Imaging stretching and displacement using gradient-index elements during the lens design process

In this study, we propose an approach to stretch or translate images using gradient-index (GRIN) elements with a rotationally symmetric shape in lens systems. In this method, the GRIN material, instead of optical surfaces, are utilized to enable a breaking of rotational symmetry for the two image translations. GRIN expression with anamorphic and tilting terms is introduced. A pair of GRIN elements in front of the given system alters the magnification in two orthogonal directions using the anamorphic terms in the expression, which realizes image stretching. A pair of GRIN elements with tilting terms is used after the given system tilts the optical path to achieve a transverse displacement of the image. The structure of the given system remains unchanged when these translations are performed. A design method for the GRIN elements is presented. Additionally, a design example is presented whose image is stretched by 1.33 times in one direction and displaced to one side of its axis to demonstrate the feasibility of the proposed approach. The approach in this study may enable novel imaging GRIN lens system designs with flexible image positions or special optical functions.