Three-dimensional super-resolution microscopy with speckle-saturated fluorescence excitation

Three-dimensional super-resolution microscopy with speckle-saturated fluorescence excitation

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Neurophotonics Laboratory UMR8250, University Paris Descartes
47 rue des Saints-Pères, 75270 Paris, France

Department of Applied Physics, The Selim and Rachel Benin School
of Computer Science Engineering, The Hebrew University of Jerusalem,
Jerusalem 9190401, Israel

E-mail address : marc.guillon@parisdescartes.fr


Nonlinear structured illumination microscopy (nSIM) is an effective approach for super-resolution wide-field fluorescence microscopy with a theoretically unlimited resolution. In nSIM, carefully designed, highly-contrasted illumination patterns are combined with the saturation of an optical transition to enable sub-diffraction imaging. While the technique proved useful for two-dimensional imaging, extending it to three-dimensions is challenging due to the fading/fatigue of organic fluorophores under intense cycling conditions. Here, we present an approach that allows for the first time three-dimensional (3D) sub-diffraction microscopy by structured illumination. We achieve this by exploiting the inherent high density optical vortices in polarized speckle patterns, as isolated intensity minima. Importantly, 3D super-resolution is achieved by a single two-dimensional scan of samples, exploiting the natural axial orthogonality of speckle illumination. Speckled structured-illumination is thus a simple approach that enables fast 3D super-resolved imaging with considerably minimized photo-bleaching.


Speckle patterns naturally appear through scattering media and are usually presented as a obstacle to imaging. However, speckle illumination allows imaging with resolutions below conventional fluorescence microscopy techniques [1, 2, 3], similarly to results obtained with fringe and grid illuminations schemes [4, 5]. Furthermore, random scalar wavefields naturally feature another property that make them suitable for super-resolution imaging: they contain a high density of optical vortices of topological charge one [6], such as those used in STED microscopy [7] to saturate stimulated emission depletion. In STED, an intense de-excitation ring of light - typically an optical vortex - is used to confine fluorescence at its center. Breaking the diffraction limit with patterned illumination indeed requires saturating an optical transition like absorption [8, 9], stimulated emission [10, 11], ground state depletion [12] or fluorescence photo-switching, generalized as RESOLFT techniques [13, 14]. Under saturated conditions, zeros of intensity are essential in order to retain contrasted fluorescence. Whether it is possible to break the diffraction barrier in fluorescence microscopy using speckle patterns has remained an open question. Moreover, although three-dimensional (3D) imaging is possible with structured-illumination in the linear excitation regime [15], 3D microscopy under saturated conditions remains a big challenge because of fading of fluorophores (and fatigue of photoswitchable probes) under intense illumination conditions.

In polarized speckle patterns focused with high numerical apertures, the axial component of the field cannot be neglected. The axial field amplitude at the center of vortices (of the transverse components) depends on their topological characteristics and on the polarization state of the beam. More in details, vortices in random waves are primarily characterized by their topological charge (), and at first order, can be described by their elliptical intensity profile around the phase singularity [16]. Vortices are then described by six parameters [17] whose geometrical ones are the eccentricity of the ellipse and its orientation. To perfectly cancel the axial field at the center of a vortex (of the transverse component), polarization with the same eccentricity and the same axes as the intensity ellipse must be chosen. In a polarized random wavefield, circular polarization thus optimizes the darkness of isotropic vortices of same handedness [47] so-ensuring isotropic power-spectrum broadening by optical saturation (theoretical and experimental demonstration can be found in supplementary materials). An illustration of the anisotropic power spectrum broadening when using linearly polarized light is shown in Fig. 5.

Here, we demonstrate the possibility to achieve three-dimensional super-resolution imaging by saturating fluorescence excitation with speckle patterns, using a custom-built speckle scanning microscope. Characterization of speckled point spread functions (SPSF) under saturated and non-saturated illumination conditions allows breaking the imaging diffraction barrier by a factor . Super-resolution imaging capabilities are demonstrated using fluorescent nano-beads and stained actin filaments. Moreover, since speckle patterns are essentially three dimensional, 3D super-resolution imaging is possible with a single two dimensional scanning of the sample. Such a compressed imaging possibility is offered by the quasi-orthogonality of speckles lying in different axial planes, even under saturated conditions.

Results

Figure 1: Principle of our speckle scanning microscope: A circularly polarized random wave-field generated by a spatial light modulator (SLM) – displaying a random phase mask – and a quarter wave-plate (QWP) is focused and scanned through an objective lens onto a fluorescent sample (a). The evolution of the fluorescence signal arising from a thin layer of beads illuminated with a speckle is plotted in (b), as a function of the exciting-pulse energy. The curve is fitted with Eq. (1). When scanning a single fluorescent nano-bead, we obtain the speckled point spread function (SPSF) for low pulse-energy () (c) and for high pulse-energy (d) (). Isolated isotropic vortices can clearly be identified as round shaped dark points (d). The power spectra of SPSFs in c and d are represented in e and f, respectively, demonstrating the power spectrum enlargement due to saturated excitation, and thus the enlarged support of the optical transfer function of the instrument. In c-f, .

The scanning speckle microscope is sketched in Fig. 1a. A spatial light modulator (SLM), conjugated to the back focal plane of the objective lens, is used to generate a fully developed speckle pattern (See Methods and Fig. 1 for a more complete description). A regular diffuser could replace the SLM but the latter allows a dynamic control of the size of the illuminating speckle pattern and thus of the the intensity at the sample plane. To get two-dimensional isotropic super-resolution, the random wave entering the objective is circularly polarized for the reasons discussed above.

The light absorption process of a typical dye may be modeled using a two-energy-level system [19]. The excitation probability of the dye typically depends on several parameters such as the fluorescence lifetime of the dye , its absorption cross-section , and the laser-pulse temporal intensity profile (width , amplitude and shape). Here, we used a sub-nanosecond pulsed laser (), shorter than the fluorescence life-time of the dye to efficiently saturate the optical transition with the minimal average power, and long enough for keeping a low multi-photon absorption probability. The repetition rate of is low enough to ensure efficient dark-state relaxation between excitation pulses and so, minimize photo-bleaching via inter-system crossing [20]. When exciting fluorescence with pulses much shorter than the fluorescence lifetime (), the fluorescence signal may be approximated by where, in the case of a stepwise pulse, ( being the quantum of excitation-light energy). For characterizing the saturation in our experiment, we illuminated a thin layer of beads with a speckle pattern and collected the average fluorescence signal. When averaging over intensity fluctuations of a fully developed speckle pattern (with probability density function ), the average fluorescence signal can be derived analytically as:

(1)

where the notation stands for spatial average. The experimental fluorescence curve shown in Fig. 1b is thus fitted with this function. The average saturation parameter can be expressed as , with the pulse energy and the pulse excitation energy for which fluorescence reaches half the maximum signal. In Fig. 1, we measured for a speckle spot.

In RESOLFT microscopy, resolution typically scales as [21]:

(2)

where is the local saturation factor. The potential resolution improvement can also be directly appreciated by plotting the power spectrum (Fig. 1e and 1f) of the speckled point spread function (SPSF) (Fig. 1c and 1d). SPSFs are obtained by scanning a single fluorescent bead. In the saturated regime, the power spectrum (Fig. 1f) spans over a larger range than in the non-saturated case (Fig. 1e), demonstrating the larger accessible spectral support of the optical transfer function for imaging application. Although the average saturation level in Fig. 1d and Fig. 1f looks modest () as compared with typical saturation levels used in RESOLFT microscopy, this value is averaged over intensity fluctuations of the speckle, meaning that locally, the saturation can be considerably higher. Importantly, the field gradient, can be large at the vortex centers where the field is minimum, ensuring efficient spatial spectrum broadening under saturated conditions. In Fig. 1d, isolated vortices of in the speckle can be clearly observed as round-shaped dark points under saturated excitation, consistent with the isotropic power spectrum broadening.

Figure 2: Speckle images of fluorescent beads taken under non-saturated (a) and saturated (d) conditions. Experimental characterization of point spread functions (b and e, respectively) allows Wiener deconvolution (c and f, respectively). For comparison, an image is also taken scanning a diffraction limited spot in the sample (g) and deconvolved (h). Line profiles in (c), (f) and (h) are plotted in (i), illustrating the improved resolution in speckle imaging mode and demonstrating super-resolution under saturated excitation conditions. In all images, and in (d), the average saturation parameter is .

Imaging with random structures raises the specifically arduous challenge of object reconstruction. Techniques have been developed to reconstruct images even when the speckles are unknown [1, 3]. Through a scattering sample, if the speckle can be translated without any change and satisfies the so-called “memory effect” [22, 23], blind deconvolution can be performed [24, 25] using phase-retrieval algorithms [48]. Reconstruction is all the more efficient when using minimization algorithms in combination with sparsity constraints, inspired by compressed sensing [27, 28, 29]. Here, in order to demonstrate that resolution improvement is solely due to the accessible power spectrum enlargement related to saturated absorption and not to the additional sparsity-promoted super-resolution enabled by the appropriate reconstruction algorithm, we first performed a simple Wiener deconvolution [30, 31] after characterizing the SPSFs with isolated -bead. In this case, the single parameter to adjust is the mean power spectral density of the noise present in experimental images. A further improvement in resolution is expected employing compressed-sensing-based reconstruction algorithms, random illumination being particularly suited for compressed-sensing [32, 33, 34].

A typical speckle image of fluorescent beads is shown in Fig. 2a and Fig. 2d under non-saturated () and saturated excitation conditions (saturation parameter ), respectively. As already pointed out in Fig. 1, even without reconstruction, the speckle image in the saturated regime exhibits thinner structures than in the linear regime. The experimental measurements of SPSFs in the non-saturated and saturated cases (Fig. 2b and Fig. 2e, respectively) are obtained using isolated nano-beads and allow Wiener deconvolution (Fig. 2c and  2f, respectively). The image retrieved from the saturated excitation condition clearly demonstrates a higher resolution power, resolving every individual bead. Clearly, some neighboring beads can only be resolved by saturating fluorescence excitation. Line profiles plotted in Fig. 2i shows a resolution down to in the saturated case, the size of the beads themselves, and a factor below the diffraction limit ( for ). An image was also taken by scanning a diffraction-limited spot in the sample, after correcting aberrations of the system thanks to the SLM (Fig. 2). For fair comparison, deconvolution with a Gaussian fit of the experimental point spread function is also shown in Fig. 2h. As expected, the resolution of this image is already outperformed by the image retrieved from the non-saturated speckle image by a factor . Saturated excitation breaks the diffraction limit and achieves super-resolution.

Figure 3: Point-scanning image of actin filaments attached on a coverslip (a) and images reconstructed from linear (b) and saturated (c) speckle images. Line profiles corresponding to the dotted lines in Figs. a, b and c are plotted in d. The NA was and in image c, the average saturation parameter was estimated to be . Raw speckle images corresponding to images (b) and (c) are shown in figures (e) and (f), respectively.

To demonstrate that super-resolution imaging is also possible with more complex objects than beads, such as biological structures, we imaged actin filaments (Fig. 3). Again, image reconstruction from the speckle image in the linear excitation regime (Fig. 3b) exhibits an improved resolution compared with regular point-scanning imaging (Fig. 3a). Saturating excitation with the speckle pattern allows resolving some structures that could not be seen by the two former techniques. Some line profiles are shown in Fig. 3d to illustrate the smaller structures observed in the saturated conditions. Here, the speckle image was deconvolved using a Richardson-Lucy (RL) algorithm [35]. RL deconvolution simply assumes Poissonian intensity statistics as a prior, and thus yields improved signal to noise ratio in the reconstruction as compared to Wiener deconvolution. A comparison between Wiener and RL deconvolution algorithms with nano-beads is shown in Fig. 6, yielding very similar results. Blind deconvolution, without prior knowledge of the SPSF, using a hybrid input output algorithm [48] such as in refs. [24, 25] can also be used, as illustrated in Figs. 7, assuming only the unknown speckle pattern is scanned in the sample, as is provided by the memory effect in scattering media[22, 23]. In Fig. 3, the actin filaments were attached on a coverslip and were thus laying in a few-nanometer-thick plane. This was important for reconstruction by deconvolution since, here, the SPSFs were measured at the focal plane of the objective only. Accurate positioning of the sample in the focal plane with a piezoelectric stage was thus critical for image reconstruction.

Figure 4: For three-dimensional (3D) super-resolution speckle imaging, a single two-dimensional (2D) scan of the object is recorded (a). The speckle images of fluorescent -beads and of actin filaments are shown in (b). Prior recording of the 3D-SPSFs (saturated and not saturated) with isolated fluorescent beads (c) and (Wiener) deconvolution of 2D-scans allows 3D reconstruction of the objects (d). The orthogonality of speckle patterns ensures axial resolution after deconvolution as illustrated in (c). The point spread function in the saturation regime is clearly smaller along the propagation axis by a factor as compared to the non-saturated case. For fluorescent beads, the saturation parameter was set to and resolution improvement is better for the bead closer to the coverslip due to index mismatch. Images taken with . Scale bars in (d): .

Saturating the speckle patterns further increases the sensitivity to axial positioning thus suggesting the possibility to improve resolution along the propagation direction also. However, super-resolution is obtained by using intense laser pulses which are incompatible with extensive 3D-scanning since inducing an accelerated photo-bleaching of fluorophores. Nevertheless, during a two-dimensional (2D) transverse scan (Fig. 4a), objects lying in different planes are scanned by uncorrelated speckle patterns. An illustration of 2D-scanned images are shown in Fig. 4b. Deconvolution by a 2D-SPSF in a given plane rebuilds the corresponding plane but not the others. Therefore, characterizing the 3D SPSF allows a plane by plane 3D reconstruction of the object. We thus recorded the 3D-SPSFs in the linear and in the saturated regimes (Fig. 4c). For a freely propagating random wave, the speckle patterns observed in transverse planes separated by more than are uncorrelated. When (Wiener) deconvolving the median plane of the SPSF by itself, a bright and sharp spot is obtained but, as expected, the spot amplitude decreases when deconvolving other planes at increasing axial separation. In the saturated regime () this separation distance is reduced by a factor (actually larger than ). The reason for axial resolution improvement with speckle pattern is related to the fact that correlations between speckle patterns are dominated by vortices and no longer by “speckle grains”. Moreover, contrary to the vortex used in standard STED configurations, the trajectory of vortices in 3D speckle patterns are not aligned with the optical axis [36, 37]. Since the axial correlation length is shorter in the saturated regime, super-resolution is also provided along the propagation axis of the beam. The result of the plane-by-plane deconvolution of a sample of fluorescent beads is shown in Fig. 4d in the saturated and non-saturated case, clearly demonstrating super-resolution along the propagation axis. Axial resolution improvement is better for the bead closer to the coverslip for two main reasons: - first, the index mismatch between the coverslip and the mounting medium (PVA) introduces aberrations to the speckle away from the coverslip - second, the speckle spot was focused at the coverslip, thus exhibiting higher intensities and smaller speckle grains near the coverslip. Saturating fluorescence excitation by speckle patterns thus allows 3D super-resolution imaging. Finally, we also demonstrate 3D image reconstruction of actin filaments over a -axial range by two-dimensional speckle scanning. 3D imaging by 2D scanning is possible if the object is uncorrelated with the speckle pattern. During the reconstruction, out of plane objects contribute to a background noise that would prevent imaging too faint object in densely fluorescent 3D objects. However, noise could potentially be filtered out using simple blurring or more efficiently by adding sparsity constraints to the sought-for object, so-resulting in an optical sectioning capability. It is expected that the technique provides similar sectioning performances to those demonstrated by dynamic speckle illumination microscopy [38].

Discussion

We have demonstrated that speckle patterns can be used to achieve 3D super-resolution microscopy beyond the diffraction limit with only a 2D raster scan. Despite the vectorial nature of light waves which disallow perfect zeros of intensity in a random light-wave structure, resolution could be obtained for a NA of . The resolution improvement is all the more appreciable in the case of low NAs. For large NAs the contribution of the axial field component prevents saturating too much the optical transition. For low NAs, resolution improvement can be huge but the laser intensity required to reach nanometric resolutions may cause too much background signal and/or photobleaching. The presented results were obtained with a very simple and inexpensive system, where the microscope objective could be easily changed by a condenser of high NA, whose optical properties are poor but which can efficiently collect the fluorescence signal. In our experiments, the background signal originating from the optics and the immersion oil was more critical than photobleaching.

Here, we saturated absorption but super-resolution speckle imaging could also be performed saturating other optical transitions such as stimulated emission, which would reduce background signal and photobleaching as compared to our saturated excitation conditions since using red-shifted light as an intense laser beam. In this case, speckle patterns having inverted intensity contrast [39] could be used for the excitation and the de-excitation speckle patterns. Taking advantage of the so-called “memory effect” [22, 23], it should be possible to use our technique through a strongly scattering sample and even tissues [40, 41].

Finally, for the sake of experimental physical evidence, we chose simple numerical deconvolution techniques for image reconstruction (Wiener, Richardson-Lucy and blind iterative phase retrieval algorithm). However, adding sparsity constrains about the sample would definitely further improve image reconstruction. Such processing is all the more efficient that speckles constitute a quasi-orthogonal basis which allows efficient compressed sensing for image acquisition [32, 33, 34]. Most of objects can indeed be described with a sparse set of modes - provided an adapted basis is used - which in general is strongly incoherent (i.e. “orthogonal”) with random patterns such as speckles [42].

Methods

The speckle-scanning microscope

A complete scheme of the experimental setup can be found in Fig. 1. For saturated speckle imaging, the laser source is a Q-switched laser diode delivering , pulses at (Teem Phononics, Meylan, France, STG-03E-120). A fully developed speckle pattern is generated by a SLM (Hamamatsu Photonics, LCOS, X10468-01) conjugated to the back focal plane of a microscope objective (Olympus, Tokyo, Japan, , NA , UPLanSApo, Oil) which focuses the beam into the sample. A quarter wave-plate is placed right before the objective lens to polarize the illuminating beam circularly. In addition, an iris placed before the SLM allows controlling the NA of the speckled beam. Finally, the speckle pattern is scanned transversely in the sample by a pair of galvanometric mirrors and fluorescence is then collected through the same objective lens and sent to a photo-multiplier tube (Hamamatsu Photonics, H10721-20). A pinhole limiting the field of view to was placed in an intermediate image plane between the objective and the photomultiplier tube to minimize background signal.

Deconvolution

In Figs. 2 and 8, a Wiener deconvolution was performed. The speckled point spread functions were characterized by scanning isolated fluorescent beads. For deconvolution of speckle images of actin filament, the saturated SPSF was estimated from an image of a fluorescent nano-bead although the dye used is different. However, it appears that the uncertainty about the exact saturation level is not critical for reconstruction. The pixelated raw speckle image was centered in a matrix with doubled dimensions and the additional border was padded. Wiener deconvolution was then performed in the Fourier domain. The noise parameter was adjusted visually in order to optimize the compromise between resolution improvement and the signal to noise ratio.

Samples

Fluorescent beads were spin-coated in a PVA matrix on a coverslip and mounted in an anti-fade mounting medium (Fluoromount, Sigma-Aldrich). Samples of actin were prepared on coverslip functionnalized with myosin 1b [43, 44]. The monomer was first polymerized in a KCl buffer solution. Then it was labeled with a concentration of Alexa Fluor 546 phalloidin (Invitrogen) in excess before deposition on the coverslip [45]. Finally, the sample was mounted in Fluoromount after several washes. Samples of beads in 3D were prepared by simply drying a colloidal suspension of beads with PVA. The actin filaments in 3D were mounted in Fluoromount.

Acknowledgements

The authors thank Julien Pernier for providing the actin filaments and the staining protocol, and Laura Caccianini, Madjouline Abou Ghali and Dany Khamsing for helping with samples. This work was supported by grants from the Région Ile-de-France, the French-Israeli Laboratory ImagiNano, and the Centre National de la Recherche Scientifique.

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Supplementary Materials

s.1 Experimental Setup

Figure 1: Complete scheme of the speckle scanning microscope. The laser power is modulated using a half-wave plate () and a polarizing beam splitter (BS). The laser beam then passes through a beam-expander (BE) before illuminating the spatial light modulator (SLM) which generates the speckle. The SLM is conjugated to a pair of galvanometric mirrors and to the back focal plane of the microscope objective. A quarter wave-plate () circularly polarizes the impinging beam in order to achieve isotropic transverse super-resolution.

s.2 Optical saturation and photobleaching

Figure 2: Characterization of the excitation curve of the fluorescent nano-beads. The raw signal (red circles) contains both the contribution of the bead fluorescence and the background. The latter may be characterized in the absence of fluorescent bead (blue crosses). Subtracting the background to the raw signal gives the excitation curve of the fluorescent nano-bead (green x-crosses). For this experiment, a cluster of fluorescent beads was illuminated with a speckle pattern and the fitting curve thus takes into account the statistics of the intensity distribution.
Figure 3: Linear evolution of photo-bleaching rate with laser intensity. Here single fluorescent nano-beads were photobleached under illumination by a focused spot of .

s.3 Axial field modulation at the center of vortices by polarization control

In a polarized random wavefield focused with a lens, the vortices of the transverse components coincide. Here, we discuss how the axial field at the center of the these vortices depends on the polarization. Without loss of generality, let us choose Cartesian coordinates centered on a given optical vortex and aligned with the main axes of its characteristic ellipse. At first order development, the transverse field may then be written:

(3)

where are the semi-minor and semi-major axes of the ellipse, is the topological charge of the vortex, the angle of the polarization ellipse with respect to the -axis and the relative phase between the and components of the transverse field. Using the Maxwell-Gauss equation (), we obtain the axial field in the paraxial approximation:

(4)

where is the wavenumber. The axial field is thus canceled at the vortex center of if the beam is elliptically polarized with the same handedness (), the same ellipticity () and the same orientation as the vortex. Since vortices in random waves contain a broad statistical distribution of ellipticities, intensity cannot be canceled at all vortices at once. The critical role of the polarization state, and thus of the axial field, at high NA and high saturation levels was demonstrated in [47] and is illustrated in Fig. 4. Typically, a linearly -polarized beam minimizes intensity at vortices strongly elongated along the dimension, and circular polarization minimizes intensity at vortices of same handedness [47]. An illustration of anisotropic power spectrum broadening is shown in Fig. 5. For imaging application, optimization of isotropic vortices is preferable in order to obtain isotropic super-resolution in the transverse plane.

Figure 4: Effect of the axial field on the saturated fluorescence signal. In each image, the green and magenta images are obtained using the same random phase mask on the SLM for excitation, but having right and left-handed circular polarizations, respectively. Changing the handedness of circular polarization mostly modulates the axial field. Here, the contrast of images have been inverted as compared to usual representation of intensities, in order to better visualize the contribution of the axial field. Bright pixels thus code for the dark regions of the speckle which are crucial for super-resolution imaging. From left to right, the saturation parameter is increased. The significant difference observed between the green and the magenta image observed at large saturation parameters demonstrates the high sensitivity to the axial field. Images taken using a speckle spot, with .
Figure 5: Effect of the polarization state of the speckle pattern on the power spectrum enlargement of the speckle scanning fluorescent image. Circular polarization (a) provides isotropic power-spectrum enlargement while a vertically polarized speckle pattern minimizes the axial field at vortices strongly elongated along the vertical direction, thus enlarging the power spectrum along the horizontal direction (b). Power spectra obtained using and an average saturation parameter .

s.4 Wiener and Richardson-Lucy deconvolution

Figure 6: Comparison of reconstructed images shown in Fig. 2 using Wiener deconvolution (a) and Richardson-Lucy deconvolution (b). Richardson-Lucy deconvolution improves the signal to noise ratio.

s.5 Blind deconvolution by a phase-retrieval algorithm

Figure 7: Images of the two nearby fluorescent beads obtained with and shown in Fig. 8. Speckle image in the linear excitation regime (a) and in the saturated regime (b). In the linear regime, the image is reconstructed by Wiener deconvolution (c). Image reconstruction from the saturated speckle image is performed by an iterative phase retrieval algorithm [48] (d). The average saturation parameter in b and d is like in Fig. 8. The image in d should be compared to Fig. 8.

Blind phase retrieval was performed (Fig. 7) using a continuous hybrid input output (CHIO) algorithm as described in [48]. First, the raw speckle image was periodized using the “edgetaper” Matlab function. This function performs a linear interpolation at boundaries of the image and thus allows avoiding artifacts related to fast Fourier transform of data having non-periodic boundary conditions. Next, a difference of Gaussian filter is applied to the auto-correlation: a low-pass Gaussian filter is applied to remove noise from data, and a high-pass Gaussian filter removes the large zero-frequency component due to the power spectrum of the speckle pattern itself. This difference of Gaussian filter with zero mean also allows equilibrating the balance between high spatial frequencies and low spatial frequencies for optimal reconstruction. Finally, iterations of the CHIO algorithm are run.

s.6 NA-independent resolution

Figure 8: Reconstructed super-resolution speckle images (Wiener deconvolution) of fluorescent beads for different NAs. For , and , the saturation parameters are , and , respectively. The scale bar is the same for all images.

We now discuss and analyze the limit of super-resolution imaging with saturated speckle patterns. On the one hand, optical saturation improves resolution thanks to the presence of optical vortices of (the transverse field).Circular polarization is chosen in order to minimize the axial field at the center of isotropic optical vortices of same handedness. On the other hand, even in this configuration, saturation degrades the contrast because the intensity at the center of the vortices does not perfectly vanish. The optimal image, featuring the best achievable resolution, should thus be recorded for a saturation level corresponding to the proper balance between resolution improvement – requiring high saturation levels – and excitation contrast – incompatible with too high excitations.

The intensity at the vortex centers follows an exponentially decaying probability density function due to the contribution of the axial field [47]:

(5)

where is the ensemble average of at the center of vortices of . To achieve super-resolution imaging, the fraction of highly contrasted vortices (remaining dark) in the speckle pattern should be high enough.As a contrast criterion, we may consider that fluorescence is not significantly excited at the center of a vortex if, there ; which limits the saturation level. The fraction of highly contrasted vortices in the population of vortices sharing the same handedness as the polarization is obtained by integrating Eq. (5) over the aforementioned interval:

(6)

For , the fraction of vortices remaining dark (i.e. satisfying ) is among the population of same handedness and among those of opposite handedness (deduced from Eq. 3 in reference [47]). We may then arbitrarily set as a limit for performing super-resolution imaging. For a top-hat shaped illumination pupil, where is the refractive index of the immersion medium of the objective and the space average intensity of the speckle pattern [49, 47]. We thus get as a maximum saturation parameter:

(7)

Finally, if resolution improves as Eq. (2) (with replaced by ) we get that the utmost achievable resolution is (assuming ). Interestingly, this limit does not depend on the NA of the imaging lens. In our experimental conditions, Eqs. (2) and (7) combined together yield . In practice, we could obviously get super-resolution slightly beyond this limit – actually limited by the bead size – suggesting that our theoretical estimate is pessimistic. The reason why Eq. (2) under-estimates the super-resolving ability of speckles may be because it involves the average saturation factor while local saturation factors in a speckle pattern can be much larger. Local high saturation can thus provide super-resolution information with, apparently, high enough signal.

An illustration of the results we obtained using three different NAs is shown in Fig. 8. Resolutions obtained in point-scanning mode as well as after reconstruction from speckle images in the linear and saturated regimes are presented. We observe that both for and , the beads are resolved in the saturated regime and , respectively). For the , with a saturation parameter of , it was not possible to reach this same resolution because of the too high pulse energy that would have been required: to get a given resolution with a given NA, the saturation factor typically scales as: . An average saturation parameter larger than would thus have been required but this was not possible. Using too high energy pulses has, indeed, two drawbacks: - First, it increases photo-bleaching (which increases almost linearly with pulse energy as shown in Fig. 3) and thus reduces the statistics of the speckle image - Second, it increases the background signal coming from optics and thus degrades the signal to noise ratio (see Fig. 2).

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