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Abstract
Demodulation of steep abrupt heights is a challenging task in interferometry because scattering from the steep region is typically weak. In this paper, we enhance the steepness measurement of steep abrupt heights in a phase-contrast image by applying a fringe thinning process before demodulation with the Morlet wavelet transforms. We demonstrate the proposed method with two steep abrupt heights of 200 µm and 30 mm. Use of this method improves the steepness substantially as compared with conventional measurements.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In metrology, the steepness of a film thickness edge is an important measure useful in many applications such as lithography [1–2]. Many techniques are already available to measure the steepness via extracting the phase information interferometrically. These techniques include phase extraction through the Fourier transform (FT) [2–7] or the phase shifting (PS) algorithms [8–11]. The FT and model-based methods require the assumption that the surface is continuous to unwrap the ambiguous phase, therefore limiting their application [12]. The PS algorithms must be designed for certain frequencies, requiring foreknowledge of the interference pattern frequency [12]. Moreover, the PS algorithms require at least three images to extract the phase of the object being tested [10–11]. Some attempts employing the Morlet wavelet transforms (MWT) to demodulate deformed interferograms of non-steep abrupt heights and smooth surfaces are reported [13–16]. However, no comparable effort was performed to demodulate deformed interferograms of steep abrupt heights. The reason is that demodulation of weak fringes in steep region of the steep abrupt heights produces fake phase in the steep region. In this paper, we propose to use a fringe thinning process [17–18] of the deformed interferogram before demodulation with the MWT. Here, we used the wavelet transform (WT) for demodulation to avoid the side lobes effect that is generated when the Fourier transform (FT) is used. The Morlet mother wavelet is used here because it provides better localization in both spatial and frequency domains. Also, the Gaussian window function used in the Morlet wavelet transform is the optimal window shape [13–16]. The proposed method is applied to demodulate two steep abrupt heights of 200 µm and 30 mm. The flat fielding with apodized apertures technique [19–20] is applied to increase the signal-to-noise-ratio (SNR) of the deformed interferogram by around 16% before appling the fringe thinning process. The obtained thinning fringes of the enhanced deformed interferogram are then demodulated by the MWT method. The extracted phase is ambiguous, so a phase unwrapping algorithm is required to unwrap the ambiguous phase. We first applied the Itoh’s algorithm [21] and Ghiglia’s algorithm [22]. However, we observed that the phase is violated due to noise as seen in Fig. 9. To address this violation, we used the graph cuts algorithm [23]. We observed a significant improvement in steepness measurement when the fringe thinning process [17–18] is applied to the enhanced deformed interferogram before demodulation with the MWT method. We attribute the improvement in the steepness measurement to the proposed thinning process which covers the steep region in the steep abrupt heights with clear thinning fringes. These thinning fringes when convolved with the MWT produce true phase in the steep region and consequently enhances the steepness measurement of the film thickness edge.
2. Principle of the MWT with simulation
The MWT method is described in detail in Refs. [13–16]. Shortly, the Morlet wavelet is a plane wave modulated by a Gaussian function, and it is expressed as:
where α is a fixed spatial frequency in the range of 5 to 6. Daughter wavelets ψb,a are built by translation on the x-axis by b as a location parameter and by a as a dilation parameter of the mother wavelet ψ(x) as given byThe interferogram is reconstructed row by row to extract the phase information of the object being tested. A row of the interferogram f(x) is projected onto the daughter wavelets as:
The obtained wavelet transform is a two-dimentional (2D) complex array. We simulated a steep abrupt height of 256 pixels in Z axis as shown in Fig. 1(a) and demodulated it by the MWT method. The steep abrupt height is generated by using the Mesh of Heaviside function in MATLAB. Figure 1(b) shows the deformed interferogram of 512 × 512 pixels. The intensity equation of the deformed interferogram is expressed as:
where f0 = 1/16 is the spatial frequency of the carrier, ϕ(x, y) = Mesh (Heaviside (x) multiplied by 256), and x & y are coordinate axes. The Morlet wavelet transform of each row of Fig. 1(b) is calculated using Eq. (3) and this produces a two-dimensional complex array. We show a row 256 as an example to see how it is reconstructed by the MWT. The extracted intensity of this row [solid red line in Fig. 1(b)] is plotted against pixel location as shown in Fig. 1(c). The modulus of the array is estimated and shown in greyscale in Fig. 1(d) as a scale ranged from 1 to 64 with a resolution of 1 against Translation in pixels. The phase ϕ of the wavelet transform is then calculated as
Fig. 1. (a) Simulated object of step height of 256 pixels. (b) Deformed interferogram. (c) 1D intensity distribution along the red line of (b). (d) The modulus of the 2D complex array of (c). (e) The angle of (d). (f) The selected phase along the red line of (e).
The previous procedures have applied to the all 512 rows of Fig. 1(b) and the extracted wrapped phase map of Fig. 1(b) is shown in Fig. 2(a). The wrapped phase is then unwrapped by the graph cuts method [23] to remove the 2π ambiguity. The spatial carrier is subtracted from the unwrapped phase and the subtraction result is shown in Fig. 2(b). The phase error between the true phase [Fig. 1(a)] and the extracted phase [Fig. 2(b)] is shown in Fig. 2(c). As seen in Fig. 2(c), the MWT method works well in phase extraction with uncertainty of ± 1 arbitrary unit.
Fig. 2. (a) Wrapped phase map of Fig. 1(b). (b) Unwrapped phase map of (a) by the graph cuts method after subtraction from the spatial carrier. (c) 3D Phase error [difference between Fig. 1(a) and Fig. 2(b)].
3. Experimental results
Schematic diagram of the optical setup used to generate the deformed interferogram is shown in Fig. 3. A light beam from incoherent LED source of center wavelength λ = 635 nm (bandwidth (FWHM) ≅ 15 nm) is collimated by a collimating lens (f = 200 mm) and incident on a non-polarizing cube beam splitter (400 -700 nm).
The beam splitter divides the beam into two copies: a reference beam and an object beam. The reference beam illuminates the reference, which is a flat mirror of λ/20 flatness, and returns to the beam splitter. The object beam illuminates the object and returns to the beam splitter. The reflected beams from the reference and the object interfere at the beam splitter and constitute the deformed interferogram, which is captured by a black-and-white CCD camera (1024 × 76b pixels with 4.65 µm x 4.65 µm pixel size) via an imaging lens (NA = 0.3, 10X). Here, we used the interferometer to generate the deformed interferograms of two step heights of 200 µm (object1) and 30 mm (object2). The object1 which is Ultra Thick Step Height Standard (UTSHS) is from VLSI Company, while the object2 is a gauge block of 30 mm fixed on a flat mirror of λ/20 flatness. Here, the measured height is not the real height whereas a single wavelength is used. Figure 4(a) shows the 3D intensity distribution of the dark frame captured when there is no illumination through the optical system (the laser is off). Figure 4(b) shows the 3D flat frame (non-uniformity of illumination) image captured by blocking the object arm when the laser is on. Figure 5(a) shows the captured deformed interferogram (512 × 512 pixels) of the object1 with SNR = 16.97 dB. The SNR of the deformed interferogram is calculated directly by using the function (SNR) in MATLAB. The deformed interferogram is then corrected by the flat fielding with apodized apertures technique [19–20] using the equation IC = W (IR-IB)/ (IF-IB), where W is the average pixel value of the corrected flat field frame, IR is the raw interferogram, IF is the flat frame image [Fig. 4(b)], and IB is the thermal noise image [Fig. 4(a)].
Fig. 4. (a) 3D intensity distribution of the dark frame. (b) 3D intensity distribution of the flat frame.
Fig. 5. (a) Raw interferogram of object1 before correction with the flat fielding with apodized apertures technique. (b) Wrapped phase map of (a). (c) Correction of (a) with the flat fielding with apodized apertures technique. (d) Wrapped phase map of (c). (e) 1D phase profile along the red line of (d). (f) Unwrapped phase map of (d) by the graph cuts method.
Figure 5(c) shows the correction of Fig. 5(a) using the flat fielding with apodized apertures method. The corrected deformed interferogram (without fringe thinning) is then numerically processed with the MWT to extract the wrapped phase map of the object being tested. The corrected deformed interferogram of Fig. 5(c) shows SNR = 19.69 dB. The difference in SNR is 19.69 - 16.97 = 2.72 dB, which provides a percentage of 16% increase in the SNR in the corrected interferogram by the flat fielding with apodized apertures technique.
Figure 5(b) shows the extracted wrapped phase map of Fig. 5(a) using the MWT method. The yellow rectangle of Fig. 5(a) shows the steep region of the step height of object1, which is 200 µm. As seen in Fig. 5(b), the phase at this region is violated. The phase is significantly improved as seen in Fig. 5(d) when the interferogram is corrected by the flat fielding and apodized apertures technique. Figure 5(e) shows the 1D wrapped phase distribution along the red line of (d). Figure 5(f) shows the unwrapped phase map of Fig. 5(d) with the graph cuts method [23]. To extract the absolute phase information of the object1, the spatial carrier should be subtracted from Fig. 5(f). Figure 6(a) shows the off-axis reference wave (without the phase of object1). The corresponding intensity distribution is expressed as
with f0 = 0.91. The modulus of the 2D complex array of the intensity distribution along the red line of Fig. 6(a) is shown in Fig. 6(b). The wrapped phase map of Fig. 6(a) is shown in Fig. 6(c). Figure 6(d) shows the unwrapped phase map of Fig. 6(c) by using the graph cuts method. The absolute phase map of the object1 is then calculated by subtraction of Fig. 5(f) from Fig. 6(d) and the result of subtraction is shown in Fig. 7(c). Figure 7(e) shows the extracted phase map of Fig. 5(c) through the Fourier transform (FT). The extracted phase map of Fig. 7(e) is extracted as follows: the fast Fourier transform is applied to Fig. 5(c), three spectra are obtained. One spectrum is selected in the spatial frequency domain. After the spatial filtering step, the digital reference wave is simulated for the centering process. Then, the final reconstructed object is obtained by adjusting the values of kx and ky [4].Fig. 6. (a) Off-axis reference wave (without the phase of object1). (b) The modulus of the 2D complex array of the intensity distribution along the red line of (a). (c) Wrapped phase map of (a). (d) Unwrapped phase map of (c) by the graph cuts method.
Fig. 7. (a) Unwrapped phase map of Fig. 5(a) with graph cuts method. (b) 3D phase in the red rectangle of (a). (c) Unwrapped phase map of Fig. 5(c) with graph cuts method. (d) 3D phase in the red rectangle of(c). (e) Unwrapped phase map of Fig. 5(c) through the FT. (f) 3D phase in the red rectangle of (e).
As seen in Fig. 7(b), the steepness of the object1 is not accurate, which confirms the importance of using the flat fielding with apodized apertures technique before demodulation with the MWT method. We compared demodulation with MWT and demodulation with FT via calculating two factors from Fig. 7(d) and Fig. 7(f). The first is the step phase thickness and the second is the standard deviation $\boldsymbol{\sigma}$.
The first factor which is the step phase thickness of object1 is calculated from Fig. 7(d) and Fig. 7(f) as T1 = S1-S2 radians in Fig. 7(d) and T2 = S3-S4 radians in Fig. 7(f). Table 1 shows the phase thickness of object1 extracted at different positions from Fig. 7(d) and Fig. 7(f), respectively. As seen from the results listed in the Table 1, the Morlet wavelet transforms extracts the phase information of the object precisely and it outperforms the FT technique in this respect. Moreover, the steepness (steep region) is enhanced significantly as compared with that extracted through the FT. The second factor which is the standard deviation $\boldsymbol{\sigma}$ is calculated from the black rectangle S1 on phase surface of Fig. 7(d) and the the black rectangle S3 on phase surface of Fig. 7(f).
Table 1. Phase thickness of the object1 in radians extracted by the MWT and the FT method.
Figure 8 shows two profiles of $\boldsymbol{\sigma}$ calculated from the averages of 21 lines in the black rectangle S1 on phase surface of Fig. 7(d) and in the the black rectangle S3 on phase surface of Fig. 7(f), respectively. The average standard deviation estimated by the MWT is estimated to be in the range of 0.1536 radians, while the average standard deviation estimated by the FT is estimated to be in the range of 0.2809 radians.
Fig. 8. Profiles of $\boldsymbol{\sigma}$ calculated from 21 lines in black rectangles of Fig. 7(d) and Fig. 7(f). The average standard deviation estimated by the MWT is estimated to be in the range of 0.1536 radians, while the average standard deviation estimated by the FT is estimated to be in the range of 0.2809 radians.
Figure 9(a) and Fig. 9(b) show the unwrapped phase map of Fig. 5(d) by the Itoh’s algorithm [21] and the Ghiglia’s algorithm [22]. As seen in Fig. 9(a) and Fig. 9(b), the phase is violated. We claim that the violation of phase may be due to discontinuity of the true phase surface of the step height, or due to the wrapped phase is still noisy. This violation in phase is significantly addressed when the graph cuts algorithm [23] is used as shown in Fig. 7(c).
Fig. 9. Unwrapped phase map of Fig. 5(d) with the (a) Itoh’s algorithm, and (b) Ghiglia’s algorithm.
4. Fringe thinning for steepness enhancement
For large step samples, the extracted phase at the steep region is violated due to weak fringes. To address this violation, we propose to Skelton [17–18] the fringes of the deformed interferogram before demodulation with the MWT method. Skelton or fringe thinning is a property to extract the central intensity of the interferogram fringes. The thinning of the fringes can be extracted through the x-axis, or the y-axis, or both (x-y) axes. Here, we extracted the thinning of the fringes through (x-y) axes because it is more accurate [17–18]. Figure 10(a) shows siumlation of four fringes using Eq. (6) at y = 0 and f0 = 0.992. Figure 10(b) shows the thinning of Fig. 10(a) through (x-y) axes. We applied the fringe thinning process on Fig. 5(c) and the results are shown in Fig. 11(a).
Fig. 11. (a) Skelton of Fig. 5(c). (b) Wrapped phase image extracted by the MWT method. (c) Zoomed wrapped phases in the white rectangle of Fig. 5(d). (d) Zoomed wrapped phases in red rectangle of (b).
As seen in Fig. 11(a), the steep region is covered with traces of skeleton fringes and hence makes the MWT method extracts the phase map of the steep region without violations as shown in Fig. 11(b). Figure 11(c) and Fig. 11(d) show the zoomed wrapped phases inside the white rectangle of Fig. 5(d) without fringe thinning and the red rectangle of Fig. 11(b) with fringe thinning, respectivelly. As seen in Fig. 11(c), the phase inside the steep region is violated and appears like noise. The violated phase inside the steep region is addressed by the proposed method as seen in Fig. 11(d). The proposed method truly enhances the steepness of the steep abrupt heights in phase-contrast image. The horizontal strip lines in Fig. 11(d) may be due some scratches on the surface of object1. We applied the same procedures on the deformed interferogram of object2, which is a gauge block of 30 mm fixed on a flat mirror of λ/20 flatness. Figure 12(a) shows the deformed interferogram of object2. Figure 12(b) shows the Skeleton of Fig. 12(a).
Fig. 12. (a) The deformed interferogram of object2. (b) Skelton of (a). (c) Wrapped phase image of (a) extracted by the MWT method. (d) Wrapped phase image of (b) extracted by the MWT method.
As seen in Fig. 12(b), the steepness region is filled with traces of the Skelton fringes. Figure 12(c) and Fig. 12(d) show demodulation of Fig. 12(a) and Fig. 12(b) by the MWT method, respectively. As seen in Fig. 12(c) and Fig. 12(d), the violated phases at the steep region in Fig. 12(c) is rather addressed by the proposed method as seen in Fig. 12(d). Figure 13 shows the flow chart of the algorithm that is used to demodulate the deformed interferogram using the MWT.
5. Conclusion
In this paper, we enhanced the steepness measurement of steep abrupt heights in phase-contrast image by applying a fringe thinning process before demodulation with the Morlet wavelet transforms. We applied this method to reconstruct two step heights of 200 µm and 30 mm with no phase violation in the steep region. We attribute the improvement in steepness measurement to the proposed fringe thinning process which covers the steep region with traces of skeleton fringes. These traces when convolved with the Morlet wavelets produce true phases (not noise due to weak scattering) in the steep region. Future work includes use a combination of the proposed method with Hermit polynomial for further enhancement in the steepness of the large step samples.
Disclosures
The author declares no conflicts of interest.
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