Experimental and numerical study on stimulated Brillouin scattering in a graded-index multimode fiber Aldo Minardo,1,* Romeo Bernini,2 and Luigi Zeni1,2 1
Department of Industrial and Information Engineering, Seconda Università di Napoli, Via Roma 29, 81031 Aversa, Italy 2 Istituto per il Rilevamento Elettromagnetico dell’Ambiente Consiglio Nazionale delle Ricerche, Via Diocleziano, 328–80124, Naples, Italy * [email protected]
Abstract: Numerical and experimental results on stimulated Brillouin scattering (SBS) in a graded-index multimode silica fiber are reported. The Brillouin Gain Spectrum (BGS) is shown to strongly depend on the pump and probe modal content. By use of a numerical model, the BGS at varying launching conditions of both pump and probe beams is computed. Numerical results show that intramodal and intermodal SBS contribute to the overall BGS. Experiments confirm the numerical predictions. ©2014 Optical Society of America OCIS codes: (290.5900) Scattering, stimulated Brillouin; (060.4370) Nonlinear optics, fibers.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
M. Nikles, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997). Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing Brillouin gain spectrum in single-mode fibers,” J. Lightwave Technol. 22(2), 631–639 (2004). A. H. McCurdy, “Modeling of stimulated Brillouin scattering in optical fibers with arbitrary radial index profile,” J. Lightwave Technol. 23(11), 3509–3516 (2005). A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005). W. W. Ke, X. J. Wang, and X. Tang, “Stimulated Brillouin scattering model in multi-mode fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 20(5), 901610 (2014) B. Ward and M. Mermelstein, “Modeling of inter-modal Brillouin gain in higher-order-mode fibers,” Opt. Express 18(3), 1952–1958 (2010). A. Li, Q. Hu, and W. Shieh, “Characterization of stimulated Brillouin scattering in a circular-core two-mode fiber using optical time-domain analysis,” Opt. Express 21(26), 31894–31906 (2013). K. Y. Song and Y. H. Kim, “Characterization of stimulated Brillouin scattering in a few-mode fiber,” Opt. Lett. 38(22), 4841–4844 (2013). K. Y. Song, Y. H. Kim, and B. Y. Kim, “Intermodal stimulated Brillouin scattering in two-mode fibers,” Opt. Lett. 38(11), 1805–1807 (2013). A. Mocofanescu, L. Wang, R. Jain, K. D. Shaw, P. R. Peterson, and A. Gavrielides, “Experimental and theoretical investigations on stimulated Brillouin scattering (SBS) in multimode fibers at 1550 nm wavelength,” Proc. SPIE 5581, 654–661 (2004). A. Mocofanescu, L. Wang, R. Jain, K. D. Shaw, A. Gavrielides, P. R. Peterson, and M. Sharma, “SBS threshold for single mode and multimode GRIN fibers in an all fiber configuration,” Opt. Express 13(6), 2019–2024 (2005). A. Fotiadi and E. A. Kuzin, “Stimulated Brillouin scattering associated with hypersound diffraction in multimode optical fibers,” presented at Quantum Electronics and Laser Science Conference, Anaheim, Calif, June 2–7 1006, paper QFC4. G. A. Decker, “Method of fabricating an optical attenuator by fusion splicing of optical fibers, US Patent no. 4557556, 1985. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
1. Introduction Stimulated Brillouin scattering (SBS) has been widely studied in single-mode silica fibers [1– 3]. In pump-probe experiments, the amplification of the optical wave at lower frequency (the Stokes beam) depends on the spectral shift from the pump beam, according to the so-called
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Brillouin Gain Spectrum (BGS). The peak position of the BGS, known as Brillouin frequency shift (BFS), depends on the effective refractive index and effective acoustic velocity in the fiber. In single-mode fibers, if neglecting fiber birefringence, there is only one effective optical refractive index, while each acoustic mode supported by the fiber gives rise to a different BFS, in virtue of the different effective acoustic velocity. Therefore, the overall BGS will be the superposition of various Lorentzian spectral shapes, each one peaked at a different BFS and weighted by the so-called acousto-optic effective area [4], which is evaluated from the spatial overlap between acoustic mode and optical mode field distributions. In multimode fibers, the picture becomes more complicated, as many optical and acoustic waves are generally involved. Actually, each pair of counter-propagating optical modes may interact through an acoustic mode travelling along the same direction of the pump wave, contributing to the overall BGS. Each contribution is weighted by a factor depending on the overlap integral between the three modes [5]. The SBS interaction between two identical optical modes is named intramodal SBS, while the interaction between different optical modes is known as intermodal SBS [6]. Recently, the SBS in few-mode fibers (FMFs) has been studied, both numerically [5,6] and experimentally [7–9]. The results show that the BGS depends on the pair of optical modes involved. Also, it has been reported that intermodal SBS gain is of the same order of magnitude [9], or even larger [6] than intramodal SBS gain. The study of SBS in multimode fibers (MMF) has been mostly focused on the evaluation of the SBS threshold [10–12]. It was shown that good agreement between numerical and experimental thresholds can be found, if using for threshold calculation an effective mode radius reff < rc, where rc is the radius of the fiber core. It was also found that the gain coefficient for lower-order optical modes strongly exceeds the gain of the higher-order modes [13]. In this work, we characterize the SBS in a graded-index multimode (GI-MMF) silica fiber, having a core diameter of 50 μm and a numerical aperture (NA) of 0.2. At the optical wavelength of 1.55 μm, the fiber supports several tens of optical modes, and several hundreds of acoustic modes. As, in general, each pair of optical modes interacts through a different set of acoustic modes and with a different efficiency, the overall BGS will depend on the pump and probe optical modal content. A numerical procedure based on finite-element analysis (FEA) has been applied in order to predict the BGS for any distribution of the pump and probe optical field. Numerical results are also compared to BGS measurements. In order to acquire the BGS of the multimode fiber for different pump and probe fields, the measurements have been performed upon offset-splicing the GI-MMF at both ends with two short sections of singlemode fibers (SMF). The experiments confirm that the BGS of the multimode fiber strongly depends on the modal content of the two interacting beams. In particular, it is shown that, increasing the energy associated to high-order optical modes (HOM), the BGS broadens inhomogenously, mainly as a consequence of the larger number of acoustic modes involved. In the next section, we present the numerical method employed to estimate the BGS in a multimode fiber. In section 3, the numerical spectra evaluated at different launch conditions are compared to experimental results. Conclusions will follow. 2. Numerical method Firstly, the optical modes of the GI-MMF were computed by use of a finite-element analysis (FEA). A parabolic refractive index profile was chosen for simulation, with the maximum refractive index nco = 1.4713 at the core axis, and a refractive index ncl = 1.458 across the cladding. A number of 55 guided optical modes were found by use of the commercial solver COMSOL Multiphysics, including degenerate modes. For each mode, we extracted the profile associated to the main component of the electric field, as well as the corresponding effective refractive index. As a next step, another FEA was carried out to calculate the displacement field and the BFS associated to each acoustic mode supported by the same fiber. To this aim, the 2-D
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scalar mechanical equation was solved by assuming a longitudinal acoustic velocity distribution as described by the following relation [4]: Vl ( r , θ ) = Vl 0 (1 − 7.8 Δn ( r , θ ) ncl ) (1) where Vl0 = 5944 m/s is the longitudinal acoustic velocity into the cladding, and Δn = n(r,θ)ncl. By assuming for the acoustic wave a propagation constant equal to the sum of the propagation constants of the two interacting optical modes (i.e., the SBS resonance condition), the BFS associated to each acoustic mode is found. For the simulations, the BFS of the various acoustic modes were found by imposing the phase-matching condition between the acoustic mode and the propagation constant of the fundamental (LP01) optical mode. In order to compute the BFS corresponding to a specific acoustic mode and a generic pair of optical modes, we applied the following formula [9]: BFSi , j , k = BFSi ,1,1 ( neff , j + neff , k ) ( 2neff ,1 ) (2)
where i is the index referring to the acoustic mode, j and k are optical modes indexes, and neff,i is the effective refractive index of the i-th optical mode. Equation (2) tells us that, for a specific acoustic mode, higher-order optical modes lead to a smaller BFS. As the number of guided acoustic modes in the analyzed fiber is very large (several hundred), it was not possible to take into account all of them due to computational limitations, rather only the first 151 acoustic modes were solved for. As it will be seen later, the computed set of acoustic modes is sufficient to describe accurately the BGS in our experimental conditions. The computed optical and acoustic modes were then processed so as to construct a three-dimensional array of Brillouin gain coefficients, in which each element refers to a specific pair of optical modes and a specific acoustic mode. In particular, for each triad of modes the Brillouin gain was computed by using the following formula [5]: Gi , j , k = g i , j , k f i * ( r , θ ) g *j ( r ,θ ) hk ( r , θ ) drdθ
2
(3)
where gi , j , k = ( c μ0γ e2 ) ( 2λρ 0ε 0 n 2 BFSi , j , k Γ ) , c and μ0 are the light velocity and magnetic
permeability in vacuum, γe is the electrostriction constant, λ is the signal wavelength, ρ0 is the material density, ε0 is the vacuum permittivity, n the refractive index, Γ is the acoustic wave damping factor, f is the acoustic mode, g and h are the two optical modes, and the domain of integration is the fiber cross-section. Before computation of integrals, all modes were normalized so as to have unitary area. In case of intramodal SBS, Eq. (3) reduces to the inverse of the acousto-optic effective area [4]. Also, as in our computations we consider purely real optical and acoustic profile basis functions, the coefficients expressed by Eq. (3) are such that Gi,j,k = Gi,k,j. At the end of this process, a 151 × 55 × 55 Brillouin gain coefficient array was computed for the investigated fiber. For any chosen pair of optical modes, the related BGS is computed as the sum of the spectral-density contributions from each mode [2]: 1 N BGS j , k (υ ) = i =ac1 Gi , j , k (4) 2 υ − BFSi , j , k 1+ Δυ B 2 where ν is the pump-probe spectral shift, Nac is the number of acoustic modes and ΔνB is the bandwidth of each Lorentzian spectral profile. In our computations, we assumed ΔνB = 35 MHz. In virtue of the cylindrical symmetry of the analyzed fiber, the interaction between a generic couple of optical modes, LPxa - LPyb, occurs through two acoustic family modes: A(x ± y)n with n = 1, 2, 3, etc. In other words, the acoustic modes must have an azimuthal mode number equal to the sum (or the difference, if not negative) between the azimuthal mode numbers of the two optical waves. As a first example, let us consider the case in which both pump and probe fields coincide with the fundamental mode (LP01) of the GI-MMF. In this case, SBS will only occur through
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axially symmetric acoustic modes. Within the computed set, we found 9 radial modes (A01 A09). We report in Fig. 1(a) the normalized Brillouin gain coefficients computed by use of Eq. (3). Apparently, the largest contributions come from the first three acoustic radial modes (A01-A03), while for the last six modes the gain is below 1% of maximum gain. 1
1
(a)
0.8
0.8
0.7
0.7
0.6 0.5 0.4 0.3
0.5
0.1
A03
A04
A05
A06
A07
A08
0 10.3
A09
A03
0.3
0.1 A02
A02
0.4
0.2
A01
A01
0.6
0.2
0
(b)
0.9
Brillouin gain [n.u.]
Brillouin gain coefficient [n.u.]
0.9
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
Pump-probe frequency shift [GHz]
Acoustic mode
Fig. 1. Brillouin gain coefficients (a) and Brillouin gain spectrum (b) computed for the LP01LP01 optical mode pair.
The corresponding BGS is shown in Fig. 1(b) (solid blue curve). The same figure also highlights the individual contributions from A01, A02 and A03 (dashed black curves). It is seen that scattering from high-order acoustic modes is responsible for some asymmetry of the BGS, which is broader on the right side. Note that the spectral distance between adjacent resonances (~27 MHz) is smaller than the bandwidth of each resonance, so the various contributions overlap each other giving rise to full width at half maximum (FWHM) of the BGS equal to 45 MHz. Let us now analyze the interaction between the fundamental optical mode, LP01, and the second-order optical mode LP11. In this case, the interaction is mediated by acoustic modes with unitary azimuthal index (A11 – A18 in our mode set). As the LP11 is anti-symmetric, we must consider both angular orientations in order to derive an azimuth-independent Brillouin gain coefficient. Practically, we compute the sum of the Brillouin gain coefficients associated to the LP01-LP11a interaction and the ones associated to the LP01-LP11b interaction, with LP11a and LP11b denoting the two possible angular orientations of the degenerate mode LP11. The Brillouin gain coefficients are shown in Fig. 2(a). Note that, as the gain rapidly decreases when increasing the radial mode number, the contribution from higher-order acoustic modes (A19, A1-10, etc.) can be safely considered as negligible. 0.8
1
(a)
(b)
0.9 A11
0.8
0.6
Brillouin gain [n.u.]
Brillouin gain coefficient [n.u.]
0.7
0.5 0.4 0.3 0.2
0.7 A12
0.6 0.5 0.4
A13 0.3 0.2
0.1 0
0.1
A11
A12
A13
A14
A15
Acoustic mode
A16
A17
A18
0 10.3
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
Pump-probe frequency shift [GHz]
Fig. 2. Brillouin gain coefficients (a) and Brillouin gain spectrum (b) computed for the LP01LP11 optical mode pair.
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The corresponding BGS is shown in Fig. 2(b), together with the contributions from the three acoustic modes A11, A12, and A13. We note that, while the spectral distance between adjacent acoustic resonances is the same as the previous case (~27 MHz), the resulting BGS is now broader (58 MHz), due to the larger gain associated to high-order acoustic modes. As a final example, we analyze the LP11-LP11 interaction. Still, we calculate the gain by summing over all the possible combinations of optical mode angular orientation. Within our mode set, we found 17 acoustic modes with azimuthal index equal to 0 (A01 - A09), or 2 (A21 – A28). The gain coefficients, reported in Fig. 3(a), suggest that the contribution from the acoustic modes not included in the computed set (i.e., with higher radial index) can be safely considered as negligible. The BGS is shown in Fig. 3(b), together with the individual contributions from the most efficiently excited acoustic modes. The FWHM of the BGS is now 81 MHz. The larger bandwidth should be attributed to the increasing number of acoustic modes involved. Actually, in this case two acoustic mode families come into play: the one with azimuthal index equal to 0 and the one with azimuthal index equal to 2. Both families contribute with similar gain to the overall spectral density (see Fig. 3(a)). 1
0.7
(a)
0.6
0.5
0.4
0.3
(b)
0.8
Brillouin gain [n.u.]
Brillouin gain coefficient [n.u.]
0.9
0.2
0.7
A21
0.6 0.5 A02
0.4
A22 A03
0.3 0.2
0.1
0
0.1 0 10.3
A01 A02 A03 A04 A05 A06 A07 A08 A09 A21 A22 A23 A24 A25 A26 A27 A28
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
Pump-probe frequency shift [GHz]
Acoustic mode
Fig. 3. Brillouin gain coefficients (a) and Brillouin gain spectrum (b) computed for the LP11LP11 optical mode pair.
As a final remark, we should observe that, in real cases, it is difficult to selectively launch individual modes in a multimode fiber. On the other hand, the model described can be equally employed to calculate the BGS resulting from a generic combination of pump and probe fields. Actually, provided that the modal content of pump and probe beams is known, the contributions associated to any pair of optical modes can be summed, properly weighting each contribution on the basis of the energy of each mode. More specifically, let us suppose that the amplitude of pump and probe fields within the MMF can be expressed as: E p ( r , θ ) = j =1 Aj f j ( r , θ )
(5a)
Es ( r , θ ) = k =1 Bk f k ( r ,θ )
(5b)
N
N
with N representing the number of MMF modes, f are the basis functions, A and B their amplitudes. In the model, we assume that no mode mixing occurs along the MMF, i.e. the modal content of pump and probe beams is uniform through the whole MMF length. Within the undepleted pump approximation, and neglecting fiber loss, the total BGS can be calculated as: BGS (ν ) = j =1 B j k =1 Ak × BGS j , k (υ ) N
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N
(6)
Received 4 Mar 2014; accepted 9 May 2014; published 11 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017480 | OPTICS EXPRESS 17484
where BGSi,j are the terms defined in Eq. (4). Therefore, when more than one mode is present in either the pump or probe field (or both), the BGS will be the result of both intramodal (j = k) and intermodal (j≠k) SBS between the modes involved. 3. Experimental results
In this section, we report a number of experimental results aimed to demonstrate the accuracy of the proposed model. Experiments were performed on a 50 m, graded-index multimode fiber with a core diameter of 50 μm and a nominal numerical aperture of 0.2. During the measurements, the fiber was left on its spool, and care was taken in order to avoid any perturbation which may alter the modal distribution of the optical fields along the fiber. The BGS measurements were taken by use of the setup schematically shown in Fig. 4. All fiber components, except from the fiber under test (FUT), were single-mode. A diode laser emitting a 1.55 μm was employed as the laser source. The light is split in two arms by means of a polarization maintaining beam splitter, with the lower arm acting as the pump branch, and the upper arm serving as the probe branch. On the upper arm, an intensity modulator, driven at a radiofrequency close to the BFS of the FUT, generates two sidebands by suppressed-carried double sideband (SC-DSB) modulation. The sideband at higher frequency is successively filtered out by a fiber Bragg grating (FBG) bandpass filter, while the one at lower frequency acts as the probe beam. In the lower branch, the pump beam is modulated in order to realize a synchronous lock-in detection. Both pump and probe beams are optically amplified before being launched into the FUT. The photodiode features a bandwidth of 125 MHz and a gain of 90 dB.
Fig. 4. Experimental setup for BOFDA measurements. DFB-LD, diode laser; FBG, fiber Bragg grating; EDFA, Er-doped fiber amplifier. EOM, electro-optic modulator; LIA, lock-in amplifier; PD: photodetector.
As previously discussed, the BGS of a multimode fiber depends on the pump and probe modal content. In order to achieve a stable excitation of pump and probe fields, the GI-MMF was fusion-spliced at both ends with two short sections of single mode fibers (SMF), which are connected to the setup by use of FC/APC connectors. For each test, the fusion splices were performed with a different value of intentional offset. The estimation of the BGS was performed by first calculating the coupling coefficients between the (eventually displaced) SMF mode and the GI-MMF modes, through 2-D overlap integrals. Assuming identical splices at both MMF ends, and denoting with Ci the coupling coefficient between the SMF mode and the i-th MMF mode, we compute the BGS as: BGS (ν ) = j =1 C j k =1 Ck × BGS j , k (υ ) N
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N
(7)
Received 4 Mar 2014; accepted 9 May 2014; published 11 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017480 | OPTICS EXPRESS 17485
where N is number of MMF modes and BGSi,j are the terms defined in Eq. (4). On the other hand, we must also consider that each probe mode passes again through the MMF-SMF splice before being detected. Therefore, we must further multiply the contribution from each probe mode with the corresponding coupling coefficient. The observed BGS will then be given by: BGSobserved (υ ) = j =1 C 2j k =1 Ck × BGS j , k (υ ) = j =1 k =1 BGS eff j , k (υ ) N
N
N
N
(8)
where BGS eff j , k denotes the effective contribution of each pump-probe modal pair, given by: ac 2 BFS eff j , k (υ ) = C j Ck i =1 Gi , j , k
1
N
υ − BFSi , j , k 1+ Δυ B 2
(9)
2
1 0.9
(a)
1
0.7
Brillouin gain [n.u.]
Coupling coefficient
0.8
0.6 0.5 0.4 0.3 0.2
Numerical BGS Experimental BGS
(b)
0.8
0.6
0.4
0.2
0.1 0
LP01
LP02
LP03
LP04
GI-MMF optical mode
LP05
0 10.38
10.4
10.42 10.44 10.46 10.48
10.5
10.52 10.54 10.56 10.58
Pump-probe frequency shift [GHz]
Fig. 5. (a) Coupling coefficients between SMF and GI-MMF computed in case of zero offset between the two fibers; (b) Numerical and experimental BGS of the GI-MMF in case of zero offset between the two fibers.
The first experiment was carried out after splicing the MMF to the SMF pigtails with no intentional offset. Due to axial symmetry of the excitation fields, only axially symmetric optical modes of the GI-MMF are excited in this particular case. By computing the overlap integrals between the SMF mode and the 5 radial modes supported by the GI-MMF, we achieve the coupling coefficients shown in Fig. 5(a). We see that most of optical energy (~90%) is coupled to the fundamental mode of the GI-MMF. As expressed by Eq. (8), the overall BGS is computed by summing the contributions associated to the various LP0x modes, including intermodal SBS between LP modes with different radial mode number. The result is shown in Fig. 5(b). The same figure also reports the experimental BGS, acquired with the setup shown in Fig. 4 and scanning the pump-probe frequency shift at a step of 4 MHz. Although not excellent, the agreement between numerical and experimental data is remarkably good, also considering the various model uncertainties (refractive index profile, excitation pattern, fiber inhomogeneities). The BFS of experimental and numerical BGS is ~10485 MHz. We note that the BGS is very similar to the numerical spectrum computed previously for the LP01-LP01 interaction (Fig. 2). This is easily explained, by considering that the optical energy of both pump and probe waves is mostly carried by the LP01 mode. Consequently, the SBS interaction in the present case mainly occurs through the first three radial acoustic modes (A01, A02 and A03). As a second test, the same GI-MMF was spliced again with the two SMF pigtails, this time by setting the fusion splicer so as to perform 3-dB attenuated splices. Fusion splicers realize attenuated splices by intentionally applying a lateral offset between the two fibers, with an offset typically ranging from 5 μm to 60 μm [13]. In order to estimate the lateral offset for a definite splice loss, we employed the formula proposed by Marcuse in Ref [14],
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according to which the power transmission coefficient through a fiber splice with an offset d is expressed as: d 2 T = exp − ω
(10)
where ω is the width parameter of the Gaussian beam propagating through the splice. As the fusion splicer applies the offset supposing to operate with single-mode fibers, we use ω = 10.4 μm in Eq. (10). This results in a lateral offset of 8.5 μm for a 3-dB splice loss. Therefore, we computed the coupling coefficients between the GI-MMF guided modes, and the SMF mode displaced by 8.5 μm from the fiber axis. We underline that, while the coefficients have been calculated for an SMF mode displaced along the x-direction, the final BGS is independent of the chosen direction, as optical and acoustic degenerate modes are always considered in both angular orientations. The computed coupling coefficients are represented in Fig. 6(a), where, for clearness, we have included only those greater than 0.01. As in this case the input field was not axially symmetric, also non symmetric GI-MMF optical modes were excited. The numerical BGS, computed by use of Eq. (8), is compared to the experimental spectrum in Fig. 6(b). We see that, the broadening of the BGS caused by offsetsplicing is well captured by the model. In particular, the full width at half maximum (FWHM) of the numerical BGS increases from 46 MHz to 88 MHz, while the corresponding experimental values are 47 MHz and 82 MHz, respectively. 0.4
(a)
0.35
1
Numerical BGS Experimental BGS
(b)
Brillouin gain [n.u.]
Coupling coefficient
0.3 0.25 0.2 0.15
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0.6
0.4
0.1
0.2
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LP01
LP02
LP21a LP21b LP31b LP12a LP41b
GI-MMF optical mode
0 10.3
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
Pump-probe frequency shift [GHz]
Fig. 6. (a) Coupling coefficients computed between SMF and GI-MMF in case of 8.5 μm lateral offset between the two fibers (only GI-MMF modes with a coupling coefficient larger than 0.01 were included in the graph); (b) Numerical and experimental BGS of the GI-MMF in case of 8.5 μm lateral offset between the two fibers.
The BGS broadening can be explained in terms of the larger number of acoustic modes involved in this case. By limiting our analysis to the two most energetic modes (LP11a and LP01), these two modes interact via acoustic modes having unitary azimuthal mode number as regards intermodal SBS, and via acoustic modes with azimuthal mode number equal to 0 and 2 as regards intramodal SBS. In general, we found 16 acoustic modes leading to a Brillouin gain above a threshold set to the 1% of maximum gain. It is also interesting to observe that the BFS of both experimental and numerical spectra is ~10500 MHz, i.e. about 15 MHz higher than the BFS of the previous test. Actually, the peak position of the BGS is the result of the superposition of several Lorentzian shapes, each centered at a different frequency. Therefore, the peak position of the BGS changes based on the relative weight of each contribution. In our conditions, the blue shift of the BGS is due to the fact that major contributions arise from higher-order acoustic modes, which resonate at higher frequencies.
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0.16
0.14
Coupling coefficient
0.12
0.1
0.08
0.06
0.04
0.02
0
LP31bLP12aLP13a LP03 LP02 LP32bLP11aLP22aLP31bLP21aLP41bLP51bLP21bLP42bLP61b LP04 LP23aLP23b LP01 LP71b
GI-MMF optical mode
Fig. 7. Coupling coefficients computed between SMF and GI-MMF in case of 15 μm lateral offset between the two fibers. Only GI-MMF modes with a coupling coefficient larger than 0.01 were included in the graph.
The final test was carried out after splicing the GI-MMF to the SMF pigtails, this time setting the fusion splicer so as to perform 10-dB attenuated splices. By using again the formula expressed by Eq. (10), the estimated offset is now 15 μm. For such an offset, the calculation of the coupling coefficients reveals that up to 20 optical MMF modes are excited with a coupling coefficient higher than 0.01 (see Fig. 7). The resulting BGS is compared to the experimental one in Fig. 8. Still, we observe an overall agreement between numerical and experimental data. In particular, the FWHM of the numerical BGS is 103 MHz, while being 109 MHz for the experimental BGS. Still, we attribute this broadening to the larger number of acoustic modes involved. Note also that the two BGS are now centered at ~10505 MHz.
Numerical BGS Experimental BGS
Brillouin gain [n.u.]
1
0.8
0.6
0.4
0.2
0 10.3
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
10.75
10.8
Pump-probe frequency shift [GHz]
Fig. 8. Numerical and experimental BGS of the GI-MMF in case of 15 μm lateral offset between the two fibers.
Determining the set of acoustic modes involved in this case is not a simple task, due to the large number of possible intermodal and intramodal interactions between the optical modes
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represented in Fig. 7. Potentially, all the 151 computed acoustic modes are involved, as well as higher-order acoustic modes not included in the computed set. In order to determine the most influent acoustic modes, we first computed the factor C 2j Ck Gi , j , k appearing in Eq. (9), for each triad of modes. Summing this factor over all the optical modes, we determined the overall contribution of each acoustic mode. The results are plotted in Fig. 9, in which all the computed modes with an azimuthal mode number ranging from 0 to 5 are included (j = 0 corresponds to the axially symmetric acoustic modes). We see that the interaction strength of acoustic modes decreases with azimuthal and/or radial mode number, with a larger slope associated to the azimuthal mode number. In other terms, the larger is the azimuthal index (i.e. the number of azimuthal oscillations), the slower the interaction strength decreases with the radial mode number (i.e. with the number of peaks along the r direction). On the other hand, we also observe that the computed set of modes includes all modes for which the interaction strength (normalized to the maximum strength) is as high as to ≈-11 dB. Therefore, we can safely conclude that the computed set is sufficient to describe accurately the overall SBS density spectrum in our experimental condition, as it is also confirmed by the agreement with experimental data (see Fig. 8).
Brillouin gain coefficient [dB]
-45
-50
j=0
-55
j=1 j=2 j=3
-60
j=4 j=5
-65
-70
1
2
3
4
5
6
7
8
9
Radial mode number Fig. 9. Brillouin gain coefficient associated to each acoustic mode of the GI-MMF, when the latter is excited at both ends by an SMF spliced with a 15 μm offset.
4. Conclusions
A numerical model based on the calculation of acousto-optic overlap integrals was applied to estimate the BGS in a graded-index multimode fiber. The results show that in these fibers the BGS shape is strongly dependent on the modal content of both pump and probe beams. Also, it was shown that increasing the number of optical modes involved, the SBS occurs with an increasing larger number of acoustic modes, leading to a broadening of the observed BGS. A number of experiments have been carried out, substantially confirming the numerical predictions. We believe that these results are potentially useful in understating the SBS behavior of multimode fibers, with implications in various fields such as sensing, spatialmode division multiplexing, and Brillouin fiber laser modeling.
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Received 4 Mar 2014; accepted 9 May 2014; published 11 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017480 | OPTICS EXPRESS 17489
Department of Industrial and Information Engineering, Seconda Università di Napoli, Via Roma 29, 81031 Aversa, Italy 2 Istituto per il Rilevamento Elettromagnetico dell’Ambiente Consiglio Nazionale delle Ricerche, Via Diocleziano, 328–80124, Naples, Italy * [email protected]
Abstract: Numerical and experimental results on stimulated Brillouin scattering (SBS) in a graded-index multimode silica fiber are reported. The Brillouin Gain Spectrum (BGS) is shown to strongly depend on the pump and probe modal content. By use of a numerical model, the BGS at varying launching conditions of both pump and probe beams is computed. Numerical results show that intramodal and intermodal SBS contribute to the overall BGS. Experiments confirm the numerical predictions. ©2014 Optical Society of America OCIS codes: (290.5900) Scattering, stimulated Brillouin; (060.4370) Nonlinear optics, fibers.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
M. Nikles, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997). Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing Brillouin gain spectrum in single-mode fibers,” J. Lightwave Technol. 22(2), 631–639 (2004). A. H. McCurdy, “Modeling of stimulated Brillouin scattering in optical fibers with arbitrary radial index profile,” J. Lightwave Technol. 23(11), 3509–3516 (2005). A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005). W. W. Ke, X. J. Wang, and X. Tang, “Stimulated Brillouin scattering model in multi-mode fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 20(5), 901610 (2014) B. Ward and M. Mermelstein, “Modeling of inter-modal Brillouin gain in higher-order-mode fibers,” Opt. Express 18(3), 1952–1958 (2010). A. Li, Q. Hu, and W. Shieh, “Characterization of stimulated Brillouin scattering in a circular-core two-mode fiber using optical time-domain analysis,” Opt. Express 21(26), 31894–31906 (2013). K. Y. Song and Y. H. Kim, “Characterization of stimulated Brillouin scattering in a few-mode fiber,” Opt. Lett. 38(22), 4841–4844 (2013). K. Y. Song, Y. H. Kim, and B. Y. Kim, “Intermodal stimulated Brillouin scattering in two-mode fibers,” Opt. Lett. 38(11), 1805–1807 (2013). A. Mocofanescu, L. Wang, R. Jain, K. D. Shaw, P. R. Peterson, and A. Gavrielides, “Experimental and theoretical investigations on stimulated Brillouin scattering (SBS) in multimode fibers at 1550 nm wavelength,” Proc. SPIE 5581, 654–661 (2004). A. Mocofanescu, L. Wang, R. Jain, K. D. Shaw, A. Gavrielides, P. R. Peterson, and M. Sharma, “SBS threshold for single mode and multimode GRIN fibers in an all fiber configuration,” Opt. Express 13(6), 2019–2024 (2005). A. Fotiadi and E. A. Kuzin, “Stimulated Brillouin scattering associated with hypersound diffraction in multimode optical fibers,” presented at Quantum Electronics and Laser Science Conference, Anaheim, Calif, June 2–7 1006, paper QFC4. G. A. Decker, “Method of fabricating an optical attenuator by fusion splicing of optical fibers, US Patent no. 4557556, 1985. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
1. Introduction Stimulated Brillouin scattering (SBS) has been widely studied in single-mode silica fibers [1– 3]. In pump-probe experiments, the amplification of the optical wave at lower frequency (the Stokes beam) depends on the spectral shift from the pump beam, according to the so-called
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Brillouin Gain Spectrum (BGS). The peak position of the BGS, known as Brillouin frequency shift (BFS), depends on the effective refractive index and effective acoustic velocity in the fiber. In single-mode fibers, if neglecting fiber birefringence, there is only one effective optical refractive index, while each acoustic mode supported by the fiber gives rise to a different BFS, in virtue of the different effective acoustic velocity. Therefore, the overall BGS will be the superposition of various Lorentzian spectral shapes, each one peaked at a different BFS and weighted by the so-called acousto-optic effective area [4], which is evaluated from the spatial overlap between acoustic mode and optical mode field distributions. In multimode fibers, the picture becomes more complicated, as many optical and acoustic waves are generally involved. Actually, each pair of counter-propagating optical modes may interact through an acoustic mode travelling along the same direction of the pump wave, contributing to the overall BGS. Each contribution is weighted by a factor depending on the overlap integral between the three modes [5]. The SBS interaction between two identical optical modes is named intramodal SBS, while the interaction between different optical modes is known as intermodal SBS [6]. Recently, the SBS in few-mode fibers (FMFs) has been studied, both numerically [5,6] and experimentally [7–9]. The results show that the BGS depends on the pair of optical modes involved. Also, it has been reported that intermodal SBS gain is of the same order of magnitude [9], or even larger [6] than intramodal SBS gain. The study of SBS in multimode fibers (MMF) has been mostly focused on the evaluation of the SBS threshold [10–12]. It was shown that good agreement between numerical and experimental thresholds can be found, if using for threshold calculation an effective mode radius reff < rc, where rc is the radius of the fiber core. It was also found that the gain coefficient for lower-order optical modes strongly exceeds the gain of the higher-order modes [13]. In this work, we characterize the SBS in a graded-index multimode (GI-MMF) silica fiber, having a core diameter of 50 μm and a numerical aperture (NA) of 0.2. At the optical wavelength of 1.55 μm, the fiber supports several tens of optical modes, and several hundreds of acoustic modes. As, in general, each pair of optical modes interacts through a different set of acoustic modes and with a different efficiency, the overall BGS will depend on the pump and probe optical modal content. A numerical procedure based on finite-element analysis (FEA) has been applied in order to predict the BGS for any distribution of the pump and probe optical field. Numerical results are also compared to BGS measurements. In order to acquire the BGS of the multimode fiber for different pump and probe fields, the measurements have been performed upon offset-splicing the GI-MMF at both ends with two short sections of singlemode fibers (SMF). The experiments confirm that the BGS of the multimode fiber strongly depends on the modal content of the two interacting beams. In particular, it is shown that, increasing the energy associated to high-order optical modes (HOM), the BGS broadens inhomogenously, mainly as a consequence of the larger number of acoustic modes involved. In the next section, we present the numerical method employed to estimate the BGS in a multimode fiber. In section 3, the numerical spectra evaluated at different launch conditions are compared to experimental results. Conclusions will follow. 2. Numerical method Firstly, the optical modes of the GI-MMF were computed by use of a finite-element analysis (FEA). A parabolic refractive index profile was chosen for simulation, with the maximum refractive index nco = 1.4713 at the core axis, and a refractive index ncl = 1.458 across the cladding. A number of 55 guided optical modes were found by use of the commercial solver COMSOL Multiphysics, including degenerate modes. For each mode, we extracted the profile associated to the main component of the electric field, as well as the corresponding effective refractive index. As a next step, another FEA was carried out to calculate the displacement field and the BFS associated to each acoustic mode supported by the same fiber. To this aim, the 2-D
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scalar mechanical equation was solved by assuming a longitudinal acoustic velocity distribution as described by the following relation [4]: Vl ( r , θ ) = Vl 0 (1 − 7.8 Δn ( r , θ ) ncl ) (1) where Vl0 = 5944 m/s is the longitudinal acoustic velocity into the cladding, and Δn = n(r,θ)ncl. By assuming for the acoustic wave a propagation constant equal to the sum of the propagation constants of the two interacting optical modes (i.e., the SBS resonance condition), the BFS associated to each acoustic mode is found. For the simulations, the BFS of the various acoustic modes were found by imposing the phase-matching condition between the acoustic mode and the propagation constant of the fundamental (LP01) optical mode. In order to compute the BFS corresponding to a specific acoustic mode and a generic pair of optical modes, we applied the following formula [9]: BFSi , j , k = BFSi ,1,1 ( neff , j + neff , k ) ( 2neff ,1 ) (2)
where i is the index referring to the acoustic mode, j and k are optical modes indexes, and neff,i is the effective refractive index of the i-th optical mode. Equation (2) tells us that, for a specific acoustic mode, higher-order optical modes lead to a smaller BFS. As the number of guided acoustic modes in the analyzed fiber is very large (several hundred), it was not possible to take into account all of them due to computational limitations, rather only the first 151 acoustic modes were solved for. As it will be seen later, the computed set of acoustic modes is sufficient to describe accurately the BGS in our experimental conditions. The computed optical and acoustic modes were then processed so as to construct a three-dimensional array of Brillouin gain coefficients, in which each element refers to a specific pair of optical modes and a specific acoustic mode. In particular, for each triad of modes the Brillouin gain was computed by using the following formula [5]: Gi , j , k = g i , j , k f i * ( r , θ ) g *j ( r ,θ ) hk ( r , θ ) drdθ
2
(3)
where gi , j , k = ( c μ0γ e2 ) ( 2λρ 0ε 0 n 2 BFSi , j , k Γ ) , c and μ0 are the light velocity and magnetic
permeability in vacuum, γe is the electrostriction constant, λ is the signal wavelength, ρ0 is the material density, ε0 is the vacuum permittivity, n the refractive index, Γ is the acoustic wave damping factor, f is the acoustic mode, g and h are the two optical modes, and the domain of integration is the fiber cross-section. Before computation of integrals, all modes were normalized so as to have unitary area. In case of intramodal SBS, Eq. (3) reduces to the inverse of the acousto-optic effective area [4]. Also, as in our computations we consider purely real optical and acoustic profile basis functions, the coefficients expressed by Eq. (3) are such that Gi,j,k = Gi,k,j. At the end of this process, a 151 × 55 × 55 Brillouin gain coefficient array was computed for the investigated fiber. For any chosen pair of optical modes, the related BGS is computed as the sum of the spectral-density contributions from each mode [2]: 1 N BGS j , k (υ ) = i =ac1 Gi , j , k (4) 2 υ − BFSi , j , k 1+ Δυ B 2 where ν is the pump-probe spectral shift, Nac is the number of acoustic modes and ΔνB is the bandwidth of each Lorentzian spectral profile. In our computations, we assumed ΔνB = 35 MHz. In virtue of the cylindrical symmetry of the analyzed fiber, the interaction between a generic couple of optical modes, LPxa - LPyb, occurs through two acoustic family modes: A(x ± y)n with n = 1, 2, 3, etc. In other words, the acoustic modes must have an azimuthal mode number equal to the sum (or the difference, if not negative) between the azimuthal mode numbers of the two optical waves. As a first example, let us consider the case in which both pump and probe fields coincide with the fundamental mode (LP01) of the GI-MMF. In this case, SBS will only occur through
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axially symmetric acoustic modes. Within the computed set, we found 9 radial modes (A01 A09). We report in Fig. 1(a) the normalized Brillouin gain coefficients computed by use of Eq. (3). Apparently, the largest contributions come from the first three acoustic radial modes (A01-A03), while for the last six modes the gain is below 1% of maximum gain. 1
1
(a)
0.8
0.8
0.7
0.7
0.6 0.5 0.4 0.3
0.5
0.1
A03
A04
A05
A06
A07
A08
0 10.3
A09
A03
0.3
0.1 A02
A02
0.4
0.2
A01
A01
0.6
0.2
0
(b)
0.9
Brillouin gain [n.u.]
Brillouin gain coefficient [n.u.]
0.9
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
Pump-probe frequency shift [GHz]
Acoustic mode
Fig. 1. Brillouin gain coefficients (a) and Brillouin gain spectrum (b) computed for the LP01LP01 optical mode pair.
The corresponding BGS is shown in Fig. 1(b) (solid blue curve). The same figure also highlights the individual contributions from A01, A02 and A03 (dashed black curves). It is seen that scattering from high-order acoustic modes is responsible for some asymmetry of the BGS, which is broader on the right side. Note that the spectral distance between adjacent resonances (~27 MHz) is smaller than the bandwidth of each resonance, so the various contributions overlap each other giving rise to full width at half maximum (FWHM) of the BGS equal to 45 MHz. Let us now analyze the interaction between the fundamental optical mode, LP01, and the second-order optical mode LP11. In this case, the interaction is mediated by acoustic modes with unitary azimuthal index (A11 – A18 in our mode set). As the LP11 is anti-symmetric, we must consider both angular orientations in order to derive an azimuth-independent Brillouin gain coefficient. Practically, we compute the sum of the Brillouin gain coefficients associated to the LP01-LP11a interaction and the ones associated to the LP01-LP11b interaction, with LP11a and LP11b denoting the two possible angular orientations of the degenerate mode LP11. The Brillouin gain coefficients are shown in Fig. 2(a). Note that, as the gain rapidly decreases when increasing the radial mode number, the contribution from higher-order acoustic modes (A19, A1-10, etc.) can be safely considered as negligible. 0.8
1
(a)
(b)
0.9 A11
0.8
0.6
Brillouin gain [n.u.]
Brillouin gain coefficient [n.u.]
0.7
0.5 0.4 0.3 0.2
0.7 A12
0.6 0.5 0.4
A13 0.3 0.2
0.1 0
0.1
A11
A12
A13
A14
A15
Acoustic mode
A16
A17
A18
0 10.3
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
Pump-probe frequency shift [GHz]
Fig. 2. Brillouin gain coefficients (a) and Brillouin gain spectrum (b) computed for the LP01LP11 optical mode pair.
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The corresponding BGS is shown in Fig. 2(b), together with the contributions from the three acoustic modes A11, A12, and A13. We note that, while the spectral distance between adjacent acoustic resonances is the same as the previous case (~27 MHz), the resulting BGS is now broader (58 MHz), due to the larger gain associated to high-order acoustic modes. As a final example, we analyze the LP11-LP11 interaction. Still, we calculate the gain by summing over all the possible combinations of optical mode angular orientation. Within our mode set, we found 17 acoustic modes with azimuthal index equal to 0 (A01 - A09), or 2 (A21 – A28). The gain coefficients, reported in Fig. 3(a), suggest that the contribution from the acoustic modes not included in the computed set (i.e., with higher radial index) can be safely considered as negligible. The BGS is shown in Fig. 3(b), together with the individual contributions from the most efficiently excited acoustic modes. The FWHM of the BGS is now 81 MHz. The larger bandwidth should be attributed to the increasing number of acoustic modes involved. Actually, in this case two acoustic mode families come into play: the one with azimuthal index equal to 0 and the one with azimuthal index equal to 2. Both families contribute with similar gain to the overall spectral density (see Fig. 3(a)). 1
0.7
(a)
0.6
0.5
0.4
0.3
(b)
0.8
Brillouin gain [n.u.]
Brillouin gain coefficient [n.u.]
0.9
0.2
0.7
A21
0.6 0.5 A02
0.4
A22 A03
0.3 0.2
0.1
0
0.1 0 10.3
A01 A02 A03 A04 A05 A06 A07 A08 A09 A21 A22 A23 A24 A25 A26 A27 A28
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
Pump-probe frequency shift [GHz]
Acoustic mode
Fig. 3. Brillouin gain coefficients (a) and Brillouin gain spectrum (b) computed for the LP11LP11 optical mode pair.
As a final remark, we should observe that, in real cases, it is difficult to selectively launch individual modes in a multimode fiber. On the other hand, the model described can be equally employed to calculate the BGS resulting from a generic combination of pump and probe fields. Actually, provided that the modal content of pump and probe beams is known, the contributions associated to any pair of optical modes can be summed, properly weighting each contribution on the basis of the energy of each mode. More specifically, let us suppose that the amplitude of pump and probe fields within the MMF can be expressed as: E p ( r , θ ) = j =1 Aj f j ( r , θ )
(5a)
Es ( r , θ ) = k =1 Bk f k ( r ,θ )
(5b)
N
N
with N representing the number of MMF modes, f are the basis functions, A and B their amplitudes. In the model, we assume that no mode mixing occurs along the MMF, i.e. the modal content of pump and probe beams is uniform through the whole MMF length. Within the undepleted pump approximation, and neglecting fiber loss, the total BGS can be calculated as: BGS (ν ) = j =1 B j k =1 Ak × BGS j , k (υ ) N
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N
(6)
Received 4 Mar 2014; accepted 9 May 2014; published 11 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017480 | OPTICS EXPRESS 17484
where BGSi,j are the terms defined in Eq. (4). Therefore, when more than one mode is present in either the pump or probe field (or both), the BGS will be the result of both intramodal (j = k) and intermodal (j≠k) SBS between the modes involved. 3. Experimental results
In this section, we report a number of experimental results aimed to demonstrate the accuracy of the proposed model. Experiments were performed on a 50 m, graded-index multimode fiber with a core diameter of 50 μm and a nominal numerical aperture of 0.2. During the measurements, the fiber was left on its spool, and care was taken in order to avoid any perturbation which may alter the modal distribution of the optical fields along the fiber. The BGS measurements were taken by use of the setup schematically shown in Fig. 4. All fiber components, except from the fiber under test (FUT), were single-mode. A diode laser emitting a 1.55 μm was employed as the laser source. The light is split in two arms by means of a polarization maintaining beam splitter, with the lower arm acting as the pump branch, and the upper arm serving as the probe branch. On the upper arm, an intensity modulator, driven at a radiofrequency close to the BFS of the FUT, generates two sidebands by suppressed-carried double sideband (SC-DSB) modulation. The sideband at higher frequency is successively filtered out by a fiber Bragg grating (FBG) bandpass filter, while the one at lower frequency acts as the probe beam. In the lower branch, the pump beam is modulated in order to realize a synchronous lock-in detection. Both pump and probe beams are optically amplified before being launched into the FUT. The photodiode features a bandwidth of 125 MHz and a gain of 90 dB.
Fig. 4. Experimental setup for BOFDA measurements. DFB-LD, diode laser; FBG, fiber Bragg grating; EDFA, Er-doped fiber amplifier. EOM, electro-optic modulator; LIA, lock-in amplifier; PD: photodetector.
As previously discussed, the BGS of a multimode fiber depends on the pump and probe modal content. In order to achieve a stable excitation of pump and probe fields, the GI-MMF was fusion-spliced at both ends with two short sections of single mode fibers (SMF), which are connected to the setup by use of FC/APC connectors. For each test, the fusion splices were performed with a different value of intentional offset. The estimation of the BGS was performed by first calculating the coupling coefficients between the (eventually displaced) SMF mode and the GI-MMF modes, through 2-D overlap integrals. Assuming identical splices at both MMF ends, and denoting with Ci the coupling coefficient between the SMF mode and the i-th MMF mode, we compute the BGS as: BGS (ν ) = j =1 C j k =1 Ck × BGS j , k (υ ) N
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N
(7)
Received 4 Mar 2014; accepted 9 May 2014; published 11 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017480 | OPTICS EXPRESS 17485
where N is number of MMF modes and BGSi,j are the terms defined in Eq. (4). On the other hand, we must also consider that each probe mode passes again through the MMF-SMF splice before being detected. Therefore, we must further multiply the contribution from each probe mode with the corresponding coupling coefficient. The observed BGS will then be given by: BGSobserved (υ ) = j =1 C 2j k =1 Ck × BGS j , k (υ ) = j =1 k =1 BGS eff j , k (υ ) N
N
N
N
(8)
where BGS eff j , k denotes the effective contribution of each pump-probe modal pair, given by: ac 2 BFS eff j , k (υ ) = C j Ck i =1 Gi , j , k
1
N
υ − BFSi , j , k 1+ Δυ B 2
(9)
2
1 0.9
(a)
1
0.7
Brillouin gain [n.u.]
Coupling coefficient
0.8
0.6 0.5 0.4 0.3 0.2
Numerical BGS Experimental BGS
(b)
0.8
0.6
0.4
0.2
0.1 0
LP01
LP02
LP03
LP04
GI-MMF optical mode
LP05
0 10.38
10.4
10.42 10.44 10.46 10.48
10.5
10.52 10.54 10.56 10.58
Pump-probe frequency shift [GHz]
Fig. 5. (a) Coupling coefficients between SMF and GI-MMF computed in case of zero offset between the two fibers; (b) Numerical and experimental BGS of the GI-MMF in case of zero offset between the two fibers.
The first experiment was carried out after splicing the MMF to the SMF pigtails with no intentional offset. Due to axial symmetry of the excitation fields, only axially symmetric optical modes of the GI-MMF are excited in this particular case. By computing the overlap integrals between the SMF mode and the 5 radial modes supported by the GI-MMF, we achieve the coupling coefficients shown in Fig. 5(a). We see that most of optical energy (~90%) is coupled to the fundamental mode of the GI-MMF. As expressed by Eq. (8), the overall BGS is computed by summing the contributions associated to the various LP0x modes, including intermodal SBS between LP modes with different radial mode number. The result is shown in Fig. 5(b). The same figure also reports the experimental BGS, acquired with the setup shown in Fig. 4 and scanning the pump-probe frequency shift at a step of 4 MHz. Although not excellent, the agreement between numerical and experimental data is remarkably good, also considering the various model uncertainties (refractive index profile, excitation pattern, fiber inhomogeneities). The BFS of experimental and numerical BGS is ~10485 MHz. We note that the BGS is very similar to the numerical spectrum computed previously for the LP01-LP01 interaction (Fig. 2). This is easily explained, by considering that the optical energy of both pump and probe waves is mostly carried by the LP01 mode. Consequently, the SBS interaction in the present case mainly occurs through the first three radial acoustic modes (A01, A02 and A03). As a second test, the same GI-MMF was spliced again with the two SMF pigtails, this time by setting the fusion splicer so as to perform 3-dB attenuated splices. Fusion splicers realize attenuated splices by intentionally applying a lateral offset between the two fibers, with an offset typically ranging from 5 μm to 60 μm [13]. In order to estimate the lateral offset for a definite splice loss, we employed the formula proposed by Marcuse in Ref [14],
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according to which the power transmission coefficient through a fiber splice with an offset d is expressed as: d 2 T = exp − ω
(10)
where ω is the width parameter of the Gaussian beam propagating through the splice. As the fusion splicer applies the offset supposing to operate with single-mode fibers, we use ω = 10.4 μm in Eq. (10). This results in a lateral offset of 8.5 μm for a 3-dB splice loss. Therefore, we computed the coupling coefficients between the GI-MMF guided modes, and the SMF mode displaced by 8.5 μm from the fiber axis. We underline that, while the coefficients have been calculated for an SMF mode displaced along the x-direction, the final BGS is independent of the chosen direction, as optical and acoustic degenerate modes are always considered in both angular orientations. The computed coupling coefficients are represented in Fig. 6(a), where, for clearness, we have included only those greater than 0.01. As in this case the input field was not axially symmetric, also non symmetric GI-MMF optical modes were excited. The numerical BGS, computed by use of Eq. (8), is compared to the experimental spectrum in Fig. 6(b). We see that, the broadening of the BGS caused by offsetsplicing is well captured by the model. In particular, the full width at half maximum (FWHM) of the numerical BGS increases from 46 MHz to 88 MHz, while the corresponding experimental values are 47 MHz and 82 MHz, respectively. 0.4
(a)
0.35
1
Numerical BGS Experimental BGS
(b)
Brillouin gain [n.u.]
Coupling coefficient
0.3 0.25 0.2 0.15
0.8
0.6
0.4
0.1
0.2
0.05 0
LP11a
LP01
LP02
LP21a LP21b LP31b LP12a LP41b
GI-MMF optical mode
0 10.3
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
Pump-probe frequency shift [GHz]
Fig. 6. (a) Coupling coefficients computed between SMF and GI-MMF in case of 8.5 μm lateral offset between the two fibers (only GI-MMF modes with a coupling coefficient larger than 0.01 were included in the graph); (b) Numerical and experimental BGS of the GI-MMF in case of 8.5 μm lateral offset between the two fibers.
The BGS broadening can be explained in terms of the larger number of acoustic modes involved in this case. By limiting our analysis to the two most energetic modes (LP11a and LP01), these two modes interact via acoustic modes having unitary azimuthal mode number as regards intermodal SBS, and via acoustic modes with azimuthal mode number equal to 0 and 2 as regards intramodal SBS. In general, we found 16 acoustic modes leading to a Brillouin gain above a threshold set to the 1% of maximum gain. It is also interesting to observe that the BFS of both experimental and numerical spectra is ~10500 MHz, i.e. about 15 MHz higher than the BFS of the previous test. Actually, the peak position of the BGS is the result of the superposition of several Lorentzian shapes, each centered at a different frequency. Therefore, the peak position of the BGS changes based on the relative weight of each contribution. In our conditions, the blue shift of the BGS is due to the fact that major contributions arise from higher-order acoustic modes, which resonate at higher frequencies.
#207647 - $15.00 USD (C) 2014 OSA
Received 4 Mar 2014; accepted 9 May 2014; published 11 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017480 | OPTICS EXPRESS 17487
0.16
0.14
Coupling coefficient
0.12
0.1
0.08
0.06
0.04
0.02
0
LP31bLP12aLP13a LP03 LP02 LP32bLP11aLP22aLP31bLP21aLP41bLP51bLP21bLP42bLP61b LP04 LP23aLP23b LP01 LP71b
GI-MMF optical mode
Fig. 7. Coupling coefficients computed between SMF and GI-MMF in case of 15 μm lateral offset between the two fibers. Only GI-MMF modes with a coupling coefficient larger than 0.01 were included in the graph.
The final test was carried out after splicing the GI-MMF to the SMF pigtails, this time setting the fusion splicer so as to perform 10-dB attenuated splices. By using again the formula expressed by Eq. (10), the estimated offset is now 15 μm. For such an offset, the calculation of the coupling coefficients reveals that up to 20 optical MMF modes are excited with a coupling coefficient higher than 0.01 (see Fig. 7). The resulting BGS is compared to the experimental one in Fig. 8. Still, we observe an overall agreement between numerical and experimental data. In particular, the FWHM of the numerical BGS is 103 MHz, while being 109 MHz for the experimental BGS. Still, we attribute this broadening to the larger number of acoustic modes involved. Note also that the two BGS are now centered at ~10505 MHz.
Numerical BGS Experimental BGS
Brillouin gain [n.u.]
1
0.8
0.6
0.4
0.2
0 10.3
10.35
10.4
10.45
10.5
10.55
10.6
10.65
10.7
10.75
10.8
Pump-probe frequency shift [GHz]
Fig. 8. Numerical and experimental BGS of the GI-MMF in case of 15 μm lateral offset between the two fibers.
Determining the set of acoustic modes involved in this case is not a simple task, due to the large number of possible intermodal and intramodal interactions between the optical modes
#207647 - $15.00 USD (C) 2014 OSA
Received 4 Mar 2014; accepted 9 May 2014; published 11 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017480 | OPTICS EXPRESS 17488
represented in Fig. 7. Potentially, all the 151 computed acoustic modes are involved, as well as higher-order acoustic modes not included in the computed set. In order to determine the most influent acoustic modes, we first computed the factor C 2j Ck Gi , j , k appearing in Eq. (9), for each triad of modes. Summing this factor over all the optical modes, we determined the overall contribution of each acoustic mode. The results are plotted in Fig. 9, in which all the computed modes with an azimuthal mode number ranging from 0 to 5 are included (j = 0 corresponds to the axially symmetric acoustic modes). We see that the interaction strength of acoustic modes decreases with azimuthal and/or radial mode number, with a larger slope associated to the azimuthal mode number. In other terms, the larger is the azimuthal index (i.e. the number of azimuthal oscillations), the slower the interaction strength decreases with the radial mode number (i.e. with the number of peaks along the r direction). On the other hand, we also observe that the computed set of modes includes all modes for which the interaction strength (normalized to the maximum strength) is as high as to ≈-11 dB. Therefore, we can safely conclude that the computed set is sufficient to describe accurately the overall SBS density spectrum in our experimental condition, as it is also confirmed by the agreement with experimental data (see Fig. 8).
Brillouin gain coefficient [dB]
-45
-50
j=0
-55
j=1 j=2 j=3
-60
j=4 j=5
-65
-70
1
2
3
4
5
6
7
8
9
Radial mode number Fig. 9. Brillouin gain coefficient associated to each acoustic mode of the GI-MMF, when the latter is excited at both ends by an SMF spliced with a 15 μm offset.
4. Conclusions
A numerical model based on the calculation of acousto-optic overlap integrals was applied to estimate the BGS in a graded-index multimode fiber. The results show that in these fibers the BGS shape is strongly dependent on the modal content of both pump and probe beams. Also, it was shown that increasing the number of optical modes involved, the SBS occurs with an increasing larger number of acoustic modes, leading to a broadening of the observed BGS. A number of experiments have been carried out, substantially confirming the numerical predictions. We believe that these results are potentially useful in understating the SBS behavior of multimode fibers, with implications in various fields such as sensing, spatialmode division multiplexing, and Brillouin fiber laser modeling.
#207647 - $15.00 USD (C) 2014 OSA
Received 4 Mar 2014; accepted 9 May 2014; published 11 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017480 | OPTICS EXPRESS 17489
