Structural length-scale sensitivities of diffuse reflectance in continuous random media under the Born approximation

Abstract

Which range of structures contribute to light scattering in a continuous random media such as biological tissue? In this Letter, we present a model to study the structural length-scale sensitivity of scattering in continuous random media under the Born approximation. The scattering coefficient μ_s, backscattering coefficient μ_b, anisotropy factor g, and reduced scattering coefficient ${\mu }_s^{\ast }$ as well as the shape of the diffuse reflectance profile are calculated under this model. For media with a biologically relevant Henyey-Greenstein phase function with g ~ 0.93 at wavelength λ = 633 nm, we report that ${\mu }_s^{\ast }$ is sensitive to length-scales from 46.9 nm to 2.07 μm (i.e. λ/13 to 3λ), μ_b is sensitive from 26.7 nm to 320 nm (i.e. λ/24 to λ/2), and the diffuse reflectance profile is sensitive from 30.8 nm to 2.71 μm (i.e. λ/21 to 4λ).

Elastic light scattering provides a valuable tool to detect and quantify sub-diffractional structures even if they cannot be resolved by a conventional imaging system. However, the limits of the sensitivity of light scattering to different structural length-scales in a continuous random media (e.g. biological tissue) have not yet been fully studied. In this Letter, we present the methodologies used to study the length-scale sensitivities of the scattering parameters μ_s, μ_b, g, and ${\mu }_s^{\ast }$ as well as the diffuse reflectance profile in continuous random media.

Consider a statistically homogeneous random medium composed of a continuous distribution of fluctuating refractive index, n(r⃗). We define the excess refractive index which contributes to scattering as n_Δ(r⃗) = n(r⃗)/n_o − 1, where n_o is the mean refractive index. Since n_Δ(r⃗) is a random process, it is mathematically useful to describe the distribution of refractive index through its statistical autocorrelation function B_n(r_d) = ∫n_Δ(r⃗)n_Δ(r⃗ − r_d) dr⃗.

One versatile model for B_n(r_d) employs the Whittle-Matérn family of correlation functions [1, 2]:

$$B_n(r_d)=A_n·{\left(\frac{r_d}{l_c}\right)}^{\frac{D-3}{2}}·K_{\frac{D-3}{2}}\left(\frac{r_d}{l_c}\right),$$ (1)

where K_ν (·) is the modified Bessel function of the second kind with order ν, l_c is the characteristic length of heterogeneity, A_n is the fluctuation strength, and D determines the shape of the distribution (e.g. Gaussian as D → ∞, decaying exponential for D = 4, and power law for D < 3). Importantly, when D = 3 this model predicts a scattering phase function which is identical to the commonly used Henyey-Greenstein model.

All light scattering characteristics can be expressed through the power spectral density Φ_s. Under the Born approximation, Φ_s is the Fourier transform of B_n [2,3]:

$${\mathrm{\Phi }}_s(k_s)=\frac{A_n{l}_c^3\mathrm{\Gamma }(\frac{D}{2})}{{\pi }^{3/2}{2}^{(5-D)/2}}·{(1+k_s^2{l}_c^2)}^{-D/2},$$ (2)

where $k_s=2ksin\frac{\theta }{2}$ and k is the wavenumber.

In order to study the sensitivity of scattering to short length-scales (lower length-scale analysis), we perturb n_Δ(r⃗) by convolving with a three dimensional Gaussian:

$$G(\overrightarrow{r})={\left(\frac{16\pi ln(2)}{W^2}\right)}^{3/2}·exp\left(\frac{-4ln(2)}{W^2}{\overrightarrow{r}}^2\right),$$ (3)

where W is the full width at half maximum. Conceptually, G(r⃗) represents a process which modifies the original medium by removing ‘particles’ smaller than W. Using the convolution theorem, this modified medium can be expressed as $n_{\mathrm{\Delta }}^l(\overrightarrow{r})={\mathcal{F}}^{-1}[\mathcal{F}[n_{\mathrm{\Delta }}(\overrightarrow{r})]·\mathcal{F}[G(\overrightarrow{r})]]$, where indicates the Fourier transform operation and the superscript l indicates that lower frequencies are retained.

The autocorrelation of $n_{\mathrm{\Delta }}^l(\overrightarrow{r})$ can then be found as:

$$\begin{array}{l}{B}_n^l(r_d)={\mathcal{F}}^{-1}[{\mid \mathcal{F}[n_{\mathrm{\Delta }}(\overrightarrow{r})]\mid }^2·{\mid \mathcal{F}[G(\overrightarrow{r})]\mid }^2]\\ =4\pi {\int }_0^{\infty }{\mathrm{\Phi }}_s^l(k_s)\frac{k_{s}sin(k_{s}r)}{r}{dk}_s,\end{array}$$ (4)

where ${\mathrm{\Phi }}_s^l(k_s)$ is the power spectral density for $n_{\mathrm{\Delta }}^l(\overrightarrow{r})$ and can be computed as:

$${\mathrm{\Phi }}_s^l(k_s)=\frac{A_n{l}_c^3\mathrm{\Gamma }(\frac{D}{2})}{{\pi }^{3/2}{2}^{(5-D)/2}}·\frac{exp\left(\frac{-k_s^2{W}_l^2}{8ln(2)}\right)}{{(1+k_s^2{l}_c^2)}^{D/2}}.$$ (5)

We note that Eq. 4 has no closed form solution, but can be evaluated numerically.

Figure 1 demonstrates the functions described by Eqs. 4 and 5 for varying values of W_l using a $B_n^l(r_d)$ with D = 3, l_c = 1 μm, and wavelength λ = 633 nm. This corresponds to a biologically relevant Henyey-Greenstein function with anisotropy factor g ~ 0.93. For increasing W_l, $B_n^l(r_d)$ shows a decreasing value at short length-scales (Fig. 1a). The point at which $B_n^l(r_d)$ deviates from the original B_n(r_d) corresponds roughly to the value of W_l. The lower value of $B_n^l(r_d)$ at short length-scales corresponds to decreased intensity of ${\mathrm{\Phi }}_s^l(k_s)$ at higher spatial frequencies after Fourier transformation (Fig. 1b). To study the sensitivity of scattering to large length-scales (upper length-scale analysis), we employ the same model as above but filter larger particles by evaluating $n_{\mathrm{\Delta }}^h(\overrightarrow{r})={\mathcal{F}}^{-1}[\mathcal{F}[n_{\mathrm{\Delta }}(\overrightarrow{r})]·(1-\mathcal{F}[G(\overrightarrow{r})])]$, where the superscript h indicates that higher frequencies are retained. The autocorrelation of $n_{\mathrm{\Delta }}^h(\overrightarrow{r})$ can then be found as:

$$\begin{array}{l}{B}_n^h(r_d)={\mathcal{F}}^{-1}[{\mid \mathcal{F}[n_{\mathrm{\Delta }}(\overline{r})]\mid }^2·{\mid 1-\mathcal{F}[G(\overrightarrow{r})]\mid }^2]\\ =4\pi {\int }_0^{\infty }{\mathrm{\Phi }}_s^h(k_s)\frac{k_{s}sin(k_{s}r)}{r}{dk}_s,\end{array}$$ (6)

where

Fig. 1.

Lower length-scale analysis for W_l = 0, 10, 50, and 100 nm with D = 3, l_c = 1 μm, and λ = 633 nm. The normalized (a) $B_n^l(r_d)$ and (b) ${\mathrm{\Phi }}_s^l(k_s)$. In each panel the arrow indicates increasing W_l.

$${\mathrm{\Phi }}_s^h(k_s)=\frac{A_n{l}_c^3\mathrm{\Gamma }(\frac{D}{2})}{{\pi }^{3/2}{2}^{(5-D)/2}}·\frac{{\left(1-exp\left(\frac{-k_s^2{W}_h^2}{16ln(2)}\right)\right)}^2}{{(1+k_s^2{l}_c^2)}^{D/2}}.$$ (7)

Figure 2 shows the functions described by Eqs. 6 and 7. For decreasing W_h, $B_n^h(r_d)$ exhibits a decrease at larger length-scales (Fig. 2a). These alterations lead to a decreased intensity of ${\mathrm{\Phi }}_s^h(k_s)$ at lower spatial frequencies (Fig. 2b). As a way to visualize the continuous media represented by the above equations, Fig. 3 provides example cross sectional slices through n_Δ(r⃗), $n_{\mathrm{\Delta }}^l(\overrightarrow{r})$, and $n_{\mathrm{\Delta }}^h(\overrightarrow{r})$ for D = 3, l_c =1μm, and W_l = W_h = 100 nm.

Fig. 2.

Upper length-scale analysis for W_h = ∞, 10, 5, and 1 μm with D = 3, l_c = 1 μm, and λ = 633 nm. (a) $B_n^h(r_d)$ where the dashed lines indicate locations in which the curve is negative. (b) ${\mathrm{\Phi }}_s^h(k_s)$. In each panel the arrow indicates decreasing W_h.

Fig. 3.

Example media with D = 3 and l_c = 1 μm. (a) n_Δ(r⃗). (b) $n_{\mathrm{\Delta }}^l(\overrightarrow{r})$ and (c) $n_{\mathrm{\Delta }}^h(\overrightarrow{r})$ for W_l = W_h = 100 nm.

Implementing the above methods, we now define a number of measurable scattering quantities. First, the differential scattering cross section per unit volume for unpolarized light σ(θ), can be found by incorporating the dipole scattering pattern into Φ_s(k_s):

$$\sigma (\theta )=2\pi k^4(1+{cos}^2\theta ){\mathrm{\Phi }}_s(k_s).$$ (8)

The shape of σ(θ) can be parameterized by the scattering coefficient μ_s, the backscattering coefficient μ_b, and g [4]:

$${\mu }_s=2\pi {\int }_{-1}^1\sigma (cos\theta )dcos\theta ,$$ (9)

$${\mu }_b=4\pi ·\sigma (\theta =\pi ),$$ (10)

$$g=\frac{2\pi }{{\mu }_s}{\int }_{-1}^{1}cos\theta ·\sigma (cos\theta )dcos\theta$$ (11)

Conceptually, μ_s is the total scattered power per unit volume, μ_b represents the power scattered in the back-ward direction per unit volume, and g describes how forward directed the scattering is. Finally, the effective transport in a multiple scattering medium is expressed by the reduced scattering coefficient ${\mu }_s^{\ast }={\mu }_s·(1-g)$.

Figure 4a shows percent changes in the above scattering parameters under the lower length-scale analysis for a $B_n^l(r_d)$ with D = 3, l_c = 1 μm, and λ = 633 nm. With increasing W_l, each parameter decreases from its original value. For μ_s, the decrease occurs because scattering material is removed from the medium. For 1 − g and μ_b, the decrease occurs as a result of reduced backscattering (see Fig. 1b). For ${\mu }_s^{\ast }$, the decrease is a combination of the previous two effects.

Fig. 4.

Percent change in scattering parameters with varying values of W_l and W_h for D = 3, l_c = 1 μm, and λ = 633 nm. (a) Lower and (b) upper length-scale percent changes. The dotted line indicates the ± 5 % threshold.

To provide specific length-scale sensitivity quantification, we focus on the parameters most relevant to reflectance measurements: ${\mu }_s^{\ast }$ for samples within the multiple scattering regime and μ_b for samples within the single scattering regime. Defining a 5% threshold (a common significance level in statistics) the minimum length-scale sensitivity (r_min) of ${\mu }_s^{\ast }$ and μ_b equals 46.9 nm (~λ/13) and 26.7 nm (~λ/24), respectively. Thus, measurements of ${\mu }_s^{\ast }$ and μ_b provide sensitivity to structures much smaller than the diffraction limit. Interestingly, r_min is smaller for μ_b than ${\mu }_s^{\ast }$. This can be understood by noting that k_s is maximized in the backscattering direction (i.e. θ = π) and so provides the most sensitivity to alterations of B_n(r_d) at small length-scales (see Fig. 1).

Figure 4b shows percent changes in the scattering parameters under the upper length-scale analysis. With decreasing W_h, μ_s decreases because scattering material is removed from the medium. For 1 − g, an increase occurs due to a reduction in the forward scattering component. Combining these two opposing effects, the maximum length-scale sensitivity (r_max) for ${\mu }_s^{\ast }$ equals 2.07 μm (~ 3λ). For μ_b, a very small value of W_h is needed in order to alter backscattering. As a result, r_max for μ_b is only 320 nm (~λ/2).

In order to study the length-scale sensitivity of the diffuse reflectance profile we performed electric field Monte Carlo simulations of continuous random medium as described in Ref. [5]. Here, we display the distribution measured with unpolarized illumination and collection, P_oo(r). P_oo(r) is the distribution of light that exits a semi-infinite medium anti-parallel to the incident beam and within an annulus of radius r from the entrance point. It is normalized such that ${\int }_0^{\infty }{P}_{oo}(r)dr=1$.

Figure 5a shows P_oo under the lower length-scale analysis for a B_n(r_d) with D = 3, l_c = 1 μm, and λ = 633 nm. With increasing W_l, the value of P_oo is decreased within the subdiffusion regime (i.e. $r·{\mu }_s^{\ast }<1$). This decrease can be attributed in part to the decreased intensity of the phase function in the backscattering direction (see Fig. 1b). For $r·{\mu }_s^{\ast }>1$, a range which is essentially insensitive to the shape of the phase function, P_oo remains largely unchanged. Figure 5b shows similar results for the upper length-scale analysis. In order to perform a sensitivity analysis, we calculate the maximum percent error at any position on P_oo relative to the original case. Applying a 5% threshold once again, we find that r_min = 30.8 nm (~λ/21) and r_max = 2.71 μm (~ 4λ).

Fig. 5.

Monte Carlo simulations of P_oo with D = 3, l_c = 1 μm, and λ = 633 nm. (a) Lower length-scale analysis for W_l = 0, 30, 60, and 90 nmm. Arrow indicates increasing W_l. (b) Upper length-scale analysis for W_h = ∞, 10, 2, 0.5 μm. Arrows indicate decreasing W_h.

Finally, we note that the exact values of r_min and r_max depend on the shape of B_n(r_d). The values given above provide an estimate assuming a correlation function shape which is widely used and accepted for modeling of biological tissue (Henyey-Greenstein). Figure 6 illustrates the dependence of r_min and r_max on the shape of B_n(r_d), assuming the Whittle-Matérn model and using ${\mu }_s^{\ast }$ as an example. As either D or l_c increases, B_n(r_d) shifts relatively more weight to larger length-scales and away from smaller length-scales. As a result, both r_min and r_max increase monotonically with D and l_c.

Fig. 6.

(a) r_min and (b) r_max for ${\mu }_s^{\ast }$ with different shapes of B_n(r_d) and λ = 633 nm.