Phase noise estimation based white light scanning interferometry for high-accuracy surface profiling

Long Ma; Yuan Zhao; Xin Pei; Yu-zhe Liu; Feng-ming Sun; Sen Wu

doi:10.1364/OE.451746

1. Introduction

The surface metrology in micro-nano scale has been increasing in demand in the past decades and follows rapid developments in applications of various science research and industrial fabrications [1]. White light scanning interferometry (WLSI), as a powerful technique for surface profilometry, is extensively applied in various measurements, such as real-time phase measurements [2], dispersion measurements [3], film thickness measurements [4], etc. WLSI uses low-coherence white light source as illuminator and recovers the surface by detecting the position of zero optical path difference (ZOPD) from the recorded correlogram [5], which enables WLSI to achieve theoretical unlimited measurement range with interferometric resolution [6].

In most cases, WLSI works in stable and vibration-isolation conditions. Unfortunately, during the scan, scanner non-linearity, optical system imperfections and unpredictable disturbances are still inevitable and will distort the correlogram by increasing the noise level in both the envelope and the phase map [7,8]. For several classical correlogram demodulation algorithms such as frequency domain analysis (FDA) method [9], Fourier method [10] and Hilbert method [11], this will bring extra errors to the measurement and therefore significantly decrease the measurement accuracy and robustness, especially for wafer level measurements.

In recent years, in order to overcome the above problem, many efforts have been dedicated to increase the accuracy of WLSI under the corruption of noises. Some achievements have been made through modeling the scan step error, where a series of iterative estimation must be implemented to extract the actual scan step length before the correlogram being processed [12–14]. These methods could be applied on most cases of scan non-linearity, however, it is only effective when the scanner noise is predominant compared with other error sources. Unfortunately, this condition is not typical in practice, especially when high precision PZT, or Motor scanners were employed. There are also some reported works to adopt artificial intelligence (AI), for instance, convolutional neutral network, or recurrent neutral network, to compensate the distortion of the correlogram as a whole, without identifying the type of error sources [15–17]. However, due to special requirements on the size of training data set and the training process itself, the AI-assist methods are only applicable for off line measurements. In addition, the reconfigurations of the AI logic are still quite inefficient once the measurement condition changed. Another solution for the problem is to combine phase distribution into the process of recovering the surface, given the fact that phase map is insensitive to intensity noises compared with that of the correlogram envelope. Such algorithms including noise-correction based Hilbert method [18], enhanced-steps phase shifting algorithm [19] and windowed-Fourier-filtering based envelope extraction method [20], according to the given results, these algorithms can successfully improve the measurement accuracy through the analysis of phase information. But if we investigate the phase distribution of correlogram in detail, the apparent non-linearity will still be found. This indicates that when introducing the phase extraction, it is necessary to perform some kind of denoising in the first step in order to ensure the phase accuracy. Otherwise, the same problem will still remain by nature, only except for happening on the phase map instead of on the envelope. However, at present, there are still not enough efforts dedicated to the research of phase noise and to perform phase noise elimination effectively.

In our approach, a novel signal processing algorithm is proposed for white light scanning interferometry to improve the measurement accuracy by removing the noise existed in phase map. Based on the assumption of Gaussian distribution of the noises in spectrum domain, the noise on phase map is firstly modeled and the corresponding statistics features are analyzed. The phase noise distribution is then estimated in discrete time, and a coefficient is defined to evaluate the consistency of the estimated phase noise map with the actual one by applying least square estimation (LSE). The optimal phase noise map is acquired when the coefficient reaches minimum. By removing phase noise map, the surface can be recovered more accurately. To further investigate the performance of the new method, a nano-scale step height standard with calibration value 9.5nm±1.0nm is tested, the measurement demonstrates that the measurement repeatability of the proposed method is only 1.9%. And the shape of the leading edge of an aero-engine blade is also measured to verify the effectiveness of the algorithm for industrial applications. Some measurement comparisons between the new method and the AFM are also provided.

2. Methodology

2.1 Statistics analysis of phase noise

Figure 1 illustrates the principle of phase noise analysis. According to the properties of broad spectrum, the correlogram of WLSI could be considered as Gaussian envelope modulated on sinusoidal carrier, as shown in Eq. (1),

(1)$$I = {I_0} + g(z - h)\cos (\frac{{4\pi }}{{{\lambda _0}}}(z - h) + \varphi )$$

where I is the typical correlogram in theory, I₀ denotes the background illumination, g is the envelope which is generally Gaussian distributed, λ₀ is known as the central wavelength of the utilized light source, z is the scan position, h is the surface height and φ is the additional phase which keeps constant during measurement. Then in spectrum domain, the positive part of the correlogram could be derived after Fourier transform, which is marked by the red frame in Fig. 1(a), as illustrated by Eq. (2) in discrete time,

(2)$$F{[I]_{\rm {po}}}({\omega _n}) = \frac{1}{2}G({\omega _n} - \frac{{4\pi }}{{{\lambda _0}}}){e^{ - j({\omega _n}h - \varphi )}}$$

where F[·]_po denotes the positive part of correlogram after Fourier transform, G presents the Fourier transform of envelope g, ω_n is the angular wavenumber of n-th harmonic in the selected positive part of spectrum, e is the natural logarithm and j denotes the imaginary unit. From Eq. (2), the relationship between phase and angular wavenumber could be retrieved as Eq. (3),

(3)$$\left\{ {\begin{array}{l} {\Phi ({\omega_n}) = \varphi - {\omega_n}h}\\ { - [{{\omega_1},\ldots ,{\omega_N}} ]h = [{\Phi ({\omega_1}) - \varphi ,\ldots ,\Phi ({\omega_N}) - \varphi } ]} \end{array}} \right.$$

where Φ(ω_n) presents the phase without noise at ω_n, N is the total index of angular wavenumber inside the selected positive part. Here if we define A = [ω₁, …, ω_N], and in the same way B = [Φ(ω₁) –φ, …, Φ(ω_N) –φ] as the phase map after removing the initial phase, Eq. (2) could then be simplified into –Ah = B, from which the theoretical linear relationship between A and B is obvious. Nevertheless, in practice, due to the unpredictable disturbances, certain deviations will be directly transmitted into phase distribution and bring additional noise to the recovered surface, once the phase analysis algorithms like FDA method are directly applied without some kind of phase denoising. The noisy correlogram after the Fourier transform could be formulated as Eq. (4),

(4)$$F[{I_{\rm a} }]({\omega _n}) = F[I]({\omega _n}) + ({\alpha _n} + j{\beta _n})$$

where F[·] denotes the Fourier transform of correlogram, I_a is the recorded noisy correlogram. α_n+jβ_n indicates the complex noise term in spectrum domain, where α_n and β_n could be both assumed Gaussian distributed with zero mean. As shown in Fig. 1(b), the real part and imaginary part of the phase at ω_n are actually affected by α_n and β_n, respectively. In general cases, as a result that α_n and β_n are relatively lower than the spectral amplitude |F[I_a](ω_n)| at each n, the actual phase map could be retrieved approximately as the sum of the first two terms after Taylor expansion, as given by Eq. (5),

(5)$${\Phi _{\rm a}}({\omega _n}) = \arg (F[{I_{\rm a} }]({\omega _\textrm{n}})) \approx \Phi ({\omega _n}) - \frac{{{\mathop{\rm Im}\nolimits} \{{F[{I_{\rm a} }]({\omega_n})} \}{\alpha _n}}}{{{{|{F[{I_{\rm a} }]({\omega_n})} |}^2}}} + \frac{{Re \{{F[{I_{\rm a} }]({\omega_n})} \}{\beta _n}}}{{{{|{F[{I_{\rm a} }]({\omega_n})} |}^2}}}$$

where Φ_a is the actual phase distribution, |·| indicates the modulus operation, Re{·} and Im{·} denote the real part and imaginary part of one complex value, respectively. Next, separate the noise term from Eq. (5) and define E(n) to present its distribution in actual case at ω_n, written as Eq. (6),

(6)$$E(n) = \frac{1}{{|{F[{I_{\rm a} }]({\omega_n})} |}}\left( { - \frac{{{\mathop{\rm Im}\nolimits} \{{F[{I_{\rm a} }]({\omega_n})} \}}}{{|{F[{I_{\rm a} }]({\omega_n})} |}}{\alpha_n} + \frac{{Re \{{F[{I_{\rm a} }]({\omega_n})} \}}}{{|{F[{I_{\rm a} }]({\omega_n})} |}}{\beta_n}} \right) = \frac{{{\varepsilon _n}}}{{|{F[{I_{\rm a} }]({\omega_n})} |}}$$

where ɛ_n is defined to describe the linear composition of α_n and β_n, and could be also regarded as Gaussian distribution with zero mean, ɛ_n/|F[I_a](ω_n)| denotes the corresponding equivalent phase noise term at ω_n. The distribution of phase noise is converted as E = [E(1), …, E(N)] in the following analysis. Here we should note that Eq. (6) gives the statistics features of the complex noise term in more simplified form, and makes the phase noise possible to be estimated in the following steps.

Fig. 1. Critical procedures of the proposed method. (a) Statistics analysis of phase noise. (b) Phase affected by the noise in spectrum domain. (c) Estimation of phase noise map.

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2.2 Estimation of phase noise map

Define B_a as the actual phase map which is corrupted by the noise in Eq. (6) with initial phase removed, the relationship between B_a and B could be given by Eq. (7),

(7)$${B_{\rm a} } = B + E$$

According to Eq. (3), after removing the phase noise E from B_a, the surface could be consequently recovered more accurately in theory, as given by Eq. (8),

(8)$$h = - B\frac{{{Y^T}}}{{{{(A{Y^T})}^T}}} ={-} {B_{\rm a} }({I_\textrm{N}} - {B_{\rm a} }^IE)\frac{{{Y^T}}}{{{{(A{Y^T})}^T}}}$$

where Y is the matrix defined as [1, …, 1]_N, (·)^T is the transpose operation, (·)^I presents pseudo inverse and I_N is the unit diagonal matrix with dimension N. Here it should be emphasized that although E might not be acquired accurately in practice, Eq. (8) can still be solved through the estimation of E, defined as E_e, as expressed in Eq. (9),

(9)$$\left\{ {\begin{array}{c} {{C^T} = ({I_N} - {B_{\rm a} }^I{E_e})\frac{{{Y^T}}}{{{{(A{Y^T})}^T}}}}\\ {H ={-} {B_{\rm a}}{C^T}} \end{array}} \right.$$

where C is the equivalent distribution directly decided by E_e, H is the corresponding surface height after eliminating E_e. In order to reconstruct H more accurately, E_e must be the optimal estimation of E, in other words, E_e must keep good consistence with the actual case.

Regroup Eq. (9) according to Eq. (3), B can be estimated by Eq. (10),

(10)$${B_{\rm e}} ={-} AH = A{B_{\rm a}}{C^T}$$

where B_e is the estimation of B. As illustrated by Fig. 1(c), E_e can then be expressed by the difference between B_a and B_e, given by Eq. (11).

(11)$${E_e} = {B_{\rm a}} - {B_{\rm e}} = {B_{\rm a}}({I_N} - {C^T}A)$$

Define coefficient D to present the difference between E and E_e through the least square estimation (LSE), as written by Eq. (12),

(12)$$D = \frac{1}{N}{||{E - {E_{\rm e}}} ||_2} = \frac{1}{N}{\left( {\sum\limits_{i = 1}^N {{{\left|{E(i) - \sum\limits_{k = 1}^N {{B_{\rm a}}(k)[{{I_N}(k,i) - C(k)A(i)} ]} } \right|}^2}} } \right)^{{\textstyle{1 \over 2}}}}$$

where ||·||₂ indicates 2-norm operation. The optimal E_e could be determined when D reaches minimum, which means the best statistical consistency between E_e and E is achieved. According to Eq. (12), C could be regarded as the only factor which directly decides D, therefore, it could be quite practicable to perform estimation with C being changed, as described by the left top part in Fig. 1(c). And the minimized D could be located as shown in the bottom plots of Fig. 1(c).

By employing unconstraint optimization to locate the minimum D, the optimized C is obtained by Eq. (13),

(13)$$C = \left[ { - \frac{{{{|{F[{I_{\rm a} }]({\omega_1})} |}^2}\omega_1^2}}{{\sum\limits_{\textrm{i} = 1}^N {{{|{F[{I_{\rm a} }]({\omega_i})} |}^2}\omega_i^2} }}, \ldots , - \frac{{{{|{F[{I_{\rm a} }]({\omega_N})} |}^2}\omega_N^2}}{{\sum\limits_{\textrm{i} = 1}^N {{{|{F[{I_{\rm a} }]({\omega_i})} |}^2}\omega_i^2} }}} \right]$$

It is noticed that C could be regarded as a set of weight coefficients, where the weight ratio is decided by the spectral amplitude. H is then given by Eq. (14).

(14)$$H ={-} {B_{\rm a}}{C^T} ={-} \frac{{\sum\limits_{\textrm{i} = 1}^N {{{|{F[{I_{\rm a} }]({\omega_i})} |}^2}{\omega _i}({\Phi ({\omega_i}) - \varphi } )} }}{{\sum\limits_{\textrm{i} = 1}^N {{{|{F[{I_{\rm a} }]({\omega_i})} |}^2}\omega _i^2} }}$$

According to Eq. (14), through the weighted averaging operation of C on the actual phase map B_a, the correction of phase noise is then performed. Compared with h directly derived from Eq. (3), it is obvious that Eq. (14) successfully removes the noise existed in the phase map, and will improve the measurement accuracy and robustness.

3. Simulations

To solidify our analytical results, the proposed method is tested under different conditions of phase noises within the framework of simulation. Five cases of phase noise are selected and statistically generated with standard deviations of 0.1rad, 0.2rad, 0.3rad, 0.4rad and 0.5rad. The capability of the proposed method in phase noise elimination is analyzed by detecting the ZOPD position of the correlograms, where 1000 times of simulations are performed for each case of phase noise to provide more convincing results. Then the proposed method is investigated by comparing with the least square (LS) method [21] which is comprehensively applied for surface height extraction in white light interferometry through least square fitting of phase map in spectrum domain.

Figure 2 presents the corresponding simulation results. Figure 2(a) is the simulated correlogram without noise, while Fig. 2(b) denotes the correlogram in case of the standard deviation of phase noise 0.5 rad. Compared with Fig. 2(a), it is noticed that the detailed shape of correlogram is quite sensitive to phase noise. Figure 2(c) shows the comparisons of phase map retrieved from Fig. 2(a) and 2(b), respectively, where the non linearity of noisy phase map are clearly observed. By calculating the root mean square error (RMSE) of the detected ZOPD position in each case, the simulation comparisons between the proposed method and the LS method are derived as Fig. 2(d). When the variance of phase noise is 0.1rad, the RMSE of the proposed method and the LS method are both less than 5nm. However, it is noticeable that the RMSE of the proposed method is significantly lower than that of LS method whose RMSE will keep increasing with the noise level. Even at the noise level of 0.5 rad, the RMSE of the proposed method is only 2.8 nm, while that of LS method is almost 25nm. The improvement of the new method compared with the LS method on phase noise elimination is then verified.

Fig. 2. Simulations of the proposed method. (a) Correlogram without phase noise. (b) Correlogram affected by the phase noise with standard deviation of 0.5 rad. (c) Comparisons of phase map of the correlograms in Fig. 2(a) and 2(b). (d) Comparisons of RMSE of ZOPD position.

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4. System setups

Figure 3 shows the white light profiler developed by us and the utilized atomic force microscope (AFM). Figure 3(a) displays the optical configuration and corresponding photograph of the white light profiler in this work. A halogen lamp (OSL2, Thorlabs Inc.) is chosen as illuminator. A beam splitter is used to divide the lights emitted from the illuminator, and a CCD camera serves as a detector of interferogram. As a linear scanner, the piezoelectric ceramic transducer (PZT) from PI Co. can drive the system vertically along optical axis with the travel range of 100µm. A 10× Mirau objective lens is employed, aiming at increasing the compactness of system. Figure 3(b) gives the photograph of the AFM (ICON) from Bruker Co. used to provide comparisons.

Fig. 3. System setups of our developed white light profiler and photograph of the AFM from Bruker. (a) Schematic and photograph of the proposed white light profiler. (b) Photograph of the AFM from Bruker.

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5. Experiments

5.1 Nanostructure measurements

In order to testify the capability of the proposed method in removing the phase noise, a nano-scale step height standard of VLSI Inc. with certificated value 9.5 ± 1.0 nm (Calibration provided by US National Institute of Standards and Technology) is taken as the specimen. Figure 4 demonstrates the effect of phase noise elimination. Figure 4(a) shows the correlogram retrieved from the selected region of the step, referred by the red frame in the snapshot with interference fringe pattern. The ZOPD position is precisely located at 3425.3 nm. Figure 4(b) and 4(c) present the actual phase map and the phase map after noise elimination, respectively. The regress coefficient r² in Fig. 4(b) after linear fitting is 0.843, whereas that of Fig. 4(c) is 0.997, which exhibits obvious improvement in linearity. Figure 4(d) shows the estimated phase noise originates from phase map by the proposed method, approximately ranging from -0.02rad to 0.04rad.

Fig. 4. Analysis of phase noise elimination of the captured correlogram. (a) Correlogram captured by the proposed profiler. (b) Actual phase map retrieved from Fig. 4(a). (c) Phase map after eliminating the estimated phase noise. (d) Estimated phase noise map.

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Figure 5 displays the measurements for the step. Figure 5(a) to 5(d) are the results using the proposed profiler. The recovered 3D structure after tilt removing is given in Fig. 5(a), where the cross section along Y=8µm and Y=30µm referring to the red lines are displayed in Fig. 5(b), and the one along X=8µm marked by green dash line is given by Fig. 5(c). To retrieve the height of step explicitly, ISO-28178/70 is applied, where three parts of step profile marked in Fig. 5(c) are used to evaluate the step morphology. Moving the profile 3 from X=0.16µm to X=63µm along green arrow with the interval of 0.16µm in Fig. 5(a), the step height variation is plotted in Fig. 5(d), where the heights of all the 400 retrieved profiles lay within the range from 8.5nm to 10.5nm. Selecting the profile at X=30µm, the step heights after ten repeat measurements are statistically listed in Table 1. It is obvious that all the repeat measurements have good consistency and agreement with the certificated uncertainty. The repeatability of the proposed method is also verified by evaluating the measurement standard deviations, which is only 1.9% compared with the nominal height.

Fig. 5. Measurement results of step height standard. (a) Recovered 3D structure of step using the proposed profiler. (b) Retrieved cross section at pixel Y=8µm and Y=30µm. (c) Retrieved cross section at pixel X=8µm. (d) Step height variation along green arrow. (e) Retrieved cross section using the AFM from Bruker with the deviations compared with profile 3.

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Table 1. Results and repeatability of ten measurements

View Table

In addition, the marked region of step in Fig. 4(a) is measured by AFM from Bruker to provide measurement comparisons. Figure 5(e) gives one reconstructed cross section retrieved by AFM. The height of the step in Fig. 5(e) is calculated as 9.36 nm, which is in range of the certificated value and close to the result using the proposed profiler. Moreover, selecting the bottom region from Y=0µm to Y=20µm, the profile deviations between profile 3 and profile 4 are described in the bottom part of Fig. 5(e). It is obvious that all the deviations lay within ±0.8 nm, which means the noise level of step measured by the proposed profiler is very close to that of AFM.

Furthermore, the step height standard is also tested by LS method to provide comparisons with the proposed method in this work. Figure 6(a) presents the recovered 3D structure using LS method. Compared with Fig. 5(a), more noise occurs and seriously affected the shape of reconstruction due to the fact that no steps have been taken to eliminate phase noise. Figure 6(b) gives the comparisons of cross section along X=8µm. From Fig. 6(b), it is obvious the morphology of step from LS method cannot be recovered as accurately as that of the proposed method. Selecting the range form Y=0µm to Y=20µm, the corresponding profile deviations are displayed in Fig. 6(c). Here it is easy to find that the maximum deviation is approximately 4 nm, which is almost half of the step height. This will definitely bring extra errors to the measurement and make LS method not applicable in nano-scale inspections.

Fig. 6. Measurement results of step using LS method. (a) Recovered 3D structure of step. (b) Profile comparisons along X=8µm. (c) Profile deviations from Y=0µm to Y=20µm.

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5.2 Metal surface measurements

The leading edge of a compressor blade from civil aero-engine is selected to verify the effectiveness of the proposed method in industrial inspection. The leading edge, as a very important part of blade, has kept suffering from the high pressure air flow and will gradually lead to aerodynamic shape degradation. Hence, the measurement of leading edge is of great importance in evaluating the blade aerodynamic characteristics.

Figure 7 illustrates the result of leading edge measurement. Figure 7(a) gives the compressor blade to be tested, where the protective painting is partially dissolved, as shown in the zoomed view of the red frame. Figure 7(b) displays the fringe pattern recorded by CCD camera of the area marked by blue frame, which is quite subtle due to the shape complicity. Figure 7(c) shows the corresponding correlogram. It could be obviously found that the signal is severely distorted by the scattering happened on the leading edge, and leads to the coherence peak position more difficult to be located. The left top dotted frame in Fig. 7(c) presents the phase comparison before and after denoising, where the non-linearity of the actual phase map can be clearly found. The estimated phase noise map is shown in the right top of Fig. 7(c), resulting from the very noisy correlogram, the phase non-linearity exceeds ±5rad. This is quite challenging for most white light interferometric algorithms, but it is still the common case in industrial or in-situ measurements. However, by applying the proposed method, the leading edge is successfully recovered, where the obvious curved morphology and degraded area can be clearly observed, as shown in Fig. 7(d) and Fig. 7(e). The profile comparisons from four repeat scans referring to the red dashed line at X=10µm in Fig. 7(e) are given in Fig. 7(f) as well, where the good profile consistency could be easily observed. The corresponding Roughness (Ra) values after removing the leading edge profile in Fig. 7(f) are calculated as 1322.63nm, 1324.61nm, 1329.57nm and 1325.38nm, respectively. The great potential of the proposed method in industrial application is therefore verified.

Fig. 7. Measurement results of blade leading edge. (a) Tested aero-engine blade. (b) Captured interferogram by the proposed profiler. (c) Analysis of phase noise elimination of the captured correlogram. (d) Recovered 3D structure using the proposed method. (e) Height axial view of the recovered leading edge. (f) Cross section comparisons along X=10µm from 4 measurements.

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6. Conclusion

In this work, we proposed a novel white light signal processing algorithm which can improve the measurement accuracy by attenuating the phase noise. The noise existed in phase map is first investigated in spectrum domain and modeled mathematically based on its statistics features. By estimating the phase noise distribution using LSE, the relationship between surface height and phase map is redefined more accurately. After deploying the proposed method on the new designed white light profiler, a step height standard is tested, where the shape of step is successfully recovered with the repeatability of 1.9%. The leading edge of an aero-engine blade is further scanned with high measurement consistency, and the utility of the proposed method in industrial application is verified. Our work reveals that the phase noise is of great importance in improving the measurement capability of WLSI, and provides a novel insight to solve the problem for phase noise estimation and correction.

Funding

National Key R&D Program of China (2021YFF0600903); Aeronautical Science Foundation of China (20200056067001).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Index	Step height/nm	Index	Step height/nm	Mean Height	Repeatability
1	9.34	6	9.41
2	9.72	7	9.51
3	9.64	8	9.46	9.49nm	0.17nm
4	9.54	9	9.53		1.9%
5	9.23	10	9.54

Phase noise estimation based white light scanning interferometry for high-accuracy surface profiling

Abstract

1. Introduction

2. Methodology

2.1 Statistics analysis of phase noise

2.2 Estimation of phase noise map

3. Simulations

4. System setups

5. Experiments

5.1 Nanostructure measurements

5.2 Metal surface measurements

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (1)

Equations (14)

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