## Abstract

Stimulated emission depletion (STED) microscopy is a versatile imaging method with diffraction-unlimited resolution. Here, we present a novel STED microscopy variant that achieves either increased resolution at equal laser power or identical super-resolution conditions at significantly lower laser power when compared to the classical implementation. By applying a one-dimensional depletion pattern instead of the well-known doughnut-shaped STED focus, a more efficient depletion is achieved, thereby necessitating less STED laser power to achieve identical resolution. A two-dimensional resolution increase is obtained by recording a sequence of images with different high-resolution directions. This corresponds to a collection of tomographic projections within diffraction-limited spots, an approach that so far has not been explored in super-resolution microscopy. Via appropriate reconstruction algorithms, our method also provides an opportunity to speed up the acquisition process. Both aspects, the necessity of less STED laser power and the feasibility to decrease the recording time, have the potential to reduce photo-bleaching as well as sample damage drastically.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Far-field fluorescence microscopy represents a well-established method in the life sciences. Due to diffraction, the resolution of conventional light microscopy is limited to about half the wavelength ($\lambda$) in the focal plane. This constraint can be surpassed by techniques like e.g. STED (stimulated emission depletion) or RESOLFT (reversible saturable optical fluorescent transition) microscopy, which employ a targeted switching scheme to precisely define a sub-resolved area where fluorophores are in a specific state [1]. In STED microscopy, the excitation focus is therefore superposed with a typically doughnut-shaped light pattern, which inhibits fluorescence by stimulated emission wherever its intensity is non-zero [2–4]. STED microscopy provided a resolution of up to 20 nm in biological samples and has gained a lot of recognition in the last decades [5–7].

The acquisition speed in STED microscopy is usually limited by the number of fluorescence photons that is also directly linked to the narrowing of the fluorescent spot. Thus, compromises between resolution increase and acquisition speed have to be made in many cases [8], although the parallelization of STED and RESOLFT techniques has shown up an opportunity to lower acquisition times [9–12]. Furthermore, STED microscopy requires focal peak laser intensities that are in the range of $GW/cm^{2}$ as the effective focal volume is decreased through saturated depletion. These high intensities may induce photo-stress and photo-damage to the fluorophores [1,5,12]. Concepts about reducing the required intensities are typically based on switching between molecular states exhibiting longer lifetimes [3], consequently leading to additionally increased image acquisition times.

The novel approach presented in this paper maintains the basic concept of STED microscopy but provides significant benefits. Concentrating the available STED laser power along a single direction allows us either to increase the achievable resolution at the same STED laser power or to reduce the light dose delivered to the sample while preserving the resolution. A homogeneous 2D resolution is obtained by subsequently reassembling sub-images with different high-resolution axes. This procedure is reminiscent of tomographic approaches. Therefore, we refer to our method as tomographic STED microscopy (tomoSTED).

## 2. Experimental setup

Our implementation of the (tomo)STED microscope is illustrated in Fig. 1. Fluorescence excitation and depletion are performed by two electronically synchronized pulsed laser systems, one running at 640 nm with a pulse width of 90 ps (*PicoQuant*, Germany) and the other running at 775 nm with a pulse width of 600 ps (*Onefive GmbH*, Switzerland). Both lasers operate at a repetition rate of 20 MHz and their relative pulse delay is electronically optimized for maximum depletion efficiency. The STED laser is phase-modulated by a spatial light modulator (SLM; *Boulder Nonlinear Systems*, USA). By generating a so-called off-axis hologram, phase-modulation of the entire beam is guaranteed. While the excitation laser is kept circularly polarized via a quarter-wave plate (*B. Halle Nachfl. GmbH*, Germany) for most efficient excitation of fluorescence, the polarization state of the STED beam can be freely adjusted by using a combination of two Pockels cells (*Leysop Ltd.*, UK) with their fast axes enclosing an angle of 45$^{\circ }$. Thereby, the beam polarization can be adapted to the imprinted phase pattern. The excitation and depletion light are combined by a dichroic mirror (*AHF analysentechnik*, Germany), before they pass through a beam scanner (*Abberior Instruments*, Germany) and are focused into the sample by an objective lens (UPLSAPO 100XO NA 1.4, *Olympus*, Japan). Fluorescence light is captured by the same lens, spectrally separated from the excitation and depletion pathways and coupled into a multimode fiber that serves as a confocal pinhole of 1.2 airy units. The fluorescence is split by an integrated 50/50 fiber splitter (*Thorlabs, Inc.*, USA) such that each of the subsequent two single photon counting devices (*Perkin Elmer*, USA) detects a stochastically independent signal, which can then be analyzed by Fourier correlation methods. However, for image visualization and image reconstruction the signal of both detectors is typically added. Synchronization of scanning and gating of detection is performed with a data acquisition card (*National Instruments corp.*, USA) and the software *Imspector* (*Abberior Instruments* and *MPI for Biophysical Chemistry*, Germany). The software provides a line trigger signal for a self-made *LabVIEW* (*National Instruments corp.*, USA) routine, which controls both the SLM and the Pockels cells. Data reconstruction and simulations are performed with custom written Matlab code (*MathWorks*, USA).

Due to the application of the SLM for phase modulation and the Pockels cells for polarization adaptation, switching between different focal depletion patterns is done purely electronically without any moving parts, which ensures that the microscope alignment is not changed. For a homogeneous resolution increase in all lateral dimensions, the STED beam is circularly polarized and imprinted with a circular phase ramp resulting in a doughnut-shaped focal depletion pattern (2D STED) [13]. One-dimensional focal depletion patterns are realized by applying a binary phase pattern with half-dividing zones [14]. The polarization direction of the beam is chosen such that it is parallel to the separating line of the phase zones. For tomoSTED measurements, individual images with different orientations of one-dimensional depletion patterns are captured. The switching rates of the SLM and the Pockels cells allow for a line by line rotation of the depletion pattern with a maximum of 200 Hz. This implies that each line within an image is repetitively recorded with each depletion pattern orientation before the next line is scanned in the same manner. In this way, artifacts due to sample movement or fluorophore bleaching are minimized. In order to avoid artifacts due to the scanning parameter dependent offset of the scanner [15], identical (square) pixel sizes and dwell times were used for all orientations of the depletion pattern.

## 3. Results

#### 3.1 Basic considerations and depletion pattern properties

The focal spot of a confocal microscope can be reasonably well described by a 3D Gaussian function with specific full width at half maximums (FWHM). In STED microscopy, the fluorescence inhibition depends especially on the product of the saturation factor $\zeta$ and the steepness of the depletion pattern [5]. The depletion pattern describes the STED light distribution in the focal plane and its shape depends in particular on the phase pattern imprinted on the STED beam. If the pattern corresponds to a circular phase ramp, the STED light distribution renders a doughnut shape [13]. A binary phase pattern with half-dividing zones results in a 1D valley of depletion [14] (see also Figs. 2(a) and 2(b)). In the vicinity of the focal spot, both patterns can be well approximated by either a 2D or a 1D parabolic function [5] and their steepnesses are defined as the coefficients in their parabolic approximation [5]. In case of the 1D depletion pattern, the light is concentrated within a smaller area as compared to the depletion doughnut; consequently its pattern steepness is higher.

The modeling of the effective point spread functions (PSFs) and corresponding FWHMs for 1D and 2D STED patterns is detailed in Appendix A. It turns out that, for an oil immersion objective lens with a numerical aperture of 1.4, the pattern steepness of the 1D depletion pattern is $\sim 1.85$ times higher (Figs. 2(c) and 2(d)) allowing theoretically for a $\sim 1.36$ times higher resolution along a single direction at equal saturation factor (i. e. equal STED light power). Alternatively, the same effective FWHM along this direction can be achieved at $\sim 1.85$ times lower STED light power. Note that the lateral peak distances of the 2D and 1D pattern are very similar (Fig. 2(c)). The increased 1D pattern steepness is therefore mainly due to the increase of the peak height and not to a reduced distance, which has been reported for standing waves used as 1D line patterns in parallelized STED microscopy [16].

To verify this model, we imaged 25 nm sized fluorescent microspheres with our tomoSTED setup (Fig. 1), utilizing the doughnut-shaped as well as the 1D depletion pattern (see Figs. 2(e) and 2(f) for exemplary measurements), and determined the effective resolution as a function of the saturation factor $\zeta$ (Fig. 2(g) and Appendix B (Figs. 7 and 8)). The measured FWHMs follow the inverse square root law for both depletion patterns and depict an experimental ratio of the pattern steepness of $\sim 2.58$. This is even better than theoretically predicted ($\sim 1.85$) and corresponds to a $\sim 1.6$ times higher resolution along a single direction at equal saturation factor or to the same resolution at $\sim 2.58$ lower STED light power.

This deviation between experiment and theory may suggest unfavorable adjustments of our 2D STED when compared to our 1D STED implementation. However, the alignment of the unrendered STED beam is identical in both cases as the same beam path is employed (Fig. 1), leaving only imperfect polarization states and phase mask implementations. An investigation of the depletion patterns (Fig. 9 in Appendix B) demonstrates that less than 0.2 % of the STED light for the 1D as well as for the 2D implementation are not phase modulated. This barely effects the fluorescence signal in measurements and indicates a precise alignment of the depletion pattern in both cases.

#### 3.2 Image reconstruction and resolution

To obtain a 2D super-resolved image when utilizing a 1D depletion pattern, a series of sub-images with varying high-resolution directions has to be collected. Therefore, the 1D depletion pattern has to be rotated along a rotary axis that is perpendicular to the focal plane and intersects with the focal center. Since in both the 1D and the 2D implementations we scan with square pixels of identical size, the pixel dwell time for each sub-image has to be reduced by the number of sub-images ($N$), as compared to 2D STED, in order to keep the overall image acquisition time constant. In principle, a standard tomographic image reconstruction approach can be used to derive a final image from such projections (Appendix C, Fig. 10). As this neglects information contained along the respective low-resolution directions, it is not specifically tailored to tomoSTED. Therefore, we present three alternative image reconstruction methods that are better suited for tomoSTED microscopy: the sum value method, the maximum Fourier value method and an adapted penalized Richardson-Lucy method (see also Appendix D, Fig. 11). All three methods take the oversampling in the direction of the lower resolution within the sub-images into account by means of a convolution with a correspondingly aligned mean filter of length $k$.

The sum value (SV) method is the simplest approach to analyze tomoSTED data. By simply adding all recorded sub-images with different directions of resolution enhancement, an isotropically super-resolved image is obtained. The effective PSF is the superposition of all sub-images’ PSFs and contains high-resolution information, but unfortunately it prioritizes low spatial frequency components and therefore generates images with a distinct confocal background.

The maximum Fourier value (MFV) method, on the contrary, is performed in Fourier space. The sub-images with different directions of resolution enhancement are Fourier transformed, providing high spatial frequencies along the respective super-resolved axes. Subsequently, all individual Fourier transforms are compared for every spatial frequency coordinate; the corresponding component with maximal amplitude is kept and attributed to a synthetic Fourier transform at the same coordinate. By doing so, the largest contribution to a spatial frequency and therefore the highest resolution enhancement obtainable from the sub-images is retained. By means of the inverse Fourier transform of the synthetic Fourier transform, an isotropically super-resolved image is obtained. The MFV method does not prioritize low spatial frequency components, does not remove noise from the data and does not amplify high spatial frequency components as it would typically be the case for a deconvolution. Further, it does not require knowledge about the PSFs of the sub-images. We therefore regard the MFV method as a simple way to produce raw data equivalent tomoSTED images. However, it neglects large parts of the data, in particular if too many projections are recorded.

Taking into account the predicted shapes of the effective STED PSFs of each sub-image allows to use a standard maximum likelihood estimation, the Richardson-Lucy (RL) algorithm [17,18], for image reconstruction. We use an adapted penalized Richardson-Lucy (APRL) algorithm with regularization parameter $\alpha$ for tomoSTED image reconstruction, which is described in detail in Appendix B. It combines the overall available information as it estimates the unknown positive object by maximizing the joint likelihood for all sub-images at the same time.

Besides the image reconstruction method, the number of recorded sub-images also determines the image quality of tomoSTED microscopy. In general, we deem a number of rotations $N={N_{\textrm {opt}}}{}=\pi /2\cdot k$ (with rotation angle step size $\pi /N$) optimal, as it provides a reasonable coverage of the diffraction-limited spot (Fig. 3(a)). Here, $k$ is the resolution enhancement factor, i.e. the ratio of confocal FWHM and 1D STED FWHM $k=\Delta {_{\textrm {conf}}} / \Delta {_{\textrm {STED,1D}}}$.

To analyze the imaging properties of tomoSTED microscopy, we calculated the optical transfer functions (OTFs) for SV and MFV reconstructed simulated images (Appendix A). Figure 3(b) shows the OTF for 2D STED as well as the effective tomoSTED OTFs for both reconstruction methods. A typical resolution enhancement ($\Delta {_{\textrm {STED,2D}}}=50$ nm, $\Delta {_{\textrm {conf}}}=250$ nm, $k{_{\textrm {2D}}}=5$) was chosen for the 2D STED case. Following our analysis of the 1D STED pattern, this corresponds to a resolution enhancement of $\Delta {_{\textrm {STED,1D}}}=31$ nm, $k{_{\textrm {1D}}}=8$ in the tomoSTED case at equal STED laser power and to an optimal number of rotations of ${N_{\textrm {opt}}}{} \approx 12$. The accentuation of low spatial frequency components for SV can be avoided by using the MFV reconstruction. At spatial frequencies corresponding to smaller dimensions than $\Delta {_{\textrm {STED,2D}}}$, both reconstruction algorithms result in tomoSTED OTFs with a higher spatial frequency transmission as compared to 2D STED, indicating that the better performance of the 1D depletion pattern can be extended to all lateral directions by tomoSTED.

In Fig. 3(c), a comparison between tomoSTED (MFV and APRL reconstruction) and 2D STED is shown on simulated data for an artificial object consisting of concentric rings of variable distance. In the radial add-up of the resulting RL/APRL estimates for 2D STED raw data and tomoSTED with $N={N_{\textrm {opt}}}{}$, one can see the superior detection of the true object by tomoSTED up to distances of the concentric rings below 40 nm, while the 2D STED reconstruction shows comparable results only at distances larger than 50 nm. It again corroborates the improved 2D resolution of tomoSTED albeit the effect is less than predicted by the ratio of the different pattern steepnesses alone. We confirmed the consistency of these findings with experimental results by analyzing images of fluorescent microspheres recorded with tomoSTED and 2D STED (Appendix E, Fig. 12).

#### 3.3 Cell imaging

Fluorescence microscopy in life science applications is typically performed at a resolution adapted to the cellular structures to be imaged, since a compromise between resolution, imaging speed, signal-to-noise ratio (SNR) and bleaching rate has to be made. Therefore, tomoSTED enables imaging of cellular structures at a significantly reduced light dose while preserving the resolution as well as the SNR. By this means, photo-damage and photo-bleaching can be significantly decreased.

To demonstrate the suitability of our method to biological imaging, we recorded vimentin structures in fixed Vero cells with both STED variants under identical conditions, in particular equal effective acquisition times. Only the STED laser power for tomoSTED has been reduced to half as compared to 2D STED to ensure the same 2D resolution. As we opted for $\sim$65 nm resolution, an artifact-free tomoSTED reconstruction should be possible with ${N_{\textrm {opt}}}{} \approx 6$ rotations.

An overview of the results obtained via RL and APRL deconvolution for 2D STED and tomoSTED, respectively, is illustrated in Figs. 4(a) and 4(b). The RL deconvolved confocal image of the same field of view is shown at the left-hand side of Fig. 4(a). As depicted in Fig. 4(b), the tomoSTED result is of the same quality as the 2D STED image. Some features are even better discernible.

As illustrated in Fig. 4(c), the effective resolution is analyzed for both imaging modalities by a Fourier correlation (FC) approach (Appendix B), computed on the raw data for 2D STED and on each raw sub-image for tomoSTED. For a correlation threshold of 0.2, the analysis yields a resolution of 64 nm in 2D STED and 67 nm in tomoSTED along the super-resolved direction. Noteworthy, the Fourier correlation strongly depends on the fluorescence signal [19]. Yet, it provides very similar results (deviation <10 %) although the pixel dwell time in each of the six tomoSTED sub-images amounts to only $\frac {1}{6}$ of the dwell time exerted in 2D STED.

As shown in Fig. 4(d), we investigated not only the reconstruction attainable with $N={N_{\textrm {opt}}}{}$ but also with $N=N_{\mathrm{opt}} / 2$ for the same data set (see Appendix F, Fig. 13, for corresponding tomoSTED sub-images). Similarly to the results of the simulations, the MFV reconstruction of angularly twofold undersampled data does not introduce significant artifacts to the final image, but exhibits a slightly reduced resolution. As already indicated in Fig. 3(c) the APRL reconstruction seems to be able to retrieve missing angular information. Although the reconstruction for $N={N_{\textrm {opt}}}{}$ provides sharper images, the APRL image for $N=N_{\mathrm{opt}} / 2$ still outperforms 2D STED (Fig. 4(e)). At this point, the potential of tomoSTED to speed up acquisition becomes obvious. Taking only half the number of optimal pattern orientations still results in adequate images without relevant losses in quality or resolution and translates directly to a doubled acquisition speed.

We used half the STED laser power but the bleaching impact on fluorophores is not self-evident. Thus, we analyzed the bleaching rates for crimson fluorescent microspheres as well as for Abberior STAR 635P stained vimentin structures within Vero cells. Both specimen showed an analogous bleaching behavior, which is exemplary illustrated for the cell sample. Figure 5(a) shows two series of frames that were respectively recorded with one of the two STED implementations. While the images fade significantly for 2D STED, a factor two lower bleaching rate is observed for tomoSTED. This is especially evident from the graphs shown in Fig. 5(b) showing the normalized courses of available fluorescence intensity identified on individual filament structures. An exponential decay fit to the data yields the mean inverse bleaching rate, specifying after how many recorded frames the intensity is reduced to $\frac {1}{e}$. The result for tomoSTED ($13.3$ frames for $N={N_{\textrm {opt}}}{}$) amounts to approximately twice the value for 2D STED ($7.7$ frames). The graph also shows a curve for using only $N=N_{\mathrm{opt}} / 2$ orientations. Halving the number of orientations and therefore the total acquisition time decreases the bleaching rate accordingly.

To demonstrate the ability of tomoSTED to image cellular processes, we performed time-lapse experiments on Silicon Rhodamine stained microtubule filaments within living human embryonic fibroblasts. For a first comparison, we recorded a series of ten frames of similar areas of the same sample with both STED implementations. Figure 6(a) shows an overlay of the corresponding first, fifth and tenth RL-deconvolved frame recorded under 2D STED imaging conditions. The movement of individual fibers is clearly visible, while intensity profiles of raw data unravel feature sizes of approximately 50 nm (Fig. 6(b)). The APRL deconvolved tomoSTED frames (Fig. 6(c)) show no significant quality difference to the 2D STED set, although they were imaged with only half the STED light power and within half the overall acquisition time. The line profiles depicted in Fig. 6(d) confirm this impression as the diameters of the filaments in the respectively oriented tomoSTED sub-image are also in the $50$ nm range. Subsequently, we additionally halved the pixel dwell time for each sub-image and recorded another cell repeatedly for 38 minutes (Appendix G, Fig. 14, and Visualization 1). Overview images of the entire cell taken before and after tomoSTED recording (data not shown) showed no evidence for bleaching.

## 4. Discussion

We have demonstrated that tomoSTED microscopy features an approximately 1.5 times higher resolution than classical 2D STED microscopy at equal STED laser power, since the depletion patterns employed concentrate the light within a smaller area as compared to the doughnut pattern and are therefore much steeper along the relevant direction. The experimentally found increase in pattern steepness of $\approx 2.58$ exceeds our theoretical estimation. To the best of our knowledge, this is not due to deficiencies in our experimental implementation and certainly has to be subject to further investigations. In this context, we would like to note that each resolution curve on its own is well described by the theory. The discrepancy only arises when the curves are compared on the basis of the independently determined saturation factor, which has never been done before. Possible causes could be photo-physical effects of the dye molecules such as rotational diffusion or resonant energy transfer, since our two STED patterns exhibit different states of polarization (linear vs. circular). Possible studies of the phenomenon could include measurements of resolution curves using other fluorophores and/or other sample geometries, e.g. DNA origami structures instead of densely labeled fluorescent microspheres.

2D super-resolved images are created by a suitable reconstruction of multiple sub-images recorded with different high-resolution directions and can also be performed without prior knowledge of the microscope’s optical properties. Importantly, high spatial frequency information is recorded considerably more efficient than in 2D STED microscopy, an effect that can also be proven in simulations on the combination of STED with line-scanning confocal microscopy [20]. This will allow tomoSTED to unravel new details within specimens where the attainable resolution is currently signal limited. Our preferred reconstruction method requires, however, prior knowledge about the PSFs of the sub-images. This might make the performance of tomoSTED a bit less robust than the classical 2D STED approach. Interestingly, using only half the optimal number of rotations still results in apparently artifact-free APRL reconstructions. For sufficiently sparse objects, this might hint at capabilities of the chosen algorithm to fill up the missing spatial frequency information very effectively. In any case, this allows to double the imaging speed.

Equally important is the possibility to keep the resolution unchanged but to halve the STED laser power, leading to a twofold decrease of the bleaching rate. Although this finding can not necessarily be generalized, a reduction of the applied light dose is in any case beneficial, especially for live cell imaging [21]. Noteworthy, the reduction in STED laser power is intrinsic to tomoSTED, implying that other methods to reduce the applied light dose, e.g. RESCue [22], FastRESCue [15], MINFIELD [23] or DyMIN [24], could be implemented to reduce photobleaching even further. We also demonstrated motion artifact-free imaging of cellular processes. In tomoSTED, the pixel dwell time per sub-image is reduced at least by the number of recorded directions with respect to 2D STED and the depletion pattern can be readily rotated line by line. Thus, the local imaging speed is at least as high as in 2D STED implementations and time-lapse experiments should be possible for all applications where 2D STED is feasible [25,26]. Within one rotation of the depletion pattern, tomoSTED explores the entire confocal volume. Therefore, applications beyond pure imaging, such as tracking, structure tracing and fluorescent correlation spectroscopy [27], are conceivable.

In terms of instrumentation, the difference between tomoSTED and 2D STED is quite minor and lies e.g. in the way the SLM is controlled. Additionally, tomoSTED requires a device for controlling the polarization state of the STED beam. Therefore, our method should be easily applicable to any STED microscope already using an SLM for phase pattern generation. It may be beneficial for tomoSTED microscopy to additionally modify the intensity distribution in the pupil of the objective lens: By using higher intensities at the edge of the pupil instead of a homogeneous or Gaussian illumination, the 1D pattern steepness can be increased in the order of 50%, while the power is kept constant. However, the resulting gain in resolution is accompanied by an increased experimental complexity. The basic principle of tomoSTED is independent of the type of switching event and is applicable to all super-resolution techniques based on targeted switching. tomoSTED microscopy may benefit from a combination with image scanning or re-scanning methods, which can increase the resolution in the confocal direction [28,29]. Thereby, the amount of usable spatial frequencies for the tomoSTED reconstruction would be enlarged, presumably improving the quality of the final image. It is also conceivable that the tomoSTED principle can be extended to obtain three-dimensionally super-resolved images by utilizing phase masks that tilt the depletion pattern with respect to the focal plane [13]. This has the potential to reduce the radiation exposure during a 3D scan of the sample approximately fivefold and could lead in conjunction with twofold undersampling to a bleaching rate that is one order of magnitude lower than with 3D STED microscopy.

## Appendix A. Theory of tomographic STED microscopy

## Depletion pattern properties and resolution

We assume that the excitation and STED light is applied as temporally separated pulses with a pulse duration much shorter than the fluorescence lifetime. Photo-bleaching, intermediate dark-states or re-excitation of the dye by the STED light are neglected (simple two-level system) and dye molecules are assumed to rotate fast enough to average the orientation of the molecular transition dipole relative to the polarization of the excitation and STED light. Under these conditions, the fluorescence suppression factor $\eta$, which describes the amount of remaining fluorescence when STED light is applied relative to the emission without any STED light, is [30]

with the cross-section for stimulated emission $\sigma {_{\textrm {STED}}}$ and the photon fluence per area and STED light pulse $J{_{\textrm {STED}}}$. The remaining fluorescence decreases exponentially with the applied strength of the STED light. The dye-specific saturation fluence per pulse, where half the fluorescence is depleted, is denoted by Typically, the fluence per pulse is not measured directly. Instead, the power of the STED light beam $P{_{\textrm {STED}}}$ is readily accessible. The total number of photons per STED light pulse $n{_{\textrm {STED}}}$ is then given byIn the calibration measurement (see Appendix B), the procedure to determine the saturation power $P{_{\textrm {sat}}}$ is described. Using a not phase-modulated focal light distribution $h{_{\textrm {cal}}}(x,y)$ and scanning over very small beads, the saturation power is defined as the STED light power required to suppress the fluorescence at the center of the bead by half.

Under the conditions stated above, the effective PSF of the pulsed STED microscope $h{_{\textrm {eff}}}$ is the product of a confocal PSF and the fluorescence suppression by the STED light.

The confocal PSF can well be approximated by a symmetrical 2D Gaussian peak in the focal plane.For large enough saturation factors, the FWHM of the effective central spot of the STED microscope is much smaller than the used wavelengths and only the shape of the fluorescence suppression in the vicinity of the focal spot determines the shape of the central spot. In this region, the focal distribution $h_{\textrm {STED,2D}}$ of the STED light in the classical 2D STED case can be well approximated by a 2D parabola [5]. Likewise, the 1D STED PSF $h_{\textrm {STED,1D}}$ can be approximated by a 1D parabola.

Combining these relations gives relatively simple expressions for the 1D and 2D STED cases.

## 2D resolution in tomographic STED microscopy

To obtain a two-dimensionally super-resolved image, a series of sub-images with varying high-resolution directions has to be provided. To this end, the 1D depletion patterns are rotated with an axis of rotation that is perpendicular to the focal plane and intersects with the focal center. The images taken at different 1D depletion pattern orientations have to be combined in order to obtain a high-resolution estimation of the unknown object. We compare a synthetic optical transfer function that results from combining $N$ rotated images with the classical 2D STED optical transfer function. For this comparison, the total acquisition time for one 2D STED image and $N$ tomoSTED sub-images are kept equal. First, we define resolution increase factors.

A simple approach at combining all rotated sub-images consists of taking the pointwise maximum of the amplitude of the Fourier transform of all sub-images and inversely Fourier transform the result (see Appendix B). The synthetic OTF $\textrm {tomoOTF}_{\textrm {MFV}}$ will then be the point-wise maximum of the optical transfer functions $\textrm {OTF}{_{\textrm {1D}}}$ rotated each by an angle $\phi =j\pi /N$ with $j=0,..,N-1$.

Function $\max$ depicts here the component whose amplitude is maximal for all rotation angles $\phi =j\pi /N$. Clearly, there will be favorable and unfavorable directions. The favorable directions are equivalent to $\textrm {OTF}{_{\textrm {1D}}}$ along the u-axis and the unfavorable directions are equivalent to $\textrm {OTF}{_{\textrm {1D}}}$ along a line of angle $\pi /(2N)$ with the u-axis. Please note, that $\textrm {tomoOTF}_{\textrm {MFV}}$ is normalized by the number of rotations $N$ to include the $N$ times lower signal acquisition time per sub-image, i.e. the more rotations are acquired the less signal is contained in each rotation. We recommended a number of rotations of which corresponds to filling half the circumference of a circle of diameter $\Delta {_{\textrm {conf}}}$ with the size of the minor axis of the PSF ($\Delta {_{\textrm {STED,1D}}}$). This is a compromise between a too low number of rotations resulting in a strong difference between favorable and unfavorable directions and possibly imaging artifacts and a too large number of rotations that will decrease the strength of $\textrm {tomoOTF}_{\textrm {MFV}}$.While the maximum Fourier transform method does not require knowledge about the exact shape of the PSFs of the sub-images, a large part of data is disregarded. If the shapes of the PSFs of the sub-images are known, a synthetical OTF can be constructed as the sum of the amplitudes of the OTF of each rotated sub-image. This is achieved here simply by adding the sub-images (see Appendix B). In the limit of continuous rotation, the performance of tomoSTED and the synthetical OTF become isotropic and we define the OTF of the summation $\textrm {tomoOTF}_{\textrm {SV}}$ as the orientational average of $\textrm {OTF}{_{\textrm {1D}}}$.

## Appendix B. Material and methods

## Setup characterization

**Determination of saturation power** To estimate the saturation factor $\zeta$, the saturation power $P{_{\textrm {sat}}}$, indicating the STED laser power for which the fluorescence is suppressed by a factor of two, must be determined (see also Appendix A). For this, the fluorescence intensity of 25 nm sized beads is measured as a function of the STED laser power ${P_{\textrm {STED}}}{}$, while the STED beam is not phase-modulated, implying that the determination of $P{_{\textrm {sat}}}$ is independent of the used STED variant. The remaining fluorescence intensity for each ${P_{\textrm {STED}}}{}$ is normalized to the intensity in a confocal image. To compensate for bleaching effects, the confocal signal is determined by averaging over measurements that have been taken before and after STED acquisition. Statistics is performed on at least 50 beads per data point. The decay of the fluorescence intensity for increasing STED laser power is depicted in Fig. 7. With an exponentially decaying fit function the saturation power can be determined to ${P_{\textrm {sat}}}{}=0.28$ mW. The saturation factor is then calculated as $\zeta ={P_{\textrm {STED}}}{}/P{_{\textrm {sat}}}$.

**Bead size correction** If the size of the object is in the range of the PSF, its size needs to be taken into account when estimating the optical resolution or evolution of the fluorescence signal. The recorded data sets for the estimation of the resolution (Fig. 2) as well as for the determination of the relative peak intensity (Fig. 9) are therefore corrected for a bead diameter of 25 nm. Respective functional dependencies are depicted in Fig. 8. For different STED laser powers, the bead images are expressed as a convolution of solid filled spheres with 3D Gaussian PSFs of a certain FWHM. Both quantities, the FWHM of the bead image as well as the FWHM of the 3D Gaussian PSF, are normalized by the bead diameter.

**Resolution** The resolution enhancement is quantified by measuring the narrowed PSF as a function of the saturation factor $\zeta$ on fluorescent beads (Fig. 2.) The FWHM of the bead images is determined via 1D Gaussian fits to intensity profiles of single beads while averaging over at least 10 beads per direction, $x$ and $y$. To consider the respective high-resolution axis, the depletion pattern in tomoSTED is either oriented vertically or horizontally. In either case, the PSF FWHM is estimated by correcting the image FWHM for a bead size of 25 nm (see above for bead size correction).

**Quality of depletion patterns** In both STED variants, the quality of the depletion patterns is evaluated by determining the relative peak intensity in dependence on the applied STED laser power (Fig. 9). For this, 25 nm sized fluorescent beads are again measured for increasing values of $\zeta$. The imaging parameters are identical as for the measurements to determine the resolution. For single beads, the intensity in the STED image is normalized by the average intensity of two images without STED illumination taken before and after the STED recording effectively canceling any photobleaching effects. In both STED variants, at least 20 beads per data point are averaged. For the tomoSTED variant, the horizontal and the vertical depletion pattern orientations are considered. In either case, the derived relative peak intensity is corrected for a bead size of 25 nm (see above for bead size correction).

## Image reconstruction

**Remarks on the maximum Fourier value method** The MFV method consists of point-wise taking the value with maximum amplitude at each point of the Fourier-transformed sub-images and of a final inverse Fourier transformation to obtain the 2D super-resolved image. Due to discontinuities in the assembled synthetic Fourier transform, resulting from the maximum operation, this inverse Fourier transform contains negative values. In the presented intensity profiles comparing 2D STED raw data and MFV processed images of tomoSTED, the full data range is included. Only for visualization, the negative values are set to zero. The relative occurrence of negative valued pixels (mostly in the background areas) are for the in Fig. 3(c) depicted simulation results 4.2 % for $N=N{_{\textrm {opt}}}$, 3.9 % for $N=N{_{\textrm {opt}}}/2$ and 3 % for $N=N{_{\textrm {opt}}}/4$ and account for less than 1 % of the total signal (sum of negative values divided by sum of positive values) in each case. The relative occurrence of negative valued pixels are 7.4 % and 4.4 % for the MFV images in Fig. 4(d) for $N{_{\textrm {opt}}}$ and $N{_{\textrm {opt}}}/2$, respectively. In both cases the negative pixels account for $\sim$ 1 % of the total signal.

**Adapted penalized Richardson-Lucy method** The Richardson-Lucy (RL) deconvolution is a nonlinear, iterative maximum likelihood estimation (MLE) method assuming Poissonian noise and a non-negative object distribution [17]. It is a widely used deconvolution method for microscopy images, often used together with some form of regularization [31,32]. It has been expanded to jointly deconvolve multiple varying images of the same object corresponding to a situation where one has a different PSF for each image [18]. We adapt the standard RL algorithm to tomoSTED with multiple sub-images each with a different direction of resolution enhancement to obtain an MLE estimate of the unknown object $o$ underlying all the sub-images $y_i$ with $i=1..N$. Optionally, a regularization term can be included. Our adapted penalized Richardson-Lucy (APRL) method computes the update factor $F_j$ in the iteration $j$ as average of the RL update factors for each image

## Fourier correlation analysis

The Fourier Ring Correlation (FRC) method is a versatile tool in image analysis since the effective resolution is directly accessed from experimental data [19,33]. The correlation of the Fourier transforms of two statistically independent recordings of the same sample is averaged on a ring with spatial frequency $q$. The spatial frequency at which the FRC falls below a threshold is interpreted as effective resolution estimate. In tomoSTED, the OTFs of the sub-images do not show a circular symmetry (Appendix A). Therefore, we generalize FRC to a locally averaged Fourier correlation (FC) defined as

## Sample preparation and imaging

**Fluorescent beads** A fluorescent microspheres solution (FluoSpheres carboxylate-modified microspheres, 25 nm diameter, crimson fluorescent (625/645); *Life Technologies*, USA) is diluted 1:20,000 with purified water and given into an ultrasonic bath for 10 min to reduce clustering effects. Subsequently, the mixture is incubated for 10 s on a glass coverslip, which is coated with Poly-L-lysine (0.1% (w/v) in $H_20$; *Sigma-Aldrich*, USA) for adhesion. Unattached beads are removed by vacuum suction. The sample is embedded in self-mixed Mowiol for index-matching, mounted onto a glass slide and sealed with nail polish.

Bead measurements for setup characterization are performed with an excitation power of 2.5 $\mu W$, a pixel size of 10 nm and a pixel dwell time of 100 $\mu$s.

**Fixed cells** Fixed cell measurements shown in Figs. 4 and 5 and Appendix F (Fig. 13) are performed on methanol-fixed Vero cells, which were immunostained for vimentin filaments according to standard protocols. After blocking with a blocking solution (BS: 2 % Bovine Serum Albumin (BSA) in Phosphate Buffered Saline (PBS, pH 7.4)), the cells are treated with a primary antibody (anti-vimentin produced in mouse, 1:50 in BS; *Sigma-Aldrich*, USA) for 1 hour at room temperature. Several washing steps are followed by an incubation time of 1 hour for the secondary antibody (Abberior Star 635P goat anti-mouse, red fluorescent (635/651), 1:50 in BS; *Abberior*, Germany) at room temperature. To remove unattached fluorophores, the sample is repeatedly washed before it is embedded in Mowiol.

A STED laser power of 35 mW and 70 mW is used for tomoSTED and for 2D STED imaging, respectively. Furthermore, the excitation power amounted to 5 $\mu$W and the pixel size to 15 nm. The pixel dwell time in 2D STED amounts to $30\,\mu$s, while the pixel dwell time per sub-image in tomoSTED is set to $5\,\mu$s.

**Living cells** The recordings of living cells shown in Fig. 6, Appendix G and Visualization 1 are performed in human embryonic fibroblasts, which were cultured and seeded on coverslips according to standard protocols. For specifically staining the microtubular network, a Silicon Rhodamine (SIR) based fluorophore is used, which provides high cell membrane permeability and specificity [34,35]. At any incubation step of the following staining procedure, the human embryonic fibroblasts are kept at 37 °C and 5 % CO$_2$. SiR-Tubulin (*Spirochrome*, Switzerland) is added at a concentration of 500 nM to the cell culture medium (Dulbeccos modified eagle medium (DMEM) + GlutaMAX (*Life Technologies*, USA) with 10% FBS (*Biochrom*, Germany)). After an incubation time of four hours, the cells are washed with DMEM and subsequently mounted on a cavity slide, which encloses a volume of DMEM$^{\textrm {gfp}}$-2 (*evrogen*, Russia) with the coverslip. This configuration does not only provide a sufficient amount of cell medium, the modification of DMEM moreover increases the photostability of the fluorophore.

The measurements of living cells depicted in Fig. 6 are performed with an excitation power of 0.5 $\mu W$, a STED laser power of 10 mW in 2D STED and 5 mW in tomoSTED and a pixel size of 20 nm. The pixel dwell time per sub-image is set to 10 $\mu$s and amounts to one sixth of the pixel dwell time in 2D STED (60 $\mu$s).

For the measurement shown in Appendix G and Visualization 1, the pixel dwell time per sub-image is set to 5 $\mu$s, all other parameters remain unchanged. The acquisition time for 50 frames with typical frame sizes of 20$\times$15 $\mu$m is in the range of 30 minutes.

## Appendix C. Reconstruction by Radon transform back projection

## Appendix D. Comparison of image reconstruction methods

## Appendix E. Comparison of 2D STED and tomoSTED images of fluorescent microspheres

## Appendix F. Raw data of vimentin measurement shown in Fig. 4(d)

## Appendix G. Long-term tomoSTED measurement

## Funding

Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC 2067/1- 390729940, EXC 171, FZT 103).

## Acknowledgments

The authors thank Jaydev Jethwa for assistance with the high voltage control of the Pockels cells and Ellen Rothermel and Tanja Gilat (all MPIbpc, Göttingen) for providing fixed cells.

## Disclosures

JRK: Laser-Laboratory Göttingen e.V. (P), CG: Laser-Laboratory Göttingen e.V. (P), AE: Laser-Laboratory Göttingen e.V. (P), Abberior Instruments GmbH (I,C).

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