Diopter Focal Lengths and Powers

In photography, a "diopter" is a supplementary lens that is inserted between a camera lens and the subject to alter the focussing distance and magnification ratio. Depending on its application, it could range from a single symmetric element up to a complicated multi-element optically-corrected unit. The power of such a lens is a signed number having a magnitude equal to the inverse of the lens' effective focal length. When the focal length is in meters, the power has units of diopters. Positive and negative powers correspond, respectively, to converging and diverging lenses. For example, a "+4 diopter" is a converging lens with a focal length of 250 mm. The indicated lens power is usually that appropriate for use in air. What happens if the lens is immersed in some other medium, such as water?

Consider a simple diopter consisting of one bi-convex element with radius of curvature 'r' and index of refraction n(g). When surrounded by a medium having index of refraction 'n', its focal length 'f' is given by the lens maker's formula:

f = (r/2)*n/[n(g) - n]

So, if n(w) and n(a) are the indices of refraction of water and air, respectively, the corresponding in-water and in-air focal lengths are

f(w) = (r/2)*n(w)/[n(g) - n(w)]

f(a) = (r/2)*n(a)/[n(g) - n(a)]

Dividing and simplifying, we get

f(w)/f(a) = [n(g) - n(a)]/[n(g)*n(a)/n(w) - n(a)]

Since n(a)=1.0 and n(w)=1.33, this becomes

f(w)/f(a) = [n(g) - 1]/[0.75*n(g) - 1]

As long as the index of refraction n(g) of the lens is greater than that of water, the lens' in-water focal length will always be greater than its in-air focal length. When used underwater it will therefore cause less bending of light rays and will require increased lens-to-subject distances. Since lens power varies inversely as the focal length, the ratio of in-water power P(w) to the in-air power P(a) is

P(w)/P(a) = f(a)/f(w)

The following table shows the results corresponding to some typical values of the lens index of refraction.

n(g)    f(w)/f(a)  P(w)/P(a)
----------------------------
1.5       4.0        .25    
1.6       3.0        .33    
1.7       2.5        .40    
1.8       2.3        .43

Note that a lens made of optical glass (n(g)=1.5) will have its focal length increased by a factor of four! Accordingly, if it's a +4 diopter lens in air, it becomes a +1 diopter lens underwater.