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Light Transport in Tissue


Reflection Moments R0, R1, and R2

B.3 For light incident on a medium with a greater index of refraction, no critical angle exists. Consequently, the Equations (B.10)-(B.12) are straightforward numerical integrals in which theta varies from 0 to $\pi/2$. When light travels into a less dense medium, a critical angle exists and the equation for calculating R0, R1, and R2 must be modified. Since $R(\theta)=1$ when $\theta > \theta_c$, R0 becomes
\begin{displaymath} R_0 = \int_0^{\theta_c} R(\theta)\sin\theta\,d\theta+ \int_{\theta_c}^{\pi/2} \sin\theta\,d\theta \end{displaymath} (B24)

so
\begin{displaymath} R_0 = \mu_c+\int_{\mu_c}^1 R(\mu)\,d\mu \end{displaymath} (B25)

where $\mu=\cos\theta$ and $\mu_c=\cos\theta_c$. Similarly, the other two moments become
\begin{displaymath} R_1 = \mu_c^2+2\int_{\mu_c}^1 R(\mu)\mu\,d\mu \end{displaymath} (B26)

and
\begin{displaymath} R_2 = \mu_c^3+3\int_{\mu_c}^1 R(\mu)\mu^2\,d\mu \end{displaymath} (B27)

Depending on the index of refraction ratio, either Equations (B.10)-(B.12) or (B.25)-(B.27) were used to numerically evaluate the reflection moments. An adaptive eighth-order quadrature algorithm which yields numerical results accurate to any specified tolerance [15] was implemented. In the table below the accuracy is $\pm0.001$. Values for R0 and R2 do not exist in the literature, however R1 values were identical to those of Orchard [46] and Ryde [53,54].


Table B.1: The Fresnel reflection moments R0, R1, and R2. Light is incident from a medium with an index of refraction of ni and transmitted into a medium with index nt. The moments in this table are accurate to 0.001.
${n_i\over n_t}$ R0 R1 R2
0.50 0.260 0.161 0.134
0.55 0.241 0.137 0.108
0.60 0.224 0.116 0.087
0.65 0.207 0.097 0.068
0.70 0.190 0.081 0.053
0.75 0.173 0.066 0.040
0.80 0.156 0.053 0.029
0.85 0.136 0.040 0.019
0.90 0.113 0.027 0.011
0.95 0.081 0.015 0.004
1.00 0.001 0.000 0.000
1.05 0.324 0.107 0.036
1.10 0.440 0.195 0.089
1.15 0.518 0.271 0.144
1.20 0.578 0.337 0.199
1.25 0.626 0.395 0.252
1.30 0.665 0.445 0.301
1.35 0.698 0.490 0.347
1.40 0.726 0.530 0.390
1.45 0.750 0.565 0.429
1.50 0.771 0.597 0.465
1.55 0.789 0.626 0.499
1.60 0.805 0.651 0.529
1.65 0.820 0.675 0.558
1.70 0.833 0.696 0.584
1.75 0.845 0.716 0.608
1.80 0.855 0.733 0.631
1.85 0.864 0.749 0.652
1.90 0.873 0.764 0.671
1.95 0.881 0.778 0.689
2.00 0.888 0.791 0.705

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.