1. Introduction

In vivo microscopy has become an indispensable tool in modern biology [1–3]. The high spatial and temporal resolutions and the noninvasiveness are greatly desired in a variety of applications. As many interesting biological events happen inside the tissue, we need imaging methods that can image deep within the tissue. Currently, the most successful and widely adopted method remains the multi-photon microscopy [3]. In particular, the two-photon excited fluorescence microscopy (TPM) is the gold standard and perhaps the most popular method for deep tissue fluorescence imaging.

To image at large depth, we can choose the high NA objective with low magnification to collect as much as possible the scattered fluorescence light. We also need to use a PMT of high quantum efficiency and large etendue. In addition, the optics between the objective and the PMT need to be well designed to ensure all the light that exits the objective’s pupil is collected by the PMT. Long excitation wavelength also helps to improve the imaging depth although the fluorescence label eventually constrains the choice of wavelength [4].

Another important parameter we can control is the optical wavefront [5–25]. Although the refractive index of biological tissue is very similar to that of water, the value in general is slightly higher. For the lowest order, in which we treat the tissue as a uniform piece of material, there will be spherical aberration that can elongate the focal volume and reduce the peak intensity. The focus distortion by spherical aberration can be rather dramatic for high NA excitation (NA≥1). In highly turbid tissue, low order wavefront correction is far from optimal as it can only provide modest focus quality improvement. To greatly improve the imaging performance at large depth, high order correction is needed. Towards such a goal, our team developed Iterative Multi-Photon Adaptive Compensation Technique (IMPACT) that can rapidly measure and correct high order wavefront distortion [26, 27]. Recently, we employed IMPACT to perform high resolution imaging of mouse neocortex at large depth and to image the neocortex through the intact skull of adult mice [27]. As the bone in the skull was highly turbid, the image appeared as random speckles if conventional methods were used. It was the higher order correction that made the high resolution imaging possible. The major drawback of high order correction is the rather limited correction FOV. Although tiling can help form a larger image [26], for many important applications, such as the calcium imaging of mouse brain, it is still greatly desired to have a simultaneous correction over a large plane. This is currently the burning problem of high order wavefront correction.

In most if not all of the microscopes that incorporate wavefront corrections, the wavefront corrector is positioned at the rear focal plane (pupil plane) of the objective lens. During the laser beam scanning, the correction wavefront is translated across the sample, which works well only if the wavefront distortion is translation invariant as in the case of spherical aberration. For the complex 3D refractive index inhomogeneity, the wavefront distortion is spatially varying and therefore the wavefront correction becomes uncorrelated with the tissue during the translation, which limits the correction FOV.

A better configuration is to image the wavefront corrector onto the tissue. If the corrector is conjugated to the tissue, their relative position does not change during the beam scanning. As the refractive index variation is 3D in nature, we would need multiple 2D correctors conjugated to different layers of tissue. Such a multi-conjugate configuration was originally developed for astronomical adaptive optics [28] and was recently applied to high resolution retina imaging [29]. Using multi-conjugation for microscopy was also studied for simple refractive index profiles by using the ray tracing method [20]. In this work, we use simulation to investigate the performance of multi-conjugation for handling highly complex random refractive index distribution and compare multi-conjugation with pupil plane configuration for high order wavefront correction.

2. Simulation method

In the simulation, we assumed that the tissue comprised 96 distortion layers. They were uniformly distributed with ten micron spacing. Therefore, the incident laser beams need to travel 960 µm before reaching the focal plane. The NA of the objective was 0.8. The average refractive index of the media was 1.33 and the laser wavelength was 920 nm. We used the Fourier space propagator method to calculate the wave propagation between the adjacent distortion layers [30]. Basically, we Fourier transformed the E field$E(x,y,z)$ from the spatial domain to the spatial frequency domain where we multiplied the E field by the Fourier space propagator ($e^{i{k}_{z}d}$). An inverse Fourier transform yielded the E field$E(x,y,z+d)$ in the spatial domain. Each distortion layer was treated as a phase mask with a random spatial profile. One example of the phase mask was given in Fig. 1(a). Due to the large dimension (5.6 mm wide, 0.96 mm deep), we performed 2D simulation to shorten the computing time. We assumed that the incident light intensity at the objective pupil plane was uniform. The diffraction limited focus profile is shown in Fig. 1(b). After propagating through the 960 µm thick tissue, the focus was far from optimal, as shown in Fig. 1(c).

 

Fig. 1 (a) The phase profile of a portion of one distortion layer. (b) The focus intensity profile in clear medium (n = 1.33). (c) The focus intensity profile after the light propagated through the 960 µm thick tissue.

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Figure 2(a) shows the proposed multi-conjugate system. The translation invariant distortion was corrected by the pupil plane corrector. Multiple relay lenses were used to image the wavefront corrector onto the right plane and very importantly in the right order. Figure 2(b) shows a much simpler configuration, in which all the correctors were still imaged to the right locations but in the wrong order. For highly turbid media, it is very important that the last corrector is for correcting the top tissue layer. After they cancel each other, the second last corrector cancels the subsequent tissue layer, and so on. The compactness of Fig. 2(b) is appealing but it does not work at all for the highly turbid tissue studied in this work. This problem has also been discussed in the astronomical AO literature [28].

 

Fig. 2 (a) Multi-conjugate configuration. A corrector (grey color) was placed at the pupil plane to correct translation invariant distortion. Multiple correctors were imaged to the corresponding tissue layers in the right order. (b) In a simpler setup, the relay lenses directly imaged the tissue volume onto the corrector volume. The axial position of each corrector was right but the light traveled through the corrector in the wrong order.

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To determine the phase values of each corrector, we let the laser beam propagate from the diffraction limited focus through the tissue section and then propagate back (apply reversed propagator$e^{-i{k}_{z}d}$) in clear medium (n = 1.33) to the corrector plane and recorded its electrical field$E_1(x)$, as shown in Fig. 3(a). This procedure essentially converted the accumulated wavefront distortion through the tissue section into a phase mask on the corresponding corrector plane. We also let the beam directly propagate from the focus to the corrector plane in clear medium and recorded the phase profile${\varphi }_1(x)$. If the tissue is distortion free like the clear medium, ${\varphi }_1(x)=\arg [E_1(x)]$. If we define$E_2(x)=E_1(x)\exp [-i{\varphi }_1(x)]$, the corrector phase should be$-\arg [E_2(x)]$. In other words, the phase difference between the distorted beam and the distortion free beam should be put on the corrector. As shown in Fig. 1(a), the laser beam only converged on a finite portion of the corrector. To determine the phase values over extended range on the corrector, we repeated the above mentioned procedure for a number of focal locations, as shown in Fig. 3(b). After the measurements, we obtained a summation$E_2(x)={\displaystyle \sum _{i=1}^N{E}_{1i}(x)}\exp [-i{\varphi }_{1i}(x)]$, where N was the number of focal locations. We then displayed$-\arg [E_2(x)]$onto the corrector. In this way, we obtained an averaged phase profile weighted by the amplitude of the E field. We call this procedure wavefront averaging. Each corrector was conjugated to the middle plane of its corresponding tissue section. For example, the correctors were located at 160, 480, and 800 µm away from the focal plane in a triple-conjugate system and each corrector handle da 320 µm thick tissue section.

 

Fig. 3 (a) Determine the electric field at the corrector plane. We let the laser beam travel from the diffraction limited focus through the tissue section and then propagate back in clear medium (n = 1.33) to the corrector that was located at the middle plane of the tissue section. (b) Determine the corrector phase values over extended range. We repeated the measurement for multiple focal locations such that the laser beams covered a larger portion of the corrector.

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For the pupil plane correction, we let the beam propagate from the focus through the entire sample (from 0 to 960 µm) and obtained the distorted E field. As the corrector was located at the pupil plane, we need to propagate the distorted E field at 960 µm to the pupil plane. To get there, we let the distorted wave propagate back in clear medium(apply reversed propagator$e^{-i{k}_{z}d}$) to the focal plane where we performed a Fourier transform to convert the E field from the focal plane to the pupil plane. For averaging over multiple focal locations, we used the same procedure as that of the multi-conjugate configuration, in which we calculated the summation of the field and then got the averaged phase values.

3. Results

In the first study, we used the same tissue profile and the same spatial range for wavefront averaging (41focal locations, 3.42µm between adjacent locations, 137 µm overall wavefront measurement range),increased the corrector number from 1 to 12 and recorded the focus peak intensity over a 200 µm spatial range. The results are summarized in Fig. 4. With 12 correctors, the focus intensity at the center of the FOV was close to the value obtained in a clear medium (n = 1.33, focus peak intensity = 1). Compared with the case without wavefront correction, the focus intensity was improved by a factor of 17. The intensity improvement decreased when we reduced the corrector number. However, the correction FOV was nearly constant and agreed well with the range of the wavefront averaging (137 µm). Interestingly, even with just one corrector (single-conjugation), we could still obtain a ~twofold Strehl ratio, which was equivalent to a fourfold two-photon or an eightfold three-photon excitation improvement. In comparison, the pupil plane correction was not effective because the pupil plane phase profiles for well separated focal locations were uncorrelated. The measurements over the 41 focal locations essentially averaged out the correction.

 

Fig. 4 Correction FOV of multi-conjugate system with different numbers of correctors (bold number in the plot). For comparison, we plot the peak intensity over a 200 µm range for multi-conjugation (MC), flat wavefront without correction (flat), and pupil plane correction (pupil).

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In the next study, we set the corrector number to four and varied the spatial range of wavefront averaging. The results are summarized in Fig. 5. Similar to the data in Fig. 4, the correction FOV agreed well with the range of wavefront averaging. When we shrunk the range, the focus intensity increased. If the range was sufficiently small as in Fig. 5(d), pupil correction became effective.

 

Fig. 5 Multi-conjugate correction with different range of wavefront averaging. The bold number in each plot is the range of averaging in µm.

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From the data in Fig. 4(a), we know that the single-conjugate configuration can outperform the pupil plane correction. Therefore, for small FOV imaging as in Fig. 5(d), perhaps single-conjugation can do a decent job. In Fig. 6, we compare single-conjugation with pupil plane correction for small FOV imaging. The single-conjugation nearly doubled the Strehl ratio obtained by the pupil plane correction, as shown in Fig. 6(a). We further increased the phase amplitude of the tissue phase masks to make the tissue more turbid and show the comparison in Fig. 6(b). The Strehl ratios of single-conjugation and pupil plane correction were both reduced but the single-conjugation still yielded a twofold improvement.

 

Fig. 6 (a) Compare single-conjugation with pupil plane correction for small FOV imaging. The phase amplitude of the tissue phase masks in (b) was higher than that in (a) to further reduce the Strehl ratio and make the tissue more turbid. The range of wavefront averaging was 34 µm in both plots.

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The need for multi-conjugation arises from the spatially varying refractive index distribution. How many conjugation layers are needed would therefore depend on how fast the spatial variation happens. In the simulation, we can control the tissue property and investigate this question. In all the simulation we have discussed so far, we used a low pass filter to limit the phase mask’s feature size to 10 µm, comparable to the size of a cell. In the next study, we changed that number to 5 µm and 20 µm. The results are summarized in Fig. 7. We adjusted the phase amplitude of the tissue phase mask such that the Strehl ratio without correction was near 0.1. We set the wavefront averaging range and therefore the correction FOV to 100 µm. We chose different corrector number such that we could obtain a ~threefold peak intensity enhancement over the 100 µm FOV. For tissue of 5, 10 and 20 µm feature size, we need to use four, two and one corrector layers to achieve comparable level of enhancement.

 

Fig. 7 Wide FOV correction for different tissue. The range of wavefront averaging was 100 µm and the goal was to have a threefold peak intensity enhancement. (a) The tissue feature size was 5 µm and quadruple-conjugation was used. (b) The peak intensity was normalized to the value without correction to show the enhancement factor. (c) and (d) are for tissue of10 µm feature size with double-conjugation. (e) and (f) are for tissue of 20 µm feature size with single-conjugation.

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4. Discussion and conclusion

The simulation suggests that the widely adopted pupil plane correction is far from optimal for wide FOV high order wavefront correction. In most cases, it fails to provide focus enhancement. In comparison, the multi-conjugate configuration consistently provides greater focus intensity. Even for the simplest case of single-conjugation, it still outperforms the pupil plane correction. Therefore for practical implementation, we should use the pupil plane correction to handle only the spatially invariant wavefront distortion and use the conjugation plane to handle the spatially varying distortion.

Given the same number of conjugation layers, the enhancement could increase if we constrained the wavefront averaging range. After all, we still used 2D correctors to handle the 3D tissue sections. A wider wavefront averaging range means that the angular variation of the beam that travels through the same position on the 2D corrector is greater. This averaging can reduce the effective enhancement and is nonetheless needed for wide FOV imaging.

How well the multi-conjugate configuration works and how many correction layers are needed would depend on the nature of the tissue. As shown in Fig. 7, it is challenging to do wide FOV correction for tissue whose refractive index profile varies rapidly over space. More conjugation layers are needed to provide the same level of enhancement. However, it is worth to note that a factor of three Strehl ratio gain can be translated to a ninefold two-photon signal improvement, which could make a huge difference in both quality and speed for practical imaging applications.