Page 145 - Fiber Bragg Gratings

122 Chapter 4 Theory of Fiber Bragg Gratings

the electric field described in complex notation is

and the induced polarization vector is also similarly defined.
Using Maxwell's equations,

where J is the displacement current, and using Eq. (4.1.1) in Eq. (4.1.9)
and with J = 0, we get

Taking the curl of Eq. (4.1.8) and using Eqs. (4.1.2H4.1.5) and the
time derivative of Eq. (4.1.10), the wave equation is easily shown to be

Using Eq. (4.1.3) and (4.1.4) in (4.1.11), we arrive at

or

4.1.1 Waveguides
The next step in the analysis is to introduce guided modes of the optical
fiber into the wave equation. The modes of an optical fiber can be described
as a summation of/ transverse guided mode amplitudes, A^z), along with
a continuum of radiation modes, A p(z] [2], with corresponding propagation
constant, £}/& and fip,