Page 144 - Fiber Bragg Gratings

4.1 Wave Propagation 121

The straightforward transfer matrix method provides high accuracy
for modeling in the frequency domain. Many representative varieties of
the types and physical forms of practically realizable gratings may be
analyzed in this way.

4.1 Wave Propagation

The theory of fiber Bragg gratings may be developed by considering the
propagation of modes in an optical fiber. Although guided wave optics is
well established, the relationship between the mode and the refractive
index perturbation in a Bragg grating plays an important role on the
overall efficiency and type of scattering allowed by the symmetry of the
problem. Here, wave-propagation in optical fiber is introduced, followed
by the theory of mode coupling.
We begin with the constitutive relations

where e 0 is the dielectric constant and /U.Q is the magnetic permeability,
both scalar quantities; D is the electric displacement vector; E is the
applied electric; B and H are the magnetic flux and field vectors, respec-
tively; and P is the induced polarization,

(1)
The linear susceptibility ^ is in general a second-rank tensor with
two laboratory frame polarization subscripts ij and is related to the per-
mittivity tensor e^ with similar subscripts as

Assuming that the dielectric waveguide is source free, so that

and in the absence of ferromagnetic materials,