Dielectric Guide with Curved Axis and Truncated Parabolic Index

A dielectric guide in which the refractive index decreases with parabolic law away from its axis acts as a lens-like medium. 1,2 The transmission through it is known even if the axis is not straight 3 and if the parabolic decrease is different in two orthogonal directions4 (astigmatic guide). Though extremely useful in many respects the parabolic medium is not realizable since it has ever-decreasing refractive index away from the axis and this in turn produces an untenable physical result. Thus though we know that in any realizable dielectric guide with curved axis, radiation losses are inevitable, 5 the modes in the parabolic medium with curved axis can have no radiation loss since the refractive index tending towards infinity far away from the axis prevents it. A more realistic model is achieved by truncating the parabolic index distribution. We begin, in Section II, studying the two dimensional guide, Fig. la, in which the index profile, Fig. lb, varies as a truncated parabolic function along the x axis and is independent of y while outside of the guide the index is uniform. Later, this guide is modified in such a way that along y, the index profile is either rectangular, Fig. 2a, or another truncated parabolic function, Fig. 2b. The first of these guides has the index distribution of the dielectric thin-film guide proposed in Ref. 6 as a low-loss, easy-to-support ribbonlike guide for millimeter and optical waves. It has also the configuration of a possible guide for integrated optics.7 This guide, with curved axis has been analyzed in Ref.