Page 149 - Fiber Bragg Gratings

126 Chapter 4 Theory of Fiber Bragg Gratings

Substituting the modes in Eq. (4.1.14) into (4.2.4) provides the follow-
ing relationship:

Ignoring coupling to the radiation modes for the moment allows the
left-hand side of Eq. (4.1.13) to be expanded. In weak coupling, further
simplification is possible by applying the slowly varying envelope approxi-
mation (SVEA). This requires that the amplitude of the mode change
slowly over a distance of the wavelength of the light as

so that

2
Expanding the second term in Eq. (4.2.5), noting that a> /u, 0e 0e r = /3^,
and combining with Eq. (4.2.7), the wave equation simplifies to

Here, the subscript t on the polarization P grating,t reminds us that the
grating has a transverse profile. Multiplying both sides of Eq. (4.2.8) by
£* and integrating over the wave-guide cross-section leads to