The weak point of the parabolic mirror is in dealing with rays coming in parallel but oblique to the axis. Figure 67 shows the situation plainly enough. The reflected rays a′, a″ no longer meet in a point at the focus F but inside the focus for parallel rays, at f forming a surface of aberration. The practical effect is that the image rapidly deteriorates as the star passes away from the axis, taking on an oval character that suggests a bad case of astigmatism with serious complications from coma, which in fact is substantially the case.
Fig. 67.—Aberration of Parabolic Mirror.
Even when the angular aperture is very small the focal surface is nevertheless a sphere of radius equal to one half the focal length, and the aberrations off the axis increase approximately as the square of the relative aperture, and directly as the angular distance from the axis.
The even tolerably sharp field of the mirror is therefore generally small, rarely over 30′ of arc as mirrors are customarily proportioned. At the relative aperture usual with refractors, say F/15, the sharp fields of the two are quite comparable in extent. The most effective help for the usual aberrations[14] of the
mirror is the adoption of the Cassegrain form, by all odds the most convenient for large instruments, with a hyperboloidal secondary mirror.
The hyperboloid is a curve of very interesting optical properties. Just as a spherical mirror returns again rays proceeding from its center of curvature without aberration, and the paraboloid sends from its focus a parallel axial beam free of aberration, or returns such a beam to an exact focus again, so a hyperboloid, Fig. 68, sends out a divergent beam free from aberration or brings it, returning, to an exact focus.
Such a beam a, a, a, in fact behaves as if it came from and returned to a virtual conjugate focus F′ on the other side of the hyperbolic surface. And if the convex side be reflecting, converging rays R, R′, R″, falling upon it at P, P′, P″, as if headed for the virtual focus F, will actually be reflected to F′, now a real focus.
