Page 169 - Fiber Bragg Gratings

146 Chapter 4 Theory of Fiber Bragg Gratings

profile of the refractive index perturbation is uniform. However, as has
been described by the general mode-coupling constants of Eqs. (4.2.5) and
n(4.2.6) dissimilar mode orders that normally cannot couple owing to the
orthogonality relationship [Eq. (4.1.15)] are allowed to couple when the
transverse profile of the refractive index is nonuniform. This applies
equally to copropagating modes. In this case, coupling may normally occur
between

1. Copropagating orthogonal polarizations, e.g., (HE^) xy <-> (HE n)y tX
(LP oljX and the LP olj,). A uniform grating profile is necessary for
good efficiency. To allow coupling between these modes, the grating
is written at 45° to the principle birefringent axes of the fiber (see
Section 4.5 and Chapter 6).

2. (LP ol) x>y <->• (LP VIJ) x>y Here, the transverse profile of the grating
strongly influences the strength of the coupling. With a uniform
profile, the coupling is zero for v + 0.
3. Coupling to the radiation field E p (as with LPGs). Since the radia-
tion field is evanescent in the core of the fiber and oscillatory in
the cladding, coupling can be strongly influenced if a grating ex-
tends into the cladding as well. The latter diminishes the overlap
integral between the guided lowest-order mode and the radiation
modes, while an asymmetric transverse grating profile can en-
hance the interaction with odd modes.
Following the analysis developed in Section 4.2 and 4.3, the mode
coupling equations for copropagating modes are

but with the phase-mismatch factor

and the dc self-coupling constant for each of the modes,