1.

Introduction

Nonlinear optics has been proved to be a powerful tool for biomedical studies.¹ Since its first demonstration by Denk ² in 1990, two-photon fluorescence microscopic imaging has been widely applied to a variety of biological imaging tasks in aspects of physiology, morphology, and cell-cell interactions, etc.^{1, 2, 3} More recently, nonlinear microscopies based on radiative second-harmonic generation (SHG) have dramatically emerged as a viable microscope imaging contrast mechanism for visualization of cell and tissue structures and functions.^{4, 5, 6, 7, 8, 9}

The nonlinear optical effect known as SHG was recognized in the earliest days of laser physics. Two decades ago, it was successfully demonstrated through a microscope by Gannaway and Sheppard¹⁰ and was first applied¹¹ in biological tissue in 1986. SHG is commonly called frequency doubling and the second-harmonic light emerging from the material is at precisely half the wavelength of the light entering the material.

SHG has been widely investigated both experimentally and theoretically. Second-harmonic signals from collagen fibrils have been quantitatively explored based on theoretical physical models^{12, 13, 14} and the biophysical features characteristic of collagen structures can be resolved from the experimental second-harmonic microscopic images on the basis of the SHG phase-matching fundamental principles.¹⁵ SHG from both single and arrays of spherical particles have been theoretically explored or experimentally observed.^{16, 17, 18, 19, 20} Meanwhile, the second-harmonic (SH) cross section, as well as the average cosine of the SH scattering angle is theoretically calculated in a dilute suspension of spherical particles confined within a very thin slab²¹ (the thickness of the slab $d⪡\text{the}$ scattering length $l$ of the turbid medium). The angle-resolved pattern of the SH light scattering from suspensions of different sizes of centrosymmetric microsize polystyrene spheres with surface-absorbed dye (malachite green) has been experimentally measured.²² However, as we have noticed, all the researchers have focused on the generation mechanism study of SH signals in either an isolate or a suspension of particles. The scattering effect of the turbid medium itself in which the SH signals generate and go through has not been paid much more attention. As our intension is directed to the SH microscopic imaging studies in the biological tissue, a kind of highly scattered media, the studies of the SH signals through a turbid medium becomes absolutely necessary.

Because biological tissue is usually composed of scatters with a size variation from nanometers to a few micrometers,²³ the dominant effect caused by these scatterers is Mie scattering. To understand the effect of multiple scattering in multiphoton fluorescence microscopy, one usually uses the Monte Carlo simulation method based on Mie scattering because the conventional image theory based on Fourier optics is not applicable.^{24, 25, 26} So far the established Monte Carlo method in multiphoton microscopy is in fluorescence applications.^{27, 28, 29} This paper establishes a Monte Carlo simulation model for SH microscopic imaging through turbid media.

The paper is organized as follows. Section 2 describes the simulation model of the SH microscopic imaging through a turbid medium. In Sec. 3, we demonstrate the simulation results of the angle-resolved SH signal distribution and the signal level $\eta$ under different constraints of the scattered numbers $\left(n_s\right)$ . The effect of the parameters such as the thickness $d$ of the turbid media, the numerical aperture (NA) of the objective, and the particles size $\left(\rho \right)$ that form the turbid medium are investigated. Discussion and conclusions are presented in Sec. 4.

2.

Simulation Model for SH Microscopic Imaging through a Turbid Medium

Figure 1 shows the schematic diagram for SH microscopic imaging through turbid media in a transmission mode. It is supposed that the excitation light is focused through an imaging objective with $\mathrm{NA}=n_{\omega }\sin \varphi$ on to a focal plane. An SH signal of a certain strength and direction within the focal spot are generated and then propagate through a thickness $d$ in the turbid medium. The focal depth $f_d$ is the distance between the turbid medium’s surface and the focal plane. In this paper, we choose $f_d\approx 0$ (imaging near the surface) to avoid the influence coming from the polarization change of the excitation light due to scattering. The turbid medium is supposed to be a slab formed by microsize scattering particles $\left(\rho \right)$ .

Fig. 1

Schematic diagram of imaging through a turbid medium in a transmission-mode SH microscope, where $f_d$ is the focal depth, and $d$ is the thickness of the turbid medium. The coordinate system defining the SHG emission direction is also shown. The shaded part demonstrates the membrane surface, which can be approximated to be planar at the length scales considered. The focused excitation beam propagates in the $+z$ direction. The resultant SHG signal is mostly confined to an interaction area schematically depicted as a dotted ellipsoid and radiates in the directions defined by $\theta$ and $\phi$ .

The Monte Carlo simulation method for multiphoton fluorescence microscopic imaging through the turbid medium is adopted and the basic principle is the same as that described elsewhere.^{27, 28, 29} However, for SH microscopic imaging, the implementation is much more complicated than that in fluorescence imaging. Unlike the process of two-photon-excited fluorescence (TPF), an SHG process is a coherent scattering process, in which the scattered photons must satisfy a phase-matching constraint, and thus produces highly directed radiation rather than isotropic emission.

According to a simplified biophysical model, the total power distribution of SHG emitted from the cell membrane labeled by Di-6ASPBS styryl dye is (the details can be found in the Appendix³⁰):

Second, the simulation starts from the beginning. This time, for each excitation photon, a direction $(\theta ,\phi )$ is produced according to $\theta =\pi \zeta$ , $\phi =2\pi \zeta$ (where $\zeta$ is a uniformly distributed random number) and SH signals will generate with the weighting factor $g_p⟨I_{\omega }\left(t\right)⟩∕\left(R\tau \right)$ and the efficiency factor ${\alpha }_{\mathrm{SHG}}=1∕2{\Theta }_y(\theta ,\phi )N^2{\sigma }_{\mathrm{SHG}}$ at that direction $(\theta ,\phi )$ . Once it is generated, the propagation of the SH light follows the same way as that detailed in Ref. 29 for TPF.

The excitation wavelength is assumed to be $800\mathrm{nm}$ and the SH emitted is therefore assumed to be $400\mathrm{nm}$ . The excitation power after the objective is $10\mathrm{mw}$ . The collective NA of the objective in this transmission model is supposed to be always the same as that of the excitation objective. According to the Mie scattering theory,³¹ the corresponding parameters of anisotropy value $\left(g\right)$ , the scattering cross section $\left({\sigma }_i\right)$ and the scattering mean-free-path length $\left(l\right)$ of the scatterers (at a concentration of $0.5∕\mu {\mathrm{m}}^3$ ) in the simulated turbid medium are shown in Table 1 .

Table 1

Scattering parameters of the turbid medium under 2p excitation.

The concentration of the molecular surface density $N_s$ is assumed to be ${10}^5∕\mu {\mathrm{m}}^2$ and the first hyperpolarizability of the dye molecule is $\beta \approx 1\times {10}^{-47}\mathrm{C}{\mathrm{m}}^3{\mathrm{V}}^{-2}$ .

Two indicators, the angle-resolved SH signal distribution on the emitted plane along the $x\text{-}y$ direction and the signal level $\eta$ collected immediately after the emitted plane, are adopted in this paper. The signal level $\left(\eta \right)$ is defined as the number of SH photons collected by the detector under different maximum scattering number limits $\left(n_s\right)$ and normalized by the number when no scattering exists (or the thickness $d\approx 0.0\mu \mathrm{m}$ ). Here, we should clarify that we use the limit of $n_s⩽5$ to implicate that the SH signals are dominated by ballistic photons, which means the basic features of the photon, such as the polarization, the coherence, etc., are kept at a maximum degree; the limit of $n_s⩽10$ includes the effect from both ballistic and snakes SH photons; while $n_s⩽300$ means all the effects including the multiple scattered photons are considered.

3.

Angle-Resolved SH Signals and Signal Level $\eta$ through Turbid Media

3.1.

Effect of the Thickness $d$ of the Turbid Media

The thickness $d$ of the turbid medium is important when we decide to explore the signal changes through a turbid medium as it directly influences the number of the scattering events, thus inducing the direction change of the photons and the collection efficiency. Therefore, in this paper, the effect of the thickness $d$ is first investigated.

Figures 2a and 2b show the angle-resolved distributions of the total SH signals $(n_s⩽300)$ under $\mathrm{NA}=1.2$ in a turbid medium formed by the scattering particle size of $\rho =0.6\mu \mathrm{m}$ under $d\approx 0.0\mu \mathrm{m}$ and $d=30.0\mu \mathrm{m}$ , respectively. From the comparison of the Figs. 2a and 2b, we can see that as the $d$ increases, the angle distribution of the SH signals becomes disperse along both the $x$ and $y$ directions. However, the lobe effect induced by the Gouy shift³¹ becomes unobvious because of the multiple scattering.

Fig. 2

Angle-resolved distributions of the SH signals under $\mathrm{NA}=1.2$ in the turbid medium formed by scattering particles of a size of $\rho =0.6\mu \mathrm{m}$ at (a) $d\approx 0.0\mu \mathrm{m}$ and (b) $d=30.0\mu \mathrm{m}$ .

Figure 3 shows the change of the signal level $\eta$ as the increase of the thickness $d$ under different maximum scattering number limits $\left(n_s\right)$ , respectively. It is shown that as the $d$ increases, the collected SH signals decrease under all three conditions. In practice, we are much more interested in the SH signals that are composed of ballistic photons (the solid line in Fig. 3 presents for the ballistic SH signal level $\eta$ versus $d$ ), since the fewer scattering events they experience, the less carried information they lose. It is shown that as the thickness $d$ reaches $30\mu \mathrm{m}$ , the SH signals made up of ballistic photons almost disappear. In another words, the penetration limit of ballistic SH signals is $30\mu \mathrm{m}$ under the condition described. After an approximate penetration depth of $10\mu \mathrm{m}$ , the signal level $\eta$ drops to half the height of the total signals.

Fig. 3

Signal level $\eta$ of the SH signals under $\mathrm{NA}=1.2$ in the turbid medium formed by the scattering particles with size of $\rho =0.6\mu \mathrm{m}$ along the thickness $d$ of the turbid medium.

3.2.

Effect of the Objective NA

The size of the objective NA is proved to play a role in multiphoton fluorescence microscopic imaging.³² In this paper, its effect on the SH microscopic imaging is thus investigated.

Figures 4a and 4b demonstrate the angle distribution of the total collected SH signal $(n_s⩽300)$ under $\mathrm{NA}=0.6$ under thicknesses $d\approx 0.0\mu \mathrm{m}$ and $d=30.0\mu \mathrm{m}$ , respectively. It is clear from a comparison of Fig. 4 with Fig. 2 that NA has a significant effect on both the angle distribution and the signal strength of SH signals. As NA becomes smaller, the angle distribution becomes narrower, while many more SH signals are collected. According to a relationship³¹ of ${\theta }_{\text{peak}}∕{\theta }_{\mathrm{NA}}$ to NA, where ${\theta }_{\mathrm{NA}}={\sin }^{-1}(\mathrm{NA}∕n_{\omega })$ , in our situation, when $\mathrm{NA}=1.2$ and $\mathrm{NA}=0.6$ are used, the peak angle of the lobes of SH emission will locate at ${\theta }_{\text{peak}}\approx 50\mathrm{deg}$ and $\phi \approx \pi ∕2$ and ${\theta }_{\text{peak}}\approx 20\mathrm{deg}$ and $\phi \approx \pi ∕2$ , respectively. This provides the explanation for why at a low NA, the distribution of the ballistic SH signals are narrow (Fig. 4). Meanwhile, because ${\omega }_z$ and ${\omega }_x$ are¹ inversely proportional to NA, as the NA becomes low, the effective total number of molecules $N=(\pi ∕2){\omega }_z{\omega }_x{N}_s$ that contribute to the generation of SH light becomes large. Therefore, the power of SH light becomes strong (Fig. 4). Figure 5 shows the corresponding signal level $\eta$ versus $d$ under three scattering number constraints. We notice that NA has no obvious effect on the penetration limit regarding the ballistic SH signals if the same turbid medium is provided (see solid lines in Figs. 3 and 5). However, it does influence the collected snake and multiple scattered SH signals, which can be realized from a comparison of the dotted and the dashed lines in Figs. 3 and 5. For a small NA (Fig. 5), the collected snake and multiple scattered SH signals have a greater rate of decrease as $d$ increases than that for a large NA, which means a small NA has a suppression function to snake and multiple scattered SH signals.

Fig. 4

Angle-resolved distributions of the SH signals under $\mathrm{NA}=0.6$ in the turbid medium formed by scattering particles of a size of $\rho =0.6\mu \mathrm{m}$ at (a) $d\approx 0.0\mu \mathrm{m}$ and (b) $d=30.0\mu \mathrm{m}$ .

Fig. 5

Signal level $\eta$ of the SH signals under $\mathrm{NA}=0.6$ in the turbid medium formed by the scattering particles with a size of $\rho =0.6\mu \mathrm{m}$ along the thickness $d$ of the turbid medium.

3.3.

Size Effect of the Spherical Particles $\left(\rho \right)$ in the Turbid Medium

The size effect of the spherical particles $\rho$ that form the turbid medium is also part our investigation scope in this paper, since different scatterer sizes have different scattering cross sections $\left({\sigma }_s\right)$ and anisotropy values $\left(g\right)$ according to Mie scattering, which may directly influence the behavior of SH light during the turbid medium. In this paper, two sizes of particles are chosen as our objects. One is the larger one $\left(L\right)$ with a diameter of $0.6\mu \mathrm{m}$ and the other is the smaller one $\left(S\right)$ with a diameter of $0.3\mu \mathrm{m}$ .

Figures 6a and 6b demonstrate the total angle-distribution of SH signals $(n_s⩽300)$ under thicknesses of $d\approx 0.0\mu \mathrm{m}$ and $d=30.0\mu \mathrm{m}$ , respectively. It is shown that the angle-distribution has no obvious difference under thicknesses of $d=30.0\mu \mathrm{m}$ compared with that for $d\approx 0.0\mu \mathrm{m}$ . Figure 7 is the signal level $\eta$ versus $d$ under three scattering number $\left(n_s\right)$ constraints. We noticed that the signal level $\eta$ under three scattering number $\left(n_s\right)$ constraints also has a slight difference along the whole thickness $d$ and the drop of the signal level $\eta$ is very low. This is because the small scattering particles have a much smaller cross section $\left({\sigma }_s\right)$ than that of large particles (Table 1). Under the same concentration, the mean-free path length $\left(l\right)$ is much longer than that of large particles, which leads to fewer scattering events at a given thickness $d$ . Thus, the penetration limit is much deeper than that in turbid medium with large scattering particles.

Fig. 6

Angle-resolved distributions of the SH signals under $\mathrm{NA}=1.2$ in the turbid medium formed by scattering particles of a size of $\rho =0.3\mu \mathrm{m}$ at (a) $d\approx 0.0\mu \mathrm{m}$ and (b) $d=30.0\mu \mathrm{m}$ .

Fig. 7

Signal level $\eta$ of the SH signals under $\mathrm{NA}=1.2$ in the turbid medium formed by the scattering particles with a size of $\rho =0.3\mu \mathrm{m}$ along the thickness $d$ of the turbid medium.

4.

Discussion and Conclusion

The fluorescence signals to the excitation intensity $I_{\omega }$ have the relationship of $p_n={\alpha }_n{I}_{\omega }^n$ ( $n=1,2,3$ corresponding to $1p$ , $2p$ , and $3p$ excitation). In our previous investigation, the model of multiphoton fluorescence microscopic imaging through the turbid medium based on the Monte Carlo simulation method was successfully established,^{27, 28, 29} where fluorescence photons are generated by a uniform random generator with the weighting factor of $I_{\omega }$ and the fluorescence efficiency factor ${\alpha }_n$ . In the case of an SH microscopic imaging condition, the phase matching between the SHG and the excitation fields largely prevents SHG propagation from the forward direction, and forces it to propagate off-axis to form two well-defined lobes. In this case, the SH photons generated are no longer as uniformly randomized as these of fluorescence photons, instead, the probability of the generation of the SH photons in each direction are determined based³¹ on the angular structure parameter ${\Theta }_y(\theta ,\phi )$ . Therefore, ${\alpha }_{\mathrm{SHG}}=0.5{\Theta }_y(\theta ,\phi )N^2{\sigma }_{\mathrm{SHG}}$ at direction $(\theta ,\phi )$ as the efficiency factor is implemented by Monte Carlo simulation to get the SH photons.

To efficiently demonstrate our model, in this paper, we simplified our condition of $f_d\approx 0.0\mu \mathrm{m}$ . As imaging at a nonzero focal depth $f_d$ in the turbid medium, the effect of the polarization change of the excitation beam due to the scattering must be considered. This may greatly increase the complication of the problem, and it becomes another interesting topic calling for further investigation.

To summarize, we demonstrated the simulation model based on the Monte Carlo method for dealing with SH signals through a turbid medium for the first time. The angle-resolved distribution and signal level $\eta$ under various conditions were explored. Simulation results reveal that the use of the small objective NA results in a narrow angle distribution and strong SH signals. The turbid medium composed of large scattering particles has a strong influence on the angle distribution and the signal level $\eta$ of SH light, which results in a low penetration limit in the turbid medium for ballistic SH signals, which is approximate $30\mu \mathrm{m}$ .

5.

Appendix

It is assumed that SHG is produced from Di-6ASPBS styryl dye molecules labeled on the cell’s membrane surface. A tightly focused driving field under a microscope is polarized in the $\widehat{y}$ direction and the molecules in the membrane are perfectly oriented in the $\widehat{y}$ direction and strictly uniaxial.

Under this condition, we can assume that the amplitude of the excitation light field with frequency $\omega$ (wavelength $\lambda$ ) has a Gaussian profile in both the axial and the lateral directions about the focal center and that its phase near the focal center progresses linearly. That is,³⁰

According to Ref. 30, the total power distribution of SHG is