Page 339 - Modern Optical Engineering The Design of Optical Systems

318 Chapter Thirteen

will determine the number of reflections and the number of reflected
source images.

13.8 Fiber Optics
A long, polished cylinder of glass can transmit light from one end to
the other without leakage, provided that the light strikes the walls of
the cylinder with an angle of incidence greater than the critical angle
for total internal reflection. The path of a meridional ray through such
a cylinder is shown in Fig. 13.22. The geometric optics of meridional
rays through such a device are relatively simple. For a cylinder of
length L, the path traveled by the meridional ray has a length given by
L
Path length (13.23)
cos U′
and the number of reflections undergone by the ray is

path length L
No. reflections tan U′ 1 (13.24)
(d/sin U′) d
where U′ is slope of the ray inside the cylinder, d is the cylinder dia-
meter, and L its length. For the light to be transmitted without reflec-
tion loss, it is necessary that the angle I exceed the critical angle

n 2
sin I
c n
1
where n 1 is the index of the cylinder and n 2 the index of the medium
surrounding the cylinder. From this one can determine that the
maximum external slope of a meridional ray which is to be totally
reflected is
1
2
sin U n n 2 (13.25)
n 1 2
0
This “acceptance cone” of a cylinder is often specified as a numerical
aperture; by rearranging Eq. 13.25, we get
NA n sin U n n 2 (13.26)
2
0 1 2

Figure 13.22 Light is transmit-
ted through a long polished
cylinder by means of total inter-
nal reflection.