We present an angular thin-lens formula giving the angle of refraction β for arbitrary values of the angle of incidence α. With this formula, we find analytical results for the focal length f of a thin-lens system. The number of lenses n and their focal lengths f<inf>1</inf>,f<inf>2</inf>,...,f<inf>n</inf> are abitrary, as are the mutual distances D<inf>12</inf>,...,D<inf>(n-1)n</inf> between the lenses. All these results are exact, i.e. not restricted to small or even paraxial angles. In the literature, the 2-lens and 3-lens versions (the last one without proof or derivation) are known . We present the general result for n lenses and for (the positions of) its principal planes.
Bruls, G. J. C. L. (2015). Exact formulas for a thin-lens system with an arbitrary number of lenses. Optik, 126(6), 659–662. https://doi.org/10.1016/j.ijleo.2015.01.035