1. Introduction

This paper presents the effects of modal dispersion, through the numerical aperture (NA), and mode size on the SBS threshold of single mode and multimode fibers in an all fiber configuration. In the past threshold was studied for focused pump beams in a pulsed format [1, 2, 3] with some recent work using focused cw pump beams [4, 5, 6, 7]. Reference [4] sets the foundation for the following since they include the effects of inhomogeneous broadening due to waveguiding. However, our approach to threshold differs in two respects from Kovalev, et al. [4]. First, we replace the core area by the optimum Gaussian mode area. Our second difference is experimental. In our experiment we use an all fiber configuration where the pump is launched from single mode fiber pigtail attached to the diode laser. Thus, free space propagation is eliminated. We show that the theory and experiments closely agree. We also find that our approach applies to experiments where the pump is focused into single mode and multimode fibers [2, 5, 7]; we will discuss these experiments in a moment. Finally, we limit our experiments to SBS generators [8] where the fiber ends are cleaved at an angle of about 80 to eliminate feedback. Some characteristics of Stokes beams with and without feedback are discussed in Ref. [1].

We begin by reiterating the development of reference [4]. In their paper they include inhomogeneous broadening of the SBS spectrum resulting from the angular confinement of the pump by up to twice the critical angle. These authors [4] apply their formula to several different fibers and find that the measured linewidths agree with their predictions. However, we are interested in the effects of the area on SBS operation. Thus, in the following we extend their work by measuring the threshold of multimode GRIN fibers and single mode fibers in an all fiber network, and compare these results with theory. This type of network is highly desirable in an industrial setting, for example.

In closing this section we note that reference [4] is not the first attempt to include the effects of pump and Stokes modal dispersion on the increase in the threshold. In the year previous to Kovalev [4] publication Tei, et al. [2] and Tsuruoka, et al. [3] in a joint effort approached the same problem from the standpoint of waveguide modes.

2. Experiment

The experimental setup is shown in Fig. 1. The pump source is a single mode narrow-linewidth (150 KHz), 5W fiber laser operating at a wavelength of 1550nm. A single mode circulator couples the pump into the SBS fiber and outcouples the backwards-propagating Stokes radiation. We investigate SBS in two different fibers. The first is a single-mode fiber (Corning SMF-28) of length 1 km. The second is a multimode GRIN fiber (Corning 50/125) of length 4.4 km. Both fibers are butt-coupled to the pump source using FC/APC connectors. These connectors are used throughout the optical chain to minimize reflection of light. In order to assess the effectiveness of coupling between multimode fibers and single mode fibers we measured the transmission between two short fibers. We found that single mode to multimode has a transmission of 98%, and that the transmission of multimode to single mode is 60%.

Prior to the SBS experiments we investigated the excited mode structure in the multimode fiber. This was done by butt-coupling a short (2 meter) length of MM fiber to the pump source and imaging the fiber end onto a CCD array camera. The spot had a 1/e ² radius of 7.8µm and an M² of 1.15. Thus, only the lowest-order mode in the multimode (MM) fiber is excited under butt coupling. Using the same technique the mode radius of the single-mode (SM) fiber was found to be 4.9µm. These results agree with the theoretical predictions, see Table 1 and the following section.

SBS thresholds were measured for both fibers. Table 1 shows the measured and calculated thresholds and mode waists; we will discuss the theoretical calculations in a moment. The mode waists agree within 1%. Additionally, for the single mode fiber we measure the reflectivity R_SM=P_c/P_p(0) as a function of the input pump power P_p(0), see Fig. 2, where P_c is the Stokes power exiting circulator. For the multimode fiber we calculate the reflectivity R_MM=P_s(0)/P_p(0)=(P_p(0)-P_p(L))/P_p(0) from P_p(z) which is the pump power measured at both ends of the fiber at z=0, and z=L with P_s(L)=0. Note, that R_MM depends only on the measured pump inside the fiber at z=0, L, and hence does not represent a conjugation property. Further, we determine the multimode fidelity from F_MM=P_c/P_s(0)=P_c/P_p(0)/R_MM where P_c is the Stokes power exiting the circulator. F_MM depends on the Stokes power P_c measured external to the fiber compared to P_p(0) which is internal to the fiber. Thus, F_MM represents a measurement of phase conjugation. This definition of phase fidelity is consistent with the standard definition based on overlap integrals. These two quantities are shown in Fig. 2 as a function of input pump power. In both figures we also show, as the solid line, our theoretical calculations for the threshold and the reflectivity R=P_s(0)/P_p(0) obtained from the well known plane wave model [11].We note that for the single mode fiber the reflectivities agree within 3% and for the multimode GRIN fiber the agreement in reflectivities is within 10%. We now discuss the theory which incorporates the effects of the mode area and inhomogeneous gain which we used for the curves in Fig. 2.

 

Fig. 1. Schematic for measuring SBS performance.

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Table 1. Summary of calculated and measured radius and threshold for single mode and multimode fibers.

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3. Theory

In the cited paper[4] the SBS gain coefficient g_i for inhomogeneous broadening due to waveguiding is derived as

 

Fig. 2. Upper figure: single mode fiber reflectivity for a length of 1.0km; lower figure: multimode reflectivity and fidelity for a length of 4.4km. The horizontal axis is input pump power, in watts, and the g₀IL product.

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Also they show that the FWHM of the inhomogeneous spontaneous Brillouin spectrum is

In these equations the frequency shift of the SBS field is given by F_B(ϕ)=2n_cov sin(ϕ/2)/λ, v is the velocity of sound, λ is the pump wavelength in vacuum, n_co is the core index-of-refraction, and ϕ is the backward Stokes angle with respect to the pump. The on-axis frequency is F₀=F_B(π), and the frequency at the critical angle is $F_c=F_B(\pi -2{\theta }_c)=2n_{\mathit{co}}v\sqrt{{1-\left(NA\right)}^2⁄n_{\mathit{co}}^2}⁄\lambda$ where the complement of the critical angle is θ_c=sin^-1((NA)/n_co). Kovalev and Harrison [4] use the values: Γ₀=36MHz for the Brillouin linewidth, λ=1.06µm, v=5.96km/s, n_co=1.46 and the bulk gain g ₀=5×10^-11m/W for their calculations. The above equations are applied to step index fibers where all ray directions are assumed to be equally probable up to the critical angle. This assumption is based on the uniformity of the index profile which is not valid for GRIN fibers. They find that the bandwidth in Eq. (2) compares favorably with their experiment.

In our experiments we are interested in the SBS threshold in fibers. For this we assume the standard threshold equation

where g_i(f_S) is the inhomogeneous gain at the SBS return frequency f_S. However, Eq. (3) differs from previous work in that we replace the core area by the optimum Gaussian mode area of radius ω. The effective length remains as l_eff=(1-exp(-αl))/α, where α=0.046/km.

For our experiment we use: v=5960m/s, λ=1.5µm, n_co=1.4682, F ₀=2n_cov/λ, g ₀=2.5×10^-11m/W this gain is half of the bulk gain value given in reference (10). Since our wavelength is λ=1.5µm the value of Γ₀ is determined from the scaling Γ₀=38.4/λ ²MHz [10]. This gives 36MHz for λ=1.06µm and 17MHz for Erbium at λ=1.5µm.

We now consider the fiber mode structure. Because of the uncertainty in the launching conditions the exact fiber mode structure is unknown. In order to estimate the mode area we turn to the work of Marcuse [9]. He assumes the field distribution is Gaussian, composed of more than the lowest mode, from which he derives an optimum Gaussian mode radius ω for both step index and GRIN fibers. He finds for a step index fiber of core radius a, the optimum beam radius ω is given by

and for GRIN a fiber the optimum beam radius is given by

Note that these waists correspond to the e ^-2 value of the measured intensity. In these equations the V-number is V=ka(NA)_SI, k=2π/λ, where (NA)_SI is the step index numerical aperture. For the GRIN fiber the numerical aperture is $\left(NA\left(r\right)\right)={\left(NA\right)}_{\mathit{SI}}\sqrt{1-r^2⁄a^2}$.

We consider two cases: single mode step index fiber and multimode GRIN fiber. For the single mode step index fiber the core radius is a=4.0µm, (NA)SI=0.14, thus V=2.34 and Eq. (4) yields a mode radius of r=4.5µm. The fiber length is l=1km. For the GRIN fiber $\left(NA\left(r\right)\right)=.2\sqrt{1-r^2⁄a^2}$, and the V-number is V≈20. Thus, Eq. (5) gives a mode radius ω=8µm for a core radius α=25µm. The length is l=4.4km. Next we consider the radial dependence of the GRIN numerical aperture and we show how it affects Eq. (1). We first note that the distribution of Stokes frequencies, or all ray directions, are not equally distributed within θ_c as was previously assumed for the step index fiber [4]. Rather than include the radial distribution in the detailed convolution, see Ref. (4), we replace the NA in Eq. (1) with the spatial average $\overline{\mathit{NA}}={\left(\mathit{NA}\right)}_{\mathit{SI}}⁄2$. This approach is approximate, however our calculations and experiments agree, see Table 1. Note that $\left(NA\left(r\right)\right)={\left(NA\right)}_{\mathit{SI}}\sqrt{1-r^2⁄a^2}$ which leads to the spatial average $\overline{\mathit{NA}}=.6{\left(\mathit{NA}\right)}_{\mathit{SI}}$, which justifies our guess of 1/2 in the reduction of the GRIN fiber numerical aperture. The decrease in the numerical aperture leads to a reduction in the threshold since the inhomogeneous broadening narrows, see Eq. (2). Evaluation of Eq. (1) gives the gain g_i(F ₀)=5.64×10^-12m/W for the single mode fiber, and a gain of g_i(F ₀)=9.93×10^-12m/W for the GRIN fiber; these are reduced from g ₀=2.5×10^-11m/W. Also, Eqs. (4),(5) for the optimum Gaussian radii leads to the same conclusion. For large V, and we have V=20, the step index fiber has a waist determined by ω/α=.65 while for the GRIN fiber $\omega ⁄a\approx \sqrt{2⁄V}=.3$.3 Thus, again, the threshold is reduced for the GRIN. In fact, if both of these effects are included the step index fiber has a threshold of about 15 times larger than a GRIN fiber of the same core radius and the same (NA)_SI.

Table 1 summarizes the experimental and theoretical results. The calculated beam mode sizes ω are obtained using Eqs. (4),(5). The corrected threshold uses Eq. (3). However, the uncorrected threshold retains the same mode size, but uses only the gain g ₀ to illustrate the effect of dispersion. If, on the other hand, the core area is used instead of the mode area, along with g ₀, the threshold increases to 400mW.

We end this section by noting that not only does Eq. (3) fit our experiment satisfactorily but it also agrees with focused pump experiments. Cotter’s [10] experiment measures a threshold of 5.6mW for a 13.6km single mode fiber at a wavelength of λ=1.32µm. Our simulation gives within 15% agreement with a threshold of 6.5mW. The experiments of Harrison et al. [5] offer another comparison. They measure a threshold of 70mW and we simulate a threshold of 77mW for a 4.23km multimode fiber at wavelength of λ=1.06µm. This brings us to an experiment by Chen, et al. [7]. Here they measure a threshold of 14mW and we calculate 16mW for a 7.2km single mode fiber cw pumped at λ=1.31µm. All of these experiments use a focused pump.

Our theoretical approach to SBS threshold in fibers has been to modify the bulk equations. We made two changes which brought the fiber theory in agreement with experiments. For the GRIN fiber we modified the radial dependent numerical aperture by introducing an average numerical aperture in the inhomogeneous gain equation. The second alteration, applicable to both single mode and multimode fibers, was to replace the core radius by the optimum Gaussian radius. For the step index fiber the experiment and theory agree to within a few percent and for the GRIN fiber the agreement is within 10%.