Page 153 - Fiber Bragg Gratings

130 Chapter 4 Theory of Fiber Bragg Gratings

On the LHS of Eq. (4.2.21), the rate of variation of either A v or B^ is
determined by the mode order /u, or v of the electric field ^* >l)t chosen as
the multiplier according to the orthogonality relationship of Eq. (4.1.15).
This was shown in Eq. (4.2.9) for the case of the single field. Once the
term on the LHS has been chosen, the next question is the choice of the
terms on the RHS. Before this is examined, we consider the terms on the
RHS in general.
The RHS of Eq. (4.2.21) has two generic components for both A and
B modes as

where the first exponent must agree with the exponent of the generated
field on the LHS of Eq. (4.2.21) and has a dependence on the dc refractive
index change, Arc. The reason is that any other phase-velocity dependence
(as for other coupled mode) will not remain in synchronism with the
generated wave. The second term on the RHS has two parts. The first
one is dependent on the phase-synchronous factor,

The mode interactions that can take place are determined by the
right-hand sides of Eqs. (4.2.21) and (4.2.22). Two aspects need to be taken
into account: first, conservation of momentum requires that the phase
constants on the LHS and the RHS of Eq. (4.2.22) be identical [Eq. (4.2.23)]
and so influences the coupling between copropagating or counterpropagat-
ing modes. Secondly, the transverse integral on the RHS of Eq. (4.2.22),
which is simply the overlap of the refractive-index modulation profile and
the distributions of the mode fields, determines the strength of the mode
interactions. Let us first consider the conservation of momentum, other-
wise known as phase matching.

4.2.2 Phase matching

We begin with Eq. (4.2.23) in which the phase factor is the sum or differ-
ence between the magnitude of the driving electric-field mode propagation