Summary
The image of a point by a lens that is not diffraction limited is often described by its geometrical spot size, defined to be the rms spot radius (not diameter). Although this quantity does not indicate the fractional energy in the spot, it has an intuitive appeal and its square is widely used in optimization merit functions. Calculating the spot size accurately is thus a matter of considerable importance in optical design software. Forbes, in a 1988 paper[1], considers several computational methods and concludes that for lenses having circular pupils, quadrature methods provide superior accuracy to those based on ray tracing through rectangular pupil grids.
Forbes’ results only apply to systems that have a plane of symmetry. Fortunately, it is possible to extend quadrature methods to handle systems without symmetry, and such an extension has recently been added to OSLO. The extended methods still provide superior accuracy and faster evaluation for many systems, when compared to grid methods.
Discussion
Traditionally, the geometrical rms spot size of the image from a single object point has been computed by tracing several rays through different aperture points and treating the intersections of the rays with the image surface as a random distribution. The rms spot size is defined as the square root of the variance of the distribution. In order that the spot size accurately represent the energy distribution, the aperture points for the rays must be chosen carefully. Rays should be spaced so that each ray represents an equal element of solid angle. Various patterns have been used, including circular patterns and square grids, neither of which is particularly accurate. In addition, the reference point on the image surface must be taken as the centroid of the distribution, not the intersection of the chief ray, so that asymmetries in the image are properly accounted for.
Forbes[1] considered the spot size to be defined by an integral equation that represents the limit that would be obtained if the number of rays traced approached infinity. Then, if the object point is located at a distance h from the axis, and the pupil intersection point is described using polar coordinates, one has the following expression for the mean square spot size.
In practice, the integral needs to be evaluated numerically, since it is not possible to trace an infinite number of rays. The question then becomes one of how to most efficiently evaluate the integral. Forbes proposed the standard numerical technique of Gaussian integration, and showed that because of the symmetry of an optical system having a circular pupil, that n sampling points would produce an exact solution for the spot size of a system having aberrations of a degree not exceeding 2n - 1. Thus, for example, if we have a system with circular symmetry whose aberrations are known to be less than ninth order, we can determine the spot size exactly using only 4 sampling points! The sampling points correspond to ray intersection points in the aperture, so the Gaussian integration method shows where rays should be traced. In addition, each ray must have a prescribed weight. The sampling points and weights for Gaussian integration can be computed, and are tabulated in standard references such as Abramowitz & Stegun [2].
If polar coordinates are used to describe the aperture, the sampling points for the angular portion of the integral occur at equal angular intervals, while the sampling points for the radial integral occur at tabulated points within the integration range. For Gaussian integration, there are no sampling points at the end of the interval. This is somewhat awkward, because neither the marginal ray nor the chief ray are used in computing the spot size, although these rays are often traced to obtain other data (e.g. vignetting, distortion). There are two alternate forms of quadrature integration that do produce sampling points at the ends of the integration range. Radau quadrature uses a sampling point at one end of the range, and Lobatto quadrature uses sampling points at both ends of the range. Lobatto quadrature is used as the default method in OSLO, since it uses both the central and edge aperture rays from each defined field point.
The original Forbes work was restricted to systems having plane symmetry and circular pupils. Under these conditions, the method is highly accurate, as the plot in the Design Note indicates. Rays need be traced through only half the pupil, owing to the plane symmetry. If this symmetry is broken, it is necessary to trace rays using the full pupil, and the optimum sampling points and weights must be adjusted. This feature has been recently added to OSLO, which now allows quadrature methods to be used for tolerance calculations that do not assume plane symmetry.
Sinclair & McLaughlin [3] studied systems with vignetted pupils, and obtained errors that were typically several per cent, similar in accuracy to results obtained using aperture coordinates set up using square grids. Typically, pupils are either circular or nearly so, so the Forbes Gaussian integration scheme is quite generally useful, but it is well to remember that it makes no claim of accuracy for systems having non-circular pupils. In addition, of course, it is important to remember that the spot size itself is a single number that provides information about the size but not the distribution of energy in an image.
1. G.W. Forbes, J.Opt.Soc.Am. A5, 1943-1956 (1988).
2. M. Abramowitz & I.A. Stegun, "Handbook of Mathematical Functions", Table 25.4, Dover Publications (1965).
3. D.C. Sinclair & P.O. McLaughlin, Proc. SPIE 1049, p55 (1989).