Page 186 - Fiber Bragg Gratings

4.7 Radiation mode couplers 163

By integrating the scattered contributions from each part of the grat-
ing separated by SR, the scattered field, E(R,(f>,<p,\} may be derived by
neglecting the angular dependence on <p and </> and follows as

The above result is consistent with Fraunhofer diffraction theory [3],
and we note that it is in the same form as scattering due to the polarization
response of a material. We note that far away from the grating,

The result in Eq. (4.7.5) neglects secondary scattering, so that it is
implicitly assumed that the incident radiation is the primary cause for
the radiation. This may be justified for STGs, since it is the aim of the
exercise to consider radiation loss to the exclusion of reflection by proper
choice of blaze angle, and because the radiation field is only weakly bound
to the core.
We are now in a position to calculate the propagation loss of the
incident radiation. The power scattered as a function of length of the
grating described in Eq. (4.7.1) and into even azimuthal mode orders can
be described as

a' is a loss coefficient, which is dependent on the wavelength, the trans-
verse profile of the grating, and the incident field and is equivalent to
the overlap integral of Eq. (4.3.6). The incident field therefore decays as

The contribution due to the oscillating term within the exponent becomes
insignificant for large z, and the power decays as

From Eq. (4.7.9) follows the approximate decay of the incident electric
field,