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Abstract
Coherence scanning interferometer (CSI) enables 3D imaging with nanoscale precision. However, the efficiency of such a system is limited because of the restriction imposed by the acquisition system. Herein, we propose a phase compensation method that reduces the interferometric fringe period of femtosecond-laser-based CSI, resulting in larger sampling intervals. We realize this method by synchronizing the heterodyne frequency with the repetition frequency of the femtosecond laser. The experimental results show that our method can keep the root-mean-square axial error down to 2 nm at a high scanning speed of 6.44 µm per frame, which enables fast nanoscale profilometry over a wide area.
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1. Introduction
Full-field optical interferometry is a basic technique for high-precision 3D imaging with a clear traceability path [1–3]. The essence of optical interferometry is the accurate extraction of the full-field interference phase [4], which is typically realized by the temporal phase shifting method because of its high lateral resolution and strong robustness. This method has found broad applications such as in profilometry [4,5] and quantitative phase imaging [6–9], among other technologies [10,11].
However, full-field optical interferometry is associated with some challenges in the measurement with large height variations. When the phase difference between adjacent pixels is greater than 2π, the measurement results are affected by phase ambiguity. Numerous methods can be used to resolve the fringe order of the interference phase [12–15], among which coherence scanning interferometer (CSI) based on a broadband light source is a practical and precise method [16], where the fringe order is determined by scanning the optical path length of one of the interfering arms and locating the position with the maximum interferometric contrast. High-density topography information can be obtained in one scan by using a camera with a high pixel density. However, as required by Nyquist’s law, the step length of such scans is typically less than 1/8 of the central wavelength of the light source. On the other hand, an area-array detector is limited by the communication throughput rate, which cannot consider both high pixel density and high frame rate. For instance, an industrial camera with 1-megapixel resolution can achieve a maximum speed of only 120 fps (8-bit) at a bandwidth of 1 Gbps. Hence, for measurement with a wide depth range, CSI requires a long time to acquire a large number of interferograms to determine coherence positions [17], leading to problems including low dynamic performance, environmental sensitivity, and large computation load due to big data.
Many methods have been proposed to improve the dynamic performance of CSI. The most straightforward method consists in upgrading the hardware bandwidth or decreasing the number of pixels [18,19]. However, even with these methods, reconstructing high-quality surface profiles still requires a large amount of data. This results in low data utilization efficiency, which is the main limiting factor for the dynamic performance. In profilometry and refractive measurement, a downsampling method has been used to reduce the number of interferograms required by CSI [20–22]; however, the contrast of the downsampled signal is sensitive to the exposure time of each frame [23,24]. This is because the interference intensity continues to change rapidly during exposure, and the signal of each frame is the integral of the light intensity over the exposure time. A high signal-to-noise ratio (SNR) signal requires a high receiving light intensity and short exposure time, which limits the practical application of downsampling. The quality of the downsampled signal can be improved by phase compensation [25]. However, because the phase compensator and scanning mechanism are independent of each other, it is impossible to accurately obtain the optical wavelength corresponding to the fringe frequency, which is only suitable for coherence positioning without absolute phase information, resulting in its precision typically in the order of tens to hundreds of nanometers.
The mode-locked femtosecond laser has the characteristics of wide spectrum and high coherence, which allows accurate pulse scanning in a large field of view (FOV) and wide range [17,26,27]. More importantly, it can be locked to a frequency standard with other microwave equipment, which is convenient to obtain accurate external modulation and enables traceable phase compensation. In this paper, we present femtosecond laser-based CSI with dynamic phase compensation that amplifies the carrier period of the time-domain interference signal without changing the measurement information it carries. Phase compensation is realized by a pair of acousto-optic modulators (AOMs). As the interference pulse interval is scanned at a uniform speed, the AOMs introduce a reverse heterodyne frequency to the interference signal, resulting in a uniform phase compensation synchronized with the scanning. Because both the heterodyne frequency and repetition frequency of the femtosecond laser are traced to the same frequency standard, a high measurement accuracy is realized over a wide depth range. The experimental results showed that our method can achieve 3D imaging with nanoscale axial precision and significantly fewer frames than those required by conventional CSI. Our method overcomes the measurement speed limitation imposed by the frame rate in CSI-based full-field 3D imaging, enabling high-efficiency nanoscale profilometry over a wide area.
2. Principle
2.1 Femtosecond laser-based CSI with phase compensation
The measurement principle is based on an asymmetric interferometer with unequal arms with optical path difference 2D [28], as shown in Fig. 1(a). A mode-locked laser with a center wavelength λc of 1558 nm and bandwidth of 46 nm is used as the light source to emit a pulse train with a repetition frequency fr of 63.804 MHz, which is determined by the optical length Lc of the cavity, whereas the optical spacing of the pulse train is always equal to Lc. When 2D is exactly equal to m times the pulse interval, the mth pulse overlaps and interferes with the 0th pulse. If Lc changes, the centers of the two pulses that initially intersected become misaligned. At this point, the relationship between the interference signal intensity and the pulse misalignment length Dm is as shown in Fig. 1(c), and such a cross-correlation interferogram can be obtained by directly scanning Lc (or fr).
Fig. 1. Schematic of the Lc scanning interferometer with a phase compensator (PC). a, Principle of the optical system; b, no output from PC; c, optical interference signal obtained without PC; d, compensation phase continuously outputted by PC; e, optical interference signal obtained with PC; f, sampled signal at different exposure (exp.) times with PC; g, sampled signal at different exposure times without PC.
Under the restriction of Nyquist’s law, the minimum sampling interval should be significantly lower than the carrier frequency. We assume that a phase compensator (PC) is added in the proposed system to one arm of the interferometer, which provides real-time phase compensation. The value of PC is controlled to change periodically and linearly with the scanning of Lc (as shown in Fig. 1(d)). Thus, the carrier frequency of the interference signal can be arbitrarily adjusted by PC, as shown in Fig. 1(e). It is difficult to implement a phase compensator for this function, but can be done in a special case: The phase of PC can vary with time at a uniform speed when Lc is controlled for uniform scan, which is equivalent to introducing a heterodyne frequency fh using a frequency shifter. At this point, for a Gaussian envelope pulse, the time-domain interferogram I can be expressed as:
Using a PC, an interference signal can be defined with a longer carrier period. Its Fourier transform (FT) spectrum, shown in Fig. 1(e), translates that in Fig. 1(c). From the principle of the FT spectrometer, the fringe spectrum can be mapped to the optical frequency [26,29]. Therefore, we can customize the mapping of the optical frequency on the FT spectrometer by adjusting the heterodyne frequency according to the scanning speed, frame rate, and spectral width, thus overcoming the sample-rate-constraint in spectrum utilization. During this process, although the fringe interval is changed, its interference phase sensitivity remains consistent with the original spectrum. In the full-field measurement, it is additionally necessary to consider the response of the camera to optical signals. In previous attempts, when the coherence scanning speed was directly increased, a very short exposure time was required to maintain interference contrast, as shown in Fig. 1(f). Since the interference signal does not change significantly during exposure, the measured signal with phase compensation is insensitive to the exposure time, as shown in Fig. 1(g).
2.2 Profilometry
Next, we introduce how this method is applied to surface profilometry. Figure 2(a) shows the optical schematic of the measurement system. The light source is a mode-locked femtosecond laser (MLL), whose fr is servo-controlled to be locked to the signal generator 1 (SG1). The light emitted from MLL is first divided into two channels by a 50:50 coupler: one is outputted directly from a fiber collimator (FC2), and the other is outputted from another collimator (FC1) after a long fiber delay. Dispersion compensation is performed in both the optical fibers. The two beams outputted from the two collimators are orthogonally polarized, passing separately through an acousto-optic modulator (AOM1/AOM2) and then combined by a polarization beam splitter (PBS1). The modulation frequency loaded onto the AOMs is derived from SG2 and is referenced to the same rubidium atomic clock as SG1. The AOMs shift towards the same direction but with a slightly different frequency, thus forming the heterodyne frequency fh in Eq. (1) to reduce the carrier frequency. After filtering through a pinhole, the orthogonal polarized beam is expanded coupled into the Twyman–Green interferometer composed of PBS2, quarter wave plates (Q1 and Q2), and a polarizer (P). Moreover, an imaging lens (L3) images the surface of the sample on the CCD.
Fig. 2. Schematic of the proposed 3D surface interferometer. a, Experimental setup. MLL: mode-locked femtosecond laser; Rb: Rubidium atomic clock; SG: signal generator; LP: low-pass filter; MIX: mixer; PI: proportional integral servo controller; AMP: voltage amplifier; FC: fiber collimator; AOM: acousto-optic modulator; PBS: polarization beam splitter; L: lens, where L1 and L2 form a pinhole filtering beam expanding system, and L3 images the surface of the sample on the CCD; Q: quarter wave plate; P: polarizer; CCD: charge-coupled device camera; b, An interference frame of a step specimen captured by a CCD camera during Lc scanning; c, Time-domain interferograms of the selected pixels in b; d, Interference spectrogram of selected pixels.
The relationship between the repetition frequency fr and cavity length Lc of the mode-locked femtosecond laser is fr = c/Lc [30]. Therefore, we can use fr feedback to control Lc for a linear scan. According to the previous principle, each pixel of a CCD camera can be considered an FT spectrometer as long as its frame rate is stable. The optical path difference between the reference mirror and the conjugate point of the pixel on the sample will be recognized in the FT spectral phase (note that the points near the real conjugate points in a certain range along the beam direction are approximated as conjugate points, due to the limitations of the pixel size and the system aperture size). Figure 2(b) shows the measurement results of a 10 µm standard step structure obtained using the proposed method. Two pixels (Pixels 1 and 2) corresponding to points at different heights in the structure are selected for demonstration. After fine-tuning the position of the reference mirror to make the CCD observe the interference, Dm undergoes corresponding changes by driving Lc to perform linear scanning. At the same time, AOMs introduce a fixed heterodyne frequency, and the CCD collects the interferograms continuously. Since the optical path of the measurement pulses are different, the pulse alignment times between the two pixels are also different, as shown in Fig. 2(c).
Figure 2(d) shows the Fourier spectrum of the interference signal recorded by the two pixels. Based on the scanning range of the optical cavity length ΔLc, sampling time T, and heterodyne frequency fh, the relationship between the Fourier frequency f and optical frequency fopt. can be deduced as fopt. = (f + fh)·T·c/(m·ΔLc). Since the interference light collected by the two pixels comes from the surface reflection of the same material, the amplitude spectra (yellow-filled area) are consistent. Moreover, because of their different optical paths, the phase spectra of the two pixels are different.
In the phase spectrum, if only the central frequency f0 of the spectrum is selected for measurement (the position of the dotted line in Fig. 2(d)), it is equivalent to a single-wavelength phase-shifting interferometer. If the interference phase ϕ0 is measured at that frequency, the height difference ΔDPM between the two measurement points can be expressed as:
Here, φ represents the interference phase difference, and f represents the optical frequency. We call Eq. (3) the slope-method (SM) solution. The minimum value of ∂f depends on the scan range L of Dm. The non-ambiguity range (NAR) of the SM solution, NARSM = c/(2∂fmin) = L/2, is greater than that of the PM solution NARPM = c/(2f0) = λ0/2, where λ0 represents the wavelength corresponding to f0. After removing integer multiples of 2π between the neighboring phases, a weighted linear fit (see Supplement 1: Algorithm for the detailed algorithm) to the phase data yields the SM and PM solutions from the slope and intercept of the fit, respectively. When the accuracy of ΔDSM reaches λ0/4, the value of N0 in the PM solution can be uniquely determined as follows:
3. Experimental results
3.1 Experimental setup
The experimental setup is consistent with that shown in Fig. 2(a). The center frequency of the light source is 192.4 THz, and the full width at half maximum bandwidth is 5.7 THz, which corresponds to a transform-limited pulse duration of 78 fs. After considering device loss, diffraction, and spatial filtering, the maximum total optical power of illumination on the sample is approximately 300 µW. The fr value of the light source is approximately 63.804 MHz, and the optical path difference between the two arms of the interferometer, introduced through the optical fiber, is 79.93 m, so that m = 17. After transmitting through the optical fiber, the pulse width of the measurement arm is 140 fs, and that of the reference arm is 105 fs, both measured by an autocorrelator (pulseCheck NX50, APE). The change in Lc is realized by two piezoelectric ceramics arranged in series, with a total stroke of 50 µm. With reference to a rubidium atomic clock, the laser fr can be accurately read and converted to Lc, thus realizing a closed-loop control of the cavity length. The modulation frequencies of the AOMs (MGAS80-B4-A1, AA Opto-Electronic) are fAOM1 = 80 MHz + fh and fAOM2 = 80 MHz. Both, along with fr, are referenced to the same rubidium atomic clock. The interferograms are obtained using a high-speed infrared camera (G-033, Allied Vision) with a maximum frame rate of 166 fps at 640 × 512 resolution and 14-bit depth. In the experiments reported in this section, the Dm-scanning range L = 618 µm. The imaging lens L3 has focal length 75 mm, magnification 1/2.2, and effective aperture 30 mm. Its numerical aperture (NA) is 1/16, so the diffraction limit is 0.61λc/NA = 15.2 µm, which matches the CCD pixel size of 15 µm.
3.2 Verification of the hybrid method
We first measured a MEMS device with a step height of 1 µm to demonstrate how the PM and SM solutions are combined. The step height exceeds the NARPM; therefore, only precise but ambiguous results can be obtained by PM, as shown in Fig. 3(a), and a spatial-phase-unwrapping algorithm (such as Itoh’s method [32]) cannot be used to obtain the correct surface profile. Figure 3(b) shows the measurement result obtained by the SM solution; it exhibits a large noise due to the error amplification of the phase difference [16]. After substituting both the SM and revealed PM solutions into Eq. (4) to obtain the real N0 value, a hybrid solution is obtained, as shown in Fig. 3(c). Clearly, this method effectively blends the measurement range of SM and the precision of PM.
Fig. 3. Measurement results of a MEMS device (top row), with profile views along the red line (bottom row). a, PM solution; b, SM solution; c, Hybrid solution.
3.3 Precision analysis
Our measurement principle suggests that the hybrid measurement precision depends on the PM, which presupposes that the SM solution lies within the NAR of PM. To quantitatively evaluate the advantages of the proposed method, we characterize the performance of the system in terms of the RMS precision and error rate using a gold-coated plano mirror. The same specimen is measured once by PM, and the deviation distribution of this measurement can be obtained by differentiating the measured surface from the calibration surface. Therefore, we can define the RMS precision as the standard deviation of the measurement deviation in the entire FOV. The error rate is defined to assess SM and related to the reliability of N0 at the transition from the SM solution to the PM solution. For example, in a single measurement of the mirror, an absolute deviation between the SM and PM solutions (SM deviations) greater than c/(4f0), results in an N0 error at that point. The error rate is used to assess SM and is related to the ratio of the number of points with N0 error to the total number of points in the entire FOV.
The specimen was illuminated by a femtosecond laser attenuated to a power of approximately 20 µW, to simulate a low-light detection scenario, where a single-frame exposure time of 40 ms is required to ensure an appropriate interference intensity. The flatness of the mirror is better than 32 nm, which is much smaller than the wavelength; therefore, it is unnecessary to consider the phase wrapping of PM. We first acquired the complete interferograms without phase compensation (fh = 0 Hz) at a frame rate of 15 fps over 100 s. A calibration surface was obtained after averaging the results of 10 measurements.
Next, phase compensation was applied (fh = 38.5 Hz), and the measurement was completed with only 48 frames (corresponding to a scanning speed of 6.44 µm/frame). As a control, a downsampling measurement [20] was also performed using the same camera parameters without phase compensation. The above-mentioned parameters were optimized to avoid the influences caused by frequency aliasing. Figure 4 shows the measurement results of the mirror obtained using the three measurement methods. Among them, Figs. 4(a) and 4(b) show the time-domain interference signals and their frequency spectra of the same pixel. As expected, continuously varying interference fringes during the exposure time results in a very low contrast of the downsampled signal (the middle column in Fig. 4, for a detailed discussion on downsampling and exposure time, see Supplement 1: Limitation of exposure duration on the downsampling method); nevertheless, the phase-compensated interference signal can still maintain a high contrast. However, due to the reduction in the number of sampling points, high-frequency noise is aliased with the signal, thus reducing the SNR of the spectrum. The presence of systematic errors, such as nonlinearity in phase shift introduced by environmental disturbance, weakens the impact of SNR improvement on the precision. Figures 4(c) and 4(d) show the measurement error distributions of the PM and SM deviations, respectively. Compared with the normal method, the deviations in both the PM and SM solutions obtained by directly downsampling are significantly larger. After phase compensation, reducing the same number of measurement frames, the RMS precision of the PM solution remains largely unchanged, whereas the deviation in the SM solution deteriorates by a factor of approximately 1.5. At this point, the error rate of the SM solution is only 0.03% (see Fig. 4(e)), which can be considered feasible in most cases. This measurement results can be described as follows: When the SM yields satisfactory results (e.g., with a standard deviation of less than 100 nm, corresponding to a SNR of about 14 dB), the system can achieve nanoscale measurement precision by hybrid method. However, if the SM does not perform well, the application of the hybrid method becomes infeasible. It can be further verified in Supplement 1: Precision analysis results with different parameters. Precision analysis results with different parameters that the measurement precision is independent of the phase compensation and is only influenced by the number of sampling points.
Fig. 4. Measurement results of a plano mirror obtained using different methods with an illumination power of 20 µW and exposure time of 40 ms. a, Interference signal acquired from the same pixel; b, FT spectrum of a; c, Deviation in the measurement results from the calibration surface obtained by PM; d, Deviation in the measurement results from the calibration surface obtained by SM; e, Histogram of d.
3.4 Examples of 3D surface profilometry
As representative examples, 3D profiles of a simple gauge-block assembly and a 2022 Beijing Winter Olympic Games coin were reconstructed, as shown in Fig. 5, to demonstrate the applicability of the present method. Both measurement results are obtained using the hybrid method. The heterodyne frequency fh was set to 38 Hz, and the sampling was performed for 10 s at a frame rate of 10 fps. After removing the edge area with large aberrations, the displayed FOV appears as a circle of approximately 16 mm in diameter.
Fig. 5. Demonstration of dynamic phase compensation-based profilometry. a, 3D surface profile imaging of a three-gauge-block assembly (heights of 1.01, 1.03, and 1.08 mm); b, Sectional profiles along the black line shown in a; c, Sectional profiles along the red line shown in a; d, Image of the coin, where the red circle is the FOV with a diameter of 16 mm; e, Intensity distribution of the light reflected from the specimen surface; f, Reconstructed 3D profile; g, Sectional profiles along the cut lines shown in f; h, Pixel-recorded time-domain interference signals from logo indentations and rough surfaces, respectively; i, FT spectrum obtained from h. (See Visualization 1 for the scanned interferograms video)
The gauge-block assembly shown in Fig. 5(a) is composed of three standard gauge blocks with heights of 1.01, 1.03, and 1.08 mm. Referring to the 1.01-mm gauge block, two steps with heights of 20.499 and 70.014 µm were obtained and calibrated by a white light interferometer (Contour GT-IM, Bruker). The measurement results of the heights of these two steps are 20.574 (residual error is 0.37%) and 70.280 µm (residual error is 0.38%), respectively (see Figs. 5(b) and 5(c)), which is consistent with the accuracy of the white light interferometer (0.75%). After 15 repeated measurements, the mean of the measured standard deviations for all the points on the 20 µm step is 5.2 nm, and that on the 70 µm step is 7.3 nm. The increase in the uncertainty with the measurement height is due to the nonlinear scan caused by the low bandwidth of our fr controller. This can be rectified in the design of the light source.
The coin shown in Fig. 5(d) is special in that its central part is a frosted surface with a smooth logo engraved on it; therefore, the intensity of the illumination light returned from its surface has a high full-field contrast. The intensity distribution of the light returned from the coin surface can be obtained by blocking the illumination of the reference arm, as shown in Fig. 5(e). The ratio of the light intensity of the smooth surface to that of the rough surface is close to 8:1, which brings significant challenges to the measurement. Figure 5(f) shows the measurement result, where the rough surface and the smooth engraved logo can be clearly distinguished in the height map. Because the phase compensation improves the interference contrast, we are able to make better use of the camera dynamic range to capture weak signals. Figure 5(g) corresponds to the height undulation of the two cut lines shown in Fig. 5(f). The height difference between the logo and the surface can be distinguished from the red cut line, and the height undulation within the logo is relatively flat, whereas the undulation on the rough surface has a fluctuation of 3 µm level. Two pixels on the logo and on the rough surface are selected from the red line in Fig. 5(g), and their time-domain interference signals are shown in Fig. 5(h). Evidently, the engraved logo has a greater light intensity and coherence due to the specular reflection, and the pixels here have a higher SNR, which can also be seen in the spectrum in Fig. 5(i). The SNR of Pixel 2 is almost 4.3 dB lower than that of Pixel 1, reaching 10.5 dB. Based on the previous analysis shown in Fig. 4, the SM solution for this pixel has a certain error rate and is unsuitable for transition to PM. After applying the hybrid method, the measurement precision of the smooth logo is consistent with the PM, achieving a nanoscale level. However, in the frosted surface, the presence of low SNR results in a precision that aligns with the SM, ranging from micrometers to sub-micrometers.
4. Discussion and conclusion
The above results demonstrate the ability of the proposed system to perform nanometer-scale full-field 3D imaging with few sampling points while maintaining precision. At this point, the high-frequency noise is aliased with the effective spectral information and inevitably affects the measurement precision. Nevertheless, the precision of the SM solution is significantly more sensitive to it. Therefore, the lower limit of the number of measurement points depends largely on the degree of deterioration of the SM solution.
The femtosecond-laser-based CSI can be employed for large FOV measurements exceeding ten millimeters. This is because the large-aperture plane waves can be easily obtained, and interference signals can be acquired even if the imaging magnifications of the reference arm and the measurement arm are different [33]. Furthermore, the two interferometric arms in the spatial optical path do not need to be equal, and complex interferometric objective designs are not required for large fields of view [34,35]. Similar to traditional CSI and PSI, the vertical resolution of the system presented in this paper is also independent of the FOV. Therefore, this technology is particularly suitable for the inspection of large size, less structurally complex precision surfaces [17,36].
In practice, the system applicability depends on the NA and depth of focus of the imaging lens L3. For the samples with large-scale step structures or surfaces with large radii (e.g., integrated circuits and microlens arrays), a small NA can be selected to meet the demand, which also results in a wide measurement depth of field and measurement FOV. The measurement of rough surfaces with a wide FOV is inaccurate because the lens L3 cannot collect all the diffracted light from the surface [37]. In addition, the restriction of NA on depth of field can be well improved by using the focusing objective reported by Pavliček et al. [38]. In order to obtain the accurate profile of the rough surface, an imaging lens with a large NA can be used; however, in this case, the imaging depth of field will be narrow (can be estimated as λ/NA2), and the results should be compensated by modeling when the surface undulations are large (the lateral resolution may be reduced). The CSI system based on mechanical phase shifting, such as white light interferometer, does not have the problem of depth of focus. Unfortunately, the current technology seems unable to equip it with a phase compensator, because it requires a phase compensator and scanning mechanism to be traced back to the same wavelength to ensure correct phase compensation in the whole scanning range.
Overall, we achieved precise dynamic phase compensation by utilizing the characteristics of frequency traceability in both femtosecond laser coherence scanning and acousto-optic frequency modulation. Our method enables fast, wide-ranging, nanoscale measurement for CSI-based full-field 3D imaging. Compared with previous CSI techniques, the measurement efficiency is increased almost losslessly by a factor of tens. This suppresses the effects of environmental perturbations on the measurement results and enables dynamic nanoscale profilometry on a large scale. More significantly, this method can be extended to all full-field measurement techniques based on coherence scanning, which is promising for time-sensitive applications.
Funding
National Natural Science Foundation of China (51835007, 92150104).
Acknowledgments
The authors kindly thank Prof. Lijiang Zeng of Tsinghua University for fruitful discussions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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.
Supplemental document
See Supplement 1 for supporting content.
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