2.1.

Finite-Difference-Time-Domain Method

Since the development of the FDTD method,¹³ it has been utilized to solve Maxwell’s equation numerically in the time domain to describe the complex behavior of light interaction and widely applied in solving the scattering issue in biological fields.^14–16 In this study, we have utilized the R-Soft software with fullwave mode based on the FDTD algorithm. The FDTD algorithm is the computation process for implementing time and space derivatives of the electric and magnetic fields in Yee’s cell.¹⁷ Solve the Maxwell’s curl equations by first discretizing the equations with central differences in time and space and these equations can be further numerically solved by the software. To estimate the optical scattering properties of ${\mathrm{TiO}}_2$ beads with different diameters and embedded in thin-films, we align the scattering direction along with the line connecting the ${\mathrm{TiO}}_2$ beads, and they are separated by the IPS, which varies from 0 to 1000 nm in an interval of 250 nm. The model used for FDTD is shown in Fig. 1(a): the wavelength of the incident plane wave is ${\lambda }_0=853\mathrm{nm}$, while the refractive index of mesoporous ${\mathrm{TiO}}_2$ beads is $n=2.5$^18,19 and assumed to be homogeneous and isotropic in distribution. The polarization of the incident wave is set as TE polarized. The grid size of calculation is set as 0.005 nm, time step is 0.005 s, and the incident wave is a plane wave in the enclosed area, respectively. With a perfectly matched layer (PML), it is composed of several grid points as the edge of the domain for the enclosed area, which can absorb the incident lights and result in no reflection.

Fig. 1

(a) General setup of FDTD simulation for computing the phenomenon of light scattering effect. (b) The schematic diagram of ${\mathrm{TiO}}_2$ for different diameters embedded in a unit cell ($\#/{\mathrm{m}}^3$).

In order to obtain the far-field scattering patterns, the virtual box with appropriate PML boundary is set to enclose the scattering objects (mesoporous ${\mathrm{TiO}}_2$ beads) and launch a plane wave inside an enclosed area, which is shown in Fig. 1(a). In the meantime, there are only scattering fields from objects propagating beyond the enclosed launch boundary, and unscattered light will be absorbed at the PML boundary. For obtaining the scattering phase function, the corresponding far field can be calculated using the near-field to far-field transformation from the simulation data in software.²⁰

2.2.

Preparation of Mesoporous TiO₂ Beads Samples

Mesoporous ${\mathrm{TiO}}_2$ bead samples with average diameters ($D$) of 20, 150, 300, and 500 nm were prepared by the sol–gel process. Initially, the ${\mathrm{TiO}}_2$ powder weight percentage is 13 wt. % (US Research Nanomaterials, Inc.) mixed with different percentages of isopropyl alcohol and DI water. Adding polyethylene glycol to the mixture, the ${\mathrm{TiO}}_2$ pastes were obtained after more than 1 week of stirring, and the ${\mathrm{TiO}}_2$ pastes prepared on glass substrates were heated by an oven at 450°C for 30 min to evaporate the solvent. In the process, the ${\mathrm{TiO}}_2$ were assumed to be homogeneously distributed. The final volume fraction of ${\mathrm{TiO}}_2$ bead samples was fixed as 0.03 for ${\mathrm{TiO}}_2$ beads with diameters varied from 20 to 500 nm, respectively. Afterward, the samples of mesoporous ${\mathrm{TiO}}_2$ bead films on glass substrates were obtained.

For a suspension of colloidal system, the samples of ${\mathrm{TiO}}_2$ beads with volume fraction of 0.03 for different diameters are assumed to be homogeneously distributed; the schematic diagram of ${\mathrm{TiO}}_2$ beads in a unit cell ($1\mu {\mathrm{m}}^3$) for different diameters from 20 to 500 nm is shown in Fig. 1(b). The IPS parameter $\mathrm{IPS}=2r[{(\frac{{\varphi }_m}{\varphi })}^{\frac{1}{3}}-1]$ is defined as the distance from the surface of one particle to the other [the line of connection between the two through the center of each bead, as shown in Fig. 4(a)], which changes according to the volume fraction in the colloidal system.²¹ The maximum volume fraction is ${\varphi }_m$ with ${\mathrm{TiO}}_2$ samples, while $\varphi$ is the volume fraction of ${\mathrm{TiO}}_2$ bead samples, and $r$ is the radius of ${\mathrm{TiO}}_2$ beads. With the volume fraction of 0.03, the values of IPS are 25, 187, 373, and 623 nm for select ${\mathrm{TiO}}_2$ diameters between 20 and 500 nm.

2.3.

Extended Huygens-Fresnel Theory (Multiple Scattering) and Signal Analysis

First, as an assistant method, the FDTD simulation process was utilized to calculate the electromagnetic near field and then transformed to a far-field scattering pattern, which is also called the phase function $p_{\text{dep}}(\theta )$, for different IPS between mesoporous ${\mathrm{TiO}}_2$ beads with different diameters from 20, 150, 300, and 500 nm. Here, the far-field $p_{\text{dep}}(\theta )$ is indicated as the probability density function of photons for every scattering angle ($\theta$),²² and the anisotropy factor $g_{\text{dep}}$ is the integral of phase function $p_{\text{dep}}(\theta )$ over all scattering angles, which can be expressed as

Second, based on EHF theory, the amplitude of experimental OCT signals versus depth $z$ can be modulated as the function of two main flexible parameters for $g$ and ${\mu }_{\mathrm{s}}$. Once the multiple scattering effect appears, the right-hand side of point spread function (PSF) can be changed by slope variation at a larger axial depth as compared to that of a signal with only single scattering effect.²³ In practice, the most common method to date to obtain the optical properties has been to analyze the axial A-scan signals based on EHF model at the right-hand side of PSF with increasing depth. Except for the fixed optical system parameters, the resulting heterodyne signals $⟨S^2(z)⟩$ [Eqs. (3) and (4)] in depth can be mainly modulated by two competing factors, the anisotropy factor ($g=⟨\cos {\theta }_{\mathrm{rms}}⟩$, ${\theta }_{\mathrm{rms}}$ is average scattering angle) and the scattering coefficients (${\mu }_{\mathrm{s}}$).²³ In addition, the fixed parameters in the system include the power $P_{\mathrm{R}}$ and $P_{\mathrm{S}}$ measured at the reference and sample arms; the quantum conversion efficiency $\alpha$ of the CCD; the backscattering cross-section ${\alpha }_{\mathrm{b}}$, and the $1/e$ irradiance radii $w_H$ and $w_S$ for the absence and presence of scattering; the refractive index for medium $n$; the radius of the sample beam at the focal plane ${\omega }_0$; and the focal length of objective lens $f$, as follows:

In the EHF theory, it should be noted that it is not easy to predict the optical coefficients of $g$ and ${\mu }_{\mathrm{s}}$ at the same time due to the fact that they are competing parameters. Therefore, using the other method to obtain one of them is a general solution to approach. After obtaining the $g_{\text{dep}}$ of mesoporous ${\mathrm{TiO}}_2$ beads with different diameter sizes for various IPS from the FDTD method, introducing the value of $g_{\text{dep}}$ for specific ${\mathrm{TiO}}_2$ beads samples, the corresponding scattering coefficient ${\mu }_{\mathrm{s}}$ can be further derived for the PSF of A-scan signal based on EHF theory.

2.4.

Optical Coherence Tomography Setup

The schematic diagram of the SD-OCT system using an 853-nm super-luminescent diode as the source is shown in Fig. 2(a). A single mode fiber (Thorlabs 780HP) is used to direct the light into the reference and sample arms. This OCT system has a numerical aperture of 0.042, with an objective lens focal length of 17.9 mm. The interferometer signal strikes a transmission grating (1200 lines/mm) and the objective lens then collects and focuses the diffracted light onto the CCD (e2V, 2048 pixels, 28-kHz line rate). Each A-scan profile (in depth) was averaged over 1000 measurements and subtracted by the reference arm intensity, to obtain the interferometer signal without the DC term.

Fig. 2

(a) Schematic diagram of SD-OCT system at the central wavelength of 853 nm, where the mesoporous ${\mathrm{TiO}}_2$ films with different average diameters are placed on the sample arm. (b) Schematic diagram of the sample shows the scattering effects of mesoporous ${\mathrm{TiO}}_2$ scatter films occur, with incident lightwave illuminated on the samples.

3.1.

Scattering Phase Function of Single TiO₂ Bead Simulated by FDTD

The optical behavior of a single TiO₂ bead with different diameters has to be confirmed prior to the simulation of multiple beads, with the FDTD method analysis. Figure 3 shows the plots of angular-dependent scattering patterns of a single ${\mathrm{TiO}}_2$ bead with diameters of 20, 150, 300, and 500 nm, obtained by using the FDTD method, and is compared with the results from the Mie scattering model.^24–26 These comparison results show that both methods are consistent and verify that the settings in the FDTD method are appropriate. The resultant values of $g_{\text{dep}}$ obtained for single ${\mathrm{TiO}}_2$ bead with diameters from 20, 150, 300, and 500 nm are 0.003, 0.174, 0.525, and 0.327, respectively.

Fig. 3

Far-field scattering patterns of single ${\mathrm{TiO}}_2$ bead calculated from Mie scattering model (red solid line) and compared with FDTD method (green dash line) for different diameters of (a) 20, (b) 150, (c) 300, and (d) 500 nm.

3.2.

Interparticle Spacing Effects of Near-Field Scattering Patterns

To study the multiple scattering effects that occur within high density tissues, multiple ${\mathrm{TiO}}_2$ beads (for smaller IPS in Sec. 2.2) are simulated, with near-field scattering patterns as shown in Fig. 4(b). In analyzing actual OCT signals, the presence of multiple scattering makes it difficult to acquire the corresponding anisotropy factors $g_{\text{dep}}$.^27,28 We simplify the scatter geometry to in-line configuration of ${\mathrm{TiO}}_2$ beads separated by different IPS in order to investigate the scattering phenomenon. In Fig. 4(a), the electrical field patterns for in-line ${\mathrm{TiO}}_2$ beads with a diameter of 300 nm are shown; the IPS is varied from 0 to 1000 nm in intervals of 250 nm. The light propagates in the upward direction and is incident onto the beads from underneath, as shown in Fig. 4. According to the results of far-field scattering patterns, except for the main lobe that resulted from the Mie scattering effect, the presence of side-lobes was found by considering the effect of multiple scattering. This needs to be noted when the IPS is increased.

Fig. 4

(a) The modeling geometric structure schematic of mesoporous in-line ${\mathrm{TiO}}_2$ beads embedded in a unit cell with different diameters ($D$) of 20, 150, 300, and 500 nm, which show the various IPS between adjacent ${\mathrm{TiO}}_2$ beads. (b) Scattering patterns of near-field electrical field magnitude of in-line ${\mathrm{TiO}}_2$ beads for 300 nm separated by IPS. The light is incident from the bottom in each subfigure.

3.3.

Interparticle Distance Effects-Dependent Phase Function

To approach the realistic $g_{\mathrm{dep}}$ [Eq. (1)], the far-field angular phase function [$p_{\text{dep}}(\theta )$] of in-line ${\mathrm{TiO}}_2$ beads was further simulated using the FDTD method. Figure 5 shows the effect of IPS on the angular scattering patterns for mesoporous ${\mathrm{TiO}}_2$ beads with diameters of 20, 150, 300, and 500 nm. In each case, we observe adjusting the value of $P$ ($P=\mathrm{IPS}+\mathrm{D}$), where $P$ is the period for in-line ${\mathrm{TiO}}_2$ beads, which can be defined as the sum of IPS and the diameter of ${\mathrm{TiO}}_2$ bead. Changing of $P$ induces variation on the phase function [$p_{\text{dep}}(\theta )$] [Figs. 5(a)–5(d)], including the contribution ratio between backward and forward scatterings, and the appearance of side-lobes. Apart from the characteristic nature of Mie scattering [$p_{\text{Mie}}(\theta )$], shown for each diameter size of ${\mathrm{TiO}}_2$ bead, the incident wave also leads to scattering field interference through the propagation path.

Fig. 5

Far-field angular scattering patterns [phase function $p_{\text{dep}}(\theta )$] against variable IPS from 0, 250, 500, 750, and 1000 nm of ${\mathrm{TiO}}_2$ beads for different diameters of (a) 20, (b) 150, (c) 300, and (d) 500 nm.

For the 150-nm-diameter ${\mathrm{TiO}}_2$ beads, the appearance of side-lobes can be found as the IPS is increased beyond 750 nm. When IPS is smaller than 500 nm, the scattering patterns are similar to Mie scattering ones; these can be maintained except for the main-lobe energy leakage with increasing IPS, without diffraction associated side-lobes. The side-lobes can be detected at scattering angles of 52 deg and 128 deg. The two lobes are symmetric with respect to the main lobe, and when reaching the condition of $P>{\lambda }_0/2$, the interference phenomenon induced by phase delay between adjacent ${\mathrm{TiO}}_2$ beads [see Fig. 4(b)] produces a grating diffraction effect.^29,30 The interference pattern can be likened to a multiple-slit interference pattern from the geometric arrangement of in-line ${\mathrm{TiO}}_2$ beads [Fig. 4(a)], by analogizing the diameter of ${\mathrm{TiO}}_2$ beads as the width of the slit, and the ${\mathrm{TiO}}_2$ bead periodicity as the grating period. The detected intensity was simulated by the function which can be expressed³¹ as $I=I_{0}P(\theta )[\frac{{\sin }^2(\alpha )}{{\alpha }^2}]{\cos }^2(\beta )$, where $I_0$ is the intensity of incident wave, $\alpha$ is the signal of single slit diffraction, where $\alpha =\frac{1}{2}kD\sin \theta$, and $\beta$ is the signal of the wave interference by phase delay, where $\beta =\frac{1}{2}kP\sin \theta$. The minor side-lobes can be approximated by using the above equations at the angles of 48.1 deg and 131.4 deg with some small, but acceptable within margin, amount of errors, as compared with the data from the FDTD method. As shown in Figs. 6(a)–6(d), the $g_{\text{dep}}$ factors of ${\mathrm{TiO}}_2$ beads with different diameters from 20 to 500 nm versus the IPS from 0 to 1000 nm were simulated and calculated by the FDTD method, which integrated the phase function overall the scattering angles. In addition, the curves as the function of IPS for these samples were fitted with the B-spline function.³² With the experimental volume fraction condition of 0.03, the corresponding $g_{\text{dep}}$ values extracted from the curves in Figs. 6(a)–6(d) are 0.0025, 0.1685, 0.3181, and 0.3866 for ${\mathrm{TiO}}_2$ bead diameters of 20, 150, 300, and 500 nm, respectively. Introducing the specific IPS values of 25, 187, 373, and 623 nm for ${\mathrm{TiO}}_2$ diameters between 20 and 500 nm with the volume fraction of 0.03 into FDTD method, shows the slight difference within 5% error deviation as compared to the curve of $g_{\text{dep}}$ as simulated in Figs. 6(a)–6(d). Therefore, the curves of $g_{\text{dep}}$ versus IPS can be further used in scattering analysis, and the scattering coefficients (${\mu }_{\mathrm{s}}$) of different samples can be obtained by inserting the $g_{\text{dep}}$ factors into EHF model. Therefore, the trend of IPS-dependent $g_{\text{dep}}$ for different diameters of ${\mathrm{TiO}}_2$ beads can be determined. As illustrated in Figs. 6(a) and 6(b), the ${\mathrm{TiO}}_2$ with smaller diameters for 20 and 150 nm show $g_{\text{dep}}$ factor results similar to those from the Mie scattering effect, which implies that the variation of IPS affects weakly for smaller ${\mathrm{TiO}}_2$ beads. However, when increasing the size of ${\mathrm{TiO}}_2$ up to 300 and 500 nm, the IPS-dependent $g_{\text{dep}}$ factor deviates from the Mie scattering effect due to the diffraction and interference phenomena between adjacent ${\mathrm{TiO}}_2$ beads. With this, an approximation of the anisotropy factor ($g_{\text{dep}}$) of each ${\mathrm{TiO}}_2$ bead sample can be made using the OCT system, and can be further introduced into the EHF theory to find the most fitted scattering coefficient (${\mu }_{\mathrm{s}}$).

Fig. 6

The curves of simulated anisotropy factors ($g_{\text{dep}}$) as a function of variable IPS with the point at IPS of 0, 250, 500, 750, and 1000 nm for ${\mathrm{TiO}}_2$ beads with different diameters of (a) 20, (b) 150, (c) 300, and (d) 500 nm. The dash lines are the $g$ factors calculated based on a Mie scattering model, and the red star shows the $g$ factor of experimental condition with an error value of 5% as compared to the value on the curve.