2.1.

Simulation of Gaussian Foci/Photonic Nanojets in COMSOL

COMSOL Multiphysics software works based on the finite element method (FEM) for various physics and engineering applications. Based on electromagnetic theory, a 2-D FEM simulation was designed. The simulation model is illustrated in Fig. 1(a). With scattering boundary conditions (black lines), simulations were carried out in a rectangular area ($30\times 60\mu {\mathrm{m}}^2$), which includes a water region and a microsphere region. The medium blue region is water, defined by a refractive index $n_1=1.33$. A dielectric microsphere (green region) defined by a refractive index ($n_2$) was embedded in water. The materials used for microspheres in the simulation are fused silica ($n_2=1.46$), polystyrene ($n_2=1.60$), and TiO2–BaO–ZnO glass ($n_2=2.20$). We simulated a focusing laser beam by a TM 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 ${\mathrm{e}}^{-2}$ radius of ($\lambda /2.35\mathrm{NA}$) in intensity,³² where $\lambda$ (532, 850, or 1064 nm) is incident wavelength, and NA (0.5, 0.7, or 1.0) is the numerical aperture of the objective. The beam waist was always centered laterally in the simulation area ($x=0$), but the $y$-position varied for different simulations. In the model settings, a smaller mesh element (20 nm) was specified in the Gaussian focus/photonic nanojet region. Outside this region, a larger mesh element (200 nm) was used. The computational time for calculating a $30\times 60\text{-}\mu {\mathrm{m}}^2$ surface plot was 9 s. Several simulations were carried out by changing the diameter of the microsphere ($D$), the incident wavelength ($\lambda$), the numerical aperture of the objective (NA), and the refractive index of the microsphere ($n_2$). Table 1 summarizes the FWHM values of the Gaussian foci and nanojets for various cases. Table 1 is discussed in detail in Sec. 3.1.

Fig. 1

Schematic diagram of the two-dimensional simulation geometry for (a) photonic nanojet generation using finite element method (FEM) in COMSOL, and (b) photoacoustic microscopy imaging using k-wave toolbox in MATLAB. Here, $n_1$ and $n_2$ are the refractive indices of water and microsphere, respectively. UST: ultrasound transducer (point detector). $D$: diameter of the microsphere. $d$: distance between the targets.

Table 1

Full width half maximum (FWHM) of Gaussian focus (fg) and photonic nanojet (fp) for different incident wavelengths (λ) in water (n1=1.33). (a) D=5  μm, NA=1.0; (b) D=10  μm, NA=0.7; (c) D=15  μm, NA=0.5. Here, D: diameter of the microsphere, n2: refractive index of micropsphere, NA: numerical aperture of the light focusing lens. Both the FWHM values and the FWHM divided by the incident wavelength (λ) are given in the table. The symbol ‘X’ in indicates that a nanojet was not observed in these cases.

2.2.

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

Figure 1(b) shows the simulation geometry used for the simulation of B-scan ORPAM images’ acquisition. For simplicity, all simulations were done in 2-D without losing any generality. A computational grid of $251\times 1501\text{pixels}$ (the white region of size $5\times 30\mu \mathrm{m}$, $20\mathrm{nm}/\text{pixel}$) was used. A perfectly matched boundary layer to satisfy the boundary condition was used for the generation of the forward PA data. A point detector (the red dot) was used for the simulations and was placed at a distance of $26\mu \mathrm{m}$ from the target center. The center frequency and the bandwidth of the detector were taken as 125 MHz and 70%, respectively. The transducer parameters are chosen to match the transducers typically used in an ORPAM system,³⁷ however, instead of a large focusing transducer, a point detector was used to simplify the simulations. Other transducers with lower or higher center frequencies (and bandwidths) can also be used in the simulation. The axial resolution in PAM is related to the bandwidth of the ultrasound transducer and will not affect the lateral resolution of the imaging system. The lateral resolution is primarily determined by the focusing of light in the ORPAM imaging system.

The scanning region was restricted to $101\times 1501\text{pixels}$ (the orange region of size $2\times 30\mu {\mathrm{m}}^2$). For forward PA simulation, the time step chosen was 0.05 ns with a total of 600 time steps. The simulations assumed a sound speed of $1500\mathrm{m}/\mathrm{s}$ in the medium. For simplicity, the medium was considered acoustically homogeneous and there was no absorption or dispersion of sound. From the simulated pressure waves, PA waves were down sampled five times to mimic a data acquisition system with $4\text{GigaSamples}/\mathrm{s}$. In all the cases, 1% noise was added with the data, resulting in a 40-dB signal-to-noise ratio. The PAM scan head (in simulation the light source and the detector pair) was moved along the $x$-direction with a step size of 20 nm, resulting in a total of 101 A-lines per B-scan. The A-line data were stacked after taking Hilbert transform to form a B-scan PAM image. The computational time for each A-line was 2 min 10 s.

Three numerical phantoms were used for the study. The first phantom was a line target with a $0.02\text{-}\mu \mathrm{m}$ width along $x$-axis and a $1\text{-}\mu \mathrm{m}$ length along $y$-axis ($\text{width}\ll$ expected lateral resolution of the imaging system, making it a line target). This was used as a line object to measure the point spread function along the lateral direction. The second object consisted of two square targets with a side length $1\mu \mathrm{m}$ separated by a distance ($d$) of $0.5\mu \mathrm{m}$. The third one consisted of two square targets with a side length $1\mu \mathrm{m}$ separated by a distance ($d$) of $0.2\mu \mathrm{m}$. In all the phantoms, the targets were binary, therefore, outside the targets there was no initial pressure rise and inside the target the initial pressure rise was 1.

3.1.

Gaussian Foci and Photonic Nanojets in Water Medium

As discussed in Sec. 2.1, simulations for the incident wavelengths (${\lambda }_1=532\mathrm{nm}$, ${\lambda }_2=850\mathrm{nm}$, and ${\lambda }_3=1064\mathrm{nm}$), sphere diameters ($D=5$, 10, $15\mu \mathrm{m}$), and numerical apertures ($\mathrm{NA}=1.0$, 0.7, and 0.5) were carried out. The FWHM of Gaussian focus (fg) and photonic nanojet (fp) for different incident wavelengths ($\lambda$) in water ($n_1=1.33$) are shown in Table 1. Both the FWHM values and the FWHM divided by the incident wavelength ($\lambda$) are shown in this table. The FWHM values are significantly affected by all the parameters $n_2$, NA, $D$, and $\lambda$. From the table, it is clear that the FWHM values of the photonic nanojets are much smaller than the corresponding Gaussian foci. It can be seen that in all cases the FWHM has decreased when $n_2$ is increased (1.46 to 2.2) or NA is increased (0.5 to 1.0) or $\lambda$ is decreased (1064 to 532 nm). These simulations indicate that it is possible to achieve super-resolution imaging in water with the aid of microspheres. The symbol “X” in the table indicates that a nanojet was not observed in these cases. A few representative fluence maps from COMSOL simulations are shown in Fig. 2. For a better view, a $10\times 15\mu {\mathrm{m}}^2$ area around the Gaussian foci/photonic nanojet is shown here. Figure 2(g) shows the normalized line profiles along with the FWHM values. The fluence maps generated for Gaussian foci and photonic nanojets were then used for k-wave simulation in MATLAB for the ORPAM light excitation source.

Fig. 2

Numerical simulation of Gaussian foci and photonic nanojets using FEM in COMSOL: Gaussian beam foci for (a) ${\lambda }_1=532\mathrm{nm}$, $D=5\mu \mathrm{m}$, $\mathrm{NA}=1.0$, (c) ${\lambda }_2=850\mathrm{nm}$, $D=10\mu \mathrm{m}$, $\mathrm{NA}=0.7$, (e) ${\lambda }_3=1064\mathrm{nm}$, $D=15\mu \mathrm{m}$, $\mathrm{NA}=0.5$, photonic nanojets for (b) ${\lambda }_1=532\mathrm{nm}$, $D=5\mu \mathrm{m}$, $\mathrm{NA}=1.0$, $n_2=1.6$, (d) ${\lambda }_2=850\mathrm{nm}$, $D=10\mu \mathrm{m}$, $\mathrm{NA}=0.7$, $n_2=2.2$, (f) ${\lambda }_3=1064\mathrm{nm}$, $D=15\mu \mathrm{m}$, $\mathrm{NA}=0.5$, $n_2=2.2$. (g) Normalized line scan profiles along the lines A and B indicated on Figs. 2(e) and 2(f), respectively. COMSOL simulations were carried out in $30\times 60\text{-}\mu {\mathrm{m}}^2$ area, however, $10\times 15\text{-}\mu {\mathrm{m}}^2$ area around the focus is shown here.

3.2.

Optical Resolution Photoacoustic Microscopic B-Scan Imaging with Photonic Nanojet in k-Wave

First, the line target numerical phantom with a 20-nm width and 1000-nm length was scanned. Figures 3(a) and 3(b) show the 2-D B-scan PAM images of the line target excited with a normal Gaussian focused beam with FWHM 1164 nm [Fig. 2(e)] and excited with a photonic nanojet with FWHM 305 nm [Fig. 2(f)], respectively. It is evident that the lateral resolution has improved with the photonic nanojet excitation. Figure 3(c) shows the normalized lateral line profiles along the lines indicated on Figs. 3(a) and 3(b). From the line profiles, the lateral resolution of the ORPAM system was quantified to be 1135 and 355 nm, respectively (very close to the values measured in COMSOL from the excitation beam profiles). The slight difference is due to the grid size (20 nm) and the nonideal line target (line width 20 nm) used in the k-wave simulation.

Fig. 3

k-wave simulation of B-scan PAM image with a line absorber target object [20-nm line width, length 1000 nm]. B-scan image obtained with (a) Gaussian focus light [shown in Fig. 2(e)] excitation, (b) photonic nanojet [shown in Fig. 2(f)] excitation. (c) Normalized line scan profiles (point spread function) along the dotted lines A and B indicated on 3(a) and 3(b), respectively.

The second numerical phantom where two square targets with side lengths $1\mu \mathrm{m}$ and seperated by a distance $d=0.5\mu \mathrm{m}$ were imaged next. Figure 4(a) shows the ORPAM B-scan image of the targets obtained by exciting with normal the Gaussian-focused beam [Fig. 2(e)]. The targets were not resolved in this case beacuse of the poor lateral resolution (the optical spot size is larger than the speration beteen the targets). Whereas in Fig. 4(b) the targets were clearly resolved when ORPAM B-scan images were generated with photonic nanojet excitation [Fig. 2(f)]. Figure 4(c) shows the normalized line scan profiles along the lines indicated on Figs. 4(a) and 4(b). Once again, the line profiles in Fig. 4(c) confirmed the resolving ability of a separation of a 500-nm gap with the photonic nanojet excited ORPAM.

Fig. 4

k-wave simulation of PAM B-scan images of two square (size $1\mu \mathrm{m}$) targets separated by a gap of 500 nm. B-scan image with (a) Gaussian focus [shown in Fig. 2(e)] excitation, (b) photonic nanojet [shown in Fig. 2(f)] excitation. (c) Normalized line scan profiles along the dotted lines A and B indicated on 4(a) and 4(b), respectively.

Next, similar exercises were repeated with a smaller gap of 200 nm between the two square target objects to determine the smallest gap the ORPAM system will be able to resolve. The excitation light source used was the smallest photonic nanojet that was simulated with the set of parameters in this study [FWHM 208 nm Fig. 2(b)]. Figure 5(a) shows the ORPAM B-scan image and Fig. 5(b) shows the normalized profile along the line indicated on Fig. 5(a). The targets are resolved in this case as expected, demonstrating the ability of breaking through the diffraction-limited lateral resolution in ORPAM with the help of photonic nanojets to achieve super-resolution.

Fig. 5

k-wave simulation of PAM B-scan images of two square (size $1\mu \mathrm{m}$) targets separated by a gap of 200 nm. (a) B-scan image with photonic nanojet [shown in Fig. 2(d)] excitation. (b) Normalized line scan profile along the dotted line indicated on (a).

The super-resolution ORPAM modality presented in this work promises to enhance the lateral resolution beyond the diffraction limit. This approach may revolutionize the fundamental biological studies involving cellular structures, vasculature, and microcirculation systems. Melanoma arises from deadly skin cancer melanocytes, and so on.^16,38 Currently, the diagnosis of melanoma is done based on invasive biopsy and inaccurate visual inspection. Super-resolution ORPAM (being an in vivo, label-free, noninvasive technique) will help us to diagnose melanoma in the early stages, a key to successful treatment. Tumor angiogenesis, a hallmark of cancer, is presently imaged either in vivo by clinical methods at low resolution or ex vivo by nonclinical methods at high resolution.^16,39 Super-resolution ORPAM, having high endogenous contrast, can image angiogenic microvessels in vivo. However, the drawback with this approach is limited penetration depth compared to the conventional ORPAM. The generated nanojets are limited in depth, as observed from the simulations. One can play with various parameters such as the diameter of the microsphere, the refractive-index of the material of the microsphere, etc., to move the focusing spot (nanojets) at various depths from the surface of the microsphere. We can also use some form of $z$-scanning to increase the depth of imaging. We may not reach the depth penetration of conventional ORPAM, however, it can be significantly improved. In the future, we will work on experimentally implementing the super-resolution ORPAM and applying it to cellular imaging. Improving the penetration depth of super-resolution ORPAM will also be a challenging task to overcome in the future.