handbook of modern sensors

4.7 Fiber Optics and Waveguides 141
(A)
(B)
Fig. 4.16. Optical fibers: (A) a step-index multiple fiber; (B) determination of the maximum
angle of entry.
from a medium having a refractive index n is subject to the limitation of an angle of
total internal reflection. In a more general form, light may pass to another medium
(cladding) having refractive index n 1 ; then, Eq. (4.23) becomes
( n1
)
0 = arcsin . (4.33)
n
Figure 4.16A shows a profile of the index of refraction for a single fiber with
the cladding where the cladding must have a lower index of refraction to assure a
total internal reflection at the boundary. For example, a silica-clad fiber may have
compositions set so that the core (fiber) material has an index of refraction of 1.5 and
the clad has an index of refraction of 1.485. To protect the clad fiber, it is typically
enclosed in some kind of protective rubber or plastic jacket. This type of the fiber
is called a "step index multimode" fiber, which refers to the profile of the index of
refraction.
When light enters the fiber, it is important to determine the maximum angle of entry
which will result in total internal reflections (Fig. 4.16B). If we take that minimum
angle of an internal reflection 0 = 3 , then the maximum angle 2 can be found
from Snell’s law:
⎛√
⎞
⎜
n 2 − n 2 1 ⎟
2(max) = arcsin ⎝ ⎠ . (4.34)
n
Applying Snell’s law again and remembering that for air n ≈ 1, we arrive at
sin in(max) = n 1 sin 2(max) . (4.35)
Combining Eqs. (4.34) and (4.35), we obtain the largest angle with the normal to the
fiber end for which the total internal reflection will occur in the core:
(√ )
in(max) = arcsin n 2 − n 2 1
. (4.36)