Imaging deep inside biological tissues at a high resolution is a long sought-after goal in microscopy. This task is extremely challenging as inhomogeneities in the refractive index scatter light, preventing the formation of a sharp focus. This problem can be overcome by shaping the wavefront of the incident light to counteract the scattering. Recent progress in wavefront shaping has enabled sub-wavelength resolution imaging through turbid media in several proof of concept experiments [1].

In wavefront shaping, two main classes of approaches can be distinguished: feedback-based wavefront shaping [2] and optical phase conjugation [3]. Feedback-based methods depend on the detection of the intensity at a desired focus location, either directly [4] or indirectly through the use of a guide star [5]. By iteratively maximizing this feedback signal, a sharp focus is formed.

An alternative approach to focus light inside turbid media is optical phase conjugation. Here, the optimal wavefront is obtained by measuring the scattered field coming from a guide star inside the medium. Subsequently, a focus is formed by playing back the conjugate of this field using a phase conjugating mirror [5]. Many types of guide stars have been used, including second harmonic generation [6], fluorescence [7], and moving particles [8,9].

The use of guide stars, however, poses severe limitations. First of all, it is not always feasible to embed guide stars or to obtain a signal with a sufficient signal to noise ratio (SNR) to form a high-quality focus. Also, when multiple guide stars are too close together, special care needs to be taken to focus on exactly one of them [10–14]. Even more importantly, once a correction is found, it is only valid for a small region around the guide star [15], so only objects in its close vicinity can be imaged. A notable exception here is the use of ultrasound tagging to generate a virtual guide star at any desired location [16–18]. Unfortunately, here the resolution is limited by the size of the ultrasound focus, which is of the order of tens of micrometers. More advanced methods allow for a sharper focus, yet they require a much longer acquisition time [19]. To summarize, the problem of tightly focusing light at an arbitrary location inside a scattering medium is still unsolved.

Here, we introduce a third class of wavefront shaping methods, which we call model-based wavefront shaping. In model-based wavefront shaping, the optimal wavefront is computed numerically using a digital model of the sample. The microscopic refractive index model of the sample, which serves as the input for the calculations, is obtained from the image data itself. With this method, no guide stars are required, and the light can be focused anywhere, provided that an accurate refractive index model can be constructed.

The concept of model-based wavefront shaping is illustrated in Fig. 1. First, we generate a refractive index distribution model by imaging the superficial region of the scattering sample and applying a priori knowledge about the materials in the sample. Once the refractive index model is generated, it is possible to compute the wavefronts required to focus light anywhere inside the sample by performing a virtual (digital) phase conjugation experiment: we place a “virtual guide star” in our model and simulate the propagation of light from that point to outside the sample. This calculated field is phase conjugated and then constructed experimentally with a spatial light modulator (SLM). As with ordinary phase conjugation, the conjugated field will propagate through the scattering sample and form a sharp focus at the location of the virtual guide star.

 

Fig. 1. Principle of model-based wavefront shaping microscopy. Step 1: the superficial region of the sample is imaged, and a 3D model of the refractive index is generated from the image data. Step 2: the model is used to compute the wavefront required to focus at the desired location. Step 3: the computed wavefront is constructed with a spatial light modulator (SLM), resulting in a sharp focus.

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As a proof of concept of our method, we demonstrate enhanced imaging of 500 nm fluorescent beads dispersed in polydimethylsiloxane (PDMS). The sample has a rough surface that acts as a light diffuser, severely degrading the quality of the image of the beads. The sample was submersed in a watery dye solution to aid in visualizing the surface. The sample was placed in a two-photon fluorescence excitation microscope (TPM) with an SLM conjugated to the back-pupil plane of the microscope objective (see Supplement 1 for materials and methods).

 

Fig. 2. Scattering compensation using model-based wavefront shaping. Maximum intensity projection of the 3D image data acquired (a) using conventional TPM without correction, (b) using feedback-based wavefront shaping, and (c) using model-based wavefront shaping. The scattering surface is located at a depth of $z = 0$. The wavefront corrections associated with four sub-stacks are also displayed. It is clear that model-based wavefront shaping works over the entire depth of interest, whereas the feedback-based method fails when noise dominates the feedback signal.

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We performed three imaging experiments to demonstrate the feasibility and robustness of our technique. In the first experiment, we used conventional TPM (with no correction for scattering) to image the fluorescent beads. In the second experiment, we used current state-of-the-art feedback-based wavefront shaping to suppress the scattering introduced by the sample. We imaged the beads after applying the correction obtained from feedback-based wavefront shaping. In the third experiment, the beads were imaged after applying the correction obtained from our new model-based wavefront shaping method.

Figure 2(a) shows the maximum intensity projection of the three-dimensional (3D) image stack acquired using conventional TPM. We combined thirteen 3D sub-stacks to cover the depth ($z$ axis) range from 42 µm to 325 µm through the scattering layer. Each 3D sub-stack consists of 41 frames with a total volume of ${26.2}\;\unicode{x00B5}{\rm m} \times {25.6}\;\unicode{x00B5}{\rm m} \times {21.7}\;\unicode{x00B5}{\rm m}$. It is clear that the intensity of the image decreases rapidly as a function of distance from the scattering layer (located at $z = 0$).

Figure 2(b) shows the maximum intensity projection after feedback-based wavefront shaping using a state-of-the-art Hadamard algorithm [20] with 256 input modes. For every 3D sub-stack, only a single optimal wavefront was used, which was found by optimizing the feedback signal from a single fluorescent bead located at the center. It is clear from Fig. 2(b) that the intensity from the beads is higher compared to Fig. 2(a). However, the method fails to optimize the focus after a depth of 175 µm, where the SNR of the feedback signal is too low to find a correction wavefront.

For our new method, we first used the microscope to acquire a 3D intensity image of the sample surface, using a fluorescent dye in the water to visualize the rough interface. From this image, we constructed a 3D refractive index model of the PDMS–water interface. Next, we used a beam propagation method [21] to simulate light propagation from a point source located at the center of each 3D sub-stack (see Supplement 1 for technical details).

Figure 2(c) shows the results of model-based wavefront shaping. The fluorescent beads are visible all the way to the maximum depth of 325 µm, which is approximately twice the depth reached by feedback-based wavefront shaping. It is clear from the image that model-based wavefront shaping is consistently successful over the full depth range. Moreover, we can freely pick the position where we want the correction to be optimal (the center of each 3D sub-stack in our case), without relying on the presence of a guide star.

The optimized and computed wavefronts corresponding to four different sub-stacks are shown in Figs. 2(b) and 2(c). It can be seen that model-based wavefront shaping finds an accurate correction wavefront, which becomes more complex with increasing depth. This result is a clear improvement over feedback-based wavefront shaping, where the number of SLM segments in the correction wavefront is limited by the SNR and optimization time [22].

Figure 3 depicts the two-photon signal as a function of depth before and after compensating for the scattering. Without compensation, the two-photon signal decreases rapidly as a function of depth. With increasing depth, the area of the scattering surface that is illuminated by the focusing beam increases quadratically, causing the excitation intensity in the focus to decrease as ${z^{- 2}}$ [23]. When we also take into account an exponential attenuation, we find that the two-photon signal $S$ decreases as $S \propto {z^{- 4}}\exp(-2z/\ell)$, where $\ell$ is the attenuation length. The exponential decay can be attributed to the scattering and absorption by fluorescent beads and scattering by small imperfections (i.e., microscopic bubbles) in the bulk of the sample. A good fit to the experimental data is obtained with $\ell = 295\;{{\unicode{x00B5}{\rm m}}}$ (blue dashed line in Fig. 3).

 

Fig. 3. Two-photon fluorescence signal as a function of depth inside the sample. Blue diamonds, signal without correction; green squares, signal after feedback-based wavefront shaping; red circles, signal after model-based wavefront shaping. The dotted black line shows the average background level of the images. The dashed blue and red lines correspond to the theoretical expectations.

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The feedback-based method successfully enhances the image intensity until about a depth of 175 µm but fails to improve the focus at larger depths because of the drop in SNR. The maximum signal enhancement over the uncorrected case is less than a factor of five. On the other hand, model-based wavefront shaping works over the entire depth of the 3D image and shows a 21-fold increase in intensity at the deepest optimized point. It is to be noted that the signal slowly decreases with depth even after correction. As before, this decay is attributed to the absorption and scattering of light in the bulk of the sample. The red dashed line in Fig. 3 shows that the remaining decay follows $S \propto \exp(-2z/\ell)$, with $\ell = 295\;{{\unicode{x00B5}{\rm m}}}$, as before. Therefore, we can conclude that $1/{z^4}$ decay factor is compensated completely, and only slow exponential decay due to scattering and absorption in the bulk remains.

It can be seen that deep inside the sample the intensity of the image degrades towards the edges in Fig. 2(c). This is because the scanning range of the optimized focus is limited to the range of the memory effect [15]. The scanning range may be improved further by conjugating the SLM to the PDMS–water interface [24] or by finding different wavefront corrections for different positions in the sample.

This work introduces a new class of wavefront shaping methods combining TPM imaging and light propagation modeling to mitigate scattering in a robust way. We demonstrate the feasibility of our method in a proof of principle imaging experiment through a light-diffusing layer. The main advantage of our technique over other methods is that it does not require any guide star for finding the optimal wavefront. Therefore, practical limitations associated with guide stars (like limited field of view, low SNR, number of optimized modes, photo-bleaching of the guide star, etc.) do not play a role at all.

The primary step in our method is the generation of a refractive index model. Whereas we used a single light-diffusing layer in this proof of concept experiment, our technique could be extended to arbitrary turbid samples, as long as a refractive index model can be found. In principle, any refractive index model that is closer to the exact sample should improve the quality of the wavefront correction. This model can be acquired by selectively labeling the fluorescence image data with known refractive index information, as we did here. We envision that other techniques, such as optical diffraction tomography, optical coherence tomography, ptychography, and structured illumination microscopy [25–28], may be used in order to relax the requirement for a priori knowledge about the sample.

Extending model-based wavefront shaping to arbitrary turbid samples poses extra challenges in terms of numerical simulations and system alignment. The beam propagation method used in our technique only considers forward-scattered light and is therefore only valid in the paraxial regime. One may use a recently developed numerical method, which was shown to accurately simulate light scattering in large inhomogeneous media [29]. Furthermore, a more precise mapping between the simulated correction wavefront and the SLM pixels is required for more complex media.

To summarize, model-based wavefront shaping can perform near-perfect correction for scattering introduced by the diffusing layers in microscopy. It opens up a promising platform for integrating with the aforementioned refractive index modeling techniques, eliminating the need for guide stars altogether.