Generalization of the Abbe Sine Law in Geometric Optics Available to Purchase

This paper presents relatively unknown, though not new, theorems applicable to a real axially symmetrical optical system. It treats the situation where rays leaving a particular axial object point O in object space are assumed to image perfectly at axial image point O′. A ray through O at angle α with the axis passes through O′ at angle α′ with the axis. The Abbe and Herschel conditions state the required functional relationship between α and α′ to ensure that rays from P image perfectly into P′, when P is infinitesimally displaced from O perpendicular to, or parallel to the axis, respectively. The formulas derived here give the detailed variation of $β1$⁠, $β2$⁠, and γ in terms of the functional relation between α and α′, independent of the further specification of the system. They are derived using Fermat's theorem and the second law of thermodynamics. Here $β1$⁠, $β2$⁠, and γ represent, respectively, the meridional (primary) lateral magnification, the sagittal (secondary) lateral magnification, and the longitudinal magnification relative to small displacements from O. The variation of $β1$ and $β2$ with α specifies the coma figure, while the variation of γ gives the longitudinal spherical aberration for an axially displaced object point.