Single-shot super-resolution quantitative phase imaging allowed by coherence gate shaping

Far-field fluorescent super-resolution techniques such as stimulated emission depletion,¹ structured illumination microscopy,² photoactivated localization microscopy,³ and stochastic optical reconstruction microscopy⁴ have become, over recent years, a standard in biomedical imaging. These methods produce images with spatial resolution reaching values way below the diffraction limit of light. The techniques mentioned above exploit sub-diffraction limited imaging of non-linear specimen responses achieved by labeling with fluorescent dyes or quantum dots. Artificial labeling is also popular for providing a high degree of specificity. However, several studies have shown that labeling changes the behavior of the studied biological specimen.^5,6 Therefore, label-free imaging techniques are a more appropriate choice in many biomedical applications. No need for labeling also allows for studying artificial micro and nanostructures.^7,8 Nonetheless, breaking the diffraction limit in label-free imaging techniques is more challenging because of the missing non-linear specimen response.⁹ 

Quantitative phase imaging (QPI) has established an irreplaceable role among label-free imaging techniques thanks to its capability to quantitatively measure morphology and intrinsic specimen contrast with nanoscale sensitivity.¹⁰ The possible super-resolution QPI will satisfy the need for quantitative observation of previously unresolved specimen features and allow increasing the space-bandwidth product (SBP),¹¹ crucial for high-throughput studies. High SBP is important in identifying rare events, for example, in drug discovery,¹² cancer-cell biology,^13,14 or stem-cell research.¹⁵ The recent development of automated data analysis and classification by artificial intelligence^16,17 exaggerates this ever-increasing demand for high-resolution quantitative data. So far, the proposed approaches to QPI super-resolution are based on oblique illumination,^18,19 structured illumination,²⁰ and speckle illumination,^21,22 which, combined with post-processing, provide synthetic images with an effectively enlarged numerical aperture (NA). These synthetic aperture methods enhance resolving power by essentially multiplexing the spatial-frequency content of the object spectrum into an unused degree of freedom in the system, sacrificing acquisition speed, quantitative information accuracy, or a field of view (FOV).

Recent advances in superoscillatory hotspot creation^23–25 that allowed the development of novel approaches to coherent label-free super-resolution microscopy could also be adopted for QPI. However, current implementations of superoscillations also sacrifice some of the valuable microscope properties similar to the synthetic aperture methods. Band-limited fields containing superoscillations oscillate locally faster than the highest Fourier component. When carried over to optical imaging, this means that the focal spot can be made much smaller than allowed by the Abbe–Rayleigh limit. This was first investigated in 1952 by di Francia,²⁶ but only recently have these principles been applied to practical microscopy.^27,28 A superoscillatory sub-diffraction limited focal hotspot can be produced, for example, by coherently illuminating a specially designed mask of concentric annuli of varying complex transmission and widths.²⁷ The concentric annuli mask design can push the central hotspot radius significantly beyond the diffraction limit, but at the cost of high-intensity sidelobes,²⁷ which degrade the image quality in standard wide-field imaging. An alternative approach to amplitude and phase modulation is the application of light states with spatially structured polarization, such as the focusing of radially and azimuthally polarized Laguerre–Gaussian beams.^29,30 The pioneering experimental research utilizing superoscillations initially demonstrated the super-resolution imaging only in a very small FOV²⁷ dictated by the distance of the first high-intensity sidelobe. To remove the FOV constraint, Rogers et al.²⁸ combined confocal detection with superoscillatory illumination. They create the super-resolution image thanks to the coherent illumination pattern with a sub-diffraction limited central hotspot and strong sidelobes. Subsequently, confocal detection eliminates the image distorting sidelobe effects at the cost of scanning the illumination pattern. Despite the great potential for resolution improvement, intensity imaging does not apply to most biological and other weakly scattering specimens and lacks quantitative information. Implementation of similar principles in QPI is thus a desirable yet challenging task due to the complexity and susceptibility of interferometric systems.

In this paper, we propose a method that does not have to sacrifice any of the favorable microscope properties to achieve super-resolved QPI. To the best of our knowledge, we show for the first time that partially coherent broad-source interferometers are capable of single-shot widefield super-resolution imaging by shaping the so-called coherence gate.³¹ Our approach is based on the fact that the point spread function (PSF) of the partially coherent system is a product of the shaped coherence-gating function¹⁹ (CGF) and the function describing the diffraction-limited image spot (Airy pattern). We shape the CGF by manipulating the illumination in the conjugated source plane similarly to the superoscillatory hotspot creation techniques. The product of the superoscillatory CGF with the Airy spot created by the objective in the object arm minimizes the sidelobes in the unbounded region while the CGF central peak delivers the super-resolving power. The minimization of sidelobes and resolution improvement co-occur in the entire field of view and allow single-shot widefield imaging. The imaging thus resembles confocal detection but with parallel filtration of all image points in the field of view. The images maintain quantitative phase information and extend the potential of superoscillations toward the QPI.

We first demonstrate the effects of the superoscillatory CGF using simulated data. Then, due to the highly aberrated pupil plane of our experimental setup, we focus in the experimental part on a limiting case between the superoscillatory and super-resolution CGF. In both situations, the hotspot width is below the Rayleigh criterion. The distinction criterion between the super-resolution function and the superoscillatory one was proposed by Huang et al.²⁴ (we provide more details on the definition of the superoscillatory and super-resolution focal spot in the supplementary material). We create the CGF in this limiting case by using a simple amplitude annular mask, which proves experimentally robust. We demonstrate experimentally QPI resolution enhancement using only the limiting case, but the principle of our method is extendable to the superoscillatory focal spot region, promising higher resolution improvement. An experiment with a phase resolution target shows a resolving power improvement of 19%, and we show practical feasibility by applying the proposed method to the imaging of biological specimens.

The proposed principles generally apply to various partially coherent interferometric systems. Without loss of generality, we will further describe the optical setup and theoretical framework of the used coherence-controlled holographic microscope³² (CCHM), commercially available as the Telight Q-Phase. The optical setup (see Fig. 1) is an adaptation of the Mach–Zehnder interferometer. It consists of an object and reference arm containing two optically equivalent microscope systems. This holographic setup guarantees off-axis hologram formation in the interference plane (IP) for broad sources of an arbitrary degree of coherence. The possibility of using partially coherent sources is provided by the diffraction grating (DG; transmission phase grating with groove frequency 150 mm⁻¹, blazed at 760 nm for the first diffraction order) implemented in the reference arm according to principles proposed by Leith and Upatnieks.³³ In our system, an LED (LED Engin LZ4-00R208, peak wavelength at 660 nm, power up to 2.9 W) is used for illumination to provide a spatially broad incoherent source, and the illuminating light is made quasi-monochromatic after passing the interference filter (IF) with a central wavelength of 660 and 10 nm full width at half maximum. The source is imaged by a pair of achromatic doublets (simplified as L in Fig. 1; focal lengths 63.5 and 350 mm) through a beam splitter (BS) to the front focal planes of the condensers (C; Nikon LWD condenser lenses, 0.52 NA, with adjustable aperture stop). This plane in object and reference arms and respective condenser properties can be described according to Ref. 34 by the pupil functions P_Co(K_t) and P_Cr(K_t), respectively, where K_t = (K_x, K_y) is the transverse wave vector of a plane wave behind condensers. The coordinates of K_t are proportional to the respective source point (pupil-plane) coordinates. For this reason, pupil properties can be characterized by a function of K_t. We use reduced wave vector notation |K| = 1/λ, where λ is the wavelength of light, and K = (K_t, K_z) = (K_x, K_y, K_z), where $Kz=|K|2−|Kt|2$⁠. We modulate the condenser pupil planes to produce the sub-diffraction limited coherence gate, as explained in Sec. III. The fundamental image properties also depend on the parameters of the object and reference arm objective lenses (O; Nikon Plan Fluorite Objectives, 10x/0.3 NA/16 mm WD) in combination with tube lenses (TL; Nikon, focal length 200 mm), characterized by the pupil functions P_Oo(K_t) and P_Or(K_t). Stepper and piezo motors provide fine adjustment of the microscope optical components, which we use for the measurement of the coherence-gating function. The holograms are recorded in IP using an Andor Zyla 4.2 sCMOS camera.

As shown in Fig. 1, we place the phase or amplitude mask in one or both of the front focal planes of the condensers. We designed the masks to shape the CGF when imaging with 10x/0.3 NA objective lenses. In simulations, we assume the phase mask is composed of concentric annuli, with the phase shift being either 0 or π radians. We also carried out simulations with the amplitude mask subsequently used in experiments. The amplitude mask is a single annulus cut by a laser cutter into a metal sheet. An inner circle of the annulus has a diameter of 16.4 mm. The outer circle diameter is about 18 mm, but more importantly, the pupil diameter in the front focal plane of the condensers is limited by the aperture stop to ∼17.3 mm (corresponding to 0.30 condenser NA).

To obtain the sub-diffraction limited resolution of QPI $argΓq;Δq=(0,0)$⁠, systems’s PSF $hq=poqG*q$ must have the central peak radius below the diffraction limit. To maintain quantitative phase information in the image, the sidelobes of the PSF must also be negligible. Numerous studies^24,26,27,35 have shown that a superoscillatory focal spot can be created by coherently illuminating a phase or amplitude mask composed of concentric annuli of different widths and complex transmission. Superoscillations are then formed by constructive and destructive interference near the focal spot. As we use partially coherent illumination in our microscope system, it is not possible to create the superoscillatory focal spot observable in the field’s intensity by interference as proposed for coherent light. However, we can adopt the principles normally applied to coherent systems and shape the system’s PSF, the product of $G*q−Δq$ and $poq$⁠, by altering one or both of these functions. By modulating the pupil function $POoKt$ of the object-arm objective, we can affect $poq$⁠, but as Eq. (2) suggests, we have more options for $Gq$⁠, because this function can be shaped by modulating one or more pupil functions $PCoKt$⁠, $PCrKt$ and $POrKt$ of the condensers and the reference-arm objective, respectively. It is also experimentally easier to modulate the condenser pupil planes. Therefore, we will focus on shaping the CGF. However, similar results can be achieved by shaping $poq$⁠, or both at the same time. Equation (2) describing CGF formation shows that $Gq$ can be shaped similarly to coherent imaging even though the plane waves $exp2πiKt⋅q$ superposed in Eq. (2) are mutually incoherent. The $PCo*KtPCrKtPOrKt$ dictates whether these plane waves are constructively or destructively superposed. This allows us to use approaches designed for coherent imaging even in a system operating with partially coherent light. The expression in Eq. (2) is in fact van Cittert–Zernike theorem,³⁶ which describes the relationship between the mutual coherence function (CGF in our case) and the modulation of the pupil plane for partially coherent broad source illumination. As we can control the constructiveness of the plane wave superposition, theoretically, it should be possible to create observable superoscillations in partially coherent systems. However, not in the field’s intensity but in the mutual coherence of two fields (in our case, the CGF), hence the need for the interferometric system.

For demonstration, we simulate the imaging and calculate the PSFs for three cases with different CGF shapes: first, the diffraction-limited case, when a full unmodulated condenser aperture is assumed; second, the limiting case of the superoscillation, when the amplitude mask with narrow annulus is used and the CGF is represented by the Bessel function $J0(2πKNACq)$⁠; and third, the case with a superoscillatory CGF produced by three-zone phase modulation. For all three cases, we assume that the $poq$ function is the Airy pattern for NA_O = NA_C = 0.30, and this function is represented in Figs. 2(a)–2(c) by yellow dashed curves. The CGF $Gq$ for the diffraction limited case is also the Airy pattern [see the red dashed curve in Fig. 2(a)]. The CCHM PSF [the product of $poq$ and $Gq$] is in Figs. 2(a)–2(c) depicted by solid purple curves. The CGF described by $J02πKNACq$⁠, shown in Fig. 2(b), can be produced in the Köhler arrangement by an annular incoherent source with an infinitesimally narrow annulus and a radius corresponding to the condenser numerical aperture NA_C. Our phase modulation approach [Fig. 2(c)] to the creation of superoscillatory CGF is inspired by the results from Ref. 35. We assume modulation of $PCrKt$ by concentric annuli with phase modulation being either 0 or π. We have found by a few adjustments and visual evaluation that a superoscillatory CGF can be created by a three-zone annular modulation produced in the following manner: two circles with radii corresponding to 0.35NA_C and 0.72NA_C define the geometry of the three zones, while the phase modulation is 0 for the inner-most and outer-most zones, and the middle annulus has the phase shift of π radians. The full width at half maximum of the central peak and the first zero value of the PSF define the system’s resolving power. The three-zone phase modulation and annular amplitude modulation (annular source) of the pupil function $PCrKt$ produce the CGF with the sub-diffraction limited central peak at the cost of stronger sidelobes [see the red dashed curves in Figs. 2(b) and 2(c)]. It is important to note that even though these functions themselves are superoscillatory, if the PSF with such strong sidelobes is used directly for imaging, it produces unwanted image artifacts that corrupt the improved resolving power.³⁷ However, as we demonstrate, the partially coherent systems provide an elegant way to suppress the sidelobe effects.

The Airy spot created by the objective in the object arm has a broad central peak with weak side lobes. As shown in this section, the PSF of the system (the solid purple curves in Fig. 2) is the product of the CGF and the Airy pattern. In both annular source [Fig. 2(b)] and phase-modulated [Fig. 2(b)] condenser pupil cases, the CGF central peak dictates the sub-diffraction limited properties of the focal spot, and sidelobes are attenuated by weak sidelobes of the Airy pattern distribution. Therefore, these approaches should provide sub-diffraction limited powers and deliver single-shot super-resolution images.

We performed imaging simulations comparing three cases corresponding to Figs. 2(a)–2(c) to evaluate the phase imaging performance. We simulated the phase resolution target imaging as a coherent convolution of its complex transmission function by the calculated PSFs, and the simulated phase images are shown in Figs. 3(a)–3(c). The insets in Figs. 3(b) and 3(c) show the potential experimental design of the masks producing the simulated modulation corresponding to Figs. 2(b) and 2(c). The smallest resolved element in the diffraction-limited case is element 5 from group −2. In the superoscillatory case [Fig. 3(c)], the smallest resolved element is number 1 from group −1, and for the annular pupil [Fig. 3(b)], this element can be considered resolved with very poor contrast. The cross-sections of the features of element 6 from group −2 in Fig. 3(d) show that this element is not resolved in a diffraction-limited image but well resolved in both the annular pupil and superoscillatory cases. However, the superoscillatory PSF produces an image with significantly better contrast. The feature width of element 1 from group −1 is ∼20% lower than the feature width of element 5 from group −2. We can conclude that the resolution improvement is slightly less than 20% because the lines of element 1 from group −1 have very poor contrast. As can be seen by comparing Figs. 3(b) and 3(c) and the cross-sections in Fig. 3(e), the contrast is better in the superoscillatory case. We expected the resolution for the case with superoscillatory CGF to be better because the annular pupil produces the limiting case CGF between superoscillatory and sub-diffraction limited ones. Even though we achieve higher resolving power with superoscillatory CGF, the overall image quality of the superoscillatory case is lower due to incomplete sidelobe attenuation. It is important to note that, for simplicity of demonstration, we have not used any sophisticated methods to optimize the condenser pupil function. Generation of superoscillatory hotspots with state-of-the-art parameters usually employs iterative and computationally expensive procedures such as particle swarm,²⁷ genetic algorithm,³⁸ or phase retrieval³⁹ optimizations. We expect to achieve higher resolution improvement and better phase image quality by employing one of these methods.

Additionally to spot size, the superoscillatory focal spot design always involves optimizing the ratio of the central peak and the sidelobe intensities.²⁹ Without taking this into account, practical applications of superoscillatory focusing for imaging are not possible due to the poor signal-to-noise ratio. A similar principle applies also to optimizing the parameters of the CGF. To reconstruct QPI from holograms with reasonable phase quality, the hologram contrast must be higher than the noise levels. When phase modulation of the pupil planes is used, the amplitude of the CGF is redistributed from the central peak to the sidelobes due to the destructive interference of light from the object and reference arm. The hologram contrast is proportional to the central peak amplitude of the PSF $hq$⁠. Therefore, one must consider the achievable hologram contrast when designing the superoscillatory CGF. We have discovered that for the combination of high-quality phase and highest resolution improvement, it is important to optimize the whole product of $poq$ and $Gq$⁠, not only CGF $Gq$⁠. Consequently, the objective function for an optimization procedure must be defined differently than for a standard intensity imaging system. One can easily deduce that the optimal solutions found for fluorescence and confocal microscopy do not apply to the proposed case.

The creation of the superoscillatory CGF requires a very precise design of the phase modulation of $PCrKt$⁠. This is easily achieved in simulations when unaberrated pupils are assumed. However, we have to account for aberrations in real experimental systems and compensate for them while also providing the modulation for CGF shaping. Aberrations can be perceived in the context of the theory outlined in this section as modulations of pupil functions $PCoKt$⁠, $PCrKt$⁠, and $POrKt$ in Eq. (2). As the creation of superoscillations is very susceptible to even subtle deviations from the designed phase shift provided by the phase mask, the aberrations prevent us from using simple symmetric phase masks in real systems. Due to the difficulty of measuring and compensating for aberration in our system, we chose to utilize the amplitude mask in experiments instead of the phase mask.

This equation shows that the complete MCF measurement $ΓBΔq$ with no objects present in both arms provides us with information about the CGF as a function of Δq. Therefore, we will use the measurement described by Eq. (3) to directly evaluate the CGF shape created by the designed mask.

We experimentally demonstrate the feasibility of the principles proposed in Sec. III utilizing the optical setup with 10x/NA_O = 0.30 objectives and the condenser aperture set to NA_C = 0.30. Our initial efforts to take advantage of the phase modulation provided by simple phase masks similar to the one in the inset of Fig. 3(c) have shown that aberrations in our system prevent the CGF from being shaped as designed. However, amplitude modulation by an annular mask [shown in Fig. 3(b) and the design parameters in Sec. II] has proven relatively robust to the aberrated pupils. Therefore, we used it in the presented experiments. We placed two identical amplitude masks into the reference and object arms to balance the light powers in the arms in order to achieve a better contrast of holographic fringes. The total power fraction that is transmitted to the specimen through the mask can be calculated as a ratio of the transparent mask area to the full aperture area. As stated in Sec. II, the diameter of the inner circle of the amplitude annulus is 16.4 mm, and the effective condenser aperture diameter in the front focal plane of the condenser is 17.3 mm. Therefore, the ratio of the light transmitted to the light incident on the mask is ∼0.1. Even though 90% light loss seems significant, our LED source is powerful enough to compensate for that. In experiments with the amplitude mask, we operated the source at about 10% of its maximum power, while the camera exposure times did not exceed tens of milliseconds.

First, we evaluate whether the amplitude modulation provides us with a CGF resembling the designed shape of $J0(2πKNACq)$⁠. We measured the complete MCF for a case with [Fig. 4(b)] and without [Fig. 4(a)] the mask placed in the front focal plane of condensers, i.e., for annular and full aperture. We measured the complete MCF by acquiring and reconstructing a hologram for each reference arm objective position from a predefined grid. The grid of Δq positions for each CGF measurement was the same, and we used a 41 × 41 grid centered at Δq = 0 with a 0.3 µm spacing. We display in Fig. 4 the normalized modulus of the MCF for a FOV point $q=0,0$ μm. Comparing the measured CGF in Figs. 4(a) and 4(b) with the corresponding simulated CGF in Figs. 4(c) and 4(d), we see the effects of the aberrated pupils (mainly due to the off-axis holographic setup). The measurement with full condenser apertures in Fig. 4(a) shows clear signs of a primary coma aberration. We can conclude that the annular aperture is not very susceptible to aberrations, as there is a notable agreement between the measured [Fig. 4(b)] and simulated [Fig. 4(d)] CGF profiles. We have fitted the Airy function to the measured CGF amplitude, shown in Fig. 4(a), with significantly better sampling than the measured data. Then, we determined the full width at half maximum (FWHM) of the central peak to be 1.60 µm. Similarly, we fitted the data obtained for the case with the annular aperture, shown in Fig. 4(b), with the J₀ function and determined the FWHM to be 1.08 µm. The measurement in Fig. 4(b) and the FWHM values show that the CGF created by the amplitude mask has a central peak narrower than the one of the Airy pattern. Therefore, the CGF is, in this sense, sub-diffraction limited.

The complete measurement of the MCF, as shown in Fig. 4, can be used to assess the optical system aberrations. This indicates that we could design and manufacture a phase mask that would simultaneously compensate for aberrations and provide the modulation needed for superoscillatory CGF. However, we decided to postpone these efforts for follow-up work as the current experimental setup limits the practical feasibility of this approach. To obtain the superoscillatory response provided by a specifically manufactured asymmetric mask, one must place it precisely in the correct position. Several degrees of freedom (parameters) must be set to optimize the phase mask position: axial, x-y transversal, and two angular positions. For this, we would need automatic alignment with a feedback loop. The amplitude mask is easier to align as we can partially see its effect in the intensity image formed by light from a single microscope arm. However, the phase mask effect is not visible in the intensity image. We can take advantage of the complete MCF measurement described by Eq. (3) to assess the phase mask effect and its position. Unfortunately, this measurement in the current setup takes tens of minutes. Therefore, it is currently unsuitable for implementing it into a necessary automatic alignment procedure with a feedback loop.

We imaged a phase resolution target with both the full and annular condenser apertures to assess the improvement in the resolving power. The full (unmodulated) aperture phase image in Fig. 5(a) represents the diffraction-limited image. The phase image in Fig. 5(b) is obtained using the amplitude mask, which we refer to as a super-resolution image. The insets in both images show details of the smallest resolved features in each image. A visual comparison of Fig. 5(a) with Fig. 5(b) shows a clear resolution improvement. The smallest resolvable features in the diffraction-limited image [Fig. 5(a)] marked by the number 20 are 0.65 µm wide. This means the resolution with the full aperture is ∼1.3 µm. Whereas the smallest resolvable features in the super-resolution image [Fig. 5(b)] are marked by the number 22 and are 0.53 µm wide. The improvement of the spatial resolution to about 1.06 µm is a gain of ∼19%.

Next, we show the performance of our method when used to image complex specimens such as rat embryo fibroblasts in Fig. 6. The presented experiment involved LW13K2 cells from a cell line of spontaneously transformed rat embryo fibroblasts LW13 of the inbred strain Lewis. Cells were cultivated at 37 °C in a humidified incubator with 3.5% CO₂ in standard Minimum Essential Medium Eagle with Hanks’ salts supplemented with 10% fetal bovine serum, 20 µM gentamicin, and 2 mM L-glutamine. Subsequently, the cells were fixed using 4% formaldehyde in phosphate-buffered saline for 20 min, then washed and incubated in phosphate-buffered saline. Again, we imaged the specimen with and without the annular amplitude masks in the front focal condenser planes. The QPI in Fig. 6(a) experiences super-resolution throughout the entire FOV and can be obtained from a single hologram measurement. Having a large FOV and sufficient resolution for cell segmentation or even observation of intracellular processes is crucial when monitoring, for example, the motility of live cancer cells.⁴⁰ Comparing the sections of diffraction-limited [Figs. 6(b) and 6(d)] and super-resolution [Figs. 6(c) and 6(e)] images, we see that our method provides the improved resolution required in many applications in addition to the large FOV.

In this paper, we have presented a method for single-shot label-free super-resolution QPI in holographic microscopes with partially coherent illumination. Our solution to overcoming the diffraction limit is straightforward to implement because it does not require any changes to the microscope’s optical system. The proposed method relies on the intrinsic partially coherent illumination properties giving rise to the coherence-gating. We propose that by introducing a phase or amplitude modulation of the planes conjugated with the light source, e.g., the front focal plane of the condenser, we can generate sub-diffraction limited CGF. We demonstrate for the first time theoretically and in numerical simulations a superoscillatory CGF shaped by phase and amplitude modulation. Due to experimental challenges, we chose to experimentally show the proposed principles using modulation provided by an amplitude mask, which has proven more robust to optical aberrations than phase masks. We demonstrated almost 20% resolving power improvement in phase imaging of the model specimen and complex objects such as cancer cells. However, the theoretical spatial resolution improvement is not in principle limited, and we expect to obtain significantly over 20% resolution gain with more sophisticated modulation techniques. For example, a spatial light modulator can be introduced into the optical setup to provide simultaneous compensation of pupil aberrations and the modulation needed to create the superoscillatory CGF.

We envision our method delivering an easily implementable super-resolution QPI, particularly suitable for high-throughput biomedical applications. Further extension of the CGF shaping theory beyond the limits of the scalar approximation will allow reaching an unprecedented spatial resolution of QPI. The possibility to monitor a large FOV in real-time with spatial super-resolution and very high quantitative information quality can significantly impact cancer research,^14,41 as previously unseen intracellular processes can now be observed. Furthermore, our work satisfies the need for time-series high-quality datasets required for rapidly developing automated analysis using artificial intelligence.^16,42

See the supplementary material for a detailed derivation of the equations in Sec. III, a definition of the superoscillatory and super-resolution focal spot that is assumed throughout this article, and additional information about simulations producing some of the presented data.

The work was supported by the Grant Agency of the Czech Republic (Grant No. 21-01953S), the specific research grants of Brno University of Technology (Grant Nos. FSI-S-20-6353 and FSI-S-23-8389), and the MEYS CR (Large RI Project No. LM2023050 Czech-BioImaging). We thank Veronika Jůzová for help in the preparation of biological samples.

R.C. is a co-author of patents covering Q-Phase (EA 018804 B1, US 8526003 B2, JP 5510676 B2, CN102279555A, EP 2378244 B1, and CZ302491) and a recipient of related royalties from Telight.

Miroslav Ďuriš: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Petr Bouchal: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Radim Chmelík: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.