Page 143 - Fiber Bragg Gratings

120 Chapter 4 Theory of Fiber Bragg Gratings

matter, and the application of the transfer matrix method [10] provides
a clear and fast technique for analyzing more complex structures.
Another technique for solving the transfer function of fiber Bragg
gratings is by the application of a scheme proposed by Rouard [11] for a
multilayer dielectric thin film and applied by Weller-Brophy and Hall
[12,13]. The method relies on the calculation of the reflected and transmit-
ted fields at an interface between two dielectric slabs of dissimilar refrac-
tive indexes. Its equivalent reflectivity and phase then replace the slab.
Using a matrix method, the reflection and phase response of a single
period may be evaluated. Alternatively, using the analytical solution of
a grating with a uniform period and refractive index modulation as in
the previous method, the field reflection and transmission coefficients of
a single period may be used instead. However, the thin-film approach does
allow a refractive index modulation of arbitrary shape (not necessarily
sinusoidal, but triangular or other) to be modeled with ease and can
handle effects of saturation of the refractive index modulation. The disad-
vantage of Rouard's technique is the long computation time and the lim-
ited dynamic range owing to rounding errors.
The Bloch theory [14,15] approach, which results in the exact eigen-
mode solutions of periodic structures, has been used to analyze complex
gratings [16] as well. This approach can lead to a deeper physical insight
into the dispersion characteristics of gratings. A more recent approach
taken by Peral et al. [17] has been to develop the Gel'Fand-Levitan-Mar-
chenko coupled integral equations [18] to exactly solve the inverse scatter-
ing problem for the design of a desired filter. Peral et al. have combined
the attributes of the Fourier transform technique [19,20] (useful for low
reflection coefficients, since it does not take account of resonance effects
within the grating), the local reflection method [21], and optimization of
the inverse scattering problem [22,23] to present a new method that allows
the design of gratings with required features in both phase and reflection.
The method has been recently applied to fabricate near "top-hat" reflecti-
vity filters with low dispersion [24]. Other theoretical tools such as the
effective index method [25], useful for planar waveguide applications,
discrete-time [26], Hamiltonian [27], and variational [28], are recom-
mended to the interested reader. For nonlinear gratings, the generalized
matrix approach [29] has also been used. For ultrastrong gratings, the
matrix method can be modified to avoid the problems of the slowly varying
approximation [30].