Page 164 - Fiber Bragg Gratings

4.2 Coupled-mode theory 141

however, a cladding boundary is present as shown by the innermost dashed
circle, then the radiated modes only have allowed y8-values. This radiation
may be viewed as the modes of a waveguide with a core of refractive index
n ciad and a radius equal to the fiber-cladding radius surrounded by an infi-
nite cladding of air/vacuum refractive index. The diagram then acquires a
set of circles with radii n vacuum <n p< n ciad representing discrete cladding
modes, similar to those for guided modes, at the points of intersection with
the dispersion curves. The radiation mode fields are slightly modified by
the presence of the high-index fiber core.
Coupling is also possible to the forward-radiating modes and fields.
This requires a different grating, known as a long-period grating, which
has a much longer period than a Bragg grating, since the momentum of
the mode does not change sign (as in forward-to-backward coupling).
The phase matching for the generalized case of the tilted grating for
copropagating coupling is shown in Fig. 4.7. The form of the diagram is
similar to Figs. 4.5 and 4.6.
For phase matching, the movement of the n g arrowhead for the LPG
is opposite to that of the STG. We begin with an LP 0i mode with the
propagation index arrow n LPoi pointing in the +z direction. The grating
n g starts at the tip of the guided mode arrow, inclined at 6 g = 0 to the
fiber axis. The wavelength at which radiation is first emitted is when the
tip of the grating vector from point A intersects the tangent to the cladding
mode circle (dashed vertical line). This point represents the longest-wave-
length LP 01 mode that has a propagation constant equal to the cladding
index and has the lowest angle. Light is coupled to radiation modes within
the radiation zone as n g is moved to the left and the. LP 01 mode is "cut
off' at the radiation angle, d r Therefore, 6 r is the angular spread of the
radiated fields. Mode coupling is only possible if there is phase matching
to specific modes. Note that this wavelength approaches oo, since the
fundamental guided mode effective index approaches n ctad. The wave-
length vs angle has the opposite dependence of the STG, i.e., long wave-
lengths exit at the largest angle in the LPG, while it is the shortest
wavelengths in the STG. The first Bragg wavelength reflection (very weak)
is at the short wavelength side of the LPG radiation loss spectrum, while
it is on the long wavelength side of the STG radiation loss spectrum. The
spectrum of the LPG is "reversed" around the Bragg wavelength.
Figure 4.8 shows various types of phase-matched interactions possible
with different types of gratings.