Long-period grating theory

Photoinduced long-period fiber gratings (LPG) with periods L_LPG = 10² - 10³ 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 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:

b2 - b1 = 2pk/L,

(1)

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

(neffcore - neffclad) LLPG = lLPG,

(2)

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 single-mode fiber, only HE₁₁ 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 ~ 10⁴ at n_ext = 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):

(3)

where a is the core radius, Е_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 inter-modal 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 energy-normalized 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 НЕ₁₂, 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 [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):

R(z) = cos2(z(h2 + d 2)1/2) + d 2 sin2(z(h2 + d 2)1/2)/(h2 + d 2) S(z) = h2 sin2(z(h2 + d 2)1/2)/(h2 + d 2),

(4)

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);

d = pDneffDl/(l2LPG) = (p/L) (Dl/lLPG)

(5)

is the normalized frequency, which describes the deviation from the exact synchronism (2); h is the coupling coefficient defined by a relation:

h = CpDnmodI/lLPG;

(6)

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:

R(z) = cos2(hz)         S(z) = sin2(hz).

(7)

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:

.

(8)

Equation (8) allows one to obtain Dl₀ 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. Near-field 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 near-field 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

1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, J. E. Sipe, T. Erdogan, "Long-period fiber gratings as band-rejection filters", OFC'95, PD4-2, 1995.
2. T. Tamir, ed., "Integrated Optics", Vol.7 of Topics in Applied Physics, Springer-Verlag, 1975.
3. T. Erdogan, "Fiber grating spectra", J. Lightwave Technol., 15, pp. 1277-1294, 1997.
4. S. A. Vasiliev, E. M. Dianov, A. S. Kurkov, O. I. Medvedkov, V. N. Protopopov, "Photoinduced in-fibre refractive-index gratings for core-cladding mode coupling", Quantum Electron., 27, pp. 146-149, 1997.
5. T. Erdogan, "Cladding-mode resonances in short- and long-period fiber grating filters", J. Opt. Soc. Am. A, 14, pp. 1760-1773, 1997.
6. H.-G. Unger, "Planar optical waveguides and fibers", Oxford, Clarendon Press, 1978.
7. T. Erdogan, D. Stegall, "Impact of dispersion on the bandwidth of long-period fiber-grating filters", OFC'98, OSA Techn. Dig. Series, 2, pp. 280-281, 1998.
8. S. A. Vasiliev, E. M. Dianov, O. I. Medvedkov, V. N. Protopopov, D. M. Costantini, A. Iocco, H. G. Limberger, R. P. Salathe, "Properties of the cladding modes of an optical fibre excited by refractive-index gratings", Quantum Electron., 29, pp. 65-68, 1999.