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We present a practical algorithm for designing an aperture field (source) that propagates along a predefined generic beam trajectory that consists of both convex and concave sections. We employ here the mechanism that forms the well-known Airy beam in which the beam trajectory follows a smooth convex caustic of the geometric optics rays and generalize it for a class of beams that are referred to as "caustic beams" (CBs). The implementation is based on "back-tracing" rays from the predefined beam trajectory to the source's aperture to form its phase distribution. The amplitude is set in order to form a uniform smooth amplitude of the CBs along their trajectories. Several numerical examples are included.

An efficient method for computing the problem of an electromagnetic beam transmission through deep periodic dielectric gratings is presented. In this method the beam is decomposed into a spectrum of plane waves, transmission coefficients corresponding to each such plane wave are found via Rigorous Coupled Wave Analysis, and the transmitted beam is calculated via inverse Fourier integral. To make the approach efficient for deep gratings the fast variations of the transmission coefficients versus spatial frequency are accounted for analytically by casting the summations and integrals in a form that has explicit rapidly varying exponential terms. The resulting formulation allows computing the transmitted beam with a small number of samples independent of the grating depth.