Effective Focal Length of a Lens and Plane Port

What is the effective focal length of a lens used underwater behind a plane port? A thin bi-convex lens of focal length F in air produces an inverted real image as shown in the figure. The ratio of image height to object height, cd/ab, is the called the magnification 'm'. From the similar triangles abf and cdf this ratio is also fc/fa, or

m = F/(O - F)



Underwater, refraction at the plane port bends light rays so that the object appears to be 25% closer to the port than measured. Assuming that the separation of port and lens is small compared with the object distance, the effective underwater magnification is therefore F/(0.75*O - F). With the object at a distance O in air, we need to know what focal length F' produces this magnification. Clearly, it must satisfy the equation

F'/(O - F') = F/(0.75*O - F)

Solving for F' we get

F' = 1.33*F

The effective underwater focal length is 33% greater than in air..