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),
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