Page 168 - Fiber Bragg Gratings

4.4 Codirectional coupling 145

perturbation beyond that region. These conditions result in the following
analytical solution for the amplitude reflection coefficient:

where

and

A few points regarding Eqs. (4.3.9)-(4.3.13) are worth mentioning.
First, for reflection gratings that have a constant period A, the variation
in the phase d<ftz)ldz = 0. Second, at precise phase matching, A/3 = 0,
and the ac coupling constant K ac is a real quantity. Finally, the power
2
reflection coefficient is \p\ ,

in which Eq. (4.3.13) has been used to simplify the result. Noting from
Eq. (4.3.14) that a can be real or imaginary, the following regimes may
be identified:

1. a is real when \K ac\ > 8 and Eqs. (4.3.11) and (4.3.14) apply.
2. a is zero when |/c ac| = S.
3. a is imaginary when K ac < S and Eqs. (4.3.11) and (4.3.14) trans-
form to

and

4.4 Codirectional coupling

In a multimode fiber, coupling can occur between orthogonally polarized
modes of the same order, or to cladding modes (LPG) if the transverse