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Abstract
Spectroscopic ellipsometry is a powerful tool for characterizing thin film, polarization optics, semiconductors, and others. Conventional approaches are subject to restrictions of mechanical instability and measurement speed. The complex locking scheme of previous dual-comb spectroscopic ellipsometry belies its practicability. We present and demonstrate here dynamic spectroscopic ellipsometry based on a simplified phase-stable dual-comb system, which could realize the online dynamic measurement of optical properties of materials. A precision of 1.31 nm and a combined uncertainty of 13.80 nm (k = 2) in the thickness measurement of thin-film samples has been achieved. Moreover, the dynamic performance of the system is investigated under a high data acquisition rate (1 kHz) with a dynamic resolution of ellipsometric parameter better than 0.1 rad.
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1. Introduction
Fast and high-precision film thickness measurements have played a significant role in high-tech and multi-functional devices’ semiconductor and display industries. As the thickness and uniformity in such products directly influence the performance capabilities, it has to be real-time measured and monitored with high-precision approaches in the manufacturing process of films [1–4]. Moreover, the monitoring of dynamic process such as chemical vapor deposition (CVD), molecular beam epitaxy (MBE), etching, oxidation, thermal annealing, liquid phase processing and so on also plays a significant role in both scientific research and industrial development and requires real-time determination in less than millisecond scale [1,4]. Ellipsometry is an optical metrology technique that utilizes polarized light to characterize the thickness of thin films and optical constants of both layered and bulk materials [5,6]. An extensive range of the applications of ellipsometry (from nanoelectronics to biotechnologies), as well as the availability of up-to-date ellipsometric instrumentation, make optical ellipsometry a very appealing and available tool [7–11]. The key feature of ellipsometry is that it measures the change in polarized light upon light reflection on a sample or light transmission by a sample, and the measured ellipsometric parameters ($\Psi$ and $\Delta$) represent the amplitude ratio $\Psi$ and phase difference $\Delta$ between light waves known as p- and s-polarized light waves. Conventional spectroscopic ellipsometry (SE) using polarization modulation obtains ellipsometric parameters from optical intensity measurements, which can be divided into extinction type and photometric type based on its operating principle [1,12,13]. However, a rotating polarizer and compensator equipped with a dispersive spectrometer limit the mechanical stability. The SE based on phase modulation by photoelastic or electro-optic modulators (EOMs) bears the drawbacks of wavelength or temperature dependency [12,14–17]. Interferometric ellipsometer enjoys several advantages over intensity-based ellipsometry, among which are rapid and precise alignment and an intrinsically broadband nature. Nevertheless, such systems generally use modified Michelson interferometers in which the temporal fringes of spatially separated p- and s-polarized components are generated by methods such as mechanical scanning, wavelength modulation of a semiconductor laser diode or the beat frequency [18,19]. As such, the speed of measurement is restricted, and the low visibility fringes for measurements at the Brewster angle also result in setbacks in determining ellipsometric parameters accurately.
Since their invention in the late 1990s, optical frequency combs (OFCs) have become an indispensable tool in optical metrology [20,21]. Dual-comb spectroscopy (DCS) is continuously arousing researchers’ enthusiasm in the field of measurement due to its ultra-high resolution, high accuracy, broadband spectroscopy with rapid data acquisition [22–26]. It typically consists of two OFCs with slightly different repetition rates, allowing the acquisition of interferogram with an ultra-wide time span without mechanical scanning to deduce a highly resolved optical spectrum by mapping with a radio-frequency (rf) sub-comb. In addition, both the amplitude and phase spectra can be obtained through directly decoding from the interferograms of two OFCs, as the DCS is based on Fourier transform spectroscopy. In general, a phase-stable dual-comb source is key to most dual-comb spectroscopy applications. For instance, with a phase-stable dual-comb source, the mode frequency could be accurately determined in gas absorption spectroscopy measurement and fiber sensing, and the phase difference between the two arms of a Michelson interferometer can be measured in absolute distance measurement with nanometer precision [23,25,27,28]. Previously, polarization-sensitive DCS schemes that employ a combination of DCS and a rotating compensator or an EOM have been proposed by Sumihara et al. for precise polarization measurement of individual frequency-comb teeth or determining the anisotropic optical responses of materials [29,30]. Due to the use of mechanical elements and phase modulators, the convention limits are not completely overcome and the configuration for achieving a phase-stable dual-comb source is complicated. Minamikawa et al. have effectively constructed a polarization-modulation-free dual-comb spectroscopic ellipsometry (DCSE) system that directly and simultaneously obtains the ellipsometric parameters from mode-resolved OFC spectra [31]. Such an approach benefits from the immunity to mechanical vibrational noise, thermal instability, and polarization-wavelength dependency. Yet, it uses an ultra-stable continuous wave (CW) laser to stabilize the repetition rates of the dual-comb for their mutual coherence, and the carrier phase drift of interferograms are phase-compensated, which complicates the system and degrades its compactness and flexibility. Although a phase-stable dual-comb interferometer can also be realized by adopting a free-running CW laser to extract the relative noise between the two combs and implementing digital post-compensation, the post-compensation method is challenging to realize real-time in-situ measurement due to a large amount of computation [32–37]. On the other hand, DCSE is one potential candidate to satisfy the considerable needs for online and dynamic monitoring of film process and antibody-antigen reaction in biosensing, as it can boost the data acquisition rate up to kilohertz. For example, dynamic SE provides an ideal tool in characterizing thin-film structures that show a complicated structural evolution with growing film thickness, as the measurement is carried out repeatedly during the film structural evolution [14].
We propose the dynamic dual-comb spectroscopic ellipsometry (d-DCSE) with a simplified phase-stable dual-comb system, which is likely to miniaturize and more suitable for industrial in-situ applications, such as rapid determination of ultrathin layers widely used in microelectronics. The feasibility of a simplified phase-stable SE based on the dual-comb noise model is theoretically analyzed and experimentally verified, avoiding the difficulties of obtaining phase-stable DCS by mutual locking or digital post-phase-compensation. Experimental results show a precision of 1.31 nm and an expanded uncertainty of 13.80 nm ($k=2$) in determining the thickness of thin-film samples under a spectral range of 10 nm. Though the precision is at the same level of commercial polarization-modulation-based SEs, our system provides three orders of magnitude faster improvements in measurement speed. In addition, the dynamic performance of the simplified d-DCSE with a normal-incidence transmission configuration is investigated, and the measurement results are consistent with the simulation results, which again proves the system’s reliability.
2. Experimental setup and calibration
2.1 Experimental setup and principles
Figures 1(a) and 1(b) illustrate the optical setups of our simplified d-DCSE. Two home-made erbium-based mode-locked fiber lasers are employed as the signal (Comb1) and local comb (Comb2) oscillators, whose repetition rates ($f_{\rm r1}=$56.09 MHz, $f_{\rm r2}=$56.0901 MHz) and offset frequencies ($f_{\rm o1}$, $f_{\rm o2}=$10.56 MHz) are independently locked by feedbacking the laser cavities and injection current of pump lasers via the f-2f interferometers, respectively (see Fig. 1(c)) [38]. The polarization states of Comb1 and Comb2 are set to linear polarizations at $-45^{\circ }$ in a coordinate system where p-polarized direction is assumed to be X axis and s-polarized direction is the Y axis. Comb1 is incident on the sample in the reflection or transmission configuration and is then spatially overlapped with Comb2 using a beamsplitter (BS). The generated interferograms of the s-polarization and p-polarization components ($I_{\rm s}$ and $I_{\rm p}$) are individually detected by a polarizing beamsplitter (PBS) and InGaAs photo detectors (PDs). Another two polarizers (P$_{3}$ and P$_{4}$) before the PDs are applied to enhance the extinction ratio of the two perpendicular components. Herein, the ellipsometric parameters of each sample are evaluated by decoding the amplitude ($|I|$) and phase (${\rm arg}(I$)) spectra after Fourier transforming $I_{\rm s}$ and $I_{\rm p}$ after a calibration process.
Fig. 1. Schematic of setups and phase difference noise $\delta \Delta \phi _{\rm c}$. (a) Reflection configuration. Q$_{1-2}$, quarter-wave plate; H$_{1-2}$, half-wave plate; M, mirror, BPF, band-pass filter. (b) Transmission configuration. (c) Scheme of the dual-comb light source used. The $f_{\rm r}$ and $f_{\rm o}$ of the two combs are independently locked. (d) Carrier phase difference $\Delta \phi _{\rm c}$ under three locking conditions without averaging. The $\Delta f_{\rm r}$ is 100 Hz as in subsequent film thickness measurements.
As for the theory of the dual-comb ellipsometry, let’s consider the optical longitudinal mode $f_1$ of Comb1 with number $n_1$ and $f_2$ of Comb2 with number $n_2$, then the two electric field after P$_1$ and P$_2$ can be written as:
The phases $\beta _{i{\rm }\hbox{-}{\rm PD}x}$ signify the propagation phase from Comb$i$ to PD$x$ ($i=1,2$, $x=\rm {p,s}$). $P_{3,4}$, $BS$, $M$, $S$, $R(\alpha _{\rm P3,4})$ represent the Jones matrices of polarizers P$_3$ and P$_4$, 50/50 beamsplitter, mirror, sample, and coordinate rotation matrices at the angle $\alpha$ (counterclockwise), which can be expressed as
Then, Eqs. (3) and (4) can be expressed as follows after substituting Eq. (5) into them:
The photo current produced at PD$x$ is $I_{x} \sim |E_{\rm {PD}x}|^{2}$. In the time domain, it contains beats at the harmonics of the axial frequencies and at all possible combination frequencies, $f_1 \pm f_2$. If we neglect the detector spectral response function and use a filter to take out the ac component within $f_{\rm r}/2$, the generated $k$th mode of the sub-comb in the rf domain ($f_{\rm rf}(k$)) can be written as:
2.2 Effects of noise
Next, we consider the effects of repetition rate noise ($\delta f_{\rm r}$) caused by the instability of cavity length for carrier phase difference noise ($\delta \Delta \phi _{\rm c}$) between $I_{\rm s}$ and $I_{\rm p}$, where $\Delta \phi _{\rm c}=\phi _{\rm c}^{\rm s}-\phi _{\rm c}^{\rm p}$. The standard deviation (STD) of $\Delta \phi _{\rm c}$ within 1 s is adopted to evaluate the $\delta \Delta \phi _{\rm c}$. The $\Delta \phi _{\rm c}$ here is the phase difference of rf carrier between $I_{\rm s}$ and $I_{\rm p}$, and any spectral mode near the center frequency can be selected as the carrier frequency $f_{\rm c}$ in Comb1. In dual-comb systems, as the multiple heterodyne process combines the frequency noise of two OFCs into one, the frequency noise in rf sub-comb can be approximately expressed as Eq. (10) [32,39]:
Thus, as the phase difference noise in the rf domain is directly inherited from the optical frequency domain, the phase difference noise $\delta \Delta \phi _{\rm rf}(k)$ in the rf domain and corresponding $\delta \Delta \phi (n_1)$ in the optical frequency domain can be depicted as the integral of $\delta f_{\rm rf}(k)$ within a time range [40]:
Fig. 2. Dual-comb polarization-separated Michelson interferometer. The measuring mirror is mounted on a translation stage to offer the delay. Mr: reference mirror, Mm: measuring mirror.
As Fig. 3(a) shows, the $\delta \Delta \phi _{\rm c}$ increases from -0.029 rad to 1.97 rad (1.10 rad) with $t_{\rm s}-t_{\rm p}$ changing from $-2.4$ ns to 1779 ns (-918 ns) (corresponding optical path difference from 12.81 $\mathrm{\mu}$m to 9.5 mm (4.9 mm)), indicating that the $\delta \Delta \phi _{\rm c}$ is sensitive to $t_{\rm s}-t_{\rm p}$. The inset in Fig. 3(a) presents the $\delta \Delta \phi _{\rm c}$ when $t_{\rm s}-t_{\rm p}$ is between $-35$ ns and 35 ns, where the minimum value of 0.029 rad is mainly limited by intensity noise as discussed in [40]. As the $|t_{\rm s}-t_{\rm p}|$ increases, the $\delta \Delta \phi _{\rm c}$ also increases as $\delta \Delta \phi _{\rm c}$ is the integral of multiple of $\delta \Delta f_{\rm r}$ as Eq. (11) and the intensity noise can be ignored in these cases. Figures 3(b)–3(d) plot the $\Delta \phi _{\rm c}$ with 1000 updating periods at different time delays. Compared with the jitter of $\Delta \phi _{\rm c}$ near $2\pi$ when the time delay is 1080 ns, the jitter of $\Delta \phi _{\rm c}$ only distributes from $-0.53$ rad to $-0.35$ rad when the $t_{\rm s}-t_{\rm p}$ is $-2.478$ ns.
Fig. 3. (a) The $\delta \Delta \phi _{\rm c}$ when tuning the $t_{\rm s}-t_{\rm p}$. The data length is 1 s with 1000 IGM pairs, and the carrier wavelength is 1568.21 nm (corresponding to 191.3 THz in optical frequency). (b) $\Delta \phi _{\rm c}$ at point i when $t_{\rm s}-t_{\rm p}$ is $-2.478$ ns. (c) $\Delta \phi _{\rm c}$ at point ii when $t_{\rm s}-t_{\rm p}$ is $-918.8$ ns. (d) $\Delta \phi _{\rm c}$ at point iii when $t_{\rm s}-t_{\rm p}$ is $1080$ ns.
As the orthogonally polarized light shares most common path in ellipsometry setup, the integral time is intrinsically short (one set of data is 0.59 ns as an example), thus the $\Delta \phi _{\rm c}$ is theoretically stable with a relatively simple independent locking scheme. As shown in Fig. 1(d), phase difference results ($\Delta \phi _{\rm c}$) under three locking conditions (i.e., intracavity-EOM-based high-speed mutual phase-locking and digital noise compensation with independent locking (more details can be found in our previous publications [32,41]), and easy independent-locking) in the setup of Fig. 1(a) indicate the consistency with the averaged $\Delta \phi _{\rm c}$ and $\delta \Delta \phi _{\rm c}$ within 1 s, which again verify the phase stability is sufficient for ellipsometry measurement without complex locking schemes.
2.3 System calibration method
As a matter of fact, ${\rm tan}(\Psi )$ and $\Delta$ derived from Fresnel formula are functions of wavelength ($\lambda$), the refractive index of the substrate, films and air, the thickness of films, and angle of incidence ($\theta _0$). In our experiment, the wavelength is obtained by mapping the rf to optical frequency after Fourier transforming $I_{\rm s}$ and $I_{\rm p}$, and the refractive index values are known. To measure the film thickness, the ellipsometric parameters of the system (i.e. $\beta _{\rm 1}\hbox{-}{\rm PDp}-\beta _{\rm 1}\hbox{-}{\rm PDs}-\beta _{\rm 2}\hbox{-}{\rm PDp}+\beta _{\rm 2}\hbox{-}{\rm PDs}$ mentioned above) and incident angle need to be calibrated beforehand. We first calibrate the ellipsometric parameters of the system using a single-side highly-polished silicon wafer (Si) as shown in Fig. 4(a). Moreover, the samples used in the experiments are also coated on single-side polished silicon wafers. Although silicon is partially transmitted in the near-infrared band around 1550 nm, the reflection of the transmitted light on the unpolished side has little effect on the signal-to-noise ratio of the measured signal due to its low reflectivity. In addition, even if double-polished silicon wafers are used in our experiment, as the thickness of the silicon wafer is about 525 $\mathrm{\mu}$m, the optical path difference between the reflected light on both sides is calculated to be 3.75 mm considering the silicon’s refractive index (around 3.5) and the incident angle of oblique incidence ($45^{\circ }$). Hence, the interferograms generated by the reflected light from both sides and Comb2 can be separated in the time domain, and the influence of the reflected light on the uncoated side of the silicon wafer can be excluded through data segmentation.
Fig. 4. (a) System’s initial phase difference and amplitude ratio calibrated by a silicon wafer. (b) The measured $\Delta$ of three different samples with the nominal thickness of 90 nm (red), 200 nm (blue), and 280 nm (green) and the theoretical values at different incident angles (mesh surfaces). (c) RMSE between the measured $\Delta$ of the three samples and the theoretical values at different incident angles.
After that, the ellipsometric parameters of thin-film samples are obtained according to the phase difference and amplitude ratio of the PDs and the calibration results. Then, three different SiO$_{2}$ thin-film samples with the known thickness (determined by a commercial ellipsometry W-VASE using a spectral range 700-1700 nm and resolution 1 nm) deposited on polished silicon base plates are applied to calibrate $\theta _0$. Figure 4(b) presents the measured $\Delta$ values of the three samples (colored lines) as well as the calculated $\Delta$ of the $\theta _0$ changing from $43^{\circ }$ to $51^{\circ }$ (in increments of $0.1^{\circ }$) on the basis of equations (2.94) in [1]. We can see that the measured values of the three samples with the nominal thickness 90 nm, 200nm and 280 nm fit well with the theoretical values near the incident angle. Next, we calculate the deviation between the parameters measured in a spectral range of 1558-1568 nm and the theoretical parameters under different incident angles, and $\theta _0$ is acquired when the root-mean-square error (RMSE) is minimal. The graph plotted on Fig. 4(c) indicates the calibrated incident angle of $47.8^{\circ }$. The maximum deviation of ellipsometric parameter $\Delta$ is less than 0.1 rad at the incident angle $47.8^{\circ }$. Subsequently, the film thickness of other samples is measured by the calibrated system. Note that the incident angle of each sample should remain unchanged. We put an iris in the distance to ensure the reflected light from the samples can pass through it every time.
3. Results
A total of six samples with the nominal thickness of 28 nm, 90 nm, 200 nm, 280 nm, 500 nm and 1000 nm are utilized for the ellipsometric evaluation of thin films, including three used to calibrate $\theta _0$. The measured ellipsometric parameter $\Delta$ and its trend within 10 nm spectral range are consistent with the theoretical values calculated at $\theta _0=47.8^{\circ }$. We measure the film thickness using the Levenberg-Marquarelt algorithm by minimizing the RMSE of the experimental and theoretical values of $\Delta$ over the spectral range of 10 nm. The thickness is determined by evaluating the local minimum of $\Delta$ within 100 nm of the theoretical estimate. The same calibration and measurement methods are likewise used in other spectral ranges (1568-1573 nm and single wavelength 1572.4 nm). As the repetition rate difference $\Delta f_{\rm r}$ in these experiments is set to 100 Hz, 100 pairs of interferograms can be obtained within 1 s. We use the averaged value and STD of the 100 measurements in 1 s to represent the thickness ($t$) results and precision ($\sigma$). For comparison, these samples are measured 20 times with a commercially available SE (W-VASE) in the spectral range 1558-1568 nm and resolution 1 nm. To ensure the consistency of the measurement conditions, the commercial SE is calibrated at only one incident angle and a narrow spectral range in the comparison experiment. As can be seen from the results exhibited in Table 1 and Fig. 5, measurement precision is improved by using a wider spectral bandwidth, and our proposed method has higher precision over the same spectral bandwidth compared to the commercial SE by virtue of the higher spectral resolution of DCS method. As the measurement position of each sample may be different when using the commercial SE and our proposed system, there are differences in the measured film thicknesses between the commercial SE and our approach. Note that the same point of each sample is measured when using the single wavelength and 5 nm spectral range, because they use different spectral bands of the same set of data. It is also necessary to note that the measuring rate of our method is three orders of magnitude greater than the traditional SE.
Fig. 5. Measured thickness and precision (error bar) under different spectral ranges compared with a commercial SE. Inset is enlarged at 500 nm film thickness measurement results.
Table 1. Measured thickness and STD ($\boldsymbol {t \pm \sigma }$ (nm)) by the proposed method and the commercial SE. 2mm
As an experimental demonstration of transparent medium and dynamic measurement, we perform the ellipsometric evaluation of an achromatic quarter-wave plate (QWP) and a Soleil-Babinet compensator (SBC) in the normal incidence transmission configuration, both of which are designed for infrared. Figure 6 showcases the ellipsometric parameters of the QWP and SBC with a variety of rotation angles and relative wedge distances within 10 nm spectral range. The slow axis of the QWP is calibrated to be perpendicular to the linear polarization of P$_1$, and the fast axis of the SBC is calibrated along the axis of X-polarization at 1560 nm before use. The system’s initial ellipsometric parameters are calibrated by QWP’s rotating angle at 0 deg when measuring QWP and the SBC at 0 mm when measuring the SBC. Throughout the $360^{\circ }$ rotation of the QWP, four peaks in ${\rm tan}(\Psi )$ and two peaks in $\Delta$ are observed, which are consistent with the incidence-angle-dependent ${\rm tan}(\Psi )$ and $\Delta$ behavior of a uniaxial crystal. Moreover, the flat structure of ${\rm tan}(\Psi )$ means that the amplitude ratio of perpendicularly polarized light is insensitive to the relative wedge distance of the SBC. In contrast, it shows a linear relationship between $\Delta$ and the relative wedge distance, which is the behavior of a phase retarder relative to X- and Y-polarizations. In addition, the measurement results are in good agreement with the theoretical estimation at 1563 nm, and the simulation is based on the models of birefringent crystal properties and the calibrated full-wave displacement of SBC. As shown in Fig. 7, the RMSE between measured and simulated results of the $\Psi$ and $\Delta$ of QWP is 52.4 mrad and 48.6 mrad , and the RMSE between the measured and simulated of the $\Psi$ and $\Delta$ of SBC is 41.1 mrad and 28.5 mrad, respectively.
Fig. 6. Ellipsometric evaluation of a QWP ((a) $\Delta$ and (b) ${\rm tan}(\Psi )$) and SBC ((c) $\Delta$ and (d) ${\rm tan}(\Psi )$) by the transmission configuration.
Fig. 7. Measured ellipsometric parameters of the QWP ((a) and (b)) and SBC ((c) and (d)), all of which are in good agreement with that of simulated results.
The dynamic ellipsometry performance is evaluated by manually rotating the QWP and changing the relative wedge distance of the SBC due to the limitations of experimental conditions. In order to improve the measurement speed, the repetition rate difference ($\Delta f_{\rm r}$) of 1 kHz is again applied. Figure 8 shows the dynamic ellipsometric parameters of QWP and SBC at 1567.8 nm, where Figs. 8(a) and 8(b) illustrate the dynamic process of QWP within 4 s when it resonates around 10 deg (blue curves) and 5 deg (red curves). Compared with Fig. 6, ${\rm tan}(\Psi )$ and $\Delta$ here have the same period during the QWP oscillation. Figure 8(c) plots the dynamic process of SBC when we linearly change the relative wedge distance (4–6 mm), and oscillate at a certain wedge distance (0.9 mm). Besides, frequency peaks in Fig. 8(d) further demonstrate the dynamic measurement capability of the system. We can also conclude that the dynamic resolution of $\Delta$ measurement is better than 0.1 rad from the dynamic results of SBC. Note that the ellipsometric parameters in our experiment change slowly due to manual adjustment, whereas our system is able to measure the oscillating process with frequency $\Delta f_{\rm r}/2$ according to the Nyquist sampling theorem, because the ellipsometric parameters can be output at a speed of $\Delta f_{\rm r}$.
Fig. 8. (a), (b) Dynamic performance of the QWP when resonating around 10 deg (blue) and 5 deg (red), gray lines indicate the static condition. (c) Dynamic change of $\Delta$ of the SBC. (d) Normalized amplitude after Fourier transforming the oscillation results. The vertical axis is zoomed in to ignore the DC component.
4. Discussion and conclusions
For reliability evaluation of the thickness measurement using the proposed system, we perform the uncertainty evaluations on the thin-film samples according to the Guide to the Expression of Uncertainty in Measurement (GUM) [42,43]. The thickness measurement uncertainty of the thin films is determined from two major uncertainty components related to measurement reproducibility and the spectrum analysis process shown in Table 2. The experiments and calculations are performed in the 1558-1568 nm spectral range. Firstly, measurement reproducibility can be calculated from the repeatability and long-term stability components in line with several International Organization for Standardization guidelines [44–46]. The uncertainty for the repeatability is estimated as the averaged measurement STD of 1 s data length of the six samples, which is 1.31 nm. Through repeated measurements every 40 minutes over 4 hours (see Fig. 9), the uncertainty for the long-term stability is estimated to be 6.45 nm (the averaged STD of the mean values within 4 h of the six samples), which results in a standard uncertainty of 6.58 nm for measurement reproducibility. Secondly, the uncertainty related to the data analysis is estimated from the theoretical model of ellipsometry. The uncertainty related to the incident angle is calculated to be approximately 1.77 nm under the assumption that $\theta _0$ can be determined within $\pm 0.1^{\circ }$. The uncertainty stemming from the refractive index accuracy is calculated to be 1.07 nm, referring to the refractive index errors reported in [47]. Consequently, the standard uncertainty related to the analysis process is calculated to be around 2.07 nm. As a result, the combined uncertainty of the thickness measurement of the thin films is estimated to be 13.80 nm ($k=2$).
Fig. 9. The six samples’ measurement results (the average of seven measurements) and the STD of the seven measurements ($t\pm \sigma$) to evaluate the long-term stability.
Table 2. Uncertainty budget of the thickness measurement of thin-film samples using the proposed simplified d-DCSE system. 4mm
According to the uncertainty analysis above, the most dominant uncertainty factor in thickness measurements is the long-term stability, meaning that the measurement locations may differ due to the repeated loading and unloading of the samples, which can be improved by using an ellipsometry arm and a sample table to keep the measurement position unchanged. To further enhance the repeatability, a broader spectrum is suggested to be used based on the experimental results in Table 1 and can reduce the random error of measurement. The uncertainty of the data analysis process can also be decreased by optimizing the processing algorithm and expanding the spectrum as $\theta _{0}$ can be determined more accurately [48,49]. To be clear, the local minimum is converged during the regression analysis due to the limited spectral range, and the initial iteration value close to the nominal value is selected. In the future, the accuracy of the iterative solution will also be improved after expanding the spectral range. Fortunately, the spectrum expansion methods in the near-infrared band have been significantly developed, such as using high nonlinear fiber, photonic crystal fiber, microstructured fiber, and so on [50 –52].
We next discuss the trade-off relationship between precision and measurement speed. In our proposed system, the measurement speed is equal to $\Delta f_{\rm r}$. Broader spectrum can enhance the precision, yet the available spectral range ($\Delta \upsilon$) is limited by the band-pass sampling theorem in dual-comb systems as $\Delta \upsilon \leq f_{\rm r}^{2}/(2\Delta f_{\rm r})$. To achieve higher precision at fast measuring speed, OFCs with high repetition rates ($f_{\rm r}$) can be used, such as microresonator combs or semiconductor-based combs with repetition rates from 10 s of gigahertz into the terahertz [53–55]. For example, in the case of d-DCSE with $f_{\rm r}$ of 10 GHz and $\Delta f_{\rm r}$ of 1 MHz, $\Delta \upsilon$ is increased up to 50 THz, corresponding to a wavelength range of $1560 \pm 200$ nm at the acquisition time 1 $\mathrm{\mu}$s per point. In this way, there is still room for enhancement of the data acquisition rate and the spectral bandwidth by the selection of OFCs. Additionally, high spectral resolution is beneficial to improve the precision in SE. Although we do not utilize every longitudinal mode of Comb1, the present spectral resolution (0.025 nm) is sufficient to measure most wavelength-dependent materials, as most samples do not indicate sharp spectral features. At the measuring speed of 1 kHz (corresponding wavelength range $1560 \pm 6.4$ nm), the precision of the thin-film thickness can still reach 5.32 nm by our system, as shown in Fig. 10.
Fig. 10. Measurement results and the STD (error bar) ($t\pm \sigma$) of the samples when $\Delta f_{\rm r}=1$ kHz and the spectral range 5 nm. The averaged STD is estimated to be 5.32 nm.
In summary, our proposed simplified d-DCSE provides a fast and high-precision ellipsometry scheme and is more flexible and industrial-friendly, as it requires no complex mutual locking and post digital noise-corrections. Because the repetition rates and offset frequencies of our dual-comb source utilize conventional low-bandwidth independent locking, this simplifies the locking system of the existing dual-comb spectroscopic ellipsometry, reduces the requirements on the use of the environment, and saves the time of data processing. In this article, the theoretical analysis and experiment results demonstrate that the proposed d-DCSE can be operated under the independent locking of repetition rates and offset frequencies of the dual-comb source. As a quantitative evaluation of the measurement performance, the high-precision thin-film thickness determination (1.31 nm) and high reliability (expended uncertainty 13.80 nm ($k=2$)) verify the feasibility of the system. The dynamic ellipsometric results under the transmission setup also manifest the high-speed measurement performance of the proposed d-DCSE. In short, our method provides a convenient and efficient solution for fast dynamic dual-comb spectroscopic ellipsometry instrumentation and practicability.
Funding
National Natural Science Foundation of China (92150104, 51835007).
Acknowledgments
The authors thank Takeo Minamikawa at Tokushima University and Xiuguo Chen at Huazhong University of Science and Technology for the fruitful discussion and comments on the work.
Disclosures
The authors declare no conflicts of interest.
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.
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