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

doi:10.1364/OL.37.005220

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

Andrew J. Radosevich, Ji Yi, Jeremy D. Rogers, and Vadim Backman »View Author Affiliations

Optics Letters, Vol. 37, Issue 24, pp. 5220-5222 (2012)
http://dx.doi.org/10.1364/OL.37.005220

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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 $μ_{s}$ , backscattering coefficient $μ_{b}$ , anisotropy factor $g$ , and reduced scattering coefficient $μ_{s}^{*}$ 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 $λ = 633 nm$ , we report that $μ_{s}^{*}$ is sensitive to structural length-scales from 46.9 nm to 2.07 μm (i.e., $λ / 13$ to $3 λ$ ), $μ_{b}$ 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 $4 λ$ ).

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

μ_{s}

μ_{b}

g

, and

μ_{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 (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}) = \int n_{Δ} (r⃗) n_{Δ} (r⃗ - r_{d}) d r⃗

One versatile model for

B_{n} (r_{d})

employs the Whittle–Matérn family of correlation functions [1

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

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

B_{n} (r_{d}) = A_{n} \cdot {(\frac{r_{d}}{l_{c}})}^{\frac{D - 3}{2}} \cdot K_{\frac{D - 3}{2}} (\frac{r_{d}}{l_{c}}),

(1)

where

K_{ν} (\cdot)

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 \to \infty

, 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

Φ_{s}

. Under the Born approximation,

Φ_{s}

is the Fourier transform of

B_{n}

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

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

Φ_{s} (k_{s}) = \frac{A_{n} l_{c}^{3} Γ (\frac{D}{2})}{π^{3 / 2} 2^{(5 - D) / 2}} \cdot {(1 + k_{s}^{2} l_{c}^{2})}^{- D / 2},

(2)

where

k_{s} = 2 k \sin (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

n_{Δ} (r⃗)

by convolving with a three-dimensional Gaussian:

G (r⃗) = {(\frac{16 π l n (2)}{W^{2}})}^{3 / 2} \cdot \exp (\frac{- 4 l n (2)}{W^{2}} {r⃗}^{2}),

(3)

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

n_{Δ}^{l} (r⃗) = F^{- 1} [F [n_{Δ} (r⃗)] \cdot F [G (r⃗)]]

, where

F

indicates the Fourier transform operation and the superscript

l

indicates that lower frequencies are retained.

The autocorrelation of

n_{Δ}^{l} (r⃗)

can then be found as

B_{n}^{l} (r_{d}) = F^{- 1} [{| F [n_{Δ} (r⃗)] |}^{2} \cdot {| F [G (r⃗)] |}^{2}] = 4 π \int_{0}^{\infty} Φ_{s}^{l} (k_{s}) \frac{k_{s} \sin (k_{s} r)}{r} d k_{s},

(4)

where

Φ_{s}^{l} (k_{s})

is the power spectral density for

n_{Δ}^{l} (r⃗)

and can be computed as

Φ_{s}^{l} (k_{s}) = \frac{A_{n} l_{c}^{3} Γ (\frac{D}{2})}{π^{3 / 2} 2^{(5 - D) / 2}} \cdot \frac{\exp (\frac{- k_{s}^{2} W_{l}^{2}}{8 l n (2)})}{{(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 \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

Φ_{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_{Δ}^{h} (r⃗) = F^{- 1} [F [n_{Δ} (r⃗)] \cdot (1 - F [G (r⃗)])]

, where the superscript

h

indicates that higher frequencies are retained. The autocorrelation of

n_{Δ}^{h} (r⃗)

can then be found as

B_{n}^{h} (r_{d}) = F^{- 1} [{| F [n_{Δ} (r⃗)] |}^{2} \cdot {| 1 - F [G (r⃗)] |}^{2}] = 4 π \int_{0}^{\infty} Φ_{s}^{h} (k_{s}) \frac{k_{s} \sin (k_{s} r)}{r} d k_{s},

(6)

where

Φ_{s}^{h} (k_{s}) = \frac{A_{n} l_{c}^{3} Γ (\frac{D}{2})}{π^{3 / 2} 2^{(5 - D) / 2}} \cdot \frac{{(1 - \exp (\frac{- k_{s}^{2} W_{h}^{2}}{16 l n (2)}))}^{2}}{{(1 + k_{s}^{2} l_{c}^{2})}^{D / 2}} .

(7)

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)

Φ_{s}^{l} (k_{s})

. In each panel the arrow indicates increasing

W_{l}

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

Φ_{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_{Δ} (r⃗)

n_{Δ}^{l} (r⃗)

, and

n_{Δ}^{h} (r⃗)

for

D = 3

l_{c} = 1 μm

, and

W_{l} = W_{h} = 100 nm

Fig. 2. Upper length-scale analysis for

W_{h} = \infty

, 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 curves indicate locations in which the curve is negative. (b)

Φ_{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_{Δ}^{l} (r⃗)

, and (c)

n_{Δ}^{h} (r⃗)

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})

σ (θ) = 2 π k^{4} (1 + \cos^{2} θ) Φ_{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

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

μ_{s} = 2 π \int_{- 1}^{1} σ (\cos θ) d \cos θ,

(9)

μ_{b} = 4 π \cdot σ (θ = π),

(10)

g = \frac{2 π}{μ_{s}} \int_{- 1}^{1} \cos θ \cdot σ (\cos θ) d \cos θ .

(11)

Conceptually,

μ_{s}

is the total scattered power per unit volume,

μ_{b}

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

μ_{s}^{*} = μ_{s} \cdot (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 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. 1(b)]. For

μ_{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 μm

, and

λ = 633 nm

. (a) Lower and (b) upper length-scale percent changes. The dotted line indicates the

\pm 5 %

threshold.

To provide specific length-scale sensitivity quantification, we focus on the parameters most relevant to reflectance measurements:

μ_{s}^{*}

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

μ_{s}^{*}

and

μ_{b}

equals 46.9 nm (

\sim λ / 13

) and 26.7 nm (

\sim λ / 24

), respectively. Thus, measurements of

μ_{s}^{*}

and

μ_{b}

provide sensitivity to structures much smaller than the diffraction limit. Interestingly,

r_{\min}

is smaller for

μ_{b}

than

μ_{s}^{*}

. 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 4(b) 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

μ_{s}^{*}

equals 2.07 μm (

\sim 3 λ

). 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 (

\sim 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

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

]. Here, we display the distribution measured with unpolarized illumination and collection,

P_{o o} (r)

P_{o o} (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_{o o} (r) d r = 1

Figure 5(a) shows

P_{o o}

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_{o o}

is decreased within the subdiffusion regime (i.e.,

r \cdot μ_{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 \cdot μ_{s}^{*} > 1

, a range that is essentially insensitive to the shape of the phase function,

P_{o o}

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_{o o}

relative to the original case. Applying a 5% threshold once again, we find that

r_{\min} = 30.8 nm

(

\sim λ / 21

) and

r_{\max} = 2.71 μm

(

\sim 4 λ

Fig. 5. Monte Carlo simulations of

P_{o o}

with

D = 3

l_{c} = 1 μm

, and

λ = 633 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}

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

μ_{s}^{*}

as an example. As either

D

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

μ_{s}^{*}

with different shapes of

B_{n} (r_{d})

and

λ = 633 nm

Acknowledgments

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]

OCIS Codes
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(290.7050) Scattering : Turbid media
(290.5825) Scattering : Scattering theory

ToC Category:
Scattering

History
Original Manuscript: October 11, 2012
Manuscript Accepted: November 14, 2012
Published: December 13, 2012

Virtual Issues
Vol. 8, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Andrew J. Radosevich, Ji Yi, Jeremy D. Rogers, and Vadim Backman, "Structural length-scale sensitivities of reflectance measurements in continuous random media under the Born approximation," Opt. Lett. 37, 5220-5222 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=ol-37-24-5220

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References

P. Guttorp and T. Gneiting, “On the Whittle–Matérn correlation family,” National Research Center for Statistics and the Environment, Technical Report Series (2005).
J. D. Rogers, İ. R. Çapoğlu, and V. Backman, Opt. Lett. 34, 1891 (2009). [CrossRef]
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).
A. J. Radosevich, J. D. Rogers, İ. R. Çapoğlu, N. N. Mutyal, P. Pradhan, and V. Backman, J. Biomed. Opt. 17, 115001 (2012). [CrossRef]

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