Multiphoton microscopy (MPM) has gained enormous popularity in a wide range of biological imaging applications, primarily because of its capacity to penetrate deep in scattering tissues.1^–5 Within the past few years, the speed scaling in MPM has been intensely pursued owing to the demand for fast monitoring of biological dynamics, revealing the cellular phenotypes, early diagnosis of diseases, high-throughput screening, etc.1^,6 Typical MPM platforms rely on a laser scanning strategy: performing transverse ($x\text{-}y$) scanning with galvanometric mirrors (GMs), then translating the objective or sample for axial ($z$) scanning.7^,8 However, the three-dimensional (3D) acquisition speed is largely hindered by the $z$-scan rate, resulting from the high mass of objects. Given the current video rate lateral scan techniques and the compatibility for raster scanning systems, scaling the temporal resolution of MPM is largely resorting to developing fast $z$-scan techniques.

Various strategies have been promoted for realizing high-speed axial scans, especially within the last 10 years. The existing approaches can generally be classified into three groups. The first one is projection imaging relying on the extended depth of field (DOF), usually provided by the non-diffracting beams (Bessel and Airy beams, etc.).9^–13 However, this method sacrifices the axial resolution by yielding two-dimensional (2D) projections of a 3D structure, restricting it best-suited for sparse samples. The second group of strategies is near-simultaneous multiplane imaging, enabled by spatiotemporal multiplexing of laser pulses, which maps different imaging planes in a temporal sequence.14^–16 The overall volume rate can be scaled down to a 2D scan rate, as all the depths are scanned within each lateral pixel. However, the maximum number of imaging planes is limited by the laser repetition rate and fluorescence lifetime, avoiding crosstalk among successive foci. In addition, the intensity distributions of different laser spots are usually difficult to control along the $z$-axis, owing to repeated beam splitting in the optical loop. Finally, high-speed detection electronics are necessary for the individual measurement of the nanosecond-separated fluorescence signals, which requires intricate and expensive modifications to current MPM systems. The third approach is more straightforward, that is, to axially speed up the focus movement. Various inertia-free methods have been exploited in place of conventional mechanical scans, including electrically tunable lens,17 tunable acoustic gradient index lens,18 deformable mirror,19 and recently mirror-based axial scan.20 Their main advantages are the readily available components and high compatibility with current MPM systems, i.e., just inserting a simple module, then utilizing the original laser source, scanning mirrors, and data acquisition system. The overall volume rate is equivalent to $1/[N/f_{2\mathrm{D}}+(N-1)/f_z]$, where $N$ is the number of imaging planes, and $f_{2\mathrm{D}}$ and $f_z$ are the 2D (transverse) and axial scan rates, respectively. Relying on the techniques, the axial scan rates can reach kHz levels.20^–22 However, the overall speed cannot be increased infinitely. According to the above equation, we can deduce that the upper boundary of the volume rate is $f_{2\mathrm{D}}/N$ when the axial scan is fast enough. As such, we asked ourselves if this boundary could be pushed further.

To address this challenge, the simplest approach we envisioned is to leverage the concept of compressive sensing (CS), a well-established and efficient sub-sampling theory, to greatly reduce the axial measurements (smaller value of $N$).23^–26 With this method, the speed boundary imposed by the Nyquist criterion can be possibly broken. For all CS techniques, signal coding is of utmost importance.27 Here, the precondition to implementing CS imaging is the power to encode the beam with structured illumination patterns, which then serve as a measurement basis set.28 CS has been widely implemented in microscopy; however, it is primarily confined in the transverse $(x\text{-}y)$ imaging plane29^,30 because the transverse coded beam is easily created by 2D beam modulators through pixel-by-pixel mapping, such as using liquid-crystal spatial light modulator (SLM) or digital micromirror device (DMD).

To realize multiplane compressive imaging in MPM, the key prerequisite is the coding of beam intensity along the $z$-axis. Traditionally, the control of the beam axial intensity is based on the coherent superposition of multiple beams in the propagation direction, by designing the amplitude and phase under Fourier transform.31^,32 Nonetheless, one of the challenges is that the beam amplitude and phase are hard to be simultaneously modulated by a single phase-only SLM. In addition, as an inverse problem, the method requires complex and time-consuming iterative algorithms. Thus, the Fourier transform approach is primarily suitable for structural beams with low requirements of axial intensity variation, such as the beam intensity reallocation31 or periodic structure.33 In contrast, axial beam coding requires a much higher freedom of beam control, to realize an intensity jump between “0” and “1” in the $z$-axis. Until now, no known technology empowers the beam with arbitrary binary intensities along the propagation direction. Therefore, a simple technology to directly encode the axial beam with off-the-shelf phase-only SLM is required.

In this paper, we present two technical breakthroughs to push the $z$-scan speed boundary in two-photon microscopy (2PM): one method for encoding the beam with binary intensities axially; based on this, CS sampling is adapted in axially compressive imaging. First, we demonstrate a novel and straightforward technique to fill the void of beam coding in the $z$-axis. The beam intensity can be controlled between “0” and “1” along the propagation direction, and then axially structured illumination patterns following the one-dimensional (1D) Hadamard matrix can be realized. Furthermore, this coding method is capable of easily controlling the number, spatial position, and power ratio of the “1” intensities. Second, we demonstrate the axial-coded point spread function (AC-PSF) in 2PM. A complete set of AC-PSFs with five imaging planes is demonstrated experimentally with cellular-scale lateral ($\sim 600\mathrm{nm}$) and axial (2 to $4\mu \mathrm{m}$) resolutions. As a result, we present the implementation of AC-PSFs in 2PM for multiplane compressive imaging, termed arbitrary illumination microscopy with encoded depth (AIMED). We optimize the compression ratio (CR) of AIMED by designing the coded beam and selecting coding combinations to balance the imaging quality and speed. We perform the AIMED technology to image brain tissues with five and eight planes in experiments, demonstrating a CR up to 60%, and further demonstrate approximately eight times speed increase for 47 imaging planes in simulation.

Beam coding usually refers to the modulation of beam intensity in the transverse plane. Specifically, the transverse plane of the beam can be divided into several small sections in space, each section can be regarded as a pixel, and then the intensity of each pixel is capable of being switched between “on” (“1” intensity) and “off” (“0” intensity) states. It is easy to realize transverse beam coding, as the modulation plane of the DMD/SLM is the same as the transverse plane of the beam. The beam illuminates on the micromirror array of the DMD, and each micromirror will independently control one transverse section of the beam. In this way, structural illumination patterns in the $x\text{-}y$ plane can be created, as shown in Fig. 1(a1). Guided by this, the concept of axial beam coding is to make a one-to-one mapping from different sections of the modulator to multiple $z$-positions of the beam during its propagation, and each section can be independently controlled for varying the axial intensities. The ideal realization is shown in Fig. 1(a2), where the beam propagates with sequential and randomly distributed “1” and “0” intensities along the $z$-axis.

CS imaging performs sampling using a basis that is not necessarily incoherent with the spatial properties of the image, including the random, Hadamard, Fourier, or wavelet basis, then realizing image reconstruction with a computationally fast algorithm. Thus, after having the power of coding the beam intensity axially, the realization of axial illumination basis patterns is also a key procedure. Figure 1(b) illustrates the Hadamard basis patterns for beam coding both in the $x\text{-}y$ plane and along the $z$-axis. The Hadamard basis patterns which are the standard orthonormal basis of binary (black and white) have horizontal and vertical characteristics.34 They are generated by a Hadamard matrix. Their elements consist of only $+1$ and $-1$. Usually, the rows/columns of the Hadamard matrix are reshaped into a 2D square array to form the basis measurement patterns, and the 2D square array is used for coding the transverse beam. In contrast, as the axial beam is under a 1D manner, the rows/columns can be directly borrowed from the Hadamard matrix to form the orthogonal patterns. Taking a 16-order Hadamard matrix as an example in Fig. 1(b), the upper nine rows in the Hadamard matrix can be directly regarded as the axial-coded beam. In this way, an axially structural illumination basis is achieved.

The axial-coded beam can be regarded as a sequence of focus spots with arbitrary intensity distributions along the $z$-axis. To produce the foci, we apply the quadratic phase modulation (QPM) based on a phase-only SLM. The principle of beam coding in the $z$-axis is demonstrated in Fig. 1(c). First, to achieve the one-to-one mapping, the modulation plane of the SLM is divided into several sections, and each section corresponds to an axial position of the beam. When the incident beam illuminates on the SLM screen, an independent phase mask is loaded into each section, which then guides the light to converge at a certain position along the $z$-axis. In this way, the “1” intensities of the axial-coded beam are created. As the QPM can easily control the focal length of the focus, the position of the “1” intensity is capable of being arbitrarily located. As shown in Fig. 1(c), the position of beam spots is evenly distributed along the propagation direction of the beam. To create the “0” intensity, a grating mask is used to replace the original quadratic phase mask to deflect the light away from the beam axis. Consequently, no light converges at the original axial position, and this empty region can be regarded as the “0” intensity. Based on this, axial coding can be realized by independently controlling the “1” and “0” intensities at different $z$-positions.

In practical applications, two aspects of this technique are optimized. First, each independent phase mask is evenly distributed on the SLM screen, as shown in Fig. 1(d), producing a diffraction-free focal spot (see Fig. S2 in the Supplementary Material). Second, to fully utilize the power of the incident beam, the SLM section corresponding to the “0” intensities is also reused by distributing its power to other “1” intensities, as shown in Fig. 1(e). Furthermore, the total energy of the light can be arbitrarily allocated among the “1” intensities; that is, except for evenly distributing the power, the focal spots can also follow other intensity ratios, such as an increasing or decreasing intensity distribution, as shown in Fig. 1(f).

Figure 2 shows the numerical simulation of the axial-coded beam based on the Fresnel diffraction theory.35 The first two rows of Fig. 2(a) demonstrate the arbitrarily coded beam with 5 and 10 foci. The corresponding codes are “10111” and “1001110011,” as shown in Fig. 2(b). The overall axial lengths of the two beams are set to be the same, indicating the flexible design of the focus numbers and positions. Here, the peak intensities of each spot of the two beams are optimized to be the same based on iterative algorithms (see Figs. S3–S6 in the Supplementary Material and Video 1, AVI, 356 KB [URL: https://doi.org/10.1117/1.AP.7.4.046010.s1]). The evenly distributed intensity is illustrated in Fig. 2(b). In the last two rows of Fig. 2(a), intensity-increasing and intensity-decreasing distributions are demonstrated, both following exponential variations (${1.3}^n$ and $1.3^{-n}$). It indicates our beam-coding technique allows arbitrary energy allocation among the foci. This feature will benefit MPM with desired energy allocation at targeted imaging planes, for example, giving more power at deeper imaging positions to compensate for scattering loss.36^,37

The PSFs of axial-coded beams are compared under different imaging modalities, including confocal (one-photon), two-photon, and three-photon microscopies. As shown in Fig. 2(c), the size of the excited fluorescence spot greatly decreases in two- and three-photon microscopies compared with the one-photon case, and the corresponding beam spot sizes are shown in Fig. 2(d). Furthermore, the side lobes at the gap region (between two successive foci) and around the spot are significant in one-photon microscopy, but they can be ignored in two- and three-photon cases, as shown in Figs. 2(c) and 2(e). The reduced beam size and eliminated side lobes are attributed to the excitation nonlinearity in MPM. This feature in MPM greatly benefits laser scanning microscopy with improved resolution and contrast, and it is especially critical for the AIMED technique to pinpoint the imaging planes for accurately resolving the 3D image.

Figure 3(a) shows a schematic representation of our experimental setup. The main component is a phase-only SLM, inserted into the optical path of a conventional laser scanning 2PM. The axial-coded beam produced by the SLM is then scaled and delivered by the 4f system to the focal plane of the objective. A series of vertically distributed beam foci is produced after the objective, and the intensity of each focus can be switched on and off. In this way, the AC-PSF is created, performing an instantaneous axial scan for multiple desired planes, as shown in the zoomed image of Fig. 3(a).

Figure 3(b) shows a technique comparison between the conventional 2PM and the AIMED. To obtain the 3D image of a structure, the conventional 2PM is based on the layer-by-layer scan using the Gaussian PSF (G-PSF), and the axial scan time is equal to the number of imaging planes. In contrast, AIMED utilizes a series of AC-PSFs to simultaneously scan multiple selected planes then with the assistance of CS algorithms to resolve the depths. As shown in Fig. 3(b), for a structure with five targeted depths, five axial scans are required for a typical 2PM, whereas the axial scan can be reduced to three scans with the AC-PSFs. Thus, the 3D imaging speed is increased by 40%, corresponding to an image CR of 60%. Furthermore, according to the sampling theory of the CS technique, the CR will be further decreased when more imaging planes are acquired. The CS theory for AIMED is illustrated in Sec. 4.

The bridge between the axial-coded beam and AIMED is the realization of the AC-PSF. Figure 4(a) demonstrates a complete set of AC-PSFs for imaging five planes in a laser scanning 2PM. The set consists of AC-PSFs with one to five foci. In total, 31 AC-PSFs can be used for axially structural illumination in AIMED (based on the calculation of $2^5-1$, and the excluded one is the PSF with five “0” intensities). Here, the G-PSFs can also be regarded as a group of specific AC-PSF; that is, only a single “1” intensity exists along the DOF. From the coding matrix consisting of five G-PSFs, it is clear that the matrix is full-rank, indicating a single solution for resolving the five planes.

Figure 4(b) demonstrates the axial intensity distribution of the AC-PSF coded with “11111.” The overall DOF is $\sim 48\mu \mathrm{m}$, and the spot separation is $\sim 10\mu \mathrm{m}$. Here, an iterative algorithm has been applied to make the spot intensity evenly distributed among the “1” intensities. As shown in Fig. 4(b), the normalized intensities of the five spots are all above 0.95; thus, the corresponding intensity variation is within 5%, indicating an excellent intensity control capacity of our beam coding technique. Figure 4(c) illustrates the spatial intensity distributions of the G-PSFs coded with “10000,” “01000,” “00100,” “00010,” and “00001.” Comparing Figs. 4(b) and 4(c), the beam spots are located at the same spatial position, indicating the accurate plane targeting when different AC-PSFs acquire the information from the same layer. Figure 4(d) shows the resolution provided by the AC-PSF in 2PM. Here, a different focal region is selected, and the overall DOF is $\sim 20\mu \mathrm{m}$. The lateral resolution is $\sim 600\mathrm{nm}$ with slight variation along the DOF. The axial resolution of each layer ranges from 2 to $4\mu \mathrm{m}$, demonstrating a decreasing trend along the DOF. The resolution is sufficient for sub-cellular imaging applications. It is needed to mention that the overall DOF can be flexibly controlled by selecting the axial focal region. The lateral resolution remains nearly unchanged by the axial coding, reaffirming that the use of segmented pupil regions on the SLM does not significantly degrade the lateral PSF.

To demonstrate the 3D imaging performance of AIMED, we apply the AC-PSFs for five layers to scan a mouse brain slice. Figures 5(a) and 5(b) exhibit the comparison of the imaging results of the mouse brain neurons between the AIMED-reconstructed image and the Gaussian-scanned image. The coding combination of the scanning beam for AIMED is a randomly generated $5\times 3$ matrix, indicating the sample is scanned by three AC-PSFs (01100, 10010, and 00111) while targeting five imaging planes. The $5\times 3$ matrix follows two principles. (1) The number of “1” intensities in each column is 2 or 3, i.e., approximating the half value of the total layers ($5/2=2.5$). This operation aims to balance the illumination power for different planes. (2) There must be at least one “1” intensity for each row of the matrix to ensure every layer is capable of being scanned. For a conventional Gaussian scan, the scanning beams can be regarded as a $5\times 5$ diagonal matrix. It indicates the imaging planes are scanned continuously, corresponding to the Nyquist criterion. Thus, the CR for five imaging planes in AIMED is calculated to be 60% ($3/5=0.6$), implying that the imaging speed of the volumetric structure is improved by 40%.

As shown in Figs. 5(a) and 5(b), the neurons with different spatial densities are examined, where the two CS images both illustrate good structural similarity image measurement (SSIM) and peak signal-to-noise ratio (PSNR) compared with the corresponding standard Gaussian images. It can also be confirmed by the comparisons of the color-coded depth and neuron details in Figs. 5(a) and 5(b). For the sparsely distributed neurons [Fig. 5(a)], the reconstructed CS image demonstrates better performance than that of the denser neurons, where the corresponding overall SSIM is up to 0.9227 and the PSNR is 34.4 dB. It should be highlighted that the total power illuminated in the sample in AIMED is less than the Gaussian scan. In AIMED, the total power $P_{\text{total}}$ illuminated on the SLM is divided into two or three layers, that is the “1” intensities in the $5\times 3$ matrix. Thus, for each layer, the illumination power is either $P_{\text{total}}/2$ or $P_{\text{total}}/3$. In contrast, for the Gaussian scan, the illumination power at each layer is set to $P_{\text{total}}/2$. The low power requirement of AIMED is beneficial to reduce the risk of photobleaching and photodamage in fluorescent bio-samples. More promisingly, as pointed out in Figs. 5(a1) and 5(a2), the CS image exhibits better contrast and resolves more details in some layers. For example, the head of the dendritic spine is resolved in Fig. 5(a1) while it is missing in Fig. 5(a2), as pointed out by the white arrows; for the dendritic shafts, as circled by the white boxes, their contrasts are much higher in Fig. 5(a1) than that in Fig. 5(a2). This performance is mainly attributed to the multiple illuminations of a series of AC-PSFs, which gain exposure time at certain planes.38Figure 5(c) exhibits the comparisons of the CS and Gaussian images at different planes for Fig. 5(a). The corresponding PSNR and SSIM are shown in Fig. 5(d). The layer-by-layer comparisons [Fig. 5(c)] and high image qualities [$\mathrm{SSIM}>0.9$ and $\mathrm{PSNR}>30\mathrm{dB}$, Fig. 5(d)] indicate excellent reconstruction accuracies of the AIMED technique.

To confirm the robustness of the AIMED technique, the mouse brain neuron was also scanned by the other randomly generated $5\times 3$ matrix. Figures 5(e1), 5(e2) and 5(f1), 5(f2) demonstrate the image comparisons in the $x\text{-}y$ plane, and the corresponding images in the x-z plane are also compared in Figs. 5(e3) and 5(f3). The main goal of AIMED is to resolve the image depth while maintaining a good image quality and similarity to the ground truth. The image reconstructions in the axial direction from AIMED illustrate outstanding agreements with the Gaussian results, as shown in Figs. 5(e3) and 5(f3). Both the axial locations of small structures and the preservation of details in the CS images show good consistency with the standard Gaussian results, as indicated by the white circles. The slight fragmentations of the neuron bodies in the CS images are mainly due to their lack of sparsity, as the CS algorithm usually works better in sparse regions. Optimizing the reconstruction algorithm and selecting a proper projection basis is possible to further enhance the image quality. Here, the illumination power at each layer in the Gaussian scan is even higher ($P_{\text{total}}$), which is doubled as that in Figs. 5(a) and 5(b), whereas the power for each “1” intensity in AIMED is still either $P_{\text{total}}/2$ or $P_{\text{total}}/3$. The excellent similarities between the CS and Gaussian images confirm the low power requirement of the AIMED technique.

The CR in CS sampling determines both the image quality and speed. To examine the influence of CR in the AIMED technique, we utilize different CRs to reconstruct the 3D images for eight imaging planes. Figures 6(a)–6(c) demonstrate the CS-reconstructed 3D neuron images with different coding matrices, and Figs. 6(d) and 6(e) show the Gaussian-scanned results. In Fig. 6(a), the coding combination of the AC-PSFs is a randomly generated $8\times 5$ matrix, indicating the eight depths are scanned by five AC-PSFs, and then the CR of the CS image is calculated to be 62.5%. For the CS images of $\mathrm{CR}=75\%$ [Fig. 6(b)] and 87.5% [Fig. 6(c)], one or two AC-PSFs are added to the $8\times 5$ matrix to produce $8\times 6$ and $8\times 7$ matrices, respectively. Comparing the SSIMs and PSNRs in Figs. 6(a)–6(c), both indicators show a slight increase (SSIM, $\sim 0.95$ to 0.96; PSNR, $\sim 41$ to 42 dB), which can be regarded to be constant. Here, we calculate the SSIM and PSNR with the Gaussian image under the illumination power of $P_{\text{total}}/2$ [Fig. 6(e)], as it retains more details than that under $P_{\text{total}}/3$ [Fig. 6(d)]. The results indicate that the quality of the reconstructed image in AIMED can be maintained when lowering the CR to speed up the axial scan.

On the other hand, the low power requirement of AIMED is confirmed again for eight imaging planes. As there are four “1” intensities for each AC-PSF in Figs. 6(a)–6(c), the illumination power for each layer is $P_{\text{total}}/4$. In contrast, the laser power for the Gaussian scan is set to be either $P_{\text{total}}/3$ or $P_{\text{total}}/2$. Comparing Figs. 6(a)–6(c) with Figs. 6(d) and 6(e), the CS images resolve much more details than the Gaussian image with higher power ($P_{\text{total}}/3$) and approach the quality of the Gaussian image with doubled illumination power ($P_{\text{total}}/2$), indicated by the ultrahigh SSIM and PSNR. The excellent AIMED performance is mainly attributed to the increased exposure time at each imaging layer.

Here, we demonstrate a novel approach to realize multiplane compressive imaging in MPM that empowers conventional laser scanning microscopies with rapid axial scanning capacity. The presented method is based on two technical breakthroughs: first, we propose a concept to axially encode the beam with binary intensities; second, CS sampling theory is introduced to the axial direction in MPM based on the AC-PSFs.

For axial beam coding, we propose a straightforward method to control the beam intensity variation along the $z$-axis, that is, adjusting the “on” and “off” states of multiple foci along the beam propagation direction. In this way, the axial beam can be switched between “0” and “1” intensities to achieve axially structured illumination patterns. This method is convenient for controlling the position, density, and number of the “1” intensities, which avoids solving the complex converse-solving problem of finding the phase and amplitude of the light field. More importantly, it is also hard to conduct the amplitude modulation based on the commonly used phase-only SLM. In contrast, for this technique, a single phase-only SLM is sufficient to realize the axial beam coding, based on the QPM in the one-to-one mapped sections on the SLM screen. Although there are “0” intensities in the coded beam, the laser power for these positions is not waisted. To fully utilize the illumination power from the laser source, the energy originally designed to the “0” intensities is dynamically reallocated to the “1” intensities in different coded beams. Furthermore, based on the power redistribution capacity, the axial beam can also follow arbitrary intensity distributions (e.g., exponential functions) by iterative algorithms. This feature can be used to give more power to the targeted imaging planes, such as for deeper penetration depths,39 and realizing novel 3D imaging technologies.40 Benefiting from the nonlinear excitation, the MPM empowers the axial-coded beam with a smaller beam size and fewer side lobes, making the AIMED technique with much better image contrast and 3D reconstruction accuracy. Although the gap intensities among successive axial foci are around 25% of the focal intensity [Fig. 4(b)], it is less problematic in AIMED: first, most of the beam codings for actual imaging in AIMED are sparse, such as “10010,” “01010,” or “10011010” (see Figs. 5 and 6 for biological samples), rather than “11111” or “11111111,” to avoid high interference among successive axial foci in the gap region; second, random beam coding combination in a set further reduces the influence of gap intensities when they work together; third, the CS reconstruction algorithm is aware of the PSF structure (including any side lobes) and mathematically separates the contributions from each plane based on the known coding patterns. In summary, the axial beam coding is a significant complement to the traditional transverse beam coding. Although in this work, the axial beam coding is designed for AIMED, we foresee the axial coding concept will benefit lots of investigations and applications, for example, optical manipulation and communication.

We developed the AIMED imaging system based on the axial beam coding technique and CS theory. The AC-PSFs can provide a lateral resolution of $\sim 600\mathrm{nm}$ and an axial resolution of 2 to $4\mu \mathrm{m}$, sufficient to achieve subcellular imaging. The experimentally demonstrated full set of the coded beam with five depths indicates the capacity of AC-PSFs serving as the basis of the CS sampling. According to the comparison between the CS-reconstructed and standard Gaussian images, the AIMED technique can not only achieve excellent image qualities, including the resolved depth accuracy and neuron details, but also promisingly exhibit better image contrast in some modalities, owing to the multiple illuminations for certain imaging planes. The CR of AIMED in this work can be increased up to 60%, indicating a nearly doubled 3D imaging speed compared with conventional Gaussian scanning. More importantly, fewer scans in the AIMED technique means less laser power illuminating the samples, which greatly benefits protein-labeled neuron imaging by significantly reducing the risks of photo-bleaching and photodamage. It is notable that AIMED is perfectly suitable for brain neuron imaging: as the reconstruction accuracy of CS is largely related to the sparsity of the information, the inherent sparsely distributed neurons in the brain match well with the applicability of the AIMED technique. The AIMED is also a plug-and-play technique for current laser scanning microscopy. Compared with other axial scan techniques reported recently, which usually have major modifications to the current imaging systems, AIMED only requires an off-the-shelf SLM inserted as a mirror in the optical path, which saves much effort.

In conclusion, the combination of the axial beam coding and CS sampling enables a fast $z$-scan of MPM in an innovative way, which is different from previously reported approaches confined by full axial sampling. To optimize the AIMED technique, several aspects can be considered in future work, including achieving longer AC-PSFs with more foci, selecting optimal coding combinations for CS algorithms, and achieving a balance between image quality and speed. To increase the axial numbers of foci, laser sources with high peak power and optimization of axial beam energy are required. Optimal design of the pupil segmentation on the SLM is needed to maintain a good quality of the foci, especially when the number of foci is largely increased. Specifically, it would be useful to explore optimizing the phase patterns to further increase the focus contrast, i.e., lower the between-foci intensity. For the phase mask partitioning strategy, each focus gets a progressively smaller fraction of the aperture when the number of foci grows very large, which requires proper partitioning design. To improve the imaging DOF, a proper axial position near the focal region of the objective should be considered to balance the DOF and imaging resolution. For the CS theory, a dense sampling contributes to a higher CR, that is, the longer AC-PSFs with more foci will enable a much faster 3D imaging speed (see Figs. S12–S14 in the Supplementary Material and Video 5, AVI, 5.12 MB [URL: https://doi.org/10.1117/1.AP.7.4.046010.s5] for 47 imaging planes, where the 3D imaging speed is improved by approximately eight times). The techniques developed in this work are not only feasible for MPM systems but also practical to other microscopic modalities, including confocal, coherent Raman, and photoacoustic imaging, for speeding up the volumetric 3D imaging in a sub-sampling way. Therefore, the powerful AIMED technology for rapid $z$-scan provides a significant prospect in the entire field of advanced 3D microscopy imaging applications.

Figure S1 in the Supplementary Material shows the experimental setup for AIMED. The excitation light was from a customized mode-locked fiber laser (1065 nm, 30 MHz, and 180 fs) for 2P excitation. The axially coded beam was generated by the SLM (Holoeye, PLUTO). The focal lengths of the lens are $f_1=200\mathrm{mm}$, $f_2=250\mathrm{mm}$, $f_3=125\mathrm{mm}$, $f_4=80\mathrm{mm}$, and $f_5=150\mathrm{mm}$. A water-immersion objective (60×, 1.2 NA, UPlanSpo, Olympus, Tokyo, Japan) was used to focus the excitation light into the sample. The $x\text{-}y$ laser scanning was achieved by a pair of GMs (Cambridge Technology, Hyderabad, India, 6220H). The excited fluorescence signal was collected in epi-direction and detected by a photomultiplier tube (PMT, Hamamatsu, H10723-20). The residual excitation beam was cleaned up by two short-pass filters (Semrock, Rochester, New York, United States, BSP01-785R-25) before the PMT. The electrical signal from the PMT was subsequently digitized by an analog-to-digital converter card (National Instruments PCI-6110, 5 MS/s). The control of the entire imaging system, including laser scanning, data collection, and image reconstruction, was realized by a MATLAB-based multifunctional program.

The images in MPM were collected at a 2D frame rate of $\sim 0.4\mathrm{Hz}$ over a field of view of $150\mu \mathrm{m}\times 150\mu \mathrm{m}$ [Figs. 5(b) and 5(f) and Fig. 6] or $75\mu \mathrm{m}\times 75\mu \mathrm{m}$ [Figs. 5(a), 5(c), and 5(e)] with $512\times 512\text{pixels}$. The 2D frame rate here is majorly limited by the speed of the galvanometric mirrors. In the imaging experiments, the total laser power ($P_{\text{total}}$) after the objective was $\sim 90\mathrm{mW}$. Image reconstruction time: based on a standard workstation (AMD EPYC 7302P 16-Core Processor 512 GB RAM), we observed that the reconstruction time for a typical imaging set is $\sim 30\mathrm{s}$. The images in the x-z plane [Figs. 5(e3) and 5(f3)] were plotted by Volume Viewer in ImageJ without interpolations.

Fluorescence microspheres: Fluorescence beads ($1\mu \mathrm{m}$, F8819, Life Technologies Ltd., Carlsbad, California, United States) were used to obtain the data in Figs. 4(a)–4(c). Fluorescence beads (200 nm, F8809, Life Technologies Ltd.) were used to obtain the data in Fig. 4(d).

Frontal cortex sample: Thy1--YFP H line (Stock No. 003782) mice were purchased from the Jackson Laboratory. Animals were perfused transcranially with PBS followed by 4% PFA. Harvested brain samples were post-fixed overnight at 4°C in 4% PFA. Brain samples were sectioned into $50\text{-}\mu \mathrm{m}$-thick coronal brain sections. Layer V pyramidal neurons expressing yellow fluorescent protein (YFP) in the frontal cortex were imaged. All experiments were approved and performed in accordance with the University of Hong Kong Committee on the Use of Live Animals in Teaching and Research guidelines.

To generate the axial-coded beam, the phase mask loaded on the SLM can be written as $$P_{\mathrm{SLM}}=\sum _{n=1}^N[P(f_n)],$$where $N$ is the number of focal spots and $f_n$ is the focal position. When a Gaussian beam with amplitude $G$ is incident on the SLM screen, the beam can be regarded as a sum of several independent regions, which is written as $$G=\sum _{n=1}^N(G\times {\omega }_n),$$where ${\omega }_n$ is the weight of each independent phase $P(f_n)$ occupying the whole phase mask. The beam modulated by the phase mask is $$\begin{gathered} \sum _{n=1}^N\{G\times {\omega }_n\times \exp [iP(f_n)]\}=\sum _{n=1}^N\{\{G\times \exp [iP(f_n)]\}\times {\omega }_n\} \\ =\sum _{n=1}^N[F(f_n)\times {\omega }_n], \end{gathered}$$where $F(f_n)=G\times \exp [iP(f_n)]$ is the light field of the divided beam in each section. The expression indicates multiple foci can be produced from $f_1,f_2,\dots ,f_n$, and the total energy can be allocated among the foci, which is controlled by ${\omega }_n$. Specifically, the number and locations of the foci can be flexibly set. The “0” intensities in the axial-coded beam can be achieved by making ${\omega }_n=0$.

To acquire the $N$-planes of the sample, a measurement matrix ($\mathrm{\Phi }$) is indispensable. The process can be written as $$P=\mathrm{\Phi }Q,$$where $Q$ is a vector with the size of $N\times 1$, which denotes the depth information, and the size of $\mathrm{\Phi }$ is $M\times N$. For the conventional axial imaging method with a Gaussian beam, $M=N$, and $\mathrm{\Phi }$ is an identity matrix. It means that the depth information will be acquired layer by layer. If the number of scans $M<N$, the depth information is no longer complete. Now, the CS theory provides a new way to retrieve depth information for $N$-planes with less scan times ($M$). The theory relies on the sparsity of the signal, which can be perfectly satisfied by the neuron imaging in MPM, as the neurons usually distribute sparsely in the brain, especially along the axial direction. Thus, the basis matrix (for making a sparse representation of $Q$) is not necessary in AIMED, and the sensing matrix is the same as the measurement matrix. Here, a random binary matrix with a size of $M\times N$ is used as the measurement matrix, and it is decomposed into multiple $N\times 1$ vectors, which are used for encoding the beam. In the experiment, the scan of $M$ times will be achieved, and then $P$ is obtained. The solution of $Q$ is not accurate and definite in conventional theory, because it is an underdetermined problem. However, it is solvable if $Q$ is sparse. Ideally, the solution is determined by $$Q=\arg \min {\Vert Q\Vert }_{l_0},\text{subject to}P=\mathrm{\Phi }Q,$$where $\Vert ·{\Vert }_{l_0}$ is the 0-norm. However, it is a nondeterministic polynomial–complete problem, which is difficult to find a solution. Instead, the 1-norm $\Vert ·{\Vert }_{l_1}$ solution can be used in most practical problems, and it is equivalent to $\Vert ·{\Vert }_{l_0}$ solution. As a result, the problem can be solved by the optimization of $$Q=\arg \min \Vert Q{\Vert }_{l_1},\text{subject to}P=\mathrm{\Phi }Q.$$

It is a convex optimization problem and can be solved with a linear program. In this work, it is achieved by the basis pursuit method. The algorithm can be seen in Fig. S9 in the Supplementary Material.

Acknowledgment. This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region of China (Grant Nos. HKU 17210522, HKU C7074-21G, HKU R7003-21, HKU 17205321, and HKU 17200219); the Innovation and Technology Commission of the Hong Kong SAR Government (Grant Nos. MHP/073/20, MHP/057/21, and Health@InnoHK program); the National Natural Science Foundation of China (Grant No. 62305274); the Natural Science Foundation of Xiamen City, China (Grant No. 3502Z202371001); and the Fujian Provincial Natural Science Foundation (Grant No. 2024J01056).

Xin Dong received his BS degree and the master’s degree from the Huazhong University of Science and Technology (HUST) in 2016 and 2019. He worked as a research assistant at the Wuhan National Laboratory for Optoelectronics, HUST, from 2019 to 2020. He is currently a PhD candidate at the Department of Electrical and Electronic Engineering, the University of Hong Kong. His research interests include fiber nonlinearities, structured illumination, fluorescence imaging, and optical neural networks.

Hongsen He received his PhD in 2022 from the University of Hong Kong, specializing in biophotonics and laser-based imaging. He is currently an assistant professor at Xiamen University. His research centers on advanced imaging lasers (miniaturized solid-state lasers, all-fiber ultrafast lasers) for various microscopy platforms (photoacoustic, multiphoton). By fusing with novel imaging techniques (axial multiplexing, spatiotemporal modulation), he aims to redefine the role of lasers from passive light generators to active drivers of imaging innovation.

Minghui Shi received her BS degree in optical information science and technology from Beijing University of Technology (BIT), Beijing, China, in 2021. She is currently a PhD candidate at the Department of Electrical and Electronic Engineering, the University of Hong Kong, Hong Kong, China. Her research interests include mid-infrared laser applications and fluorescence imaging.

Cora S. W. Lai is an associate professor in the School of Biomedical Sciences, the University of Hong Kong (HKU). She obtained her bachelor and doctoral degrees at HKU. She later joined the Langone NYU Medical Center for postdoctoral training. She has been working on intravital imaging of the mouse central nervous system in learning and memory, particularly in studying synaptic plasticity in fear associative learning and different psychiatric disease animal models with dendritic spine pathology.

Kevin K. Tsia is currently a professor in the Department of Electrical and Electronic Engineering, and the Program director of the Biomedical Engineering Program, at the University of Hong Kong. His research interest covers a broad range of subject matters, including ultra-fast optical imaging for imaging flow cytometry and cell-based assay; high-speed in-vivo brain imaging; computational approaches for single-cell analysis. He is a co-founder of start-up company commercializing high-speed microscopy technology for clinical diagnostic applications.

Kenneth K. Y. Wong is currently a professor in the Department of Electrical and Electronic Engineering, the University of Hong Kong. He is the PI of OMEGA HKU (Optical Management and Engineering Group @ HKU), advancing the technological frontier of photonics systems and photonic signal processing technology to bring synergy among different applications. His research focuses on Novel Optical Generation; Photonic Parametric Processing; Ultrafast Optical Fiber Communication and Imaging (Spectroscopy, Microscopy and Tomography).