Boundary reflections and caustics

In Eq.(8) the factor $\phi_c$ represents the phase shift induced by caustics, which can be written as

$$\phi_c = \frac{\pi}{2} n_c ,$$ (10)

where $n_c$ is the number of times that there is a sign change in $q(s)$ [4]. The reflection decay is given by

$$\phi_r = \displaystyle{\prod\limits_{i=1}^{n_r}} R_i ,$$ (11)

where $n_r$ represents the total number of reflections and $R_i$ is the reflection coefficient. Surface reflections are characterized by either $R_i = -1$ (vacuum over surface) or $R_i = 1$ (rigid over surface). Bottom reflections are characterized by a complex reflection coefficient, which depends on frequency $R_i = R_i(\omega )$. With $n_r = 0$ there is no decay in either phase or amplitude due to reflections, which implies that $\phi_r = 1$.