# What is the math behind the smartphone fish eye lenses

I found out information that these add-on fish eye lenses for smartphones have a focal length of about 2mm. I bought a lens and tried it with several different phones and it worked. I found out that these phones have different focal lengths (3.85mm and 5mm). My question is what is the mathematic behind this optical combination (fish eye lens + phone lens). These are my calculations: $\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}$; I assume that distance between both lenses is 1mm. The image of the fish eye lens is the object for the second lens so: $\frac{1}{4}=\frac{1}{-1}+\frac{1}{d_i}$, or $d_i=0.8\rm{mm}$. So the focal point of the total system has changed from 4mm (without fish eye lens) to 0.8mm. Am I wrong?

In practice, however, the object you are taking a picture of is likely to be at a distance $d\gg 2\rm{mm}$ away, and your math is a good approximation. You should keep in mind however that in general two lens systems cannot be simplified as a single lens with one focal point.