Page 167 - Fiber Bragg Gratings

144 Chapter 4 Theory of Fiber Bragg Gratings

To find a solution, the following substitutions are made for the forward
(reference) and backward propagating (signal) modes [32]:

Differentiating Eq. (4.3.8) and substituting into Eqs. (4.3.3) and
(4.3.7) results in the following coupled-mode equations:

The physical significance of the terms in brackets is as follows: K dc influ-
ences propagation due to the change in the average refractive index of
the mode, as has already been discussed. Any absorption, scatter loss, or
gain can be incorporated in the magnitude and sign of the imaginary part
of K dc. Gain in distributed feedback gratings will be discussed in Chapter
8. There are also two additional terms within the parentheses in Eqs.
(4.3.9) and (4.3.10), the first one of which, A/3/2, is the detuning and
indicates how rapidly the power is exchanged between the "radiated"
(generated) field and the polarization ("bound") field. This weighting factor
is proportional to the inverse of the distance the field travels in the
generated mode. At phase matching, when A/2 = 0, the field couples to
the generated wave over an infinite distance Finally, the rate of change
of (f> signifies a chirp in the period of the grating and has an effect similar
to that of the detuning. So, for uniform gratings, dfldz = 0, and for a
visibility of unity for the grating, K ac = K dc/2.
The coupled-mode Eqs. (4.3.9) and (4.3.10) are solved using standard
techniques [33]. First the eigenvalues are determined by replacing the
differential operator by A and solving the characteristic equation by equat-
ing the characteristic determinant to zero. The resultant eigenvalue equa-
tion is in general a polynomial in the eigenvalues A. Once the eigenvalues
are found, the boundary values are applied for uniform gratings: We
assume that the amplitude of the incident radiation from — oo at the input
of a fiber grating (of length L) at z = 0 is R(Q) — 1, and that the field S(L)
= 0. The latter condition is satisfied by the fact that the reflected field
at the output end of the grating cannot exist owing to the absence of the