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, $n(\mathrm{r⃗})$. We define the excess refractive index which contributes to scattering as $n_{\mathrm{\Delta }}(\mathrm{r⃗})=n(\mathrm{r⃗})/n_o-1$, where $n_o$ is the mean refractive index. Since $n_{\mathrm{\Delta }}(\mathrm{r⃗})$ is a random process, it is mathematically useful to describe the distribution of refractive index through its statistical autocorrelation function $B_n(r_d)=\int n_{\mathrm{\Delta }}(\mathrm{r⃗})n_{\mathrm{\Delta }}(\mathrm{r⃗}-r_d)\mathrm{d}\mathrm{r⃗}$.

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 $n_{\mathrm{\Delta }}(\mathrm{r⃗})$ by convolving with a three-dimensional Gaussian:

The autocorrelation of $n_{\mathrm{\Delta }}^l(\mathrm{r⃗})$ 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 $D=3$, $l_c=1\mathrm{\mu m}$, and wavelength $\lambda =633\mathrm{nm}$. 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 $n_{\mathrm{\Delta }}^h(\mathrm{r⃗})={\mathcal{F}}^{-1}[\mathcal{F}[n_{\mathrm{\Delta }}(\mathrm{r⃗})]·(1-\mathcal{F}[G(\mathrm{r⃗})])]$, where the superscript $h$ indicates that higher frequencies are retained. The autocorrelation of $n_{\mathrm{\Delta }}^h(\mathrm{r⃗})$ can then be found as

 

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

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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 $n_{\mathrm{\Delta }}(\mathrm{r⃗})$, $n_{\mathrm{\Delta }}^l(\mathrm{r⃗})$, and $n_{\mathrm{\Delta }}^h(\mathrm{r⃗})$ for $D=3$, $l_c=1\mathrm{\mu m}$, and $W_l=W_h=100\mathrm{nm}$.

 

Fig. 2. Upper length-scale analysis for $W_h=\infty$, 10, 5, and 1 μm with $D=3$, $l_c=1\mathrm{\mu m}$, and $\lambda =633\mathrm{nm}$. (a) $B_n^h(r_d)$ where the dashed curves indicate locations in which the curve is negative. (b) ${\mathrm{\Phi }}_s^h(k_s)$. In each panel the arrow indicates decreasing $W_h$.

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Fig. 3. Example media with $D=3$ and $l_c=1\mathrm{\mu m}$. (a) $n_{\mathrm{\Delta }}(\mathrm{r⃗})$, (b) $n_{\mathrm{\Delta }}^l(\mathrm{r⃗})$, and (c) $n_{\mathrm{\Delta }}^h(\mathrm{r⃗})$ for $W_l=W_h=100\mathrm{nm}$.

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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 ${\mathrm{\Phi }}_s(k_s)$:

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 $D=3$, $l_c=1\mathrm{\mu m}$, and $\lambda =633\mathrm{nm}$. 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.

 

Fig. 4. Percent change in scattering parameters with varying values of $W_l$ and $W_h$ for $D=3$, $l_c=1\mathrm{\mu m}$, and $\lambda =633\mathrm{nm}$. (a) Lower and (b) upper length-scale percent changes. The dotted line indicates the $\pm 5\%$ threshold.

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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 ($r_{\min }$) 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, $r_{\min }$ 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 ($r_{\max }$) 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, $r_{\max }$ 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 $D=3$, $l_c=1\mathrm{\mu m}$, and $\lambda =633\mathrm{nm}$. With increasing $W_l$, the value of $P_{oo}$ is decreased within the subdiffusion regime (i.e., $r·{\mu }_s^{*}<1$). This decrease can be attributed in part to the decreased intensity of the phase function in the backscattering direction [see Fig. 1(b)]. For $r·{\mu }_s^{*}>1$, 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 $r_{\min }=30.8\mathrm{nm}$ ($\sim \lambda /21$) and $r_{\max }=2.71\mathrm{\mu m}$ ($\sim 4\lambda$).

 

Fig. 5. Monte Carlo simulations of $P_{oo}$ with $D=3$, $l_c=1\mathrm{\mu m}$, and $\lambda =633\mathrm{nm}$. (a) Lower length-scale analysis for $W_l=0$, 30, 60, and 90 nm. Arrow indicates increasing $W_l$. (b) Upper length-scale analysis for $W_h=\infty$, 10, 2, 0.5 μm. Arrows indicate decreasing $W_h$.

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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 that 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 ${\mu }_s^{*}$ 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 ${\mu }_s^{*}$ with different shapes of $B_n(r_d)$ and $\lambda =633\mathrm{nm}$.

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