The Limits of Light Diffusion Approximation 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 Approximation 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 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, and (1983)

Typical measurement configuration I0I0 IRIR spectrograph  I R /I 0  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 I R /I 0   valid  Signal-to-noise ratio degrades as  increases noise

The 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 I Let U s (r) be the scattered energy density. Let U d (r) be the direct energy density. Let k d, A be constants of the media. Let S i (r) be the internal source function. Then we study the diffusion-like equation:

Review of Diffusion Theory II Let D be the diffusion coefficient. Let  a  be the absorption coefficient. Let  s  be the scattering coefficient. Let g be the mean cosine of the scattering angle. Let v be the speed of light in the media.

Review 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:

Uniform Media The model presupposes that a homogeneous Gaussian beam of light is focused onto a uniform media with diffusion coefficient D 0 and absorption coefficient  . Define k 2 =   / D 0. Then we study the Helmholtz equation:

Source 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 II Let z 0 >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:

Source as a Glowing Cone III

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

The Solution on the x,y – plane I The inner part  z tan(  ),

The Solution on the x,y – plane II The outer part  z tan(  ),

Intensity due to a single shell

Intensity due to a solid cone

The Monte Carlo method

Comparison of methods Here is a log-log plot:

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