Development of a beam propagation method to simulate the point spread function degradation in scattering media

Abstract

Scattering is one of the main issues that limit the imaging depth in deep tissue optical imaging. To characterize the role of scattering, we have developed a forward model based on the beam propagation method and established the link between the macroscopic optical properties of the media and the statistical parameters of the phase masks applied to the wavefront. Using this model, we have analyzed the degradation of the point-spread function of the illumination beam in the transition regime from ballistic to diffusive light transport. Our method provides a wave-optic simulation toolkit to analyze the effects of scattering on image quality degradation in scanning microscopy. Our open-source implementation is available at https://github.com/BUNPC/Beam-Propagation-Method.

The quality of images obtained in scanning microscopy, including confocal and multi-photon microscopy, is fundamentally determined by the point-spread function (PSF) [1]. When imaging thick biological samples, the PSF is a function of the illumination geometry as well as the sample’s scattering properties, which in turn limits the instrument’s resolution, signal-to-noise ratio, and maximum imaging depth. Thus, quantifying the PSF of an illumination beam in the presence of scattering is of great interest for deep tissue scanning microscopy. In the diffusive wave transport regime, the effects of scattering can be well described by the particle-picture-based methods including the diffusion equation and Monte Carlo simulations. However, in the transition regimes (i.e., from ballistic to diffusive wave transport), the PSF has not been fully characterized, which is relevant for most scanning microscopy applications. One effect of scattering is the reduction of the effective numerical aperture (NA) [2]. But the effective NA description based on single scattering breaks down when the anisotropy factor g is large, resulting in multiply scattered photons remaining in the vicinity of the ballistic photons. Another issue is that particle-based light scattering methods fail to capture the small-scale (~ wavelength) features such as the speckle patterns induced by wave interference. An accurate wave-optic forward model that incorporates tissue scattering is desired, which will also spur the advancement of wavefront engineering [3], computational microscopy [4,5], and machine-learning-based techniques [6,7].

Among the various wave-picture-based methods, the beam propagation method (BPM) is relatively fast and simple to implement [8,9]. This method, initially developed for light propagation within fibers [10], has been extended to model wave scattering in biological tissues [11]. Although the BPM has the drawback of ignoring the effects of backscattering, it has been shown to be particularly effective in predicting wavefronts in the image plane for anisotropic scattering media such as biological samples, where forward scattering dominates [5,12–14]. In all these existing efforts, the BPM was used as a heuristic model to capture the net effect of scattering, while lacking an explicit recipe relating the macroscopic scattering properties of the media and the microscopic parameters used in the BPM simulations. This limits the utility of the BPM, since it is often desirable to quantify the effect of scattering on imaging metrics, such as the PSF, with pre-defined scattering properties of the medium. Recently, Yang et al. has made progress by estimating the transport mean free path ℓ* from the asymptotic behavior of the ratio between the direct-current (DC) and alternating-current (AC) components of the transmitted waves for a collimated incident beam [15]. However, ℓ* characterizes scattering only in the diffusive regime, where the direction of a photon is fully randomized relative to its incident direction [16]. In contrast, both the scattering mean free path ℓ_s and the anisotropy parameter g are the primary macroscopic parameters of interest in the ballistic and transition regimes instead of ℓ* = ℓ_s/(1 − g). The relation between these macroscopic parameters of the scattering medium and the BPM parameters has not been fully explored.

In this Letter, we establish a comprehensive relation between the microscopic statistical parameters of the BPM to the macroscopic scattering properties of the medium. We obtain a scaling law for the physical parameters of interest and extend the existing discrete BPM models using a fixed sampling configuration [14,15] to the continuous medium limit. Using this model, we have simulated wave propagation for different imaging depths z, scattering mean free paths ℓ_s, anisotropy factors g, and NAs. Our modeling results are first validated against the analytical solutions for the broadening of the waist of a collimated Gaussian beam due to scattering approximated with the radiative transfer equation (RTE) in the limit of a small NA (NA = 0.1) and a large g (g = 0.99) [17]. PSFs are then obtained for a focused beam with higher NA (NA = 0.5) illumination, relevant for scanning microscopy, under different g and z/ℓ_s, where analytical solutions are not available. We find that in the transition regime, the PSF degrades faster in media with larger g. This is contrary to the common notion that a larger g with a higher probability of forward scattering, which facilitates energy penetration, should provide better PSF quality.

The main idea of the BPM is to model the scattering medium as a series of parallel planar layers of phase masks, as illustrated in Fig. 1(a). At each phase mask plane, the local wavefront is multiplied by a spatially varying random phase term e^iφ(x,y). The medium is assumed to be uniform between neighboring phase masks, and the propagation of the wavefront is computed via the angular spectrum method [8]:

$$E\left(k_x,k_y,z+d\right)=E\left(k_x,k_y,z\right)e^{in\sqrt{k^2-k_x^2-k_y^2}d},$$ (1)

where d is the distance between neighboring phase masks, n is the refractive index, λ is the wavelength, k = 2π/λ is the wave-number, and k_x, k_y are its x, y components, respectively. The angular spectrum representation is

$$E(x,y,z)=\frac{1}{{(2\pi )}^2}\int \int E(k_x,k_y,z)e^{i(k_{x}x+k_{y}y)}\text{d}{k}_x\text{d}{k}_y.$$ (2)

A useful input wavefront profile at z = 0 is the Gaussian beam focused at (0, 0, z₀):

$$E(x,y,0)=e^{-(x^2+y^2)/w^2}{e}^{-ik(x^2+y^2)/[2(z_0+z_R^2/z_0)]},$$ (3)

where w₀ = λ/(πNA) is the beam waist at z₀, $w=w_0\sqrt{1+z_0^2/z_R^2}$, and $z_R=w_0^2/\lambda$ is the Rayleigh range.

Fig. 1.

(a) Illustration of the beam propagation model. (b) Parameters of the phase masks. σ_p is the standard deviation of the seed phase φ_p at each pixel. The seed phase profile is convolved with a spatial Gaussian profile with width σ_x. (c), (d) Examples of phase masks with σ_p = π/10, λ = 500 nm, a = λ/4, σ_x = λ/2 (c), and σ_x = 2λ (d).

The statistical parameters for generating the random phase masks control the scattering properties of the medium. Each phase mask is modeled on a square grid with pixel size a. A seed phase φ_p(x, y) is assigned at each pixel, drawn from a Gaussian distribution with zero mean and standard deviation (std) σ_p, which determines the strength of the global phase variations. Each randomly seeded phase profile is smoothed by convolving with a Gaussian phase bump with a std of σ_x, which determines the spatial extent of the local phase variations. The final random phase profile is re-normalized to keep the same std σ_p before and after convolution [see Fig. 1(b)]. Evanescent components with $\sqrt{k_x^2+k_y^2}>k$ are ignored. Example phase masks with different σ_x are shown in Figs. 1(c) and 1(d). Unlike existing BPM models that use real-space phase masks with the pixel size matching the speckle size [15], we use σ_x to vary the spatial correlation properties of the medium, independent of the pixel size a. Our procedure allows any arbitrary choice of layer distance d, which overcomes the limitation of the sampling parameters in existing discrete models [14,15].

The main features of our model include: (1) σ_p is only a function of the layer distance d and scattering mean free path ℓ_s, in which ℓ_s determines the magnitude of the phase variations; and (2) the anisotropy factor g is controlled by σ_x, since g is related to the spatial correlation of the phase variations.

First, we investigate the relation among ℓ_s, σ_p, and d. Considering a medium with a total thickness z having N = z/d layers of phase masks with separation distance d, we characterize the total accumulated phase variation at z by its std of the wavefront σ_p(z). By modeling the phase distribution as Gaussian random variables and the fact that the std of the sum of N identically distributed random variables is proportional to $\sqrt{N}$ times the std of each variable, we arrive at the following relations: ${\sigma }_p\left(z\right)/{\sigma }_p\left(d\right)=\sqrt{N}=\sqrt{z}/\sqrt{d}$ and ${\sigma }_p\left(z\right)\propto \sqrt{z}$, where σ_p(d) denotes the phase variation induced by a single mask. We further conjecture that after N_s = z/ℓ_s scattering events, the phase is fully randomized with a std of π, i.e., σ_p(N_sℓ_s) = π. Combining this with the previous relations, we obtain ${\sigma }_p\left(z\right)=\sqrt{z/\left(N_s{\ell }_s\right)}\pi$. Replacing z with layer distance d, the relation between σ_p and ℓ_s, d for a single phase mask is

$${\sigma }_p\left(d\right)=\sqrt{d/\left(N_s{\ell }_s\right)}\pi .$$ (4)

The constant N_s is independent of d and ℓ_s, and can be found by matching the input ℓ_s used in Eq. (4) and the fitted ℓ_s obtained from the numerical simulation. Specifically, in particle-picture-based methods for simulating photon migration, such as Monte Carlo methods, ℓ_s is determined by the inverse of the decay rate (i.e., μ_s = 1/ℓ_s, the scattering coefficient) along the photon launching direction of a collimated beam. However, in wave models, estimating the decay rate for a collimated incident beam is complicated by wave diffraction. In addition, multiply scattered photons that fall along the propagation direction will decrease the measured decay rate, which is impacted by the beam size and g [18]. Thus, to estimate ℓ_s without these confounding effects, we instead simulate a plane wave propagating along the z direction. To find the scattering coefficient, we measure the DC component I_DC of the intensity as a function of depth z by taking the k_x = 0, k_y = 0 contribution from the angular spectrum of the wavefront at each depth. Examples of ln(I_DC) versus z are plotted in Fig. 2(a) for different σ_p = π/5, π/10, π/20, with d = 10 μm while keeping σ_x = λ/2 fixed. As expected, the ln(I_DC) versus z curves follow straight lines, whose decay rates μ_s are fitted and found to be 40 mm⁻¹, 10 mm⁻¹, and 2.5 mm⁻¹, respectively. We thus find the constant N_s to be N_s = 10 ≈ π², which matches the input ℓ_s in Eq. (4) with the fitted decay rate ℓ_s = 1/μ_s. With N_s determined, we repeat the same procedure for different values of σ_x and plot the input and measured μ_s in Fig. 2(b). Most importantly, all the results fall on the same line, which further validates Eq. (4) and our conjecture that μ_s is independent of σ_x. We have used λ = 500 nm, n = 1, a = λ/4 in the model to generate all the figures in this paper. The same results are found for n = 1.33, wavelength ranging from 500 nm to 1300 nm, and pixel size from a = λ/10 to λ/4.

Fig. 2.

(a) DC component of the intensity I_DC as a function of depth z for phase mask separation d = 10 μm and σ_p = π/5, σ_p = π/10, σ_p = π/20, with σ_x = λ/2 fixed. The decay rates obtained from fitting are 0.04 μm⁻¹, 0.01μm⁻¹, and 0.0025 μm⁻¹ respectively. (b) Measured decay coefficient μ_s and the input μ_s for various values of σ_x with N_s = 10 in Eq. (4). The black dashed line indicates where the measured μ_s and input μ_s match. (c) Anisotropy factor g as a function of σ_x for different d values with z = ℓ_s fixed.

Next, we characterize the relation among g, σ_x, and σ_p. As an intrinsic property of the scattering medium, the value of g should be independent of the layer thickness d. To achieve this, scattering angles are measured at a depth of single scattering mean free path in the medium, i.e., z = ℓ_s, with an input plane wave. In addition, unlike existing discrete models, our approach ensures that g does not depend on the std of a single mask σ_p, since at z = ℓ_s, the std of the phase distribution is a constant $\sqrt{1/N_s}\pi \approx 1$. The transverse wave-number at (x, y) is obtained from the gradient of the phase of the wavefront ϕ(x, y) as k_x = dϕ(x, y)/dx, k_y = dϕ(x, y)/dy. Accordingly, the scattering angle θ is obtained by $\cos \left(\theta \right)=\sqrt{k^2-k_x^2-k_y^2}/k,$ and the anisotropy factor is computed as g = 〈cos θ〉 [14]. The relation between g and σ_x is plotted in Fig. 2(c) for different layer distances d with the total thickness z = ℓ_s fixed. We see that g is independent of d with z = ℓ_s fixed, as expected. From this curve, one can choose σ_x based on the desired value of g. Note that this calculation of g is biased towards large g values due to the lack of backscattering in the BPM. However, we expect the relation is sufficiently accurate to model wave propagation in biological tissues in the few scattering limits, since the majority of the reported values of g satisfy g > 0.9 [3,18].

With optical properties of scattering media characterized in our BPM model, we now analyze the impact of scattering on the PSF of the illumination beam for scanning microscopy. Wave propagation in scattering media falls into three regimes: the ballistic regime z < ℓ_s, transition regime ℓ_s ≤ z < ℓ*, and diffusive wave transport regime z ≥ ℓ* [19]. Within the ballistic regime, the degradation of the PSF is due mainly to optical aberrations from the focusing optics or slow variations in the refractive index of biological tissues, where the PSF broadening due to scattering is negligible as observed experimentally [20]. In the diffusive regime, the PSF formalism is no longer valid, and the wave scattering properties can be analytically solved using the diffusion equation. The transition regime, where the PSFs are impacted by scattering but the speckle patterns are not yet fully developed, is poorly understood and is of main interest here. In the small scattering angle approximation of the propagation of a Gaussian beam with NA < 0.2 and g > 0.9, the intensity at the focal plane still follows a Gaussian profile expressed as I(x) ~ e^−2x2/w2, as shown in Figs. 3(a) and 3(b). The broadening of the PSF width w has been calculated using the RTE and is shown to be proportional to $\sqrt{z/{\ell }_s}$ when z ≫ ℓ_s [17]. To validate our BPM model, we simulate the beam width w of a Gaussian focus with NA = 0.1, scattering anisotropy factor g = 0.99, z = 100 μm, and various ℓ_s. The results of w versus z/ℓ_s from our model are consistent with the theoretical predictions as shown in Fig. 3(c).

Fig. 3.

(a) The Gaussian intensity profile I(x) normalized by the peak intensity with no scattering I₀ at the focal plane for NA = 0.1, g = 0.99 with z/ℓ_s = 1, 2, 5, each averaged over 20 phase mask configurations. (b) Examples of Gaussian fitting for relatively large z/ℓ_s. The black solid curves are fitting results. The values of w are 37.18, 15.78 μm for z/ℓ_s = 50, 10 respectively. The intensity profile is normalized by the peak intensity I_p. (c) Gaussian waist w obtained from the BPM simulation compared with the calculations using RTE theory.

In scanning microscopy, larger NA values are commonly used, and analytical solutions from the RTE are no longer valid. Using our method, the PSFs can be simulated for a wide range of NA and g values. Results are plotted in Fig. 4 for NA = 0.5, z = 100 μm, and various ℓ_s. Mostly importantly, the PSF can no longer be approximated by a single Gaussian profile, as seen in the example in Fig. 4(a). Significant contributions from the diffuse background are observed and expected to degrade PSF quality. Naively, one would expect that with fixed ℓ_s, the PSF should degrade more slowly for larger g because of the larger transport mean free path ℓ* = ℓ_s/(1 - g) facilitating greater penetration depth. However, as seen in Fig. 4(b), the ratio between the peak intensity I_p and the estimated diffuse background peak level I_db is lower for larger g values. Also, the full width at half-maximum (FWHM) of the PSF is larger for greater g when z/ℓ_s < 20, as shown in Fig. 4(c). The local contrast metric I_p/I_db used here is a proxy to the signal-to-background ratio commonly used to characterize image quality, where I_p/I_db = 2 determines the depth at which the FWHM has a sharp increase. This sharp increase happens at z/ℓ_s ~ 5 for g = 0.99 and z/ℓ_s ~ 13 for g = 0.8, 0.9. At greater depths of z/ℓ_s > 20, the FWHM is dominated by the diffuse background when the I_p/I_db approaches 1. The diffuse background FWHM measured relative to I_db is smaller for larger g as expected, as shown in Fig. 4(d). From the above analysis, we conclude that the PSF quality of the illumination beam degrades faster for larger g in the transition regime measured by both degraded local contrast and resolution.

Fig. 4.

PSF degradation of the illumination beam due to scattering in the transition regime. (a) Example of I(x) for g = 0.9, ℓ_s = 12.5. The peak intensity I_p and diffuse background level I_db are indicated. (b) Ratio of I_p and I_db for g = 0.8, 0.9, 0.99. (c) FWHM of the primary focus and (d) diffuse background FWHM (FWHM_db) for g = 0.8, 0.9, 0.99. All the results are obtained with 50 random configurations to average out speckle fluctuations from wave interference.

In summary, we have established a continuous medium version of the BPM model with the optical scattering properties μ_s and g fully characterized. Using this model, the PSFs for various NA Gaussian beams and sample scattering properties are simulated. We find that the PSF of the illumination beam degrades faster for larger g when the scattering mean free path ℓ_s is fixed in the transition regime from ballistic to diffusive wave transport, contrary to the conventional expectation that greater forward scattering would facilitate light penetration and maintain image quality. The current model works for large g values only when forward scattering dominates. Higher-order moments and other details of the phase function are also ignored, which will be studied in the future by incorporating other forms of power spectral densities using different local phase smoothing functions [21]. Nonetheless, we expect that this model will be useful to guide further developments of optical microscopy techniques to overcome scattering such as non-degenerate two-photon microscopy [22], and three-photon microscopy [23].