Design of graded-index lenses in the superposition eyes of scarab beetles

Abstract

Superposition-image quality in the clear-zone eye depends in the first instance on the optical characteristics of the lens elements in each ommatidium. The optical design strategy of the two lens elements, a thick corneal facet and an underlying crystalline cone, in the scarab eye is reported. The formation of a good superposition image at the rhabdom layer in the eye demands that the lens elements be precisely arrayed, virtually free of optical aberrations, and that each lens pair function as an afocal (telescopic) lens system with an internal intermediate focal plane. The optical properties of the corneal facet were examined by a variety of means. The isolated corneas of most scarab species focused good quality images of a distant object. Cardinal-point analysis of the intact corneal lens revealed that the back focal point of the lens lies just proximal to the inner corneal surface, many micrometres distal to the rhabdom layer, and the position of the principal planes suggested that the corneal lens had internal lens-cylinder properties. This was confirmed by the examination of the focusing power of transverse lens slices of known thickness; the power of the corneal lens slice was a function of its thickness. Interference refractometry of corneal sections revealed that the facet is a graded-refractive-index (g.r.i.) lens in the great majority of more than 40 scarab species examined. The position of the back focal point is achieved in a thick corneal lens by (i) the presence of a g.r.i. lens, best developed in the proximal corneal region, where it consists of a g.r.i. lens cylinder capped by a g.r.i. lens hemisphere, and (ii) the loss of front facet curvature in the homogeneous distal corneal region. In situ , the back focal point lies deep within the crystalline cone. Since the quality of the superposition image depends on the exact location of the intermediate-image plane in the crystalline cone, this position was determined from a comparative analysis of cone shape, experimental observations, and theoretical modelling of the cone. Four observations, namely the presence of a waist in the crystalline cone of many species, the back focal distance of the isolated cornea when the refractive index (r.i.) of the medium in the back focal space approximated that situ, the presence of screening pigment around specific regions of the crystalline cone and the position of the intermediate-image plane in the exocone of a passalid beetle eye, all suggested that the intermediate focus lies in the waist region. The proximal region of the crystalline cone was modelled on the basis of its known g.r.i. lens properties. The model used comprised a radial g.r.i. lens cylinder with a parabolic profile in r.i., terminating in a g.r.i. lens hemiellipsoid. Dimensions and r.i. distribution in the model were based on values from real cones. The model cone focused an incident parallel beam to a point within the cone corresponding to the waist region in real cones. For beams at angles as great as 20° to the optic axis, aberrations in the model cone are small, and restricted to the most peripheral rays. A homogeneous hemiellipse of similar dimensions has severe aberrations for beams at an angle to the optic axis. The model predicts that the ommatidial optics are diffraction-limited; the spread of rays leaving the proximal cone tip due to diffraction at the small exit aperture of the cone (for all aperture diameters) is broader than that due to lens aberrations. Consequently, tolerance exists to optical imperfections in the lens components and their spacing. A tolerance in the position of the intermediate focal plane of + 2-3 pm was calculated. Lens design is strongly correlated with the daily activity pattern of the scarab species under consideration. The corneal facets of nocturnal and crepuscular species are wide with little individual facet curvature; such ‘glacial’ corneas are completely transparent. The crystalline cone is large and well developed. In diurnal species, the corneal facets are narrower, with strong individual curvature, and the corneal lens cylinders are often lined with a brown screening pigment. The crystalline cones of diurnal scarabs are frequently strongly waisted or greatly reduced in size. Pigment surrounding the cone waist serves as a field stop limiting the angular acceptance of the ommatidial optics. The waist limits the number of ommatidia that can contribute to the superposition image and therefore determines the maximum aperture of the eye. This aperture is greatest in nocturnal species with little or no waist constriction in the crystalline cone. Most scarab clear-zone eyes are of the eucone type (separate crystalline cone). However, in the Passalidae and bolboceratine and pleocomine Geotrupidae, the crystalline cone is replaced by a corneal g.r.i. lens extension, the exocone, that serves as an optical analogue of the crystalline cone.