## Abstract

Measurements of the interommatidial angle (<latex>$\Delta\phi$</latex>) and facet diameter (D) of the same ommatidia in a number of insects and crustaceans with large eyes have been related to the effective intensity at which the eye functions by the following theory. The highest spatial frequency which the eye is able to reconstruct as a pattern is limited by the interommatidial angle <latex>$\Delta\phi$</latex>, which is the sampling angle, because two ommatidia are required to cover each cycle of the pattern. At the same time, the absolute modulation of light in the receptors caused by the pattern depends on three interdependent factors. (a) The theoretical minimum angular sensitivity function, which has a width of <latex>$\lambda$</latex>/D at the 50% level. The wavelength <latex>$\lambda$</latex> is taken as 0.5 <latex>$\mu$</latex>m. This component is not only the limiting angular resolving power of the lens: it reduces modulation caused by all patterns, with greater loss at higher spatial frequencies. Larger lenses increase resolution and sensitivity. (b) The effective light catching area of the rhabdom. This is the angular subtense of the rhabdom area (the receptor) as seen in the outside world (i.e. subtended through the posterior nodal point of the lens), and is the equivalent of the grain size in a film. Large receptors favour sensitivity at the expense of resolution. (c) The F value or focal ratio f/D, as in the camera, where F is the distance from the focal plane to the posterior nodal point. Larger F values increase sensitivity. The modulation resulting from these three factors is then set so that it exceeds the noise caused by the random arrival of photons at each ambient intensity. From this, the optimum value of the product D<latex>$\Delta\phi$</latex> (known as the eye parameter) can be calculated for eyes adapted to any ambient intensity. The same result is reached by a recent theory of Snyder, Stavenga & Laughlin (1977) who calculate the value of D<latex>$\Delta\phi$</latex> which allows the eye to reconstruct the maximum number of pictures despite photon noise. The eye parameter (divided by half the wavelength) is the ratio of the highest spatial frequency passing the lens to the highest spatial frequency reconstructed by the eye. Compound eyes should have larger facets and interommatidial angles than predicted by diffraction theory alone because photon noise must be exceeded at all intensities. Theoretically, D<latex>$\Delta\phi$</latex> lies in the range 0.3 to 0.5 for bright light insects and is increased to 2.0 or more for those active in dim light. As well as depending on intensity, D<latex>$\Delta\phi$</latex> should depend on factors such as the typical angular velocity and level of intensity discrimination at which the eye is used. As D<latex>$\Delta\phi$</latex> can be measured from the outside of the eye, the theoretical predictions can be compared with measured values. Most of this paper consists of maps of the values of D<latex>$\Delta\phi$</latex> for eyes of a variety of arthropods from different habitats. The maps are made by a new convention in which the minimum theoretical field of each ommatidium is placed in angular coordinates as a circle of diameter <latex>$\lambda$</latex>/D with centre on the axis at the place where it lies on the eye. To do this, a value of <latex>$\lambda$</latex> must be assumed. Every fifth facet is taken on most maps with the fields magnified five times. The overlap or separation of these circles of diameter <latex>$\lambda$</latex>/D shows the local value of D<latex>$\Delta\phi$</latex> for any direction for each part of the eye. The method is independent of horizontal and vertical axes, of directions of facet rows, of regularity of facets and of eye radius. The problem of mapping a surface with double curvature upon a flat sheet was solved approximately by working with strips taken along the eye surface. Equal distances on the map then represent equal angles in any direction; with this projection there are no poles, and axes are arbitrary. Maps of D<latex>$\Delta\phi$</latex> reveal that compound eyes differ according to the intensity of light they normally encounter; eyes of animals which are active in bright light have smaller values of D<latex>$\Delta\phi$</latex>. The smallest value of about 0.3 is found in the forward facing acute zone of the sand wasp Bembix, which hovers while hunting in bright sunshine. The absolute limit set by diffraction of 0.25 (for square facets) is approached but never reached. Values of D<latex>$\Delta\phi$</latex> up to about 2.0 or even 4.0 are found in crepuscular animals which have apposition eyes. The interpretation is that the values of D and <latex>$\Delta\phi$</latex> are the result of a compromise between contrast sensitivity and resolution. An increase in aperture provides increased modulation and therefore increased sensitivity, but the additional angular resolving power which comes with the increased aperture is not used because sensitivity is also enhanced by an increase in the receptor size. The ommatidium then detects only the lower spatial frequencies (wider stripes) from the range which passes the lens. In an ommatidium optimized for any but the highest known intensities, both <latex>$\Delta\phi$</latex> and D (and therefore D<latex>$\Delta\phi$</latex>) are larger than they would have to be if set at the diffraction limit. The maps also reveal that many apposition compound eyes have one or more regions of smaller <latex>$\Delta\phi$</latex>, called acute zones. None of the eyes are spherically symmetrical; all have gradients of both D and <latex>$\Delta\phi$</latex>, and all have regions where <latex>$\Delta\phi$</latex> varies in different directions on the eye surface. The acute zone is usually forward looking, but is upward looking in some insects which catch prey or mate in flight. In addition, some dragonflies have a lateral acute zone. In the acute zone the facet pattern is always more regular than elsewhere. Some acute zones, as in the locust (Locusta), the mantid Orthodera and the ghost crab Ocypode, are formed by reduction in <latex>$\Delta\phi$</latex> with little compensatory increase in D. Others such as the native bee Amegilla, the dragonfly Austrogomphus, and the mantid shrimp Odontodactylus have larger values of D which match the decreasing <latex>$\Delta\phi$</latex> towards the centre of the acute zone, so that D<latex>$\Delta\phi$</latex> remains constant. Others again, particularly the wasp Bembix, the dragonflies Hemicordulia, Orthetrum and several mantids, show an increase in D which is insufficient to compensate for the large decrease in <latex>$\Delta\phi$</latex>, so that D<latex>$\Delta\phi$</latex> is smaller in the acute zone. A reduced D<latex>$\Delta\phi$</latex> in the acute zone may imply that it requires brighter light or more time than the rest of the eye in order to make full use of its increased sampling density. Mapping the regional differences of the theoretical resolving power of the ommatidia, and of the potential spatial resolution of different parts of the eye is only the first step towards understanding the functions of the different eye regions. The anatomical basis of the optics, the actual field sizes of receptors as measured physiologically, the part played by binocular overlap, the regional differences in the mechanisms of integration behind the eye, and the patterns of behaviour that are dependent on each eye region, remain to be elucidated.