Page 224 - Fiber Bragg Gratings
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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.
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