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First published online May 1, 2006
Journal of Experimental Biology 209, 1904-1913 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02223

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Partial coherence and other optical delicacies of lepidopteran superposition eyes

D. G. Stavenga

Department of Neurobiophysics, University of Groningen, NL 9747 AG Groningen, The Netherlands

View larger version (38K): [in a new window]   Fig. 1. Diagram of an ideal refracting superposition eye in the dark-adapted state. Light entering an eye first passes the corneal facet lenses (c) and subsequently the crystalline cones (cc). Proximal to the array of crystalline cones is the clear zone (cz) and the layer of rhabdoms (rh). A beam of light parallel to the optical axis of an ommatidium is focused on the rhabdom of the central ommatidium. Sheets of screening pigment surround the crystalline cones. In many species, tracheolar tapeta and/or screening pigment isolate the rhabdoms from each other. The set of facets that contributes to the superposition image of a distant point source is called the superposition aperture.

View larger version (25K): [in a new window]   Fig. 2. Schematic light paths in a superposition eye (modified from Land, 1984). (A) Focal point at the retinal surface. r is the radius of curvature of the distal surface of the rhabdom layer, w is the clear zone width, i.e. the distance of the rhabdom layer surface to the proximal tips of the crystalline cones, and c is the thickness of the crystalline cone and facet lens layer. R=r+w+c is the radius of the corneal outer surface, or the eye radius, and p=r+w is the radius of curvature of the proximal cone tips. is the angle of incidence of a ray through the vertex of a facet, and the angle of that ray with the ommatidial axis after having passed the facet lens and crystalline cone is the exit angle ß. The central ray is defined by =ß=0. The oblique ray travels a distance q in the clear zone before intersecting the central ray at point P, which ideally coincides with the tip of the central rhabdom. (B) Focal point proximal to the retinal surface. The focal point P* is then located at a distance r* from the center of curvature of the eye and w* from the proximal cone tip. The distance traveled across the clear zone by the oblique ray is q*. The difference in optical path lengths of the two rays in point P* is u+nq*–nw*=u+n(q*–w*), where u is the path length difference between the ray with incident angle, , and the central ray, when reaching the corneal surface, and n is the refractive index of the eye tissue.

View larger version (23K): [in a new window]   Fig. 3. Angular magnification, m, as a function of the exit angle, ß, calculated for the moths and skippers listed in Table 1 with the condition that the superposition eyes are ideal, that is, all incident light rays from a distant point source are assumed to converge at the tip of one and the same rhabdom (see Fig. 1). For species abbreviations, see Table 1.

View larger version (16K): [in a new window]   Fig. 4. Defocus distance for two moth species as a function of the exit angle, with angular magnification m=1.32 for E. kühniella and m=2.6 for P. tristifica.

View larger version (52K): [in a new window]   Fig. 5. Facet lens lattice with numbers, where the facets are classified according to their distance from the central facet, number 1. The facets 1, 2, 4, 6, 9, etc. are the facets of adjacent ommatidia in a meridional section (indicated by arrows). Rays parallel to the central ray (the ray through the vertex of facet 1) that travel through equal-numbered facets have the same optical path length difference with the central ray.

View larger version (18K): [in a new window]   Fig. 6. Optical path length difference and coherence length of the diurnal moth P. tristifica and the nocturnal moth E. kühniella. The numbers adjacent to the large symbols indicate facets in a meridional row (Fig. 5). The optical path length difference between rays passing through the facets marked by the number 2 and a central ray passing through facet 1, is smaller than the coherence length of light for the green-sensitive photoreceptors (lc=2.8 µm). This also holds between rays passing through facet pairs 7 and 8, 13 and 14, and 16 and 17.

View larger version (25K): [in a new window]   Fig. 7. Fraunhofer diffraction patterns of facet annuli for the diurnal moth P. tristifica, where the facet lens diameter is 24 µm, for monochromatic light with wavelength 530 nm. The light flux density at the corneal level is 1 W µm–2. (A) The light distribution in the focal plane of the facet lens–crystalline cone system, assuming a focal distance of 200 µm. The irradiance due to the single, central facet is low and spread out. The peak irradiance resulting from the sets of 6 facets with numbers 2, 3, 4 and 6 is 36 times higher than that of the central facet, and the peak irradiance due to light from the set of 12 facets with number 5 is 144 times higher. (B) Normalized light patterns plotted as a function of the angle. The angle is the lateral distance of a divided by the focal distance. The width of the light distribution patterns reduces with increasing radius of the annuli.

View larger version (16K): [in a new window]   Fig. 8. Diagram of a facet lens with obliquely incident light. A light ray parallel to the ray through the lens vertex, arriving at the facet lens margin has an angle of incidence m= +l, where l=arctan(Dl/2Rl), with Dl the facet lens diameter and Rl the radius of the facet lens front surface.

View larger version (20K): [in a new window]   Fig. 9. Diagram of two types of corneal nipples, with parabolic and sinusoidal cross-section. The corneal nipples are spaced at a distance of d=220 nm, and they have a height h=250 nm. The paraboloid is chosen so that the area at its base equals the area of the unit lattice cell, and has at the base a radius . The radius of the sinusoid at the base is r0=d/2=110 nm.

View larger version (13K): [in a new window]   Fig. 10. The effective refractive index for paraboloid and sinusoidally shaped nipples calculated for a corneal nipple layer. For the paraboloid nipples, the effective refractive index decreases virtually linearly from the facet lens substrate (amplitude 0 nm) to the nipple tips (amplitude 250 nm), but for the sinusoidal nipples the decrease is more abrupt.

View larger version (21K): [in a new window]   Fig. 11. The reflectance of the facet lens surface with paraboloid and sinusoidal nipples as a function of the angle of incidence and polarization, for light of wavelength 530 nm. The value of the refractive index of the facet lens medium is 1.52 (Vogt, 1974). (A) The reflectance for TE waves. (B) The reflectance for TM waves. For a smooth surface, that is in the absence of nipples, the reflectance for TM waves falls to zero at the Brewster's angle =arctan(1.52)=56.7°. Both types of nipples strongly reduce the reflectance at angles of incidence below 50°, but at larger values the reflectance rapidly rises. A Brewster's angle only exists in the absence of nipples.