In this and the following two problems we will use Fermat’s principle to derive laws gover...

In this and the following two problems we will use Fermat’s principle to derive laws governing paraxial image formation by spherical mirrors.

Consider an object point O in front of a concave mirror whose center of curvature is at the point C. Consider an arbitrary point Q on the axis of the system and using a method similar to that used in Example 3.3, show that the optical path length Lop(= OS + SQ) is approximately given by

  (90)

where the distances x, y and r and the angle θ are defined in Fig. 3.32; θ is assumed to be small. Determine the paraxial image point and show that the result is consistent with the mirror equation

   (91)

where u and v are the object and image distance and R is the radius of curvature with the sign convention that all distances to the right of P are positive and to its left negative.