System of N Thin Coaxial Lenses

In geometrical optics, in a system of two thin coaxial lenses, there are several standard formulas, including “𝟏𝑭=𝟏𝒇𝟏+𝟏𝒇𝟐−𝒅𝒇𝟏𝒇𝟐”. The purpose of this paper is to generalize these formulas to the case of a system of an arbitrary number of thin lenses. In particular, this paper proves that the focal length Fn of a system of n thin coaxial lenses is given by 𝟏𝑭𝒏=Σ{(−𝟏)𝒎Π[(Σ𝟏𝒇𝒓𝒔𝒂𝒔𝒓𝒔=𝒂𝒔−𝟏+𝟏 𝟎=𝒂𝟎<𝒂𝟏<⋯<𝑎𝒎<𝒂𝒎+𝟏=𝒏;𝒅𝒏=𝟏 )𝒅𝒂𝒔]𝒎+𝟏𝒔=𝟏}𝒏−𝟏𝒎=𝟎, where, fr is the focal length of the rth lens, and dr is the distance between the rth lens and (r+1)th lens. For a fixed value of m, all combinations of values of the a’s (satisfying the condition “0 = a0 < a1 < … < am < am+1 = n”) are taken in the inner sum.