1. Introduction

Owing to the diffraction barrier in the imaging system, the resolving ability of conventional far-field fluorescence microscopy has long been limited. Therefore, its performance in nanoscale science has been inevitably restricted [1, 2]. Though electron microscope techniques can enhance the spatial resolution, they cannot be applied to observing living sample and still do not break the diffraction barrier. Several techniques have been proposed to overcome the diffraction limit [3–8]. Among these techniques, fluorescence emission difference microscopy (FED) provides an approach to perform investigations at nanoscale by means of subtraction, which can be used to analyze biological samples without the requirement of special dyes.

In FED, the zero and 0–2π vortex phases are alternately loaded on the spatial light modulator (SLM) to generate the corresponding solid and doughnut excitation spots [9]. The final image is obtained by the subtraction of two scanning images modulated by two different beams. Previous studies have demonstrated that a sharp effective point spread function (PSF) could be created as a result of subtraction, and the resolution can be enhanced by increasing the contained high-frequency components compared to the conventional method [7]. Numerical analyses of FED have been completed to enhance its performance in some aspects [10–12]. Although the resolving ability of conventional FED cannot rival those of stimulated emission depletion microscopy (STED) or stochastic optical reconstruction microscopy (STORM), it is straightforward to implement in various optical systems because of its simple setup and lack of specimen constraints. Previous studies have proved its practical performance of resolution and contrast enhancement in Second-harmonic imaging microscopy (SHIM) [13], two photon fluorescence (TPF) microscopy [14], and Coherent Anti-Stokes Raman Scattering (CARS) microscopy [15]. In addition, compared with other methods that are not constrained by the specimen, FED has the advantage of fewer recording times and straightforward processing over structure illumination microscopy (SIM) [16]. Also it has the advantage of freedom from a large quantity of detectors and ease of acquiring a signal over Array scan [17].

However, the previous FED has risks of deformation and information loss when the subtraction factor becomes higher, which results from a mismatch of the two acquired confocal images. In the previous system, the doughnut excitation pattern is obtained by modulating a circularly polarized beam with a 0–2π vortex phase plate. The outer contour of the doughnut excitation spot is much larger than that of the solid illuminating pattern. Thus, the effective excitation point spread function (PSF) after subtraction will have some negative side-lobes. In order to obtain a better result, the negative values are reset to zero during the reconstruction process, which leads to a certain degree of information loss and may degrade the image’s quality. The information loss is considerably more severe when the sample becomes dense, which strongly limits the feasibility of FED. The key solution to this problem is to discover methods to generate a solid spot and doughnut spot with approximately equal outer contours. In previous studies, Rong et al. established a cylindrical vector beam FED system in which an azimuthally polarized beam and a radially polarized beam create a shrunken hollow spot and an extended solid spot, respectively [18]. You et al. enhanced the FED performance by extending the generated solid spot through modulating a circularly polarized beam with the same 0–2π vortex with an opposite vortex direction corresponding to the doughnut generating method [19]. Ma et al. improved the FED system by using a photon reassignment method that we call virtual fluorescence emission difference microscopy (vFED) [20]. The extended solid spot is obtained by the photon reassignment of a hollow spot, which realizes the FED method with only one scan of the hollow excitation pattern.

Though the three methods mentioned above decrease the deformation of the FED results to a certain degree, the cut-off spatial frequency remains the same, restricting the further improvement of the resolution. Saturation methods have been used to reduce the size of the resulting PSF for several decades, such as STED and saturated excitation microscopy (SAX) [21]. STED uses the depletion of fluorescence emission upon saturation to reduce the size of the detected PSF. SAX enables super-resolution simply by modulating the excitation intensity at a single frequency and demodulating the emission intensity at harmonic frequencies, where the extracted nonlinear components contribute to high-resolution images. The previous work of Liu et al. introduces this phenomenon to the vFED method (we call it svFED) [22] by taking a cut-top approximation of the saturated fluorescence effect. The nonlinear effect shrinks the center of doughnut spot and consequently widens the cut-off spatial frequency of the resulting PSF, achieving a resolving ability better than any of the FED methods mentioned above. It is obvious that svFED improves the imaging performance compared to the nonsaturation FED. However, because of the pixel reassignment procedure, the acquired solid spot in svFED has a risk of being too flat compared with the direct saturation method. As a result, positive side-lobes appear. Our new discovery reveals that svFED can be optimized to decrease the positive side-lobes by changing the distance between the adjacent detectors at a certain saturation intensity [22]. Currently, in actual situations, it is difficult to implement because the distances of the detectors are located and the illumination power always needs adjustment as well. Moreover, the theoretical model in the study of Liu et al. could be more concrete and use specific samples.

In this study, the saturation effect in FED is further discussed. Based on a point detector, we present a simple but effective method to exploit the saturation phenomenon with FED (called sFED) to obtain a considerably higher resolution result with the least deformation. By using a saturated solid excitation spot, a flattened and extended solid focal spot is obtained. Meanwhile, by using a saturated doughnut excitation spot, a center-shrunken doughnut focal spot is obtained. Because of the nonlinear effect caused by the saturation of the fluorescence emission at high illumination intensities, both the solid and hollow focal spots contain high spatial frequency components. The center-shrunken doughnut focal spot indirectly improves the resolving ability of FED with notable performance. Simulations of a specific sample of the fluorescent molecule rhodamine 6G are introduced [23]. Additionally, experiments on nanoparticles and biological samples are performed to verify the feasibility in biological observations. Imaging results of 40 nm fluorescent nanoparticles show that sFED can discern two peaks with a distance of 108 nm at the excitation wavelength of 633 nm.

2. Theory

2.1 Saturation effect in confocal microcopy

The FED system is normally built upon a confocal microscopy system with simple modifications [24–26]. To thoroughly understand the effects of saturated excitation in our method, we first analyze the saturation model in confocal microscopy.

The effect of saturation is illustrated by calculating a photo-physical model with a five-level molecular electronic state. The model is based on rhodamine 6G as the fluorescent molecule excited by 532 nm CW laser light with an exposure time of 0.24 ms for irradiance conditions. The absorption cross-section is 2.7 × 10⁻¹⁶ cm² and the quantum efficiency is ~0.95. Simulation parameters and saturation relationship of rhodamine 6G are from reference [23].Populations of the S_n and T_n states are neglected. We determine the fluorescence intensities resulting from various excitation irradiances through solving the rate equations describing the population of fluorescent molecules in each state. The calculated relationship between the emission and excitation intensities is shown in Fig. 1.

 

Fig. 1 Saturation effect in fluorescence emission from rhodamine 6G molecules calculated using the five-level molecular electronic state model. The black dashed line denotes the excitation-emission curve with photo-bleaching ignored. The solid line denotes the excitation-emission curve with photo-bleaching considered. The blue line segment denotes the linear response portion, the red line segment denotes the nonlinear portion where the saturated effect is evident, and the green line segment denotes the portion where bleaching appears to be evident and cannot be ignored. The dashed black line represents the simulated data without any bleaching reaction.

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The effective PSFs of confocal fluorescence microscopy with saturated excitation can also be calculated. A confocal fluorescence microscopy PSF $h_c$ is given as [9]:

where$h_{em}$ and $h_{\det }$ are the excitation and detection PSFs, respectively, $p(x,y)$ denotes the 2D transmission function of the pinhole $h_{exc}$, and $h_{\det }$ is obtained by calculating the vectorial diffraction theory$\xi (i)$ represents the excitation probability.

Under nonsaturation situations, $\xi (i)$ is merely a constant related to the essence of the sample, such as rhodamine 6G. $h_{em}$is proportional to the excitation PSF, $h_{exc}$ and the relationship corresponds to the linear response portion of the excitation-emission curve in Fig. 1.

While under saturation situations at high excitation intensities, $\xi (i)$is related to the essence of the sample and the illumination intensity $I$. $h_{em}$ is no longer proportional to$h_{exc}$, the relationship corresponds to the nonlinear region in Fig. 1.

Moreover, as we can see Fig. 1, the simulated data indicate that bleaching becomes evident when the incident intensity exceeds 250 kW/cm². Notably, the instant rhodamine 6G is generated from its essential elements, the actual numerical saturation and photobleaching relationship with illumination intensities vary among different samples.

2.2 Principle of saturated fluorescence emission difference microscopy

According to the concept of FED, two confocal images are generated and then passed through a subtraction process. One image is illuminated by a solid pattern, and the other is illuminated by a doughnut pattern with a dark spot at its center. The subtraction process is given by:

where $I_{sFED}$, $I_{ssolid}$, and $I_{sdoughnut}$ are the normalized intensity distribution of$sFED$ image, saturated confocal image, and saturated doughnut confocal image, respectively, and $r$ represents the subtractive factor. The value of $r$ should be set properly according to both the excitation intensity and the ratio of non-normalized $soli{d}_{\mathrm{int}ensity}/doughnu{t}_{\mathrm{int}ensity}$ (called $q$). Normally, it has been calculated with vectorial diffraction theory that under the same power incident onto the SLM, the maximum intensities of solid spot and laterally hollow spot are 3.54 (arbitrary units) and 1.03, respectively [27]. In the simulations part of this paper, for a uniform simulation, the value of q is set to 1.As we have referred in introduction part, due to the mismatch of doughnut and solid profiles, negative values are inevitable in subtraction, therefore the results are accompanied by the negative values. In order to present an acceptable image with only positive pixels, the negative values of imaging results are all set to 0. Considering that this operation is at the risk of discarding useful signal, one aim of this paper is introducing saturated solid PSF to better match the profile of doughnut one, thereby counteracting the negative deformations even brought by saturation doughnut PSF. The excitation intensity in the following sections denotes the peak power density the excitation PSF. Given this situation, a higher $r$ decreases the full width at half maximum (FWHM) and induces negative artifacts. In most former studies, $r$ is empirically set to 0.7. In the simulations as well as experiments of this paper, the selection of subtraction is determined by comparing the confocal image and resulting FED image after subtraction. As we continuing adjusting the subtraction factor with self-developed software, meanwhile artificially evaluating the trade-off of resulting resolution enhancement and information loss through comparison, we obtain the roughly optimum imaging result of FED and sFED. Noting that a numerical study of the subtraction factor has been completed by Wang et al. [10], which analyzes the value of $r$ in different situations and concludes that the optimum value for this coefficient is strictly dependent on sample type and imaging mode. In addition, since regarding the subtraction factor of every pixel in a whole acquired image to be equal is a little bold, recent study by Korobchevskaya et.al. presents an algorithm optimized the value of $r$pixel by pixel, in their study, $r$ is weighted and adjusted pixel-by-pixel with respect to the intensity relationship of the two subtraction components [28]. The research gives inspirations of quantitative and meticulous manipulation of subtraction factor.

2.3 Point spread function and optical transfer function of the saturated FED

Assume that the numerical aperture of the objective is 1.4 and the magnification is 100X, the peak wavelength of the fluorescence is 532 nm. Based on Eqs. (1) and (2) and the result in Fig. 1, the effective 2D PSFs are calculated in Fig. 2. Figures 2(a)–2(c) shows the confocal PSFs with excitation intensities of 3 kW/cm², 30 kW/cm², and 100 kW/cm². Figures 2(d)–2(f) and Figs. 2(g)–2(l) denote the doughnut and FED PSFs at the corresponding intensities. The negative values in the FED results are not set to zero. As we can easily see in the figures, the saturation effect widens the profiles of the solid spots and shrinks the center of doughnut spots as the excitation intensity increases. Meanwhile, when taking a carefully look at Figs. 2(d)–2(f), profiles of the doughnut spot are slightly narrowed due to the saturation effect. Combining the narrowed doughnut spot profiles and the expanded solid profiles leads to a better match of the two subtraction components, resulting in distinctly lower negative values in Fig. 2(i) than that in Fig. 2(g), thus achieving a nearly non-negative (in other words, nondeformation) difference value. In addition, the saturated doughnut spots contribute to the resolving enhancement, which is demonstrated by the narrowed PSFs in Figs. 2(g)–2(i). Previous FED work has validated that the valley of the negative side-lobe can be −0.2 or even larger without too much distortion of the image [19]. In our work, a better match of the solid and doughnut images enables us to achieve a PSF of even smaller size than the conventional FED at the expense of fewer deformations.

 

Fig. 2 Saturated PSFs of normalized intensity distributions at corresponding excitation intensities. Side lengths of each square are 2λ. (a)–(c) Confocal solid PSFs at 3 kW/cm², 30 kW/cm², and 100 kW/cm², respectively. (d)–(f) Confocal doughnut PSFs at the corresponding irradiances. (g)–(i) FED PSFs at corresponding irradiances. The subtraction factors are all set to 0.72.

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To take a closer look at the enhancement of the FWHM of the obtained PSF and profile-match of the subtraction components, linear profiles of normalized intensities are extracted from Fig. 2 and illustrated in Fig. 3. It is obvious that the two curves representing the subtraction components in Fig. 3(a) are poorly matched, and the negative side-lobes are consequently very large. However, as the intensities increase, the two curves tend to be well matched in Figs. 3(b) and 3(c), and the negative side-lobes consequently decrease, which agrees with the discussions in the above paragraph. Moreover, the narrowed FED images validate the enhanced resolving ability as well. In Fig. 3(d), which illustrates of PSFs of confocal microscopy, FED, and sFED at 100 kW/cm², the direct comparison gives a distinct validation. Exactly, since the degree of photobleaching and the resolution enhancement cannot be quantified in the same dimension, in the simulation model of rhodamine 6G, there is no absolute intensity that represent the best point where we can get the optimum result. As can be seen in Fig. 1, the photobleaching becomes a little obvious when the incident power surpasses 100 kW/cm², and more evident at the point of 250 kW/cm² at where continuing increasing the incident power will conversely decrease the emission intensity. In addition, as referred in other saturated super-resolution methods such as STED or ESSat, the decrease of the FWHM of the doughnut center versus the incident power obey a square-root law [29,30], which means that considering the shape of solid spot, parameters such as $r$, as well as the environment factors, the resolution enhance with the intensity increase from 100 kW/cm² to 250 kW/cm² is no more than $\sqrt{2}$ . Therefore, we take the intensity of 100 kW/cm² to represent the relative optimized simulation intensity of rhodamine 6G.

 

Fig. 3 (a) Linear profiles of solid, doughnut, and FED results at the illumination intensity of 3 kW/cm². (b) Linear profiles of the corresponding components at the illumination intensity of 30 kW/cm². (c) Linear profiles of the corresponding components at the illumination intensity of 100 kW/cm². The black, red, and green curves in (a)–(c) denote the linear profile of confocal, doughnut, and FED PSFs, respectively. (d) Linear profiles of confocal microscopy (black line, consistent to the ones in (a)-(c)), in conventional FED (red line, obtained without illumination intensity considered, with the subtraction factor of 0.72) and sFED (blue line, obtained from the red profile in (c)). (e) Normalized effective OTF profiles analogous to Fig. 3(d).

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It is noted that the FWHM of the PSF in sFED is much sharper than that of confocal microscopy and FED images in Fig. 3(a). The disappearance of positive side-lobes would dramatically improve the imaging quality as well.

Optical transfer functions analogous to Fig. 3(d) are calculated by performing the two-dimensional (2D) Fourier transform of the corresponding calculated PSFs. The spatial frequency is normalized by$\text{2}\pi NA/\lambda$. Calculation results in Fig. 3(e) show that the OTF of FED is broadened to include higher spatial frequencies with higher intensities. The cut-off frequency of sFED is obviously higher than that of FED and contains more high frequency components, indicting more details can be detected. It can be easily derived that the cut-off frequency of FED is approximately 3.5 units, while the counterpart of sFED is about 10 units.

From the above calculations, we preliminarily confirm that the saturated effect of both solid and doughnut spots can be used with the FED method to further improve the resolving performance.

3. Simulations

In this simulation, basic parameters including, but not limited to wavelength and numerical aperture (NA), are all consistent with counterparts in the theory section. The process of simulation is described Fig. 4. As can be seen from the Fig. 4, when the doughnut or solid illumination PSFs imposed on the sample, the saturation effect results in the deformation of emission PSFs. After the detecting process expressed in Eq. (1), we obtain the system PSF of saturation doughnut and solid PSFs. Then the imaging procedure is processed by convoluting the system PSFs with the simulated sample. Finally, through the subtraction procedure, we obtained the sFED result.

 

Fig. 4 Flow chart of simulation process of sFED.

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According to the process, a sample composed of a 25-point array shown in Fig. 5(a) is imaged to verify the improvements in lateral resolution and to test the deformation levels. The distance between the adjacent points is 0.2λ with the side length of the square being 0.08λ. The white points have twice the intensity of the orange ones. It is obvious that the confocal image in Fig. 5(b) can hardly separate the 25 points. Figure 5(c) shows the FED results of the white points at the margin being distinguishable; however, the excess subtraction and limited size of the PSF lead to the disappearance of the orange points, causing information loss and deformation in the reconstructed image. The sFED result in Fig. 5(d) shows a good performance in discerning the white and orange points, where all points are completely resolved. Notably, the profiles of imaging results of white points are much larger than that of orange ones, this consequence is attributed to the not perfectly match of the solid and doughnut PSFs. Though the consistency of the size of the points in Fig. 5(d) are still not perfect, when compared to the results in Fig. 5(c), the above results demonstrate the capability of sFED to enhance the lateral resolution with decreased deformation.

 

Fig. 5 Imaging results of a 5 × 5-point array sample. (a) Designed array sample box, side lengths of the squares are 0.08λ, and the distance between adjacent points is 0.2λ. Normalized intensity imaging results with (b) confocal, (c) FED (r = 0.75), and (d) sFED (r = 1) with an excitation intensity of 100 kW/cm².

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Then, in order to demonstrate the enhancement of the resolving ability of sFED in complex samples, a sample of computer-simulated cell microtubules shown in Fig. 6(a) is used to verify the resolving ability in imaging biological samples and to demonstrate the extent of deformation of sFED. Thirty cell microtubules of 20-nm-width are simulated as randomly distributed in the 6λ × 6λ scope. The subtraction factor for FED, and sFED are set equal to the counterparts in Fig. 5. Figure 6(b) shows the vague imaging results of the sample with the confocal method, where the microtubules at the lower right corner can barely be distinguished. The imaging results of FED in Fig. 6(c) enable more details to be discerned, while inducing information loss that corresponds to the negative side-lobes of the red curve line in Fig. 3(d). Compared with the above two methods, the sFED result in Fig. 6(d) shows the best resolving ability with the minimum unresolved area. Some microtubules stuck together at the lower right corner in Fig. 6(b) can be clearly separated. Meanwhile, the evident information loss of FED is well avoided by this method. Though some intense near microtubules are undistinguishable, an sFED image still performs much better than the former methods. Magnified views of the regions indicated by boxes in Figs. 6(a)–6(d) provide a comprehensive comparison of imaging abilities, which further verifies the superiority of sFED.

 

Fig. 6 Simulation results of a sample of computer simulated microtubules, side-lengths of the scope are 6λ. (a) Microtubules sample. Normalized intensity imaging results with (b) the confocal technique, (c) conventional FED technique (r = 0.75), and (d) sFED technique at the excitation intensity of 100 kW/cm². (r = 0.8). (g)–(h) Magnified views of the regions indicated by the boxes in (a)–(d) with the corresponding color of the outer contour.

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4. Experiments

4.1 Proposed configuration of sFED

The schematic of the sFED system is depicted in Fig. 7. After being collimated by a collimating lens (CL1), the CW laser beam (Coherent CL 2000, 633 nm) becomes P polarized through a polarizing beam splitter (PBS). A half wave plate (HWP1) placed ahead of the PBS is utilized to tune the light intensity illuminated on the sample. In order to fit the polarization demand of the SLM, HWP2 is utilized to compensate the tolerance introduced by the misplacement of optical devices and adjust the incident polarization angle into precise P polarization. The quarter-wave plate (QWP) changes the linearly polarized beam into a circularly polarized beam. A 1.4 NA oil immersion objective lens (OB, Olympus Uplan SApo 100x/1.4 Oil) strongly focuses the circularly polarized beam and collects the fluorescence emitted from the sample. The Galva mirror (Thorlabs GVS002) aids realization of two-dimensional fast scans of the sample. A dichroic mirror (DM, Chroma ZT647 rdc-uf3) reflects the illumination while transmitting the fluorescence, which is then converged by a converging lens (CL2) after spatial filtering by a pinhole and coupled into a multi-mode fiber (MMF, M31L02, Thorlabs). Finally, the fluorescence will be detected by an avalanche photon diode (APD, SPCM-AQR-16-FC, PerkinElmer). The scanning of the Galva mirror, combined with the zero and 0–2π vortex phases that are alternately loaded on the Spatial light modulator (SLM, model 512-633, Boulder Nonlinear Systems, Inc.), help obtain 2D confocal images of the different modes.

 

Fig. 7 Schematic of the sFED system.

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4.2 Experiments on fluorescent nanoparticles

A test experiment of 100 nm fluorescent nanoparticles (T7279, TetraSpeck Microspheres, 0.1 µm, fluorescent blue/green/orange/dark red) is first performed to examine the relationship of PSFs to illumination powers. Figures 8(a)–8(f) present the confocal and doughnut images of nanoparticles in a 3.84 × 3.84 µm area. Texts that show the illumination powers of each subfigure are inserted at the top right corner, measured by an optical power meter at the entrance pupil of the objective lens. Notably, the illumination power ($P$) have a certain relationship with the illumination intensity ($I$) regarding the illumination area ($A$), given by:$P\text{=}A\text{*}I$. For instance, at ${\lambda }_{\text{excitation}}\text{=633nm}$, an oil immersion lens of NA = 1.4 yields $A\text{=2}\text{.7}\times {\text{10}}^{\text{-9}}{\text{cm}}^2$as the doughnut area. Then the illumination power of 4 µW in Fig. 8(a) corresponds to an illumination intensity of 1.5 kW/cm². It is noteworthy that when looking at the color bars beneath Figs. 7(a) and 7(b), the peak detection of photon counts in Fig. 8(b) is 539, which is nearly two fold greater than that (270) in Fig. 8(a). The two-fold increase of count detection shows a great linearity with the illumination powers, which corresponds to the relationship in the linear region in Fig. 1. However, the continuing increase in illumination power will suffer obvious saturation effects. A four-fold power enhancement from Figs. 8(b) to 8(c) is accompanied by an increase of peak photon counts of only ~160%. When comparing Figs. 8(c) and 8(d), the same power increase will only render an amount of ~140%.

 

Fig. 8 (a)–(d) Confocal results at the corresponding powers indicated on the top-right corner, (e)–(f) doughnut results at the corresponding powers indicated on the top-right corner, (i)–(l) comparison of an extracted confocal image, an FED image of a solid at illumination power of 4 µW subtracted by a doughnut at illumination power of 2.8 µW, a doughnut saturated FED image of a solid at illumination power of 4 µW subtracted by a doughnut at illumination power of 717 µW, and a saturated FED (sFED) image of a solid at illumination power of 128 µW subtracted by a doughnut at illumination power of 717 µW, respectively. The position of the single spot corresponds to the position indicated by the light blue box in (a). (m) Line profiles of confocal, FED, and sFED spots indicated by a green double arrow line in (i). Subtracted factors are set equally to 0.9 for a fair comparison. Pixel size is 15 nm, and the per dwell time is 0.1 ms.

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As shown in the figures, when the illumination power increases, the profiles of the confocal spots widen; meanwhile, the centers of the doughnut spots shrink. In Fig. 8(a) and 8(e), the outer profiles of the confocal spots are much narrower than those of the doughnut, while in Figs. 8(d) and 8(h), though the profiles of the doughnut spots widen a little, the broadening of solid spots is more evident, which yields a better match of the two subtractors. In order to characterize the deformations, none of the negative pixels in subtraction results is set to zero. As can be seen in Figs. 8(i) and 8(j), the resulting PSF of FED is much sharper than that of confocal microscopy. Though the doughnut saturated FED result in Fig. 8(k) perform even better for PSFs, the negative values appear distinct, which can be as high as −0.95, rendering the risk of suffering severe deformations. Then, the performance of sFED effectively settles the problem because of the increased power of the confocal image, the negative values in Fig. 8(l) are distinctly lower than those in both Figs. 8(j) and 8(k), and most values of the negative pixels are well below −0.25. Moreover, the acquired PSF is narrowed as well. As can be seen in Fig. 8(m), line profile across the bead in Fig. 8(i) shows a FWHM of 105 nm for sFED,which reduces the PSF FWHM by 2.5 fold comparing with Confocal. However, because the existing negative values for FED, this reduced FWHM cannot exactly represent the resolution improvement, and it is mentioned here just for reference. A more convincible resolution criterion would be characterized by the minimum discernable distance of two peaks of nanoparticle images, which will be discussed in the next section.

Next, to evaluate the ability of resolution enhancement in sFED, 40 nm fluorescent nanoparticles (F8789-FluoSphere Carboxylate-Modified Microspheres, 0.04 µm, dark red (660,680)) are imaged in Fig. 9. Most of closely spaced nanoparticles that are not resolved in the confocal reference image are clearly discernible in the following FED and sFED images. The green, blue, and yellow boxes in Fig. 9(a) and Figs. 9(e)–9(h) indicate the highlighted regions we compared. When it comes to the areas indicated by these colored boxes, the performance of FED can no longer rival that of sFED. Three groups of nanoparticles all appear in one spot in confocal microscopy and FED due to the limited resolving ability, whereas, in the sFED image, each group of spots is resolved into two spots. Even in Fig. 9(g), where we implement an iterative image restoration (Richardson-Lucy) algorithm on the raw FED data, the undistinguishable spots still cannot be resolved, while in Fig. 9(h), the deconvoluted sFED image with the same algorithm further separates these spots and improve the contrast and resolution. Moreover, compared to conventional FED, the signal to noise ratio (SNR) is improved by the saturation, as shown in Figs. 9(j1) and 9(j2). The FED histogram suffers a large quantity of sharp peaks, which represent the isolated noise in the processed figure, while in the sFED histogram, the problem is well settled and values of the histogram drop more smoothly. The key reason of this enhancement comes directly from a better match of the two subtraction profiles. A main concern for this sFED approach is the practical extent of photo-bleaching. Figure 9(k) shows the re-acquired saturated confocal image after the entire cycle of Figs. 9(a)–9(d). The result evidences a good persistence of photons by sFED. As can also be seen by the red boxes in Figs. 9(c) and 9(k), the loss of peak fluorescence photons is smaller than 11%, which strongly supports the photo stability of our method. Finally, normalized intensity profiles confocal microscopy (Fig. 9(a)), FED (Fig. 9(e)), and sFED (Fig. 9(f)) of raw data are performed in Fig. 9(l) to evaluate the resolution. The confocal profile as well as the FED profile features only one obvious peak with a broad flat top, while the sFED profile has two peaks separated by a distance of 108 nm, approximately $\lambda /6$, which evidences the powerful resolving ability of sFED.

 

Fig. 9 Experimental results of 40 nm nanoparticles. (a)–(d) Sequentially acquired confocal images at an illumination power of 6.7 µW, doughnut images at an illumination power of 3.6 µW, saturated confocal images at an illumination power of 100 µW, and saturated doughnut images at an illumination power of 140 µW. (e) FED image obtained by subtracting (a) and (b). (f) sFED image obtained by subtracting (c) and (d). (g)–(h) Deconvoluted images of (e) and (f) with the Richardson-Lucy algorithm. (i1)–(i3) OTFs of confocal microscopy, FED, and sFED. (j1)–(j2) Histogram of FED, sFED. (k) Re-acquired saturated confocal image at an illumination power of 100 µW after the entire cycle of image acquisition from (a) to (d). (l) Line profiles of confocal, FED, sFED images indicated by the light blue line in (a). Notably, in order not to blur the corresponding spots in Figs. 9(e) and 9(f), positions of the normalized intensity profiles are only indicated by one light blue dashed line in Fig. 9 (a). Pixels size are 15 nm, and the per dwell time is 0.1 ms. The subtraction factors of both FED and sFED are set equally to 1 for a fair comparison.

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4.3 Experiments on biological cells

In Fig. 10, we image the vero cells (Vero MeOH, Vim-S635P), the sample was once affiliated with the commercial STED system sold by Abberior Instruments. Unlike the fluorescent nanoparticles, the cell sample appears more sensitive to illumination power, and the saturation effect occurs at comparatively low illumination powers.

 

Fig. 10 Experimental results of vero cells. (a)–(d) Confocal image at the illumination power of 0.67 µW, FED image obtained by subtracting the confocal image at 0.6 µW and doughnut at 0.6 µW, doughnut saturated image obtained by subtracting the confocal image at 0.6 µW and saturated doughnut at 4 µW, sFED image obtained by subtracting the confocal image at 3 µW and doughnut at 4 µW, respectively. (e1)–(e4) Enlarged views of regions indicated by the white boxes in (a)–(d), (f1)–(f4) deconvoluted results of the magnified regions with an inserted colorbar for the above 2D figures. (g) Line profiles across the indicated line in the confocal, FED, and sFED images. Pixel size is 20 nm, and the per dwell time is 0.1 ms. All the subtraction factors are set equally to 0.89 for a fair comparison.

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It is clear that the results in Figs. 10(b)–10(d) improve both the contrast and resolution of cell images compared with that of the confocal reference image. It can also be seen that in the sFED image more details are discernable, and some samples that stuck together in the FED image are distinguished in the latter sFED image. Besides, because of the improved illumination intensities and saturation effect, the contrast of the processed image is also improved by the saturated methods. Samples analogous to the dim region at top right corner of FED are clearly seen in the sFED image. Although the doughnut-saturated image in Fig. 10(c) shows a narrowed image, some sections of samples are missing in the image, suffering from evident information loss degrading the imaging quality.

Enlarged views in Figs. 10(e1) and 10(e4) enable a more intuitive comparison of results. Then, the Richardson-Lucy algorithm is further applied on these magnified results, the corresponding figures are Figs. 10(f1) and 10(f4), where the sFED image, which discerns more details with virtually no deformations, evidences a dramatic increase in the resolution. Highlighted regions with the green boxes prove this as well. Additionally, as indicted by the green arrows in Figs. 10(f3) and 10(f4), the results verify the decrease of information loss via the sFED method. Meanwhile, the deconvoluted results perfectly coincide with the simulations in section 4. Depicted in Fig. 10(g), line profiles across the sample images reveal a discernable distance of 175 nm. Considering that the density of the sample is larger than any of the samples used in the previous FED experiments, the new sFED system achieves the best experimental performance among all of the published FED results.

5. Discussion and conclusion

This paper introduces and validates a simple sFED concept to further improve the transverse resolution and eliminate FED deformations. By exploiting saturation phenomenon with both solid and doughnut excitation beams, better-matched profiles and center-shrunken doughnut focal spots indirectly improve the resolving ability. A specific theory concerning the concrete characteristics of samples is analyzed. Subsequently, detailed and thorough experiments are performed to verify the method.

It is noted that the results in this paper have not reached the ultimate sFED limit. Illumination power imposed on samples can be systematically studied and optimized. Continued increases in the irradiance will undoubtedly enhance the resolution, while the accompanied photo-bleaching would limit the enhancement. However, further modifications, such as introducing a pulsed laser as well as the bleaching-stubborn fluorescent dyes in STED or other saturation based methods, would alleviate it.

The experiment in this paper demonstrates that with the nonlinear effect, which broadens the cut-off frequency, the method of FED can exactly surpass the diffraction limit (approximately $\lambda \text{/6}$). Practically, among the past explorations to overcome the diffraction limit, a main direction is firstly utilizing mathematical or physical process methods to fill the high frequency part within the diffraction limit, e.g. SIM, or STED without saturation of doughnut beam, and then truly surpass it with the saturation effect, e.g. SSIM [31], and conventional STED. The concrete demonstration enables us to further explore saturation effect in FED. We believe that sFED, combing with the past studies regarding FED, enable more comprehensive uses in further study, e.g. 3d-sFED, sFED combing with quantitative study of subtraction factors such as IWS etc. In addition, specific implements such as large scope and multilayer biological observations are also included in our aims.

Overall, the study in our paper provides a comprehensive view of optional enhancement of the resolving ability in conventional and previously proposed FED methods by increasing the excitation intensity. The improved resolution and decreased deformations greatly widen the feasibility of the FED method, especially in dense samples. Concerning the unprecedented results, easy-post processing, and nonconstraint of dyes, we are confident that sFED can be further explored and widely used, thus facilitating biological observation.