Refraction at a spherical interface (1)

A curved interface intersects the optical axis at $(x=0,y=0)$. The interface has a radius $R$ centered at $C$. For $R>0$ the interface is convex and for $R<0$ the interface is concave. The index of refraction is $n_1$ to the left of the interface and $n_2$ to the right of the interface. A light ray (red) leaves an object $o$ on the left of the interface and strikes the interface at point $P$. Part of the ray is reflected and part is refracted. The reflected ray is blue and the refracted ray is red. The angles between the red rays and the gray normal to the interface obey Snell's law $n_1\sin\theta_1 = n_2\sin\theta_2$.

The angle $\theta_1 = \angle oPC$ is calculated by taking the inner product,

If a light ray only makes a small angle $\phi_0$ to the optical axis and $|y_p|$ is small compared to $|R|$, then when the ray reaches the interface it will bent to a new angle, \[ \begin{equation} \large \phi_{1} = \frac{(n_1-n_2)y_p}{Rn_2}+\frac{n_1}{n_2}\phi_0. \end{equation} \]