Fake Airy functions and the asymptotics of reflectionlessness

Abstract

Two classes of analytic refractive-index profile P2(z, in ), whose reflection coefficients r are zero for all values of a parameter in , are studied as in to 0. The aim is to understand why r=0 rather than r varies as exp(-1/ in ) as for generic profiles. The authors find that reflectionlessness is a consequence of the fact that transition points of P2 (zeros or poles in the complex z plane) form tight clusters (whose size vanishes with in ) which can be regarded neither as coalesced nor well separated. Expansion near a cluster yields the local wave not as the usual Airy function, whose Stokes phenomenon generates reflection, but as Bessel functions of half-integer order (fake Airy functions) which are exactly trigonometric functions with no Stokes phenomenon and so no reflection.