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Abstract
Through-focus scanning optical microscopy (TSOM) is a high-efficient, low-costed, and nondestructive model-based optical nanoscale method with the capability of measuring semiconductor targets from nanometer to micrometer level. However, some instability issues resulted from lateral movement of the target and angular illuminating non-uniformity during the collection of through-focus (TF) images restrict TSOM’s potential applications so that considerable efforts are needed to align optical elements before the collection and correct the experimental TSOM image before differentiating the experimental TSOM image from simulated TSOM image. An improved corrected TSOM method using Fourier transform is herein presented in this paper. First, a series of experimental TF images are collected through scanning the objective of the optical microscopy, and the ideally simulated TF images are obtained by a full-vector formulation. Then, each experimental image is aligned to its corresponding simulated counterpart before constructing the TSOM image. Based on the analysis of precision and repeatability, this method demonstrates its capability to improve the performance of TSOM, and the promising possibilities in application of online and in-machine measurements.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Through-focus scanning optical microscopy (TSOM) is a model-based optical metrology method [1] which has been proved promising in such areas of the semiconductor and nanotechnology industry as i) mask defect inspection [2], ii) three-dimensional (3D) measurement for Fin field-effect transistors (FinFET), high aspect ratio(HAR) features, and other structures based on 22 nm nodes [3–5], and iii) determination of nanoparticle sizes [6], with advantages of high-efficiency and low-cost [7]. A 2D image called the TSOM image comes from a series of through-focus (TF) images and is key to the TSOM method. 3D geometric information of the target is indirectly extracted by matching the experimental TSOM image to the simulated TSOM image; therefore, it is highly sensitive to interruptions during TF images collection, such as the mechanical noise of lateral movements of the target and the optical noise of angular asymmetry of Kohler illumination [8]. The mechanical and optical noise is proved to decrease the measurement accuracy and limits the range of potential applications of the TSOM method, such as those pertaining to online and in-machine conditions [9–12].
Some attempts have been expended on such noise. As for lateral vibrations, often in subpixel level, Attota increased the exposure time of the camera to average the effect of the displacement [9]. Although the fidelity of the image is promoted, this technique sacrifices efficiency, which is important in the application of high-volume manufacturing (HVM). Angular asymmetry of Kohler illumination is reported to exist in many commercial inspection devices, such as optical microscopes and wafer inspection tools, and leads to typical tilt patterns in TSOM images [13,14]. Ryabko et al. [15] investigated an all-optical device with multiple focal points that corresponded to different wavelengths which were produced by a grating and a lens with chromatic aberration. Similarly, Lee et al. [16] added a tip/tilt mirror and a wavefront sensor into the imaging optical path of the microscope to self-adaptively compensate the effect of the optical aberration. However, both of these techniques proposed critical demands of hardware level and render the system more complex and expensive.
To overcome these limitations and promote the accuracy of the TSOM method, a new technique is developed in this study known as “corrected TSOM method with Fourier transform.” The idea came from that a two dimensional (2D) image can be separated into two 2D complex matrices – known as amplitude matrix and phase matrix through Fourier transform. The amplitude matrix denotes the magnitude of the image while the phase matrix is used to analyze the position of image features to retrieve or reconstruct the image [17]. In this proposed method, before constructing the TSOM image, both simulated and experimental image are separated into amplitude and phase matrices, respectively. The corrected image comes from utilizing inverse Fourier transform to the pointwise product of amplitude matrix from the experimental image and phase matrix from the simulated image. Compared with the previous endeavors described above, this technique is able to achieve the same tasks with the use of an image process at a software level instead of introducing changes to the TSOM system. Furthermore, the proposed method is almost vibration-free that will aid its potential applicability to online and in-machine measurements.
2. Principle
2.1 TSOM system and nanoscale samples
An overview of an established in-house TSOM system is shown in Fig. 1(a). In this system, the illumination part consists of a light emitting diode (LED) source, two objective lens (OB1 & OB2) and two achromatic lenses (CL & L1). The imaging part consists of another two achromatic lenses (L2 & L3) and shares OB1 and L1 with the illumination part. A pinhole (PH) is placed between OB2 and CL to produce the small illumination numerical aperture (INA). The polarized illumination can be obtained by setting a polarizer (P) in the collimated illumination path. The relevant parameters of the system is present in Table 1.
Fig. 1. Schematic view of the system and the sample. a) Experimental system. The abbreviated names of the components are: S (stage), OB (objective lens), PZ (piezo positioner), RM (reflective mirror), L (lens), BS (beam splitter), PH (pinhole), CL (collimated lens), and P (polarizer). b) An isolated Au line deposited on a smooth Si substrate with line height (LH), line width (LW), and sidewall angle (SWA). c) Nominal in-focus optical image of the sample.
Table 1. Parameters of the optical elements
It should be noted that, even if all the optical elements are tightly mounted with each other, and OB1 is scanning along the optical axis by the piezo positioner PZ, the lateral movement of the target in pixel and subpixel level owing to the vibrations or deviations from the optical axis along the scanning direction reported in the previous literature [13,16] is inevitable in our experimental environment. In addition, a small deviation in micro-scale to the ideal position of any aperture of the illumination part in the optical system may leads to angular asymmetry of Kohler illumination. Therefore, careful design and optimization of the optical system are still likely unable to totally eliminate the optical and mechanical noise, which will receive further discussion in the latter sections.
In this paper, an Au line target with nanoscale parameters and its nominal in-focused optical image are depicted in Figs. 1(b) and (c). The Au line is deposited on the smooth Si substrate. The target contains three geometric parameters, that is line height (LH), line width (CD), and sidewall angle (SWA). The sample is measured with an atomic force microscope (AFM) and a scanning electron microscope (SEM) before, and the magnitudes of LH, CD and SWA are found to be equal to 102 nm, 754 nm, and 90°, respectively. The AFM and SEM results are used to be a “gold standard” to justify the accuracy improvement of the correction method in this paper. For simplification, only the LH, whose magnitude is far beyond the half of illumination wavelength, is regarded as the floating parameter to be measured here, and the other two parameters are set as the inputs of the simulation model.
2.2 Procedure for simulating TF images
Careful analysis and quantification of INA and CNA of the optical microscope is critical for simulation. In Kohler illumination, each point within the field of view (FOV) receives a light cone which consists of a series of plane waves with identical magnitudes and different propagating directions [18]. Suppose that the side lengths of the numerical region (where the calculation takes place) along the $\textrm{X}$ and $\textrm{Y}$ axes are ${L_x}$ and ${L_y}$, respectively. The wave vector of the illuminating light can then be expressed as,
2.3 Fourier transform Correction
An in-focus image can be expressed as a convolution of the scalar field at the object with a point spread function which is equal to the Fourier transform of the imaging pupil:
where ${\tilde{E}_{geo}}$ is the ideal image of the target without the diffraction, and where ${P_{CNA}}$ is the pupil function of the optical microscope.Each point of the TF image plane receives a converged sphere wave with different wavefronts from those of the in-focus image plane on the exit pupil. In this case, the only difference is the extra phase factor with respect to the TF distance $\textrm{z}$ according to following term,
The Fourier transform of Eq. (13) is,
Figure 2 presents a simple example of Fourier transform correction. For clarification, an artificial movement with the magnitude of 5 pixels is added to the one of the experimental TF image, as demonstrated by Fig. 2(a). The simulated TF image at the same focus height as Fig. 2(a) without any noise is depicted in Fig. 2(b). The central white bars in these two images are diffraction pattern, and denote the approximate position of the target. A red dotted line highlights the movement of the target. Fourier transform is then introduced onto the experimental and simulated TF image to separate the amplitude and phase matrices, respectively. Figure 2(d) depicts the amplitude matrix of the experimental TF image while Fig. 2(e) depicts the phase matrix of the simulated TF image. The corrected TF image at the same focus height is obtained after combining the above amplitude and phase matrices, as demonstrated in Fig. 2(c). With the assistant of the dotted line, the target is moved back to the central position, which is similar to the simulated TF image.
Fig. 2. An example of Fourier transform correction for the image. a) An experimental TF image with the lateral movement of 5 pixels for visibility. b) The corresponding simulated image at the same focus height with the experimental image. c) The corrected experimental TF image after performing Fourier transform method. d) The amplitude matrix derived from Fig. 2(a). e) The phase matrix derived from Fig. 2(b).
2.4 Workflow of TSOM image construction
Figure 3 highlights the workflow of TSOM image construction. The details are listed below.
Fig. 3. Workflow of constructing a TSOM image. a) TF images are located at their focus position. b) A region of interest (ROI) of each TF image is selected, and is depicted by a blue rectangular window. c) The intensity profiles is extracted along the width of ROI. d) A typical TSOM image is obtained by stacking the intensity profiles along the respective focus position.
Step 1: Two series of experimental TF images are collected by the camera during the objective (OB1 in Fig. 1(a)) scanning process; one with the target and the other with the smooth Si substrate only. Each TF image is normalized by its mean intensity. Then the normalized substrate image is subtracted from the normalized target image at the corresponding focus height (Fig. 3(a)). In addition, the simulated TF images with and without the target are obtained as introduced in section 2.2, and are also normalized in the same way.
Step 2 For each normalized TF image, a region of interest (ROI) is chosen with a rectangular window (the blue box in Fig. 3(b)).
Step 3: The intensity profile of each ROI (Fig. 3(c)) is extracted by averaging along the box-width at each focal height.
Step 4: The raw TSOM image is constructed by stacking the intensity profiles at their respective focus height.
Step 5: the final TSOM image is constructed by interpolating and smoothing the raw TSOM image (Fig. 3(d)).
3. Results and Discussion
3.1 Results of correction
A typical TSOM image with angular asymmetry of Kohler illumination and lateral movement of the target is presented in Fig. 3(d). Correspondingly, the simulated TSOM image of the target with 754 nm LW, 102 nm LH, and $90^\circ $ SWA is demonstrated in Fig. 4(a). By comparing this two images, it can be summarized two characteristics of the pattern in Fig. 3(d). i) As the target in FOV receives angular asymmetric illuminating wave, the scattered intensity is also asymmetric about the central axis, even the target itself is geometrically symmetric, as Fig. 3(c) depicted. ii) Lateral movement of the target leads to small dislocation of the diffraction pattern in every TF image. This leads to distinct ripple-like edge of the pattern, which is evident at the upper part of Fig. 3(d). From these two characteristics, it can be implied that the asymmetric pattern and ripple-like edge are resulted from optical and mechanical noise, respectively.
Fig. 4. Qualitative results of Fourier transform correction. a) Simulated TSOM image at 102 LH. b) Experimental TSOM image after correction. c) Uncorrected DTSOM image calculated by uncorrected TSOM image subtracting simulated TSOM image. d) Corrected DTSOM image calculated by corrected TSOM image subtracting simulated TSOM image. All of the MAVs of the TSOM and DTSOM images are included in the Figure.
Correction with Fourier transform is perform to each TF image before extracting the intensity profile from the ROI as introduced in section 2.4. The amplitude matrix provided by experimental image multiplies the phase matrix provided by simulated image, and then the consequent product is transferred into the spatial domain to generate the corrected TF image. Figure 4(a) demonstrates the simulated TSOM image with 102 nm LH and Fig. 4(b) demonstrates the experimental TSOM image of the target after correction. It can be observed that the symmetric pattern is retrieved in the corrected TSOM image and the pattern edge is more straight than the uncorrected one.
In addition, a differential TSOM (DTSOM) image is obtained by directly subtracting the simulated TSOM image from the experimental TSOM image. The DTSOM image highlights the difference between the simulated and the experimental TSOM images. A metric called mean absolute value (MAV) is used to quantify this difference, which is calculated as following,
where n is the total number of pixels of the DTSOM image. Figure 4(c) is the DTSOM image from the uncorrected experimental TSOM image while Fig. 4(d) from the corrected experimental TSOM image. By comparing both MAVs of these two DTSOM images, it can be concluded that the corrected TSOM image is more similar to the simulated TSOM image than the uncorrected one, for the reason that it has smaller MAV magnitude. This proves potential validity of this method to correct the optical and mechanical noise.The effect of Fourier transform correction is also quantitatively evaluated by magnitudes of the lateral deviation of the diffraction pattern of the target from the experimental TF image to the simulated TF image, as well as the similarity between these two images.
Figure 5(a) demonstrates the lateral movements of the target pattern in the experimental TF images at different focus height in respect to that in the simulated TF images in the condition with or without correction. The lateral movements are quantified through performing the phase correlation method [20] onto the TF images. Some lateral movements are pixel level and the others are subpixel level. The results show that the Fourier transform correction successfully decrease the movements to some extent. Besides, the Fourier transform correction is also compared with the conventional correlated correction method. Figure 5(a) shows remarkably larger movement for TF images after performing correlated correction. This implies invalidity of the conventional method for the reason that it may introduce unacceptable errors when attempting to retrieve the movement by manipulating the rectangular box to choose ROI with interpolating the image in pixel level. The same results are shown again in terms of similarity, which is quantified by correlation between the two TF images, as demonstrated in Fig. 5(b). The similarity for the experimental TF images with Fourier transform correction are better than that without correction. The results for conventional correlated correction is still worse for the same reason above.
Fig. 5. a) Magnitudes of lateral movement and b) correlation between uncorrected images and corrected images with Fourier transform (FT) method or with conventional correlated correction (CC) method.
3.2 Accuracy and repeatability analysis
Before the measurement takes place, a simple matching library with nine simulated TSOM images is constructed numerically by inputting nine different magnitudes of LH ranging from 90 nm to 106 nm to the model. According to the definition of the MAV, the LH of the target is extracted at the point where the minimal MAV of the DTSOM image emerges, and the larger deviation of LH it is, the larger the MAV of the DTSOM image is. Besides, it should be noted that as the experimental TSOM image is used to match every simulated TSOM image of the library, the Fourier transform correlation should be performed every time before comparison.
Figures 6(a) and (b) demonstrate the matching results in terms of MAV with and without correction. The magnitude of the measurand should be extracted from the global minimum of the domain defined by the matching library . For the TSOM image without correction, the minimal MAV locates at the point where LH is 94 nm. However, two local minimums and one maximum emerge in the domain. This adds difficulty to exclude the possibility for the authentic global minimal to locate out of the domain. In contrast to Fig. 6(a), the results corrected with Fourier transform, are able to be fitted with a convex function, as depicted in Fig. 6(b). The global minimum is easy to be searched where LH is 97.9 nm. Compared with the AFM result of 102 nm, it can be concluded that the accuracy of TSOM method is improved after correction.
Fig. 6. Plots of magnitudes of MAV versus LH a) without correction, and b) with correction. The minimal MAV and the corresponding LH are highlighted in the figure.
Repeatability of the method can be quantified by the magnitudes of mean value (MV) and standard deviation (STD) of a series of successive results of the measurand. It is also affected by the optical and mechanical noise. In this paper, ten successive measurements with Fourier transform correction are performed in the same environment to the same target of the isolated Au line with the same optical microscope. A data set consists of these ten measuring results
According to the distribution of the spots in Fig. 7, the magnitude of MV of the data set is calculated as 97.5 nm. A dotted line is introduced to the figure to highlight the MV. The magnitude of STD is calculated as 0.6 nm. These results prove that the Fourier transform corrected TSOM method still retains the nanoscale accuracy for the error is less than 5 nm and good consistency as the magnitude of STD is less than 1 nm.
Fig. 7. Figure shows the distribution characteristic of the repetitive measuring results. The magnitudes of mean value and standard deviation are 97.5 nm and 0.6 nm, and are marked in the figure. The dotted line highlights the mean value.
4. Conclusion
A TSOM method with Fourier transform correction for the angular asymmetry of Kohler’s illumination and lateral vibrations is proposed in this study. In principle, before extracting optical intensity to construct TSOM image, each experimental and simulated TF images are separated into 2D amplitude and phase matrices. The corrected experimental TF image is obtained by using inverse Fourier transform to the product of the amplitude matrix from the original experimental TF image and the phase matrix from the simulated TF image at the same focus height. Compared with conventional correction method, it is revealed that the proposed correction has the ability to effectively decrease the lateral movement of the image pattern due to the optical and mechanical noise. In addition, by reducing the effects of the noise with the proposed correction, the optimized TSOM method demonstrates a better performance with improved accuracy compared with the uncorrected results, and retains high repeatability and consistency.
As it is difficult to align all components of the optical system at their theoretical position, and prevent the system from the interaction with the environment, the optical inhomogeneity and mechanical movement are usually inevitable during the measurement. The proposed Fourier transform correction is easy to perform and manipulated in total software level. It is hoped that the proposed correction will develop TSOM method to become a preferable and competitive measuring technique in online and in-machine applications of nanotechnology.
Funding
National Natural Science Foundation of China (51675033); National Key Research and Development Program of China (2019YFB2006703); National Science Fund for Distinguished Young Scholars (61725501).
Acknowledgements
The authors acknowledge Prof. Wei Tao of Shanghai Jiao Tong University for sample preparation and Shuran Liu for figure drafting.
Disclosures
The authors declare that they have no conflict of interest.
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