## Abstract

White-light scanning interferometry is an effective and widely used technology for measuring the microscopic three-dimensional morphology of an object. However, it is easily affected by external disturbances and appears to have a non-uniform sampling problem, which reduces the measurement accuracy. In this study, an effective correction algorithm is presented, in which a Hilbert transform and a correlation analysis of the white light interference envelope curves, as well as the simulated ideal interference signal envelope, are employed for a robust and high precision signal correction. In addition, the proposed method is at least 4 times as accurate as a traditional method and achieves a high repeatability, which is analyzed through a simulation and contrast experiments.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Interferometry is a widely applied technology for non-destructive, non-contact, whole-field optical metrology [1–5]. It can handle both reflective and speckled surfaces [6–10], and is commonly used in 3D shape and roughness characterizations of engineering and biological objects [11]. Single wavelength phase shifting interferometry (PSI) offers excellent vertical resolution and sensitivity. However, its unambiguous step-height measurement range is limited to half-a-wavelength (λ/2). The techniques used to extend the measurement range include a multiple-wavelength [12–18] and white-light scanning interferometry (WLSI) [19–22]. A multiple-wavelength technique requires two or three laser wavelengths for the surface profiling, which makes the system bulky and expensive, and the coherent laser light can generate unwanted speckle noise, affecting the measurement accuracy. WLSI is a more ideal tool for the optical metrology of reflective samples, and achieves a sub-micron lateral resolution and sub-nanometer vertical resolution [23]. It makes use of the short coherence length of the white-light source, which is inexpensive and does not require a long movement range of the optical path modulation or a complex structured system.

In WLSI, the optical path difference between the two interference beams is changed by moving the reference or measuring arm, and a high contrast fringe occurs when the optical path difference is close to zero. In addition, the 3-D plot of the axial positions of the zero optical path difference (ZOPD) along the optical axis represents the surface of the test object. Thus, a precise movement of the measuring and interference arms is important. However, some inevitable factors, such as an unstable voltage, external vibration and low accuracy of the moving unit, may cause nonlinear movement of the measurement arm, which is the source of the non-uniform sampling problem. This is also a common and less focused problem in the practical application of WLSI, which needs to be addressed. Some common white-light interference methods, such as a centroid method and a Fourier transform method (FTM) cannot effectively correct the error. In 2016, Huazhong University of Science and Technology proposed a method based on a correlation analysis of WLI envelop curves and a multi reference phase position method [24], which is used to optimize the white-light interference algorithm and improve the robustness and precision of the surface recovery.

In this study, we propose a method for correcting the non-uniform sampling problem in WLSI, in which a Hilbert transform and envelop correlation calculation is used. In a Hilbert transform based envelope substitution method (HT-ESM), the Hilbert transform (HT) approach is utilized to extract the envelope curve of the white-light interference signal, and based on a correlation analysis between the white light interference envelope curve and the simulated ideal interference signal envelope curve, a corresponding signal substitution is applied to obtain a correction signal. We then extract the surface height position from the splicing signal. With this method, we attempt to obtain an accurate result despite the presence of noise and a non-uniform sampling during an actual measurement, and compare the proposed method with a peak location method (PLM) and FTM to show the effectiveness, which is verified through a simulation and a comparison experiment.

## 2. Theoretical analysis

A typical calculation method of WLSI, such as a centroid or FTM, is based on the interference signal without a correction of the signal sampling problem. However, owing to the existence of disturbances such as mechanical vibrations and optical noises, as well as an aliasing of the zero-order fringes, the reference position is often incorrectly determined, and the recovery accuracy is reduced obviously. Compared with the current algorithms used in WLSI, the novel surface recovery algorithm proposed in this study is based on a Hilbert transform and the envelope correlation value. Based on an envelope correlation analysis, the algorithm can be robust, thereby ensuring the level of accuracy.

#### 2.1. Ideal white-light scanning interference signal simulation

When applied to dimensional metrology, broadband light interferometry provides a zero optical difference to measure the characterization of an object. The measurement system, which contains an interference microscope objective, a CCD camera, a positioning device (piezoelectric ceramic transducer (PZT)), and an imaging component, will create a series of interferograms along the scanning direction. The interference intensity field can be derived as follows:

*z*is a vertical scanning position along the optical axis, $h({x,y} )$ indicates the height of the position of the object, ${z_0}$ is the length of the reference arm, ${\lambda _c}$ is the central wavelength of the light source, and $\phi ({x,y} )$ is the initial phase at a position on the object, $\Delta \lambda $ is the bandwidth of the light source and ${l_c}$ describes the coherence length of the light source [25,26]. Different intensity values for individual pixels along the scanning direction are provided, in which the zero optical path difference can be theoretically located based on the maximum intensity. Nonetheless, the interference intensity will actually be affected by an external disturbance when in application. Thus, the proposed algorithm corrects the non-uniform sampling problem, compensates the error induced by the external environment disturbance, and further improves the measurement accuracy.

According to Eq. (2), when the movement step *z* is constant, an ideal white-light interference signal can be constructed, which is only related to ${l_c}$ and ${\lambda _c}$. In addition, from Eq.(3), ${l_c}$ is only dependent on the light source wavelength and waveband, which means that a light source has only one ideal interference signal, and the ideal signal is used to splice and correct the actual signal.

#### 2.2. Interference signal envelope curve solution by a Hilbert transform

The Bedrosian theorem is used in the calculation of the interferogram envelope, which states that the Hilbert transform of the product of two functions with non-overlapping spectra equals the product of the low-frequency function by the Hilbert transform of the high-frequency function [27]. In the case of the interferogram described by Eq. (2), the low-frequency function is the interferogram envelope $\textrm{exp}\left[ { - {{\left( {\frac{{z - h - {z_0}}}{{{l_c}}}} \right)}^2}} \right]$ and the high-frequency function is the cosine function.

In addition,$\; I^{\prime}(z )$ is the Hilbert transform of function $I(z )$, with $z\epsilon ({ - \infty , + \infty } )$, is as follows [28]:

We can determine the envelope through Eq. (6), where $\hat{I}(z )$ is called the envelope of a Hilbert transform.

One important characteristic of a Hilbert transform is as follows [26,29]:

when ${\omega _c}$ is greater than the bandwidth of the low-frequency signal $I(z )$. Using Eqs. (2), (7) and (8), two Hilbert transforms are applied as follows:The envelope of the white-light scanning interference signal can be derived from Eqs. (2), (6), and (10) as Eq. (11), and the simulated interference signal and its envelop curve are shown in Fig. 1.

#### 2.3. Signal correction process through envelope correlation calculation

Using vertical scanning interference and interferogram sampling with a scanning step of $\Delta \textrm{z}$, discrete scanning interference signals corresponding to a point on the object are obtained, and its envelope curve is extracted to match that from the ideal interference signal obtained from the light source.

As shown in Fig. 2(a), the upper envelope is from an ideal signal from the simulation via Hilbert transform calculation, which is called the reference signal, and the envelope below is from the actual measurement signal, called the match signal, in addition, the sequence covering the envelope characteristic is constructed as an envelope sequence with n discrete points. We normalize the reference signal and extract the part of envelope signal larger than a certain threshold as the reference sequence, which is the sequence that the reference window covers, as shown in the Fig. 2(a). The sampling step number corresponding to the first point of the reference sequence is ${r_0}$, the intensity value of each point is ${I_{ri}}$, where *i* is the serial number of the reference sequence, $i{\; } = {\; }1,2,3,4{\; }.{\; }.{\; }.{\; }n$.

Taking a sequence of the same length from the match signal envelope as the match sequence and the envelope intensity of each point ${I_{\textrm{mi}}}$, we extract the match sequence from the starting point of the interference signal, and then extract the matching sequence step by step with the same length as the reference sequence, the correlation coefficient of the matching sequence and the reference sequence can be obtained through the following correlation formula

When a series of matching sequences are taken with a gradual increase in the step number, their correlation coefficients with the reference sequence are obtained as shown in Fig. 2(b), and an optimal matching position corresponding to the maximum correlation coefficient can be found. The reference sequence is used as an ideal signal for correlation analysis with the matching sequence and to find the part with the best correlation with the reference sequence in the matched signal. Therefore, as long as the information of the reference sequence can be extracted, the selection of its position coordinates is arbitrary. So, the coordinates of the correlation curve in Fig. 2(b) is the coordinates position corresponding to the matching signal, and the position of the reference sequence is independent of the coordinates of the correlation curve.

The parameters of the optimal matching sequence are defined as follows:

${M_{pm}}$—the step number of the first point in the optimal matching sequence,

${I_{pmi}}$—the envelope intensity at each point in the optimal matching sequence,

$i$—the serial number of each sampling point in the optimal matching sequence, where $i{\; } = {\; }1,2,3,4{\; }.{\; }.{\; }.{\; }n$.

The ideal white-light interference signal starting from position of ${r_0}$ is extracted to replace the measurement interference signal starting from the position of ${M_{pm}}$. In addition, the splicing signal is the corrected white-light interference signal.

A schematic flow chart of the proposed method is shown in Fig. 3, and the original inference signal collected by the CCD is filtered by a Gaussian filter. At the same time, an ideal interference signal is simulated according to the wavelength and bandwidth of the light source through Eqs. (2) and (3), and the envelope curves of the collected and simulated signals are obtained through HT approach. A correlation calculation of the envelope curves is conducted, and the position of the maximum correlation value within the measurement signal is assumed as the starting point of the match signal envelope and replaces the signal with the ideal signal to realize the signal correction. In addition, the ZOPD position of the correction signal is extracted to calculate the surface height values. Moreover, the above steps are repeated to measure the surface of the sample.

## 3. Simulation results

To test the correction effect of the proposed method in a non-uniform sampling problem, we established a step model through a numerical simulation, which the height of the step is 1µm. The source of incident light is a broadband light from 500nm to 600nm, in which the central wavelength is 550nm. The scanning step is 20nm, and we artificially add non-uniform sampling points (NUSPs) through a random increase or decrease from the sampling interval length $\Delta z$ at a few random longitudinal scanning positions.

In Fig. 4(a) is the interference signal with NUSPs and its HT-based envelope curve, which is realized by randomly selected a finite number of sampling positions and changed the step distance $\Delta \textrm{z}$, and Fig. 4(b) is the ideal signal according to the light source. To further improve the accuracy of the calculation, we normalized the two envelope curves. In addition, the correlation calculation above the two signals is shown in Fig. 4(c), with the maximum correlation coefficient ${M_{pm}}$ determined as the start point, replacing the interference signal with NUSPs with an ideal signal, the splicing signal of which is shown in Fig. 4(d).

A comparison of the surface recovery is shown in Fig. 5 and we compared the proposed HT-ESM with the PLM and FTM. In addition, the average surface height error of the three methods is shown in Table 1. Among the three methods above, the HT-ESM demonstrates the best recovery effect and smoothest shape, and the average height error is 2.093nm to the 1000nm height in the simulated step, which is far less than the errors of the PLM and FTM.

## 4. Experimental results and discussion

To further confirm the validity and feasibility of the new method, a white-light interferometer was developed in our laboratory. Its light source is a broadband LED with central wavelength of 576nm. As shown in Fig. 6(a), incident light passes through the collimating lens, and illuminates the BS in the form of parallel light. The light beam then passes through a Mirau interference object (Nikon, 20x), and the interferograms are collected using a CCD (GEV-B1410M-SC000, IMPERX). With the sample driven by the PZT (PI, P-721, PIFOC) at scan step of 72nm, the total scanning length is 10µm, and 120 frame interferograms are taken in a single measurement. The test sample is a standard step plate, the height of which is 460 ± 3nm. Figure 6(b) shows interferograms of the standard step surface collected by the CCD, and in order to improve the accuracy, we adjusted the pitch and tilt of the sample platform to make its tilt angle as small as possible, which there was only a pair of light and dark stripes in the whole field of vision.

To improve the accuracy and reliability of the experiment, we conducted six measurements on the step plate, and obtained six groups of test sets, which contain 120 interferograms in each set. In order to improve accuracy of the measure result, we used spline interpolation to achieve the precision of 0.01nm. And the experimental results as shown in Table 2, the difference in mean height among the six groups of data from HT-ESM is 18.72nm to the 460nm, the error of which is far less than the results calculated by the PLM and FTM, and thus the accuracy is at least 4 times higher than that of a traditional method. In addition, the standard deviation of the proposed method also indicates its good repeatability. The process of the calculation as shown in Fig. 7, which illustrates the correction method more intuitively

The shape recovery result from a group of data in the above dataset are shown in Figs. 8(a), 8(b) and 8(c), which are the recovery shapes from PLM, FTM, and proposed method respectively, and it is apparent that the HT-ESM achieves the best result, the surface of which is smooth and the edge is sharp. From Fig. 8(d), the cross-section profile of the three methods at Y = 400, the PLM result shows the worst flatness, and has numerous burrs, almost all of which are greater than 0.28µm; the FTM result has fewer burrs than PLM result, although, the burr range is still no less than 0.25µm; by contrast, in the HT-ESM result, the upper and lower step surfaces are extremely smooth, with only a few burrs in the upper and lower border areas.

## 5. Conclusions

To determine the non-uniform sampling problem in a practical WLSI measurement, a new method is proposed in this study, through which a Hilbert transform and envelope curve correlation are calculated. In terms of the Hilbert transform properties, we extract the envelope curves of the measurement interference signal and the ideal interference signal obtained from the light source through a simulation. In addition, an envelope correlation analysis was conducted to correct the interference signal. In the simulation and experiment section, we compared our method with traditional methods, such as PLM and FTM, the results of which show that the accuracy of the proposed method is at least 4-times higher than that of the traditional method and the recovery shape has achieved a significant improvement, which means that the proposed method has good correction effect.

## Funding

National Natural Science Foundation of China (61905131); China Postdoctoral Science Foundation (2018M630773); Natural Science Foundation of Shandong Province (ZR2019QF013); Opening Project of CAS Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics & Technology (CAS-KLAOT-KF201804); Jiangsu Key Laboratory of Spectral Imaging and Intelligence Sense (3091801410413).

## Acknowledgments

We would like to thank the National Natural Science Foundation of China (NSFC) (61905131), Shandong Provincial Natural Science Foundation (ZR2019QF013), the China Postdoctoral Science Foundation (2018M630773), the Opening Project of CAS Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics & Technology (CAS-KLAOT-KF201804) and the Open Project Program of Jiangsu Key Laboratory of Spectral Imaging & Intelligent Sense (No.3091801410413) for their financial support.

## Disclosures

The authors declare no conflicts of interest.

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