Page 154 - Fiber Bragg Gratings

4.2 Coupled-mode theory 131

constant /3 V and the phase factor of the perturbation. The resultant /3 p is
the phase constant of the induced polarization wave. This is the propaga-
tion constant of a "boundwave" generated by the polarization response of
the material due the presence of sources. For there to be any significant
transfer of energy from the driving field amplitude A v to the generated
fields on the LHS of Eq. (4.2.22), the generated and the polarization
waves must remain in phase over a significant distance, z. For continuous
transfer of energy,

Equation (4.2.24) then describes the phase-matching condition. A phase
mismatch A/2 is referred to as a detuning,

Including Eq. (4.2.23) in (4.2.25), we get,

If both (3 V and /^ have identical (positive) signs, then the phase-matching
condition is satisfied (A/3 = 0) for counterpropagating modes; if they have
opposite signs, then the interaction is between copropagating modes.
Identical relationships for co- and counterpropagation interactions
apply to radiation mode phase matching. A schematic of the principle of
phase matching is shown in Fig. 4.2.
Finally, energy conservation requires that the frequency to of the
generated wave remains unchanged.

4.2.3 Mode symmetry and the overlap integral
The orthogonality relationship of Eq. (4.1.15) suggests that only modes
with the same order IJL will have a nonzero overlap. However, the presence
of a nonsymmetric refractive index modulation profile across the photosen-
sitive region of the fiber can alter the result, allowing modes of different
orders to have a nonzero overlap integral. The reason for this fundamental
departure from the normalization of Eq. (4.1.15) is the nonuniform trans-
verse distribution of sources, giving rise to a polarization wave that has
an allowed odd symmetry. This is graphically displayed in Fig. 4.3: A
driving fundamental mode (LP 01, IJL = 0) electric field, £„ interacts with