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The geometric optical characteristics of Morse lens and its inside-out version

In this letter, we will explore the geometric optical characteristics of Morse lens. We derive the general trajectory formula and find that parts of them in generalized Eaton/Luneburg lenses are coincided to circles. Moreover, we find a new kind of combined lenses that can perform perfect images on the basis of Morse lens. Our work provides new ideas for Morse lens, other traditional lenses and their combinations.