Page 208 - Photonics Essentials an introduction with experiments

Optical Fibers and Optical Fiber Amplifiers

202 Advanced Topics

around the central axis, and z the length of the fiber. The Laplacian
operator has the following form in cylindrical coordinates:

2 1 1 2 2
2
= + + + (9.5)
r 2 r r r 2 2 z 2
We can deal with the z dependence of the problem by substituting a
trial solution for the z component that looks like a simple sinusoidal
wave. That is,
E(z, t) = Ae (j t–i t) (9.6)

This leaves us with an equation in r and that describes the behavior
of the electric field in the circular cross section of the fiber:

2 1 l 2
2
2
E(r, ) + E(r, ) + k – – E(r, ) = 0 (9.7)
r 2 r r r 2
A similar equation can be written down for the magnetic field.
Because the fiber has a circular cross section, the variable is
quantized following the same reasoning as that of de Broglie in Chap-
ter 2. The number l can only be an integer indicating how many peri-
ods of the wave are found when you complete a full circle around the
fiber cross section.
This equation has been solved by many people, and the solutions
are Bessel functions. Bessel functions are specially designed to de-
scribe waves constrained by circular geometries, like the vibrations of
a drum, for instance. Although they do not appear on your calculator
keyboard like the sine and cosine functions, they make life much easi-
er for describing these kinds of situations. In the radial direction, they
oscillate with declining amplitude. We will not solve the equation, be-
cause what you would really like to know is not what the electric field
looks like, but rather the relationship between k, , and l. This rela-
tionship is determined by the boundary conditions.
The boundary conditions are determined by conditions of continu-
ity of the electric and magnetic fields at the interface between the
core and the cladding where there is a discontinuity in the index of
refraction. This leads to a somewhat tedious exercise in algebra, the
chief benefit of which is to bring the core diameter of the fiber into
the problem, for the discontinuity in the index of refraction occurs
when r = d/2. An important parameter involves the ratio of the fiber
core diameter to the wavelength of light. This is called the V param-
eter:
d 2 d
n 2
1 –
V =
=
NA (9.8)
n 1

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