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Abstract
A novel super-resolution imaging technique based on the minimum photon flux (MINFLUX), can achieve nanometer-scale localization precision and sub-5-nm imaging. However, aberrations can affect the localization performance and degrade the quality of reconstructed images. In this study, we analyze the effects of different low-order aberrations on the MINFLUX system through both theoretical limits and Monte Carlo methods. We report that 1) defocus and spherical aberration have little effect on 2D localization performance, whereas astigmatism and coma have significant negative effects; 2) system aberrations that can be measured in advance cause changes primarily in the magnitude and angular uniformity of localization precision, whereas sample-induced aberrations that cannot be a priori introduce large biases and reduce localization accuracy.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The emergence of super-resolution optical microscopy, such as stimulated emission depletion (STED) microscopy [1], structured illumination microscopy [2], and single-molecule localization microscopy (SMLM) [3,4], has broken through the limits of optical diffraction. Recently, a newly proposed super-resolution imaging technique based on minimum photon flux (MINFLUX) has achieved approximately 1-nm localization precision and sub-5-nm imaging, essentially breaking new ground for observing the dynamics, distribution, and structures of macromolecules in living cells and beyond [5].
In super-resolution microscopy, aberrations cannot be avoided owing to the complexity of the optical setups. In such setups, low-order aberrations can have dominant effects [6]. They change the beam wavefront and effective point spread function (PSF) regarding aspects such as size, contrast, and symmetry, and thus are detrimental to the performance of the localization and final imaging.
Aberrations are caused primarily by two aspects [7]: 1) mechanical errors in the optical system, for example, a machining error of the optical components, or a not well-aligned optical system; and 2) refractive index mismatches, including the non-uniform refractive index of the specimen itself, and the refractive index difference between the sample and the immersion medium of the objective lens. The first category represents system aberrations; these can be reflected whole. Most system aberrations can be effectively corrected using optical path adjustment and adaptive optics. The latter category represents sample-induced aberrations; these are generated in the optical path of the objective-sample section and are difficult to measure and compensate for in advance.
Aberrations in typical super-resolution imaging techniques can affect system performance in many ways. In a STED microscope, based on its confocal scanning microscopic architecture, such aberrations cause the focus of the excitation beam to diffuse, thus leading to increased background fluorescence interference [8]. However, understanding the effects of aberrations on the depletion beam is more crucial for determining the imaging properties of STED microscopes. The presence of different aberrations affects the shape, peak intensity, zero intensity, and full width at half maximum (FWHM) of the depletion beam, thereby deteriorating the image quality. In the process toward the final image reconstruction with an optical system of a standard SMLM, aberrations have major effects on the data acquisition and localization procedures [6]. They distort the PSF of the system and reduce the fluorescence intensity, consequently decreasing the signal-to-noise ratio and spatial resolution [9].
As a novel concept in super-resolution imaging, MINFLUX has a localization principle that is different from the conventional fitting localization, and the effects of optical aberrations on the localization of MINFLUX super-resolution microscopy have not been previously studied. In this study, we discussed the effects of some common low-order aberrations on the MINFLUX localization performance. In addition to illustrating the simulated aberrations on the pattern of the system PSF, the theoretical limits of the localization precision under various aberration conditions are compared. The different effects of system aberrations and sample-induced aberrations on the localization are also considered. Based on simulations of the optical aberrations and by measuring their effects on the localization of the MINFLUX microscopy, it is possible to assess the imaging quality or to perform targeted optimization of aberrations in experimental operations, thereby providing technical support for further breakthroughs in super-resolution microscopy performance.
2. Simulation methods
Two-dimensional MINFLUX is usually achieved by using a vortex phase mask in the excitation path to produce a donut-shaped focal spot. The excitation beam must be deflected in a particular pattern to enhance localization precision. This pattern is called the targeted coordinate pattern (TCP) and usually consists of the vertices of an equilateral triangle and its center [10]. According to the photon statistics, the number of detected photons n follows a Poisson distribution with a mean λ. This mean value λ of the Poisson distribution is considered proportional to the corresponding excitation intensity I(r) in case of unsaturated fluorescence. Clearly, the first step in assessing the effects of aberrations on the 2D MINFLUX is to calculate the distribution of excitation intensity in the focal plane.
To derive the intensity distributions in the focal plane affected by different aberrations, the vector diffraction theory (i.e., the Debye diffraction integral) was applied in this study. This is because the microscope objectives in MINFLUX systems usually have a high numerical aperture (NA), and there are strict requirements on the polarization of the incident beam [11].
We considered the complex amplitude of an input beam in the form given as
where A1(θ) represents the amplitude variation of the collimated beam; θ is the focusing angle; and $\phi $ is the azimuthal angle.In a focal system [12], the field distribution in the focal plane is given as
The wave aberration function represents the deviation of the actual wavefront from the ideal wavefront when an aberration exists, and can be written as
The phase aberration $\psi $ can be further expressed as
where Zi is the normalized Zernike polynomial for expressing the different types of aberrations [13]; pi is the expansion coefficient of the polynomials, and it represents the magnitude of the corresponding root mean square (RMS) aberration in units of rad.In this study, four common and typical low-order Zernike modes: defocus, oblique astigmatism, vertical coma, and primary spherical aberration at various strengths were considered. It is worth noting that the different orientations of the aberrations, such as horizontal and vertical coma, only introduce rotations to the results and do not affect any conclusion of this study (Sec. 1 of Supplement 1).
Using the above equations, the aberrated system PSFs were generated. Subsequently, the Cramér-Rao bounds (CRBs) were derived directly from these aberrated PSFs. The CRB expresses a lower bound on the variance of the unbiased estimators of a deterministic parameter; it also represents the theoretical optimal localization precision of the system. As the CRB of 2D position is a covariance matrix, we chose its arithmetic mean ${\widetilde \sigma _{CRB}}$ as evaluation metrics, for the convenience of visualization. In this study, the effects of each aberration on the CRBs were investigated. Therefore, the theoretical localization precision was calculated for different aberration conditions.
Finally, the effects of different types of system aberrations and sample-introduced aberrations on the localization were considered using Monte Carlo methods. The photon count generation method was the statistical method in [14], to ensure that each localization had the same total number of photons. The photon counts along with the corresponding PSFs were processed using the maximum likelihood estimator to retrieve localizations [5]. After performing 1000 simulated localizations for each sampling point in the region of interest (ROI), the localization performance at that sampling point under the specific aberration conditions was obtained by observing the distribution, degree of concentration, and directionality of the simulated localization points.
The coordinates of the estimated positions were compared with their corresponding sample points (ground truths) [15]. The localization accuracy was obtained by calculating the Euclidean distance between the mean value of the estimated positions and the ground truth. The localization precision was characterized by the standard deviation of the estimated positions from their mean. To visualize the distribution of the estimated position, a covariance ellipse was constructed for each sampling point. The covariance ellipse contained 95% of the estimated positions, representing the 95% confidence interval. In this case, the Euclidean distance between the center of the ellipse and ground truth indicated the accuracy, where the distance from a point on the ellipse to its center indicated the radial precision, and the direction and eccentricity of the ellipse reflected the angular uniformity of the localization.
The simulations used a 638-nm excitation wavelength, with a total number of N photons (N = 500) used in a single localization. An NA of 1.4 and signal-to-background ratio (SBR) of 10 were used throughout the simulations, consistent with the actual experimental conditions. As noise is an important problem in super-resolution microscopy, the simulation localization results for different aberrations under different SBR conditions are discussed in Sec. 2 of Supplement 1.
3. Results and discussion
3.1 System point spread function (PSF)
As shown in Fig. 1, different aberrations have intuitive effects on the PSF of the system, thereby affecting subsequent localization performance. Herein, the RMS aberration strength was taken as pi = 0.2 rad, and all the light intensity distributions in Fig. 1 were uniformly normalized.
Fig. 1. 2D schematic of the targeted coordinate pattern (TCP) with different aberrations: (a) without aberration, (b) defocus, (c) oblique astigmatism, (d) vertical coma, and (e) primary spherical aberration; all the aberrated imaging results share the same simulative parameters: root mean square (RMS) aberration strength pi: 0.2 rad; scale bar: 700 nm.
Evidently, defocus had a negligible effect on the shape of the PSF but caused decreases in the edge sharpness and peak intensity, as shown in Fig. 1(b). Oblique astigmatism broke the symmetry of the system PSF, thus leading to a serious distortion of the structure of the pattern; the donut circle was split into two parts and the central hollow was tilted and stretched, as shown in Fig. 1(c). Vertical coma caused a slight shift in the pattern in the y-direction, thus leading to a significant inhomogeneity in the light intensity distribution, and the contrast of the upper edge decreased, as shown in Fig. 1(d). Primary spherical aberration did not affect the shape of the PSF and only reduced its peak intensity. It seems to have a smaller impact compared with defocus, as shown in Fig. 1(e). With the same strength, the effects of oblique astigmatism and vertical coma aberrations on the system PSF were more significant.
3.2 Cramér–Rao bound (CRB)
The localization precision and imaging quality of the system was evaluated under different conditions by discussing the extreme values and spatially distributed uniformity of the CRB arithmetic mean ${\widetilde \sigma _{\textrm{CRB}}}$ in the ROI. The strengths pi = 0.15, 0.3, 0.45 rad were selected to compare the effects of the different types of aberrations on the CRB. As shown in Fig. 2, the yellow dots represented the centers of the excitation beam at four exposures, and the diameter L of TCP was set to 100 nm (white dashed line).
Fig. 2. 2D theoretical limits of the MINFLUX system: (a) without aberration, (b) defocus, (c) oblique astigmatism, (d) vertical coma, and (e) primary spherical aberration; all the results share the same simulative parameters: L: 100 nm; N: 500 photons; signal-to-background ratio (SBR): 10; scale bar: 20 nm.
When the aberration strength was small (pi = 0.15 rad), the CRBs were not very different from the case without an aberration. By gradually increasing the aberration strength of oblique astigmatism, the shape of the high-precision area in the center was stretched, and the highest achievable precision decreased significantly. Moreover, the low-precision area at the edge shrank toward the center, and the precision at the edge significantly worsened, as shown in Fig. 2(c). As for vertical coma, the high-precision of the center remained essentially the same, with only a slight upward shift in the y-direction. Additionally, vertical coma decreased the precision of the lower part of the whole region and increased the overall inhomogeneity, as shown in Fig. 2(d). Adding defocus and primary spherical aberration only marginally affected the precision of the edges of the whole region and the overall impact was less severe; this is consistent with the influences of the different types of aberrations on the system PSF.
Typically, a quantitative analysis of the theoretical optimal localization precision of a system is performed for the ROI. The localization precision of the MINFLUX system is closely related to the diameter L of the TCP. The smaller the diameter L, the smaller the available localization region and the higher the precision. Therefore, L must match the ROI size.
Herein, the mean value of the CRB at the edge of the circle with the diameter dROI was chosen as the cost function for optimizing L. Numerical simulation revealed that Lopt(dROI) ≈ 1.57 • dROI under this simulation condition [16]. In Fig. 2, the solid white circles represent the ROIs at L = 100 nm. Next, the extreme and average values of the CRB for different aberration conditions in this region were analyzed, as shown in Fig. 3.
Fig. 3. Maximum, minimum, and average values of CRB in the region of interest (ROI) in the presence of aberrations: (a) maximum and minimum values of CRB with different strengths of aberrations; (b) average values of CRB with different strengths of aberrations.
From Fig. 3, the defocus and primary spherical aberration had a negligible effect on the extreme and average values of the CRB in the ROI, whereas the oblique astigmatism and vertical coma had considerable effects. When pi = 0.4 rad, compared with coma of the same strength, the minimum value of the CRB with astigmatism reached 4 nm (four-fold) and the average value was more than 5.5 nm (two-fold), with the overall precision decreasing more evidently.
3.3 System aberrations
Figure 4 shows the Monte Carlo method for locating specific sampling points in the ROI when there is no aberration. The black “+” sign represents the actual position of the fluorescent molecule (ground truth), the blue scattered dots represent the estimated localizations, the red dots represent the expectation of the estimated localizations, and the red ellipse represents the 95% confidence interval for the set of estimated localizations (95% of the estimated localization fall within this ellipse). The green dots in Fig. 4(c) represent the centers of the donut-shaped excitation beam at the four exposures (i.e., the TCP), and the sampling point spacing is 10 nm. Evidently, the closer the sampling point is to the center of the ROI, the more similar the confidence ellipse is to a circle, thus indicating a better isotropic localization precision. It has been noticed that such anisotropy was caused by the asymmetry of the TCP itself [15]. Moreover, the black “+” sign was closer to the red dot in the ROI center, thus indicating a higher localization accuracy.
Fig. 4. Monte Carlo method of localization: (a) single-point simulation at (0, 0); (b) single-point simulation at (–30, 30); and (c) simulation of multiple sampling points in ROI without aberration, scale bar: 20 nm.
As noted in the introduction, system aberrations are primarily owing to mechanical errors in the optical system itself. They can be considered in the localization process by measuring the PSF of the system in advance. The addition of system aberrations affects both the photon count distribution n and system PSF. Using an aberrated PSF to process the aberrated photon count distribution n, that is, having a prior understanding of the aberration before localization, represents a simulation of the system aberration. Based on adding different types and strengths (pi = 0.15, 0.3 rad) of system aberrations, the localization simulations of all sampling points in the ROI are shown in Fig. 5.
Fig. 5. Localization simulation of each sampling point in ROI with different system aberrations: (a) defocus, (b) oblique astigmatism, (c) vertical coma, and (d) primary spherical aberration; Scale bar: 20 nm.
Compared with the localization simulation without the aberration shown in Fig. 4(c), the defocus and primary spherical aberration had almost no effect on the magnitude and angular uniformity of the localization precision in the ROI. However, adding oblique astigmatism in Fig. 5(b), the overall confidence ellipses became noticeably larger; additionally, their directions changed, with the long axes tending to be tilted to the left. This affected the precision magnitude of the full ROI and destroyed the precision isotropy completely. Moreover, the distance between the expectation (red dot) of each sampling point and its ground truth (black “+”) increased slightly, thus indicating that astigmatism in the system aberration affect the localization accuracy of the whole ROI.
After adding vertical coma, the changes in each part of the ROI varied, as shown in Fig. 5(c). The confidence ellipses of the upper half of the ROI grew larger and their tilt directions changed, whereas the confidence ellipses of the lower half of the ROI became smaller, thus indicating that the coma unevenly affects the precision and angular uniformity. As the strength increased, the distance between the expectation and ground truth also increased; however, the systematic coma aberration led to a lower degree of decrease in the localization accuracy compared with astigmatism.
The following parameters were quantitatively compared along the profiles y = 0 in Fig. 4(c) for different types and strengths of system aberrations, with sampling point spacing of 5 nm. 1) Localization bias: the Euclidean distance between the ground truth (black “+” sign) and expectation of estimated localizations (red dot) of the molecule; 2) precision in the x and y directions: the standard deviation of the simulated x and y coordinates; and 3) angular uniformity of the localization: the eccentricity of the confidence ellipse. Figure 6 shows the localization bias, precision, and confidence ellipse eccentricity as a function of the x-coordinate for each point on y = 0 with different system aberrations.
Fig. 6. Localization bias, precision, and confidence ellipse eccentricity as a function of x-coordinate for each point on y = 0 with increasing levels of system aberrations (pi = 0.05, 0.25, 0.45 rad): (a) defocus, (b) oblique astigmatism, (c) vertical coma, and (d) primary spherical aberration.
The comparison of the bias of each aberration in Fig. 6 revealed that regarding the system aberrations, the defocus, coma, and spherical aberrations introduced negligible bias (all were below 1 nm). The bias introduced by the astigmatism was slightly larger but was also within the acceptable range (less than 2 nm when pi = 0.45 rad). On the profile line, the precision difference between the x and y directions changed with the distance of the sampling point from the center of the region, first decreasing and then increasing. It reached the closest value at x = 0 (exactly the center of view), where the confidence ellipse eccentricity was the smallest.
The defocus and primary spherical aberration had almost no effect on the precision, whereas the vertical coma slightly increased the standard deviation on y = 0 of the localization. With increasing strength, the astigmatism evidently reduced the localization precision, as shown in Fig. 6(b) in the precision graph: the precision curves with different astigmatism strengths are clearly separated.
Additionally, defocus and primary spherical aberration had little effects on the angular uniformity of the localization at each point. On the line y = 0, vertical coma had a small effect as well; however, evident from Fig. 5(c), the addition of the system vertical coma aberration caused a significant change in the eccentricity of the confidence ellipse and localization precision on the line x = 0 (Sec. 3 of Supplement 1). The eccentricity plot in Fig. 6(b) shows that astigmatism evidently affects the angular uniformity, thus decreasing the isotropy of the localization.
3.4 Sample-induced aberrations
The sample-induced aberration in the MINFLUX system is related primarily to the refractive index mismatch; this arises between the objective and specimen. This aberration cannot be measured in advance as system aberrations; therefore, the related simulation was represented using the aberration-free PSF to process the aberrated photon count distribution n for the localization. For most MINFLUX thin-layer samples, the strengths of the sample-induced aberration tend to be smaller than the system aberration. Figure 7 shows the localization simulations of the sampling points in the ROI with different types and strengths (pi = 0.05, 0.15 rad) of sample-induced aberrations.
Fig. 7. Localization simulation of each sampling point in ROI with different sample-induced aberrations: (a) defocus, (b) oblique astigmatism, (c) vertical coma, and (d) primary spherical aberration; Scale bar: 20 nm.
Comparing Fig. 7 with the localization simulation without aberrations in Fig. 4(c) and with the simulation when adding a small magnitude of system aberrations (Fig. 5, pi = 0.15 rad), it revealed that the defocus and primary spherical aberration in the sample-induced aberrations had the same small effect on the precision and angular uniformity of localization in the ROI. However, the effects of the astigmatism and coma in sample-induced aberrations appeared to be very different from those in the system aberrations.
After adding a small amount of oblique astigmatism in Fig. 7(b), all the confidence ellipses became larger, implying a decrease in precision. Simultaneously, for each sampling point, the expectation of estimated localizations (red dot) exhibited an evident and irregular deviation from the ground truth, thus indicating that astigmatism in the sample-induced aberrations can seriously reduce the localization accuracy of the system.
After adding a small amount of vertical coma in Fig. 7(c), the shapes and directions of the confidence ellipses changed similarly to the response to the coma of the system aberration; however, the expectations of estimated localizations were significantly shifted upward for all points compared with the ground truth. This demonstrates that although the sample-induced coma also leads to a large decrease in localization accuracy, the effect has a certain regularity.
Figure 8 shows the localization bias, precision, and confidence ellipse eccentricity as functions of the x-coordinate for each point on y = 0 with different sample-induced aberrations.
Fig. 8. Localization bias, precision, and confidence ellipse eccentricity as a function of x-coordinate for each point on y = 0 with increasing levels of sample-induced aberrations (pi = 0.05, 0.15, 0.25 rad): (a) defocus, (b) astigmatism, (c) coma, and (d) spherical.
Evidently from Figs. 7 and 8, the most obvious difference between the effects of a sample-induced aberration on the localization and those of a system aberration was that with sample-induced aberrations, even a small amount of coma and astigmatism could introduce large biases (on the order of 10 nm). Vertical coma introduced a uniform localization bias for all points on the line y = 0 (corresponding to the uniformly upper confidence ellipse centers in Fig. 7(c)), whereas astigmatism introduced a non-uniform bias.
The effect of a sample-induced aberration on the precision was smaller than that of a system aberration, as shown in the precision plots in Fig. 8, where the curves under different aberration strengths overlap better than those in Fig. 6. In addition, the effect of the sample-induced aberration on the angular uniformity of the localization was similar to that of the system aberration.
Remarkably, on the y = 0 line, on the left side (x < 0), the localization precision in the x-direction was higher than that in the y-direction, whereas on the right side (x > 0), the localization precision in the y-direction was higher under both system aberration and sample-induced aberration conditions. This could be related to the locations of the excitation beam at the four exposures. In this study, the simulation considered the green dots in Figs. 5 and 7 as the centers of the excitation beam (where there are two exposures at x > 0); therefore, the right side could obtain more localization information regarding the y direction.
4. Conclusion and outlook
In this study, we demonstrated the effects of different types and strengths of low-order aberrations on the 2D localization of the MINFLUX system using theoretical limits and simulation procedures, such as an analysis of the system PSF, calculation of the theoretically optimal localization precision (i.e., CRB), and Monte Carlo method of the localization. Additionally, we compared the effects between system aberrations and sample-induced aberrations on the system localization.
Through theoretical calculations and simulation experiments, we showed that 1) defocus and spherical aberration have a small effect on the 2D localization precision of MINFLUX, whereas astigmatism and coma significantly worsen the localization performance; 2) system aberrations adversely affect the precision and angular uniformity of the localization, whereas sample-induced aberrations can generate large biases and deteriorate the localization accuracy of the system.
The impact of defocus and spherical aberration on the 2D localization precision was small, for a precision of 3 nm could still be achieved with aberrations of pi = 1 rad. In the presence of astigmatism and coma, the localization precision of the system was low as astigmatism distorted the high-precision area in the center of the whole region and considerably reduced the highest precision (the minimum value of CRB in the ROI reached 6.5 nm when the strength of astigmatism was approximately pi = 0.5 rad). With astigmatism, the low-precision area at the edge shrank to the center and the precision at the edge of the region decreased significantly. Furthermore, the angular uniformity of the localization at each point of region decreased. The effect of coma on the system was smaller than that of astigmatism as the high-precision area in the center of the region shifted slightly upward in the y-direction. With coma, the precision of the lower part was reduced, thus increasing the overall inhomogeneity of the precision of the whole region.
We hope that our results can illustrate the effects of various aberrations on different aspects of the system. That will help reveal the main types of aberrations contained in an entire system through the observed beam intensity distribution. It will be beneficial for evaluating the imaging quality or for targeted optimization of certain aberrations, providing technical support for further breakthroughs in system performance.
In the simulations above, we only considered a few low-order aberrations to analyze the corresponding effects of different types of aberrations on MINFLUX localization performance. For the simulation results of certain higher-order aberrations and combined aberrations, please refer to Sections 4 and 5 of Supplement 1. Although in most cases the expected ideal condition is without aberrations, in some occasions, aberrations can be used to collect additional dimensional information regarding the sample [17]. Furthermore, in single-molecule localization microscopy, sample-induced aberrations can be measured or corrected in advance by methods such as image-based sensorless adaptive scheme [18] and INSPR [19]. A high-order Laguerre–Gaussian vortex can also be used as the aberration correction metric in SLM-based systems [20]. These methods are expected to be introduced into the MINFLUX system in the future, correcting the large localization bias and precision reduction caused by both system and sample-induced aberrations.
Taken together, the simulations in this study were limited to the 2D case. Currently, by using a 0/π toroidal phase (“Top-hat” phase) to form high-quality 3D hollow excitation beam, and using appropriate changes of the deformable mirror curvature in the optical path to add the defocus aberration, the MINFLUX system can shift the zero-intensity point at the focal plane along the z-axis, thereby realizing 3D imaging [15]. Even though defocus and spherical aberration have little effect under 2D conditions, they can strongly affect 3D PSFs [8]. Indeed, it is worthwhile to analyze the localization of the aberrated system in 3D cases. In addition, for multicolor MINFLUX [10], the effects of chromatic aberrations should also be discussed.
Funding
National Natural Science Foundation of China (61735017, 61827825, 62125504, 62205288); Natural Science Foundation of Zhejiang Province (LD21F050002); Key Research and Development Program of Zhejiang Province (2020C01116); Fundamental Research Funds for the Central Universities (2022FZZX01-20); Zhejiang Lab (2020MC0AE01); Zhejiang Provincial Ten Thousand Plan for Young Top Talents (2020R52001); China Postdoctoral Science Foundation (2021TQ0275, 2022M712734).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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