Reduction to the 2.5D case

Let us consider an environment with the following set of conditions:

$$\mbox{$\mathbf{C}$}= \left[ \begin{array}{cc} c_{nn} & 0 \\ 0 & 0 \end{array} \right]$$

(i.e., there is no dependence on the second component of $\mbox{$\mathbf{n}$}$, and indexing is being omitted by obvious reasons) and

$$\mbox{$\mathbf{Q}$}(s) = \left[ \begin{array}{cc} q(s) & 0 \\ 0 & q_{\perp} (s) \end{array} \right]$$

so the matrix contains only diagonal elements, and we introduced the notation

$$q(s) = q_{11}(s) \hskip5mm , \hskip5mm q_{\perp}(s) = q_{22}(s) ,$$

so det $\mbox{$\mathbf{Q}$}= q(s)q_{\perp}(s)$. It follows from the system given by Eq.(3.2) that

$$p_{12}(s) = p_{21}(s) = 0 ,$$

and

$$\begin{array}{ccl} \displaystyle { \frac{d}{ds}} q = c_0 \; ... ..., & \displaystyle { \frac{d}{ds}} p_{\perp} = 0 \ \end{array}$$

where $p = p_{11}$ and $p_{\perp} = p_{22}$. The solution of the last equation is trivial and corresponds to

$$p_{\perp}(s) = p_{\perp}(0) = \makebox{constant} .$$

It follows further that

$$q_{\perp}(s) = p_{\perp}(0)I_c(s) + q_{\perp}(0) ,$$

where

$$I_c(s) = \int c_0(s) ds .$$

Recalling $\mbox{$\mathbf{n}$}$ as two-dimensional vector one gets that

$$\mbox{$\mathbf{M}$}\mbox{$\mathbf{n}$}\cdot\mbox{$\mathbf{n}$}= \frac {p(s)}{q(s)}n_1^2 + \frac {1}{I_c(s)}n_2^2 .$$ (3.6)

In the case of a point source it follows that

$$q_{\perp}(s) = \frac {\cos\theta(0)}{c(0,0,0)} I_c(s) \sim s ,$$

which indicates that the parameter is proportional to the ray path.