Gravitational lensing and light deviation is a well known subject in General Relativity. I'm wondering if the null geodesics equation could be recast into the classical lenses equation: \begin{equation} \frac{1}{p} + \frac{1}{q} = \frac{1}{f}. \tag{1} \end{equation} If it's possible, how should be defined the gravitational focal length $f$ ?
I never saw the geodesics equation written into the shape (1), even in the case of the Schwarzschild metric: \begin{equation}\tag{2} \frac{d k^{\lambda}}{d \sigma} + \Gamma_{\mu \nu}^{\lambda} \, k^{\mu} \, k^{\nu} = 0, \end{equation} where $k^{\mu} \propto \frac{d x^{\mu}}{d \sigma}$ is a null four-vector: $g_{\mu \nu} \, k^{\mu} k^{\nu} = 0$.
The weak field approximation is defined by the following metric: \begin{equation}\tag{3} ds^2 \approx (1 + 2 \phi) \, dt^2 - (1 - 2 \phi)(\, dx^2 + dy^2 + dz^2), \end{equation} where $\phi \ll 1$ is a gravitational potential (I'm using $c \equiv 1$ and metric-signature $(1, -1, -1, -1)$).
Could the geodesics equation (2) be reformulated into the shape of (1), assuming the special case of the Schwarzschild metric, or the weak field metric (3)?
EDIT: I'm thinking about this kind of rays, with their focal length dependant on the impact parameter:
