Apr 5, 2005
Long-period grating theory
Photoinduced long-period fiber gratings (LPG) with periods LLPG = 102 - 103 mm were proposed in . 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 coupled-mode theory in which only two modes are supposed to be nearly phase-matched 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 b1 and b2 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 neffcore and neffclad are effective refractive indexes of the core and cladding modes, respectively, and lLPG is the resonance coupling wavelength.
In order to get a complete set of modes HElm and EHlm (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 single-mode fiber, only HE11 mode is guided by the fiber core at l > lc (where lc is the cutoff wavelength) . Normally, a large quantity of modes (N ~ 104 at next = 1) can be guided by the cladding (stripped fiber with 125-mm 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 cross-section 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, Åcore and Eclad 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 inter-modal conversion. Its value is large only for HE1m (m > 1) cladding modes, because only these modes have a sufficiently great electric field component in the fiber core. Fig.1 shows the energy-normalized radial distributions of the electric field for some of HE1m cladding modes .
Fig.1. Radial distributions of the electric field amplitude of the cladding modes ÍÅ12, HE13, HE16
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 inter-modal 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 .
Fig.2. A typical transmission spectrum of a LPG
The solution of coupled mode equations  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 z-coordinate 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:
Dnmod is the induced index modulation amplitude of the fiber core, related with the total induced index change Dnind via relation Dnmod = Dnind/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 Dl0 in the assumption of a constant Dneff within the grating bandwidth. However, the presence of dispersion of Dneff in some cases can give a significant discrepancy between the experimental bandwidth and the calculated one. In  it was shown that the first term in Dneff expansion into a Taylor series in the vicinity of lLPG 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. Near-field intensity distributions of HE1m 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 near-field technique for a group of cladding modes . 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.
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