The purpose of this memoir is to discover an optical appliance which shall correct in a practical manner the faults in the field of a Cassegrain reflector, while leaving unimpaired its achromatism and the characteristic features of its design, which gives a focal length much greater than the length of the instrument, combined with a convenient position of the observer. The question touches an investigation by Schwarzschild as to what can be done with two curved mirrors the figures of which are not necessarily spherical. With these be corrects spherical aberration and coma, but in order to secure a flat field he is led to a construction in which the second mirror, which is between the great mirror and its principal focus, is concave, and therefore shortens the effective focal length, in place of increasing it. The deformations from spherical figures are also so great, especially for the great mirror, as to leave it doubtful whether the construction discussed could ever be the model for practicable instruments. If we keep to the Cassegrain form, spherical aberration and coma may equally be corrected by deformations of the mirrors which through large, are less extreme, but there remains a pronounced curvature of the field. For this reason I am led, in the present memoir, to consider more complicated systems produced by the interposition of systems of lenses, achromatism can be preserved completely for a single focus if there are three lenses of focal length determined when their position are given, and if all are made of the same glass. One of these lenses, which I call the reverser, is silvered at the back and replaces the convex mirror; the other two are placed close together in the way of the outcoming beam, about one third of the distance from the great mirror to the reverser; the members of this pair, which I call the corrector, are of nearly equal but opposite focal lengths, introducing very little deviation in the ray but an arbitrary amount of aberration, according to the distribution of curvatures between the two faces of each lens. All the surfaces are supposed spherical except that of the great mirror, The essential problem is to bring the necessary work into a form that will allow unknown quantities which express the distribution of curvature between the faces of each lens to be carried forward algebraically. The methods employed are those of a recent memoir by the author,* and a part of the paper is occupied in working out expressions to which this theory leads, for thin lenses, systems of thin lenses, mirrors, reversers and the like, and it may be regarded as an expansion and working illustration of that memoir. Ibis part does not lend itself to summary, When the expressions are obtained the solution proceeds in a straightforward manner, by approximation, which is somewhat complicated owing to the number of considerations which it is necessary to keep in view, but is not otherwise difficult. The solution is completed at the stage where the unextinguished aberrations are considered negligible.
Pitching Single Focus Confocal Analysis One Photon at a Time with Bayesian Nonparametrics