Imaging and transforming transmission through a medium with nonrotation-symmetric gradient index

The objective of this paper is to show that it is possible to transmit a paraxial optical image and transform through a dielectric inhomogeneous medium whose refractive index is given by n2 = n2(1)(z) + n2(0)[hz(z)x + h2(z)y - g2(z)(x2 + y2)], where n0 = n1(0), and n1, g, h1 and h2 are arbitrary functions of z. The optical image transmission, with a scaling factor F = H2(zm), m being an integer, is obtained at planes z = zm such that H1(zm) = 0 (the image condition), and the optical transform transmission is obtained at planes z = zm such that H2(zm) = 0 (the transform condition), where Hz(z) and H2(z) are two independent solutions of the paraxial ray equation H(z) + g2(z)H(z) = 0 with the initial conditions H1(0) = 0,H1(0) = 1,H2(0) = 1, and H2(0) = 0, where the point denotes the derivative with respect to z. Finally, we show that this medium can be represented by a transmittance function similar to the spherical-lens transmittance function and thus can be an element of image-forming systems.