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Abstract
We proposed a numerical and accurate quadratic phase aberration compensation method in digital holography. A phase imitation method based on Gaussian 1σ-criterion is used to obtain the morphological features of the object phase using partial differential, filtering and integration successively. We also propose an adaptive compensation method based on a maximum-minimum-average- α-standard deviation (MMAαSD) evaluation metric to obtain optimal compensated coefficients by minimizing the above metric of the compensation function. The effectiveness and robustness of our method are demonstrated by simulation and experiments.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As a phase-contrast imaging method, digital holography (DH) allows for digital recording and numerical reconstruction of the object wave for the quantitative analysis. By using a detector such as a CCD or a CMOS, the interference pattern between the object wave and the reference wave enables noninvasive and label-free imaging. However, the recording setup will induce undesired phase aberrations, including the phase tilt caused by the off-axis geometry as well as the phase curvature and high-order aberrations related to the lenses and microscopic objectives (MOs) [1–3], which must be corrected before accurate analysis of the sample.
Several methods have been utilized to compensate phase aberrations in digital holography. A physical compensation method has been used for this purpose, such as adding an additional MOs in the optical path [4,5], using a telecentric lens structure [6–8], or inserting an electrically tunable/adjustable lens in the illumination path or in the reference optical arm [9,10]. Most subsequent mathematical methods eliminate quadratic phase errors using double exposure and phase shift [11,12]. Compared with physical approaches, numerical compensation methods are more flexible for phase compensation.
Among the numerical aberration compensation algorithms, previous method has used the object-free region of the phase image as the initial phase map. Based on this, the curvature and spherical center of the quadratic phase are calculated by measuring the distance of the optical path components [13–19]. Study also showed that given knowing the phase aberrations, the primary and quadratic term aberrations can be eliminated by rotating the hologram and spectrograms [20,21]. In addition, reconstruction algorithms such as transport of intensity equation (TIE) are used to obtain the optimal reconstruction distance for phase aberration removal [22,23].
Lateral sheer, spectral analysis and least-squares algorithms are typical methods used for aberration elimination that require no prior knowledge [24–26]. Among them, the least-squares algorithm is however prone to overfitting, which can be optimized through the constrained matrix inversion and Zernike polynomials to obtain the quadratic aberration coefficients using ${l_1}$-norm, background segmentation and gradient segmentation [27–31]. Coefficient optimization methods such as deep learning and weighted least-squares can be applied to scenarios where the sample phase has a low impact [32–36]. However, the least-squares method suffers the overfitting problem. The overfitting problem is attributed to the factors including the position, shape, area and height of the sample phase [37,38]. More particularly, we observed that when the area ratio of the sample phase versus the total area of image exceeds 50%, most numerical compensation methods fail to remove the phase aberration.
In this paper, inspired by our observation, we propose a phase imitation method to achieve robust aberration removal. The method is based on the Gaussian $1\sigma $-criterion to obtain the morphological features of the object phase. The phase imitation method is specified as follows: we first partially differentiate the phase along the x and y axes, respectively, filter out the values within the $1\sigma $-criterion, and finally integrate along the x and y axes, respectively. An evaluation metric termed maximum-minimum-average- $\alpha $-standard deviation ($\textrm{MMA}\alpha \textrm{SD}$) metric is proposed to quantitatively evaluate the accuracy of quadratic aberration compensation. We also propose an adaptive compensation method to obtain an optimal compensated coefficient by minimizing the above metric of the compensation function. Our method takes into account the influence of the position, shape, size and height of the object on the solution of the quadratic coefficients, but requires no prior knowledge. Compared to the traditional least-squares algorithm, our method is more robust and can achieve a higher accuracy of phase reconstruction and effectively eliminate the quadratic phase errors. Our method is able to achieve phase reconstruction even given the sample phase area ratio exceeds 50%. The robust phase aberration removal method is able to impact phase imaging research and applications.
2. Phase imitation and compensation method
The phase distribution of the object wave is recorded by the hologram and retrieved with numerical reconstruction in DH. We express the reconstructed unwrapped total phase ${\phi _{tot}}(x,y)$ as
where ${\phi _{obj}}(x,y)$, ${\phi _{abe}}(x,y)$ and ${\phi _{noi}}(x,y)$ represent the error-free object phase (or sample phase), quadratic phase aberrations and phase noise, respectively.We assume that the quadratic phase aberration can be expressed by the Zernike polynomials, thus ${\phi _{abe}}({x,y} )$ can be written as
where ${Z_i}(x,y)$ is the Cartesian form of i-order Zernike polynomial, ${u_i}$ is its corresponding coefficient describing the degree of phase aberration.The polynomial coefficients ${u_i}$ could be acquired by solving linear equations [37] as
As the phase noise ${\phi _{noi}}(x,y)$ is small in magnitude and randomly fluctuates around 0, it has limited effects on the calculation results based on the phase integration operation in the matrix inverse transport sequence process. The least-squares algorithm is typically used to eliminate the quadratic phase aberration ${\phi _{abe}}(x,y)$. However, an overfitting makes it impossible to obtain accurate error-free phase ${\phi _{obj}}(x,y)$ [27]. We term the incorrectly compensated phase as the roughly corrected phase ${\phi _{ls}}(x,y)$, which is obtained from the traditional least-squares algorithm.
Calculation of phase ${\phi _{obj}}(x,y)$ introduces overfitting in the process of solving for polynomial coefficients, for example, if we put ${\phi _{obj}}(x,y)$ into Eq. (3) yields its corresponding coefficient $u_i^{(obj)}$, we can obtain its Zernike fitted surface $\sum\nolimits_{i = 0}^n {u_i^{(obj)}{Z_i}(x,y)}$. Thereby the relation between ${\phi _{obj}}(x,y)$ and ${\phi _{ls}}(x,y)$ can be given by
The crucial task is to compensate for ${\phi _{ls}}(x,y)$ by estimating the Zernike fitted surface $\sum\nolimits_{i = 0}^n {u_i^{(obj)}{Z_i}(x,y)}$. For this reason, we propose a phase imitation method to imitate the morphological features of ${\phi _{obj}}(x,y)$. Based on the $\textrm{MMA}\alpha \textrm{SD}$ metric, a compensation function is then used for accurate compensation of ${\phi _{ls}}(x,y)$.
The proposed phase imitation method consists of four steps. In the first step, we perform the partial differential operation on ${\phi _{ls}}(x,y)$ along the x-axis and y-axis to obtain ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$ and ${{\partial {\phi _{ls}}(x,y)} / {\partial y}}$, respectively. It is followed by representing them as histogram and Gaussian distributions (HGDs), respectively [36]. In the second step, based on the HGD, the partial differential values within the $1\sigma $ range are filtered out using the $1\sigma $-criterion (one-sigma-criterion, OSC). In the third step, integrate along the x-axis and y-axis respectively. In the fourth step, the weighted value $\gamma $ is calculated according to the integration result, and finally the sum of the weighted value $\gamma $ is performed. Hence the obtained imitation phase can be written as
The flat parts of ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$ or ${{\partial {\phi _{ls}}(x,y)} / {\partial y}}$ are equivalent to Gaussian noise with $\mu \approx 0$ and $\sigma = \sigma \{{{{\partial {\phi_{ls}}(x,y)} / {\partial x}}} \}$ or $\sigma \{{{{\partial {\phi_{ls}}(x,y)} / {\partial y}}} \}$, respectively. We filter out the above Gaussian noises using OSC operation, then the obtained imitation phase ${\phi _{imit}}(x,y)$ would mimic the phase distribution characteristics of ${\phi _{obj}}(x,y)$.
Thus, putting ${\phi _{imit}}(x,y)$ into Eq. 3 yields its correspondent coefficient $u_i^{(imit)}$, we construct the compensation function as
In order to evaluate the effectiveness of the compensation function [36], we define a $\textrm{MMA}\alpha \textrm{SD}$ metric as
From Eq. 9, a $\beta$-$\textrm{MMA}\alpha \textrm{SD}$ values are shown to describe the $\textrm{MMA}\alpha \textrm{SD}$ curve characteristics with respect to $\beta $. We could assume that the $\textrm{MMA}\alpha \textrm{SD}$ metric of the compensation function would be minimized if the overfitting is corrected, i.e., $\beta u_i^{(imit)} - u_i^{(obj)} \approx 0$. The optimal standard deviation coefficient $\alpha $ and the compensation coefficient $\beta$ can be acquired by
The effectiveness of the proposed method is demonstrated by simulations and experimental results in Section 3 and Section 4, respectively.
3. Simulation and evaluation
Figures 1(a)–1(c) show the profile features of ${\phi _{tot}}(x,y)$, ${\phi _{abe}}(x,y)$ and ${\phi _{obj}}(x,y)$, respectively, where ${\phi _{tot}}(x,y)$ of Fig. 1(a) is the sum of ${\phi _{abe}}(x,y)$ and ${\phi _{obj}}(x,y)$. The phase aberrations are simulated with 0∼5 orders Zernike polynomials in Fig. 1(b). According to the most practically used quadratic term phase distribution, the coefficients used in the simulation are given in Table 1. As shown in Fig. 1(c), a step-type phase is used with a resolution of 2592(H) × 1944(V) pixels and an area ratio of 63.73%, which is characterized by a stepped and two parabolic surfaces. The roughly corrected phase ${\phi _{ls}}(x,y)$ then is obtained using traditional least-squares algorithm [36], as shown in Fig. 1(d), which shows an obvious distortion.
Fig. 1. (a) Reconstructed unwrapped total phase; (b) quadratic phase aberration; (c) true object phase; (d) roughly corrected phase from traditional least-squares algorithm.
Table 1. Simulated aberrations using Zernike polynomials
Using the proposed phase imitation method, the true distribution characteristics of the object phases can be extracted. Figure 2 illustrates the process of acquiring the imitation phase, where ${\otimes} $ and ${\oplus} $ represent multiplication and addition, respectively. Figures 2(a1)–2(a4) show the phase imitation process along the x-axis. Figure 2(a1) show the distribution of the partial differential values of ${\phi _{ls}}(x,y)$ along the x-axis, i.e., ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$. As shown in Fig. 2(a2), we use the HGD plot for the phase analysis of ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$. Combining the distribution and HGD plot of ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$, we find that the flat part of the partial differential values is equivalent to Gaussian noises with $\mu \approx 0$ and $\sigma = \sigma \{{{{\partial {\phi_{ls}}(x,y)} / {\partial x}}} \}$. From Fig. 2(a3), the above Gaussian noises in ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$ are filtered out using the $1\sigma $-criterion and then integrated along the x-axis. As shown in Fig. 2(a4), the imitation phase in the x-axis direction mimics the step characteristics of the object phase.
Fig. 2. The process of obtaining the imitation phase. (a1) Partial differentiation along the x-axis, i.e., ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$; (b1) partial differentiation along the y-axis, i.e., ${{\partial {\phi _{ls}}(x,y)} / {\partial y}}$; (a2), (b2) are the HGD from Fig. 2(a1) and Fig. 2(b1), respectively; (a3) and (b3) are the results of Fig. 2(a1) and Fig. 2(b1) filtered by the $1\sigma $-criterion, respectively; (a4), (b4) are the phases of integration along the x- and y-axes, respectively; (c) obtained imitation phase.
Figures 2(b1)–2(b4) have the same layout as that of Figs. 2(a1)–2(a4). It shows the phase imitation process along the y-axis. As shown in Fig. 2(b4), the imitation phase in the y-axis direction mimics parabolic morphology from the object phase. Figure 2(c) shows the integral summation phase of the x and y axes with $\gamma $ calculated to be 0.664. Comparing Fig. 1(c) and Fig. 2(c), it can be observed that the imitation phase has the same phase distribution characteristics as object phase ${\phi _{obj}}(x,y)$, which proves the effectiveness of the proposed method.
Figures 3(a)–3(c) show the process of obtaining the optimized compensation phase. Based on the obtained imitation phase, the compensation function is constructed and we use the $\textrm{MMA}\alpha \textrm{SD}$ metric for the evaluation of the compensation effect to obtain the optimal coefficients $\alpha $ and $\beta $. Figure 3(a) shows the fitted phase composed by $\sum\nolimits_{i = 0}^5 {u_i^{(imit)}{Z_i}(x,y)}$. As shown in Fig. 3(b), based on Eq. 8 and Eq. 9, the compensation function is evaluated using the $\textrm{MMA}\alpha \textrm{SD}$ metric as $\alpha $ = -1, 0 and 1. In order to evaluate the accuracy of the compensation, the root mean squared errors (RMSEs) between the compensation function and the error-free object phase is calculated. As $\alpha $= 1, $\textrm{MMA}\alpha \textrm{SD}$ and RMSE have similar curve trends, i.e., they firstly decrease and then increase with the rise of $\beta $. By comparing the minimum points of the curves, $\textrm{MMA}\alpha \textrm{SD}$ and RMSE reach the minimum both at $\beta $=1.22, which proves that the $\textrm{MMA}\alpha \textrm{SD}$ metric robustly evaluates the efficiency of compensation function. The RMSE values of the least-squares algorithm and the proposed method are 3.011 and 0.081, respectively, indicating that the proposed method can improve the accuracy by about 36 times compared to the least-squares algorithm. The result after the accurate compensation is shown in Fig. 3(c). In order to demonstrate the superiority of the proposed method, we also take the phase difference along the trace line from Fig. 1(c), Fig. 1(d) and Fig. 3(c), respectively, as marked by red dashed line. The results shown in Fig. 3(d) indicates the proposed method is able to greatly improve the accuracy of phase correction compared with the least-squares algorithm.
Fig. 3. Phase compensation process and phase comparison. (a) Fitted compensated phase obtained from imitation phase, i.e., $\sum\nolimits_{i = 0}^5 {u_i^{(imit)}{Z_i}(x,y)}$; (b) comparison of $\beta$-$\textrm{MMA}\alpha \textrm{SD}$ plot and $\beta$-RMSE plot; (c) compensated phase obtained from proposed method; (d) comparison of phase profiles along the red dashed line from Fig. 1(c), Fig. 1(d) and Fig. 3(c), respectively.
4. Experimental studies
Comprehensive reflective and transmissive types samples have been tested in our experiment for phase aberration elimination. The acquired phase information is processed using the least-squares algorithm and the proposed method for comparison.
4.1 Measurement of reflective resolution plate
An approximate plane wave is incident on the CCD and interferes with a typical off-axis Michelson interference system shown in Figs. 4(a) and 4(b). The wavelength of the He-Ne laser source is 632.8 nm, and the pixel pitch of the detector is 2.2µm × 2.2µm. In a lens-free Michelson interference system, the CCD has no magnification for the reflection USAF1951 resolution plate with resolution of 1680(H) × 1540(V) pixels as shown in Fig. 4(c). Since the optical path difference between the object and reference beam is difficult to adjust to zero, the reconstructed phase distribution will contain quadratic phase aberration because of the non-ideal reference plane wave used in this experiment. Figures 4(d)–4(f) are the captured digital hologram, reconstructed wrapped phase and unwrapped phase, respectively. There is also obvious distortion introduced by the spherical wave, as shown in Fig. 4(f).
Fig. 4. Experimental setup and acquisition of the total phase. (a) Diagram of digital holographic setup; (b) diagram of experimental setup; (c) reflection USAF1951 sample; (d) recorded hologram; (e) wrapped phase obtained using convolutional reconstruction; (f) obtained unwrapped phase.
Figures 5(a)–5(h) are experimental results by using the proposed method. Figures 5(a) and 5(b) are the rough corrected phase and imitation phase, respectively. Figure 5(c) shows the compensated phase, where the overfitted phase has been eliminated and the resolution plate is thus located on a flat plane. Based on Eq. 8 and Eq. 9, we use the $\textrm{MMA}\alpha \textrm{SD}$ metric to evaluate the compensation function as $\alpha $ = -1, 0 and 1.Figure 5(d) shows that $\textrm{MMA}\alpha \textrm{SD}$ metric reaches its minimum as $\alpha $= -1 or 0 and $\beta $=1.2. In Fig. 5(e), compare to least-squares algorithm, the flatness comparison of the trace lines shows that the proposed method can obtain more accurate phase result than the least-squares method.
Fig. 5. Experimental results of the proposed method. (a) Phase map after traditional least-squares algorithm; (b) imitation phase; (c) phase obtained from proposed method; (d) $\beta$-$\textrm{MMA}\alpha \textrm{SD}$ plot; (e) comparison of phase profiles along the red dashed line from Fig. 4(f), Fig. 5(a) and Fig. 5(c), respectively.
4.2 Measurement of reflective ceramic steps
A Michelson off-axis interferometer with an optical fiber as the light source and a spherical wave transmitted to the CCD with a resolution of 1180(H) × 860(V) pixels as shown in Figs. 6(a) and 6(b). The wavelength of the semiconductor laser is 632.8 nm, the pixel pitch of the detector is 4.65µm × 4.65µm, and the lateral magnification is 2.66 times. As shown in Fig. 6(c), the selected sample is a ceramic step sample with a height difference of 20.2µm (based on the calibration of a coordinate meter). Figures 6(d)–6(f) are the captured digital hologram, reconstructed wrapped phase and unwrapped phase, respectively. Since the optical path difference between the object and reference beam is difficult to adjust to zero, the reconstructed phase distribution will contain quadratic phase aberration because of the spherical wave used in this experiment. The step area ratios of the left and right phases in Fig. 6(e) are 53.30% and 46.70%, respectively. Since the area ratio of the sample phases exceeds 50%, the commonly used numerical compensation methods such as the coefficient optimization method [31,39] and the background segmentation method [36,37] might be unable to achieve an accurate compensation due to insufficient data utilization. For this reason, the proposed method is used to accurately eliminate the aberrations in this experiment and compared using the conventional least-squares algorithm.
Fig. 6. Experimental setup and acquisition of the total phase. (a) Diagram of digital holographic setup; (b) diagram of experimental setup; (c) ceramic step sample; (d) recorded hologram; (e) wrapped phase obtained using convolutional reconstruction; (f) obtained unwrapped phase.
Figures 7(a)–7(g) are experimental results by using the proposed method. Figures 7(a) and 7(b) are the rough corrected phase and imitation phase, respectively. Based on the vertical angle of the step gutter in Fig. 7(a), we find that the residual aberration in the rough corrected phase ${\phi _{ls}}(x,y)$ is not a linearly tilt, and such residual aberration cannot be eliminated using the linear aberration elimination algorithm [31]. From the imitation phase in Fig. 7(b), we can intuitively obtain the step characteristics of the object phase. Figure 7(c) shows the phase results after compensation using the proposed method, and the phase overfitting has been effectively eliminated. The compensation function has been evaluated using the $\textrm{MMA}\alpha \textrm{SD}$ metric as $\alpha $ = -1, 0 and 1. According to the $\beta $-$\textrm{MMA}\alpha \textrm{SD}$ plot in Fig. 7(d), the results of the compensation coefficients can be obtained as $\alpha $=0 and $\beta $=0.88. As the trace lines shown in Fig. 7(e), by comparison, the excellent performance of the proposed method shows the superiority in compensation accuracy.
Fig. 7. Experimental results of the proposed method. (a) Phase map after traditional least-squares algorithm; (b) imitation phase; (c) phase obtained from proposed method; (d) $\beta$-$\textrm{MMA}\alpha \textrm{SD}$ plot; (e) comparison of phase profiles along the red dashed line from Fig. 6(f), Fig. 7(a) and Fig. 7(c), respectively.
4.3 Measurement of transmission microfluidic channel
An off-axis Mach-Zehnder interferometer combined with an inverted microscopy (Olympus CKX53) has been built [36]. The optical diagram and its experimental setup of the digital holographic microscope is shown in Figs. 8(a) and 8(b), respectively. The wavelength of He-Ne laser source is 632.8 nm. The resolution of the CCD is 2592(H) × 1944(V) with single pixel size 2.2µm × 2.2µm. The microfluidic channel was selected as the experimental sample, as shown in Fig. 8(c). The light beam emitted by laser propagates through the collimator and the beam splitter (BS1). The object beam reflecting off the mirror (M) is incident on the resolution plate sample and a 20x microscopic objective is used to image the diffracted field onto the CCD. Using a 20x objective, the reference beam with some phase curvature is incident on the CCD. The beam splitter (BS2) is set to be rotatable to adjust the off-axis recording angle. Since the optical path difference between the object and reference beam is difficult to adjust to zero, the reconstructed phase distribution will contain quadratic phase aberration. Figures 8(d)–8(f) are the captured digital hologram, reconstructed wrapped phase and unwrapped phase. According to the obtained unwrapped phase in Fig. 8(f), the traditional least-squares algorithm and the proposed method are applied to eliminate aberration.
Fig. 8. Experimental setup and acquisition of the total phase. (a) Diagram of digital holographic setup; (b) diagram of experimental setup; (c) microfluidic channel sample; (d) recorded hologram; (e) wrapped phase obtained using angular spectral reconstruction; (f) obtained unwrapped phase.
Figures 9(a)–9(g) are experimental results by using the proposed method. Figures 9(a) and 9(b) are the rough corrected phase and imitation phase, respectively. Since the microfluidic channel occupies 70.87% of the area ratio in the phase diagram, the phase in Fig. 9(a) shows a greater distortion. The phase characteristics of the error-free object phase are effectively imitated in Fig. 9(b) using the phase simulation method. Figure 9(c) shows the phase results after compensation using the proposed method, and the phase overfitting is effectively eliminated. Based on Eq. 8 and Eq. 9, the $\textrm{MMA}\alpha \textrm{SD}$ metric is used to evaluate the compensation function as $\alpha $ = -1, 0 and 1. According to the $\beta $-$\textrm{MMA}\alpha \textrm{SD}$ plot in Fig. 9(d), the results of the compensation coefficients can be obtained as $\alpha $=1 and $\beta $=1.58. Compared to the simulation in Section 3, the microfluidic channel has a larger area ratio, thus making $( - \mu - \alpha \cdot \sigma )$ term in the $\textrm{MMA}\alpha \textrm{SD}$ metric have a greater slope with the rise of $\beta $, leading to a monotonically increasing $\textrm{MMA}\alpha \textrm{SD}$ curve as $\alpha $= -1 and 0, respectively. Based on the obtained $\alpha $ and $\beta $, as the trace lines shown in Figs. 9(e) and 9(f), it can be clearly observed that the proposed method is able to robustly solve the overfitting problem compare with the conventional least-squares algorithm.
Fig. 9. Experimental results of the proposed method. (a) Phase map after traditional least-squares algorithm; (b) imitation phase; (c) phase obtained from proposed method; (d) $\beta$-$\textrm{MMA}\alpha \textrm{SD}$ plot; (e) comparison of phase profiles along the red dashed line from Fig. 8(f), Fig. 9(a) and Fig. 9(c), respectively; (f) comparison of phase profiles along the blue dashed line from Fig. 8(f), Fig. 9(a) and Fig. 9(c), respectively.
5. Discussion
The limitations of the phase imitation should be discussed. For example, phase imitation relies on differentiation, filtering, and integration. For this reason, for complex shapes or phases, errors might be accumulated in the phase integration process. For compensating the potential error accumulation, the compensation function based on the $\textrm{MMA}\alpha \textrm{SD}$ metric is used to prevent the cumulative errors.
Another problem is the computation efficiency. As the area ratio of the sample phase increases, the slope of $( - \mu - \alpha \cdot \sigma )$ term increases in the $\beta $-$\textrm{MMA}\alpha \textrm{SD}$ plot with the rise of $\beta $, leading to an increase in the value of the optimized coefficient $\alpha $. Based on the fact that the value of $\alpha $ is positively related to the area of the sample phase, by estimating the area ratio of the sample phase, we do not need to calculate the $\textrm{MMA}\alpha \textrm{SD}$ metric at $\alpha $=-1,0 and 1, thus the proposed method would further reduce the computational cost. For example, Intel Xeon CPU with 3.30 GHz and 32 G RAM was used for the research, the overall data processing of the proposed method can be completed within 0.469 second per 106 pixels.
6. Conclusion
In this study, we demonstrate a phase imitation method based on the Gaussian $1\sigma $-criterion and an adaptive phase compensation method based on $\textrm{MMA}\alpha \textrm{SD}$ metrics. Using partial differential operations and $1\sigma $-criterion filtering, the imitation phase obtained by integral summation can effectively mimic the morphological characteristics of the object phase. The $\beta $-$\textrm{MMA}\alpha \textrm{SD}$ plot is created to obtain the optimized standard deviation coefficient $\alpha $ and compensation coefficient $\beta $. By extending the fitting model to include other aberration coefficients by extending Eq. 3, high-order aberrations could be removed using our proposed technique. Simulation and experimental results suggested that the proposed method can achieve higher accuracy compared with the traditional least-squares algorithm, especially our method is suitable for scenarios with the sample phase area ratio exceeding 50%.
Appendix
- (1). Partial differentiation of the left and right sides of Eq. 4 for the x and y axes, respectively.$$\begin{array}{l} \frac{{\partial {\phi _{ls}}(x,y)}}{{\partial x}} \approx \frac{{\partial {\phi _{obj}}(x,y)}}{{\partial x}} - \frac{{\partial \left[ {\sum\nolimits_{i = 0}^5 {u_i^{(obj)}{Z_i}(x,y)} } \right]}}{{\partial x}}\\ = \frac{{\partial {\phi _{obj}}(x,y)}}{{\partial x}} - [{u_1^{(obj)} + 4xu_3^{(obj)} + 2yu_4^{(obj)} + 2xu_5^{(obj)}} ]\approx \frac{{\partial {\phi _{obj}}(x,y)}}{{\partial x}} \pm \frac{C}{{{{10}^3}}} \end{array},$$$$\begin{array}{l} \frac{{\partial {\phi _{ls}}(x,y)}}{{\partial y}} \approx \frac{{\partial {\phi _{obj}}(x,y)}}{{\partial y}} - \frac{{\partial \left[ {\sum\nolimits_{i = 0}^5 {u_i^{(obj)}{Z_i}(x,y)} } \right]}}{{\partial y}}\\ = \frac{{\partial {\phi _{obj}}(x,y)}}{{\partial y}} - [{u_2^{(obj)} + 4yu_3^{(obj)} + 2xu_4^{(obj)} - 2yu_5^{(obj)}} ]\approx \frac{{\partial {\phi _{obj}}(x,y)}}{{\partial y}} \pm \frac{C}{{{{10}^3}}} \end{array},$$where $C$ denotes an arbitrary constant. Since the difference between the partial differential values of ${\phi _{obj}}(x,y)$ and ${\phi _{ls}}(x,y)$ is a constant term at the 10−3 level, thus they have similar distribution characteristics.
- (2). Distribution characteristics of ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$ and ${{\partial {\phi _{ls}}(x,y)} / {\partial y}}$.$$\scalebox{0.92}{$\displaystyle\mu \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial x}}} \right\} \approx 0,\,\min \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial x}}} \right\} \ll{-} \sigma \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial x}}} \right\},\,\sigma \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial x}}} \right\} \ll \max \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial x}}} \right\},$}$$$$\scalebox{0.92}{$\displaystyle\mu \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial y}}} \right\} \approx 0,\,\min \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial y}}} \right\} \ll{-} \sigma \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial y}}} \right\},\,\sigma \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial y}}} \right\} \ll \max \left\{ {\frac{{\partial {\phi_{ls}}(x,y)}}{{\partial y}}} \right\}.$}$$
The flat parts of ${{\partial {\phi _{ls}}(x,y)} / {\partial x}}$ and ${{\partial {\phi _{ls}}(x,y)} / {\partial y}}$ are equivalent to Gaussian noises with $\mu \approx 0$ and $\sigma = \sigma \{{{{\partial {\phi_{ls}}(x,y)} / {\partial x}}} \}$ or $\sigma \{{{{\partial {\phi_{ls}}(x,y)} / {\partial y}}} \}$, respectively.
Funding
National Key Research and Development Program of China (2020YFE0204600); National Natural Science Foundation of China (61975112).
Acknowledgments
This project was supported by the National Key R&D Program of China (No.2020YFE0204600) and the National Natural Science Foundation of China (No.61975112).
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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