$v$ is governed by the geodesic deviation equation $$\label{gde1}
k^\alpha k^\beta \nabla_\alpha \nabla_\beta \xi^\mu = {R^\mu}_{\nu\alpha\beta} k^\nu k^\alpha\xi^\beta,$$ where ${R^\mu}_{\nu\alpha\beta}$ is the Riemann tensor.

### Sachs equation {#subsubsec:Sachs_equation}

Consider an observer with four-velocity $u^\mu$. In view of relating $\xi^\mu$ to observable quantities, we introduce the Sachs basis $(s_A^\mu)_{A\in\{1,2\}}$, defined as an orthonormal basis of the plane orthogonal to both $u^\mu$ and $k^\mu$, $$s_{A}^\mu s_{B\mu}=\delta_{AB},
\quad
s_{A}^\mu u_{\mu}=s_{A}^\mu k_{\mu}=0,$$ and parallel-transported along the geodesic bundle, $$k^\nu \nabla_\nu s_A^\mu = 0 .$$ The plane spanned by $(s_1,s_2)$ can be considered a screen on which the observer projects the light beam. The two-vector of components $\xi_A=\xi_\mu s_A^\mu$ then represents the relative position, on the screen, of the light spots corresponding to two neighboring rays separated by $\xi^\mu$.

The evolution of $\xi_A$, with light propagation, is determined by projecting the geodesic deviation equation on the Sachs basis. The result is known as the Sachs equation [@pubook; @sachs; @refbooks], and reads $$\frac{{{\rm d}}^2\xi_A}{{{\rm d}}v^2} = \mathcal{R}_{AB} \, \xi^B,
\label{eq:Sachs}$$ where $\mathcal{R}_{AB}={R}_{\mu\nu\alpha\beta}k^\nu k^\alpha s_A^\mu s_B^\beta$ is the screen-projected Riemann tensor, called optical tidal matrix. It is conveniently decomposed into a Ricci term and a Weyl term as $$\label{eq:decomposition_optical_matrix}
(\mathcal{R}_{AB}) =
\begin{pmatrix}
\Phi_{00} & 0 \\ 0 