Page 157 - Fiber Bragg Gratings

134 Chapter 4 Theory of Fiber Bragg Gratings

where a N is the Fourier amplitude coefficient of the Nih harmonic of the
perturbation. Differently shaped periodic functions have their correspond-
ing a N coefficients, which in turn influence the magnitude of the overlap
integral, and hence the strength of the mode coupling.

4.2.5 Types of mode coupling

The phase-matching condition is defined by setting A/3 in Eq. (4.2.26) to
zero. Therefore,

Equation (4.2.28) states that a mode with a propagation constant of (B^
will be synchronously drive another mode A v with a propagation constant
of ySy, provided, of course, the latter is an allowed solution to the unper-
turbed wave equation (4.1.28) for guided modes and its equivalent for
radiation modes.
The guided modes of the fiber have propagation constants that lie
within the bounds of the core and the cladding values, although only
solutions to the eigenvalue Eq. (4.1.28) are allowed. Consequently, for
the two lowest order modes of the fiber, LP 01 and LP n, the propagation
constants /3 V and fi^ are the radii of the circles 27m v/A and 27771^/A. A mode

traveling in the forward direction has a mode propagation vector K LP()l
that combines with the grating vector K grating to generate K- resuLt. Since
the grating vector is at an angle 6 g to the propagation direction, and the
allowed mode solution, K LPu is in the propagation direction, the phase-
matching condition reduces to

Under these circumstances, the process of phase matching reverses
in sign after a distance (known as the coherence length Z c) when

Consequently, the radiated LPn mode (traveling with a phase con-
stant of (3^ propagates over a distance of l c before it slips exactly half a
wavelength out of phase with the polarization wave (traveling with a
phase constant j3 v).