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I need to find a substance whose refractive index is constantly changing and directly proportional to the square root of the depth. I know I'm looking for a lot, but I'd really appreciate any ideas or advice from anyone. Thanks a lot

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    $\begingroup$ Square root or whatever, there is no such compound. In effect, you are asking for a metamaterial with peculiar properties; whether or not it can be made is a separate question. $\endgroup$ – Ivan Neretin May 27 '17 at 16:48
  • $\begingroup$ There are commercially available optical fibres with a gradient in refractive index, presumably lenses also, often called GRIN lenses. $\endgroup$ – porphyrin Jul 1 '18 at 7:50
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It's not entirely true that no such substance exists, though you won't find this property in a homogeneous substance.

It's well known that changing the concentration of solute in a solution can change the index of refraction; a concentration gradient would then provide a changing refractive index, though having this mimic a square root function would be difficult. Typically, the refractive index changes linearly with the concentration of solute, so a nonlinear function may be difficult to achieve. A similar approach could be applied to solids, forming a solid solution, though again there would be some difficulty in finding two compounds with similar lattice parameters that would allow to a continuous gradient between compositions to product the refractive index gradient.

Your best bet may be with polymers, where changing proportions of the monomers used could change the refractive index; organic dopants can also be used to effect* this gradient in polymers (see, e.g., Polym. Adv. Technol. 2014, 25, 1099). Again, the difficult part would be matching a square-root function, but you could at least have a varying gradient.

*https://xkcd.com/326/

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