Page 146 - Fiber Bragg Gratings

4.1 Wave Propagation 123

where ^ and ^ are the radial transverse field distributions of the /uth
guided and pth radiation modes, respectively. Here the polarization of the
fields has been implicitly included in the transverse subscript, t. The
summation before the integral in Eq. (4.1.14) is a reminder that all the
different types of radiation modes must also be accounted for [e.g., trans-
verse electric (TE) and transverse magnetic (TM), as well as the hybrid
(EH and HE) modes]. The following orthogonality relationship ensures
2
that the power carried in the /zth mode in watts is JA^J :

Here, e z is a unit vector along the propagation direction z. 8^ is Kro-
necker's delta and is unity for /JL = v, but zero otherwise. Note that this
result is identical to integrating Poynting's vector (power-flow density) for
the mode field transversely across the waveguide. In the case of radiation
modes, 8^ is the Dirac delta function which is infinite for /u, = v and zero
for ytt ^ v. Equation (4.1.15) applies to the weakly guiding case for which
the longitudinal component of the electric field is much smaller than
the transverse component, rendering the modes predominantly linearly
polarized in the transverse direction to the direction of propagation [1].
Hence, the transverse component of the magnetic field is

The fields satisfy the wave equation (4.1.13) as well as being bounded
by the waveguide. The mode fields in the core are J-Bessel functions and
.fiT-Besse! functions in the cladding of a cylindrical waveguide. In the
general case, the solutions are two sets of orthogonally polarized modes.
The transverse fields for the /uth jc-polarized mode that satisfy the wave
equation (4.1.13) are then given by [2]