Answer Happy

Example 4 A plane glass disk with a small thickness do is used to focus collimated light Show that for stigmatic imaging on axis (free from spherical aberration) one needs to use glass with gradually varying refractive index as a function of the distance h to the optical axis h2 2df where n, is refractive index on axis. Hint: apply Fermat's principle of stationary optical path n(h) n do

Example 4 (Solution) Perfect imaging on axis can be achieved if the optical paths of the ray at any height h to the focal point F' (AA'F') are the same and equal to the optical path along the axis [BBF'] = n,do+f, and (AA'F'] = n(h)do + A'F', where (A'F')2 = 12 + f2 We have S n²+ f² n.de + s = don(h)+ Vh?+S= n(h)=n. + de d h? 1h h Expanding +1=1+- +... = n(h) n,+ 2² 21 2dS I h2 =n d. 1 1 1 41----- =n B do

Homework Problem 4 A plane glass disk with a small thickness do is used to re-image the laser spot o into the center O' of an optical fiber (fiber coupling). Show that for perfect imaging from a finite object distance I to finite image distance l' perfect requires that glass disk has gradual variation in its refractive index such that h? n(h) zn 2df where n, is refractive index on axis and f' is the back focal length. Hint: apply Fermat's principle of stationary optical path and formula 11f 1 --- do