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 , backscattering coefficient , anisotropy factor , and reduced scattering coefficient 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 at wavelength , we report that is sensitive to structural length-scales from 46.9 nm to 2.07 μm (i.e., to ), is sensitive from 26.7 to 320 nm (i.e., to ), and the spatial reflectance profile is sensitive from 30.8 nm to 2.71 μm (i.e., to ).
© 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 , , , and 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, . We define the excess refractive index which contributes to scattering as , where is the mean refractive index. Since is a random process, it is mathematically useful to describe the distribution of refractive index through its statistical autocorrelation function .
In order to study the sensitivity of scattering to short length-scales (lower length-scale analysis), we perturb by convolving with a three-dimensional Gaussian:
The autocorrelation of can then be found as4) 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 using a with , , and wavelength . This corresponds to a biologically relevant Henyey–Greenstein function with anisotropy factor . For increasing , shows a decreasing value at short length-scales [Fig. 1(a)]. The point at which deviates from the original corresponds roughly to the value of . The lower value of at short length-scales corresponds to decreased intensity of 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 , where the superscript indicates that higher frequencies are retained. The autocorrelation of can then be found as
Figure 2 shows the functions described by Eqs. (6) and (7). For decreasing , exhibits a decrease at larger length-scales [Fig. 2(a)]. These alterations lead to a decreased intensity of 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 , , and for , , and .
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 , can be found by incorporating the dipole scattering pattern into :4]:
Figure 4(a) shows percent changes in the above scattering parameters under the lower length-scale analysis for a with , , and . With increasing , each parameter decreases from its original value. For , the decrease occurs because scattering material is removed from the medium. For and , the decrease occurs as a result of reduced backscattering [see Fig. 1(b)]. For , 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: for samples within the multiple scattering regime and for samples within the single scattering regime. Defining a 5% threshold (a common significance level in statistics) the minimum length-scale sensitivity () of and equals 46.9 nm () and 26.7 nm (), respectively. Thus, measurements of and provide sensitivity to structures much smaller than the diffraction limit. Interestingly, is smaller for than . This can be understood by noting that is maximized in the backscattering direction (i.e., ) and so provides the most sensitivity to alterations of 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 , decreases because scattering material is removed from the medium. For , an increase occurs due to a reduction in the forward scattering component. Combining these two opposing effects, the maximum length-scale sensitivity () for equals 2.07 μm (). For , a very small value of is needed in order to alter backscattering. As a result, for is only 320 nm ().
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 . Here, we display the distribution measured with unpolarized illumination and collection, . is the distribution of light that exits a semi-infinite medium antiparallel to the incident beam and within an annulus of radius from the entrance point. It is normalized such that .
Figure 5(a) shows under the lower length-scale analysis for a with , , and . With increasing , the value of is decreased within the subdiffusion regime (i.e., ). This decrease can be attributed in part to the decreased intensity of the phase function in the backscattering direction [see Fig. 1(b)]. For , a range that is essentially insensitive to the shape of the phase function, 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 relative to the original case. Applying a 5% threshold once again, we find that () and ().
Finally, we note that the exact values of and depend on the shape of . 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 and on the shape of , assuming the Whittle–Matérn model and using as an example. As either or increases, shifts relatively more weight to larger length-scales and away from smaller length-scales. As a result, both and increase monotonically with and .
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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