# Can a biconcave thick lens ever have a positive focus? I.e. if both sides have radius of curvature -/+R?

Since a thin biconcave lens has a negative focus, can a thick biconcave lens with a certain thickness have a positive focus if both sides have the same (magnitude) radius of curvature?

## 2 Answers

Gullstrand's equation says that the equivalent power of a thick lens ($P = 1/f$) is given in terms of its front and rear surface powers $P_1$ and $P_2$ by $$P = P_1 + P_2 - P_1 P_2 \frac{d}{n},$$ where $d$ is the separation between the surfaces and $n$ is the index of refraction of the lens material. For concave surfaces, $P_1$ and $P_2$ are negative, and $d$ and $n$ are always positive. Thus, the equivalent power will always be negative, which means that the equivalent focal length would always be negative as well.

Similar arguments would apply for the front and rear vertex focal lengths, using the formulas in the link above.

• Ah, never heard of that equation, thanks for your answer! – Laura Feb 9 '16 at 22:34

Finally thought this out, hopefully it's right. So for the lensmaker's equation, (n-1)[-2/R - (n-1)d/(n R^2)] = 1/f. For a positive focus, it requires thickness d to be negative, which is impossible (since for lenses, n>1)