Graded-index model of a fish double cone exhibits differential polarization sensitivity

The close apposition of the inner segments of the two cones that combine to form a double cone causes the pair of cone inner segments to guide light as a unitary structure whose transverse sections are roughly elliptical. Electron micrographs of the photoreceptors of a green sunfish (Lepomis cyanellus) retina provide evidence that the refractive index in the ellipsoid region of the inner segments of the double cones is higher in the center than at the perimeter. The hypothesis that the shape and refractive-index gradient could confer differential polarization sensitivity on double cones is examined with a two-dimensional waveguide model of a double-cone inner segment. The model has a dielectric constant that varies parabolically along the narrowest (x) dimension, leading to the index profile: n(x) = nmax[1-(x/x0)2]1/2, where nmax is the peak value of the index and x0 is a parameter specifying the rate at which the index decreases with increasing magnitude of x. A quantity, the polarization contrast, is introduced as a measure of the differential polarization sensitivity of adjacent receptors in the square mosaic of double cones in the sunfish retina. Polarization contrast is proportional to the relative difference in power absorbed by two double cones oriented with their shortest axes orthogonal to each other and stimulated by a field of uniform polarization. Polarization contrast is computed as a function of wavelength for appropriate values of nmax and x0. For normally incident light polarized parallel to one of the two axes of the double cones' cross sections, the polarization contrast is generally between 1% and 5% for wavelengths ranging from 550 to 750 nm. Over most of those wavelengths the polarization contrast of the graded-index-model double cone is approximately five times as large as that of a homogeneous-slab model of the same size and average refractive index. Additional benefits of a graded index, optical isolation of adjacent photoreceptors and antireflection at the photoreceptor entrance, are also discussed.