,
the transmission more closely resembles the manner in which microwaves
advance along waveguides.
Light striking its walls from within will be
totally internally reflected, provided that the incident angle at each
reflection is greater than 
,
where nf is the index of the cylinder or fiber. If the
fiber has a diameter D and a length L, the path length traversed
by the ray will be
.
(11.1)
The number of reflections is then given by
(11.2)
rounded off to the nearest whole number. The 
1,
which depends on where the ray strikes the end face, is of no significance
when N is large, so we will ignore it from now on. For example,
if D = 50 
m,
and if nf = 1.6, n0 = 1 and 
i
= 30
,
N is approximately 6500 reflections per meter.
Typically, a fiber core might have an index of
refraction of approximately nf = 1.62 and a cladding
index of refraction of nc = 1.52, although a range of
values is available. Notice that there is a maximum value of i,
for which the internal ray will impinge at the critical angle C.
Rays incident on the face at angles greater than max
will strike the interior wall at angles less than C.
They will be only partially reflected at each such encounter with the core-cladding
interface and will quickly leak out of the fiber. Thus max,
which is known as the acceptance angle, defines the half angle of
the acceptance cone of the fiber. We can find an analytical expression
for the angle as
(11.3)
Using Snell's law and rearranging terms, this becomes
(11.4)
The quantity n0 sin max
is defined as the numerical aperture (NA). Its square is
a measure of the light gathering power of the system. Thus, for a fiber,
(11.5)
Not all fiberoptic arrays are made flexible; for example, fused, rigid coherent fiber faceplates, or mosaics, are used to replace homogeneous low-resolution sheet glass on cathode-ray tubes, vidicoms, image intensifiers, and other devices. Mosaics consisting of millions of fibers with their claddings fused together have mechanical properties almost identical with homogeneous glass. Similarly, a sheet of fused tapered fibers can either enlarge or shrink an image, depending on whether the light enters the smaller or larger end of the fiber. Another common application of mosaics involving imaging is the field flattener. If the image formed by a lens system resides on a curved surface, it is often desirable to reshape it into a plane. A mosaic can be ground and polished on one of its end surfaces to correspond to the contour of the image and on the other to match the detector.
Consider a medium whose index of refraction is lowest at the top and increases down toward the bottom.
At the top, th index of refraction is n,
while at the bottom it is n'. Let the light have a wavelength of 0
outside the medium, a wavelength of
at the top, and '
at the bottom. Then if there are N wavefronts within the volume,
the upper arc has a length of
(11.6)
while the lower arc's length is
.
(11.7)
From analytical geometry, we know that the arc
lengths also depend on the radius of curvature, R, and on the angle
of deviation, 
,
(11.8)
and
.
(11.9)
Subtracting (11.9) from (11.8), and then using (11.6) and (11.7) yields
or, solving for ,
.
(11.10)
If the light is incident horizontally, then we
can split
into components, y
and z.
The y component, to first approximation, is given by
(11.11)
where the integral is taken over the length L traversed by the light. If this length is relatively short and the medium is homogeneous in that direction, the integral becomes trivial, and thus
.
(11.12)
From this, we see that the angle of deflection
is a function of the gradient 
.
Inserting (11.12) into (11.8) yields,
.Setting 
L
= L and solving for R,
.
(11.13)
Thus, the steeper the gradient, the shorter the radius of curvature through which the light is bent.
Gradient index fibers have an index of refraction that is highest along the axis and lower farther away from it. Normally, light travels near the center of the fiber, but if the ray enters obliquely, the ray encounters the gradient and is bent back towards the axis. The ray usually never even reaches the surface of the fiber. The advantage of a gradient index fiber is that the rays inside the fiber all have the same path length so that a pulse of light injected at one end retains its shape when it emerges at the other.
Finally, there is the transmission of images, such as is done by a flexible fiberscope. In this case, some of the fibers are used to conduct light into a cavity, while the majority of the fibers transmit the image back to the observer. Fiberscopes are used extensively in engineering as well as in medicine.
Last updated: July 23, 1997
Comments to: D-Suson@tamuk.edu