ECE532 Biomedical Optics © 1998 Steven L. Jacques, Scott A. Prahl Oregon Graduate Institute |
The time-resolved fluence rate F(r,t) [W/cm^{2}] in response to an impulse of energy U_{o} [J] and the steady-state fluence rate F_{ss}(r) [W/cm^{2}] in response to an isotropic point source of continuous power P_{o} [W] are summarized:
where the transport factors T(r,t) [cm^{-2} s^{-1}] and T_{ss}(r) [cm^{-2}] have been introduced to more carefully distinguish the source, the transport, and the fluence rate.
Note on notation: In this class, we use F_{ss}(r) rather than F(r) to especially emphasize the steady-state fluence rate from the time-resolved fluence rate. However, F(r) should be used outside this class.
The above expression for T_{ss}(r) can be obtained by integrating T(r,t)exp(-µ_{a}ct) over all time to yield the total accumulated amount of photon transport to each position r. The factor exp(-µ_{a}ct) accounts for photon absorption (recall that ct = pathlength so this expression is simply Beer's law for photon survival) and causes photon concentration to approach zero as time goes to infinity. The expression for T_{ss}(r) is derived:
Note that the final expression above has made the substitutions:
which removes the diffusion length D and introduces the optical penetration depth δ which is the incremental distance from the source that causes F_{ss}(r) to decrease to 1/e its initial value. The penetration depth δ is a parameter which is very easily understood in experimental measurements and consequently has more intuitive value to some people than D which is important from the perspective of the local step size of the diffusion process. In this class we will often use the following expression for F_{ss}(r) when we prefer to emphasize the roles of µ_{a} and δ: