 {s_j^n(\lambda )} + {\gamma _i}^n(\lambda )}}{{1 + \alpha \times {{(\lambda - \lambda _i^n)}^4}}}.$$

The center wavelengths λ_i of the corresponding mode only appear in the band width prior, which can be updated by the centroid algorithm

(9)$$\lambda _i^{n + 1} = \int {\lambda \times |s_i^{n + 1}(\lambda )|} d\lambda /\int {|s_i^{n + 1}(\lambda )|} d\lambda .$$

It should be noted that the existing NFAs can also be used to deduce the updated formula of center wavelengths, which does not change the objective function and iterative process of the proposed algorithm.

The Lagrangian multipliers γ(λ) is updated according to the following formula

(10)$$\gamma _i^{n + 1}(\lambda ) \leftarrow {\gamma _i}^n(\lambda ) + \mu \times [{I(\lambda ) - s_i^{n + 1}(\lambda )} ],$$

where μ is a noise margin parameter. For a better denoising effect, μ is generally set to 0.

The algorithm is very similar to dual ascent and the method of multipliers: it consists of an s-minimization step (8), a λ-minimization step (9), and a dual variable update (10). For more information about the conversion process, please refer to the relative literature [22]. The above process could be repeated until the convergence condition is reached

(11)$$\sum\limits_{i = 1}^2 {{{[{s_i}^{n + 1}(\lambda ) - {s_i}^n(\lambda )]}^2}/{{({s_i}^n(\lambda ))}^2}} \le \varepsilon,$$

where ɛ is the artificial terminate parameter.

4. Evaluation of the peak extraction algorithm

Based on the conventional evaluation model utilized by Liu et al. [23–26], we emphatically analyze the characteristic of the peak extraction algorithm caused by the overlapping or crossing of reflected wavelengths and compare the proposed MDA with the conventional fitting algorithms, such as GFA and SFA. During the Monte Carlo simulation process, the distribution of light source intensity is collected by spectrometer in practice. Reference to the sARS shape of experimental CCS, the reflected lights of the upper and lower surfaces are assumed to follow Gaussian distribution. Its variance can determine the FWHM of sARS and was set to 5nm, which can guarantee only when the spacing between two peaks is less than 15nm, the distribution of the reflection spectrum is independent. Meanwhile, the reflectivity of each surface is uniformly distributed between 0.5 and 1.0. We perform 20,000 trails at each given sample thickness to ensure high confidence. Figure 3 shows the decomposition results of simulation sARS obtained by MDA, which indicates the proposed method is effective. The residual error in decomposition results is almost equal to the noise of sARS, which demonstrates that MDA has the better filtering performance.

Fig. 3. Decomposition results of sARS with peak spacing of (a) 3 nm and (b) 12 nm.

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In order to facilitate subsequent analysis, there are two indicators, including the absolute peak extraction error, and the corresponding thickness error, are calculated for evaluating the performance of the proposed algorithm. The absolute error of peak extraction result is calculated by comparing the fitted result λ_i with the ideal result λ_i’, which can be written as

(12)$${e_{peak}} = |{{\lambda_i} - {{\lambda^{\prime}}_i}} |$$

As the relationship between displacement and wavelength of refractive chromatic objective is described by a linear equation, the relative error of thickness can be directly expressed as

(13)$${\delta _{thickness}} = |{({\lambda_i} - {\lambda_{i + 1}})/({{\lambda^{\prime}}_i} - {{\lambda^{\prime}}_{i + 1}}) \times 100\%} |.$$

Figure 4 shows the comparisons of peak extraction errors obtained by three algorithms at different spacing between the peaks λ₁ and λ₂, whose range is from 1nm to 15nm, while the detector noise is regarded as a Gaussian white noise with signal to noise ratio (SNR) equaling to about 5dB, and the spectrometer resolution is 1.0nm. In the proposed algorithm MDA, there are several parameters that need to be set in advance. The initial peak values λ_i obey the equidistant or random distribution in the chromatic dispersion range, the terminate parameter ɛ is normally set to 1.0e⁻¹⁸. As the key parameter to determine the width of each sub modal, the initial value of α is equivalent to 3000. It can be seen that the peak extraction errors of the three algorithms tend to be stable when the peak spacing is greater than 10nm, and the uncertainties of these three algorithms are less than 0.01nm, the corresponding expectations are 0.14nm, 0.02nm, 0.44nm, respectively. Theoretically, as the Gaussian function is regarded as a simulation model, the GFA should outperform other nonlinear fitting algorithms, such as the SFA and MDA, and the results have also come in these matches.

Fig. 4. Peak extraction performances of the three algorithms (Δλ=1.0 nm). (a) Mean value of peak extraction errors, and (b) variance of peak extraction errors.

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The corresponding histograms of peak extraction errors obtained by MDA, GFA and SFA for two different peak spacing are displayed in Figs. 5 and 6. As illustrated in Fig. 5, when the peak spacing is equal to 15nm, the peak extraction errors corresponding to the upper and lower surfaces all match basically Gaussian distribution with the same variance and mean value. But for GFA, when the peak spacing is reduced to 3nm, the errors of every surface tend to follow exponential distribution, and the peak mean error of the upper surface is approximately equal to the value of the lower surface. On the other hand, although the error distribution of the MDA and SFA algorithm has not changed, but its mean and variance have increased significantly. The main reasons for the above-mentioned changes are that the intersection of reflected light intensity between the upper and lower surfaces will affect the fitting of the distribution parameters, which causes big calculation errors in each wavelength peak. Meanwhile, the decrease of the peak distance will increase the cross area of reflected light intensity, which leads to the increase of peak extraction errors.

Fig. 5. Histograms of extraction errors for two peaks whose spacing is 15 nm. (a) MDA, (b) GFA and (c) SFA.

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Fig. 6. Histograms of extraction errors for two peaks whose spacing is 5 nm. (a) MDA, (b) GFA and (c) SFA.

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According to Eq. (13), the relative thickness errors of two peaks obtained by MDA, GFA, and SFA are shown in Fig. 7. It indicates that, with the increase of the theoretical thickness, the mean and variance of thickness errors obtained by these methods decrease gradually. In general, to ensure the thickness measurement error caused by MDA to be less than 1%, the peak spacing of the transparent plate must be greater or equal to 7nm, which is better than the others. Meanwhile, the new model has similar performance with the traditional approaches, which illustrate the thickness precision does not increase with the reduction of peak error. The major reason is that independent decomposition of sARS is not a global optimization process, which will reduce the correlation between two peaks and may decrease the accuracy of sub reflection intensity distributions.

Fig. 7. Thickness simulation results of the three algorithms. (a) Mean value of relative thickness errors, and (b) variance of relative thickness errors.

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5. Experiment and results of the thickness measurement

In the experimental CCS platform, a commercial chromatic confocal sensor head (OP2-Fc, THINKFOCUS, China) is utilized, whose maximum of measurable displacement and thickness are 400µm and 511µm, respectively. The control unit (CDS-500, THINKFOCUS, China) includes a broadband light source covering an optical spectrum of 380–780nm, and a spectrometer with a 1.0nm pixel resolution. A common multimode fiber with 50µm diameter and a matched fiber coupler are used to ensure that the wavelengths satisfying the confocal condition can converge into the public port of Y-type coupling fiber through the reflection of the object surface. Figure 8(b) shows the calibration result of displacement-wavelength relationships using polynomial fitting method, which indicates that its displacement measurement error is less than 1µm.

Fig. 8. Experimental setup for the transparent plate of thickness measurement. (a) Schematic of the experimental CCS. (b) Displacement-wavelength relationships for the chromatic dispersion model.

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To evaluate the practical performance of extraction algorithms in the case of peaks overlapping, six HCNDT polyethylene films representing different thickness ranges were measured with several repeated measurements, and their corresponding spectral intensities are shown in Fig. 9. It is obvious that the two peaks cannot be considered independent of each other when the thickness value is less than 60µm. Meanwhile, this experiment can ensure the diversity of measurement data and test the adaptiveness of the algorithm.

Fig. 9. Normalized reflected light intensity of six transparent plates obtained by CCS.

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When the standard film is directly placed on the reference plane, two peaks can be obtained to calculate the thickness value using the displacement-wavelength relationship. Meanwhile, the sample is moved by a high-accuracy piezoelectric (PZT) actuator (nPX600, nPoint, USA) to acquire different axial reflection spectrum signals within the working scope. The moving distance of the PZT was 10µm at a time. Figure 10 shows the measurement results of monolayer thin films obtained by different methods. Meanwhile, different algorithms are implemented to extract the thickness values using the same measurement data. It can be seen that the thickness will still have some significant fluctuations, which disclose that many factors, such as disperse lens, detection noise, full-width-at-half-maximum, and spectrometer resolution could affect measurement results. Since the characterization of the displacement response is rather complicated in practice, and we cannot obtain the accurate peak value of reflectance spectrum, the mean thickness of monolayer is an appropriate parameter to evaluate the performance of peak algorithm in the experiment, who can reduce the influence of other factors on the measurement results through the average value method in the axial range.

Fig. 10. Thickness measurement results of six transparent plates at different axial positions, whose theoretical values are (a) 9.7µm, (b) 23.3µm, (c) 27.4µm, (d) 37.2µm, (e) 49.5µm, (f) 59.1µm, respectively.

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The average thickness of monolayer thin films measured by different methods are provided in Table 1, whose standard values are obtained from a Heidenhain grating measuring machine with measured precision of ±0.1µm. It can be seen that the measuring errors obtained by MDA are almost always less than the GFA and SFA algorithm. The calculation error increased with the increase in the thickness of the standard sample, while the relative error decreased. When the thickness is equal to 9.7µm, the corresponding measured results were of large errors as the two sub modes are highly coincident, and the corresponding reflected light intensity can almost be regarded as containing only one peak. Meanwhile, when the thickness is more than 9.7µm, the average difference between the theoretical thickness and the average measured value are 0.3µm, 0.5µm and 0.6µm for MDA, GFA, and SFA, respectively. Compared to other fitting algorithms, the MDA has some advantages in terms of extraction accuracy with at least 40% accuracy enhancement. Besides, its performance is similar to other methods in the thickness measurement field [6,7].

Table 1. Plate thickness measurement results of three algorithms.

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Figure 11 shows the variance of measurement results of different methods with different thicknesses. The measurement error of MDA is still the smallest of the previous three methods. As simulation signal has more characteristics of sin2 model distribution and its FWHM was lower than that of actual signal, MDA has similar performance with SFA in simulation while MDA has much better performance than SFA in experiment. With the increase of the thickness distance, that is, the decrease of the overlapping area of reflected light intensity, the thickness variance of the three algorithms will gradually decrease, which is basically consistent with the simulation results in the previous section. Therefore, this procedure can be effective in most cases, and the research method is valid.

Fig. 11. Variance of thickness measurement results of the three algorithms.

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6. Conclusion

The adaptive modal decomposition method is introduced in this paper for extracting peak information from the sARS of ultra-thin samples obtained by a confocal surface topography measurement device. The simulation results show the proposed algorithm can effectively extract the peaks who are overlapped from the Gaussian spectrum signal, and its performance is better than the other two algorithms, GFA and SFA, respectively. In addition, our chromatic confocal microscope experiment has also demonstrated the effectiveness of proposed MDA and shown that the MDA can achieve approximately 40% accuracy enhancement over conventional fitting algorithms. On the other hand, it is necessary to note that the new algorithm has obvious advantage in the peak recognition, but reduces the relations among peaks, which results in no significant improvement on extraction accuracy. Therefore, the next step in this paper is to introduce the prior distribution information constraint in the objective function, or analyze the relationship between two sub reflection intensity distributions in the iterative process.

Funding

National Key Scientific Instrument and Equipment Development Projects of China (YQKFB1803-1); Special Project of Ability Improvement for Quality and Technology Supervision (ANL1822).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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True Thickness (µm)	Average Thickness (µm)			Absolute Error (µm)			Relative Error (%)
True Thickness (µm)	MDA	GFA	SFA	MDA	GFA	SFA	MDA	GFA	SFA
9.7	10.5	11.6	11.8	0.8	1.9	2.1	8.87	19.71	21.91
23.3	23.1	25.1	22.0	−0.2	1.8	−1.3	−0.72	7.56	−5.69
27.4	27.0	27.9	26.6	−0.4	0.5	−0.8	−1.47	1.60	−3.05
37.2	37.6	37.7	37.6	0.4	0.5	0.4	1.17	1.43	1.00
49.5	49.1	49.0	49.4	−0.4	−0.5	−0.1	−0.72	−0.92	−0.14
59.1	59.3	59.4	59.4	0.2	0.3	0.3	0.36	0.54	0.67

Adaptive modal decomposition based overlapping-peaks extraction for thickness measurement in chromatic confocal microscopy

Abstract

1. Introduction

2. Peak overlapping problem in ultra-thin film thickness measurement

3. Peak extraction algorithm based on modal iterative decomposition

4. Evaluation of the peak extraction algorithm

5. Experiment and results of the thickness measurement

6. Conclusion

Funding

Disclosures

References

Cited By

Figures (11)

Tables (1)

Equations (13)

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