Page 156 - Fiber Bragg Gratings

4.2 Coupled-mode theory 133

a modulated permittivity that has a uniform transverse profile. Also
shown is a polarization field that is in the LP n mode (v — 1). Examining
the transverse overlap (which is proportional to the product of the field
amplitudes and the refractive-index profile) on the left half of the core,
we find that magnitude is the same as on the right half, but they have the
opposite signs, resulting in a zero overlap. The orthogonality relationship
holds and exchange of energy is not possible between the different order
modes. If, however, the refractive index profile is not uniform across the
core (Fig. 4.3), then although the signs of the overlap in the two halves
(around a plane through the axis of the fiber) are different, the magnitudes
are no longer identical.
Thus, the overlap is now not zero, allowing a polarization wave to
exist with a symmetry (and therefore, mode-order) different from that of
the driving mode. The selection rules for the modes involved in the ex-
change of energy are then determined by the details of the terms in the
integral in Eq. (4.2.22) and apply equally to radiation mode orders.
The consequence of the asymmetric refractive index perturbation
profile may now be appreciated in Eq. (4.2.21). On the RHS, the integrals
with the electric fields of the driving field ^ and the polarization wave
g vt along with the asymmetric profile of the refractive index modulation
are nonzero for dissimilar mode orders, i.e., /JL =£ v. The magnitude of the
overlap for a particular mode combination will depend on the exact details
of the perturbation profile.

4.2.4 Spatially periodic nonsinusoidal refractive index
modulation

Note that in Eq. (4.2.21), the refractive-index perturbation can have a ±
sign in the exponent. This is a direct result of the Fourier expansion
of the permittivity perturbation. However, since it is equivalent to an
additional momentum, which can be either added to or taken away from
the momentum vector of a driving field, it may be viewed as a factor that
can take place, as already discussed.
In the general case when the refractive index modulation is not simply
sinusoidal but a periodic complex function of z, it is more convenient to
expand 8n in terms of Fourier components as