Fig. 2. Reflectance geometry. (A) Directionality. A directionally iridescent feather can be understood as an imperfect spectrally tuned mirror. Light from a point source, I, incident at angle , is reflected in a diffuse beam centred on axis R at 180°-. The width of this (approximately circular) beam at 50% of maximum intensity is . The structurally reflective surface is not always in the same plane as the feather vane, and its tilt, t, out of this plane can be calculated from the angles I and R with respect to the feather surface. Tilt was predominantly away from the proximo-distal axis. For t>0°, light from directly above the feather is directed towards its base, and for t<0° towards the apex. Unlike the iridescent reflection, the direction of non-spectrally selective specular reflection was consistent with the reflective surface being in (or close to) the plane of the feather vane. Spectral tuning of laminar interference reflectors varies with angle of incidence, (see Fig. 7 in Land, 1972). (B) The spectral location of the reflectance peak, _max, varied across the reflected beam. For a fixed observer (see Fig. 1A), _max was independent of surface orientation, O, and a linear function of the angle, E, separating I from the line of sight. i.e. _max=a-bE, where a is _max at E=0°, and b is a constant. (C) We can see why _max might be independent of O because, on the axis of reflection, the angles of incidence (I') and reflectance (R') are equal, here being . If O varies and E is fixed, then the angle of incidence and reflectance shift from by equal and opposite amounts, ±ß. Consequently, spectral shifts in _max for the incident and reflected light are approximately equal and opposite, compared with the value of _max, where angles of incidence and reflectance are equal and can be expected to cancel each other out.

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