Modes in a Sequence of Thick Astigmatic Lens-Like Focusers

One possible long distance transmission medium for optical waves consists of a periodic sequence of converging lenses. In order to negotiate unwanted hut unavoidable bends of the axis of the sequence it is necessary to space the lenses as closely as possible.1 Nevertheless, ordinary dielectric lenses exhibit substantial surface scattering, and therefore the minimum spacing between lenses depends on the tolerable transmission loss. D. W. Berreman has shown that an effective lens can be made using gas with thermal gradients,2 3 thus avoiding the .solid-to-gas transition problems. 1). W. Berreman and 8. E. Miller4 proposed a gaseous lens consisting of a tube with hot walls through which a mild gas current at lower temperature is forced to How. At any cross section the temperature increases from the center to the wall. The density and consequently 2887 2888 T H E B E L L SYSTEM T E C H N I C A L J O U R N A L , NOVEMBER 1964 the dielectric constant is then maximum on the axis and decreases radially roughly with a square law. Without the problem of scattering at the interfaces, tubular gas lenses can be closely spaced and the gaps may be comparable to the thickness of the lenses. The advent of such a new transmission medium makes it opportune and important to generalize the theory of modes in a sequence of thin lenses by determining the normal modes in an idealized structure which consists of a periodic sequence of arbitrarily thick slabs of dielectric whose dielectric constant tapers off radially with quadratic law.