We study the formation of caustic produced by refraction through conic surfaces, considering a point source placed along the optical axis at arbitrary distances from the vertex of refracting surface. We demand that the optical surface is represented by a mathematical function, which is smooth, continuous, and derivable. We implement an exact ray trace to obtain a monoparametric equation that describes a family of refracted rays, which are propagated as a function of the angle of emission from the point source for each ray. Subsequently, by using the envelope’s method, we provide an analytical equation for the caustic surface as a function of all the parameters involved in the process of refraction. We analyze the paraxial approximation assuming a very small angle about the axis of the system. Additionally, we provide a formula, which describes the conditions for total internal reflection. Finally, we present the formation of caustic surfaces by considering a liquid polymer in a rotating vessel forming a paraboloid surface, also we present the caustic surface produced by the package of a Light Emitting Diode, to produce an uniform illumination pattern.
A method for designing afocal achromatic doublet is presented. We have implemented an exact ray trace through a separated doublet lens considering a plane wavefront propagating along the optical axis. The analytic equation of both the caustic surface and the back focal length for separated doublet lenses are provided. Demanding that the back focal length tends to infinity, we impose the conditions to design afocal optical systems, obtaining sixth and fourth degree polynomials as a function of the radii of curvature. In order to produce an afocal achromatic optical system, we solve numerically a set of two nonlinear equations assuming two spectral lines. Therefore, we have two unknowns which are the curvature radii for both the front surface and the rear surface. The contribution of this work is to provide simple formulas for designing optical beam expander or reducer devices based on separated doublets.
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