Page 219 - Fiber Bragg Gratings

196 Chapter 5 Apodization of Fiber Gratings

refractive modulation amplitude profile diminishes the side lobes substan-
tially. The suppression of the side lobes in the reflection spectrum by
gradually increasing the coupling coefficient with penetration into, as
well as gradually decreasing on exiting from, the grating is called apodiza-
tion. Hill and Matsuhara [2,3] showed that apodization of a periodic wave-
guide structure suppresses the side lobes. However, simply changing the
refractive index modulation amplitude changes local Bragg wavelength as
well, forming a distributed Fabry-Perot interferometer [4], which causes
structure to appear on the blue side of the reflection spectrum of the
grating, although side-lobe amplitudes are reduced [5]. To avoid this com-
plication, the key is to maintain an unchanging average refractive index
throughout the length of the grating while gradually altering the refrac-
tive index modulation amplitude.
The alternative approach for generating a reflection spectrum that
has a constant reflectivity over a certain band and zero outside of it is to
write a sin x/x refractive index modulation envelope. From the Fourier
transform analogy, it is apparent that the grating reflection spectrum will
be a "top-hat" function. The problem, however, is to incorporate the grating
in such a way that the fringes have the appropriate phase relationship
on either side of the zeroes of the sine function. Since the induced refractive
index change is proportional to the square of the electric field amplitude
(intensity), it is always positive. The phase change can be physically
incorporated either by including a A/4 dead zone in which no grating exists
at each zero or by changing the phase of the grating abruptly, for example
in a phase mask [6] or slowly over the length of the section [7], Strictly
speaking, the Fourier transform analogy is only applicable to weak grat-
ings, as mentioned in Chapter 4. However, the principle of using the
space-frequency transform does allow the techniques to be used for the
design of gratings.
The beneficial effects of apodization are not manifest only in the
smoothness of the reflection spectrum, but also in the dispersion charac-
teristics. There are many techniques, as there are appropriate profile
functions (shading) for the refractive index modulation amplitude to
achieve the end result. However, they all rely on a single principle: keeping
the sum of the dc index change and the amplitude of the refractive index
modulation constant throughout the grating. In the following section,
several of these techniques and types of "shading" functions used for
apodization are reviewed.