IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 30 (2009) 935–943 doi:10.1088/0143-0807/30/5/002 Focusing light rays back to the vertex of a reflecting parabolic collector: the equivalent of Dionysius ear effect in optical systems R De Luca and A Fedullo DMI, University of Salerno, 84084 Fisciano (SA), Italy E-mail: rdeluca@unisait Received 16 March 2009, in final form 5 May 2009 Published 8 July 2009 Online at stacks.iop.org/EJP/30/935 Abstract A vertical light ray coming from infinity is reflected by a primary parabolic mirror M 1 having focus at F 1 . At a small distance from F 1 a secondary mirror M 2 , symmetric with respect to the vertical axis, is placed. One would like to find the analytic equation of the mirror M 2 , so that all rays from infinity, incident on M 1 , could be sent back to the vertex V 1 of the primary mirror M 1 . We denote the mirror M 2 with the name of zozzaroid, after P Zozzaro, who first proposed the problem. Recalling the famous legend about Dionysius ear, we find that M 2 has either an elliptic or a hyperbolic profile. (Some figures in this article are in colour only in the electronic version) 1. Definition of the problem When capturing sunlight [1], one first needs to confine solar radiation in a small region of space; secondly, one has to send this radiation into a black body, consisting of a cavity with a small aperture and perfectly reflecting inner walls. In this way, solar energy is in principle stored and is available for many interesting uses such as, for example, production of heated water. However, if a rather large parabolic primary mirror M 1 is used as a collector of solar radiation, the energy density on the secondary mirror M 2 can be so high to melt it. As a matter of fact, if the reflection coefficient of the latter is not very much close to unity, a large amount of energy on M 2 can first deteriorate its optical properties and then cause a transition of phase from solid to liquid of M 2 , because of a rapid temperature increase. In practice, one usually places a second confocal parabolic mirror of small dimensions in front of the primary parabolic mirror in order to reflect back the light rays coming from M 1 . In this way, a bundle of light rays, whose radius is equal to the radius r of the effective surface 0143-0807/09/050935+09$30.00 c  2009 IOP Publishing Ltd Printed in the UK 935