2.1.

Principle and Simulation

By introducing a cubic phase into a basic Gaussian beam, the generation of an Airy beam is made possible. The behavior of optical Airy wave packets can be characterized by the potential-free Schrödinger equation under paraxial conditions, following the diffraction relation,²⁵

Here, the dimensionless transverse coordinate is represented by $s=x/x_0$, an arbitrary transverse scale is denoted by $x_0$, and the electric field envelope is denoted by $\varphi$. $\xi =z/k_0^2$ denotes a normalized propagation distance. To obtain the envelope of a finite Airy beam, the normalized paraxial equation of diffraction is solved by performing an exponential aperture function, resulting in Refs. 25 and 32,

The mathematical expression of the Airy beam in normalized $k$-space is derived by taking the Fourier transform of Eq. (3), with its Fourier spectrum given as³²

Through an analysis of the propagation features of finite-energy Airy wave packets, it can be observed that the Fourier transform of function $Ai(s)\exp (as)$ results in a Gaussian beam that is modulated with a cubic phase ($k^3$). Specifically, the cubic phase spectrum for a 2D Airy beam can be represented as ${\phi }_{AB}(k_x,k_y)=[i{\beta }^3(k_x^3+k_y^3)/3]$, where $kx$ and $ky$ denote the Fourier spectrum coordinates, and $\beta$ represents the scale factor.³³ This cubic phase spectrum is shown in Fig. 1. Thus, by applying a cubic phase modulation and focusing the resulting beam with a Fourier objective lens to give a Fourier transformation, a linearly polarized Gaussian beam can be converted into an Airy beam. Supplementary Material Note 1 contains simulation results that illustrate the beam intensity distribution and propagation properties of both the Gaussian and Airy beams. Correlation analysis shows that PAM based on Airy beam excitation can extend the DOF with relatively ideal resolution. In order to further characterize the influence of different phase modulation parameters on Airy beam property, we made beam simulation maps under different scale factors $\beta$; the results are present in Fig. 1. The first row of Fig. 1(a) represents the phase patterns of an Airy beam under different scale factors, and the second row is the cross-sectional maps of the spot at the focus position. It can be observed that with the increase of the scale factor, the sidelobe of the Airy beam becomes more significant, and both have relatively consistent size distribution of the focusing spot. And then, from the third and fourth rows of Fig. 1(a), with the increase of propagation distance, Airy beams with different modulation parameters have good beam focusing property. Furthermore, we analyzed the characteristics of the beam in the propagation direction under different scale factors; the results are shown in Fig. 1(b). As can be observed from the simulation results, the propagation distance of the Airy beam increases with the decrease of attenuation, but the sidelobe becomes more and more significant. In addition, it can be found that with the increase of the scale factor, the initial beam intensity of the Airy beam becomes stronger (refer to the rightmost column of Fig. 1(b), corresponding to the yellow box area of the left column).

Fig. 1

Analysis of the influence of modulation parameters of Airy beam phase pattern on beam property. (a) The different phase patterns, cross-sectional beam-spot maps of $xy$ direction at the focus position, and cross-sectional beam-spot maps of $xy$ direction at the defocus position, respectively. (b) Left column: beam propagation property maps in the $xz$ direction that correspond to the different modulation parameters in (a); right column: zoom-in maps of the initial beam intensity distribution that correspond to the yellow region in the left column. (c) and (d) Schematic diagram of flexibly adjustable FDIR-PAM using an SLM. NDF, neutral density filter; PD, photodetector; HWP, half-wave plate; PBS, polarization beam splitter; M, mirror; L1–L4, lenses; OL, objective lens; D, diaphragm; P, polarizer; SLM, spatial light modulator; UT, ultrasonic transducer; AMP, amplifier; DAQ, data acquisition card; PC, personal computer.

2.2.

FDIR-PAM System

In the experimental setup, we employed an SLM to introduce the cubic phase modulation at the Fourier plane of the excitation path in the FDIR-PAM system. The use of an SLM allowed us to adjust the scale factor and control the cubic phase, thereby truncating the tail of an infinite Airy beam. The experimental configuration of the FDIR-PAM setup is shown in Fig. 1(c). The excitation beam of the microscopy was from a pulsed laser (10 ns, 1 KHz, and 532 nm). During the experiment, we adjusted the laser pulse energy to 379 mW using neutral density filters (NDF, NDC-50C-2, Thorlabs Inc.). The laser energy output ratio was regulated by a polarizer (P, LPVISA100-MP2, Thorlabs) and a polarizing beam splitter (PBS, PBS201, Thorlabs Inc.), dividing the pump laser beam into two paths. One path carried a power of $2.20\mu \mathrm{J}$ at 532 nm, while the other path had a power of $0.28\mu \mathrm{J}$ for triggering the photodetector. To ensure efficient conversion from a basic Gaussian beam to an Airy beam, we employed half-wave plates (HWPs, WPH10E-532, Thorlabs Inc.) to control the linear polarization direction of the excitation beam before it passed through the SLM (Holoeye, PLUTO). A lens telescope comprising L1 (focal length $f1=30\mathrm{mm}$) and L2 ($f2=100\mathrm{mm}$) was used to extend the beam size and sufficiently illuminate the SLM screen. Scanning relay lenses L3 ($f3=100\mathrm{mm}$) and L4 ($f4=50\mathrm{mm}$) were employed to match the beam size with the back aperture of an objective lens (4×, 0.1 NA, UPlanSApo, Olympus) that had a working distance of 18.5 mm. The implementation of a blazed grating in combination with an iris was used to remove spurious orders of diffraction, separating the zeroth- (unmodulated) and first-order (desired) diffraction. By employing a blazed grating alongside an iris through an SLM, adjusting the iris diameter allows us to eliminate the unwanted zeroth-order spot while maintaining the intended first-order diffraction. The pulsed laser-triggered ultrasonic waves were detected and received by a ring-shaped ultrasonic transducer (UT) fixed coaxially with the optical axis of the objective lens. The characteristics of the UT include an inner diameter of 1.2 mm, outer diameter of 4.5 mm, focal length of 4.4 mm, center frequency of 50 MHz, and a 6-dB bandwidth of 90%. The detected photoacoustic (PA) signals were amplified using two 30 dB low-noise amplifiers (ZFL-500LN+, Mini-Circuits) and digitized by a data acquisition card (DAQ) with a sampling rate of 250 MS/s (PCIE-1425, Yixing Technology). For sequential control of the PA laser trigger, data acquisition, and raster scanning, we employed a controller card (Zmotion, National Instruments) programmed using LabVIEW. Figure 1(d) shows the principle of conventional Gaussian-beam excitation PAM and Airy-beam excitation PAM, respectively. This gives the representative phase patterns and the corresponding Airy beam achieved from the optical focus plane.

3.1.

Beam Profiles and Setup Performance

To characterize the optical properties of the modulated beam, we utilized a CMOS camera (CS165MU/M, Thorlabs) positioned at the back focal plane of the excitation beam to capture beam images (refer to Fig. 2). Figures 2(a)–2(c) display the different phase patterns employed in our experiments, denoted as Gaussian, Airy#1, and Airy#2 with varying scale factors. In the absence of phase modulation, the measured spot size of the Gaussian beam (full width at half-maximum, FWHM) expanded to $\sim 26\mu \mathrm{m}$ at 1.5 mm beyond the focal plane of the objective lens, accompanied by a significant drop in laser fluence, as shown in Figs. 2(d) and 2(g) (note the difference in color bar ranges). Upon loading the cubic phase onto the SLM, finite-energy Airy beams were generated at the Fourier focal plane. The intensity pattern revealed a multilobed distribution [Figs. 2(e) and 2(f)], with the main lobe containing most of the total beam energy, consistent with the results of simulation experiments involving various Airy modulation depth parameters (Figs. 1 and S1 in the Supplementary Material). The main lobe FWHM of the Airy beams at the focus were $\sim 5$ and $3\mu \mathrm{m}$ for phase patterns #1 and #2, respectively. The scale factors of Airy beams #1 and #2 were 0.6 and 2, respectively, determining the decay rate of sidelobes along the transverse direction and the beam decay rate along the propagation direction. A larger scale factor resulted in a narrower main lobe size, higher energy carried in the main lobe with a longer DOF, and a greater total displacement of self-bending. Moreover, a smaller scale factor led to broader main lobes of the Airy beam, with the sidelobes positioned farther away from the main lobe and each other [Figs. 2(e) and 2(f)]. During the experiment, it was observed that Airy beam #2 exhibited a relatively larger DOF with more prominent sidelobes compared to Airy beam #1. In addition, the energy carried in the central-main lobe for Airy beam #2 was greater than that for Airy beam #1, with measured values of 69% and 54%, respectively [Figs. 2(k) and 2(l)]. Furthermore, the lateral resolution of FDIR-PAM was measured by imaging a USAF 1951 resolution target (Thorlabs) in Fig. 2(m). The corresponding edge spread function (ESF) was fitted according to the white dotted line of B-scan data. After taking the first derivative of the ESF, the lateral resolution was measured based on the FWHM values of the Gaussian fitting, performing a lateral resolution of $3.2\mu \mathrm{m}$ at the focal plane. As the sidelobes of the Airy beam existed and influenced the focusing light spot to some extent, the lateral resolution value was slightly greater than the size of the central-main lobes. Figure 2(n) depicted the depth-axial distribution of the lateral resolution, demonstrating that the system maintained a DOF of $926\mu \mathrm{m}$ with an invariant lateral resolution of $3.2\mu \mathrm{m}$.

Fig. 2

Beam property characterization. (a)–(c) Phase patterns of Gaussian and Airy beams generated by SLM. (d)–(f) Measured 2D beam intensity distributions for (a) focused Gaussian, (b) Airy beam #1, and (c) Airy beam #2 at the focal plane. (g)–(h) Measured beam intensity distributions of Gaussian beam and Airy beams #1 and #2 at 1.5 mm away from the focal plane, respectively. (j)–(l) Optical intensities in the sagittal plane of Gaussian beam and Airy beams #1 and #2, respectively. The white dashed line depicts the measured FWHM. The color bar gives a linear intensity distribution. (m) Lateral resolution of the FDIR-PAM system, giving the ESFs and its first derivative extracted from the corresponding B-scan of PAM map. (n) The depth-axial distribution of lateral resolution.

3.2.

Flexibly Adjusting Airy Beam in Phantom Imaging

In order to assess the feasibility of FDIR-PAM in a phantom experiment, we conducted volumetric imaging of a letter phantom fixed in a mixture of 1.6% Intralipid solution and 1% agar gel. The “XD” pattern of the letter phantom was positioned in ultrasound gel at an angle along the depth direction ($z$ axis), and raster scanning was performed on the phantom during the imaging process. Using both Gaussian- and Airy-beam modalities, we sequentially performed volumetric imaging and switched between PA excitation modalities by assigning corresponding phase patterns onto the SLM. Figures 3(a)–3(d) illustrate the PAM images of the “XD” pattern produced by the four PA excitation beams. In the MAP image of the Gaussian-beam excitation PAM [Fig. 3(a)], only one letter with a clear “X” pattern could be visualized, while the right “D” letter was missed due to the short focal length. When using PA excitation generated from a cubic phase pattern, the elongated DOF of the Airy-beam excitation PAM [Figs. 3(b)–3(d)] allowed for the visualization of both letters, where the dim appearance of the “D” letter can still be captured. Furthermore, when the cubic phase pattern of Airy beam #3 was assigned to the SLM, both “XD” letters were most clearly observed in the FDIR-PAM [Fig. 3(d)], confirming its much larger DOF of at least 2.26 mm. Line profiles at different depths were analyzed to compare the image quality between FDIR-PAM and Gaussian-beam excitation PAM [Figs. 3(e) and 3(f)], demonstrating that the PA images were more resolvable in the former of FDIR-PAM. The increased DOF, combined with consistent lateral resolution throughout the entire mapping volume, could potentially facilitate the characterization of various-layer absorber topology at a micrometer scale. These results indicate the feasibility and superiority of FDIR-PAM, which allows for flexible adjustment of the depth-axial focal length via the incorporation of an SLM.

Fig. 3

PAM images of a letter pattern phantom. (a)–(d) Photoacoustic imaging characterization of the letter phantom by Gaussian-beam excitation PAM (GB-PAM) and FDIR-PAM, respectively. The related line profiles at different depths are presented in (e) and (f). Scale bars, 1 mm.

3.3.

Zebrafish Larva and Adult Imaging

The zebrafish is an important model organism that is widely used in neuroscience, cardiovascular research, and other fields because of its transparent development and easy maintenance. The melanin and antholeucin produced by zebrafish are arranged into fixed stripes patterns when they are transported from cells to the surface of the embryo. Hence, it is of utmost importance to study the pigment characterization of zebrafish.

The in vivo experiment of zebrafish larvae was first performed to prove the superiority of FDIR-PAM in complex absorption and scattering structure. Two wild-type zebrafish larvae with $\sim 3\mathrm{mm}$ body lengths at different depths were imaged (1 week old, the Chinese Zebrafish Resource Center). Before imaging, the zebrafish was immobilized on the agarose (1%) at different depths. Figures 4(a) and 4(b) show the structural MAP PAM map results acquired with Gaussian and Airy beams, respectively, given both zebrafishes were roughly caught. Furthermore, zooming in on the yellow boxed regions corresponding to Figs. 4(a) and 4(b) clearly shows that Airy-beam excitation PAM outperformed Gaussian-beam excitation PAM, presenting a high-resolution and deep-resolved zebrafish’s head and body with Airy-beam excitation PAM, while the PA signals on the left zebrafish’s head and the right fish’s body were missing in Gaussian-beam excitation PAM, as shown in Figs. 4(c) and 4(d) and indicated by yellow dashed lines and white arrows. The B-scan images of the dashed line in Figs. 4(c) and 4(d) that correspond to Gaussian-beam excitation PAM and Airy-beam excitation PAM are displayed in Figs. 4(e) and 4(f), respectively. The areas indicated by white arrows are the pupil and head. Obviously, Airy-beam excitation PAM not only characterizes a more comprehensive topology structure of the eyeball but also captures richer depth information on the head structure. In contrast, Gaussian-beam excitation PAM fails to provide this depth information. This experimental results indicated that FDIR-PAM can provide a larger depth range of image detail with the flexibly elongated DOF.

Fig. 4

In vivo PAM characterization of zebrafishes, larvae and adults. (a) and (b) PAM images of two zebrafish larvae via Gaussian-beam excitation PAM (GB-PAM) and FDIR-PAM. (c) and (d) Zoom-in en face maps that correspond to the yellow boxes of (a) and (b), respectively. (e) and (f) Cross-sectional maps that correspond to the dashed lines in (c) and (d). (g) and (h) PAM images of zebrafish adult fish body using the Gaussian and Airy beams. (i) and (j) Zoom-in en face maps that correspond to yellow boxes of (g) and (h), respectively. (k) and (l) Cross-sectional maps that correspond to the dashed lines in (g) and (h). (m) Intensity profiles along the lines depicted in (I), (II), (III), and (IV).

In addition, an experiment was conducted to characterize adult zebrafish in vivo, as their pigment pattern formation provides valuable insights into adult form development and evolution.³⁴ Pigment patterns in zebrafish serve as highly accessible characteristics for studying the cellular basis of the adult form, as microscopic imaging allows visualization of the cells responsible for diverse patterns during development. Leveraging the strong absorption of visible light by pigmentation, label-free 532 nm Gaussian beam excitation PAM proves convenient for acquiring pigment pattern images without excessive sample operation preparation. However, when it comes to imaging irregular topology surfaces with high resolution, conventional Gaussian-beam excitation PAM encounters challenges in accurately characterizing pigment patterns on the fish body. To demonstrate the comprehensive benefits of FDIR-PAM in pigment pattern characterization, a comparison was made between Gaussian-beam excitation PAM and Airy-beam excitation PAM while imaging the adult zebrafish’s marginal areas with irregular topology surfaces. Prior to imaging, the adult zebrafish was placed under general anesthesia (tricaine: 0.5 to $1.9\mathrm{mg}/\mathrm{ml}$), and its body temperature was maintained at 28°C using a thermostatic water tank to ensure motionlessness during the imaging process. The Gaussian-beam excitation PAM image of a 3-month-old adult zebrafish body [Fig. 4(g)] exhibits blurred edges and indistinct pigmented textures in the close-up image. Moreover, important pattern details are missing, as highlighted by the yellow box in Fig. 4(g). These limitations can be attributed to the restricted DOF and significant height fluctuations on the body surface [Fig. 4(i)]. The brighter stripe in Fig. 4(g) represents the regions around the optical focal plane of Gaussian-beam excitation PAM, which exhibit better signal-to-noise ratio and resolution compared to the out-of-focus regions. In contrast, the Airy-beam excitation PAM image of the zebrafish body [Fig. 4(h)] showcases well-resolved texture features of pigment patterns that are not discernible in the Gaussian-beam excitation PAM image, as indicated by the white arrow. The close-up image shown in Fig. 4(h) clearly exhibits distinct pigment patterns, whereas the corresponding region of interest in the Gaussian-beam excitation PAM image appears blurred or even absent, as denoted by the asterisk (*) in the relevant areas [Fig. 4(j)]. Furthermore, a comparison of the cross-sectional images obtained from the respective dotted lines in Figs. 4(g) and 4(h) unequivocally demonstrates the superiority of Airy-beam excitation PAM in characterizing pigment patterns. This advantage becomes particularly evident when dealing with the uneven surface of the fish body, which possesses an irregular topology structure and necessitates a substantial DOF. To further facilitate a quantitative comparison, it is important to consider the contrast, which closely relates to the signal-to-background ratio (SBR) and profoundly impacts spatial resolution. A markedly high SBR, coupled with a superior FWHM, signifies the detail-resolving capability. Consequently, in Fig. 4(m), the measured FWHMs of (I), (II), (III), and (IV) [marked by green dashed lines in (a), (b), (e), (f), (i), and (j)] obtained through Gaussian PAM were $164.6\mu \mathrm{m}$, unmeasurable, and $40.6\mu \mathrm{m}$, respectively. However, with the implementation of FDIR-PAM, the corresponding FWHMs decreased to 87.3, 34.9, and $34.6\mu \mathrm{m}$, affirming a significant improvement in lateral and axial resolving capabilities using Airy-beam excitation PAM. Simultaneously, the SRS profile of (IV) exhibited better signal-to-noise performance with Airy beam imaging, indicating that pigmented streaks present in deeper regions of fish bodies can be effectively detected using FDIR-PAM, which offers a greater DOF.

3.4.

In Vivo Mouse Brain Imaging

Eventually, with the capability of revealing structural and functional information in a label-free manner, Gaussian-beam excitation PAM in visible light has been widely used to characterize both the brain vasculature morphology, blood flow, oxygen saturation, and oxygen metabolism function. However, the pronounced curvature of the mouse brain cortex can hinder the high-resolution mapping of the comprehensive cerebral vessels to some extent. To showcase the advantages of FDIR-PAM in characterizing in vivo brain vasculature, we performed imaging on a mouse brain with an intact skull. Prior to the experiments, the scalp of a 4-week-old BALB/c female mouse was meticulously prepared, exposing the intact skulls and creating a region of interest measuring $4\mathrm{m}\mathrm{m}\times 2\mathrm{m}\mathrm{m}$. The mouse was then anesthetized using a diluted solution of chloral hydrate ($0.2\mathrm{g}/\mathrm{kg}$) administered through intraperitoneal injection. Subsequently, the mouse’s head was secured onto a brain stereotactic apparatus to facilitate further coupling operations. All experimental procedures strictly adhered to the protocols approved by the Institutional Animal Care and Use Committee at the Chengdu Dossy Experimental Animal Company. The results revealed that conventional Gaussian-beam excitation PAM in visible light exhibited fewer blood vessels and missed some vessels located around the right edge areas of the mouse brain [Fig. 5(a)]. In contrast, Airy-beam excitation PAM clearly depicted more vessels in these edge areas, exhibiting enhanced resolution even at distances exceeding $1000\mu \mathrm{m}$ from the top surface [Fig. 5(b)]. The presence of irregular non-vessel-shaped structures in the upper corner of the images may be attributed to imperfect coupling operation procedures, leading to minor bleeding in the brain. Moreover, multiscale vessel enhancement filtering maps were performed within the region of interest [Figs. 5(c) and 5(d)], encompassing the rectangular dotted line areas in Figs. 5(a) and 5(b). Consequently, it was evident that Airy-beam excitation PAM could accurately identify a more comprehensive distribution of vasculature compared to Gaussian-beam excitation PAM, capturing a greater number of vessels (indicated by the white arrow mark within the oval area with dotted green lines). The B-scan images in Figs. 5(e) and 5(f) depict the dashed lines corresponding to Gaussian-beam excitation PAM and Airy-beam excitation PAM, respectively. Due to slight animal movement during the imaging process, the two B-scan lines did not align perfectly. Nevertheless, this discrepancy further confirmed the DOF extension capability in the depth direction. Furthermore, through the analysis of SBR intensity at specific locations of interest (I, II, III), it became evident that Airy-beam excitation PAM exhibited superior vascular-resolving capabilities, with FWHM values of 40.1 versus $65.5\mu \mathrm{m}$ and 19.3 versus $31.3\mu \mathrm{m}$ for Airy-beam and Gaussian-beam excitation PAMs, respectively. Particularly in the axial direction, Airy-beam excitation PAM clearly captured vessel details with an FWHM of $22.9\mu \mathrm{m}$ that were undetectable by Gaussian-beam excitation PAM (III). Following a comprehensive statistical analysis of zebrafish larvae, zebrafish adults, and cerebral vasculature, a notable 62% improvement in detail-resolving capability was observed on average using FDIR-PAM compared to Gaussian-beam excitation PAM. Finally, quantitative statistics were performed on vascular density in the boxed region, revealing that FDIR-PAM comprehensively and significantly captured more vasculature network information [Fig. 5(g), $P<0.05$], thereby demonstrating its ability to extend the DOF for comprehensive vasculature characterization.

Fig. 5

In vivo label-free PAM characterization of the mouse brain. (a) and (b) Imaging of brain vasculature via GB-PAM and FDIR-PAM. (c) and (d) Zoom-in en face maps with multiscale vessel enhancement filtering operator that correspond to the yellow boxes of (a) and (b), respectively. (e) and (f) Cross-sectional maps that correspond to the dashed lines of (c) and (d), respectively. FDIR-PAM reveals a greater number of blood vessels surrounding the right-edge areas of the mouse brain compared to conventional GB-PAM. (g) Intensity profiles along the lines depicted in (I), (II), and (III), as well as statistical analysis of vessel density, were conducted to compare GB-PAM and FDIR-PAM.