Maximum Transmission Coefficient Through a Smoothly Inhomogeneous Medium

We consider the problem of energy transmission from one aperture to another by the wave fields in smoothly inhomogeneous media. The study is performed in the quasioptical and scalar approximations when the aperture sizes significantly exceed the wavelength and there exists a ray connecting the aperture centers. The transmission coefficient is determined using the well-known parabolic equation, which describes the wave-beam propagation in smoothly inhomogeneous media. As in the homogeneous media, the maximum transmission coefficient is shown to be reached when the field structures at the aperture are specified in the form of special functions, namely, the prolate spheroidal angular functions. The maximum achievable transmission coefficient is determined.