Page 201 - Fiber Bragg Gratings

178 Chapter 4 Theory of Fiber Bragg Gratings

where r is the Fresnel amplitude reflection coefficient at the cladding
surface, and L b is the distance between the successive reflections at the
cladding surface.

4.8 Grating simulation

4.8.1 Methods for simulating gratings
Many of the techniques for simulating fiber Bragg gratings were intro-
duced at the beginning of the chapter [10,58,9,19]. All the techniques
have varying degrees of complexity. However, the simplest method is
the straightforward numerical integration of the coupled-mode equations
such as Eqs. (4.3.9) and (4.3.10). While this method is direct and capable
of simulating the transfer function accurately, it is not the fastest. Another
technique is based on the Gel'Fand-Levitan-Marchenko inverse scatter-
ing [58] method. This is again a powerful scheme based on integral coupled
equations but has the primary disadvantage of obscuring the problem
being solved. It has, however, the advantage of allowing a grating with
particular characteristics to be designed. Perhaps the most attractive
method is based on techniques developed for the analysis of metal wavegu-
ides by Rouard [11] and carries his name as a result. This technique,
extended by Weller-Brophy and Hall [12], works on the principle that the
waveguide may be segmented into subwavelength thin films. Standard
thin-film techniques for calculating the amplitude and phase of the trans-
mitted and reflected electric fields at each interface are propagated back-
ward from the output end of the grating. The method is slow, but is one
of the few that offers complete piecemeal control of the spatial variation
in the refractive index modulation of the grating. For example, the transfer
function of a grating with a sawtooth modulation is analyzed equally as
well as a square or sinusoidal profile. The influence on the phase and
amplitude response of the grating cannot be fully characterized if the
Fourier coefficient of the phase-synchronous term for phase matching as
shown in Eq. (4.2.27) alone is used. Thus, it is a laborious and time-
intensive computation; however, the results accurately simulate the char-
acteristics of complex-shaped grating periods with reasonable reflectivi-
ties, being limited by the rounding errors in the computation.
A fast and accurate technique is based on the T-matrix (transfer) for
calculating the input and output fields for a short section SI of the grating