The first way is by analogy. If we consider a convex bubble and a concave glass lens in air or vacuum, we can see that the interface between water and air in the bubble and the glass and air in the lens is really exactly the same: the material with the higher index (water or glass) curves in and the material with the lower index of refraction curves out. The only difference is that with the bubble, two of these interfaces are face to face and in the lens, they are back to back. Thus the convex bubble and the concave lens should behave the same.
The second way to analyze the problem is to do some ray tracing. A ray of light is bent toward the perpendicular when entering a higher index medium (like water or glass) and away from the perpendicular when exiting a higher index medium. For a concave bubble, a ray going through the center of the bubble will not be refracted at all at either transition. A ray perpendicular to the plane of the "lens" formed by the bubble will be refracted away from the center of the bubble as it enters the bubble and then only partially back toward the original direction as it exits the bubble. THe net effect is one of divergence.
The third method is to analyze a spherical lens using the lens-maker's formula. This will give a negative focal length for a convex lens of air inside water.
Answered by: Ted Dunning, Ph.D., Chief Scientist, ID Analytics
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