Does adding water on convex mirror increase its power?

Question

What is the comparison between the power of convex lens in air and in water ? What causes the change in its focal length

Answer 1

The short answer is that, if steeped in water, the optical power decreases, possibly a great deal if your lens is a crown glass (like N-BK7) with a refractive index of 1.5 or so. It doesn't matter whether the lens is concave or convex: the power is scaled down (made less negative if the lens is diverging) by the same scaling constant:

$$\frac{n_{glass}-n_{water}}{n_{g;ass}-n_{air}}$$

This is because the power of a spherical interface is given by $\Delta n/R$, where $\Delta n$ is the difference between the refractive indices on either side of the interface and $R$ the radius of curvature, with the sign of course being positive (converging) if the denser optical medium is the convex hull (you know intuitively what I mean, but if you want to get rigorous then you say that the denser medium is the convex hull if a line between any two points within the medium stays within the medium (look up the definition of a convex hull).

Think of this thought experiment. You're looking at a lens, or any glass object and you steep it in a magical liquid whose refractive index you can vary. You begin with the liquid's index at 1 and raise it slowly. When the liquid's index reaches that of the glass object, the glass object seems to disappear.

The easiest way to derive this is to make the thin lens approximation, that is that the deviation of the lens surface from the transverse plane (normal to the optical axis) is so small that you can think of the interface as a phase mask. Then you derive the delay relative to the phase of an on-optical-axis ray as a function of radial co-ordinate and compare the answer with the known field curvature expressions for Gaussian optics: see my answer here for more details.