This Demonstration shows how a plano-convex lens acts on a bundle of parallel light rays. Ideally, a collecting lens would deflect all these rays to meet at a single point. If the rays are parallel to the optical axis, this can indeed be achieved by a convex lens surface in the form of a rotating hyperbola, which was already known to Descartes, Huygens, and Newton. For parallel bundles that form a small angle with the optical axis, the concentration to a point holds only approximately. When deflected into an extended light spot, the oblique rays form surprisingly complex and beautiful patterns that can be studied by mode set to "point image".

Compared with the usual spot diagrams delivered by optical design software, the dots are connected by lines. This allows us to trace the dot-producing ray from its starting point. Since the intersections of the rays with the last lens surface are arranged to form a spiral (setting mode to "spiral" shows this), the dots in the image plane form an image of this spiral that can exhibit surprising features.

For large openings (small -numbers), some rays will undergo total reflection and thus not reach the image plane. In this case one finds the image curve interrupted. Setting mode to "lens" shows a total view, comprising the lens, some rays, and the imaging plane.

For layout reasons the light comes from below. This does not perfectly fit the intended interpretation according to which the parallel rays come from a star and the lens is the objective lens of a telescope.

The action of each control is described by a tooltip.

By setting the inclination angle different from zero we see that the ideal focusing property gets lost rapidly with increasing . As a consequence, the imaging capability even for small objects in the center of the field of view is not significantly better than that of a plano-convex lens with a spherical surface. This was not known when the unique focusing property of hyperbolic lenses was discovered. This discovery and the enthusiasm it created is brilliantly described in [1]: Aristotle already posed the problem of a perfectly focusing curve, "the anaclastic", for which a hyperbola is a solution. Kepler was the first to find this solution, although without a correct proof. Descartes was the first to give a valid proof that a plano-convex hyperbolic lens (with geometric data matching the refractive index in a manner derived by him) has the property of focusing axis-parallel rays exactly in one point. Descartes held the opinion that a telescope with such a hyperbolic objective lens would show "whether there are animals on the moon". He did not foresee the limitations caused by light dispersion and diffraction. The destructive effect of the former—the colored fringes from chromatic aberration—were known to Descartes. He thought, however, that they originate from imperfections of the manual lens grinding process and hoped to get rid of these by ingenious mechanical grinding machines that he designed. He spent much time and energy finding craftsmen willing and capable of building such machines. He thought that using machines would be the only way to make a hyperbolic surface profile with sufficient accuracy. In modern optics, hyperbolic lens surfaces are only one species among many kinds of aspherical surfaces that are omnipresent in all but the most simple photographic lenses.

Snapshot 1: a hyperbolic lens in which the hyperbola is close to its asymptotes

Snapshot 2: on axis, where the computed rays intersect the image plane; the deviation from perfect focusing is due to numerical noise

Snapshot 3: only 0.1 degrees off-axis: the computed rays form an extended light spot dominated by a coma

Snapshot 4: 0.3 degrees off axis and 4% out of focus: astigmatism becomes noticeable

Snapshot 5: where the computed rays intersect the hyperbolic lens surface

Reference

[1] D. G. Burnett, Descartes and the Hyperbolic Quest: Lens Making Machines and Their Significance in the Seventeenth Century, Philadelphia: American Philosophical Society, 2005.