Page 151 - Fiber Bragg Gratings

128 Chapter 4 Theory of Fiber Bragg Gratings

(l an
The terms within the parentheses are equivalent to x \ d e r is the
relative permittivity of the unperturbed core. The constitutive relations
between the permittivity of a material and the refractive index n result
2
in the perturbation modulation index being derived from n = e r so that

Assuming the perturbation to be a small fraction of the refractive
index, it follows that

Defining the refractive index modulation of the grating as

where ATI is the refractive index change averaged over a single period of
the grating, v is the visibility of the fringes, and the exponent term along
with the complex conjugate cc describe the real periodic modulation in
complex notation. An arbitrary spatially varying phase change of (fj(z) has
been included. A is the period of the perturbation, while N is an integer
(-00 < N < +00) that signifies its harmonic order. The period-averaged
change in the refractive index has to be taken into account since it alters
the effective index n eff of a mode.
Combining Eqs. (4.2.15) and (4.2.17), the total material polarization
is

where the first term on the RHS is the permittivity, the second term is
the dc refractive index change, and the third term is the ac refractive
index modulation. Finally, defining a new modulation amplitude by incor-
porating the visibility,

with A/i = vkn as the amplitude of the ac refractive index modulation.
Equation (4.2.19) describes the UV-induced refractive index change due
to a grating written into the fiber core. Figure 4.1 shows the refractive
index modulation for a uniform grating on a background index of the core