Page 187 - Fiber Bragg Gratings

164 Chapter 4 Theory of Fiber Bragg Gratings

where a = a'/4 is a function of wavelength only, and fyis the propagation
constant for the incident fundamental mode. The longitudinal component
of the guided mode field is small and has been neglected in Eq. (4.7.10).
The physical analogy of the STG as a distributed antenna is particu-
larly useful, equivalent to an infinite sum of mirrors, each contributing
to the light scattered from the fiber core. For small lengths, we have to
include the oscillating term in quadrature in Eq. (4.7.8), but with z > A g,
the electric field for the fundamental mode decays approximately as it
would for constant attenuation per unit length. The attenuation constant
depends on wavelength and the transverse distribution of the grating and
the incident field, but not on z. This approximate result suggests that the
filter loss spectrum should be independent of the length of the grating,
which is indeed the case.
To calculate the scattered power and the spectrum of the radiation,
we use Eq. (4.7.6) in Eq. (4.7.5) and include the grating function W grating
to arrive at

where L g is the length of the fiber grating, the constant F is given

and I L(x, L g) is obtained by integration with respect to z,

where y was defined in (4.7.2), and A/fy and A/3 fe are the forward and
backward phase mismatch factors,

where /3 cfad = 2,7m ci ad/A, and the signs are consistent with the measure-
ment of the angle, (p. The forward scattering process can easily be included