Fast holographic scattering compensation for deep tissue biological imaging.

Experimental demonstration The performance of DASH was experimentally compared to F-SHARP by measuring the TPEF signal from a cluster of quantum dots obfuscated by a highly scattering tape layer. The laser was tuned to 810 nm to maximize the two-photon signal and a power of ≤10 mW was used to avoid photobleaching. TPEF images were acquired at three intervals during the correction process (i) before the first iteration when no correction was available at measurement number m = 0, (ii) after the first iteration through the 225 modes corresponding to m = 1120, and (iii) when both algorithms have converged after 2025 measurements. During each measurement, the TPEF intensity for 5 subsequent phase steps of M n is recorded, with a total signal accumulation time per measurement of 1 ms. The resulting corrected images are shown in Fig. 2 a along with intensity profiles extending over 5 μm through the center of the corrected region along the dashed line. A striking difference can be seen after the first measurement iteration, where the DASH correction achieved over an order of magnitude higher signal enhancement than F-SHARP (here, signal enhancement is defined as the ratio of the maximum signal in the corrected image to that in the uncorrected image with the uncorrected scanning beam blocked). At this point, the DASH correction already forms a bright, well-defined focus that could be used for imaging while the F-SHARP correction has hardly improved the wavefront distortion. Furthermore, even after nine measurement iterations when both algorithms had converged, the DASH correction continued to yield a 50% higher signal enhancement than F-SHARP. Note that the color scales of the images in Fig. 2 a have been optimized for visibility, while the magnitude of the inset line profiles have been normalized to the maximum DASH corrected signal. Fig. 2 Comparison of DASH and F-SHARP. a TPEF images of a quantum dot sample in DASH (top) and F-SHARP (bottom) before correction, after one iteration, and after both algorithms have converged with intensity profiles extending over a lateral distance of 5 μm along the dashed line and the final phase masks shown as insets along with the measurement number, m. b Signal enhancement η on a uniform dye sample after each mode measurement during DASH (blue) and F-SHARP (red) algorithms. The convergence of the two algorithms was further investigated by measuring the TPEF signal from a uniform layer of dye under a highly scattering tape mask. This uniform sample allows quantification of the signal enhancement from each mode measurement, which is plotted in Fig. 2 b. This emphasizes that the DASH algorithm benefits from rapid signal increase with every measurement, enabling a signal enhancement of nearly 11x after just the first iteration. In contrast, the F-SHARP correction is only implemented at the end of each measurement iteration, leading to a step-wise signal increase. This hinders the signal growth significantly, and at the end of the first measurement iteration the TPEF signal is only slightly enhanced. At this stage, the enhancement provided by the DASH algorithm was more than a factor of 8 higher than that provided by F-SHARP. After nine iterations through the correction modes, both algorithms had converged to their optimal correction phase masks and the DASH correction performed only modestly better than the F-SHARP correction. The higher overall signal enhancement of DASH probably arises from errors introduced due to the experimental complexity of F-SHARP. Specifically, the use of an external scan mirror in F-SHARP means that the correction pattern derived from the scanned interferograms must be manually aligned to the phase mask on the SLM. In contrast, DASH has a common-path design, which means that mode wavefront M n and reference field C both originate from the same SLM pattern. Each mode M n is directly applied to the SLM during both the phase stepping interferometry and correction steps, which alleviates the need for an alignment calibration. DASH was also compared experimentally with IMPACT, yielding similar results as discussed in Supplementary Materials Section 2 .

Methods

A home-built two-photon scanning microscope with tunable femtosecond laser excitation (Spectra Physics Mai Tai DeepSee) is used for TPEF. As shown in Fig. 1 a, the excitation beam is sent through a beam splitter (BS, Thorlabs BS014) and the reflected beam is sent to an SLM (Hamamatsu X10468-07), which is imaged onto the entrance pupil of a water immersion objective lens (OL, Olympus XLUMPLFLN20XW, NA = 1) and the xy-scan galvos (Thorlabs GVS011) using 4f relays. Note that while a high resolution SLM display was used here, in general the number of SLM pixels does not need to exceed the number of corrected modes. The light transmitted through the beam splitter (about a third of the total power) is blocked while running the DASH algorithm, but for the F-SHARP experiments it is reflected off a combined tip/tilt and phase stepping scan mirror (Physik Instrumente S-325) to generate the scanning interferometer arm 38 . The laser power is increased slightly during the DASH algorithm so that the same total power is used for each approach. The excitation light is then either passed back through the beam splitter in the DASH configuration, or the scan and reference arms are interfered on the beam splitter in the case of F-SHARP. Finally, a dichroic mirror (DM) is used to direct the excitation light through the objective to the sample plane and subsequently to send the filtered fluorescence signal to the detection path where it is measured using a PMT (Hamamatsu H10682-210). The principle of the DASH algorithm is outlined in Fig. 1 a and the simulation code is included in the Supplementary Materials . The excitation beam is holographically split by the SLM into a modulated wavefront M n with an additional phase shift φ p and a reference field C i , n , where i denotes the iteration index, starting at i = 0, and n is the mode number, which is stepped from 0 to N − 1. During the first measurement, the phase of the reference field is set to zero, i.e., angle( C 0,0 ) = 0. The two fields are given a specific intensity weighting, determined by the constant f , by displaying the following phase pattern on the SLM: ssssq 1 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{{Phi }}}_{i,n,p}={rm{angle}}left(sqrt{1-f}frac{{C}_{i,n}}{| {C}_{i,n}| }+sqrt{f}{e}^{jleft({M}_{n}+{varphi }_{p}right)}right).$$end{document} Φ i , n , p = angle 1 − f C i , n ∣ C i , n ∣ + f e j M n + φ p . The choice of f can influence the convergence behavior, and its optimal value depends on both the sample structure and the signal intensity. Further details on the optimal value of f are provided in Supplementary Section 3 , but in practice a value of f ≈ 0.3 has proven to be robust for all cases that we have investigated in both simulations and experiments. The wavefront modulations M n can take the form of a broad range of basis functions. Here, a phase grating basis was chosen with M n = k x , n x + k y , n y , where x , y are the row/column pixel indices of the SLM and k x , n , k y , n are the k-vectors of grating n . Because the SLM is in a Fourier conjugate plane to the sample (conjugated to the objective pupil plane), the phase gratings effectively scan the modulated beam across the reference beam which is analogous to the scanning interferometry used in F-SHARP. Similarly to F-SHARP and IMPACT, during the initial phase measurements both the reference and modulated beams are scattered by the sample and appear as speckle patterns in the image plane. However, as the correction improves, the reference beam becomes a single bright focus, thereby improving the accuracy of the subsequent phase measurements (see Fig. 1 a). The phase shift φ p = p (2 π / P ) is stepped between subsequent measurements p = [0, 1, . . . , P − 1] of the two-photon signal intensity. The minimum value for P is three. Here, the number of phase steps P = 5 was chosen to optimize the correction with the minimum number of measurements under our experimental conditions. A phase stepping interferometry algorithm is then used to determine the phase offset ϕ i , n and amplitude weighting a i , n for mode M n in iteration i as described in the Supplementary Section 4 . Finally, this information is immediately used to update C : 2 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${C}_{i,n+1}={C}_{i,n}+{a}_{i,n},{e}^{j({M}_{n}-{phi }_{i,n})}.$$end{document} C i , n + 1 = C i , n + a i , n e j ( M n − ϕ i , n ) . This way, the conjugated scattered wavefront is built up step by step. The process is repeated until all of the modes, in our case N = 225, have been measured, at which point another iteration through the modes can begin with M 0 . The final reference field C i , N −1 of the completed iteration i acts as initial field for the following one, i.e., C i +1,0 = C i , N −1 . Apart from the fact that the DASH routine updates the correction mask after each mode measurement, there are two other notable differences to F-SHARP: Firstly, in DASH the mode wavefront M and the reference beam C are shaped by a pure phase mask while in F-SHARP the full complex field is modulated by the beam splitter and scanning mirror. The phase only light modulation used in DASH inevitably introduces a small systematic error to the measured phase. Secondly, each newly measured mode contribution is added to C instead of replacing the contribution found in the previous iteration, which was found to be more robust to errors in the phase measurement in numerical simulations because it effectively averages all previous mode measurements and increases the accuracy of the contribution.

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