Reflected Light and Ghosts in Optical Systems

Each surface in an optical system reflects a small fraction of the incident flux, which thus is lost to the image; these surfaces singly and in combination form ghost images of the light source, in various locations in and about the system. The number of first-order ghosts is equal to the number of surfaces, and the number of second-order ghosts equals the number of pairs of surfaces. The total flux through the ghosts is shown to be F0ρ(1−t2n)/1−t2 for first-order ghosts, and ρ2tn∑2n(i-1)t2(n-i)=ρ2tn1-t2[n-1-t2n1-t2]for the double reflection second-order ghosts, where n is the number of surfaces at which reflection can take place, t the transmittance, and ρ the reflectance at each surface. The left member has a simple interpretation.In the general case of s interreflections, 1<s≤n, combinational theory indicates an expression: n!(n-2)!2!+2n!(n-3)!3!+3n!(n-4)!4!+⋯+(s-1)n!(n-s)!s!for the number of ghosts, since (a) interreflection potentially can happen an unlimited number of times in any group, so that all multiplets of order less than s must be included, and (b) first reflection is prohibited at the first surface in all multiplets, because of the direction of light flow.The relations are applied to a system of triplet condensers and projection lenses, coated and uncoated in all combinations. Even though the ghost flux is minimum in the condenser-unfilmed, objective-filmed objective, it offers no advantages over the all-filmed case, because of the lower transmission.