The Sinai billiard, square torus, and field chaos

An experiment is reported in which the Sinai quantum billiard and square-torus quantum billiard are compared for field chaos. In this mode of chaos, electromagnetic fields in a waveguide are analogous to the wave function. It is found that power loss in the square-torus guide exceeds that in the Sinai-billiard guide by approximately 3.5 dB, thereby illustrating larger field chaos for the square-torus quantum billiard than for the Sinai quantum billiard. Solutions of the Helmholtz equation are derived for the rectangular coaxial guide that illustrate that transverse electric or transverse magnetic modes exist in the guide provided the ratio of edge lengths of the outer rectangle to parallel edge lengths of the inner rectangle is rational. Eigenfunctions partition into four sets depending on even or odd reflection properties about Cartesian axis on which the concentric rectangles are oriented. These eigenfunctions are uniquely determined by four coaxial parameters and two eigen numbers. Justification of experimental findings is based on the argument that the rationals comprise a set of measure zero with respect to the irrationals. Consequently, from an observational point of view, these modes do not exist, which is in accord with the reported experiment. (C) 2000 American Institute of Physics. [S1054-1500(00)00704-7].