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A Luneburg lens is a spherical lens with a gradient refractive index. It has the interesting property that light coming from focal length of infinity will be focused on the surface of the lens.

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The gradient refractive index n(r) of a Luneburg lens of radius R is calculated by the following formula:

n(r)=(2-(r/R)^2)^1/2

The refractive index at the outer layer of the Luneburg lens (r=R) is 1.

To make an Luneburg lens that operates in air would thus require layers of the Luneburg lens to have refractive indexes of 1 and above.

Transparent solids have refractive indices that start at about 1.3, so it would be impossible to make a Luneburg lens that operates in air.

I am wondering if there is a more generalised Luneburg lens formula that accounts for Luneburg lenses in a medium other than air? For example, a Luneburg lens in water.

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    $\begingroup$ if the refractivity of the "outside " is $n_0$ then the lens should have index $n_L(r)=n_0\sqrt {(2-|\tfrac{r}{R}|^2)}$ $\endgroup$
    – hyportnex
    Commented Jul 20 at 2:08
  • $\begingroup$ Nanoscale metamaterials can achieve small or even negative refractive indices. $\endgroup$
    – A Nejati
    Commented Jul 20 at 2:09
  • $\begingroup$ @ANejati but generally only for a specific wavelength $\endgroup$
    – Carl Witthoft
    Commented Jul 20 at 15:32
  • $\begingroup$ Interesting methods of manufacture: laserfocusworld.com/optics/article/14188413/… . And, as @hyportnex wrote, if your lens is embedded in a block of material, all that matters is that the Luneberg lens' index at the center by sqrt(2) times the interface (external material) index $\endgroup$
    – Carl Witthoft
    Commented Jul 20 at 15:55
  • $\begingroup$ @CarlWitthoft With any material (including glass or plastic), this type of lens only works at one wavelength, due to dispersion ('chromatic aberration'). $\endgroup$
    – A Nejati
    Commented Jul 20 at 16:41

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