Let's say I have a thin lens model of an optical system. When I have a ray that is parallel to the optical axis, the situation is quite standard - the ray refracts and passes the focal point f (see my bad drawing).
From the triangle in the picture, I can calculate the angle $\beta$ by using the formula $\tan(\beta) = y/f$ and so $\beta = \arctan(y/f)$. But what if my ray is not parallel with the optical axis? How do I calculate the angle of the refracted ray with the opt. axis $\beta'$?
I thought the ray might obey simply $\beta' = \beta + \delta$ = $\arctan(y/f) + \delta$, e.g. angle $\beta'$ could be calculated by simply adding the angle a parallel ray produces when refracted on a lens $\beta$ and an angle of deviation from being parallel with the optical axis $\delta$. On the other hand, I am not sure this approach is right. All in all, I am interested in a solution that does not involve the paraxial approximation (notice I use $\tan()$ in my equations) and I would like to know the following. How does one calculate the angle of refracting rays that are not parallel with the optical axis, in the thin lens model approximation?