Page 220 - Fiber Bragg Gratings

5.1 Apodization shading functions 197

5.1 Apodization shading functions

In filter and information theory, there are well-established functions for
capturing a signal with a given bandwidth for the required signal to noise
ratio [8]. Generally, these are known as Hamming and Hanning functions.
The distinguishing feature between the two is whether or not the function
forces the filter parameter to zero at infinite "detuning." For example, if
the window function profiling the refractive index modulation amplitude
reduces it to zero at either end of the grating, it is known as a Hanning,
and otherwise a Hamming function. There are various types of functions,
each of which results in a compromise between roll-off of the filter and
the useable bandwidth. For the numerical analysis commonly used for
the computation of the response and also to define the functions for apodi-
zation and chirp, it is convenient to define the grating function as

where the maximum ac index change is indicated by the subscripted
variable n, f A(z) and f g(z) are the apodization envelope and periodic refrac-
tive index modulation functions, and 0 < e < 2 is a parameter that controls
the level of the background, dc index change. In general the function, f^z)
includes chirp as

where A 0 is the period at the start of the of the grating, and F g is a
function that describes the spatial variation of the grating period with a
power dependence m and chirp step SA = AA/N g (where Ayl is the total
chirp of the grating). The grating is composed of N g discrete sections.
The apodisation factor f A(z~) is described by a number of commonly used
functions F g( 0) and has an argument of the form

where N A is the number of sections in the entire grating, A A is the apodisa-
tion envelope period, and <f> A is a starting phase.