The Limits of Light Diffusion The Limits of Light Diffusion ApproximationApproximation

Robert Scott Brock(1), Jun Qing Lu(1), Xin-Hua Hu(1), David W. Pravica(2)

Department of Physics,(1) Department of Mathematics,(2)

East Carolina University,

Greenville, NC 27858

The Limits of Light Diffusion The Limits of Light Diffusion ApproximationApproximation

Summary

The diffusion approximation to the radiative transfer theory has been used widely to model light propagation in turbid media. An analytical

solution to the light diffusion equation is derived for a converging laser beam. Results are compared with those of Monte Carlo method

to discuss the limitation of the approximation.

Document ID is 43834

Reflectance measurementsReflectance measurements

A method of reflectance measurement was first reported in 1983 [1]. The purpose was to determine the optical properties of turbid media basedisagreementn approximation to the radiative transfer theory.

This method has attracted active efforts for noninvasive determination of the medium properties in terms of an absorption coefficient a and a reduced scattering

coefficient s.

[1] R.A.J. Groenhuis, et al., Appl. Opt., 22, 2456-62 and 2463-7 (1983)

Typical measurement configurationTypical measurement configuration

I0 IR spectrograph

IR/I0

The fundamental flaw of the diffusion model is that it is based on the data inversion approach

The fundamental flaw of diffusion model based data inversion

approach

Diffusion model fails

when

< valid

IR/I0

valid

Signal-to-noise ratio degrades

as

increases

noise

The problems of this researchThe problems of this research

1. The degree of approximation of the diffusion model to the radiative transfer theory in a configuration of cylindrical symmetry.

2. The dependence of valid on the medium

properties.

Review of Diffusion Theory IReview of Diffusion Theory I

Let Us(r) be the scattered energy density.

Let Ud(r) be the direct energy density.

Let kd , A be constants of the media.

Let Si(r) be the internal source function.

Then we study the diffusion-like equation:

2 2( ) ( ) ( ) ( )s d s d iU r k U r AU r S r

Review of Diffusion Theory IIReview of Diffusion Theory IILet D be the diffusion coefficient.Let abe the absorption coefficient.

Let sbe the scattering coefficient.

Let g be the mean cosine of the scattering angle. Let v be the speed of light in the media.

/ 3[ (1 ) ]a sD g 2 / 3 [ (1 ) ]d a a a sk D g

Review of Diffusion Theory IIIReview of Diffusion Theory III

Let u(r) be the energy density.Let D(r) be the diffusion coefficient and (r) be the absorption coefficient of the media.Let S(r) be the source function.

Then we study the diffusions equation:

( ) ( ) ( ) ( ) ( )D r u r r u r S r

Uniform MediaUniform MediaThe model presupposes that a homogeneous Gaussian beam of light is focused onto a uniform media with diffusion coefficient D0 and absorption coefficient .

Define k2 = / D0 .

Then we study the Helmholtz equation:

2 20( ) ( ) ( ) /u r k u r S r D

Source as a Glowing Cone ISource as a Glowing Cone I

When light enters the media as a beam it begins to scatter as well as be absorbed.

Main Assumptions: A physical process is used so that a glowing source is

created and maintained in the media in the shape of a converging cone;

The cone exponentially decreases in intensity up to the vertex, where it then vanishes.

Source as a Glowing Cone IISource as a Glowing Cone II

Let z0>0 be the vertex

point;

Let >0 be the decay rate of the cone’s intensity;

Let be the slope variable.

The Source as a Conical Shell:

0( )zS r 0 0( )2

cos( )

S z

0

0

sinh[ ( )( )]

sinh[ ( ) ]

z z

z

0( cot( ) )z z

Source as a Glowing Cone IIISource as a Glowing Cone III

The Green’s Function IThe Green’s Function I

To simplify the Green’s function we work in cylindrical polar coordinates (z), and assume no – dependence.Furthermore, we assume symmetry across the x,y – plane.Thus we have a double cone source symmetric about x,y – plane.This corresponds to Neumann boundary conditions on the intensity function u(r).

The Green’s Function IIThe Green’s Function II

[2] V.A. Markel and J.C. Schotland, J. Opt. Soc. Am. A, 22, 1336-1347 (June 2001)

0 1 1 2 2( , ; , )G z z

The Solution on the The Solution on the x,yx,y – – plane Iplane I

The inner part z tan(),

The Solution on the The Solution on the x,yx,y – – plane IIplane II

The outer part z tan(),

Intensity due to a single shellIntensity due to a single shell

Intensity due to a solid cone Intensity due to a solid cone

The Monte Carlo methodThe Monte Carlo method

Comparison of methodsComparison of methods

Here is a log-log plot:

ConclusionConclusion

The equation obtained from the Diffusion approximation agrees well with the Monte Carlo simulation, although the largest disagreement occurs nearest the azimuthal coordinate.