DOI 10.1393/ncb/i2005-10224-y IL NUOVO CIMENTO Vol. 121 B, N. 1 Gennaio 2006 Third-order monochromatic aberrations via Fermat’s principle A. Marasco and A. Romano Dipartimento di Matematica e Applicazioni, Universit` a di Napoli “Federico II” Via Cintia, Naples, Italy (ricevuto il 15 Dicembre 2005; approvato il 19 Dicembre 2005) Summary. — By Fermat’s principle and particular optical paths, which are not rays, a new aberration function is introduced. This function allows to derive, without resorting to the whole Hamiltonian formalism, the third-order geometrical aberra- tions of an optical system with a symmetry of revolution. PACS 42.15.Fr – Aberrations. PACS 42.15.Eq – Optical system design. 1. – Introduction Starting from Fermat’s principle, Hamilton introduced [1] the characteristic functions to analyze the behavior of rays in a general optical system. The knowledge of one of these functions implicitly supplies the relation between the data of any object ray and the data of the corresponding image ray. However, making the above relation explicit and evaluating one of the characteristic functions of a complex optical system is a very hard task. Consequently, Seidel and Schwarzschild [2,3] defined a new characteristic function, which is called the Schwarschild eikonal. It is related to the Hamiltonian angle charac- teristic and depends on suitable nondimensional variables. The above authors succeeded in developing by this function a complete and expressive analysis of the monochromatic third-order geometrical aberrations of an optical system having a symmetry of revolu- tion around the optical axis (see also [4-9]). Another approach to the optical aberration theory is based on the aberration function (see, for instance, [6]), whose evaluation again requires the knowledge of Schwarzschild’s eikonal. The difficulties of these procedures are related to the circumstance that the quantities we are interested in are evaluated along the unknown rays. Whatever is the approach which is used to attain to the third-order aberration for- mulae, the whole Hamiltonian formalism has to be introduced. In this paper a different approach is presented which is based both on Fermat’s principle and the introduction of some auxiliary optical paths, here called stigmatic paths, which allow us to evaluate a new aberration function by a very simple procedure. c  Societ` a Italiana di Fisica 91