2.1.

COMSOL-Based Simulation for Generating Focused Gaussian Beam

We designed a two-dimensional (2-D) FEM simulation geometry based on electromagnetic theory in COMSOL. The simulation model used is illustrated in Fig. 1(a). Area of simulation ($20\mu \mathrm{m}\times 250\mu \mathrm{m}$) includes optical focusing, water medium, and microsphere. Perfect matching layer (PML) was used as a boundary condition. A focused beam was simulated by a Gaussian beam propagating along the $+y$-axis with a full width at half maximum (FWHM) beam waist diameter of $\lambda /2\mathrm{NA}$ or an $e^{-2}$ radius of $\lambda /2.35\mathrm{NA}$ in intensity,²⁶ where $\lambda$ is the incident wavelength and NA is the numerical aperture of the objective. The beam waist was always centered laterally in the simulation area $(x=0)$, but the $y$-position varies for different simulations. In our simulations, we have generated a focusing Gaussian beam with $\lambda =800\mathrm{nm}$ and $\mathrm{NA}=0.1$, so its theoretical FWHM waist diameter would be 4000 nm.

Fig. 1

(a) Simulation geometry for PNJ generation using FEM in COMSOL: FGB—focused Gaussian beam, $n0$—refractive index of water, $n1$—refractive index of microsphere, $D$—diameter of microsphere, WD—working distance, $W_p$—FWHM, and $L_p$—effective length of PNJ. (b) Simulation geometry for PA microscopic imaging using k-wave in MATLAB. UST—ultrasound transducer (point detector).

2.2.

Simulation of Photonic Nanojets in COMSOL

The FEM model shown in Fig. 1(a) was used for generating PNJs. The dielectric solid microsphere with diameter $D$ and refractive index $n1$ was immersed in water medium having refractive index $n0=1.33$. In most of the previously reported PNJ simulations, the incident light was a plane wave. But in ORPAM, excitation beam is a focused Gaussian beam. Hence, in our simulations, we have used 800 nm Gaussian beam focused using an objective with $\mathrm{NA}=0.1$ as discussed in Sec. 2.1. A sphere with $D=5\mu \mathrm{m}$ was placed inside the Gaussian focus having an FWHM diameter of 3958 nm. Spheres with different designs and different materials were tested. The materials used for the spheres were all existing materials. We have carried out simulations for the following spheres: (i) RM: It was a perfectly round, solid sphere with a diameter, $D=5\mu \mathrm{m}$. The material used for the sphere is $n1=1.45$ (fused silica), $n1=1.58$ (polystyrene), or $n1=2.2$ (${\mathrm{TiO}}_2$–BaO–ZnO glass). This design was widely used for generating PNJs.^21,27 (ii) TM: Through simulation studies, it was observed that PNJ parameters are significantly influenced just by truncating the RM.²⁸ In our simulations, a $D=5\text{-}\mu \mathrm{m}$ diameter RM made out of silica was cut by different thicknesses, $T=0.25D$, $0.5D$, $0.75D$. So, the cutting thicknesses used here are $T=1.25,2.5,3.75\mu \mathrm{m}$. (iii) MCR: Both experimental and simulation studies showed that the MCR on its shadow-side surface could modulate the PNJ generated.²⁹ In our simulations, we have used a $D=5\mu \mathrm{m}$ silica sphere decorated with 2, 3, or 4 uniformly distributed rings. The depth and width of all the rings were 1.2 and $0.25\mu \mathrm{m}$, respectively. The inner radius of the first ring was $0.5\mu \mathrm{m}$ and the distance between the adjacent rings was $0.25\mu \mathrm{m}$. (iv) MLM: It was shown that using a two-layer microsphere configuration it is possible to enhance the length of the nanojet significantly. A PNJ with FWHM length of $\sim 22\lambda (\sim 14\mu \mathrm{m})$,³¹ and a PNJ with decay length of $\sim 100\lambda (\sim 40\mu \mathrm{m})$³⁰ were achieved using a two-layer microsphere. In our simulations, we have used an MLM sphere consisting of an inner core of radius $(r1)$ and with a refractive index $(n1)$, and of a shell with outer radius $(r2)$ and with an index $(r2)$. The schematic of microspheres is shown in Fig. 2. The total diameter of the two-layer sphere was $D=5\mu \mathrm{m}$. Three different two-layered spheres were designed with existing material: (a) silica sphere $(n1=1.45)$ and BK7 layer $(n2=1.51)$, (b) BK7 sphere $(n1=1.51)$ and ${\mathrm{MgF}}_2$ layer $(n2=1.38)$, (c) lanthanum dense flint (LaSF) sphere $(n1=1.83)$, and barium flint (BaF) layer $(n2=1.60)$. (v) TMLM: The two-layer spheres discussed above were truncated to study the influence of its cutting thickness on the PNJ parameters. We found that truncating the MLM design significantly enhanced the effective length, and working distance compared to all other designs. The three different spheres used for MLM design were truncated by thickness, $T=0.25D$, $0.5D$, $0.75D$ (i.e., $T=1.25,2.5,3.75\mu \mathrm{m}$). Figure 2 summarizes the properties of PNJs generated from the five different designs discussed above. Figure 2 is discussed in detail in Sec. 3.

Fig. 2

Properties of PNJ generated by microspheres (with diameter, $D=5\mu \mathrm{m}$) of different designs when illuminated with a focusing Gaussian beam (generated in water with 0.1 NA and $\lambda =800\mathrm{nm}$) in water medium $(n0=1.33)$. The FWHM waist diameter of Gaussian focus in water, $W_g=3958\mathrm{nm}$. The highlighted ones are the best values observed in our simulations. $N$—number of concentric rings. *The PNJs used for demonstrating PA imaging in k-wave simulations, where $L_p$ is the effective length, $W_p$ is the FWHM width, WD is the working distance of the microsphere, $I_p$ is the peak intensity, and $Q$ is the quality criterion or figure-of-merit.

2.3.

Simulation of Photoacoustic B-Scan Images Using k-Wave Toolbox in MATLAB

The fluence maps of focusing Gaussian beam and PNJs were exported to MATLAB for simulating PA images using the k-wave toolbox. Figure 1(b) shows the 2-D simulation geometry used in k-wave simulations. A grid of $251\times 2001\text{pixels}$ (i.e., $12\mu \mathrm{m}\times 100\mu \mathrm{m}$) was created with $50\mathrm{nm}/\text{pixel}$. A PML was used as a boundary condition for the generation of the forward PA data. A numerical phantom as shown in Fig. 1(b) was used for the study. The phantom consists of square targets with a side length of $1\mu \mathrm{m}$, separated by a distance $1.5\mu \mathrm{m}$. Seven such targets were placed along $+y$-axis with $10\mu \mathrm{m}$ gap between each pair of targets to create an object with a depth of $60\mu \mathrm{m}$. The targets were embedded inside a nonabsorbing medium.

In our simulation, the light illumination was fixed at the center of the scanning area and the object was scanned across the light field. The scanned area was $101\times 2001\text{pixels}$ (i.e., $5\mu \mathrm{m}\times 100\mu \mathrm{m}$). The object was moved along the $x$-direction with a step size of 50 nm resulting in a total of 101 A-lines. A point detector (UST) was used to collect the A-lines and it was placed at a distance of $27\mu \mathrm{m}$ from the object. The center frequency and the fractional bandwidth of the detector were 375 or 1225 MHz and 80%, respectively. According to the simulations, 375 and 1225 MHz transducers can provide up to $\sim 1$- and $\sim 0.1\text{-}\mathrm{mm}$ depths, respectively, at $-20\mathrm{dB}$ signal attenuation.^33–35 To simplify the simulations, we used a point detector. To collect forward PA signal over a length of $100\mu \mathrm{m}$, a total of 1340 time steps with each step 50 ps was used. The PA signals were down sampled five times to mimic a data acquisition system with $4\mathrm{GS}/\mathrm{s}$ sampling rate. To every PA signal, 1% noise was added, resulting in 40 dB signal-to-noise ratio level. The A-line PA data were stacked after taking Hilbert transform to form a B-scan PAM image. The simulations assumed a sound speed of $1500\mathrm{m}/\mathrm{s}$. For simplicity, the medium was considered acoustically homogeneous and there was no absorption or dispersion of sound.

3.1.

Properties of Photonic Nanojets from Various Sphere Designs

As discussed in Sec. 2.1, a Gaussian focus was generated with $\mathrm{NA}=0.1$ and $\lambda =800\mathrm{nm}$, and its FWHM waist diameter measured to be $W_g=3958\mathrm{nm}$. As discussed in Sec. 2.2, the simulations were carried out to investigate PNJs for sphere designs RM, TM, MCR, MLM, and TMLM. The basic characteristics of PNJ are as follows: (i) effective length $(L_p)$ is the distance of the point where the nanojet intensity drops by a factor of $(1/e)$ with respect to the peak intensity $(I_p)$ in the axial profile ($+y$-axis), (ii) working distance (WD) is the distance from the sphere shadow-side surface to the point of peak intensity ($I_p$), (iii) waist of the nanojet $(W_p)$ is the FWHM of the nanojet intensity profile (along $+x$-axis) at peak intensity $(I_p)$, (iv) peak intensity $(I_p)$ is the maximum intensity of the PNJ, (v) figure-of-merit or quality criterion “$Q$” is a measure of the quality of the PNJ, it combines the basic characteristics of PNJ, and is defined as $Q=(L_p\times I_p)/W_p$.³⁶ For a given microsphere, the PNJ was optimized based on the $Q$ value of the nanojet they produced. The sphere inside the focusing Gaussian beam was moved (along $y$-axis) to generate a PNJ with highest $Q$ value. The PNJ parameters, such as the effective length $(L_p)$, FWHM waist size $(W_p)$, working distance (WD), and peak intensity $(I_p)$, and quality criterion $(Q)$ of the PNJs generated when a $5\mu \mathrm{m}$ sphere was placed inside a focusing Gaussian beam with 3958 nm waist size, are listed in Fig. 2. The summary of the table is as follows: (i) The PNJ properties are significantly affected by the design of the sphere under the same illumination conditions. (ii) FWHM widths of all the PNJs $(W_p)$ are smaller than the Gaussian focus (i.e., $W_g=3958\mathrm{nm}$). (iii) Increase in $n_1$ will reduce the PNJ width, length, and working distance. The PNJ is elongated at the expense of PNJ broadening and intensity lowering. (iv) RM design could generate a strong and narrow nanojet. With $n_1=2.2$, PNJ having peak intensity $I_p=4.87$ and width $W_p=360\mathrm{nm}$ was observed. The ultranarrrow nanojet having length $L_p\sim 1380\mathrm{nm}$ $(\sim 1.73\lambda )$ is shown in Fig. 3(a). The $n_1=2.2$ sphere generates a 360 nm spot from a 3958 nm Gaussian spot, i.e., $\sim 11$-flod enhancement in the resolution. But the drawback with the RM design is it could only generate a nanojet with short length and short working distance. (v) By cutting the silica $n_1=1.45$, sphere by $0.75D$, i.e., $3.75\mu \mathrm{m}$, the PNJ was elongated from 11,779 to 74,314 nm (i.e., $\sim 6.3$-fold enhancement in length). Figure 3(b) shows the PNJ with the length of 11,779 nm obtained by round silica sphere and its intensity plot along the line indicated as shown in Fig. 3(c). Figure 3(d) shows the PNJ with the length of 74,314 nm obtained by truncating the silica sphere by $T=0.75D$. (vi) In case of MCR design, the number of rings 2, 3, or 4 made on the round silica sphere does not influence the PNJ parameters much. However, the length and working distance are slightly better compared to the RM silica sphere $(n_1=1.45)$. (vi) By properly selecting the sphere and layer materials, the MLM design is promising to generate an ultralong nanojet. Here, the layer thickness was optimized to generate PNJ with highest $Q$ value. With $n_1=1.51$, $n2=1.38$, $r1=2.4\mu \mathrm{m}$, $r2=2.5\mu \mathrm{m}$, a $47\text{-}\mu \mathrm{m}$ long nanojet was generated. (vii) The TMLM design with $n_1=1.51$, $n2=1.38$, $r1=2.4\mu \mathrm{m}$, $r2=2.5\mu \mathrm{m}$, and cutting thickness $T=0.75D=3.75\mu \mathrm{m}$ generated PNJ with a remarkable extension of $\sim 138\mu \mathrm{m}$ ($\sim 172\lambda$, it is the longest nanojet compared to the previously reported nanojet lengths^30,31) and long working distance $\sim 26\mu \mathrm{m}$ ($\sim 32\lambda$, it is the longest working distance compared to the previously reported values^28,36). The ultralong nanojet is shown in Fig. 3(e). The fluence maps for Gaussian focus, nanojets from RM design with silica sphere $(n_1=1.45)$, and TMLM design with $n_1=1.45$, $n2=1.51$, and $T=0.75D$ are shown in Figs. 4(a)–4(c), respectively. Figure 4(d) shows the normalized FWHM line profiles showing the widths and lengths. The fluence maps for Gaussian focus and PNJs as shown in Figs. 4(a)–4(c) were then used for illuminating the object [Fig. 1(b)] in k-wave simulations in MATLAB to generate PA B-scan image.

Fig. 3

Fluence maps generated when a Gaussian beam with waist 3958 nm focused through (a) an RM with $D=5\mu \mathrm{m}$ and $n1=2.2$,   (b) an RM with $D=5\mu \mathrm{m}$ and $n1=1.45$, and (c) intensity profile along the line shown in (b). Here, $D$—diameter of the sphere $(5\mu \mathrm{m})$, WD—working distance $(\sim 5.9\mu \mathrm{m})$, $L_p$—effective length $(\sim 11.8\mu \mathrm{m})$, (d) a TM with $D=5\mu \mathrm{m}$, $n1=1.45$, and $T=0.75D=3.75\mu \mathrm{m}$, and (e) a TMLM with $n1=1.51$ and $n2=1.38$. COMSOL simulations were carried out in $20\mu \mathrm{m}\times 250\mu \mathrm{m}$ area. For clarity, the area around the nanojet is shown here. The area of images in (a) and (b) is $10\mu \mathrm{m}\times 50\mu \mathrm{m}$ and the area of images in (c) and (d) is $20\mu \mathrm{m}\times 170\mu \mathrm{m}$.

Fig. 4

Fluence maps for Gaussian focus and PNJs: (a) focusing Gaussian beam in water with $\lambda =800\mathrm{nm}$, $\mathrm{NA}=0.1$. PNJs generated when the Gaussian beam (a) focused on (b) round sphere with $D=5\mu \mathrm{m}$ and $n1=1.45$, (c) TMLM with $D=5\mu \mathrm{m}$, $n1=1.45$, and $n2=1.51$. Normalized line profiles showing (d) widths and (e) lengths. The area of images in (a)–(c) is $20\mu \mathrm{m}\times 180\mu \mathrm{m}$.

3.2.

Super-Resolution Photoacoustic B-Scan Imaging

The numerical phantom described in Fig. 1(b) was sequentially scanned by the fluence maps shown in Figs. 4(a)–4(c). Figures 5(a)–5(c) show the 2-D B-scan PAM images of the target excited with the Gaussian focused beam with FWHM 3958 nm [Fig. 4(a)] and excited with PNJs with FWHM 909 nm [Fig. 4(b)] and 1459 nm [Fig. 4(c)], respectively. The images shown in Figs. 5(a)–5(c) were acquired using 375 MHz UST. From the ORPAM image shown in Fig. 5(a), it is clear that the Gaussian beam illumination could provide information of all the targets at depth but it could not resolve the gap between the targets. It is because the Gaussian spot size $(W_g=3958\mathrm{nm})$ is larger than the separation between the targets 1500 nm. Figure 5(b) shows that the narrow PNJ (909 nm) generated by RM could resolve the gap (1500 nm) between the targets but it could not image the targets beyond $10\text{-}\mu \mathrm{m}$ depth. From Fig. 5(c), it is clear that the elongated PNJ $(\sim \mathrm{77,614}\mu \mathrm{m})$ generated using TMLM design could image all the targets and resolve the gap (1500 nm) between the targets. Here, the gap between the targets is clearly resolved, but the gap is not uniform along the depth. This is due to slight variation in the width along the length of the nanojet. Similar simulations were repeated using 1225 MHz transducer and the corresponding ORPAM images are shown in Figs. 5(d)–5(f). From these results, it is evident that the depth of super-resolution ORPAM can be enhanced with the help of ultralong PNJs.

Fig. 5

PAM B-scan images obtained by scanning the object [Fig. 1(b)] with Gaussian focus and PNJ: (a, d) images obtained with Gaussian focus shown in Fig. 4(a), (b, e) images obtained with short $(\sim \mathrm{11,779}\mathrm{nm})$ PNJ shown in Fig. 4(b), and (c, f) images obtained with ultralong $(\sim \mathrm{77,614}\mathrm{nm})$ PNJ shown in Fig. 4(c). Images were acquired using USTs (a–c) 375 MHz and (d–f) 1225 MHz.

These simulations promise that it is possible to control or tune the imaging depth, lateral resolution, and signal enhancement in the conventional ORPAM system just by changing the sphere design being used. Based on the requirement, the PNJ parameters can be optimized by choosing the appropriate sphere and refractive index. Microsphere has already proven its potential in white light microscopy²¹ by imaging 50-nm features, and in fluorescence microscopy²⁴ by imaging subcellular structures such as centrioles, mitochondria, and chromosomes, and also in Raman microscopy by imaging subdiffraction features of a Blu-ray disc.²⁶ The simulation results in our work promise that by using an appropriate microsphere, conventional ORPAM could be transformed into PAN at virtually no extra cost. The PAN can be used in many biomedical imaging applications that demand high resolution. For example, cellular structures, vasculature, and microcirculation systems, in vivo PAN, having high endogenous contrast, may help to monitor the molecular mechanism of blood vessel formation that can reveal the cause for angiogenesis. Currently, the diagnosis of melanoma is done based on invasive biopsy and inaccurate visual inspection. In vivo PAN being noninvasive, noncontact, and label-free will help us to diagnose melanoma in the early stage, a key to successful treatment. The approach may allow ORPA imaging of biological entities, such as bacteria $(\sim 1\mu \mathrm{m})$, large virus (0.05 to $0.4\mu \mathrm{m}$) particles, labeled as nucleic acids, molecules, etc. In spite of its promises, there are challenges in implementing the microsphere approach in practice. Although, other optical imaging modalities have successfully demonstrated the use of PNJ to improve the imaging resolution by breaking the diffraction limit. The experimental validation of PNJ for PA imaging is yet to be demonstrated. Our future work is to experimentally validate the usefulness of this simple technique in optical-resolution PA imaging.