Page 308 - Fiber Bragg Gratings

6.7 In-coupler Bragg grating filters 285

by the grating. The evanescent fields of the two fibers overlap and an
exchange of energy takes place.
The coupler can be analyzed by considering four modes with ampli-
tudes A! 4. Mode 1 travels in the positive z-direction (from left to right)
in the upper fiber, from z = 0 to z = L. Mode 2 also propagates along the
positive z-direction in the lower fiber, from z = 0 to z = L. Mode 3 propa-
gates in the negative ^-direction in the upper fiber, from z = L to z = 0.
Last, mode 4 also propagates in the negative ^-direction in the lower fiber,
from z — L to z = 0. The evolution of the mode amplitudes of the four
waves is then described by a set of four first-order coupled differential
equations.
There are three significant interactions that need to be considered:
1. Copropagating interactions between the modes of different waveg-
uides (modes 1 and 2 as well as 3 and 4), as they do in a normal
coupler without a grating. The overlap of the mode evanescent
fields allows an exchange of energy to take place since the guides
are phase matched. Because of symmetry considerations, a single
coupling coefficient K [Eq. (6.3.1c)l can be used.
2. Counterpropagating interaction between the modes of the same
fiber. Modes 1 and 3, and 2 and 4, interact because of presence
of the Bragg grating. The general form of the refractive index
modulation of the grating, which allows this coupling, is given by
Eq. (4.2.27). The phase mismatch is

where the moduli of the mode propagation constants for the forward- and
Counterpropagating modes 1 and 2 are

The phase-matched condition determines the optimum coupling be-
tween the forward- and backward-propagating modes when A/3 = 0, at
the Bragg wavelength. The presence of the grating promotes coupling
between modes 2 and 4 since the grating is only in the lower waveguide,
while modes 1 and 3 remain uncoupled. The coupling coefficient K ac for
Bragg reflection is given by Eq. (4.3.6),