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Abstract
Structured illumination microscopy (SIM) enables live-cell super-resolution imaging with wide field of view (FOV) and high imaging speed, but the illumination system is usually bulky. With the advantages of small structure and high efficiency, lattice patterns assisted by diffractive optical elements (DOEs) have been used for structured illumination in SIM. But it is still challenging to raise the spatial frequency of diffractive lattice patterns when using traditional DOE design method, and thus the super-resolution imaging performance is restricted. In this paper, we propose a novel design method for DOE to generate lattice patterns with spatial frequency close to the cut-off frequency. It is the first time to obtain a lattice pattern with such high spatial frequency by diffractive optics. Finally, the proposed SIM achieves a lateral resolution of 131 nm at 519 nm fluorescent light while maintaining an original size as a standard inverted fluorescence microscope by only inserting a single well-designed DOE in the illumination optical path, which may promote this compact SIM applied in super-resolution imaging field.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The spatial resolution of optical microscopy has been restricted to about half of the wavelength because of the diffraction limit [1]. However, several super-resolution methods have developed in recent years, bypassing this barrier and extending the lateral resolution to tens of nanometers [2–5]. Although not providing better resolution than other super-resolution approaches, structured illumination microscopy (SIM) is considered as one of the most promising candidates in live-cell imaging area for its advantages of wide field of view (FOV), high imaging speed, compatibility with standard labels and low phototoxicity [6–10].
In SIM, when the sample is illuminated by a periodic pattern, in mathematical way it represents the sample’s combining with the illumination pattern multiplicatively in spatial space, as well as their convolution operation in spatial frequency space. After convolved by a series of delta functions that are Fourier transforms of the periodic patterns, the high spatial frequency information unresolved before will be shifted into the low spatial frequency scope and then is made available within the band-limited imaging system. Because different spatial frequency components are mixed in the images acquired with detector, a series of low-resolution images with different angles and phases are collected to synthesize the super-resolution image. The frequency down conversion determined by the pattern period is also constrained by the diffraction limit, so linear SIM doubles the resolution of conventional microscopy [5].
Besides conventional line grid patterns, two-dimensional (2D) lattices have been employed for structured illumination because the 2D lattice patterns only require lateral phase shift and provide a better signal-to-noise ratio theoretically [11–14]. Such lattice patterns are generated mainly through interference method and a bulky 4f projection system is inevitable. The diffraction orders of a grating are diffracted onto a median plane where zero and other orders are filtered out and only the first orders are projected onto the rear aperture of the objective lens. The spatial frequency of the pattern gets higher when the first diffraction orders approach nearer opposite edges of the rear aperture.
Diffractive optical elements (DOEs) are also suitable for generating lattice patterns due to its high efficiency and design flexibility [15–17], however the spatial frequency is not easy to raise in the same way as interference method. Iterative methods such as modified Gerchberg-Saxton (G-S) algorithms as well as other non-iterative methods also have been developed for generating various lattice patterns [18–20]. Nevertheless, these methods including our previous work [21] all restrict the period of patterns no smaller than the size of Airy spot, which means the spatial frequency of lattice patterns cannot exceed 0.58 fco, where fco is the cutoff frequency and can be seen as the frequency of the fringes interfered by two ideal point sources at the edge of the entrance pupil. Although some methods have been proposed to generate sub-diffraction patterns [22–24], they only generated a single very small spot or a few periods rather than a lattice pattern. Therefore, it is necessary to propose a new method for designing DOE to raise the spatial frequency of the lattice patterns and then increase the imaging resolution.
Here, we propose a new method for DOE design to achieve high spatial frequency lattice patterns. Firstly, through a mathematical analysis we explain how to choose a propriate initial phase in detail to keep the subsequent iterative process away from divergence. Moreover, we modified the G-S algorithm to optimize the uniformity and efficiency of the lattice patterns. Using this method, we can obtain the illumination pattern with a high spatial frequency of 0.9fco. Generating such high spatial frequency lattice pattern with DOE, first reported to our knowledge, is the key factor to realize high resolution for SIM. We also further establish the system with the epi-illumination where the phase shifts are achieved through the light source movement and the DOE is mounted on the filter cube to provide a better support for oil immersion objective lens. As a result, the size of this compact illumination system is 75 mm × 75 mm × 60 mm, which is about 2 orders of magnitude smaller than that of existing commercial SIM [25]. Lastly, imaging comparisons, using both 100 nm fluorescence beads and biological specimens, indicate obvious resolution enhancement with the compact SIM system.
2. Design of DOE
Here we present an iterative method to generate high spatial frequency lattice patterns. Considering high spatial frequency of the target pattern in this case, we should reduce the sampling interval in the output plane. But as the sampling interval decreases, the number of points to be optimized increases, which makes the optimization difficult to be convergent with the GS algorithm and its existing modified versions [26]. The iterative process is sensitive to the initial phase choice, so it is possible to improve the optimization process by setting different starting points [27,28]. Instead of random phase in most iterative algorithms for designing DOE, we choose an explicit quadratic phase as the initial value to keep the subsequent iterative process away from encountering a local stagnation and assure a higher convergence speed. In what follows, we will present how to choose the propriate initial phase.
Assuming N represents the number of frequencies in the lattice pattern, for example, N is two for a square lattice and three for a hexagonal lattice. The ideal amplitude of lattice pattern can be expressed as
The fact that E(f) is phase-only and constrained by the system cut-off frequency, leads to D(f) also phase-only and frequency constrained. From this point of view, our purpose is to design a proper phase φ(r) superimposing on the pattern and keep the Fourier transform of the complex field ep(r) phase-only and band-limited. It is consistent with the flat-top beam shaping studied by some researchers before [29,30].
As described in Ref. [31], the Fourier transform of the quadratic phase is relative smooth and meanwhile band-limited when suitable parameters are set. Derived from the perspective of geometric optics, the field $\exp (i{\alpha _1}{x^2})$ with size Δx and the field $\exp (i{\alpha _2}{f_x}^2)$ with size Δfx are approximate Fourier transform pair if their parameters satisfy
Supposing fp is the target frequency for the lattice pattern, and FOV is circular with radius of r0, then the size Δfx is set as fco−fp to make those sperate Fourier spectrums fill the whole DOE region while not overlap with each other as far as possible. According to Eq. (4), the initial phase can be obtained as
The iterative process based on GS algorithm is depicted in Fig. 1. Note that some modifications are applied and will be explained in detail. The first one is the initial phase and it is clarified below. Next one is that the sampling rate of amplitude in the sample plane is reduced for describing target pattern here more precisely. The last modification is the constraint in the sample plane, compared with the direct replacement with the target amplitude, the amplitude constraint is defined as
It means the sample plane is separated into signal and noise windows by binary mask function d(r). γ is a constant between 0 to 1 for controlling the signal energy ratio. And ${\beta _{k + 1}} = \sqrt {{\beta _k}} \;({\beta _1} = 1{\textrm{0}^{\textrm{ - 9}}})$ is used for obtaining next iteration adaptive weight to accelerate the convergence speed.
With this method, we have generated both square and hexagonal lattices. The simulation result of square lattice and the corresponding spectrum are shown in Fig. 2. In the simulation, we use the same wavelength and effective numerical aperture as in the experiment, and the phase in the DOE plane is quantized with eight-step. Square lattice has nine orders, and the highest frequency in the diagonal direction we can design is very close to the cutoff frequency while in previous work this frequency is not higher than 0.58fco plotted by solid line [21].
Fig. 2. Simulation result and the spectrum of the square lattice. (a) Simulation of the square lattice and scale bar 1 µm, (b) Frequency orders in Fourier spectrum, the cut-off frequency is marked with dash line, the obtained highest frequency and the frequency realized by previous work are marked with dotted line and solid line, respectively.
Theoretically, for SIM, the hexagonal lattice can provide an isotropic resolution and a better signal-to-noise ratio than the square one [11,32]. Therefore, we choose hexagonal lattice as the illumination pattern in our SIM system. To evaluate this method, we design many DOEs to generate hexagonal lattice patterns with different spatial frequencies.
The diffraction efficiency is defined as the ratio of the signal energy within the target area over the total energy. The lost energy is shifted to the region away from the target area with our method. And the modulation depth of illumination pattern to evaluate the uniformity, is defined as the energy ratio of the highest frequency order over zero frequency order. High modulation depth assures high-quality construction for SIM, otherwise it will cause severe artifacts due to low signal-to-noise ratio raw images [33]. Therefore, we design our DOE to generate patterns while keeping the modulation depth over 80%. As shown in Fig. 3, the diffraction efficiency will remarkably decline with the increasing frequency when the ratio fp over fco (fp/fco) is larger than 0.5.
Although we have fabricated the DOE whose fp/fco is set as 0.9, the efficiency is low and the super-resolution construction is of poor quality because of high zero order intensity and inferior modulation in the experimental pattern resulting from the fabrication error. We have to use the DOE with the spatial frequency of 0.75fco to carry out the experiments, and Fig. 4 demonstrates the simulation and experimental patterns realized by such a DOE. Despite the intensity deviation, the experimental result is consistent with the simulation, especially from the detail comparison in the magnification view. Another convincing point lies that they have the same spatial frequency orders. Considering the NA, the cut-off frequency fco = 5.08 µm-1, and the highest frequency of pattern fp = 3.81 µm-1, which is consistent with the design target. The experimental efficiency is 41.4% by calculating the ratio of the energy in lattice area over the total incidence energy.
Fig. 4. Verification for DOE design. (a) Simulation result of hexagonal pattern and enlarged ROI in box. (b) experiment results of hexagonal pattern and enlarged region of interest (ROI) in box; Scale bar 1µm. (c, d) Fourier spectrums of (a) and (b) show all the frequency orders, and the cut-off frequency is marked with white dash line.
But the experimental pattern is deviated from the ideal one if we use a phase-determined coefficient matrix to separate different spectrum components similar in linear SIM, there will be artifact in the reconstruction image. Therefore, we use an iterative algorithm, termed as pattern-illuminated Fourier ptychography (pi-FP) detailed in Ref. [34], to avoid the artifact and robustly construct the super-resolution image.
3. SIM system with DOE
For the purpose of implementing the lattice as the illumination pattern, a beam modulation experimental set is established as shown in Fig. 5, and the DOE consists of 1250 × 1250 square pixels of 4µm size. In the illumination system the 488 nm laser beam coupled into a single mode fiber is collimated with a lens L1. After being modulated by the DOE, which is mounted on the filter cube (B-2A fluorescence filter cube, Nikon), the laser beam reflects from the dichroic mirror, passes through the objective (100×/1.45, Nikon) and forms a hexagonal lattice to illuminate the sample plane. The illumination system is highly compact because it generates the illumination pattern just through a single ∼1 mm thickness DOE mounted on the filter cube.
Fig. 5. The experimental setup. Illumination and fluorescence are shown in blue and green, respectively. (a) the phase in the DOE plane. (b) Simulation of hexagonal pattern in the sample plane. (c) DOE mounted on the filter cube.
Epi-fluorescence illumination is used here for further simplifying the whole system. The image system shares the same objective (100×/1.45, Nikon) with the illumination system. An emission filter is used for filtering out the excitation light (Sapphire 488, Coherent), and the fluorescence raw images are captured by an sCMOS camera (edge 4.2, pco.). Furthermore, the phase shift unit is established specially to provide a better support for oil immersion objective. The fiber port is stick to a motorized stage (C-483, PI), scanning through 25 (5 × 5) different positions to provide phase shifts. Because the objective and collimation lens L1 form a telescope system, the displacement of fiber port can be transferred with a certain scale to the sample plane. The step size of phase shift in the sample plane is 1/5fp in x direction and 2/(${5}\sqrt {3} $fp) in y direction. Considering that the highest spatial frequency of hexagonal lattice used in the experiment is 3.81µm-1 and the magnification of the telescope system is 20×, the scanning step size is 2.10µm in x direction and 2.42µm in y direction.
4. Experimental results
4.1 Quantitative resolution measurement with fluorescence beads
The resolution performance of this system is testified quantitatively by imaging a cluster of 100 nm fluorescence microspheres (F36924, ThermoFisher). Comparing to the diffraction-limited images, the resolution enhancement is clearly visible with the SIM system shown in Fig. 6. In this paper, the widefield image is the average of all scanned raw images. In addition, those raw images could be used to estimate the modulation depth. We compute the modulation depth per pixel as shown in Fig. S3. Then the modulation depth of 20 pixels, mostly the center of the beads, are averaged to estimate the modulation depth of the lattice pattern as 0.712.
Fig. 6. Imaging results of fluorescence beads. (a) Widefield image of fluorescence beads and enlarged ROI (lower right) in the boxed region, and Fourier spectrum (upper right). (b) SIM image of fluorescence beads and enlarged ROI (lower right) in the boxed region, and Fourier spectrum (upper right). (c) Intensity profiles of fluorescence beads on the selected line in (a) and (b). (d) Profiles of PSFs in widefield microscope and SIM system. Scale bar 1 µm.
As shown in the enlarged images that two adjacent fluorescence beads with a central distance of 177 nm can be resolved in the SIM system but the situation is not in widefield microscope. Intensity distributions of 20 beads are fitted to Gaussian model and averaged to approximately represent the point spread function (PSF) of the image system, and the full-width-at-half-maximum (FWHM) value of the function also backs up the resolution enhancement. Here the average size of the FWHM in the SIM system amounts 131 nm, while 240 nm in widefield microscope, meeting the diffraction limit considering the fluorescence label (Ex488/Em519) and objective (100×/1.45 NA, Nikon) used in this experiment. In addition, the Fourier spectrum of the construction image is expanded in all direction compared to that of the widefield image, which verifies the isotropic resolution improvement with hexagonal lattice pattern.
4.2 Super-resolution imaging with fixed cells
To demonstrate the SIM on biological sample we use a FluoCell Prepared Slide #1 (F36924, Thermofisher), which contains bovine pulmonary artery endothelial (BPAE) cells stained with green-fluorescent Alexa Fluor 488 phalloidin. Figure 7 presents the obtained diffraction-limited image and the SIM reconstruction. The intensity profiles on the yellow selected lines present that the SIM visualizes the separation of two neighbor filaments, which are not resolved in the diffraction-limited image. The Fourier spectrum of the construction image is expanded compared to that of the widefield image. Furthermore, we evaluate the results through analysis of circular averaged power spectral density (PSDca) [35]. This method estimates the resolvable frequency at which the spectral power density decreases to noise level, and it also supports the resolution enhancement of the SIM system, from 230 nm to 137 nm.
Fig. 7. Imaging results for tubulin of BPAE cells. (a) diffraction-limited image of tubulins, enlarged ROI (lower left), and Fourier spectrum (upper right). (b) SIM image of tubulins, enlarged ROI (lower left), and Fourier spectrum (upper right). (c) Intensity profiles on the yellow lines in (a) and (b). (d) Circular average power spectral density of image in (a) and (b). Scale bar 1 µm.
5. Discussion and conclusion
In conclusion, we use a well-designed DOE to achieve a lattice pattern with high spatial frequency approaching to the cut-off spatial frequency for illumination. The SIM system, mounting the DOE on the filter cube, avoids extra room compared to an inverted fluorescence microscope. The experiment lateral resolution is quantitatively testified by measuring FWHM of fluorescence beads and circular average power spectral density of BPAE cells. This advantage of miniaturization may help this SIM system more suitable for those imaging applications where both resolution and portability are needed.
We have experimentally demonstrated the pattern with spatial frequency of 0.75fco, and the simulation have shown spatial frequency of 0.9fco, but this high spatial frequency DOE is not applied here because of the low efficiency and poor pattern quality resulting from the fabrication error of the DOE. It may be improved by optimizing the super-resolution construction algorithm to alleviate the influence of illumination pattern. Furthermore, an ultra-high spatial frequency over the cut-off frequency still deserves to be explored for more than two-fold resolution enhancement. It suggested that multi-lobe superoscillating pattern with spatial frequency over the cut-off frequency is potential to be used in SIM [36]. Theoretically, this multi-lobe superoscillating pattern may be generated with DOE, but the key is to separate the superoscillating area with those strong lobes and increase the number of the oscillations, which will be further explored.
Funding
National Natural Science Foundation of China (Grant No. 62075112).
Acknowledgments
We acknowledge Assistant Professor Lingfeng Pan of Institute of Semiconductors, Chinese Academy of Sciences, for fabricating the designed DOEs.
Disclosures
The authors declare that there are no conflicts of interest related to this paper.
Data Availability
Raw data presented in this paper and code for analyzing the data are available in Code 1, Ref. [37].
Supplemental document
See Supplement 1 for supporting content.
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