## Abstract

Influence of cladding-mode coupling losses on the spectrum of multi-channel fiber Bragg grating (FBG) has been numerically investigated on the basis of the extended coupled-mode equations. It has been shown that there exists a reflection slope in the spectrum of both the intra- and inter-channels due to the existences of the cladding modes. This slope could be larger than 1 dB when the induced index change is about 3×10^{-3}, which makes the channels considerably asymmetric. For comparison, a 39-channel linearly chirped FBG with a channel spacing of 0.8 nm and a chromatic dispersion of -850 ps/nm has been designed and fabricated. The experimental results show good agreement with the numerical ones. Finally, one method to pre-compensate the reflection slope within the intra-channels has been proposed and successfully demonstrated.

© 2005 Optical Society of America

## 1. Introduction

Recently, multi-channel fiber Bragg grating (FBG) has attracted great interest in dense wavelength division multiplex (DWDM) system due to its identical inter-channel specifications for filtering or chromatic dispersion compensation [1–6]. For a FBG written in a standard single-mode photosensitive fiber, mode coupling occurs not only between the forward and backward core-mode, but also between the forward core-mode and the backward cladding-modes, i.e., there always exist some loss-bands on the short wavelength side of the main Bragg-band due to the core-cladding coupling [7]. This loss could be fairly large if the grating is strong enough. For a one-channel FBG, since the cladding modes are resonant at wavelengths beyond the main Bragg-band, cladding modes will have no effect on the main band of FBG itself. However, for a multi-channel FBG, since most of the cladding modes are resonant at the wavelengths within the channels of FBG, the losses result from the claddingmodes coupling will inevitably affect the grating’s performances and thus limit its utilization in DWDM system. To solve this problem, various methods to minimize the cladding-modes losses have been proposed [8–11]. Most of them are based on the use of some special fibers, which are designed with either a photosensitive or a depressed cladding for suppression of the core coupling into the cladding mode. To date, this kind of fiber has been regarded as the best selection to write a multi-channel or a broad-band FBG, since the cladding-mode loss could be restrained less than 0.2 dB for a one-channel FBG with a strength over 30 dB [12]. However, there exist two inherently negative effects when the grating is written in a cladding-mode suppression fiber. First, the fusion-splicing losses between the grating and the standard single-mode fiber (for example, SMF-28) will considerably increase due to the core-mode matching of these two fibers [9], so great effort should be taken in the process of splicing. Second, it has been empirically found that the gratings generally have larger polarization-mode-dispersion (PMD) compared with the ones written in standard photosensitive fibers. The reason for the higher PMD is most probably due to the special geometric structure of this fiber, which will in return result in a higher power penalty for the transmitted signal and limit the grating’s application in a fiber transmission system with bit-rate higher than 10 Gb/s or beyond.

Therefore, we believe that there exists a trade-off for choosing either the general photosensitive fiber or the cladding-mode suppression fiber to write a multi-channel FBG. In other hand, when a FBG with high-channel-counts is fabricated, since the demands for the induced index-change are considerably increased compared with that in one-channel FBG, it is desired to know that in what extent the coupling between the core mode and the cladding mode can be neglected even if the cladding-mode suppression fiber is used. Therefore there exists a strong demand to theoretically and systematically investigate the influence of the cladding-mode losses on the response of multi-channel FBG. Finazzi et al. [13] first addressed the effect of cladding-mode losses on a linearly chirped FBG. However, their discussion is limited to a single-channel FBG and a few cladding modes have been considered.

In this paper, investigations for the cladding mode influence on the spectrum of a multi-channel FBG with high-channel-counts have been numerically and experimentally demonstrated in the first time, to the best of our knowledge. Typically, a linearly chirped 39-channel FBG with a channel spacing of 0.8 nm and chromatic dispersion of -850 ps/nm has been designed and demonstrated, which is assumed to be written in a three-layer fiber with step-index profile. The conventional coupled-mode equations including both the core-core and core-cladding mode coupling presented by Erdogan [14] is extended to a new one, where any multi-channel FBGs based on a sampling method are included. Finally a method to alleviate the cladding-mode influence has been demonstrated.

## 2. Theory of the coupling for the core-core and core-cladding modes in a multi-channel FBG

#### 2.1 Extended coupled-mode equation for a sampled FBG

Sampling method is widely used to realize a multi-channel FBG in a limited length of photosensitive fiber. In a sampled FBG, the induced refractive index modulation Δ*n* can be expressed as

where *Δn*
_{1}(*z*) is the maximum index modulation, *z* is the position along the grating. *Λ* is the central pitch of the grating, and *ϕ*_{g}
(*z*) is the phase change for one channel grating. *s*(*z*) represents a sampling function with period *P*, which is a amplitude-only or phase-only function and can be expressed in the Fourier series

where *s*_{m}
is the *m* complex coefficient of the Fourier series. Based on the model used to theoretically describe the sampled FBG [5], the extended coupled-mode equations including the core-core and core-cladding coupling are given by

$$+i\sum _{\nu =1}^{N}\sum _{m=-M}^{M}{S}_{m}({\kappa}_{\nu}^{cl-co}\u20442){B}_{\nu}^{cl}\mathit{exp}\{-i(2\Delta {\beta}_{\nu}^{cl-co}z-{\varphi}_{g}\left(z\right)-2m\pi z\u2044P)\}$$

where *A*^{co}
and *B*^{co}
are the amplitude of the forward- and backward- core mode field, respectively. ${B}_{\nu}^{\mathit{\text{cl}}}$
is the amplitude of the *ν*th cladding mode. Δ*β*
^{co-co}=*π*·(2${n}_{\mathit{\text{eff}}}^{\mathit{\text{co}}}$
/*λ*-1/*Λ*) and Δ${\beta}_{\nu}^{\mathit{\text{cl-co}}}$
=*π*·(${n}_{\mathit{\text{eff}}}^{\mathit{\text{co}}}$
/*λ*+${n}_{\mathit{\text{eff}}}^{\mathit{\text{cl}}}$
(*ν*)/*λ*-1/*Λ*) are the detuning parameters. ${n}_{\mathit{\text{eff}}}^{\mathit{\text{co}}}$
and ${n}_{\mathit{\text{eff}}}^{\mathit{\text{cl}}}$
(*ν*) are the effective index of the core-mode and the *ν*th cladding mode, respectively. *κ*^{co-co}
and ${\kappa}_{\nu}^{\mathit{\text{cl-co}}}$
are the coupling coefficients corresponding to the core-core mode and the core-*ν*th cladding mode coupling, respectively. *N* is the number of the cladding modes to be considered here. 2*M* +1 is the number of channel for the FBG. It is noted that the couplings among the cladding modes are neglected here, since they are very small compared with those happened between core and cladding modes [14]. By using the boundary conditions,

where *L* is the grating length, Eqs. (3)–(5) can be numerically solved with the Runge-Kutta integration method, and the transmission *T*(*λ*) and reflection *R*(*λ*) of the grating are given by

#### 2.2 Coupling for core -core and core-cladding mode

Prior to the numerical calculation of Eqs. (3)–(5), one needs to determine the resonant wavelengths for core and cladding modes based on the parameters of the fiber and the central pitch of the grating. The corresponding coupling coefficients for all the core-core and core-cladding coupling need to be decided as well. For convenience, we assume that the fiber used here is a simple three-layer with step-index, and the radius and refractive index of the core and cladding are *a*
_{1}, *n*
_{1}, and *a*
_{2}, *n*
_{2}, respectively. The surrounding material is assumed to be air with index *n*
_{0}=1. In general, solution to the resonance wavelengths of the backward core-mode and the cladding modes are somewhat complicate and one can obtain them by solving the dispersion equation given in Ref. [14] with the phase matching condition Δ*β*^{co-co}
=0 and Δ${\beta}_{\nu}^{\mathit{\text{cl-co}}}$
=0. According to the coupled-mode theory, the transverse coupling coefficients for core mode (HE_{11}) and cladding modes are expressed by

where *E*^{co}
(*r*,*φ*) and ${E}_{\nu}^{\mathit{\text{cl}}}$
(*r*,*φ*) represent the transverse electrical fields of core- and cladding-mode, respectively. For convenience, we limit our calculation to the grating where the induced index-change is circularly symmetrically distributed in the transverse plane of the fiber, so nonzero coupling for core-cladding mode occurs only between the core mode (HE_{11}) and the hybrid cladding modes with azimuthal number *l*=1. Note that since the index difference between the cladding and the surrounding material (air) is rather large, the concept of linear polarization (LP) mode is not suitable here for the solution of the cladding modes. By solving the dispersion equation of (5) in Ref. [14], we have obtained two sets of cladding modes with azimuthal number *l*=1, i.e., HE_{1ν} and EH_{1ν}. Since EH_{1ν} modes are radially asymmetric and change sign when the azimuthal angle *φ* changes by 180°. From Eq. (9), it is expected that no coupling would occur between the core mode and the EH_{1ν} cladding mode, that is the reason why the effects of the even number of cladding modes reported in Ref. [14] are so small and can be neglected. Thus, for a FBG with a symmetric index-change, the mode coupling happens only between the forward and backward core-mode (HE_{11}), and between the forward core-mode and the backward cladding-modes HE_{1ν}, respectively.

In the following calculations, parameters about the radii and the refractive indexes of the core, cladding, and the surrounding material are typically chosen as: *a*
_{1}=2.65 µm, *a*
_{2}=62.5 µm, *n*
_{1}=1.46, *n*
_{2}=1.45 and *n*
_{0}=1.0. The central pitch of the grating is assumed to be 533.5 nm and the central wavelength is supposed to be 1550 nm, so the effective index for the core mode can be obtained ned as ${n}_{\mathit{\text{eff}}}^{\mathit{\text{co}}}$
=1.45267 at wavelength of *λ*
_{0}=1550 nm. Figure 1 shows the calculated results for the resonant wavelength and the corresponding refractive index of the cladding modes. The red symbols represent the resonant points. It can be seen that both the resonant wavelength and the effective index are decreased when the mode number *ν* is increased. Moreover, the interval of each mode is gradually increased. Noted that there actually exist more than 100 cladding modes. Since the wavelength span of the least 35 modes is about 40 nm, which is larger than the span of a 39-channel FBG (channel spacing =0.8 nm), it is enough to consider only the least 35 cladding modes as shown in Fig. 1(a). With Eq. (9), dependence of the normalized coupling coefficient (=${\kappa}_{\nu}^{\mathit{\text{cl-co}}}$
/*Δn*
_{1}) on the wavelength can be calculated as shown in Fig. 1(b), where the core mode is supposed to be resonant at 1550 nm.

## 3. Numerical and experimental results

A 39-channel linearly chirped FBG is designed by using the general sampling method [5], which can be used as a broad-band dispersion compensator to cover the whole C or L band of DWDM system. Figure 2 shows a typical design result obtained from Eqs. (3)–(5) where all the couplings between the core-cladding modes are assumed to be zero. As is shown that this grating is designed to have a reflection of 80 % and chromatic dispersion D_{2}=-850 ps/nm in each channels with a bandwidth of 0.24 nm. The channel spacing is 0.8 nm and whole length of this grating is 6 cm. Figure 2(b) shows a one-channel spectrum at the wavelength of 1550 nm, which is magnified part of Fig. 2(a) in the central wavelength region.

Next, we consider the cladding-mode terms in Eqs. (3)–(5). By substituting the known coupling coefficients into the Eqs. (3)–(5) and integrating them from the input port to the end of the grating, the resulted spectrum with considering the cladding-modes effect can be obtained. Figure 3(a) shows the results for a 39-channel linearly chirped FBG, where the maximum index-change is about 3×10^{-3}. Figures 3(b), (c), and (d) shows part of the spectra at the shortest wavelength side, the central wavelength region, and the longest wavelength side, respectively. The reflection for each channel is about 80%. Compared with the results shown in Fig. 2, it is found that due to the existences of the cladding modes, the spectrum in both the intra- and inter-channel become asymmetric, i. e., there exist an amplitude slope (roll-off) in the reflection spectrum. For the channels at the longest wavelength side, this slope is very small and can be neglected as shown in Fig. 3 (d). However, for those channels at the shortest wavelength side, this slope becomes large and could be larger than 1 dB as shown in Fig. 3(b), this kind of strong non-uniformity in both the intra- and inter-channel would make the multi-channel FBG not acceptable in DWDM system. For references, the cladding-mode influence on one-channel FBG has also been calculated with the same fiber. Figure 4 shows the results obtained by simply setting *M*=0 in Eqs. (3)–(5), where the index-change of Δ*n*
_{1} is 1.6×10^{-4}. From the inset, it is found that the cladding-mode loss is larger than 0.8 dB while the grating strength for the main Bragg band is about 30 dB. This result means that for one-channel FBG with strength over 30 dB, the cladding-mode loss can be larger than 0.8 dB when a standard single-mode fiber is used.

In order to alleviate the amplitude slope at short wavelength side, we propose one method using the pre-compensated apodization. Figure 5 shows the principle. The curve I in Fig. 5 (a) is the obtained spectrum within the central channel as is shown in Fig. 3(c), the curve II is the opposite of the curve I about the dot line at the central wavelength of 1550 nm. If the apodization (profile of the index-change) can be pre-designed to have a spectrum profile like the curve II, then it is expected that the resulted spectrum under the influence of the cladding-mode losses will be flat. Here, we set the curve II as the spectral target, using the discrete layer peeling method [15], it is easily for us to obtain the grating structures such as the new apodization curve. Fig. 5(b) shows the results for the index-change profiles, where the curve I corresponds to the original apodization and curve II is the obtained pre-compensated apodization. Figure 6 shows the design results for 39-channel FBG with the pre-compensated apodization (curve II in Fig. 5(b)). Compared with Fig. 3, it is found that the spectrum within the central channels region become nearly flat as shown in Fig. 6 (c), and the amplitude slopes within those channels at the shortest wavelength side become small. Although the amplitude slopes within those channels at the longest wavelength side become worse, the maximum of the in-band reflection roll-off is decreased to 0.5 dB from the original magnitude of 1 dB. In the following, we will verify that this improvement in the intra-channel spectrum is enough to make our high - channel - counts FBG available in DWDM system. However, it needs to point that the pre-compensated apodization here is obtained by only considering a one-channel spectrum in central wavelength region, so the amplitude slope throughout all the channels cannot be completely eliminated. As a more precisely method, we need to design a 39-channel FBG directly with a layer peeling method [16] rather than the sampling method, where each channel could be non-identical, i.e., the response of each channel can be designed to have the profile which is just the opposite one as shown in Fig. 3(a), then the spectra for all the inter- and intra-channel will become flat.

In order to confirm our numerical results, several 39-channel FBGs discussed above have been fabricated by using the advanced phase-mask technique [17]. Figure 7 shows the measured results of one typical FBG with the ordinary super-Gaussian apodization. As expected from the numerical calculation, there exists an amplitude slope throughout all the channels. Figure 8 shows the measuring results for another 39-channel linearly chirped FBG where the pre-compensation apodization is used. For comparison, some specifications of these two gratings shown in Fig. 7 and Fig. 8 are summarized in Table 1. It can be seen that the reflection amplitude slope (maximum value throughout all the channels) is considerably improved after the utilization of pre-compensated appodization. The magnitude is changed from 1.74 to 0.94 dB. Moreover, it can be seen that nearly identical 39 channel with a channel spacing of 0.8 nm, useable bandwidth of 0.21 nm, peak-peak group-delay ripple around 20 ps, and chromatic dispersion of about −850 ps/nm have been obtained. These results agree very well with our design data. Noted that this kind of high-quality high-channel-counts FBG has already been successful used as a dispersion compensator through full C band for transmission over 640-Km SMF fiber [18]. Details of the fabrication technique and testing processes for this kind of gratings are beyond the scope of this paper and will be given elsewhere.

## 4. Conclusions

Cladding mode influence on the spectrum of a linearly chirped FBG with high-channel-counts has been numerically investigated on the basis of the extended coupled-mode equations, where the core-core, core-cladding mode coupling and a sampling function are considered together. It has been found that due to the cladding-mode losses, there exists an amplitude slope in the spectrum of both the intra- and inter-channels if the standard photosensitive fiber is used. Numerical calculations also show that this slope could be larger than 1 dB when the induced index change is about 3×10^{-3}, which makes the intra-channel considerably asymmetric, especially for the channel at the shortest wavelength. For comparison, a 39-channel FBG with a channel spacing of 0.8 nm and a chromatic dispersion of -850 ps/nm has been fabricated and measured. The experimental results agree well with the numerical ones. Finally one method to alleviate the cladding-mode effect has been proposed and successfully demonstrated by using the pre-compensated apodization.

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