Page 224 - Fiber Bragg Gratings
P. 224
5.2 Basic principles and methodology 201
present in the source. This leads to a situation in which the visibility of
the fringes becomes a function of z. The two sets of fringes with different
periods according to Eq. (5.2.1), one for each frequency. For this simple
case, the form of the visibility function is simply the "beat" envelope, with
the fringe visibility vanishing at positions ± z from the center of the
overlap region at which the fringes from one frequency get out of phase
with those from the other. The envelope is described by the function
where
In Eq. (5.2.3), the interference term has been retained from Eq. (5.2.1),
and equal, unity UV intensity of the interfering beams has been assumed.
The first term on the RHS is identical to that of Eq. (5.2.1). However,
notice that the fringes are now modulated by a slowly varying function
with the argument dependent on AA, the difference in the wavelength
between the two frequencies of the source.
For the general case of a source with a Gaussian spectral content,
the visibility function becomes an integral over the bandwidth. The points
at which the visibility vanishes are determined by the bandwidth of the
source. For a particular length of the illuminated region, the fringes
vanish at the edges, replaced by constant UV illumination. For a uniform
intensity beam profile of the laser beam, the fringes are self-apodizing
[9], With the induced index change being proportional to the fringe ampli-
tude, the interference pattern is written into a fiber core when exposed
at this position. Figure 5.4 shows the fringes including the self-apodizing
envelope for a 100-micron-long grating. The fringe period has been chosen
for illustration purposes only. For real laser systems, it is possible to
apodize grating lengths approaching —50 mm [9].
The principle of apodisation described above is based on the moire
effect. Apodization occurs in the presence of two gratings of different
periods, without affecting the total UV dose. The envelope of such a moire
grating is a cosine function, and to alter it, for example, to a Gaussian,
a laser with the appropriate spectral shape may be used. This could be
a broadband frequency-doubled dye laser source.