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 μs, backscattering coefficient μb, anisotropy factor g, and reduced scattering coefficient μs* 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 g0.93 at wavelength λ=633nm, we report that μs* is sensitive to structural length-scales from 46.9 nm to 2.07 μm (i.e., λ/13 to 3λ), μb is sensitive from 26.7 to 320 nm (i.e., λ/24 to λ/2), and the spatial reflectance profile is sensitive from 30.8 nm to 2.71 μm (i.e., λ/21 to 4λ).

© 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 μs, μb, g, and μs* 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 nΔ(r⃗)=n(r⃗)/no1, where no is the mean refractive index. Since nΔ(r⃗) is a random process, it is mathematically useful to describe the distribution of refractive index through its statistical autocorrelation function Bn(rd)=nΔ(r⃗)nΔ(r⃗rd)dr⃗.

One versatile model for Bn(rd) employs the Whittle–Matérn family of correlation functions [1,2]:

Bn(rd)=An·(rdlc)D32·KD32(rdlc),
where Kν(·) is the modified Bessel function of the second kind with order ν, lc is the characteristic length of heterogeneity, An 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 that is identical to the commonly used Henyey–Greenstein model.

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

Φs(ks)=Anlc3Γ(D2)π3/22(5D)/2·(1+ks2lc2)D/2,
where ks=2ksin(θ/2) and k is the wavenumber.

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

G(r⃗)=(16πln(2)W2)3/2·exp(4ln(2)W2r⃗2),
where W is the FWHM. Conceptually, G(r⃗) represents a process that modifies the original medium by removing “particles” smaller than W. Using the convolution theorem, this modified medium can be expressed as nΔl(r⃗)=F1[F[nΔ(r⃗)]·F[G(r⃗)]], where F indicates the Fourier transform operation and the superscript l indicates that lower frequencies are retained.

The autocorrelation of nΔl(r⃗) can then be found as

Bnl(rd)=F1[|F[nΔ(r⃗)]|2·|F[G(r⃗)]|2]=4π0Φsl(ks)kssin(ksr)rdks,
where Φsl(ks) is the power spectral density for nΔl(r⃗) and can be computed as
Φsl(ks)=Anlc3Γ(D2)π3/22(5D)/2·exp(ks2Wl28ln(2))(1+ks2lc2)D/2.
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 Wl using a Bnl(rd) with D=3, lc=1μm, and wavelength λ=633nm. This corresponds to a biologically relevant Henyey–Greenstein function with anisotropy factor g0.93. For increasing Wl, Bnl(rd) shows a decreasing value at short length-scales [Fig. 1(a)]. The point at which Bnl(rd) deviates from the original Bn(rd) corresponds roughly to the value of Wl. The lower value of Bnl(rd) at short length-scales corresponds to decreased intensity of Φsl(ks) 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 nΔh(r⃗)=F1[F[nΔ(r⃗)]·(1F[G(r⃗)])], where the superscript h indicates that higher frequencies are retained. The autocorrelation of nΔh(r⃗) can then be found as

Bnh(rd)=F1[|F[nΔ(r⃗)]|2·|1F[G(r⃗)]|2]=4π0Φsh(ks)kssin(ksr)rdks,
where
Φsh(ks)=Anlc3Γ(D2)π3/22(5D)/2·(1exp(ks2Wh216ln(2)))2(1+ks2lc2)D/2.

 

Fig. 1. Lower length-scale analysis for Wl=0, 10, 50, and 100 nm with D=3, lc=1μm, and λ=633nm. The normalized (a) Bnl(rd) and (b) Φsl(ks). In each panel the arrow indicates increasing Wl.

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Figure 2 shows the functions described by Eqs. (6) and (7). For decreasing Wh, Bnh(rd) exhibits a decrease at larger length-scales [Fig. 2(a)]. These alterations lead to a decreased intensity of Φsh(ks) 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 nΔ(r⃗), nΔl(r⃗), and nΔh(r⃗) for D=3, lc=1μm, and Wl=Wh=100nm.

 

Fig. 2. Upper length-scale analysis for Wh=, 10, 5, and 1 μm with D=3, lc=1μm, and λ=633nm. (a) Bnh(rd) where the dashed curves indicate locations in which the curve is negative. (b) Φsh(ks). In each panel the arrow indicates decreasing Wh.

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Fig. 3. Example media with D=3 and lc=1μm. (a) nΔ(r⃗), (b) nΔl(r⃗), and (c) nΔh(r⃗) for Wl=Wh=100nm.

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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 σ(θ), can be found by incorporating the dipole scattering pattern into Φs(ks):

σ(θ)=2πk4(1+cos2θ)Φs(ks).
The shape of σ(θ) can be parameterized by the scattering coefficient μs, the backscattering coefficient μb, and g [4]:
μs=2π11σ(cosθ)dcosθ,
μb=4π·σ(θ=π),
g=2πμs11cosθ·σ(cosθ)dcosθ.
Conceptually, μs is the total scattered power per unit volume, μb represents the power scattered in the 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 μs*=μs·(1g).

Figure 4(a) shows percent changes in the above scattering parameters under the lower length-scale analysis for a Bnl(rd) with D=3, lc=1μm, and λ=633nm. With increasing Wl, each parameter decreases from its original value. For μs, the decrease occurs because scattering material is removed from the medium. For 1g and μb, the decrease occurs as a result of reduced backscattering [see Fig. 1(b)]. For μs*, the decrease is a combination of the previous two effects.

 

Fig. 4. Percent change in scattering parameters with varying values of Wl and Wh for D=3, lc=1μm, and λ=633nm. (a) Lower and (b) upper length-scale percent changes. The dotted line indicates the ±5% threshold.

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To provide specific length-scale sensitivity quantification, we focus on the parameters most relevant to reflectance measurements: μs* for samples within the multiple scattering regime and μb for samples within the single scattering regime. Defining a 5% threshold (a common significance level in statistics) the minimum length-scale sensitivity (rmin) of μs* and μb equals 46.9 nm (λ/13) and 26.7 nm (λ/24), respectively. Thus, measurements of μs* and μb provide sensitivity to structures much smaller than the diffraction limit. Interestingly, rmin is smaller for μb than μs*. This can be understood by noting that ks is maximized in the backscattering direction (i.e., θ=π) and so provides the most sensitivity to alterations of Bn(rd) 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 Wh, μs decreases because scattering material is removed from the medium. For 1g, an increase occurs due to a reduction in the forward scattering component. Combining these two opposing effects, the maximum length-scale sensitivity (rmax) for μs* equals 2.07 μm (3λ). For μb, a very small value of Wh is needed in order to alter backscattering. As a result, rmax for μb is only 320 nm (λ/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, Poo(r). Poo(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 0Poo(r)dr=1.

Figure 5(a) shows Poo under the lower length-scale analysis for a Bn(rd) with D=3, lc=1μm, and λ=633nm. With increasing Wl, the value of Poo is decreased within the subdiffusion regime (i.e., r·μs*<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·μs*>1, a range that is essentially insensitive to the shape of the phase function, Poo 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 Poo relative to the original case. Applying a 5% threshold once again, we find that rmin=30.8nm (λ/21) and rmax=2.71μm (4λ).

 

Fig. 5. Monte Carlo simulations of Poo with D=3, lc=1μm, and λ=633nm. (a) Lower length-scale analysis for Wl=0, 30, 60, and 90 nm. Arrow indicates increasing Wl. (b) Upper length-scale analysis for Wh=, 10, 2, 0.5 μm. Arrows indicate decreasing Wh.

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Finally, we note that the exact values of rmin and rmax depend on the shape of Bn(rd). 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 rmin and rmax on the shape of Bn(rd), assuming the Whittle–Matérn model and using μs* as an example. As either D or lc increases, Bn(rd) shifts relatively more weight to larger length-scales and away from smaller length-scales. As a result, both rmin and rmax increase monotonically with D and lc.

 

Fig. 6. (a) rmin and (b) rmax for μs* with different shapes of Bn(rd) and λ=633nm.

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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.

References

1. P. Guttorp and T. Gneiting, “On the Whittle–Matérn correlation family,” National Research Center for Statistics and the Environment, Technical Report Series (2005).

2. J. D. Rogers, İ. R. Çapoğlu, and V. Backman, Opt. Lett. 34, 1891 (2009). [CrossRef]  

3. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

5. A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012). [CrossRef]  

References

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  1. P. Guttorp and T. Gneiting, “On the Whittle–Matérn correlation family,” National Research Center for Statistics and the Environment, Technical Report Series (2005).
  2. J. D. Rogers, İ. R. Çapoğlu, and V. Backman, Opt. Lett. 34, 1891 (2009).
    [CrossRef]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  5. A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
    [CrossRef]

2012 (1)

A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
[CrossRef]

2009 (1)

Backman, V.

A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
[CrossRef]

J. D. Rogers, İ. R. Çapoğlu, and V. Backman, Opt. Lett. 34, 1891 (2009).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Çapoglu, I. R.

A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
[CrossRef]

J. D. Rogers, İ. R. Çapoğlu, and V. Backman, Opt. Lett. 34, 1891 (2009).
[CrossRef]

Gneiting, T.

P. Guttorp and T. Gneiting, “On the Whittle–Matérn correlation family,” National Research Center for Statistics and the Environment, Technical Report Series (2005).

Guttorp, P.

P. Guttorp and T. Gneiting, “On the Whittle–Matérn correlation family,” National Research Center for Statistics and the Environment, Technical Report Series (2005).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Mutyal, N. N.

A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
[CrossRef]

Pradhan, P.

A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
[CrossRef]

Radosevich, A. J.

A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
[CrossRef]

Rogers, J. D.

A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
[CrossRef]

J. D. Rogers, İ. R. Çapoğlu, and V. Backman, Opt. Lett. 34, 1891 (2009).
[CrossRef]

J. Biomed. Opt. (1)

A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012).
[CrossRef]

Opt. Lett. (1)

Other (3)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

P. Guttorp and T. Gneiting, “On the Whittle–Matérn correlation family,” National Research Center for Statistics and the Environment, Technical Report Series (2005).

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Figures (6)

Fig. 1.
Fig. 1.

Lower length-scale analysis for Wl=0, 10, 50, and 100 nm with D=3, lc=1μm, and λ=633nm. The normalized (a) Bnl(rd) and (b) Φsl(ks). In each panel the arrow indicates increasing Wl.

Fig. 2.
Fig. 2.

Upper length-scale analysis for Wh=, 10, 5, and 1 μm with D=3, lc=1μm, and λ=633nm. (a) Bnh(rd) where the dashed curves indicate locations in which the curve is negative. (b) Φsh(ks). In each panel the arrow indicates decreasing Wh.

Fig. 3.
Fig. 3.

Example media with D=3 and lc=1μm. (a) nΔ(r⃗), (b) nΔl(r⃗), and (c) nΔh(r⃗) for Wl=Wh=100nm.

Fig. 4.
Fig. 4.

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

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

(a) rmin and (b) rmax for μs* with different shapes of Bn(rd) and λ=633nm.

Equations (11)

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Bn(rd)=An·(rdlc)D32·KD32(rdlc),
Φs(ks)=Anlc3Γ(D2)π3/22(5D)/2·(1+ks2lc2)D/2,
G(r⃗)=(16πln(2)W2)3/2·exp(4ln(2)W2r⃗2),
Bnl(rd)=F1[|F[nΔ(r⃗)]|2·|F[G(r⃗)]|2]=4π0Φsl(ks)kssin(ksr)rdks,
Φsl(ks)=Anlc3Γ(D2)π3/22(5D)/2·exp(ks2Wl28ln(2))(1+ks2lc2)D/2.
Bnh(rd)=F1[|F[nΔ(r⃗)]|2·|1F[G(r⃗)]|2]=4π0Φsh(ks)kssin(ksr)rdks,
Φsh(ks)=Anlc3Γ(D2)π3/22(5D)/2·(1exp(ks2Wh216ln(2)))2(1+ks2lc2)D/2.
σ(θ)=2πk4(1+cos2θ)Φs(ks).
μs=2π11σ(cosθ)dcosθ,
μb=4π·σ(θ=π),
g=2πμs11cosθ·σ(cosθ)dcosθ.

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