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Abstract
To extend the axial depth of nanoscale 3D-localization microscopy, we propose here a splicing-type vortex singularities (SVS) phase mask, which has been meticulously optimized with a Fresnel approximation imaging inverse operation. The optimized SVS DH-PSF has proven to have high transfer function efficiency with adjustable performance in its axial range. The axial position of the particle was computed by using both the main lobes’ spacing and the rotation angle, an improvement of the localization precision of the particle. Concretely, the proposed optimized SVS DH-PSF, with a smaller spatial extent, can effectively reduce the overlap of nanoparticle images and realize the 3D localization of multiple nanoparticles with small spacing, with respect to PSFs for large axial 3D localization. Finally, we successfully conducted extensive experiments on 3D localization for tracking dense nanoparticles at 8µm depth with a numerical aperture of 1.4, demonstrating its great potential.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In optical imaging, point spread function (PSF) engineering stands for a powerful technique for resolution enhancement, aberration correction, particle tracking and localization, three-dimensional (3D) imaging, and depth-of-field extension. Modulating the phase of the collected light in the Fourier plane to change the shape, size, and intensity of the PSF is commonly used for encoding the desired information. Examples of this include hollow PSFs in STED microscope, double-helix, and tetrapod PSFs for particle tracking and localization [1–3], self-bending PSFs with expanded depth-of-field [4], etc. In the past decade, nanoparticle trajectory tracking and 3D precise localization have attracted broad interest, particularly for their significant performance in the life sciences, including nanomedicine development, blood-flow characterization, and microfluidic research [5]. Among all, PSF engineering is the most prominent due to its high time resolution, high localization precision, and simplicity. The common PSFs used in particle localization include the double-helix PSFs (DH-PSF) [6–11], the tetrapod PSFs [12,13], and the Airy-beam-based PSFs [4–16]. To note, the axial range of DH-PSF [17,18], elliptical PSF [19], and corkscrew PSF [20] are limited to 4µm, although some researchers have combined defocus grating to expand its axial depth at the cost of low localization precision and signal-to-noise ratio [9]. Differently, the Airy-beam-based PSFs have a large axial detection range, as the axial position increases, the two-dimensional size of the PSF rapidly expands with enhanced side lobes requiring complex image processing to obtain the exact position of the particle. In addition, large-size Airy-beam-based PSFs and saddle-point-type PSFs are not conducive to the localization and tracking of dense particles [3,21]. On the contrary, compared with the saddle-point-type PSFs and the Airy-beam-based PSFs, the DH-PSF can easily obtain the axial information without complicated calculations. However, the rotation angle of DH-PSF is limited to π radians, and the angle cannot be distinguished when it exceeds π radians. As a result, the restricted rotation angle leads to a sharp drop in localization precision when the axial range of the DH-PSF increases.
In this article, we propose a variable spacing DH-PSF with an adjustable large axial range. The PSF combines the features including the compact size of conventional DH-PSFs, high localization precision of saddle-point-type PSFs, and a large axial range of Airy-beam-based PSFs, thus bringing outstanding performances. Specifically, the operable depth range can be adjusted by parameters that balance the compromise between localization precision and operable axial range. Furthermore, the axial position is obtained more precisely by the rotation angle and the main lobes’ spacing. On this basis, we optimized the PSF using the Fresnel approximation imaging inverse operation to weaken the intensity of the side lobes and improve the transfer function efficiency of the PSF.
2. Methods
2.1 Splicing-type vortex singularities (SVS) phase mask
The vortex singularities phase [18] with the defocus phase and translation parameters can be expressed as,
Because the down-sampled phase still retains most of the original information, multiple down-sampled phases can be spliced together into a single phase of the original resolution, whereas the PSF generated by the recombined phase takes the form of a combination of the PSFs generated by multiple downsampled phases. Based on this principle, by dividing the Fourier plane into multiple partitions, multiple phases can be combined into one phase, and multiple PSFs can be combined into one PSF. The expression for phase splicing can be expressed as,
Fig. 1. Configuration of the designed Annular-zone plate phase to generate splicing-type PSF. In this example, that S takes the value of 4 and N takes the value of 16.
The number of vortex singularities and the defocusing factor of the VS phase mask determines the axial range of the SVS phase mask. The relationship between the defocusing factor α and the axial deviation from the focal plane of the objective lens is, the defocusing distance = 0.1α µm. Therefore, the axial depth of the SVS phase mask can be expressed as,
In the electromagnetic scalar approximation, the resulting PSF at the detector plane I(u,v) satisfies [22–2],
The form of the splicing-type vortex singularities (SVS) phase mask and the performance of the associated PSFs are shown in Fig. 2. The value of N used in the experiments is 460, which is computed by using the pixels of the spatial light modulator. Vortex singularities phase mask can generate DH-PSF with an axial depth of about 2 µm and a rotation of π radians. The number of vortex singularities and their relative location (ρk, θk) can adjust the spacing of the two main lobes of the DH-PSF. Therefore, we could splice the DH-PSFs with different spacing axially to expand the axial depth, which solves the shortcomings of the 2π-DH-PSF proposed previously. With the introduction of the difference in the main lobes’ spacing, the difference between θ and θ+π can be easily distinguished, leading to an advantage that the conventional 2π-PSF does not have. Figures 2(a), (b), (c), and (d) show the vortex singularities phase mask and PSF intensity distributions when the number of vortex singularities are 9, 11, 15, and 19, respectively. The SVS phase mask is generated from the defocus vortex singularities phase with different parameters according to the phase splicing method of Eq. (2), as shown in Fig. 2(e). The PSF generated by the SVS phase mask changes both the spacing and the rotation angle of the main lobe as it moves away from the focal plane of the objective which makes the DH-PSF higher distinguishability and localization precision over the entire axial range, an advantage that neither the conventional DH-PSF nor saddle-point-type PSF has.
Fig. 2. The SVS phase mask and its PSF (NA = 1.4, λ=514 nm). (a) T = 9, α=-30, the values of ρk are ±0.437R, ± 0.306R, ± 0.175R, ± 0.044R. (b) T = 11, α=-10, the values of ρk are ±0.524R, ± 0.393R, ± 0.306R, ± 0.131R, ± 0.0441R. (c) T = 15, α=10, the values of ρk are ±0.568R, ± 0.437R, ± 0.393R, ± 0.349R, ± 0.306R, ± 0.218R, ± 0.131R. (d) T = 19, α=30, the values of ρk are ±0.611R, ± 0.524R, ± 0.437R, ± 0.349R, ± 0.306R, ± 0.262R, ± 0.218R, ± 0.175R, ± 0.131R. (e) The SVS phase mask, (f) Optimized SVS phase mask.
2.2 Vortex singularities (VS) phase mask
For the vortex singularity (VS) phase mask and its PSF description more in detail. Figures 3(a) shows the PSF intensity distributions with various axial depths generated for the number of vortex singularities are 9, 11, 15, and 19, respectively. The axial detection depth of the VS phase mask increases as the T value increases, but the peak intensity of the main lobes decreases. Similar to the Fresnel zone phase mask, the axial detection depth of the vortex singularity phase becomes larger as the T value increases. Notably, the difference exists in the main lobes’ spacing of the DH-PSF generated by the Fresnel zone phase mask is approximately constant as the T value increases, while the main lobes’ spacing of the DH-PSF generated by the VS phase mask is variable, as shown in Fig. 3(c). However, for the DH-PSF generated by the VS phase mask, the rotation angle becomes rotates more quickly with depth, the intensity decreases, and the brightness and number of sidelobes increase at the position far from the focal plane of the objective lens, as shown in Figs. 3(a) and 3(d). For example, when T = 19, the rotation angle rapidly jumps from -0.228π radians to -0.356π radians between -1.2 µm and -1.3 µm in the axial position. From the results of the simulation calculations, for the value of T are 9, 11, 15, and 19, the main lobes’ spacing of the DH-PSF generated by the VS phase mask remains constant in the axial detection range from -1 µm to +1 µm, and the rotation angle is linearly related to the axial depth, as shown in Figs. 3(c) and 3(d). In addition, the main lobe was more regular in shape (See Fig. 3(a)), with fewer side lobes, and higher peak intensity (See Fig. 3(b)). Therefore, only PSFs with axial depths in the range of -1 µm to +1 µm were selected for stitching. And, we take this range as the effective region for the DH-PSF of each group of VS phase masks.
Fig. 3. The effects of the vortex singularities phase mask on the PSF. (a) The PSFs generated by the vortex singularities phase mask with different values of the parameter T. (b) Normalized intensity of these PSFs with varying amounts of defocus. (c) Main lobes’ spacing of these PSFs with varying amounts of defocus. (d) Rotation angle of these PSFs with varying amounts of defocus.
In the splicing process, we selected the part of the vortex singularity PSF with high intensity of the main lobes (-1µm—+1µm) as the effective area of the SVS DH-PSF. Figure 4(a) and Fig. 4(b) are the splicing positions of the two vortex singularity PSFs we set, respectively. Since the main lobe intensity of vortex singularities (VS) DH-PSF in Fig. 4(a) and Fig. 4(b) is similar, four main lobes appear after splicing, as shown in Fig. 4(c). In the latter position, only the main lobe of one group of PSFs will be presented due to the larger intensity difference between the two groups of PSFs. For example, the splice of Fig. 4(d) and Fig. 4(e) mainly present the main lobe of Fig. 4(d) because the strength of Fig. 4(e) is much weaker than that of Fig. 4(d).
Fig. 4. Schematic diagram of DH-PSF splicing. The VS DH-PSF obtained at T = 11 is spliced with the VS DH-PSF obtained at T = 15 as an example. Its modulation phase mask is consistent with Fig. 2. (a) is spliced with (b) to generate (c), and (d) is spliced with (e) to generate (f).
3. Optimization
The phase generated by splicing multiple phases in the way of down-sampling decreases the transfer function efficiency, and the generated PSFs are later accompanied by a strong sidelobe, as shown in Figs. 4(c) and 4(f). The generated SVS phase mask exhibits an improved performance via algorithm optimization according to a previously published paper [22]. Specifically, the algorithm is an improvement of the iterative Fourier algorithm (IFTA) or Gerchberg Saxton algorithm (GSA). For each z position, calculating a corresponding current mask was estimated by inverse Fourier-transforming and quadratic phase factor dividing that corresponds to this z position. The optimization algorithm is then carried out on the optical axis, so the relationship between the modulation function and the electric field at different depths z can be obtained from Eq. (4),
By replacing the original amplitude of the electric field distribution with the ideal amplitude |AM’| at different axial depths z, a new electric field distribution AM’’ is finally obtained. The expected SVS DH-PSF needs to focus as much energy as possible on the main lobes and minimize the side lobes, so we defined a pupil function called Mask to extract the main lobes of the SVS DH-PSF while weakening the side lobes. The number of photons selected by Mask is called the effective photon number. For the optimization of the SVS DH-PSF, we chose three groups of Mask functions. The first group of Mask functions take the diameter of the main lobes of the SVS DH-PSF as the diameter and extract the main lobes of the SVS DH-PSF exactly. The other two groups of Mask functions are obtained from Eq. (7) below, with δ taking values 0.5 and 0.7, respectively (the best range of δ is 0.5–0.8). The Mask pupil (low-pass filter) functions can be expressed as,
Substituting the newly obtained amplitude AM’’ into Eq. (5), an approximate phase PM with respect to the ideal amplitude intensity is obtained. The optimized phase after one iteration can be expressed as,
Fig. 5. optimization algorithm. (a) Schematic diagram of optimization algorithm, (b) Relation-ship between the number of effective photons and the number of iterations. (c) Transfer function efficiency of unoptimized phase mask and optimized phase mask. (d) Transfer function efficiency of SE-PSF, TA-PSF, Spiral DH-PSF, SVS-PSF.
The optimized SVS phase mask using the Fresnel approximation imaging and the inverse Fourier transform operation is shown in Fig. 2(f). The DH-PSF generated by the optimized SVS phase mask is shown in Fig. 6(a). As compared to the previously proposed 2π-DH-PSF, the DH-PSF generated in this paper is more clearly distinguished at different axial depths. The conventional 2π-DH-PSF [7] uses only the angle of the main lobes to obtain the axial position, except where the same angle corresponds to two different axial positions. The SVS DH-PSF in this paper gains the axial position information using the main lobes’ spacing and rotation angle together, whereby the distinction among PSFs with different axial positions is more obvious. For example, the PSFs, corresponding to four axial positions, Q1, Q2, Q3, and Q4, have the same main lobes’ angle but different main lobes’ spacing, as shown in Figs. 6(b) and 6(c). It should be noted that there will be four main lobes at the -2.25um, 0um, 2.25um where PSF splicing happens (See Fig. 6(a)). The relationship between rotation angle and axial depth is linear in each interval ([-4, -2.25], [-2.25, 0], [0, 2], [2,4]). Even the special position where four main lobes are present conforms to the linear relationship, as shown in the red dashed box in Fig. 6(a). It should be noted that when the PSF has four main lobes, the angles of the two sets of main lobes belong to different interval ranges. The attribution of two sets of diagonal main lobes is regular, when the angle is positive it belongs to the previous intervals, and when the angle is negative it belongs to the next intervals (See Fig. 6(b)).
Fig. 6. Performance of SVS DH-PSF. (a) The intensity distribution of SVS DH-PSF at various axial depth, (b) The rotation angle of the SVS DH-PSF at various axial positions (-0.5π∼0.5π radian range), (c) The main lobes’ spacing of the SVS DH-PSF at various axial positions.
4. Comparison
According to the calculation using the Fisher information, the Cramer-Rao lower bound (CRLB) provides a fundamental theoretical limit on localization precision obtained by all unbiased estimators. The variable can be obtained by calculating the square root of the diagonal of Fisher's information inverse matrix [24],
Fig. 7. Performance of SVS phase mask. (a) The localization precision for x directions, (b) The localization precision for y directions, (c) The localization precision for z direction.
Surprisingly, the maximum primary main lobe size of the SVS DH-PSF (Fig. 8(a)) is much smaller than that of the SE-PSF (Fig. 8(b)), the Tetrapod-PSF (Fig. 8(c)), and the COSA-PSF (Fig. 8(d)), which is crucial in applications of particle localization and tracking. In dense multi-particle localization and tracking, a more compact PSF size can reduce the overlap between PSFs and facilitate subsequent processing. Additionally, the intensity distribution of the other three PSFs near the focal plane of the objective lens is more complex, as shown in the red circular box in Fig. 8, and requires the help of complex post-processing algorithms to obtain the exact axial position. The complex post-processing algorithm is difficult to achieve real-time particle localization and trajectory tracking. SVS DH-PSF relies only on the rotation angle and spacing of the main lobe to obtain accurate axial information, without the need for complex post-process algorithms. The compact size of SVS DH-PSF helps to observe more particles at the same time.
Fig. 8. Comparison of the maximum size of the main lobes’ spacing for four typical phases. SVS phase mask (a), SE phase mask (b), Tetrapod phase mask, and COSA phase mask(d), and its PSF intensity distribution at each axial depth. The red boxes are partial enlargements. The PSF intensity distribution at the focal plane of the objective lens is shown in the red circular box.
5. Localization
In this paper, the axial depth of the nanoparticles is determined by calculating the distance and rotation angle between the two main lobes. In general, the SVS DH-PSF has two Gaussian spots, and four Gaussian spots in and around three special locations. Therefore, we use the Gaussian mixture model [25,26] to fit the Gaussian center of the obtained spot (using the fitgmdist function in MATLAB, See Supplement 1). Firstly, the image of the SVS DH-PSF is taken. Secondly, the binary image of SVS DH-PSF is obtained by processing the image (removing noise) and threshold segmentation (determining the distribution area of SVS DH-PSF on the image plane). Thirdly, the gray value of each pixel in the SVS DH-PSF distribution area is recorded as the number of occurrences of the pixel coordinates,
Prior to fitting with the Gaussian mixture model, it is difficult to determine whether the shape of SVS DH-PSF is four Gaussian spots or two Gaussian spots. So, we need to use the AIC option (an option in the fitgmdist function) in the Gaussian mixture model. AIC can determine whether the model is a four-Gaussian model or a double-Gaussian model. The coordinate set H is brought into the Gaussian mixture model for fitting. If it is a four-Gaussian model, we infer that the particle is in or around the three special locations. If it is a double Gaussian model, we may obtain the center coordinates of two Gaussian spots. Through the distance and rotation angle between the two center coordinates, we can get the exact distance between the nanoparticles and the focal plane. The transverse position coordinates of a particle can be directly obtained by averaging the center coordinates of each Gaussian spot.
Used for multi-particle localization, the center fitting algorithm may roughly define the center coordinates of the main lobes [27]. As a result, the individual particles are intercepted from the image based on that center coordinate value. The localization algorithm is simple but challenging for high-density nanoparticles and low signal-to-noise ratio image localization.
6. Experiment
The experimental setup is shown in Fig. 9(a). An oil immersion objective (NA = 1.4, oil immersion, OLYMPUS, 100×) was used in the experiments. The microscope's mercury lamp with a wavelength of 488 was used to excite the fluorescence. The optical modulation can be conveniently introduced by using a 4f system to bring the exit pupil of the objective to an external location. Moreover, the modulation phase mask was loaded on the spatial light modulator (SLM, Meadowlark Optics, 1920 × 1152), and the corresponding fluorescent beads images were obtained on the sCMOS camera (See Fig. 9). The exposure time of the sCMOS (ORCA-Flash4.0 V2, Hamamatsu) was set to 100 ms. Based on the Gerchberg-Saxton algorithm [28,29], the DH-PSF is used as an aberration indicator to correct the aberration of the system. The nanoparticles (Fluospheres 0.1µm, yellow-green FluoSpheres (505/515)) were placed in various z positions (-4 ≤ z ≤ 4µm, Δz = 0.125 µm, Nano displacement platform, OPLAN Nano Z100), and imaged for 65 frames under the phase mask modulation conditions, as shown in Fig. 9(d). According to the obtained images, we established the measured stack of the SVS DH-PSF for the localization of the particle. In addition, the depth range calibration curves of the experimental system versus the rotation angle and the main lobes’ spacing of the SVS DH-PSF are shown in Figs. 9(b) and 9(c).
Fig. 9. Calibration curves of the depth range of the experimental system with the rotation angle and main lobes’ spacing of the SVS DH-PSF. (a) Experimental setup employing an oil immersion objective, P, polarizer, Obj, Objective Lens, SLM, liquid crystal spatial light modulator, (b) The rotation angle at various axial positions, (c) The main lobes’ spacing at various axial positions, (d) The images of the SVS DH-PSF on the sCMOS when the particles are in different axial positions.
Specifically, we performed dense nanoparticles imaging experiments to compare the imaging effects of the five PSFs (Tetrapod-PSF, COSA-PSF, SE-PSF, TA-PSF, and SVS-PSF). The nanoparticles with a diameter of 100 nm were fixed by agar. Using Tetrapod-PSF, COSA-PSF, SE-PSF, and TA-PSF for dense nanoparticles localization imaging, the main lobes overlapped severely and indistinguishably, as shown in Figs. 10(a)–10(d). Compared with the four phase masks mentioned above, the individual particles in the particle aggregation region can be well distinguished using the SVS phase mask, which facilitates the subsequent 3D coordinate calculation, as shown in Fig. 10(e). Thus, the SVS DH-PSF with a smaller spatial extent has an obvious advantage in experiments with high-density particle localization, which has a higher resolution.
Fig. 10. Localization experiments of dense nanoparticles. The fluorescent microspheres with a diameter of 100 nm in agar were localized by Tetrapod-PSF(a), COSA-PSF(b), SE-PSF(c), TA-PSF(d), and SVS DH-PSF (e), respectively.
The SVS DH-PSF can maintain high localization precision over an extended axial range in three dimensions, which is demonstrated by evaluating the CRLB. In addition, an outstanding advantage of SVS DH-PSF is that the 3D position information of the particles can be accurately obtained only by the position relationship of the two main lobes, making the later data processing simpler, which is an advantage that other PSF of the same type does not have. The localization precision should be composed of two aspects, one is the theoretical localization precision, and the other is the information reconstruction precision during post-data processing. The final position information of the particle is obtained from the shape of the PSF. If the PSF has a complex shape distribution, although it may have high localization precision, it requires complex post-processing algorithms to obtain the axial position.
Particularly, we assume that the motion of the particles is purely diffusive. For a purely diffusive motion with measurement uncertainty σ at each time point, the mean-squared displacement (MSD) scale is related to δ(t) as follows [11,30],
where D is the diffusion coefficient, n is the number of dimensions, and σi is the position uncertainty in each dimension [31]. The exposure time of the sCMOS was set to 100 ms and the experimental ambient temperature was 25 degrees Celsius. Figure 11(a) shows a plot of the MSD in x, y, z, and r versus lag time averaged over the trajectories of three different nanoparticles. The trends of MSD with respect to x, y and z are all approximately linear and overlap each other within the error range. Finally, the diffusion coefficient of the particles inside the 90% concentration of glycerol solution was measured to be ∼0.0243 µm2/s, as shown in Fig. 11(a). Using the Stokes-Einstein relation for a spherical particle, we obtained a hydrodynamic radius of 109 nm for the individual nanoparticles. The error between the experimental results and the actual values was within 10%, indicating that the method can be used to accurately determine the diffusion coefficients of nanomedicines which cannot be calculated by the theoretical formulas.Fig. 11. Nanoparticles 3D position localization and trajectory tracking. (a) Plots of the mean-square displacement in x, y, and z for the average of three different single nanoparticles, A fit to the MSD for displacement r gives a diffusion coefficient of 0.0243 µm2/s, (b) Plot of the number of photons detected for each 100 ms time bin over the 10 s track, (c) Dense particle localization after Gaussian filtering and contrast adjustment, (d) 3D position reconstruction of dense particles, (e) The three-dimensional trajectory of the particles in 90% concentration of glycerin.
Figure 11(b) shows the detected photons as a function of time. Locations where the photon counter value approximates 0 indicate that the nanoparticles are bleached or diffused out of the field of view. Gaussian filtering and contrast adjustment were performed on Fig. 10(e) to yield Fig. 11(c). Then the center coordinates of the main lobes were obtained by Gaussian fitting. In this paper, the position of each nanoparticle is computed by using a Gaussian mixture model to fit the center of each PSF main lobes, then the midpoint between the centroid positions is determined to define the x, y position of the particles, and finally, the angle and spacing between the centroid positions are obtained to determine the axial position z. The 3D positions of the particles can be obtained according to Figs. 9(b)(c), as shown in Fig. 11(d). The imaging particles in 90% concentration glycerol solution were imaged using SVS DH-PSF, and the final reconstructed motion trajectory is obtained as shown in Fig. 11(e). This trace is constructed of 104 frames, which corresponds to a total trace time of 10.4s.
7. Conclusion
In summary, we successfully optimized a new point spread function that can be used for the three-dimensional localization of nanoparticles. The optimized PSF combines the features such as the compact size of conventional DH-PSFs, high localization precision of saddle-point-type PSFs, and a large axial range of Airy-beam-based PSFs. Moreover, the axial detection range of the SVS DH-PSF can be adjusted by the design parameters. In addition, the SVS DH-PSF with a smaller spatial extent can effectively reduce the overlap of particle images and realize the 3D localization of dense multi-particles. For proof of concept, SVS DH-PSF was used to reconstruct the 3D position of particles in agar and to track the particles in glycerol solution, demonstrating the effectiveness of the method. Though, SVS DH-PSF still holds the weakness, i.e that the compromised diffraction efficiency which may lead to strong background noise, current state-of-the-art already illustrates how common problems are being addressed. To impact future directions, further improvement of the diffraction efficiency of SVS DH-PSF is the next focus of our work.
Funding
National Natural Science Foundation of China (62175144); Shanghai Science and Technology Innovation Action Plan Project (20142200100); Projects of International Cooperation of Jiangsu Province (BZ2020004); The Strategic Priority Research Program (C) of the CAS (XDC07040200).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in App1, Ref. [27].
Supplemental document
See Supplement 1 for supporting content.
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