The inherent inhomogeneity of complex samples such as biological tissues induces light scattering, which limits the resolution of light focusing and poses a major hurdle to deep-tissue microscopy [1]. Adaptive optics techniques are very effective in correcting the aberrations induced by sample index mismatch and thin tissues [2–4]. However, until recently, these techniques were considered impractical for turbid and thick multiply scattering samples, where no unscattered, “ballistic” light components remain and the propagating light is diffused to form complex speckle patterns with no simple relation to the incident wavefront [5,6]. This conception changed after the 2007 work of Vellekoop and Mosk [7,8], where it was shown that high-resolution wavefront shaping can be used to create a high-intensity, diffraction-limited focus deep in the diffusive light propagation regime where essentially no ballistic photons are present, such as through a thick layer of white paint or the shell of an egg.

Diffraction-limited focusing by wavefront shaping is attained even when the number of degrees of control is much smaller than the number of scattered modes, i.e., with a correction that is far from being perfect [7–14]. Indeed, as a result, only a small fraction of the initial light energy is actually focused by this technique, but the high-contrast focus can still be very valuable for many applications, and in particular for nonlinear microscopy. The main limitation of wavefront-shaping techniques is that a direct feedback from the target point is required, either by directly observing it with a camera [8–14], or by embedding a “guide star” at the target position [12,13]. Both these requirements are invasive and are thus incompatible with most imaging applications. Very recently, wavefront shaping was combined with acoustics to obtain the feedback signal noninvasively [15–20], though with focal dimensions exceeding the optical wavelength.

Here, we show that nonlinear optical feedback, such as the one widely used for two-photon microscopy, can be used in epidetection geometry to obtain diffraction-limited focusing through strongly scattering turbid samples. Furthermore, we demonstrate that this focus can be used for nonlinear imaging (here demonstrated in two-photon microscopy) in reflection mode in a totally noninvasive manner. We show that one can focus an ultrashort pulse on an extended object hidden behind a scattering medium by optimizing the total nonlinear signal that is produced in the specimen. This technique should be adaptable to most nonlinear microscopy techniques, such as two- and three-photon fluorescence (2PF, 3PF), second- and third-harmonic generation (SHG, THG), four-wave mixing, and coherent anti-Stokes Raman scattering (CARS). We show that the optimization of a nonlinear signal of sufficient order results in a single, tightly focused spot, even when the optimized signal is collected from a very large area on the object, specifically by epidetection through the same scattering medium. Such a result cannot be obtained by optimizing a linear signal such as total intensity of fluorescence or reflected intensity. Nonlinear optimization has been shown before to correct also for the temporal distortions of ultrashort pulses propagating through scattering media [21]; thus, the generated focus is also short in temporal duration and hence intense. Furthermore, by raster-scanning this focal spot, exploiting the so-called memory effect [22,23], an image could be collected from the vicinity of the optimized focus [11,12,24]. We believe that these results form an important step toward optical focusing deep inside scattering tissues and imaging through turbid layers.

Figure 1(a) shows the setup we have used for demonstrating our technique with two-photon excited fluorescence microscopy, one of the most widely used nonlinear imaging modalities in neuroscience [25]. Pulses $\sim 100\mathrm{fs}$ long from a Ti:sapphire laser illuminate a fluorescent object after passing through a phase-only spatial light modulator. As a first experiment, an optical diffuser was used to introduce scattering and a thin, two-photon homogeneous fluorescence screen served as the distributed object. The total excited fluorescence signal is epidetected through the scattering medium with a single sensitive integrating detector. The resulting spatial distribution of the two-photon excited fluorescence on the fluorescent screen target is inspected by a camera placed on the other side of the medium, but its image is not used for the wavefront shaping optimization (see Fig. S1 in Supplement 1), an important advantage over the previous works on nonlinear signal optimization [21,26]. Figure 1(b) shows the speckle, spatially scattered fluorescence pattern recorded from the screen prior to optimization. We then used an optimization procedure based on a genetic algorithm [27] to find the SLM phase pattern that maximizes the total two-photon fluorescence signal measured in epidetection. Figure 1(c) shows the resulting optimized focus point on the screen. The progress of the total detected signal during the optimization process is shown in Fig. 1(d) and in Media S2. As can be observed in Fig. 1(c), by optimizing the total backward-scattered 2PF signal collected via the illumination optics with a large-area detector, the excitation pulses focused to a single diffraction-limited focal spot on the object, though its location is not controlled or predetermined. Note that the total enhancement in the signal is rather small (about 50% over its initial value), although the focused intensity is enhanced by a considerably larger factor. In the simple case of a planar fluorescent object and where no temporal pulse distortions are present, the total signal enhancement factor is expected to be $\sim N_{\mathrm{SLM}}^N/N_{\text{speckles}}$, where $N_{\text{SLM}}$ is the number of controlled SLM pixels, $N$ is the order of the nonlinearity ($N=2$ for 2PF), and $N_{\text{speckles}}$ is the number of speckle grains that initially illuminate the fluorescence object and that contribute to the initial signal [See Eq. (S1) in Supplement 1].

 

Fig. 1. Noninvasive focusing through scattering samples. (a) Experimental system. 100 fs pulses are sent via a spatial light modulator (SLM) and focused through a diffuser on a two-photon fluorescent screen. The fluorescence is collected via the same optics and used as a signal for an optimization algorithm. An auxiliary imaging system records the speckle image on the screen. (b) Speckle pattern recorded from the 2PF screen before optimization. (c) Optimized focus via maximization of total epi 2PF. (d) Progress of the optimization process. Scale bars, 100 μm.

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Once a sharp spot is formed, it may be raster-scanned to obtain a microscopic image of the fluorescent object hidden behind the scattering layer [11,12]. The image is collected point-by-point, in a similar manner to image formation in standard two-photon microscopy, by scanning the focal point through the sample; for example, by using the same SLM as a scanning device [12], although fast scanning mirrors can be utilized for increased speed. The field of view (FOV) of this imaging technique is limited by the range where the single SLM wavefront correction is effective, which is dictated in the diffusive light propagation regime by the optical memory effect [22–24]. This range is analogous to the isoplanatic patch size in turbulent media [28]. In a diffusive medium of thickness $L$, the HWHM angular FOV of the memory effect can be estimated by $\mathrm{\Delta }{\theta }_{\mathrm{FOV}}\approx \lambda /\pi L$ [22]. As a result, similar to other memory-effect-based techniques [11,12,24] a large FOV is available only when the target distance from the diffusive layer is considerably larger than the effective layer thickness, as is the case when imaging in an eggshell geometry or “around corners” in reflection geometry [24]. Wide-field imaging in the diffusive regime inside thick, multiply scattering tissues (i.e., at depths larger than the transport mean free path) presents a challenge. Note that, for the scanning to be effective, it is crucial to conjugate the correcting SLM and scanning optics plane with the distorting scattering layer plane [11,12,24].

An experimental demonstration of focusing and raster imaging using this technique is presented in Fig. 2. In this experiment, the fluorescent screen is replaced by a cluster of two-photon fluorescence Fluorescein crystallites on a glass cover slip, placed at a distance of $\sim 2\mathrm{mm}$ behind the optical diffuser (Newport 10° light-shaping diffuser). After the same optimization procedure, the formed focus (whose exact position on the object is unknown and depends on the initial conditions and fluorescent structure), was scanned by adding linear phase ramps on top of the SLM correction phase pattern. The image that was obtained with the optimized focus [Fig. 2(b)] is not just of higher resolution and contrast when compared to the blurred standard two-photon microscopy image, but is also approximately an order of magnitude brighter than an image obtained without optimization [Fig. 2(a)]. The true image of the object, with the diffuser removed, is shown in Fig. 2(c).

 

Fig. 2. Two-photon imaging through an optical diffuser using the memory effect. (a) Standard 2PF microscopy image of an object (cluster of Coumarin 307 crystallites) as observed through a diffuser. (b) After optimizing the total 2PF signal, a focus is formed. A bright and crisp image of the two-photon object is obtained by collecting the signal in epigeometry, while the focus is raster-scanned by adding linear phase ramps to the SLM phase pattern. (c) Transmission microscope image of the same object without the scattering medium is presented for comparison. Scale bar, 100 μm.

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The experimental results show that optimizing a nonlinear signal is a promising path to forming a diffraction-limited spot on a planar object in the diffusive light propagation regime, where no ballistic unscattered light is present. This result is related to recent results in adaptive optics, where two-photon optimization is used to refocus light at depths of an order of one transport mean free path [29], where an initial focus is present in the unoptimized field, and to nonlinear photoacoustic focus formation on a planar target [30]. However, as we show below, careful analysis of the focus formation process shows that in the general case of a three-dimensional fluorescent object located in the diffusive regime, a higher order ($N>2$) nonlinearity is required to assure focusing to a single speckle grain. For a linear signal, as in standard fluorescence imaging, energy conservation dictates that the total integrated signal would not change when the energy is focused tightly or spread over a large area if the fluorophores are distributed homogeneously in the specimen. In contrast, an integrated nonlinear signal does not obey such a conservation law, and the total integrated signal could be larger the tighter the beam is focused, depending on the order of nonlinearity. Simple geometrical considerations can help to determine the range where this mechanism could be applied.

To obtain some insight, consider first a simple planar nonlinear object producing an $N$th-order nonlinear signal, and assume that the incoming beam is scattered to a number of speckles $N_{\text{speckles}}$ on the object. The total generated signal power would be proportional to

The fact that our samples (either the uniform 2PF screen used in Fig. 1 or the sample used in Fig. 2) were thin was important for this process to work. To understand the issue with a thick nonlinear medium, it is instructive to consider a Gaussian beam focused to a waist $w_0$ inside such a thick medium, producing an $N$th-order nonlinear incoherent signal, such as N-photon fluorescence. In that case, for $N>1$ nonlinearity, the total nonlinear signal can be approximated by the signal generated inside a cylinder of volume $V$ along the beam confocal parameter $b=4\pi {\omega }_0^2/\lambda$. Under this assumption, the total generated (or detected) signal would be proportional to

It is evident that for $N=2$, e.g., 2PF, the total signal is constant, independent of $w_0$. Intuitively, the effect of the increased intensity by tighter focusing is compensated by the shrinking effective excitation volume [31]. Hence, optimization in a thick, homogenous second-order medium is not possible. Only for $N>2$, i.e., only for a nonlinearity which is higher than a second order such as the one used in 3PF microscopy [32], does the signal increase by focusing. This should be contrasted with the result of Eq. (1) for a planar nonlinear object, where any nonlinearity of $N>1$ was sufficient to ensure focusing down to a single speckle grain. We note that this fact is closely related to harmonic generation in bulk crystals and the determination of optimal focusing [33].

To complement this naïve basic analysis, we performed exact numerical simulations that produce results which are on par with this conclusion. Figure 3 shows results of simulations emulating the actual process: the speckle field was calculated by applying a random phase structure, and the nonlinear signal generated by the speckle field was integrated and used as feedback for optimization of wavefront correction using the same genetic algorithm. Note that in these simulations, as in our experiments, the field is first scattered by a thin layer and then propagated through free space to the nonlinear medium, which was homogeneous and nonscattering. The simulation results, which are averaged over 10 realizations of disorder for each case, show that 2PF does not lead to optimized focusing in thick three-dimensional objects, while 3PF worked well in all geometries.

 

Fig. 3. Results of simulations of optimization of 2PF and 3PF signals. A speckle field is simulated, and the calculated total nonlinear signal is then used as feedback in a genetic algorithm for wavefront shaping. (a) shows the resulting peak enhancement of such a simulation with three different thicknesses of the nonlinear medium. In thin media, both nonlinearities led to convergence and significant enhancement of the peak intensity. In thick media, however, only 3PF led to focusing and peak enhancement (b). Two-photon effects did not converge and left the speckle field practically unchanged (c).

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Nonlinear feedback signal has been utilized previously for temporal compression; as has been shown in previous works [21,26,34], nonlinear optimization not only focuses the pulse in space, but also compresses the pulse in time, compensating for temporal distortions and resulting in a spatiotemporal focus. We expect, therefore, that the optimized focal points here are also temporally short, although we have not verified this by measurements.

Note that the analysis and simulations assumed a homogeneous distribution of the nonlinear medium, which is of course far from being realistic and actually represents a worst-case scenario. In actual samples, where the nonlinear agent is distributed unevenly in the specimen, it might well be that the algorithm would perform better, as a concentrated region of high nonlinearity would serve as “guide stars” and help the optimization process. As an example, Fig. 4 shows focusing, through a 1 mm slice of brain tissue, on a nonlinear object placed $\sim 300\mathrm{\mu m}$ behind it. The algorithm finds an optimal focus which is, not surprisingly, located on a spot of highly concentrated fluorophore. Interestingly, we have observed that as the signal of one bright spot eventually fades due to photobleaching, the search algorithm will switch to another focal spot on another crystallite, bleaching it as well, and so on. We have confirmed in numerical simulations that two-photon nonlinearity is sufficient in many such cases where the volumetric nonlinear object is sparsely tagged or whenever there is even a weak ballistic unscattered component in the beam, on par with the recent experimental results of Tang et al. with 2PF [29], and Fiolka et al. with linear time-gated signal optimization [35]. An important advantage of our nonlinearity-based technique over linear-signal-based approaches employing either time-gated detection [35], ultrasonic tagging [15], or photoacoustics [20] is that, given a sufficient order of nonlinearity and measurement signal-to-noise, the generation of a single diffraction-limited focus is guaranteed. The main drawback of the technique is that the exact position of the focus is undetermined.

 

Fig. 4. Experimental focusing through a 1 mm thick brain tissue. An object (Coumarin 307 crystallites) is placed behind the fixed brain slice. (a) Image of the 2PF as measured by the inspection camera prior to the optimization. (b) After optimizing, the total 2PF measured through the scattering layer leads to a focal point which is actually localized on one of the strongest fluorescing crystallites. Scale bars, 100 μm.

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It is important to note that in the experiment of Fig. 4, as the target object was placed a relatively short distance from the scattering tissue compared to the tissue thickness, it was not possible to scan the focused spot over a range of more than a few micrometers, in accordance with the limitations of the memory effect.

An important practical hurdle to generating the focus deep inside volumetric fluorescent samples such as weakly tagged biological tissues is the large nonlinear background signal generated near the surface of such samples [36]. Although the global maximum for the optimized signal is obtained for the most tightly focused beam [Eq. (2)], the measurement of the small total signal enhancement [Eq. (S1) in Supplement 1] on top of the large background is expected to be a challenge in cases where the target object is not strongly fluorescing. This hurdle may be overcome by limiting the detection volume via ultrasonic tagging [15–17], photoacoustics [30], or spatial filtering [36]. The above analysis and discussion assumed an incoherent nonlinear process, in particular two- and three-photon fluorescence. In coherent processes, such as second- and third-harmonic generation and CARS, phase-matching considerations could have a significant effect. We will not discuss them here, but generally we expect the above conclusions to be valid in cases where the phase-matching length is not large.