Page 343 - Modern Optical Engineering The Design of Optical Systems

322 Chapter Thirteen

the radial distance from the center of the fiber), the ray paths are sinu-
soidal as shown in Fig. 13.25. This has two significant effects. Rays
originating from a point are brought to a focus periodically along the
fiber; thus the fiber is capable of forming an image just as a lens is.
This is the basis of the GRIN or SELFOC rod. For example, if the
index is given as a function of the radial distance r as
2
n (r) n (1 kr /2)
0
then the focal length of a rod with an axial length of t is
1
ef l
n k sin (t k )
0
and the back focus is
1
bf l
n k tan (t k )
0
The “pitch” of the sinusoidal ray path is 2 / k .
Since the focusing effect is continuous along the length of the rod,
such a device is the equivalent of the periscope system of relay and
field lenses described in Sec. 13.2. A length of rod corresponding to two
relay lenses and one intermediate field lens as shown in Fig. 13.25 will
thus produce an erect image of an area approximately equal to the rod
diameter. A row, or a double row, of such rods is the basis of compact
table top (scanning) copy machines. Obviously, a long GRIN rod can
function as an endoscope and a short rod (less than a quarter of the
length shown in Fig. 13.25) will function like an ordinary lens. This
latter is called a Wood lens.
The other significant aspect of such an index gradient is that
because the light rays travel in sinusoidal paths, they never reach the
walls of the fiber and do not depend on reflection at a low-index cladding

Figure 13.25 In a gradient index rod or fiber (GRIN or SELFOC
rod), light rays travel in sinusoidal paths because the index is
high at the center of the rod and lower at the edge. Such a rod can
form an image just as a lens does. The rod length shown is the
equivalent of two relay lenses and an intermediate-field lens.
A short length of rod will act like a single lens element, and a
longer length can act like a periscope.