Neural Network Approach for Modelling and Compensation of Local Surface-Tilting-Dependent Topography Measurement Errors in Coherence Scanning Interferometry

Abstract

The topography measurement accuracy of coherence scanning interferometry (CSI) suffers from the local characteristic of micro-structured surfaces, such as local surface slopes. A cylindrical reference artefact made of single-mode fiber with high roundness and low roughness has been proposed in this manuscript to traceably investigate the surface tilting induced measurement deviations using coherence scanning interferometry with high NA objectives. A feed-forward neural network (FF-NN) is designed and trained to model and thereafter compensate the systematic measurement deviations due to local surface tilting. Experimental results have verified that the FF-NN approach can well enhance the accuracy of the CSI for radius measurement of cylindrical samples up to 0.3%. Further development of the FF-NN for modelling of the measurement errors in CSI due to the optical properties of surfaces including areal roughness is outlined.

1. Introduction

Coherence scanning interferometry (CSI) [1,2,3] has long been utilized for surface topography measurements, owing to its outstanding features such as non-contact and fast areal measurements [4]. Nowadays, commercially available CSI instruments are capable of topography measurements of plane smooth surfaces with down to sub-nm sensitivity [5]. By means of the utilization of high NA objectives [6,7] and/or advanced imaging methods, such as the virtual annual aperture method [8] and micro-sphere assistance [9], CSI will further achieve a lateral resolution better than 200 nm.

Consequently, in recent years CSI instruments have found tremendous applications in various scientific and industrial areas for the geometrical characterization of samples with lateral dimensions down to sub-µm, and have become powerful tools used in, e.g., quality control of advanced manufactured micro-products [10,11,12], which typically have the feature size down to 10 microns [13], therefore demanding reliable and high-throughput nano-dimensional measurement methodologies.

In the meantime, it has already been noticed that CSI instruments usually demonstrate evident measurement errors in the case of the measurement of complex surfaces with relative higher varying local slopes and curvatures. And the measurement accuracy of CSI instruments for the topography measurement of three-dimensional microstructures suffers not only from instrumentation issues, e.g., the aberrations of microscope objective and misalignment [14,15], but also from the local surface features of the samples under measurement, e.g., local curvature and surface tilting/gradient [16,17,18,19]. In the case of surface topography measurement with high NA objectives, i.e., the lateral resolution of interference microscopes is far less than the local curvature of microstructures, topography measurement error caused by local surface tilting becomes one of the major error sources in interference microscopy. A process to calibrate and partially correct the slope-dependent errors has been proposed in [14,17], in which the 3D surface transfer function of a CSI system is obtained by measuring micro-spheres and it is demonstrated that CSI is capable of measuring the topography of surfaces with varying tilt with sub-nanometer accuracy. However, the 3D surface transfer function needs to be characterized in the full field of view and therefore it is very time consuming. In [20], a simulation-driven machine learning approach based on a deep neural network (DNN) has been developed for correction of slope-dependent errors in CSI based on a previously developed virtual CSI method. With this method, the DNN learns the errors’ characteristics directly from simulated surface measurements provided by virtual CSI and the trained DNN can then map the topographies with errors to the corrected topographies.

It has been observed that the emergence of artificial neural network (NN) machine learning may provide solutions to enhance the topography measurement accuracy of CSI instruments for three-dimensional microstructures by training the surface-tilting-dependent topography measurement errors; hence, in this manuscript, an NN approach has been proposed to compensate for the local surface-tilting-dependent topography measurement errors in CSI microscopy. In Section 2, a well-calibrated micro-artefact made of single-mode optical fiber is used to directly characterize the measurement errors of a commercial CSI due to surface tilting. In Section 3, an NN model is trained to model and therefore compensate for the surface-tilting-dependent topography measurement errors. The experimental results detailed in Section 4 have validated the effectiveness of the trained NN model, indicating that the measurement deviation of our CSI for measuring the radius of micro-objects has been reduced to less than ~0.3%.

2. Method for Direct Evaluation of the Surface-Tilting-Dependent Measurement Error in CSI

In the case of interference measurement of the topography of surface under testing (SUT) with local tilting, as illustrated in Figure 1, the reference beam within the interference microscope can fully illuminate the exit aperture of the microscope objective (in Figure 1 a Mirau objective is employed), and the measurement beam coming from the SUT will sweep the exit aperture, due to the surface tilting γ_s. Strong measurement errors will be yielded, when the local surface tilting becomes close to the maximum illumination angle θ_NA determined by the numerical aperture of the objective in use [17].

It is worthwhile to mention that for an SUT with nearly perfect surface properties, e.g., mirror-like surfaces of micro-optics, a ray tracing method or analytical model can be utilized to estimate the tilting-angle-dependent surface measurement error [17,19]. For relative rough SUT, the influence of the stray light on the measurement error also needs to be considered, since the microscope objective will collect back-scattered light coming from the surface with γ_s even larger than its theoretical numerical aperture [21,22]. Unfortunately, however, numerical modelling for the latter case is usually highly challenging.

As a result, the experimental determination of the surface-tilting-dependent topography measurement error in a CSI microscope has gained much interest [17,19,22]. And the typical reference artefacts in use are precision microspheres with diameters ranging from fifty microns to a few hundred microns [19,20,21,22,23,24].

2.1. Micro-Rod-like Reference Artefacts

To directly evaluate the γ_s-dependent measurement error of a scanning white light interference microscope, a new type of reference artefact made of single-mode (SM) fibers with a nominal cladding diameter of Ø125 µm is proposed in this manuscript. These commercially available SM fibers feature low cost, relatively low surface roughness, high roundness and tight diameter tolerance, especially suitable for traceable topography measurement using atomic force microscopes and tactile stylus profilometers, because of easy alignment and data analysis.

As illustrated in Figure 2, to evaluate the measurement error of an interference microscope, the top surface of the cylindrical artefact with an opening angle γ_s → θ_NA needs to be measured. In addition, as shown in Figure 2b, slight topography measurement errors can yield relatively large errors for cylinder radius measurement.

2.2. Traceable Topography Measurement of Reference Artefact Using MEMS-SPM

Reliable and traceable measurement of the topography of the proposed cylindrical reference artefact will lay a solid foundation for the evaluation and modelling of the measurement errors of an interference microscope. A self-developed micro-topography measurement platform using a microelectromechanical system (MEMS)-based scanning probe microscopic (SPM) head (MEMS-SPM) [25] for topography measurement has therefore been used.

As illustrated in Figure 3a, the MEMS-SPM head [25] consists of a set of micromachined electrostatic comb-drives [26] for capacitive displacement measurement and an integrated AFM probe gripper which can hold various AFM probes for topography measurement and also for nanomechanical measurement. The typical stiffness of the MEMS-SPM head k_MEMS used for topography measurement amounts to about 10 N/m.

A self-developed capacitive displacement measurement system [27,28] has been used to measure the displacement of the MEMS-SPM head with sub-nanometer resolution for a bandwidth of 300 Hz, yielding a force resolution of ~1 nN.

A three-axis piezo-stage (P-733, Physik Instrumente, Karlsruhe, Germany) with an x-y scanning range of 100 µm and a z-axis scanning range of 10 µm with a sub-nanometer resolution has been used to perform surface scanning of the sample fixed on the stage. After calibration, the nonlinearity of the x-y-z axes of the piezo-stage becomes typically less than 0.3‰. To further improve the performance of the scanning system, a laser interferometer has been used to in situ traceably measure the z-axis displacement of the piezo-stage.

A digital PID controller has been used to drive the z-axis of the piezo-stage for closed-loop topography measurement with a nanometer noise floor for a typical bandwidth of 40 Hz.

In the case of the topography measurement of strongly curved micro-structures, e.g., microcylinders, as shown in Figure 3b, the measurement capability of the MEMS-SPM system is mainly determined by the specifications of the AFM probe in use. The maximum measurable surface tilting γ_smax is limited by the opening angle of the AFM probe tip, and the measured local curvature R_m = R_tip + R_s, where R_s is the real local curvature, and R_tip the equivalent tip radius of the rounded AFM tip.

Typical high-NA Mirau objectives used in CSIs for topography measurement usually have the nominal numerical aperture of 0.55, corresponding to γ_max = 33.4°. Therefore, in this manuscript, diamond coated AFM probes (Adama Innovation) with a semi-opening angle θ_tip = 45° and a tip radius of 25 ± 15 nm will be used for reference measurement, which can provide adequate lateral and vertical measurement data for evaluation of the measurement errors of CSIs.

2.3. Artificial Neural Network Approach for Modelling of Surface-Tilting-Induced Topography Measurement Errors in an Interference Microscope

Preliminary analysis and experimental results in [17,19,21,23] have indicated that surface-tilting-induced measurement error ∆z is strongly nonlinear with respect to the surface local tilting γ_s.

Artificial NN [29] has long been proposed for the numerical analysis of nonlinear models and has become a powerful tool especially for the approximation and fitting of nonlinear functions [30] and image classification [31]. In this manuscript, typical feedforward neural networks [32,33] will be utilized to quantitatively model the relationship between γ_s and ∆z, once the measurement error ∆z can be traceably determined.

Figure 4 depicts the configuration of the feedforward NN model used in this manuscript. It has one hidden layer and one linear output neuron, where log sigmoidal neurons are employed in the hidden layer (1st HL) of the NN, LW¹ and LW² are layer weight matrices, b¹ and b² are corresponding bias vectors. p = {γ_s} and t = {$\Delta \widehat{z}$} are the input and output of the NN model, respectively.

To train the NN model, reliable measured surface tilting γ_s and the corresponding topography error ∆z_m will be fed input the model, a typical Levenberg–Marquardt (LM) algorithm [34,35] can then be used to determine the parameter matrices LW¹, LW², b¹ and b². As usual, 75% of the randomly chosen measurement data will be used for training the NN model, and 25% of the data will be used for validation of the trained NN. Of course, in this manuscript, a few reference artefacts will be prepared and measured by both MEMS-SPM and an interference microscope, the corresponding measurement data including surface tilting and topography error of multi-artefacts can also be used for validation of the trained NN model.

3. Results

3.1. Top Surface Topography of Reference Cylindrical Artefacts Measured by MEMS-SPM

Under the condition of a contact force of 60 nN (i.e., MEMS-SPM setpoint z₀ = 5 nm), a reference cylinder made of a piece of SM-fiber (10 mm long) with a nominal cladding diameter of 125 µm has been measured by the measurement setup detailed in Section 2.2.

Figure 5a shows one of the typical surface topography images of the reference cylinder; the cross-sectional profile at y = 51 µm has been illustrated in Figure 5b. A least-square circle fitting [36] of the measurement dataset {x, z} has been used to determine the radius of the cylinder’s profile, and the measured R_m is found to be 62.77 ± 0.05 µm. After the removal of the circular shape of the profile, the residual topography error δz is obtained as follows:

$$\delta z=z-\left[{\widehat{z}}_0+\sqrt{{\widehat{R}}^2-{\left(x-{\widehat{x}}_0\right)}^2}\right],$$

(1)

where $\widehat{R}$ is the fitted cylinder radius, and ${\widehat{x}}_0$ and ${\widehat{z}}_0$ the coordinates of the fitted cylinder center in x- and z-direction in Figure 5. For comparison, the residual error is also illustrated in Figure 5b.

It can be seen from Figure 5b that the top surface of the reference artefact demonstrates excellent circular shape, and its topography error over the whole x-scan range of 60 µm amounts to less than ±30 nm.

3.2. Top Surface Topography of Reference Cylindrical Artefacts Measured by a CSI Using a Mirau Objective with NA = 0.55

A commercial CSI has then been used to measure the top surface of the same artefact, in which a Mirau objective with a nominal objective NA = 0.55 is employed. Figure 6a shows one of the typical topography images obtained by the objective. The line profile of the cylinder measured by the CSI microscope at y = 160 µm is illustrated in Figure 6b. It can be seen that the effective lateral measurement range of this objective is limited to about ±22 µm, which is clearly smaller than the theoretical measurement range along x-axis determined by its nominal NA, i.e., ±34.4 µm, which might be caused by the hardware configuration or software filters.

Similar to the data postprocessing in Section 3.1, the measured radius of the artefact by CSI within its effective measurement range along the x-axis is found to be R_{m_CSI} = 64.13 ± 0.36 µm. And the corresponding residual roundness error determined by the interference microscope is also illustrated in Figure 6b.

3.3. Modelling and Compensation of the Surface Tilting Induced Measurement Errors in Interference Microscopy

To evaluate the surface tilting induced measurement error of this interference microscope objective, the line profiles shown in Figure 5b and Figure 6b have been numerically correlated. And the local surface tilting is evaluated as follows:

$${\gamma }_{\mathrm{s}}=\mathrm{asin}\left(\left(x-{\widehat{x}}_0\right)/\widehat{R}\right).$$

(2)

Using the profile measured by the MEMS-SPM as a reference, the line profile deviation by the interference microscope ∆z_m (=z_CSI − z_MEMS) with respect to the surface tilting within the effective measurement range is depicted in Figure 7a. It can be seen that the actual measurement errors of the CSI show strong high-order nonlinearity.

As described in Section 2.3, 75% of the data pairs {γ_i, Δz_i} are randomly selected from the measurement data of Fiber 1# shown in Figure 7a to create the training dataset for the artificial NN model. The remaining data pairs in the measurement data are later used as a validation dataset [37]. The training and validation datasets are therefore independent, but all cover the entire tilting angle range under investigation.

The artificial NN is first trained with the training dataset. The average squared difference between the measurement results and the values predicted by the NN, i.e., mean squared error MSE_Train is then analyzed. To validate the trained NN, the input data (i.e., surface tilting γ) of the validation dataset will be fed into the NN, the predicted output of the NN is then to be compared with the output data of the validation dataset, and the mean squared error over the validation dataset (i.e., MSE_Valid) is evaluated. The NN model is considered acceptable if both MSE_Train and MSE_Valid are low and comparable.

It is already known that an FF-NN with adequate hidden layers and neurons can well approximate high-order nonlinear curves. In practice, however, we expect that a desired FF-NN should provide the best modelling capability while having the fewest neurons. Figure 7b shows the modelling results of a trained FF-NN with 15 neurons in its single hidden layer, indicating that this trained neural network can already well approximate the high-order aberration of the CSI system.

To evaluate the performance of FF-NN with one hidden layer and various neurons, the measured topography z_CSI by the CSI is firstly compensated by the simulated error ∆z_NN of each trained FF-NN, i.e., z_CSI + ∆z_NN, and then analyzed by the least square circle fitting to determine the estimated cylinder radius ${\widehat{R}}_{\mathrm{m}\_\mathrm{compensated}}$. Figure 7c shows the compensated cylinder radius by FF-NN with different neurons. It can be seen that an FF-NN with neurons less than 12 or larger than 18 demonstrates relatively poor compensation results. The FF-NN with neuron numbers from 14 to 17 will have an almost equally good compensation effect. Figure 7d shows the residuals before and after the compensation using an FF-NN with 15 neurons.

3.4. Further Validation of the Trained FF-NN for Compensation of the Surface-Tilting-Induced Measurement Error in Interference Microscopy

Two additional cylindrical artefacts made of SM fiber with a nominal cladding diameter of 125 µm ± 2 µm have been measured by the MEMS-SPM and the CSI microscope with the same objective. The original and FF-NN compensated radius measurement results are summarized in

Table 1. Measured and FF-NN compensated radius measurement results.

Reference Cylindrical Artefact	Measured Radius, µm
	MEMS-SPM	CSI	CSI with Trained FF-NN Compensation
SM-Fiber 1#	62.77 ± 0.05	64.13 ± 0.36	62.73 ± 0.28
SM-Fiber 2#	62.79 ± 0.05	64.63 ± 0.18	62.94 ± 0.17
SM-Fiber 3#	64.09 ± 0.05	65.78 ± 0.27	64.18 ± 0.22

.

These results have clearly indicated that the feed-forward neural network trained with the measured data can be well utilized to compensate for the measurement error induced by local surface tilting in CSI for different cylindrical artefacts.

4. Conclusions

Although interference microscopy demonstrates high resolution for topography measurement of nearly flat surfaces, its measurement accuracy suffers from the local surface characteristics of the sample under measurement such as local surface tilting. In this manuscript, to our knowledge, a deep learning approach has been, for the first time, introduced to model the surface tilting induced measurement errors in scanning white light interference microscopy with high NA microscopic objectives.

The typical measurement conditions, parameters, and NN modelling specifications employed in this manuscript are summarized in

Table 2. The typical measurement parameters and conditions and modelling specifications employed in this manuscript.

Reference samples		Diameters (nominal value)	125 ± 1.5 µm
		length	10 mm
Surface topography measurement methods	MEMS-SPM	Minimum measurable curvature, Rmin	34.1 µm for θtip = 45o
		Line scan range	100 µm
		Data acquisition rate	10 samples/s
		Line scan speed	512 s/line
		Resolution	11.7 nm
	CSI	Areal image size	350 µm × 264 µm
		Pixels	1024 × 1360
Artificial NN modelling	NN model	Feed-through neural network: configuration	Single hidden layer
	Training period	≤20 neurons	<1 s

. Experimental investigation has indicated that the trained feed-forward neural network can well be utilized to compensate for the measurement error in topography measurements of cylindrical artefacts. After compensation, the measurement error of CSI for radius measurement of cylinders can be effectively reduced from several micrometers down to less than 0.2 µm, i.e., ∆R/R < 0.3%.

In the next step, the reference artefacts used in this manuscript will be coated with different materials to simulate the surfaces of metal work pieces fabricated by different techniques. Interference topography measurement errors of such surfaces will be traceably determined with the help of the MEMS-SPM. The FF-NN model proposed in this manuscript will be further extended to model the measurement errors of CSIs induced not only by surface tilting, but also by optical properties of the metal surfaces including their areal roughness.

Compared to traditional error compensation methods, including lookup tables and polynomial fitting, the deep learning approach proposed in this manuscript is characterized by a universal solution, ease of use, and higher accuracy in modelling higher order error components. It can be anticipated that the deep learning approach for modelling and compensation of measurement errors in CSIs can also be applied for enhancement of the measurement accuracy of similar high-resolution scanning microscopes such as scanning confocal microscopy [38] and focus variation microscopy [39].

It is worthwhile to mention that the traceably measured topography deviations for the Mirau-type microscopic objective used in this manuscript can also be utilized to evaluate and verify the usefulness of different analytical and numerical models developed for interference microscopy.