Oriental magic mirrors and the Laplacian image

The pattern embossed on the back of an oriental magic mirror appears in the patch of light projected onto a screen from its apparently featureless reflecting surface. In reality, the embossed pattern is reproduced in low relief on the front, and analysis shows that the projected image results from pre-focal ray deviation. In this interesting regime of geometrical optics, the image intensity is given simply by the Laplacian of the height function of the relief. For patterns consisting of steps, this predicts a characteristic effect, confirmed by observation: the image of each step exhibits a bright line on the low side and a dark line on the high side. Laplacian-image analysis of a magic-mirror image indicates that steps on the reflecting surface are about 400 nm high and laterally smoothed by about 0.5 mm.