To avoid sign convention confusions, assume that the
object is placed on the optic axis a distance z1 (> 0) in front of the
lens. If the GRIN lens acts like a converging lens then the image
will be formed on the optic axis a distance z2 (>0) behind the lens. Fermat's
principle tells us that the light travels along the path that takes minimum
time, i.e. the smallest possible optical path length. The corollary
to this is that all paths taken by the light (i.e. through different portions
of the lens) have the same optical path length, so that the optical path
length does not vary with the distance r from the optic axis that the light
passes through the lens. Assume that the lens is thin enough (t is
small enough) that the index of refraction does not vary over the path
the light takes through the lens at a particular value of r. Then
the optical path length for a path traversing the lens at distance r from
the optic axis is
If we make the paraxial approximation, so that r is
small compared to z1 and z2, then we can expand the
square roots and this is approximately
Fermat tell us that this is stationary, i.e. independent
of r:
For this to be true for all r, we must have
This is the imaging equation for a thin lens with the desired focal length.