Orestes N. Stavroudis

Simpler derivation of the formulas for generalized ray tracing

The equations for the refraction operation in generalized ray tracing are rederived using the Frenet equations from metric differential geometry and the idea of the directional derivative. Expressions are obtained for the principal directions and curvatures of a refracted wave front in terms of those quantities on the incident wave front and on the refracting surface. This provides the geometrical properties of the refracted wave front in the neighborhood of the traced ray.

Caustic surfaces and the structure of the geometrical image

A general solution of the eikonal equation is used to derive general expressions for both the wave fronts and the caustic surface associated with an orthotomic system of rays in a homogeneous optical medium. Both wave front and caustic can be expressed as the sum of two vectors, the first being the direction vector of the ray and the second being the gradient of the arbitrary function occurring in the general solution of the eikonal equation. This vector can be determined from a knowledge of the angle characteristic function and is closely related to Herzberger’s diapoint characterization of the geometrical image.

Geometry of the half-symmetric image

Herzberger defined the half-symmetric image as the image formed when the manifold of diapoints degenerates into a curve on the meridian plane. The caustic associated with the half-symmetric image is here analyzed in terms of the W and the C± functions discussed in an earlier paper. It is shown that two cases are obtained. In one, an aberration pattern is produced which resembles coma. In the other, an object point is imaged as a curve on the meridian plane and thus resembles astigmatism. However, true coma and true astigmatism both belong to the more general deformation error in which the diapont manifold covers a region of the meridian plane.

Generalized ray tracing, caustic surfaces, generalized bending, and the construction of a novel merit function for optical design

Generalized ray tracing is an algorithm for calculating the geometrical parameters of a wave front in the neighborhood of a traced ray. These calculations are applied, surface by surface, for each traced ray, to an optical system being designed. These calculations determine the two points of contact of each traced ray with the two sheets of a caustic surface. The caustic surfaces are, in fact, aberrated three-dimensional images of object points and therefore contain all information on the geometrical aberrations of the subject lens. Generalized bending is a procedure in which the curvature of a pair of adjacent spherical refracting surfaces, their separation, and the distance to the next succeeding or next preceding surface may be changed so that any paraxial ray is left invariant except at the two affected surfaces. In this study we show that the displacement of a caustic point caused by a generalized bending is in essentially a straight line, that the direction of the displacement is determined by which refracting surfaces are selected, and that the magnitude of the displacement is proportional to the logarithm of the bending parameter. This suggests that caustic surfaces can be used as a merit function in the optical design process, that the merit functions can be calculated by means of generalized ray tracing, and that generalized bending provides an effective means of optimizing the design when included in a feedback loop.

Two-Surface Refracting Systems with Zero Third-Order Spherical Aberration

An analysis is made of refracting systems consisting of two spherical surfaces. Solutions are found for those systems having zero third-order spherical aberration. These are in the form of four one-parameter families of functions. Expressions for third-order coma and astigmatism are derived. Parameter domains for useful solutions are indicated. A method for applying these results to optical design is described.

Canonical properties of optical-design modules

Modules are conceived as building blocks in formulating optical designs with certain useful properties. This paper provides necessary and sufficient conditions for successfully combining (coupling) modules to form such a design system. Canonical ray tracing is defined, and canonical versions of aberration coefficients are derived. These are used to obtain equations for zero values of third-order-monochromatic and primary-chromatic-aberration coefficients. Their utilization in practical problems in optical design is illustrated.

The Fifth Order

The concept of fifth-order aberrations proceeds logically and arithmetically from the idea of third-order aberrations. The latter are the third-order terms of a power-series expansion of an eikonal function. The fifth-order aberrations must then be identified with the fifth-order terms of the same power series. In principle, we can proceed to higher and higher orders of aberrations ad infinitum.

Third-order approximation of the displacement of a caustic point due to a generalized bending

The motion of a caustic point for an axial object point is studied while the lens is perturbed by a generalized bending. It is shown that, to third order, the direction of motion is constant. The magnitude of the motion is proportional to the change in the third-order aberration coefficient. An application to the optical design process is indicated.

Multimodule optical systems

Earlier work on modular optical-design techniques is extended to lens systems that consist of an arbitrary number of modules; the work is generalized to include finite-conjugate systems (copy lenses) and afocal systems as well as infinite-conjugate systems (camera lenses). All such modular designs have the property that third-order spherical aberration and astigmatism are identically zero. Equations are derived for obtaining zero values for the remaining third-order monochromatic aberrations as well as the primary chromatic aberrations. Several examples are given.

Tracing Wavefronts: Can It Be Done?

The starting point of this work is a general solution of the eikonal equation for a homogeneous, isotropic medium which provides a detailed description of the structure of a train of geometrical wavefronts and also of the associated caustic surface. This description depends ultimately on the k-function, an arbitrary function that arises in the general solution. Suppose a known train of wavefronts, with a known k-function, is incident on a refracting surface, tracing a wavefront involves determining the k-function for the refracted wavefront train. To do this a set of boundary conditions needs to be setup and solved.