A 4.3 dB stimulated Brillouin scattering (SBS) threshold suppression is measured in a passive Al-doped acoustically anti-guiding single mode optical fiber relative to that of a Ge-doped acoustically guiding single mode optical fiber. Stimulated scattering is generated by the electrostrictive acoustic wave generated in the fiber core. This acoustic excitation has a decay length Ld related to the sound absorption decay length Labs and the acoustic waveguide decay length Lwg by: Ld −1= Labs −1+ Lwg −1. The acoustic waveguide decay length Lwg is associated with the diffraction, refraction and reflection of the acoustic wave in the elastically inhomogeneous optical fiber cores. The SBS gain is proportional to the net acoustic decay length Ld and the relative SBS suppression is proportional to the ratio of the net decay lengths of the Al and Ge doped cores (LAl/ LGe). An acoustic beam propagation model is used to calculate the evolution of the complex acoustic excitations in the optical cores and determine the acoustic wave decay lengths Lwg. Model predictions for the relative SBS suppression for the two fibers are in good agreement with experimental values obtained from Stokes power and optical heterodyne linewidth measurements.
© 2009 OSA
Stimulated Brillouin scattering (SBS) can be the primary power limiting mechanism in many fiber optic applications that require narrow linewidth (<100 MHz) laser light. For example, single mode optical fiber telecommunication data links [1,2], high power rare-earth doped optical fiber amplifiers [3,4] and Raman amplifiers  can all be power limited by SBS. SBS is initiated by the spontaneous Brillouin scattering of the forward propagating laser light by density fluctuations associated with the thermally driven acoustic phonons. A fraction of the scattered light is captured by the fiber mode. The counter propagating electric fields exhibit an optical frequency difference equal to that of the acoustic phonon and create a forward propagating pressure wave by means of the electrostrictive effect. This pressure wave modulates the core refractive index and constitutes a traveling Bragg grating that leads to further light scattering, thereby generating stimulated light scattering.
Several approaches have been demonstrated for increasing the SBS threshold in optical fibers. One approach has been to vary the center frequency of the Brillouin gain along the fiber length by varying the dopant concentrations , the strain  or the temperature [8,9]. Alternatively, varying the acoustic properties of the optical fiber in the transverse direction has demonstrated significant SBS suppression levels [10–12]. Ref. 12 introduced an acoustic beam propagation method (BPM) model to design a ramp-like anti-guiding acoustic index profile in a large mode area [LMA] Yb-doped gain fiber that provides ~11 dB of SBS suppression. A recent elaboration of the acoustic BPM  and a finite element analysis  has shown this ramp structure to be nearly optimal. The purpose of this paper is to measure the SBS suppression in acoustically guiding and anti-guiding single mode fibers [SMFs] and to interpret the measured SBS suppression levels with the acoustic BPM model. The SBS threshold is usually defined in terms of a Brillouin pump power PP that generates a Stokes power PS corresponding to a chosen SBS reflectivity RSBS=PS/PP. The pump power at which the SBS reflectivity reaches the designated reflectivity level is the threshold power Pth. This threshold power is dependent upon various fiber characteristics; material properties, optical modal properties, fiber length and others. The SBS process is seeded by the spontaneous Brillouin scattering, however the growth of the Stokes power is dominated by the SBS gain coefficient gB which appears in the exponential function. Therefore, the relative SBS suppression of the anti-guiding Al-doped fiber relative to that of the guiding Ge-doped fiber is investigated by measuring the relative strengths of the gain coefficients gB and normalizing for the other fiber properties. In this manner, a meaningful comparison between the intrinsic SBS thresholds of two fibers can be made.
The report is structured in the following manner: Section 2 is the experimental section that presents the measurements of SBS gain coefficients. The SBS gain coefficient ratio, and therefore the SBS threshold suppression, is measured in three ways. In the first case, it is extracted from very low (<0.01) SBS reflectivity measurements with a high resolution optical spectrum analyzer (OSA). In the second case, it is measured from the backscattered Stokes power with a power meter (PM) at 1% and 10% SBS reflectivities and in the third case it is extracted from optical heterodyne linewidth measurements of the Stokes light with an electrical spectrum analyzer (ESA). Section 3 discusses the SBS process in the two fibers and models the relative SBS gain with an acoustic adaptation of the BPM used for optical waveguides. The BPM is used to calculate a value for the waveguide decay length Lwg of the Al-doped fiber and estimate the SBS suppression relative to that of the Ge-doped fiber. The model prediction of −4.7 dB SBS suppression is in good agreement with the measured suppression levels. These experimental and theoretical results present a good basis for the further design of SBS suppressing optical fibers. Section 4 contains a summary of the results presented in this report.
The relative SBS suppression of the Al-doped anti-guiding single mode fiber (SMF) to that of the Ge-doped guiding SMF is quantified by the ratio of the SBS gain coefficients: gB[Al]/gB[Ge]  whereEq. (1), n is core refractive index, p12 is the Pockel’s coefficient, c is the speed of light in vacuum, Vc is the sound speed in the core and ρ is the material density. The spectral with Δν, full-width at half-maximum (FWHM), is related to the lifetime τ of the electrostrictively generated acoustic disturbance by τ=1/πΔν and is also related to the acoustic decay length by Ld=Vc/πΔν. Here we distinguish between bulk electrostrictive acoustic waves that grow the SBS and the acoustic phonons, driven by thermodynamic fluctuations, which generate the spontaneous Brillouin scattering that seeds the SBS process.
The experimental arrangement is shown in Fig. 1 . The seed laser source is a distributed feedback fiber laser that provides approximately 15 mW of single frequency (linewidth ~25 kHz) light at 1083 nm to a double clad Yb-doped fiber amplifier. The amplifier output is controlled by adjusting the power of the 915 nm pump light. The amplified radiation passes through an isolator and a 1% fused coupler to the fibers-under-test (FsUT). The tap monitors both the backscattered and injected light. Two 500 m spools of SMF are investigated. The first fiber has a Ge doped core and is acoustically guiding, i.e. the optical core has an acoustic index N=V0/Vc greater than that of the optical cladding where V0 is the sound speed in the cladding. Further details and measurement results concerning the two optical fibers are presented below in Table 1 at the end of this section. The fused coupler monitors the injected Brillouin pump light and the backscattered Stokes light. The Stokes power is investigated with the OSA and the PM, and the optical heterodyne linewidth measurements are made with the ESA. The FsUT are terminated with an angle cleave to minimize Fresnel reflections for the PM and OSA measurements and are terminated with a flat cleave to provide an optical local oscillator for the heterodyne measurements.
OSA spectra are shown for the Al and Ge-doped fibers are shown in Fig. 2(a) and Fig. 2(b), respectively. At low powers, the Brillouin triplet is evident showing the elastic Rayleigh scattering peak and the two inelastic Stokes and anti-Stokes peak at the shift wavelength of ~60 pm which corresponds to a frequency shift of ~15 GHz. As the Brillouin pump power is increased the Stokes component to the spectrum increases with the onset of stimulated Brillouin scattering. The SBS reflectivity at very low reflectivities (RSBS<<1) is given by [16,17]:18], Leff is the effective interaction length ,15]:Eqs. (3) and (4), α is the optical absorption in m−1, E(r) is the optical fiber modal electric field distribution, r is the radial coordinate and <…> is a cross-sectional spatial average. The optical fibers are low-birefringent single mode fibers and are wound on identical 6 inch diameter spools. The polarization efficiency factors ηP is estimated to be 2/3  and is assumed to be the same for the two fibers. Therefore, they do not appear in the relative gain coefficient measurements.
Figure 3 is a plot of the SBS reflectivity in dB as a function of the Brillouin pump power for the guiding and anti-guiding optical fibers. The Stokes powers are extracted from the OSA spectra shown in Figs. 2a and 2b. The solid line is a least-squares fit of Eq. (2) for the SBS reflectivity RSBS to the Brillouin pump powers. The two fitting parameters, ηβS and CB are shown in the figure.The relative SBS gain coefficients, and therefore the SBS suppression, is given by:Table 1 yields an SBS gain suppression of −4.0 dB.
Figure 4 shows a plot of the backscattered Stokes power, obtained from the PM measurements, as a function of the Brillouin pump power at higher SBS reflectivities. The dotted lines correspond to backscatter reflectivities of 1% and 10%. These reflectivities are used to define the SBS threshold powers Pth. For example, 1% reflectivities are achieved in the Ge and Al doped fibers at Brillouin pump powers of 60 mW and 140 mW, respectively.The SBS threshold powers are given by :19]. A complete description of the power evolution in the optical fiber, including pump depletion, can be described by the coupled rate equations [15,17] for the Stokes power PS and the Brillouin pump power PP. A numerical solution of these coupled differential equation yields the relation between κ and the SBS reflectivity RSBS and is shown in Fig. 5 for the both the Ge doped and Al doped fibers.The numerical values for the captured spontaneous Brillouin scattering coefficient ηβS and the SBS gain efficiency CB are taken from Fig. 3. It is found that the values for κ are nearly the same for the two fibers and approach 21 for RSBS=1. The SBS gain suppression, as determined by Eqs. (6) and (7), is given by:Table 1 yields an SBS gain suppression of −4.3 and −4.4 dB for 1% and 10% SBS reflectivities, respectively. The curved dashed lines appearing in Fig. 3 are a numerical computation of the Stokes powers obtained from the coupled rate equations  for the Stokes power PS and the Brillouin pump power PP that includes pump depletion. The agreement of the numerical results is excellent for the Ge-doped guiding fiber while there is some discrepancy noted for the Al-doped anti-guiding fiber. Good agreement between the measured result and the numerical calculation can be obtained for a gain efficiency of 0.18 (m-W)−1, so these results show a 10% discrepancy.22] for a Gaussian optical mode. Hence, the back-scattered Stokes power captured by the fiber per unit length is given by: ηβS=RΩ/2. This relation yields values for ηβS of 5.6x10−9 m−1 and 6.7x10−9 m−1 for the Ge and Al-doped fibers, respectively. The order of magnitude agreement with the measured values presented in Fig. 3 is reasonable given that the calculated values pertain to plane waves and do not include the transverse optical and acoustic mode structure.
Figure 6 shows the heterodyne spectra obtained with the ESA of the FsUT. The Ge and Al doped fibers exhibit Brillouin shift frequencies of 15.525 and 16.329 GHz and SBS gain bandwidths ΔνG (FWHM) of 16 and 54 MHz at 56 and 90 mW of Brillouin pump power, respectively. Note the measurable broadening of the SBS gain in the acoustically anti-guiding fiber. The SBS gain bandwidth ΔνG is related to the spectral width Δν of the acoustic disturbance by :Eq. (10) and the measured SBS gains G and SBS gain bandwidths ΔνG. They are found to be 64 and 185 MHz for the Ge and Al-doped fibers, respectively.The SBS suppression may be calculated from the heterodyne measurements and the relation:
3. SBS mechanism, acoustic beam propagation modeling and data analysis
Stimulated Brillouin scattering has been described in monographs by Fabelinskii  and Boyd . A representation of the SBS process in a single mode fiber is presented in Fig. 7 . Light guided in the forward direction with an electric field E1, optical frequency ω1 and propagation constant β1 is scattered by acoustic phonons exhibiting an acoustic frequency Ωm and wavevector Qm where the index ‘m’ labels the relevant acoustic eigenmode or phonon. For SBS in a SMF we are primarily interested in acoustic phonons traveling along the fiber axis in the direction of the forward propagating light and spontaneous Brillouin scattering events that generate Stokes light propagating in the backward direction with electric field E2, optical frequency ω2 and propagation constant β2. The optical propagation constants and optical frequencies are related by: β1,2=ω1,2neff/c where neff is the effective index of the optical mode. Conservation of energy and momentum require that the phonon acoustic frequency is equal to the difference in optical frequencies, i.e. Ωm=ω1−ω2 and that the acoustic wavevector is Qm=β1+ β2=4πn/λ where the effective index neff is approximately equal to the core index n. The total electric field may be written as:
There are three physical mechanisms that will govern the evolution of the acoustic disturbance. First, the acoustic wave will spread due to diffraction with an angle θ~0.66Λm/a (Airy disk) where ‘a’ is the core radius. For Λm~0.4 μm and a~3 μm we find that θ~0.1 radians so that the launched acoustic wave is collimated along the fiber axis. Secondly, the acoustic frequency of the sound wave is equal to that of the Brillouin shift (15-16 GHz) and exhibits a decay length of ~25 μm due to viscous damping . And third, the inhomogeneous elastic medium in the fiber core and nearby cladding, due to the differing dopants and dopant concentrations, causes the acoustic wave to refract as it propagates along the fiber. A complete description of the dynamic acoustic disturbance generated in the optical fiber by the volumetric source term Eq. (14) is given by the inhomogeneous acoustic wave equation for an unbounded inhomogeous elastic medium . Rather than solve this wave equation, the approach taken here is to launch an acoustic wave with a transverse density fluctuation profile f(r)2 into an inhomogeneous elastic medium characterized by N(r) and calculate the evolution of the complex density fluctuation profile as it propagates along the fiber axis with a beam propagation code . Here we take advantage of the fact that the optical waves and the acoustic waves can both be described by a scalar wave equation. The acoustic index profiles for the Ge and Al doped fibers are shown in Fig. 8 . The empirical relations used are:Equation (15) is taken from Ref . for the Ge doped fiber. The dependence of Vc upon Δn for the Al doped fiber was obtained by measuring the Brillouin shift frequencies νΒ=2nVc/λ in three Al doped fibers with differing Al concentrations and measured index profiles. An acoustic index N=1 corresponds to a cladding sound speed of 5944 m/s . The inset shows the intensity distributions of the fiber modes. Note that the intensity distributions extend into regions of the fiber core that experience significant gradients in the acoustic index.
Figure 9 shows the evolution of the complex density fluctuation amplitude along the fiber length for three cases: (i) an elastically homogeneous fiber, (ii) the guiding Ge doped core fiber and (iii) the anti-guiding Al doped fiber at longitudinal distances of 0, 10, 20 and of 30 μm. The first case for an elastically homogeneous material shows some small spreading the acoustic waveform due to diffraction. In the acoustic guiding fiber, the acoustic amplitude remains within the fiber core exhibiting some refraction and reflection at the core-cladding interface. However, the energy again remains within the fiber core. In the anti- guiding case, the acoustic wave migrates out of the core region into the nearby cladding at propagation distances comparable to and less than the acoustic decay length. The amount of energy that leaves the core is significant since is scales with the radius squared.
Figure 10 shows the radial phase of the complex acoustic disturbance over a radius of 5 μm after propagating 20 μm along the fiber axis. In the elastically homogeneous case, some curvature of the phase front of ~2 radians is evident, indicating the beam spreading due to diffraction. The Ge doped fiber shows less spreading of ~1 radian due to the acoustic guiding properties of this fiber. Here beam spreading due to diffraction is compensated by the acoustic waveguide. However, the Al doped fiber shows a large phase curvature of ~10 radians representative of the strong acoustic anti-guiding properties of this fiber.
The electrostrictively generated acoustic wave Δρe will be absorbed as it propagates along the fiber length. The acoustic amplitude will decay Δρe ~exp(-z/2Labs) and the acoustic intensity will decay as |Δρe|2~exp(-z/Labs). The decay of the acoustic amplitude from its initial value is generalized in an ad hoc fashion by calculating the cross-sectional correlation function or normalized overlap integral of the complex density fluctuations at location z compared with those appearing at z=0:
Figure 11 shows plots of | γ(z)|2 as a function of propagation distance z for several situations. The solid black line shows the decay for viscous damping with a phonon linewidth of 75 MHz. This phonon linewidth is consistent with the low gain bandwidth extracted from the heterodyne measurements for the guiding Ge-doped fiber. The e−1 point corresponds to a decay length of 25.1 μm. The dotted lines show |γ(z)|2 for elastically homogeneous bulk media with the same dopant concentrations as the guiding and anti-guiding optical fibers.These lines show the effects of diffraction and exhibit e−1 levels at ~50 μm. This length scale is significantly greater than that of the viscous damping; therefore, acoustic diffraction is not expected to play a large role in determining the acoustic decay length. The solid blue line is |γ(z)|2 for the acoustic guiding structure. It exhibits some fluctuations due to the structure of the acoustic guiding refractive index N. However, it remains highly correlated over the range of the calculation and is therefore interpreted to have a correlation length that significantly exceeds that due to the viscous damping. Therefore the decay length of the acoustic guiding fiber is taken to be determined by the viscous damping. The red line shows |γ(z)|2 for the acoustic anti-guiding fiber. It shows a more rapid and pronounced decay due to the acoustic waveguide effects and the dispersal of the acoustic energy away from the central core region. The total decay length Ld therefore has contributions from the viscous damping and waveguide effects and may be written as :Eq. (1) with the spectral widths Δν expressed by the BPM-determined decay lengths Ld:Table 2 . Other physical parameters used in these calculations are shown in Table 3.
The relative SBS gain coefficients of a Ge doped step index acoustic guiding SBS an Al doped acoustic anti-guiding fiber have been measured. Stokes power and linewidth measurements demonstrate a relative SBS suppression level of ~ 4.0-4.8 dB. The SBS suppression mechanism is modeled with an acoustic adaptation of the BPM for optical waveguides. SBS suppression in the anti-guiding fiber is attributed to the decorelation of the electrostrictively generated acoustic intensity by the inhomogeous elastic medium. This decorelation is quantified by an acoustic waveguide decay length Lwg which provides an additional decay mechanism to the viscous damping decay length Labs. Agreement between the BPM model and the experimental results indicate that the BPM may be useful in the design of other SBS suppressing optical fibers.
The author thanks Man Yan for providing the Al doped fibers and helpful discussions with David DiGiovanni and Ben Ward. Andrew Yablon is thanked for many in-depth discussions and his assistance with the BPM code.
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