斜入射散射测量的非对称扩散模型

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本文探讨了一种针对斜入射反射测量的新型扩散模型——不对称扩散模型（Asymmetric Diffusion Model）。该模型是基于线源分布理论构建的，其核心在于处理混浊介质（turbid medium）中的光学特性研究，特别是吸收系数（μ_a）和减少散射系数（μ_s'）的精确测定。在斜入射光线下，当光束在离入射点负向位置1.5个传输平均自由路径（mfp'）处收集到的绝对散射反射信号可以作为模型的输入数据。作者Yaqin Chen、Liji Cao和Liqun Sun通过将实验数据与该不对称扩散模型进行拟合，实现了对混浊介质光学参数的高精度测量，达到10%的准确度。这种模型特别之处在于，即使在正向位置的数据（0-1运输平均自由路径范围内），也能通过特定的计算方法估算出μ_s'，这在实际应用中极具价值，如医学成像和材料科学等领域中的光散射分析。斜入射反射测量通常用于研究非均匀介质中的光传播，因为光线在这种情况下会经历复杂的散射和吸收过程。通过不对称扩散模型，科学家能够更深入地理解这些过程，优化仪器设计，提高测量的可靠性和精确度。此外，文中提到的方法可能有助于解决其他类似技术中遇到的难题，如光通信中的多径效应或生物组织内光传输的研究。这篇论文不仅提供了新的理论框架，还展示了在实际操作层面如何利用不对称扩散模型来提升斜入射反射测量技术的性能，对于混浊介质的光学性质研究以及相关领域的实验技术进步具有重要意义。

COL 9(10), 101701(2011) CHINESE OPTICS LETTERS October 10, 2011

Asymmetric diffusion model for oblique-inc i dence

reﬂectometr y

Yaqin Chen (



äää



)

1∗

, Liji Cao (

ùùù

ááá

ÄÄÄ

)

, and Liqun Sun (



|||

+++

)

State Key Laboratory of Precision Measurement Technology and Instruments, Tsinghua University, Beijing 100084, China

Division of Medical Physics in Radiology, German Cancer Research Center,

Im Neuenheimer Feld 280, 69120 Heidelberg, Germany

∗

Corresponding author: yaqinchen@mail.tsinghua.edu.cn

Received March 31, 2011; accepted May 9, 2011; posted on line July 11, 2011

A diffusion theory model induced by a line source distribution is presented for oblique-incidence reﬂectom-

etry. By fitting to t his asymmetric diffusion model, the absorption and reduced scattering coefficients µ

and µ

′

of the turbid medium can both be determined with accuracy of 10% from t he absolute profile of

the diffuse reﬂectance in the incident plane at the negative position –1.5 transp ort mean free path (mfp

′

)

away from the incident point; particularly, µ

′

can be estimated from the data at positive positions within

0–1.0 mfp

′

with 10% accuracy. The method is verified by Monte Carlo simulations and experimentally

tested on a phantom.

OCIS codes: 170.0170, 170.3660, 170.7050.

doi: 10.3788/COL201109.101701.

Knowledge about the optical properties, including the

absorption coefficient (µ

) and the reduced scattering

coefficient (µ

′

= µ

(1–g)), where µ

is the sc attering

coefficient and g is the anisotropy factor of scattering,

of biological tissues plays an important role for opti-

cal therapeutic and diagnostic techniques in medicine.

Over the past two decades, many techniques have been

developed to measure the optical properties of turbid

media

[1−3]

. Oblique-incidence reﬂectometry is an attrac-

tive method for determining the optical properties of

semi-infinite turbid media due to its simplicity and

accuracy

[4,5]

. To deduce µ

and µ

′

from a spatial distri-

bution o f diffuse reﬂec tance produced by obliquely inci-

dent light, Lin et al.

[5]

proposed a diffusion theory model

by using two isotropic scattering point sources–one pos-

itive source located at 1 transport mean fre e path (1

mfp

′

= 3D = (0.3 5µ

+ µ

′

)

−1

, w here D is the diffusion

coefficient) measured along the optical path determined

by Snell

′

s law below the surface and one negative image

source above the surface. As a result, the positions of

the point sources had a shift ∆x in the x direction. By

introducing this shift, the dipole-source model in Ref.

[5] can accurately model the oblique-incidence diffuse re-

ﬂectance falling outside the range of 1–2 mfp

′

from the

source. To determine µ

and µ

′

using model in Ref. [5],

the distance ∆x should be measured by finding the ap-

parent center of the symmetric reﬂectance profile at posi-

tions several mfp

′

away from the source. Apparently, this

is not good for the construction of a compact oblique-

incidence reﬂectometer. In this letter, we introduce a

different diffusion model induced by a line source distri-

bution for oblique- incidence reﬂectometry.

Figure 1 shows a schematic illustration of the geome-

try c oncerned in this letter. An infinitely narrow light

beam is incident upon a planar interface between a semi-

infinite, highly scattering medium and a non-scattering

medium with the r e fr active indices n

and n

. We set up a

Cartesian coor dinate system Oxyz on the geometry. The

origin O is the point o f entry on the medium surface, the

z axis is the norma l o f the surface pointing toward the

inside of the medium, the x axis is along the projection

of the pencil beam onto the planar surface (z = 0), and

the y axis is per pendicular to both the x and z directions,

pointing outward to form a right-handed Cartesian coor-

dinate system. In such a coordinate system, the incident

beam is located in the x-z plane, oriented at an angle

(0 6 α

< π/2) with respect to the z axis, which

is defined as the angle of incidence. When light enters

the semi-infinite medium, ignoring the scattering inter-

actions, it would be refracted to a transmission ray OL

with an angle α

to the z axis, as determined by Snell’s

law. However, light traveling in turbid media will suffer

from multiple scattering events. Only the first scatter-

ing interactions of incident photons are situated along the

unscattered-light tra ns mission path OL. In diffusion ap-

proximation, scattering is assumed to be iso tropic, thus

each first scattering site can be treated as an isotropic

point source. The stre ngth of these sourc e s will be de-

termined by the optical properties of the medium and

will exp onentially decrease with the path length l

along

the line OL measured from the incident point, similar

to the exponentially decreasing line source distribution

for normal-incidence beam in Ref. [6]. If assuming the

light beam of unit intensity and taking account the Fres-

nel reﬂection at the boundary, the source term can be

mathematically expressed as

S(x, y, z; l

) = [1 − R

Fres

(α

, n

)] ·

′

+∞

−µ

′

δ(x − l

sin α

)δ(y)δ(z − l

cos α

)dl

(1)

where δ is the Dirac delta function; R

Fres

is the Fresnel

reﬂection function

[7]

; µ

′

= 1/(3D) is the total reduced

attenuation c oefficient, where D has the same definition

as in Ref. [5], 3D = (0.35µ

+ µ

′

)

−1

Extrapolated boundary condition is used to ac c ount

for the effect of Fresnel reﬂections due to the refractive

1671-7694/2011/101701(5) 101701-1

 2011 Chinese Optics Letters

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