The fiber is assumed lossless, single
mode and weakly guiding (small refractive index difference between the cladding
and the fiber core). The electromagnetic field is supposed transverse to the
fiber axis z, with the polarization state that is conserved along the
propagation (e.g. x-polarized). Moreover, the refractive index modulation of
the grating is assumed to be homogeneous and restricted to the fiber core.
The core refractive index perturbation
n(z) is defined as (Chapter 2, §2.3.1)
|
(3-1) |
where n0 is the refractive index
of the non-perturbed fiber core, Dnac and Dndc are the "ac"
and "dc" index change amplitudes, respectively, and Ld is the design period, which is chosen in order to have a slowly
varying period phase function q(z). The forward and backward propagating field envelopes (u and v,
respectively) are mutually coupled by the coupled wave equation for weak
coupling coefficients (see Appendix C for full description)
|
(3-2) |
where d = b-p/Ld is called the wavenumber detuning (b = kneff
is the propagation constant). The function q(z) is called the coupling
coefficient and its amplitude and phase are defined as
|
(3-3) |
where h is the fraction of the
modal power contained in the fiber core.
| 