How to use the plane metamaterial lens to realize the function of the dielectric lens (such as convex lens or concave lens), or how to convert the change of the thickness of the dielectric lens into the change of the refractive index of the plane lens, and what is the corresponding relationship between the two
Basically, it is required to transform the thickness of the lens for each light ray to: first, the optical path length, and then to phase shift. And with the phase shift profile you can now design an equivalent metalens.
The fundamentals of the phase change introduced by a lens can be found in Goodman's book "Introduction to Fourier Optics".
It may help Section 3 of an introduction to the optics of metalens: "Optics of the metalens" Article Optics of the metalens
Just as an additional comment, it might be useful to (also) think about how (and why) a Fresnel lens works, since it helps remove the "lens shape" (i.e. concave or convex) implicit assumptions we tend to make when thinking about how lenses work.
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