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Abstract
Ellipsometric measurement of transparent samples suffers from substrate backside reflection challenges, including incoherent and partial superposition issues. The recently developed angle-resolved ellipsometry (ARE) can naturally eliminate the backside reflections of substrates with a micro-spot equivalent thickness or thicker; however, for thinner substrates, ARE working with general incoherent backside reflection models shows significant inaccuracy or measurement failure. In this paper, an incoherent partial superposition (IPS) model is proposed to characterize the optical superposition effect between the frontside and uncertain backside reflections from an unknown substrate. IPS introduces a cosine-like correction of the backside reflection, corresponding to the overlapping-area change of backside and frontside reflections along with incident angles. Benefiting from ARE’s wide-angle spectral imaging capability, IPS achieves single-shot measurement of thin film thicknesses on transparent substrates of unknown thickness. An ARE system was built and calibrated regarding the linear relationship between the cosine-corrected angular frequencies and substrate thicknesses. Then, commercial ITO films on glasses of different thicknesses ranging from 200 to 1000 µm were measured. Experimental results show that IPS-ARE results in a root-mean-square accuracy error of ∼1 nm in film thickness measurement and provides a ∼77% error reduction from general incoherent backside reflection models.
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1. Introduction
With the deepening application of flexible electronics in advanced display and sensing, transparent materials, such as polymers and glasses, are widely used as device substrates, benefiting from their fascinating, stable surface quality, mechanics, and physical properties [1,2]. These substrates offer mechanical support for attached films, protecting them from physical and chemical damage while also contributing to device performance. For example, polymer substrates [1,3] with high water/oxygen permeability and mechanical deformability are extensively used in wearables for high breathability and flexibility requirements. Flexible substrates also enable low-cost manufacturing processes. For example, roll-to-roll processing of ultra-thin (<200 µm) glass substrates has been developed recently for indium-doped tin oxide (ITO) layers deposition [4,5]. However, roll-to-rolls result in residual stress and thickness changes of films due to mechanical bending in processing with varying thermal expansion of material layers [6,7]. Such property changes further result in performance changes in final device products [8]. Therefore, the characterization of film structures, especially the changes in their properties during processing, is essential for electronics manufacturing.
Optical methods have shown great potential in film measurement due to their non-destructiveness, high precision, and high speed. Transmission-type methods, like transmission ellipsometry and transmission spectrometry, exhibit excellent sensitivity to substrate signals, while the signal from the coated thin films is relatively less sensitive, which limits the applications of these methods to bulk material [9]. Hence, reflection-based methods, like spectral reflectometry (SR) and spectral ellipsometry (SE), are common choices for thin-film structure measurement. SR [10] measures the reflected light intensity of a sample and determines the thickness and refractive index by fitting the reflectance, which makes it sensitive to the intensity fluctuations of the used light source. In contrast, SE [11], determining thin film parameters by measuring the change of polarization states of the reflected spectra, has shown high robustness and accuracy and has gradually become the primary method of thin film measurement. However, transparent substrates pose a challenge for these reflection-based methods. Reflections from the rear surface of the transparent substrate will also be captured and mixed with the frontside signals. This behavior complicates optical modeling and data analysis, thus reducing measurement accuracy and efficiency [12]. Generally, a substrate layer is optically thick, where backside reflections are incoherent with the reflections from the frontside. Besides, an uncertain off-axis offset and overlap between the backside and frontside reflection beams exists, specific to incident angles and substrate thicknesses. This results in an unknown portion of partial superposition of backside and frontside reflections. The incoherent partial reflection superposition depends on incident angles, the aperture configuration of a detector, as well as the thickness and optical properties of a substrate.
There are generally two approaches to address the incoherent partial superposition challenges: suppressing backside reflections or considering backside reflections using an enhanced optical model. For example, back-surface roughening [13,14] is the most common choice to suppress the backside reflection by scattering the reflected light away since it is suitable for various materials and relatively easy to perform. However, it causes damage to samples, and it is difficult to deal with thin, brittle, or soft materials. Back-surface index-matching using liquids or tapes [15,16] is another experimental approach to suppress the backside reflection by absorbing or scattering the light entering an index-matched medium. Index-matching requires a firm contact and perfect refractive index consistency between the substrate and index-matched medium, which is usually difficult to satisfy in practice. Analogously, exploiting the absorption of the substrate in a specific wavelength band can also eliminate backside reflections [17]. However, this method is valid only for special substrate materials or a unique light source, such as a UV light. A thick or wedged substrate can also spatially separate back reflections away from a detector [18]. However, the separation effect is affected by substrate thicknesses and beam spot sizes. If a substrate is thinner than the diameter of a probing beam, the separation method fails [19]. Focusing an incident beam to a small spot has been found to be a natural, beneficial solution for backside reflection separation. Liu et al. [20] focused an SE beam to a micro-spot, making a general SE able to measure thin films with a transparent substrate of hundreds of microns thicker. However, a focus configuration introduces a frustrated angular spread of incident lights, thus resulting in complicated angular calibration issues. To this end, the recently developed angle-resolved ellipsometry (ARE) is developed based on highly focused back-focal-plane (BFP) imaging and exact wide-angle spectral calibration [21,22]. The nature of focusing spectral imaging makes ARE an excellent potential for measuring thin films on transparent substrates.
Although ARE can potentially eliminate backside reflections for substrates with thickness exceeding the micro-spot size, there is still the challenge of backside reflection for samples with thinner substrates. A potential solution is building an enhanced optics model, including incoherent and partial backside reflections for ellipsometric measurement. Many researchers worked on exact backside reflection model construction, but most focused on solving optical incoherence-induced problems [23,24]. For instance, Forcht et al. [25,26] used Stokes vectors and Mueller matrices to model incoherent superposition for ellipsometric measurements. However, they ignored the changing overlaps between backside and frontside reflection beams specific to incident angles, substrate thicknesses, substrate refractive index, and system aperture. Hong et al. [27] proposed an equivalent model to describe the backside reflection caused by flexible plastic substrates using a retarder Mueller matrix, where the detection wavelength was limited to an ultraviolet-visible band, and only the Mueller components ${m_{12}}$ and ${m_{21}}$ were investigated. Xia et al. [28] introduced a model to compensate for the wavelength-dependent loss of substrate backside reflections. However, both models [27,28] have measurement accuracy and reliability shortcomings. The former ignored the backside reflection-induced oscillations of the measured signal, and a stringent selection of data was needed for data analysis. The latter was limited to quasi-normal incidence cases. Another crucial shortcoming of the above methods is that the off-axis offset between the backside and frontside beams is not considered. A beam offset results in only a portion of the backside reflection light spatially being overlapped with the frontside beam and being detected. Rappich et al. [29,30] built an overlap fraction-assisted partial superposition model for free-standing glass high-sensitivity characterization under a combination of multi-angle ellipsometric analysis and transmission measurement. However, the overlap fraction, defined as the ratio of the overlap area to the detector aperture, changes with system configurations, complicating the data analysis and making measurements inefficient, especially in unifying the overlap fraction at different incident angles and set-ups. Overall, these enhanced optical models face either reliability or efficiency shortcomings. Therefore, a novel model is urgently needed to measure thin films on transparent substrates accurately and efficiently.
To address the transparent substrate-induced measurement problem, an incoherent partial superposition (IPS) model-assisted single-shot ARE method is proposed. The IPS model simplifies and unifies the change of the overlap area with incident angles in Rouch's model and effectively combines the benefits of ARE’s wide-angle spectral measurement capability, enabling rapid measurements of thin films on transparent substrates. IPS introduces a cosine-like correction of backside reflections, which are determined by the overlap area changes of the frontside and backside-reflected light along with incident angles. After calibrating a linear relationship between the angular frequencies of cosine-like correction functions and substrate thicknesses, IPS-ARE has been validated by measuring commercial ITO films on glass substrates of different thicknesses ranging from 200 to 1000 µm. This work provides a novel backside reflection model suitable for ARE, enabling rapid and accurate measurement of thin films with transparent substrates. ARE provides a wide range of incident angles and has excellent potential for thin film measurement on transparent substrates. When measuring samples with thicker substrates (e.g., over hundreds of micrometers), ARE can naturally eliminate backside reflections at a large angle of incidence, and the IPS model degrades to a backside reflection-free model. For thinner substrates, the IPS model can compensate backside reflections. Therefore, IPS enables single-shot measurement of transparent films with various substrate thicknesses.
2. Method
2.1 Angle-resolved ellipsometry
Figure 1(a) depicts a system layout of the angle-resolved ellipsometry, which comprises three parts: an illuminating source, a measuring probe, and a detection unit. The collimated light is polarized linearly by passing through a linear polarizer and focused on the BFP of the objective lens. The polarized light, varying with azimuthal angles $\varphi $, incidents on the measurement sample with various angles of incidence $\theta $ through the objective lens. The light, reflected with the same angles, is recollected by the objective lens, and refocused on the BFP. After modulated by an analyzer (another linear polarizer), the BFP image is captured. The azimuthal change of annular BFP image data is utilized for calculating the ellipsometric parameters. To address systematic errors like light source non-uniformity, we utilize the BFP reflectance, calculated by referencing a known sample, rather than the intensity signal, for extracting the ellipsometric parameters.
Fig. 1. Principle of ARE. (a) System configuration, (b) the BFP reflectance of 20 nm SiO2 in the polar coordinate system, and (c) multi-parameter inversion by fitting ${m_{21}}\; \& \; {m_{33}}$.
The principle of BFP imaging stems from the phase-transforming properties of spherical lenses. Every radial point of the BFP corresponds to a unique reflecting angle from the sample. The angular range of $\theta $ is determined by the numerical aperture (NA) of the objective lens from $- {\sin ^{ - 1}}NA$ to ${\sin ^{ - 1}}NA$. The intensity field of BFP can be expressed as follows:
where ${S_P}$ is the Stokes vector of the light leaving the polarizer, ${\mathrm{\Gamma }_A}$ is the first row of the Mueller matrix of the analyzer, and ${M_s}$ is the Mueller matrix of the sample, ${M_T}$ represents the reflection-induced variation of the coordinate system, and R is a coordinate rotation matrix. If the polarizing angles of both linear polarizers are fixed, e.g., 0° and 45° respectively in this study, and the sample to be measured is isotropic, the intensity distribution can be rewritten in the form of:Ellipsometric parameters of the sample can be extracted from the annular data using the Fourier transform or least-square fitting [31,32]. Here by converting the BFP image from the Cartesian coordinate system to a polar coordinate system ($\theta ,\varphi $), ellipsometric parameters at full incident angles can be calculated through a single Fourier transform. A simulated BFP image of a 20 nm SiO2 thin film is illustrated in Fig. 1(a). The BFP reflectance image in Fig. 1(b) is expanded along the direction of azimuthal rotation and transformed to polar coordinates ($\theta ,\varphi $), where $\theta $ ranges from $0^\circ $ to ${\sin ^{ - 1}}NA$. Notice that the azimuthal angles $\varphi $ can be further extended to integer multiple periods, e.g., $10\pi $, for a robust calculation. A single Fourier transform allows the calculation of the Fourier coefficients in Eq. (2) at full incidence angles, thus enabling fast acquisition of the Mueller matrix elements ${m_{ij}}$. Subsequently, film parameters ${p_f}$, can be solved by fitting the angle-resolved Mueller elements to a theoretical model using the non-linear least squares multivariate regression:
2.2 Optical modeling of thin films with transparent substrates
The key to modeling the optics of thin films with a transparent substrate is how to describe the backside reflection signal. As shown in Fig. 2(a), beams reflected from the backside are displaced with respect to the frontside reflection beam, and the offset varies with the incident angle, substrate thickness, and refractive index. Consequently, the overlap, denoted as the red area in Fig. 2(a), between the reflected two beams changes accordingly. Here, the system aperture is assumed to be the same size as the frontside beam. Usually, a thin film thickness d is much less than a substrate thickness h, the off-axis displacement effect caused by coated thin films is ignored. Thus, the offset ${z_j}$ for the $j$th backside reflection beam is given by:
where ${\theta _0}$ is the incident angle and ${\theta _s}$ is calculated from Snell’s law by ${n_0}\sin {\theta _0} = {n_s}\sin {\theta _s}$. The conventional off-axis type ellipsometry, detecting frontside and backside reflected mixed signals, cannot directly separate this spatial overlap and requires aids (e.g., multi-angle analysis) to achieve the measurement [29,30]. On the contrary, ARE's wide-angle imaging capability enables the detection of the variations of offset ${z_j}$ with the incident angles, making it highly suitable for analyzing the offset-induced influence.Fig. 2. Optical models of thin films with transparent substrates. (a) Spatial separation of frontside and backside reflection beams at a constant incident angle, (b) the simulated angle-resolved reflectance of 500 µm fused silica plate (top), and the change of backside reflections with incident angles (bottom), and (c) the comparison of angle-resolved reflectance from different cases.
Figure 2(b) depicts a simulated BFP reflectance image of a 500 µm free-standing fused silica glass plate, and the two concentric circles in the figure indicate the presence or absence of reflections, with their radii corresponding to the respective cut-off angles determined by the system aperture, glass thickness, and glass refractive index. For instance, the cut-off angle denoted as ${\theta _M}$ at the radius of the inner circle determinates the maximum angle for inclusion of backside reflected light. The data enclosed within ${\theta _M}$ represent the reflected signals from both the frontside and backside, while the annular data outside ${\theta _M}$ exclusively represent the reflected signals only from the frontside. Thus, there are three scenarios in Fig. 2(b): entire backside reflection, partial backside reflection, and no-backside reflection. As shown in Fig. 2(b) (bottom), the red area, representing detected backside reflections, diminishes from a full circle at a normal incident angle as the angle increases until it disappears. Conventionally, these cases require different modeling approaches. Yet, most of the literature has typically focused on only two cases: entire backside reflection or no-backside reflection, neglecting the scenario where partial backside reflection is detected. Each radial line in the BFP image represents the angle-resolved reflectance (ARR) along the current azimuth, exemplified by $\varphi = 100^\circ $ in Fig. 2(c). The ARR calculated from the entire backside reflection (green line) and no-backside reflection (red line) are significantly different from partial backside reflection (blue line), indicating the gradual decrease in backside reflection with increasing incident angle. Thus, ARE detects changes in backside reflection at different incident angles, offering the potential to unify reflection modeling by incorporating an angular correction. The degree of the partial overlap in Fig. 2(a) is defined as the ratio of the red area ${A_0}$ to the area of the reflected beam of radius b and expressed as a fraction ${F_z}$, and ${A_0}$ is determined by calculating the overlap of two circles of radius b with center distance ${z_j}$:
Fig. 3. (a) Fraction ${F_z}$ changes with incident angles under different glass thicknesses, and (b) the linear relationship between $\omega $ and h.
When a semi-infinite thick substrate assumption is chosen, the detected reflectance ${\boldsymbol{r}_f}$ without backside reflection can be described by the transfer matrix method or the multiple-beam interferometry [34], e.g., for a simple single layer thin film:
In addition, if the first backside beam is taken as a reference, i.e., the rear surface of the substrate is set at the front focal plane of the objective lens, the frontside reflection is relatively offset, and the above equation is rewritten as:
The Mueller matrix ${M_s}$ of the film with transparent substrate can be derived from the above coherency matrix $\mathcal{R}$t.
3. Experiments
A homemade angle-resolved ellipsometer was built to verify the proposed method with an objective lens of 0.9 NA and two Glan-Taylor linear polarizers. Detailed equipment information and system calibration can be referred to in our previous work [32,36]. The carefully calibrated incident angle ranged from -61.4 to 61.4 deg. The illumination spot was around 54.91 µm measured with the assistance of 1200 lines/mm grating imaging, and the number of detected lines was around 65.9.
3.1 Calibration
The proposed IPS model provides an accurate description of the backside reflection, yet the additional factor ${f_h}$ introduces an unknown correction-frequency $\omega $. Therefore, calibration is needed to probe correction-frequency $\omega $ which is related to substrate thickness h.
As shown by the dashed line in Fig. 4(a) (top), an axial displacement ${d_f}$ can likewise produce the same offset, where ${d_f} = h\tan {\theta _s}/\tan {\theta _0}$ [37]. This means the signal variation of backside reflection with thicknesses in the BFP can be analyzed by axially moving an opaque sample, e.g., a bare silicon. The BFP signal at the focal position is treated as the ideal backside reflection (${\boldsymbol{R}_b}$), while signals at different axial positions represent partial backside reflections (${f_{{d_f}}}{\boldsymbol{R}_b}$) corresponding to various substrate thicknesses. Thus, the ratio between them can be used to calibrate the change of backside reflection. A bare silicon with a native oxide layer of 1.85 nm was used, and BFP images were captured at various axial displacements in 10 $\mathrm{\mu}\textrm{m}$ steps, starting from the focal plane. These defocus intensities were divided by the in-focus intensity to determine the change in backside reflection with an axial displacement. Figure 4(a) (bottom) illustrated the ratio images (${f_{{d_f}}}{\boldsymbol{R}_b}/{\boldsymbol{R}_b}$) of 0, 50, and 100 $\mathrm{\mu}\textrm{m}$ axial positions, the image radius decreased with increasing axial displacement, which was consistent with that the maximum angle of detected reflections decreased with sample defocusing [38]. Figure 4(b) showed the radial cross-section lines of the ratio images at different axial positions ranging from 50 to 250 $\mathrm{\mu}\textrm{m}$. Each curve was fitted by a function ${f_{{d_f}}} = S\cos \omega \theta $ in dash line, where S was an additional damping factor ($S\sim 1$) that could describe the change of normal incident intensity caused by axial displacement. These curves gradually contracted with increasing displacement, indicating that the BFP reflectance varied monotonically and regularly with defocus. The fitting results were shown in Fig. 4(c-d). The damping S gradually decreased with increasing axial displacement, while the angular frequency $\omega $ increased. Notably, in the [0, 250] $\mathrm{\mu}\textrm{m}$ displacement, there was a robust linear relationship between $\omega \; \& \; {d_f}$, with R-square of 0.9995, which was consistent with the simulation result in Fig. 3(b).
Fig. 4. Angle correction calibration. (a) Geometric diagram between $h\; \& \; {d_f}$, (b) the ratios of different axial displacements, (c) the fitting results and (d) the linear relationship between $\omega \; \& \; {d_f}$.
In Fig. 4(b), the cut-off angle ${\theta _M}$ of the BFP image decreased with increasing axial displacement, indicating that the ratio of $\tan {\theta _s}/\tan {\theta _0}$ changed during the above defocusing process and the relationship between $h\; \& \; {d_f}$ couldn't be calculated directly. An additional process was needed to calculate the relationship between $\omega \; \& \; h$. A 300 µm fused silica glass plate was measured, and the measured BFP reflectance was shown in Fig. 5(a). The measured ARR image aligned with the simulation result in Section 2.2, where ${A_b}$, within the cut-off angle ${\theta _M}$ (inner circle), represented the signal with partial backside reflection, and ${A_f}$ was the signal with no-backside reflection. The refractive index of the glass was measured by a commercial ellipsometer of 1.457 (@632.8 nm). The IPS model derived in Eq. (10) was used to fit the angular frequency $\omega $ with ${A_b}$ data, resulting in a value of 0.465, the result in Fig. 5(b) presented an excellent fit. Finally, the relationship between $\omega $ and h was established: $\omega = 0.00147\cdot h$. In Fig. 5(c), reflectance calculated by the IPS model in pink and the no-correction model from Eq. (8) in red were compared with the angle-resolved reflectance of the BFP image along $\varphi ={-} 30^\circ $ in blue. When the angle was below ${\theta _M}$, the reflectance of IPS exhibited an excellent fit compared with the entire backside reflections model, while the angle exceeded ${\theta _M}$, the reflectance from the no-backside reflection model aligned well with the measured data, confirming the absence of backside reflection in ${A_f}$.
Fig. 5. Fitting results of a fused silica glass. (a) Fitted Mueller elements by the IPS model, (b) the measured BFP reflectance and (c) comparison of reflectance between the no-correction model and IPS model.
3.2 Experiment results
Five transparent film samples with different substrate thicknesses were measured, and each sample was repetitively measured ten times for accuracy verification. The samples were single-layer ITO films coated on different glass substrates by magnetron sputtering. The glass thickness ranged from 200 to 1000 µm. The accuracy and precision of measurement were then evaluated against a commercial spectroscopic ellipsometer: SE-VM-L (Wuhan Eoptics Technology Co. Ltd., Wuhan, China). For accurate measurements, the sample's rear surface was well roughened to eliminate backside reflection effects before ellipsometry measurements, and measurement results were in Table 1. The refractive indices utilized for thickness fitting in IPS-ARE were derived from SE.
Table 1. Comparison of film thickness measurements from different models.
A sample of 1000 µm substrate (No. 5) was first measured with ARE to validate its capability in measuring transparent films with a thick substrate. In Fig. 6(a), two concentric circles were present in the image; the area enclosed by the dashed line contained backside reflections, while the region between the two circles represented the signal without backside reflections. The angle resolved-ellipsometric parameters calculated from the no-backside reflection signal were fitted using the model built in Eq. (6) and Eq. (11), yielding the results depicted in Fig. 6(b). The excellent agreement between the model's calculations and the measured data confirmed the reliability of our method.
Fig. 6. (a) Measured BFP image of sample No.5, the backside reflection was limited in the dotted circle and (b) fitting results of Mueller elements calculated from the no-backside reflection data.
Figure 7(a) presented sample images of No.1 (200 µm) and No.2 (300 µm), with their right area roughened. Figure 7(b) detected their corresponding BFP images, and radii of the inner circles in the two BFP images indicated that cut-off angles decreased with the increase of substrate thickness, which is consistent with the change of backside reflection with substrate thicknesses in Fig. 4(a). Three different fitting models were compared: the conventional model with entire backside reflections (dotted lines), the IPS model with a constant $\omega $ (dot-dash lines), and the IPS model with an unknown $\omega $ (dashed line) with data enclosed in the inner circle, fitting results of the two ITO films with 200 µm and 300 µm substrates were illustrated in Fig. 7(c) and (d), respectively. The constant $\omega $ calculated by the calibrated linear relationship were 0.294 and 0.441 for (c) and (d), respectively. The unknown $\omega $ was fitted simultaneously with the thin film thickness. Both IPS model fitting strategies provided a better fit compared to the conventional uncorrected model, indicating the superiority of our model. Additionally, although no-backside reflection signals of these two samples were limited by narrow angular spectra, preventing direct calculation as in sample No. 1, they hold the potential to be fitted using a conventional model after spectral modulation [36]. These results could complement IPS model calculations, potentially enhancing measurement accuracy.
Fig. 7. Fitting results for No.1 and No.2 ITO films. (a) Two films with right area roughened. (b) Measured BFP images of ITO films with 200 µm (left) and 300 µm (right) substrates, and their fitting results of three different models in (c) and (d), respectively.
Figure 8(a) and (b) displayed the fitting results of ten repetitive experiments for samples No.3 (400 µm) and No.4 (550 µm), respectively. Among the three fitting strategies, the conventional model (circle marker) performed worst, while the IPS model with a constant $\omega $ (diamond marker) exhibited a minor error, and the IPS model fitting both thickness d and angle frequency $\omega $ (star marker) showed the best accuracy. The three models shared the same error trend in ten repeated experiments, indicating that our model owned equivalent robustness to the conventional model. The measurement results from the three fitting strategies in Fig. 8(b) displayed an overall offset from the reference value and an inadequate accuracy compared to other samples, likely due to the narrower angular spectra of backside reflections determined by the substrate thickness.
Fig. 8. Results of ten repetitive experiments for samples No.3 (a) and No.4 (b); their substrate thicknesses were 400 µm and 550 µm, respectively.
Table 1 compared thickness results from ten repetitive experiments on ITO films with various substrates. The strategy involving the simultaneous fitting of $d\; \& \; \omega $ exhibited a minor absolute error among the three fitting approaches, displaying a root-mean-square (RMS) error of 0.543 nm. This represented a significant 77% reduction compared to the conventional method (2.367 nm). Absolute errors from ten repetitive experiments for four samples (No.1∼4) and their standard deviations (STD) were shown in Fig. 9. The STDs of thickness errors calculated by the three fitting strategies in repeated experiments with multiple samples were close, indicating their equal robustness.
Fig. 9. Absolute errors from ten repetitive experiments for four samples (No.1∼4) and their standard deviations.
The experimental results validate that IPS-ARE achieves high-robustness thickness measurements of thin films on transparent substrates with various thicknesses. In addition, the proposed method has a natural advantage for measurement challenges introduced by inhomogeneity and non-planar surfaces of flexible films. The used high NA objective lens enables micro-spot measurement, offering benefits in reducing the effect of film inhomogeneities. When the measured surface is non-flat, the detected angle-resolved signal changes. For instance, a tilted sample with an incline angle ${\theta _t}$ shown in Fig. 10(a) results in a reduction of the range of reflected angles detected within the principal plane (formed by the system optical axis and the sample surface normal), corresponding to a decrease in the maximum radius from ${r_o}$ to ${r_t}$. Consequently, the captured BFP reflectance image changes from a circle to an ellipse, as illustrated in Fig. 10(b), and the minor axis of the ellipse, represented by a bidirectional arrow, lies in the principal plane. The normal direction of the sample surface can be determined by the direction of the ellipse's minor axis (${\varphi _t}$) and the corresponding change in maximum radius (${r_o} - {r_t}$). This enables fast and accurate adjustment of the sample position to ensure that the measured surface is flat. IPS-ARE addresses the measurement challenges posed by flexible substrates, making it a promising technique for measuring flexible films.
Fig. 10. Effects of sample surface tilting. (a) Changes of reflected angles, (b) the BFP reflectance image of a tilted SiO2 sample.
4. Conclusions
In this work, we provide a novel backside reflection model suitable for ARE by introducing a cosine-like correction for the backside reflection, enabling rapid measurement of thin films with transparent substrates. The degree of overlap is determined by the variation of the detected signal with the incident angle and the NA of the objective lens. Subsequently, the IPS model is accurately established using the Mueller matrix to describe the optical response of thin films with transparent substrates. A model reliability validation experiment with a self-constructed ARE was carried out. Calibration was first conducted to establish a linear relationship between the angular frequency of the cosine-like function and the thickness of the substrate, based on which we measured five ITO films on glasses with different thicknesses ranging from 200 to 1000 µm. Results show that the proposed model results in a root-mean-square absolute accuracy error of ∼1 nm in film thickness measurement and provides a ∼77% error reduction approximately from the generally used model.
Funding
National Natural Science Foundation of China (52075206); National Key Research and Development Program of China (2023YFB4606000); Knowledge Innovation Program of Wuhan-Basic Research.
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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