How to make the Statue of Liberty disappear Basics  Fermat’s principle: Light rays take extremal optical paths, the optically shortest one.  Snell’s law: = , n is called the refractive-index  The Hooke-profile: => trajectories form ellipses Coordinate transformation (mathematically) • Transformation of space: compresses an extended region of physical space into a single point in virtual space so that anything inside the region disappears from sight • Any holomorphic map is conformal: It changes the measure of distance in space but does not alter angles between lines. • Transformation medium: Does a transformation of physical to virtual space so that the virtual position does not depend on the viewpoint. The Zhukovski transformation , |z| >>1 => w≈z, n = | | n', here: n' = 1 Branch points: points where (here: w = +2 and w=-2), are connected by the branch cut (wavy black line, left) which connects the two w-planes (Riemann sheets). The green and blue line miss the branch cut, pass through the medium as if it were empty and leave as if nothing has happened. The red ray enters the branch cut and proceeds towards the singularity of the conformal map, if nothing else prevents it (seems like a shadow for the spectator). How to prevent this: You have to put an additional refractive-index profile n* on the lower How to make the Statue of Liberty disappear Seite 1 How to prevent this: You have to put an additional refractive-index profile n* on the lower Riemann sheet such that light rays form closed loops (e.g. ellipses by the Hooke-profile). Then, light rays entering the lower sheet via the branch cut do a loop back to the upper sheet and leave in the same direction as they came from towards infinity in physical space. => Light does not reach some areas of the lower Riemann sheet, these remain hidden. • If you have something to hide you can do it on Riemann sheets. • You need a transformation medium combined with an additional refractive-index profile n* for the invisibility device. • It also works in 3 dimensions. Sources: • • • • Geometry and Light – The Science of Invisibility (Ulf Leonhardt and Thomas Philbin) Optical Conformal Mapping (Ulf Leonhardt) http://www.dict.leo.org http://www.metacafe.com/watch/806470/david_copperfield_statue_of_liberty/ Michaela Frischmann, 10.07.2012 How to make the Statue of Liberty disappear Seite 2