Page 142 - Fiber Bragg Gratings

Chapter 4

Theory of Fiber Bragg

Gratings

Wave propagation in optical fibers is analyzed by solving Maxwell's equa-
tions with appropriate boundary conditions. The problem of finding solu-
tions to the wave-propagation equations is simplified by assuming weak
guidance, which allows the decomposition of the modes into an orthogonal
set of transversely polarized modes [1-3]. The solutions provide the basic
field distributions of the bound and radiation modes of the waveguide.
These modes propagate without coupling in the absence of any perturba-
tion (e.g., bend). Coupling of specific propagating modes can occur if the
waveguide has a phase and/or amplitude perturbation that is periodic with
a perturbation "phase/amplitude-constant" close to the sum or difference
between the propagation constants of the modes. The technique normally
applied for solving this type of a problem is coupled-mode theory [4-9],
The method assumes that the mode fields of the unperturbed waveguide
remain unchanged in the presence of weak perturbation. This approach
provides a set of first-order differential equations for the change in the
amplitude of the fields along the fiber, which have analytical solutions
for uniform sinusoidal periodic perturbations.
A fiber Bragg grating of a constant refractive index modulation and
period therefore has an analytical solution. A complex grating may be
considered to be a concatenation of several small sections, each of constant
period and unique refractive index modulation. Thus, the modeling of the
transfer characteristics of fiber Bragg gratings becomes a relatively simple

119