Studies in optics. I. General coordinates for optical systems with central or axial symmetry
In previous papers1,2 the author has proposed an approach to geometrical optics different from that developed by Hamilton and his successors. The purpose of the present paper is to generalize the formulas in these papers, and to find the most general treatment of systems with central (point) symmetry and with axial symmetry. By leaving the coordinates general, subject only to the symmetry conditions of the problem, we retain the symmetry in the formulas up to the point where we desire to draw conclusions for a special problem. We can then introduce special coordinates adapted to the problem in question, and find the particular answers. The fundamental invariants of geometrical optics show no preference for either object or image side, nor for pbint or angle coordinates as variables. The different approaches suggested by Hamilton, as well as the direct approach just mentioned, are special cases of the methods developed here corresponding to special choices of coordinates. Several different choices of coordinates will be given as examples. The fundamental formulas (A, B, B', C below) are based only on symmetry conditions and on the validity of the Lagrange invariant (A). They are therefore not restricted to optical problems,3 but are also valid for problems in mechanics, hydrodynamics, and electron optics. 1. Ray tracing formulas, the Lagrangian invariant. Let us assume a ray traversing a number of optical media with refractive indexes n, «i2, n23, •••,«'. Let a(x, y, z), &'{x', y', z') be a vector from an arbitrary origin to a point on the object and image rays, respectively. Let ak(xk, y*, zk) be the vector from the same origin to the intersection point of the ray with the &th surface. Let sktk+\($ik,k+i, Vk.k+i, £k,k+i) be a vector along the ray in the medium between &th and (& + l)th surface, a vector of length equal to the refractive index nk,k+i of the medium. Let Ok be a vector perpendicular to the fcth surface at the intersection point. Its length may remain arbitrary, for the moment. The refraction law then reads: