System of N Thin Coaxial Lenses

Abstract

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= sigma{(-1)mTT[(sigma1frsasrs=as-1+1 0=a0<...<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.