The technique of calculating aberrations of homogeneous optical systems considered in Chap. 4 can be extended to the systems containing GRIN elements. However, to do this, one needs to obtain the formulae describing how aberration coefficients must be transformed when an aberrated wave propagates through an inhomogeneous medium. As with the homogeneous medium, this problem can be solved by first obtaining the transformation formulae for aberration coefficients when the wave propagates between two planes and then the formulae relating aberration coefficients at the vertex tangent plane and spherical surface. As far as the transfer from a sphere to an aspherical surface in an inhomogeneous medium is concerned, such formulae will hardly be needed because the difficulties of fabricating GRIN lenses with aspherical surfaces now outweigh the advantages of this combination. However, the development of this technique showed that the mathematics is so cumbersome that its use even for calculating low-order aberrations is practically impossible. That is why, to calculate third and lower-order aberrations of hybrid optical systems with GRIN elements, the authors employ methods using optical invariants and quasi-invariants. To calculate higher methods using optical invariants and quasi-invariants - . To calculate higher-order aberrations, the authors developed a universal method based on tracing orders. This computer method makes it possible not only to obtain ray aberrations and aberration coefficients of practically any order, but also to optimize the whole process of defining the design parameters of the hybrid optical system.
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