| May 99 issue | NewsEtc. index | Home ## Diffuse reflectance from a semiinfinite mediumMay 17, 1999, Steven L. Jacques |

When photons enter a tissue, they scatter and escape out the surface where they are observable. The escaping flux of light is called "diffuse reflectance", R_{d}. The value of R_{d} depends on the absorption coefficient of the tissue, µ_{a}, and the effective pathlength L that photons travel in the tissue.

R_{d} is uniquely related to the ratio of scattering to absorption. The more scattering events that occur before the photon is absorbed, the more likely is the photon to be returned to the surface for escape as observable reflectance. If both the absorption and the reduced scattering coefficients are doubled or halved, the R_{d} does not change because the ratio of scattering to absorption remains constant.

The diffuse reflectance from the surface of a semi-infinite homogeneous medium with absorption coefficient µ_{a} [cm^{-1}] and reduced scattering coefficient coefficient µ_{s}' [cm^{-1}] is approximated by the expression:

is the 1/e optical penetration depth. N' is defined as the ratio of reduced scattering to absorption. The factor A equals the apparent pathlength L for photon attenuation due to the absorption coefficient µ_{a}. In other words, the ensemble of photon pathlengths for escaping light observed as R_{d} can be approximated by a single pathlength L = A.

The following figure shows the behavior of R_{d} for an aqueous solution, in other words for a refractive index mismatch of 1.33:1 at the air/medium surface:

for air/water surface with refractive index mismatch 1.33:1.

which is based on calculations using the Adding-Doubling Method. The data are for various choices of choices of absorption and reduced scattering and for various choices of anisotropy, g = 0, 0.5, 0.8, 0.9, 0.95. Above R_{d} = 0.40, g has no influence on R_{d}. Below R_{d} = 0.40, the R_{d} depends on the anisotropy.

The value of A is not constant, and depends on N' and the refractive index mismatch at the air/medium surface. The value of A is roughly about 7-8 for most soft tissues, and choosing a constant value of A allows Eq. 1 to roughly mimic the behavior of R_{d}. For an accurate calculation of R_{d}, however, the appropriate value of A should be specified as a function of N':

for air/water surface with refractive index mismatch 1.33:1.

In summary, R_{d} is a function of µ_{s}'/µ_{a} = N'.

A measurement of R_{d} is perhaps the simplest measurement one can make on a semi-infinite scattering medium. The following figure illustrates how measurements on the unknown medium (M) and on a standard reflectance (M_{std}) whose reflectance is R_{std} yield R_{d}:

Once R_{d} is specified, one can calculate N':

Since A is a function of N', one must use an iterative loop, or a fitting routine, to specify both A and N' from a given R_{d}. N' and A are not independent, so we are still specifying only one unknown, N', from one independent variable, R_{d}. A simple MATLAB program illustrates such an iterative loop: N' = findNp(Rd).