Light Transport in Tissue

Basic Reflection Formulas

The relationship between the angle of incidence and angle of transmission is given by Snell's law

$\begin{displaymath} n_i\sin \theta_i = n_t\sin\theta_t \end{displaymath}$

(B1)

The angle at which total internal reflection occurs is called the critical angle is obtained by setting $\theta_t=\pi/2$

$\begin{displaymath} \theta_c = \sin^{-1}{n_t\over n_i}= \sin^{-1}{1\over n} \qquad{\rm where}\qquad n={n_i\over n_t} \end{displaymath}$

(B2)

The cosine of the critical angle $\mu_c$ is then

$\begin{displaymath} \mu_c = \cos(\sin^{-1}{1\over n})=\sqrt{1-{1\over n^2}} \end{displaymath}$

(B3)

Reflection of light at the boundary separating two media of different indices of refraction is dependent on the angle of incidence (Figure B.3). The reflection is given by the Fresnel equations which depend on the incidence angle ( $\theta_i$ ), the transmission angle ( $\theta_t$ ) and the electric field polarization [30]

$\begin{displaymath} R_\Vert = {\tan(\theta_i-\theta_t)\over\tan(\theta_i+\theta_... ..._\bot = -{\sin(\theta_i-\theta_t)\over\sin(\theta_i+\theta_t)} \end{displaymath}$

(B4)

$\begin{displaymath} T_\Vert = {2\sin\theta_t\,\cos\theta_i)\over\sin(\theta_i+\t... ...t = {2\sin\theta_t\,\cos\theta_i)\over\sin(\theta_i+\theta_t)} \end{displaymath}$

(B5)

where | indicates that the electric field is parallel to the plane of incidence and $\bot$ indicates that the electric field is perpendicular. The reflected radiance is

$\begin{displaymath} L_{\rm reflected} =\vert R_*\vert^2 L_{\rm incident} \end{displaymath}$

(B6)

where R_* equals either R| or $R_\bot$ depending on the polarization. For unpolarized light the net reflection is

$\begin{displaymath} R(\theta) = {1\over2}(R_\bot^2+R_\Vert^2) \end{displaymath}$

(B7)

$\begin{displaymath} R(\theta_i)= {1\over2}\left[{\sin^2(\theta_i-\theta_t)\over\... ...}+{\tan(\theta_i-\theta_t)\over\tan(\theta_i+\theta_t)}\right] \end{displaymath}$

(B8)

This formula is not useful for two cases. First, for normal incidence $\theta_i=\theta_t=0$ and evaluation of Equation (B.8) results in division by zero. For normally incident light the correct expression (the limit of Equation (B.8) as $\theta\rightarrow0$ ) is

$\begin{displaymath} R(\theta=0)={(n_i-n_t)^2\over(n_i+n_t)^2} \end{displaymath}$

(B9)

When the incidence angle is larger than the critical angle ( $\theta_i>\theta_c$ ), no transmitted angle exists. This is the case for total internal reflection of light and $R(\theta)=1$ when $\theta > \theta_c$ .

To implement Fresnel reflection at the boundaries in the diffusion approximation, the first few moments of the Fresnel reflection over a hemisphere are required. Defining the zeroth moment R₀ as the integral of the Fresnel reflection over a hemisphere without weighting leads to

$\begin{displaymath} R_0 ={\int_{2\pi} R(\theta)\,d\omega\over {\int_{2\pi} d\omega}} =\int_0^{\pi/2} R(\theta)\,\sin\theta\,d\theta \end{displaymath}$

(B10)

The first moment of the Fresnel reflection is obtained by including a $\cos\theta$ factor

$\begin{displaymath} R_1 ={\int_{2\pi} R(\theta)\cos\theta\,d\omega\over {\int_{2... ...ga}} =2\int_0^{\pi/2} R(\theta)\cos\theta\,\sin\theta\,d\theta \end{displaymath}$

(B11)

The second moment is found by including a factor of $\cos^2\theta$

$\begin{displaymath} R_2 ={\int_{2\pi} R(\theta)\cos^2\theta\,d\omega\over {\int_... ...}} =3\int_0^{\pi/2} R(\theta)\cos^2\theta\,\sin\theta\,d\theta \end{displaymath}$

(B12)

The reflection moments are normalized such that each is unity when $R(\theta)\equiv1$ .

A useful property governing radiance as it travels through media with differing indices of refraction is the n²-law of radiance. This law states that the ratio of the radiance over the square of the index of refraction is invariant along a light path

$\begin{displaymath} {L_1\over n_1^2} = {L_2\over n_2^2}. \end{displaymath}$

(B13)

Figure B.1 shows the physical basis for this law. As a cone of light passes from n_i into n_t the angle of the cone changes due to refraction of light at the interface. Since the same total amount of light passes across the boundary (ignoring reflection) the energy per steradian must change [48].

Finally, a result which relates R₁ for light passing from one medium to another to that for light travelling in the reverse direction is [11]

$\begin{displaymath} {1-R_1(n_i/n_t)\over n_t^2} = {1-R_2(n_t/n_i)\over n_i^2} \end{displaymath}$

(B14)

**Figure B.1:** The change in radiance passing through media of different indices of refraction. As light passes into a medium with a smaller index of refraction (n₁ to n₂) the angle of a cone will increase. The net change in radiance is equal to the square of the indices of refraction.
$\includegraphics [scale=1.000]{figA21.eps}$

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