## Abstract

Which range of structures contributes 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 ${\mu}_{s}$, backscattering coefficient ${\mu}_{b}$, anisotropy factor $g$, and reduced scattering coefficient ${\mu}_{s}^{*}$ as well as the shape of the spatial reflectance profile are calculated under this model. For media with a biologically relevant Henyey–Greenstein phase function with $g\sim 0.93$ at wavelength $\lambda =633\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, we report that ${\mu}_{s}^{*}$ is sensitive to structural length-scales from 46.9 nm to 2.07 μm (i.e., $\lambda /13$ to $3\lambda $), ${\mu}_{b}$ is sensitive from 26.7 to 320 nm (i.e., $\lambda /24$ to $\lambda /2$), and the spatial reflectance profile is sensitive from 30.8 nm to 2.71 μm (i.e., $\lambda /21$ to $4\lambda $).

© 2012 Optical Society of America

Elastic light scattering provides a valuable tool to *detect* and *quantify* subdiffractional 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 ${\mu}_{s}$, ${\mu}_{b}$, $g$, and ${\mu}_{s}^{*}$ 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, $n(\mathrm{r\u20d7})$. We define the excess refractive index which contributes to scattering as ${n}_{\mathrm{\Delta}}(\mathrm{r\u20d7})=n(\mathrm{r\u20d7})/{n}_{o}-1$, where ${n}_{o}$ is the mean refractive index. Since ${n}_{\mathrm{\Delta}}(\mathrm{r\u20d7})$ is a random process, it is mathematically useful to describe the distribution of refractive index through its statistical autocorrelation function ${B}_{n}({r}_{d})=\int {n}_{\mathrm{\Delta}}(\mathrm{r\u20d7}){n}_{\mathrm{\Delta}}(\mathrm{r\u20d7}-{r}_{d})\mathrm{d}\mathrm{r\u20d7}$.

One versatile model for ${B}_{n}({r}_{d})$ employs the Whittle–Matérn family of correlation functions [1,2]:

All light scattering characteristics can be expressed through the power spectral density ${\mathrm{\Phi}}_{s}$. Under the Born approximation, ${\mathrm{\Phi}}_{s}$ is the Fourier transform of ${B}_{n}$ [2,3]:

In order to study the sensitivity of scattering to *short* length-scales (lower length-scale analysis), we perturb ${n}_{\mathrm{\Delta}}(\mathrm{r\u20d7})$ by convolving with a three-dimensional Gaussian:

The autocorrelation of ${n}_{\mathrm{\Delta}}^{l}(\mathrm{r\u20d7})$ can then be found as

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 $D=3$, ${l}_{c}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, and wavelength $\lambda =633\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$. This corresponds to a biologically relevant Henyey–Greenstein function with anisotropy factor $g\sim 0.93$. For increasing ${W}_{l}$, ${B}_{n}^{l}({r}_{d})$ shows a decreasing value at short length-scales [Fig. 1(a)]. 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. 1(b)]. 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 ${n}_{\mathrm{\Delta}}^{h}(\mathrm{r\u20d7})={\mathcal{F}}^{-1}[\mathcal{F}[{n}_{\mathrm{\Delta}}(\mathrm{r\u20d7})]\text{\hspace{0.17em}}\xb7(1-\mathcal{F}[G(\mathrm{r\u20d7})])]$, where the superscript $h$ indicates that higher frequencies are retained. The autocorrelation of ${n}_{\mathrm{\Delta}}^{h}(\mathrm{r\u20d7})$ can then be found as

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. 2(a)]. These alterations lead to a decreased intensity of ${\mathrm{\Phi}}_{s}^{h}({k}_{s})$ at lower spatial frequencies [Fig. 2(b)].

As a way to visualize the continuous media represented by the above equations, Fig. 3 provides example cross-sectional slices through ${n}_{\mathrm{\Delta}}(\mathrm{r\u20d7})$, ${n}_{\mathrm{\Delta}}^{l}(\mathrm{r\u20d7})$, and ${n}_{\mathrm{\Delta}}^{h}(\mathrm{r\u20d7})$ for $D=3$, ${l}_{c}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, and ${W}_{l}={W}_{h}=100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$.

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 $\sigma (\theta )$, can be found by incorporating the dipole scattering pattern into ${\mathrm{\Phi}}_{s}({k}_{s})$:

*backward*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 ${\mu}_{s}^{*}={\mu}_{s}\xb7(1-g)$.

Figure 4(a) shows percent changes in the above scattering parameters under the lower length-scale analysis for a ${B}_{n}^{l}({r}_{d})$ with $D=3$, ${l}_{c}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, and $\lambda =633\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$. With increasing ${W}_{l}$, each parameter decreases from its original value. For ${\mu}_{s}$, the decrease occurs because scattering material is removed from the medium. For $1-g$ and ${\mu}_{b}$, the decrease occurs as a result of reduced backscattering [see Fig. 1(b)]. For ${\mu}_{s}^{*}$, the decrease is a combination of the previous two effects.

To provide specific length-scale sensitivity quantification, we focus on the parameters most relevant to reflectance measurements: ${\mu}_{s}^{*}$ for samples within the multiple scattering regime and ${\mu}_{b}$ for samples within the single scattering regime. Defining a 5% threshold (a common significance level in statistics) the minimum length-scale sensitivity (${r}_{\mathrm{min}}$) of ${\mu}_{s}^{*}$ and ${\mu}_{b}$ equals 46.9 nm ($\sim \lambda /13$) and 26.7 nm ($\sim \lambda /24$), respectively. Thus, measurements of ${\mu}_{s}^{*}$ and ${\mu}_{b}$ provide sensitivity to structures much smaller than the diffraction limit. Interestingly, ${r}_{\mathrm{min}}$ is smaller for ${\mu}_{b}$ than ${\mu}_{s}^{*}$. This can be understood by noting that ${k}_{s}$ is maximized in the backscattering direction (i.e., $\theta =\pi $) and so provides the most sensitivity to alterations of ${B}_{n}({r}_{d})$ at small length-scales (see Fig. 1).

Figure 4(b) shows percent changes in the scattering parameters under the upper length-scale analysis. With decreasing ${W}_{h}$, ${\mu}_{s}$ 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 (${r}_{\mathrm{max}}$) for ${\mu}_{s}^{*}$ equals 2.07 μm ($\sim 3\lambda $). For ${\mu}_{b}$, a very small value of ${W}_{h}$ is needed in order to alter backscattering. As a result, ${r}_{\mathrm{max}}$ for ${\mu}_{b}$ is only 320 nm ($\sim \lambda /2$).

In order to study the length-scale sensitivity of the spatial reflectance profile we performed electric field Monte Carlo simulations of continuous random medium as described in [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 antiparallel to the incident beam and within an annulus of radius $r$ from the entrance point. It is normalized such that ${\int}_{0}^{\infty}{P}_{oo}(r)\mathrm{d}r=1$.

Figure 5(a) shows ${P}_{oo}$ under the lower length-scale analysis for a ${B}_{n}({r}_{d})$ with $D=3$, ${l}_{c}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, and $\lambda =633\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$. With increasing ${W}_{l}$, the value of ${P}_{oo}$ is decreased within the subdiffusion regime (i.e., $r\xb7{\mu}_{s}^{*}<1$). This decrease can be attributed in part to the decreased intensity of the phase function in the backscattering direction [see Fig. 1(b)]. For $r\xb7{\mu}_{s}^{*}>1$, a range that is essentially insensitive to the shape of the phase function, ${P}_{oo}$ remains largely unchanged. Figure 5(b) 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 ${r}_{\mathrm{min}}=30.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ ($\sim \lambda /21$) and ${r}_{\mathrm{max}}=2.71\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ ($\sim 4\lambda $).

Finally, we note that the exact values of ${r}_{\mathrm{min}}$ and ${r}_{\mathrm{max}}$ depend on the shape of ${B}_{n}({r}_{d})$. The values given above provide an estimate assuming a correlation function shape that is widely used and accepted for modeling of biological tissue (Henyey–Greenstein). Figure 6 illustrates the dependence of ${r}_{\mathrm{min}}$ and ${r}_{\mathrm{max}}$ on the shape of ${B}_{n}({r}_{d})$, assuming the Whittle–Matérn model and using ${\mu}_{s}^{*}$ 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 ${r}_{\mathrm{min}}$ and ${r}_{\mathrm{max}}$ increase monotonically with $D$ and ${l}_{c}$.

This study was supported by National Institutes of Health grants RO1CA128641 and R01EB003682. A.J. Radosevich is supported by a National Science Foundation Graduate Research Fellowship under Grant DGE-0824162.

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