The compound eyes of most diurnal butterflies have a reflecting tapetum below the retina. Light that enters the eye is guided down the rhabdom, reflected by the tapetum, and then guided back up the rhabdom. The light that is not absorbed by the rhabdom is reemitted and gives rise to an eyeshine. We have measured the fraction of the incident light that is re-emitted, and also the degree to which this light retains its original polarization. The following conclusions are drawn: 1. Even at the wavelength where the eyeshine is most intense, only a few percent of the incident photons are re-emitted. 2. The tapetum acts as a plane mirror that preserves polarization. 3. The light that passes through the rhabdom in secondorder waveguide modes is depolarized to a greater extent than the light contained in first-order modes. The depolarization is expected to decrease only slightly the polarization sensitivity of the retina. 4. Theoretical modelling of the waveguide properties of the rhabdom provided a way of using depolarization measurements for estimating the refractive index of the rhabdom. The measured amount of depolarization is consistent with the dispersion of phase velocities of different second-order modes propagating in a rhabdom of refractive index 1.363.