Analytical solution of photon diffusion in finite media

[vitom001]

Homework Statement

I'm trying to derive an analytical expression for the photon backscatter flux in finite turbid media using the diffusion equation and the method of images. What I want to write is: for a given volume (x,y,z), where a coherent light source is incident on the x-y plane and z is the depth, what is the probability of a photon that enters at (xi,yi,zi=~0) reaching a particular point within the medium (x,y,z) within time (t1), before exiting the medium at a point (xo,yo,zo=0) at time (t2). This total probability I can write as the product of two probabilities since they are independent. For the incident photons I use the method of images as shown in the expression for P1, where h is the thickness of the medium (h=z_max). I have stated zi=~0 since it is actually on the order of the transport mean free path (l*) which is relatively small to the sample thickness (in this case). My question is whether it is necessary to apply this also to the expression for the backscattered photons (P2) since I am evaluating it at zo=0? Also, do I have to write the expressions with (t1) and (t2) or can I use (t) for both? Since I want to integrate the resulting expression (after differentiation at z and setting zo=0) over all time (0 to Inf), being able to substitute (t) for time would simplify the calculations.

Homework Equations

The Attempt at a Solution

P = P1*P2

 

[Asim Riaz]

P1 = exp(-mu*z - (mu/2l*)^2*(x-xi)^2 - (mu/2l*)^2*(y-yi)^2) + exp(-mu*(2h-z)- (mu/2l*)^2*(x-xi)^2 - (mu/2l*)^2*(y-yi)^2)P2 = exp(-(mu/2l*)^2*(x-xo)^2 - (mu/2l*)^2*(y-yo)^2)*[1/(4piDt)]^1/2*[exp(-(z-zo)^2/(4Dt)) - exp(-(2h-z-zo)^2/(4Dt))]Yes, it is necessary to apply the approximation zi=~0 also to the expression for backscattered photons (P2). As for time, I believe you can use (t) for both since you are integrating over all time (0 to Inf).