Updated on: Apr 5, 2005 webmaster 
Longperiod grating theoryPhotoinduced longperiod fiber gratings (LPG) with periods L_{LPG} = 10^{2}  10^{3} mm were proposed in [1]. Such gratings couple the fundamental mode with the cladding modes of the fiber, propagating in the same direction. The excited cladding mode attenuates in the coated fiber part after the grating, which results in the appearance of resonance loss in the transmission spectrum. In contrast to Bragg grating, LPG does not produce reflected light and can serve as spectrally selective absorber. The interaction of one mode of a fiber with other modes is commonly described with the help of coupledmode theory in which only two modes are supposed to be nearly phasematched and capable of resonant coupling. Based on this theory, quantitative information about the coupling coefficients and spectral properties of fiber gratings can be obtained [2, 3]. Two modes are coupled by a grating with period L, if their propagation constants b_{1} and b_{2} satisfy the phase matching condition:
where k is an integer describing the order of the grating, in which the mode coupling occurs. Calculation methods of spectral characteristics of LPGs can be found in papers [4, 5]. Below we will consider the most important relations describing the grating properties. Equation (1) for the resonant coupling of the fundamental mode and one of the cladding modes can be rewritten as
where n_{eff}^{core} and n_{eff}^{clad} are effective refractive indexes of the core and cladding modes, respectively, and l_{LPG} is the resonance coupling wavelength. In order to get a complete set of modes HE_{lm} and EH_{lm} (l and m are azimuthal and radial orders of the mode, respectively), the wave equation for a dielectric cylinder with a certain radial index distribution should be solved. In a singlemode fiber, only HE_{11} mode is guided by the fiber core at l > l_{c} (where l_{c} is the cutoff wavelength) [6]. Normally, a large quantity of modes (N ~ 10^{4} at n_{ext} = 1) can be guided by the cladding (stripped fiber with 125mm cladding diameter). Nevertheless, only some of them have a significant overlap integral I with the fundamental core mode. The integral should be taken in the fiber crosssection region, where modulation of the refractive index has been induced (for photoinduced gratings, the integration region usually coincides with the fiber core):
where a is the core radius, Å_{core} and E_{clad} are the amplitudes of the electrical field of the core and cladding modes, respectively, r and j are radial and azimuthal coordinates. The overlap integral I defines the efficiency of intermodal conversion. Its value is large only for HE_{1m} (m > 1) cladding modes, because only these modes have a sufficiently great electric field component in the fiber core. Fig.1 shows the energynormalized radial distributions of the electric field for some of HE_{1m} cladding modes [4]. Fig.1. Radial distributions of the electric field amplitude of the cladding modes ÍÅ_{12}, HE_{13}, HE_{16} These modes are linearly polarized, their intensity distributions are axially symmetric, and the number of zeroes in the radial direction is m  1. The overlap integral increases with increasing the radial mode number up to m ~ 10, which is accompanied by an increase in the intermodal coupling intensity. The latter can be seen from the transmission spectra of LPGs (Fig.2). Starting with a certain value of m, the overlap integral decreases to zero and thereafter oscillates with m, the amplitude of the oscillation tending to zero [5]. Fig.2. A typical transmission spectrum of a LPG The solution of coupled mode equations [2] in the approximation of two interacting modes traveling in the same direction and in the assumption of small amplitude of induced index modulation in comparison with the silica glass index, gives the following energy exchange law (for initial conditions R(0) = 1, S(0) = 0):
where R(z), S(z) are the normalized energies of the core and cladding modes, respectively, considered as a function of zcoordinate along the fiber axis (the beginning of the grating corresponds to z = 0);
is the normalized frequency, which describes the deviation from the exact synchronism (2); h is the coupling coefficient defined by a relation:
Dn_{mod} is the induced index modulation amplitude of the fiber core, related with the total induced index change Dn_{ind} via relation Dn_{mod} = Dn_{ind}/2, C is a constant equal to the first coefficient in the Fourier transform of the grating pitch shape. If the index profile is sinusoidal, this constant is equal to unity. For a rectangular profile, which is more typical for LPG, C = 4 sin(pd/L)/p, where d is the size of the irradiated part of the fiber within one grating period. At the exact resonance (d = 0), equation (4) gives a sinusoidal law of the energy exchange, showing a possibility of mutual energy transfer from one mode to another:
With the help of equations (4) and (5) it is possible to determine the total spectral width, which follows from the first zero of the grating spectrum:
Equation (8) allows one to obtain Dl_{0} in the assumption of a constant Dn_{eff} within the grating bandwidth. However, the presence of dispersion of Dn_{eff} in some cases can give a significant discrepancy between the experimental bandwidth and the calculated one. In [7] it was shown that the first term in Dn_{eff} expansion into a Taylor series in the vicinity of l_{LPG} is sufficient to estimate the bandwidth to an acceptable accuracy. As mentioned above, the standard diameter of silica cladding is large enough and the cladding can guide plenty of modes. An LPG allows selective excitation of an individual cladding mode. This fact makes it possible to investigate the field distribution of the cladding mode excited by the grating. Fig.3. Nearfield intensity distributions of HE_{1m} cladding modes with m = 4 (a), 5 (b), 6 (c), 7 (d), 8 (e), 10 (f). Fig.3 shows the intensity distributions measured by the nearfield technique for a group of cladding modes [8]. All the modes, as follows from theoretical consideration, have pronounced axial symmetry, which corresponds to azimuthal number l = 1. The number of rings is equal to the radial mode number m. References


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