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FEED LINE BASICS 293

bandwidth in long-haul communication fiber lines is also restricted by so-called chromatic
dispersion due to refractive index wavelength-dependence.

The reader can object our far-reaching conclusions saying that the boundary between the real
fiber core and cladding is not flat as we assumed and Snell’s law probably does not work
reasonably. It is correct but not the whole truth. Look back at Figure 6.3.1b-d. The
circumference of the smallest fiber core is *9 [µm] = 28.2 µm while the wavelength in the
core of fiber is around 1500/ [nm] = 1.5/1.458 [µm] or about 1µm. Therefore, the light
1
wavelength is the small fraction of core curvature and the incident wave practically “sees and
meets” the flat boundary.

c) Slightly different and somewhat complicated refraction
process takes place in a graded-index fiber. Figure 6.3.1c
depicts that the core refractive index gradually drops as the
radial distance increases. To simplify the consideration let
us replace the continuous radial index core distribution with

a stack of thin uniform slabs of indices , < , <
1
2
3
1
Figure 6.3.4 Fiber with step , etc. and then apply Snell’s law to each stab. Figure 6.3.4
2
refractive index and 6.3.5 demonstrate how the wave path progressively
bends toward the horizontal, comes back to some point on
the z-axis (black spots), crosses it and then continues to propagate the same way along the fiber.
The refractive index profile is typically close to so-called parabolic or more
exactly ()~�1 − ( ) where is some
2
⁄
0
0
constant exceeding the core radius to avoid too
low or even imaginary values of index. Such
graded-index fiber has highest refractive index
at the center and then gradually decreases
toward the cladding. If so, the lower modes
travel slower along the shorter path around the
core center of the peak refractive index while
Figure 6.3.5 Ray paths in graded-index the higher modes moves faster over the longer
fiber path with reduced index. Then at least in theory
and not so bad in practice, all rays leaving one
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focal point (red spots) come to the next focal points together as Figure 6.3.5 illustrates. This
refocusing effect minimizes line dispersion since all wave modes come to focal cross section
almost simultaneously. In other words, the step-index fiber is multimodal too. The periodical
refocusing reduces the modal dispersion, but does not eradicate it. As a results, the bandwidth
expends to 3.5 GHz per km.

d) The standard single mode step-index fiber has a core diameter of 9 µm and refractive index
= 1.458 while the refractive index of cladding is = 1.451 or only ~0.5% less. If so, the
1
2
critical refraction angle is about 84.3° and, loosely speaking, the propagating EM wave

travels inside core almost tangential to the core boundary. The reflections are relatively seldom
and as the rigorous mathematical analysis shows such fiber can support single hybrid wave
7 Public Domain Image, source: https://www.fiberoptics4sale.com/blogs/archive-posts/95047942-
optical-fiber-dispersion, permitted by Bill Reid, Amphenol Fiber Systems International