The procedure for our systematic comparison between DASH and a-CSA is as follows.

1) Both algorithms are initialised with an empty correction mask.

Detailed algorithm descriptions are provided in the Supplementary Material. ³

In our numerical model, we consider aberrations defined on a square grid of 64 × 64 pixels accounting for N scat 2 = 4096 scattering modes. The aberration phase masks are calculated by adding cosines of all spatial frequencies supported by the grid size with uniformly distributed random phases. The amplitudes of the cosines follow a Gaussian weighting with standard deviation σ, such that cosines with higher spatial frequencies are increasingly attenuated, as expected for most scattering scenarios found in nature. By varying σ we can hence tune the spatial frequency content of the model scatterer. For details on our specific implementation of calculating the scatter mask, the reader is referred to our Supplementary Material (Supplementary Algorithm S4).

The simulated SLM for (square) correction patterns features 32 × 32 pixels. Both the scattering mask as well as the simulated SLM are located in the BFP, in Fourier relation to the object plane, and have a side length measuring 2k ₀ NA, where k ₀ is the vacuum wavevector of light and NA = 0.7 the numerical aperture. Each set of modes is tested repeatedly for 3 iterations. In our experiments, the SLM illumination does not feature an ideal, flat-top intensity profile, but exhibits a (weak) Gaussian shape of e^−1/2 distance approximately equal to the pupil radius. To reflect the experimental situation, also in our simulations we assume a light intensity distribution across the BFP which is symmetrically Gaussian and of corresponding width. The fluorescent sample is assumed to be a homogeneous, absorption-free fluorescent layer of d _s = 10 μm thickness. ⁴ We model this sample volume by grid points on N _s distinct 2D planes with axial interspaces of d _s /(N _s − 1), and the nominal focus plane located at its center. Normally, we set N _s = 6 layers (interspaced by 2 μm), since a larger number of planes does not notably improve the accuracy of results and we have checked that the exact choice of N _s is uncritical for our conclusions. To simulate the PMT counts, we simply propagate the light field into the N _s planes, calculate the squared intensity on each grid point, and take the sum over all points.

In Section 4.4, we show simulation results at high-SNR conditions, for which photon shot noise can be neglected. For scenarios with low SNR, as discussed in Section 4.6, we simulate shot noise by varying the summed PMT counts according to Poisson statistics.

We compare DASH and a-CSA experimentally in a home-built TPEF microscope featuring a phase-only SLM (Meadowlark HSP1920-500-1200-HSP8, 1920 × 1152 pixels of side length 9.2 μm) located in a BFP-conjugate plane of the excitation path. This SLM serves two purposes in parallel: First, by displaying a “scattering” phase mask of defined scattering mean free path l_s (see Section 2) it allows to emulate the effect of a scattering medium in the light path. Second, by running our sensorless wavefront correction schemes and displaying the retrieved phase compensation patterns on top of the given scattering mask, we can test and compare the algorithm performances.

A sketch of the setup is shown in Figure 3. For excitation, we use a mode-locked Ti:sapphire laser (MaiTai DeepSee, Spectra Physics) with emission maximum set to 800 nm wavelength. The epi-TPEF is collected by a water-dipping objective (Olympus XLUMPLFLN20XW, NA = 1) and directed towards a photomultiplier tube (PMT, Hamamatsu H10769A-40) using a dichroic beamsplitter (AHF Analysentechnik, HC665LP) and an additional emission filter in front of the PMT (AHF, ET680SP-2P8). In our measurements, to operate with a more homogeneous pupil illumination, we artificially reduce the NA to about 0.7 using an aperture in a pupil-conjugate plane (not shown in Figure 3).

For our systematic comparison, the sample consists of a thin layer of rhodamine solution sandwiched between a glass slide and a coverslip of 1 mm and 170 μm thickness, respectively. Before each algorithm run, we perform an initial precorrection run to compensate for optical-system wavefront distortions such as the spherical aberrations introduced by the coverslip. Starting from this precorrection, the wavefront correction algorithms afterwards only have to correct for the artificial scatterer displayed on our SLM, ensuring compatibility with the numerically simulated correction runs.

For DASH, we typically set the intensity in the test mode to 30% of the total intensity (i.e., f = 0.3, see Supplementary Material). For a-CSA, we amplify the relative power in the test-superpixel by superposing all other a-CSA-superpixels (consisting of many physical SLM pixels) with a blazed grating of defined modulation depth [24]. Thus, for all except the test-superpixel, power is diffracted away from the optical axis and cut by an aperture as indicated in Figure 3. It is important that the mean phase of the blazed grating is nought to prevent an effect on the zero-order beam, which carries the phase mask. ⁵ Our blazed gratings have a period of 4 SLM-pixels and a modulation depth of approximately π rad, resulting in (1 − β) ∼ 50% of the laser power being dumped (measured value) and a power fraction f ≈ 1/(βN ²) in the case of N×N superpixels. Naturally, dumping power on the iris also decreases the total TPEF signal, wherefore we have to compensate by increasing the total laser power. Since in practice the available power is necessarily finite, this approach for amplitude modulation is only applicable in regimes of turbidity which do not require too many (i.e., too small) a-CSA-superpixels.

For compatibility, we conduct our experiments with the same number of test modes as in the simulations for the different regimes, i.e., 16, 64, and 256 modes for low, medium, and high turbidity, respectively. Our SLM holograms measure 560 × 560 SLM-pixels and entirely fill the reduced objective pupil. For a-CSA this means that we form 4 × 4 square superpixels (each 140 × 140 SLM-pixels) for the low turbidity, 8 × 8 superpixels (each 70 × 70 SLM-pixels) for the medium turbidity, and 16 × 16 superpixels (each 35 × 35 SLM-pixels) for the high turbidity scenario. Analogous to the simulation, DASH and a-CSA are executed for three full iterations. The results are summarized in Figure 4.

For our systematic comparison, we study three different scenarios (A, B, C) concerned with increasing levels of turbidity.

In Scenario A we study low turbidity, with an effective scatterer thickness of a single scattering mean free path, L/l_s = 1, and a spatial frequency distribution of the scatterer chosen accordingly narrow, σ = 1. An example of such a scatter mask is shown on the top left of Figure 4A. In the focal plane, such a mild scatterer typically leads to an intensity distribution which is still spatially confined and only moderately deviates from the aberration-free focus, as shown in Figure 4A, bottom left. Here, the intensity scale is normalized to the peak intensity of the aberration-free focus, such that the maximum value (typically around 0.4 for Scenario A) equals the respective Strehl ratio. In this low-turbidity case it is expected that only a small number of modes is needed for adequate compensation, wherefore we correct 4 × 4 = 16 modes. The plots in the center and right column of Figure 4A show the TPEF signal enhancement simulated numerically (center) and measured experimentally (right), respectively, for a-CSA (blue) and DASH (orange). Specifically, we plot the TPEF signal measured when the established compensation pattern is applied at the respective measurement number. ⁶ In the numerical simulations (Figure 4A, center), we observe that for a-CSA the signal initially increases fast, but then flattens off from the second iteration onwards. DASH, in contrast, exhibits a less monotonic signal evolution than a-CSA, first increasing more slowly, but ultimately surpassing the final signal level of a-CSA. We attribute the signal flattening and lesser total performance of a-CSA to the fact that its compensation patterns are inherently displayed at the native resolution of the test-mode basis (i.e., 4 × 4 superpixels in Scenario A) and therfore do not exploit the full resolution supported by the SLM (32 × 32 pixels in the simulation). This coarser spatial discretization leads to increased diffraction losses compared to DASH (which always exploits the full SLM resolution, regardless of the number of tested modes) during the course of the optimisation. Additionally, we observe that for DASH the steepest changes in signal usually occur at the beginning of each iteration. This is due to the fact that we intentionally sort our test modes in ascending order with regard to the angle between their propagation direction and the optical axis. Since small scattering angles are typically more dominant in fluorescence imaging settings, this tends to speed up the algorithm convergence. Both methods achieve comparable final Strehl ratios around 0.75 and 0.85 for a-CSA and DASH, respectively (see Supplementary Material). Our experimental measurements (Figure 4A, right) show the same general trends. When comparing absolute values of signal enhancement, it is important to keep in mind the critical dependence on the sample thickness: in our simulations, e.g., we observe for increasing the thickness as 0 → 4 μm → 10 μm (at constant fluorophore density) a tendency of decreasing final enhancements of 2.14 → 1.32 → 1.29 in case of a-CSA and 2.9 → 1.8 → 1.5 in case of DASH. Given that in our experiments the sample layer thickness is controllable and measurable only with limited accuracy, it seems fair to claim good agreement between the observed values in simulation and experiment.

In Scenario B, we assume medium turbidity with L/l_s = 3 and an intermediate contribution of modes of higher spatial frequency, σ = 3. We increase the number of correctable modes to 64 to account for this greater spatial frequency content. Before correction, the typical Strehl ratios in this scenario are on the order of 5% (Figure 4B, bottom left). Our numerical simulations (Figure 4B, center) indicate that again, a-CSA initially improves faster than DASH, but levels off at a lower final signal. The total signal enhancement achievable by both correction algorithms is higher than for the low-turbidity case; the final Strehl ratios are around 0.4 for a-CSA and 0.6 for DASH. We suspect that the initial delay of DASH in comparison to a-CSA is related to the appearance of higher diffraction orders (“ghost foci” as shown in Figure 2) stemming from the phase-only field shaping of DASH. This is one of several reasons which lead us to believe that the use of a complex field-shaping technique might bear great potential for improvement of algorithm performance in the future. In our experimental measurements (Figure 4B, right), we observe similar behaviour. Initially, a-CSA improves faster than DASH, but ultimately DASH delivers higher signal enhancement. For a-CSA, we observe a kink in the signal enhancement at the transition between two iterations (marked by the blue asterisk in Figure 4), which is not present in the simulations. We attribute this kink to the circular pupil in our experiment which cuts power from superpixels located near the corners of the square SLM pattern. Since we step through the a-CSA superpixels in order of their distance to the pupil center, this effect is most dominant towards the end of each iteration.

In Scenario C, we assume high turbidity, with L/l_s = 5 and σ = 5, where without correction typical Strehl ratios are on the order of 1%. In this scenario, we correct 256 modes. In the numerical simulations (Figure 4C, center), we observe that the initial speed advantage of a-CSA compared to DASH decreases, but the final enhancements achievable with both algorithms are even higher than in Scenario B. Comparing between the two algorithms, our simulations again deliver a better performance for DASH than for a-CSA. Final Strehl ratios are around 0.35 for a-CSA and 0.45 for DASH. These trends are well supported by our experimental data (Figure 4C, right).

Graphical animations of our simulated correction runs are provided as GIF files (see Supplementary Material). These animations show the phase patterns displayed on the SLM during the correction algorithm as well as the evolution of the focal plane intensity distribution.

Both DASH and a-CSA can offer striking quality improvements for imaging through turbid media, such as in microscopy of layers deep inside tissue. To demonstrate this, we use our TPEF scanning microscope to image microglia expressing green fluorescent protein (CX₃Cr-1^GFP, cf. Ref. [1]), located 200 μm deep inside the corpus striatum of a mouse brain slice fixed via perfusion. For this, we adjust our wavelength to the excitation maximum at 900 nm, retrieve a precorrection for optical system aberrations by focussing on microglia directly below the coverslip, and subsequently move the objective focus mechanically 200 μm down into the brain tissue.

Figure 5A  shows an example image of a microglia cell in the deep layer, recorded with only the precorrection applied. Light scattering in the brain tissue above leads to rather low contrast between structures in Figure 5A, as illustrated by the plot of signal intensity along the black dashed line, shown in the inset. Figures 5B–D show the same microglia cell after running one out of three AO correction algorithms, respectively, each for 3 iterations of 256 modes. Figure 5B shows the cell after application of the correction mask shown the upper left inset, which has been obtained by performing a-CSA with power-overhead in the test mode (f ∼ 2/256) on top of the precorrection. The improvement in signal is on the order of a factor of 2–3 across the cell body, and processes extending from the cell into its surrounding are starting to become visible, especially when a logarithmic color map is applied (upper right inset). In Figure 5C, the cell is shown after performing regular CSA (i.e., without power overhead in the test mode, f = 1/256) on top of the precorrection. The measured modulation SNR was insufficent for algorithm convergence; as is apparent, the signal quality did not improve compared to the precorrection alone (Figure 5A), or even became slightly worse. Figure 5D shows the cell after performing DASH (3 iterations of 256 modes, f = 0.3) on top of the precorrection. The signal intensity across the cell body is increased by a factor of about 5; the contrast between structures is clearly enhanced, allowing to distinguish processes extending from the cell body into the surrounding tissue.

Note that for taking the images of Figure 5 we have started from the precorrection of Figure 5A simply for illustration purposes, to disentangle the correction of (mild) optical system aberrations from the correction of scattering inside the brain tissue, and for compatibility with the dye-slide simulations and experiments. This, however, is not an experimental necessity; typically, images of a similar quality as shown in Figures 5B, D can also be obtained by performing DASH or a-CSA without any precorrection.

The practicality of every sensorless wavefront correction scheme depends on its robustness with respect to noise. It has already been shown that DASH performs well in this regard in comparison to alternative methods [1]. Let us now compare performances at low SNR conditions. The key for all such methods is being able to discern the TPEF signal modulation—caused by phase-stepping of the test beam—on top of the noise floor, since this modulation carries the relevant information. For this, the test mode must contain a significant percentage of the total light power. In the following, we use numerical simulations to compare which SNR each of the three methods DASH, ⁷ a-CSA, and regular CSA ⁸ requires to operate successfully. To this aim, we simulate shot noise in the PMT readout and then repeatedly execute the methods starting from a decreasing initial signal level. We stop repeating once the signal enhancement η achieved by a method (after 3 full iterations) drops below a certain threshold η _th. We set this threshold to η th = 0.75 η max − 1 + 1 , where η _max is the enhancement for the noise-free case (i.e., simulations as in Section 4.4). The exact choice of the threshold prefactor (0.75 in our case) is largely arbitrary and uncritical for our conclusions. We assume the same sample and investigate the same three turbidity scenarios A, B, C as discussed above.

Our simulations show that for weak scattering (A) and correction of N ² = 16 modes, DASH signal enhancement crosses threshold when the photon level (before correction) exceeds around 100 photons. a-CSA delivers comparable performance when the test-superpixel contains about 10 times more laser power than any of the other superpixels, corresponding to around 40% of the total power. Regular CSA only crosses threshold for signal levels from around 1000 detected photons onwards, i.e., about 10 times higher than required for DASH.

For medium turbidity (B) and N ² = 64 modes, DASH crosses threshold at about 100 photons collected per measurement (before correction). Comparable performance can be achieved using a-CSA if the test-superpixel amplification factor is around 50, again representing roughly 40% of the total laser power. Regular CSA crosses threshold at around 3 k photons.

Finally, for high turbidity (C) and N ² = 256 modes, DASH crosses threshold at around 400 counted photons (before correction). Again, comparable performance can be achieved using a-CSA with a test-superpixel amplification factor around 50. Regular CSA, in contrast, requires about 15 k photons.

These results are summarized in Table 1. Of course, in practice the required signal levels will also depend on the nature of the sample, wherefore these numbers can only serve as a rough orientation. For example, increasing the sample thickness will make it increasingly difficult to obtain a successful correction.

Nevertheless, from our results we can draw three main conclusions. First, as expected, regular CSA requires a much higher SNR than DASH to operate successfully. Achieving a higher SNR requires sending more optical power into the sample volume, making regular CSA unfavourable, e.g., for imaging of fragile specimens. Second, it is possible to successfully operate a-CSA at the same SNR conditions as DASH, if a sufficient amplification of the test-superpixel can be provided. However, it needs to be stressed that if the amplification is realized using a superpixel method (as in Section 4.3), discarding light from other pixels, this high performance of a-CSA comes at the price of wasting much optical power. For instance, if the hologram features N×N superpixels and the test-superpixel is supposed to contain a fraction f of the total pupil intensity, the incident laser power must be increased by a factor of (N ² − 1)f/(1 − f) to keep the total optical power in the objective pupil constant. Even for a moderate number of modes of 64, i.e., N = 8, achieving f = 0.3 requires a total power increase by a factor of 27. This steep scaling with N in practice quickly becomes unfavorable for application of a-CSA in deep tissue imaging, where typically many correctable modes are needed. Third, for DASH, in contrast, such problems do not arise, since the power in the test mode is independent of N and can be tuned conveniently through the depth of the corresponding grating on the SLM.