Page 195 - Fiber Bragg Gratings

172 Chapter 4 Theory of Fiber Bragg Gratings

of the coupling to the radiation mode (as it does to the individual cladding

modes in the counterpropagating direction). The fundamental, LP 01
guided mode can only couple to the even-order cladding modes of the same
order, LP 0n [53]. Only if there is an asymmetry in the transverse profile
will modes of different order couple. For example, in depressed cladding
fibers that support a leaky LP n mode, coupling to the LP 16 mode is
possible because of the very large overlap of the fields in the core [53],
almost as large as the LP 01 —> LP 01 modes. For fibers that support only
the Z/POI mode, a tilt in the grating allows coupling to copropagating (and
counterpropagating) modes of different order. Erdogan has shown that
the coupling constants to the radiation modes of the order v and the core
mode (LP Ql) follow the definitions of Eqs. (4.4.3) and (4.4.4). They are
[40]

The eigenvalues and the field distributions for the cladding modes
may be calculated by field matching at the boundaries as for the core for
the low-order LP Qn modes, using a procedure similar to the guided core
modes of the fiber [54]. Only coupling to the radiation modes with the
azimuthal order 1 = 1 (LP t_ 1>v type) has a nonzero integral. The equations
that describe the overlap integral, Eq. (4.7.24) of the modes for a trans-
versely uniform grating are involved and cumbersome [40]. For the v =
2, 4 modes, the field in the core is very low, and therefore contributes
little to the coupling. However, the field for the odd-numbered v modes
has high intensity and these fields dominate the coupling for the lower-
order modes. In Fig. 4.25 is shown the calculated coupling constants for
a set of 168 cladding modes for a fiber at 1550 nm, normalized to the
refractive index modulation, kn(z). The important point is that coupling
to the low-even-order modes is weak compared to the odd modes. For v
> 40, both even and odd order modes have almost identical coupling
constants, but remain <20% of the maximum possible for the odd modes.
Therefore, for many applications, it is necessary only to take account of
a maximum of first 20-30 cladding modes, especially when computing
the loss spectrum of an infinite cladding fiber (pure radiation loss).
A major difference between the STG and the LPG is shown in the
phase-matching Eq. (4.7.14). We note that the detuning A/^for a LPG is
sensitive to the difference in the propagation constants of the guided and
radiation modes. Any UV-induced change in the core index will result