Proximate ray tracing and optical aberration coefficients

Analytic-domain lens design with proximate ray tracing

“…The first implementation of proximate ray tracing, by Hopkins in 1976 [6][7][8], appears to have been done as a method of reducing the computational burden and complexity for calculating higher-order aberrations. By sampling a specific set of rays passing through the system, one is able to obtain each of the various aberration coefficients.…”

“…Hopkins [6][7][8] described proximate ray tracing as an iterative ray tracing technique in algebraic form. The basic approach is 1.…”

“…In Hopkins' presentation [6][7][8] of the proximate ray trace equations, the variables x, y, c x , and c y have only odd-order terms, and the variables z, c z , and w have only even-order terms due to his assumption of rotational symmetry about the optical axis. In order to design more general optical systems, we drop these symmetry assumptions here and use the most general form of the equations.…”

“…While it is possible to minimize the computational effort needed by providing the algorithm with a starting configuration which is as close as possible to the global solution, thereby minimizing the search space, the method for doing this relies heavily on the intuition of the designer and on tools developed in classical aberration theory (such as aberration cancellation in symmetric systems) . We present what we believe to be a new design tool which approaches the problem from a different direction, and thus may be helpful in situations where traditional tools get stuck: a ray tracing engine that provides the expression for the rays, to any desired polynomial order of approximation, as an analytical function of the system parameters. The approach is a generalization of the proximate ray tracing method developed by Hopkins [6][7][8]-a generalization which is made possible through the power of modern computer algebra systems [9] to manipulate large analytical equations.…”

“…In the discussion below, we first review Hopkins' method [6][7][8] and show how it can be readily generalized to arbitrary orders of approximation and to asymmetric systems. We then show how to construct the merit function and optimizer for designing optical systems with this approach and present three example designs.…”