Appendix B

As we are dealing with scattering along the ray and the medium has no sources, we can set B = 0 and U₂ = 0. Then the intensities $\tau$ are given by,

$$U^+ (\tau) = U_1 {\rm e}^{-k \tau} \frac {1 - r^2 {\rm e}^{-2k (T - \tau)}} {1 - r^2 {\rm e}^{-2kt}},$$ (B1)

$$U^- (\tau) = r \ U_1 \frac {{\rm e}^{-k \tau} - {\rm e}^{-k (2T - \tau)}} {1 - r^2 {\rm e}^{-2kT}},$$ (B2)

where

$$k^2 = (1 - \omega) [1 + \omega (1 - 2p)],$$ (B3)

and

$$\ \ \ \ r = \frac {k - 1 + \omega} {k + 1 - \omega}.$$ (B4)

And the emergent intensities are

$$U^+ (T) = U_1 \frac {{\rm e}^{-kT} (1 - r^2)} {1 - r^2 {\rm e}^{-2kT}},$$ (B5)

$$U^-(0) = U_1 r \frac {1 - {\rm e}^{-2kT}} {1 - r^2 {\rm e}^{-2kT}}.$$ (B6)

If we represent the reflection and transmission coefficients by r(T) and t(T) respectively, then and

$$r(T) = r \frac {1 - {\rm e}^{-2kT}} {1 - r^2 {\rm e}^{-2kT}},$$ (B7)

$$t(T) = \frac {(1-r^2) {\rm e}^{-kT}} {1-r^2 {\rm e}^{-2kT}}.$$ (B8)

We set $p=\frac {1} {2}$ (for isotropic scattering) then

$$k = (1 - \omega)^{1/2}, \ r = \frac {1 - k} {1 + k}.$$ (B9)

In the above treatment we assumed that $\omega < 1$. If $\omega = 1$, the case of pure scattering, the treatment will be different and we obtain, (for B = 0 and U₂ = 0)

$${\bf U}= \frac {U_1} {1 + T(1-p)} \left[ \begin{array} {l} 1+(T-\tau)(1-p) \\ (T-\tau)(1-p)\end{array}\right]$$ (B10)

The reflection and transmission factors are

$$r(T) = \frac {T(1-p)} {1 + T (1-p)} \rightarrow 1 \ {\rm as} \ T \rightarrow \infty,$$ (B11)

$$t(T) = \frac {1} {1 + T (1-p)} \rightarrow 0 \ {\rm as} \ T \rightarrow \infty,$$ (B12)

so that

r(T) + t(T) = 1.

which express conservation of energy.