1. Introduction

Higher-order mode (HOM) fibers have attracted recent attention due to their ability to propagate optical signals with large effective areas and high modal purity even when coiled to relatively small diameters [1–5]. These characteristics are desirable for high-power fiber lasers where non-linear processes can limit peak and average power. While the near field mode field distributions of these fibers produce rather large M² beam parameter values, they are highly spatially coherent and can be converted to near-Gaussian beams using long period gratings or phase plates [7]. These fibers have been considered for use in high-power narrow-linewidth fiber amplifiers due to the bend-resistance of the higher-order modes [3,6]. In this application, stimulated Brillouin scattering (SBS) is the main limiting process. Measurements of the Brillouin gain efficiency in the LP₀₁ and LP₀₈ modes have shown suppression in the peak Brillouin gain in the LP₀₈ case consistent with its higher non-linear effective area [6]. The fact that these fibers support multiple modes introduces the possibility that inter-modal Brillouin scattering can occur in which case the pump beam and counter-propagating Stokes beam can propagate in different transverse modes. Indeed, beam cleanup through stimulated Brillouin scattering in multi-mode optical fibers relies on this process [8].

In this paper we present finite-element calculations of the inter-modal Brillouin gain in a HOM fiber similar to those reported previously [2,6] and discuss the implications of our results for the application of HOM fibers for realizing high-power narrow-linewidth amplifiers.

2. Finite element calculation of inter-modal Brillouin gain

We make the assumption that the pump and counter-propagating Stokes beams each propagate in a single LP_0n mode. Since the thermal initiation of the SBS process leads to excitations of each transverse mode [10], it is reasonable to assume that the mode experiencing the highest Brillouin gain will grow at the expense of the others leading to a single-transverse mode Stokes beam. Following a procedure described previously [9] leads to the following equation for the evolution of the Stokes power in the absence of pump depletion:

Symmetry dictates that a pump beam in an LP_0n mode cannot scatter into an LP_mn Stokes mode in an azimuthally-symmetric fiber for$m\ne 0$as can be deduced from Eq. (2). In the case of such scattering, all terms in the integral in the numerator of this equation are azimuthally symmetric except f_s. Therefore, this integral must be zero due to the sinusoidal dependence of f_s on the azimuth angle which can be derived using the separation of variables technique to solve the scalar optical wave equation. In the limit of a uniform plane-wave acoustic disturbance the peak value of $C_B$in Eq. (2) reduces to $C_{B,\text{peak}}=g_B/A_{\text{eff}}$where $g_B$is the usual peak Brillouin gain coefficient, found to be $1.9\times {10}^{-11}$m/W for the parameters employed here, and the generalized inter-modal non-linear effective area is

Likewise, it is useful for comparison purposes to define the equivalent area $A_{eq}\equiv g_B/C_{B,\text{peak}}$for the acoustically inhomogeneous case.

Implementing a finite element method [9] leads to the following expression for the Brillouin gain coefficient

W_k are the weighting factors for the summation over the numerical integration points, K is the acoustic stiffness matrix, V is the metric to accomplish numerical integration over the fiber cross-section, ×represents elementwise multiplication, N _k are the element shape functions, J _k are the Jacobian matrices for the local to global coordinate transformation, and $E_{p,s}$are the vectors containing the values of the electromagnetic fields, defined on the nodes of the finite element mesh, characterizing the pump and Stokes transverse modes respectively. More information on the symbols used in Eq. (1-9) may be found in a prior work [9].

The principal distinction between the intra-modal and inter-modal Brillouin gain processes is the electrostrictive driving field. In the intra-modal case, the mode field intensity profile of the single participating transverse mode comprises this field. In the inter-modal case, interference between the pump beam in one transverse mode and the counter-propagating Stokes beam in the other transverse mode creates the electrostrictive driving field according to Eq. (3) or in the finite element treatment, Eq. (8). Another physical characteristic of inter-modal Brillouin scattering is the shift in the Brillouin gain peak frequency due to the difference in the effective modal refractive indices $n_{s,p}$between the two transverse modes. In this case, the local Brillouin frequency takes the form

3. Results and discussion

LP_0n modes exhibit higher inter-modal index spacing, and therefore more robust guiding, for larger values of n [2,3]. We therefore first consider the case in which the pump beam is launched in the LP₀₈ mode of a fiber similar to ones reported previously [2,6] characterized by a single-mode core with a 10 µm diameter surrounded by a pure silica cladding with an 82 µm cladding which is then surrounded by another down-doped cladding to facilitate guiding of the higher-order LP₀₈ mode. The single-mode core has an acoustic velocity 4% slower than that of pure fused silica and an optical numerical aperture of 0.09. The calculated Brillouin gain spectrum for LP₀₈- LP₀₈ scattering is shown in Fig. 1 for a pump wavelength of 1083 nm. Two distinct peaks are present corresponding to the different acoustic velocities in the core and cladding. These peaks are separated in frequency by 600 MHz or ~4% of the Brillouin frequency shift as expected based on Eq. (10) and the low frequency peak exhibits noticeably higher gain than the high frequency peak.

 

Fig. 1 Brillouin gain spectrum for LP₀₈-LP₀₈ scattering showing the electric field magnitude profile, acoustic index profile and peak acoustic displacement profiles in the insets.

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We next examine the ability of a pump beam in the LP₀₈ mode to amplify other LP_0n Stokes modes. Table 1 shows the Brillouin gain equivalent and effective areas and the Brillouin frequency shift at which the maximum gain occurs. We notice that the gain experienced by the LP₀₁ Stokes mode is 7 dB greater than the next strongest mode which happens to be LP₀₈. The mode with the least peak gain is the LP₀₂ mode 10 dB below the LP₀₁ gain. The peak gain in each of these cases is either near 15.35 GHz or 15.97 GHz depending on whether the strongest interaction occurs in the core or the cladding. For pump powers well above the SBS threshold the frequency with the highest peak gain will dominate the Stokes signal. These results suggest that as the SBS threshold is approached, the Stokes beam will consist of the LP₀₁ mode as this mode will be amplified much more strongly than any other mode.

Table 1. Properties of inter-modal Brillouin gain with an LP₀₈ pump beam

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Figure 2 shows the Brillouin gain spectrum for an LP₀₈ pump beam and a LP₀₁ Stokes beam. The high frequency peak near 16 GHz is barely noticeable due to the tight confinement of the LP₀₁ mode in the core. The low frequency peak near 15.4 GHz occurs at a frequency approximately 30 MHz higher than in the LP₀₈-LP₀₈ case due to the higher effective index of the LP₀₁ mode. The HOM fiber considered here may also be operated in lower-order LP_0n modes so if these modes exhibit reduced scattering into the LP₀₁ Stokes mode, they may be more suitable for high-power narrow-linewidth amplification. Table 2 shows the properties of the gain spectrum for the LP₀₁ Stokes beam when the pump beam is an LP_0n mode.

 

Fig. 2 Brillouin gain spectrum for LP₀₈-LP₀₁ scattering. The insets show the acoustic displacement profile at the Brillouin gain peak and a logarithmic intensity plot of the interference pattern between the two modes that drives the acoustic displacement through electrostriction.

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Table 2. Properties of inter-modal Brillouin gain with an LP₀₁ Stokes beam

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These results suggest that a tradeoff exists between robust guiding, which occurs for higher order LP_0n modes and reduced Brillouin gain which occurs for lower order LP_0n modes. For example the LP₀₄ mode which exhibits a higher non-linear effective area and robust guiding [2] may be a suitable candidate as it exhibits a maximum Brillouin gain only slightly larger than the peak gain for LP₀₈-LP₀₈ scattering. All of the gain peaks occur at frequencies near 15.4 GHz due to the tight confinement of the LP₀₁ mode within the core where the local Brillouin frequency is near this value.

The values of C _B derived from the values of A _eq listed in Tables 1 and 2 for LP₀₈-LP₀₈ and LP₀₁-LP₀₁ scattering of 0.0068 and 0.23 (m-W)⁻¹ are fairly close to experimental values of 0.0085 and 0.20 (m-W)⁻¹ obtained with a similar fiber [6]. These differences may be accounted for by differences in the material parameter g _B and the refractive and acoustic index profiles between the two fibers which give rise to different values for A _eq and A _eff. The C _B value of 0.0077 (m-W)⁻¹ for LP₀₈-LP₀₈ scattering in the experimental fiber calculated previously [6] assumed a uniform acoustic profile with $g_{\text{B}}=\text{1}\text{.3}\times {\text{10}}^{-11}$m/W and$A_{\text{eff}}=1714\mu {\text{m}}^2$.

Completely neglecting the acoustic properties of the modeled fiber yields a theoretical C_B value of 0.014 (m-W)⁻¹ for LP₀₈-LP₀₈ scattering which is almost double the observed value. The difference of approximately 10% in the maximum modeled gain for the dominant LP₀₈-LP₀₁ scattering process caused by neglecting the acoustics is insignificant by comparison. We also note that published experimental Stokes spectra [6] are consistent with the modeling results presented here. Far above threshold where these spectra were taken, the low-frequency LP₀₈-LP₀₈ BGS peak dominates the Stokes spectrum leading to a single peak occurring at a slightly lower frequency than the LP₀₁-LP₀₁ peak as observed. The expected ratio of the low and high frequency Stokes signals based on the gain coefficient ratios, the pump power, and the length of fiber [6] is approximately 40 which would make the higher frequency Stokes peak undetectable due to noise present in the measurement.

The acoustic velocity profiles of HOM fibers such as the one treated here arise naturally from the doping profiles required to establish the desired optical index profile. The core of this fiber has an acoustic index higher than that of the surrounding cladding and thus guides acoustic modes. While this has been shown to cause an increase in SBS in conventional single-mode step-index fibers [11], in the higher-order mode case, the acoustically-guiding core reduces the peak Brillouin gain. This is due to the fact that the acoustic displacement field extends beyond the core enough to shift some of the Brillouin gain to the higher cladding frequency near 16 GHz. This effect is the strongest for higher-order modes that have greater overlap with the cladding region as reflected in Tables 1 and 2.

The differences in Brillouin gain coefficients for the acoustically heterogeneous and homogeneous cases vary between 1% and 50% for the 15 total scattering processes considered here. The majority of the large differences occur in the case of an LP₀₈ pump beam. Amplifier architectures incorporating long period gratings to launch the LP₀₈ mode are most susceptible to LP₀₈-LP₀₈ scattering, highly affected by acoustic variations, due to the conversion of the LP₀₈ Stokes signal back to the LP₀₁ mode as it traverses the grating. This reinforces the need for reliable numerical methods for calculating Brillouin gain coefficients in HOM fibers.

4. Conclusion

We have presented a numerical analysis of LP₀₈-LP_0n and LP_0n-LP₀₁ inter-modal Brillouin gain for $1\le n\le 8$in higher-order-mode fibers. The peak LP₀₈-LP₀₈ Brillouin gain exhibits 15 dB suppression relative to the LP₀₁-LP₀₁case. The acoustic structure of the fiber most directly influences LP₀₈-LP₀₈ Brillouin gain resulting in a 3 dB difference in the peak gain coefficient when the acoustic structure is neglected. For an LP₀₈ pump beam, the gain is the strongest for the LP₀₁ Stokes mode, its peak gain coefficient being 7 dB higher than that for LP₀₈-LP₀₈ scattering. Preliminary experimental investigations with a Brillouin amplifier arrangement have demonstrated LP₀₈-LP₀₁ inter-modal gain in reasonable agreement with the modeling presented here. However, evidence of this inter-modal gain is absent from thermally-initiated SBS experiments presented in [6]. Further theoretical and experimental studies of thermally-initiated LP_0n-LP₀₁ scattering are therefore warranted. These studies are currently underway and their results reserved for a future work. In the absence of extensive experimental data, the modeling results presented here provide a description of intra-modal Brillouin gain in higher-order-mode fibers consistent with currently-available experimental data.

To the authors’ knowledge, a model of inter-modal Brillouin gain in acoustically heterogeneous fibers has never before been discussed in the literature. This work bridges the gap between general multi-mode SBS calculations which require numerous approximations [8] and SBS calculations assuming single-transverse-mode pump and signal propagation [9]. The modeling presented in this paper, while exercised for a simple idealized HOM fiber, provides the groundwork for a fiber design tool for HOM fibers and amplifiers that admits the possibility that inter-modal Brillouin gain determines the SBS threshold. This may lead to improved HOM fiber designs.