Page 189 - Fiber Bragg Gratings
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166 Chapter 4 Theory of Fiber Bragg Gratings
In understanding the physics of the scattering, we consider separately
the two components of the integral, the transverse phase-matching term
(Eq. 4.7.17) and the longitudinal phase-matching (pm) term which de-
pends on the detuning, A/8 6.
In the low-loss regime (a <^ A/3 b), the longitudinal pm term is simply
like the Bragg matched reflection condition, but now as a function of <p.
For all practical purposes, this term is like a delta function that is only
significant at very small angles of radiation (<p < 1°). The integral has a
term dependent on cos<p, which becomes broader and asymmetric in its
angular bandwidth as <p —» 0° and which is also inversely dependent on
the length of the grating. For typical filter lengths of a few millimeters,
we find the angular bandwidth to be —1°. The asymmetry and broadening
at small phase-matching angles have been observed in phase-matched
second-harmonic generation with periodic structures [50].
In the high-loss regime, we find that the delta function broadens but
has a width similar to that of the low-loss case. We can therefore choose
to consider the dependence of the scattered power on the longitudinal
phase matching as a very narrow filter at a given angle. Comparison of
the longitudinal term with the transverse pm condition of Eq. (4.7.17)
shows that the angular dependence of the radiation for the transverse
case varies much more slowly and may be approximated to be a constant
over the region of the longitudinal bandwidth. Figure 4.19 shows the
dependence of the longitudinal and the transverse pm as a comparison for
standard fiber and a uniform grating profile, W grating = I. The longitudinal
response for a blaze angle of 5° and the transverse response for three
blaze angles are shown.
The analytical result for the loss coefficient a has been shown to be
[511,
By normalizing the radius as p = r/a (a is the core radius),