Analytical approach of Brillouin amplification over threshold

Transcription

1 Research Article Vol. 57, No. 4 / February 208 / Applie Optics 607 Analytical approach of Brillouin amplification over threshol FİKRİ SERDAR GÖKHAN,, *HASAN GÖKTAŞ, 2,3 AND VOLKER J. SORGER 2 Department of Electrical an Electronic Engineering, Alanya Alaain Keykubat University, Kestel, Alanya, Antalya, Turkey 2 Department of Electrical an Computer Engineering, The George Washington University, Washington, DC 20052, USA 3 Department of Electrical an Electronic Engineering, Harran University, Sanliurfa 6300, Turkey *Corresponing author: fsgokhan@gmail.com Receive 29 September 207; revise 5 December 207; accepte 9 December 207; poste 20 December 207 (Doc. ID 30823); publishe 24 January 208 We report on an accurate close-form analytical moel for the gain of a Brillouin fiber amplifier that accounts for material loss in the eplete pump regime. We etermine the operational moel limits with respect to its relevant parameters an pump regimes through both numerical an experimental valiation. As such, our results enable accurate performance preiction of Brillouin fiber amplifiers operating in the weak-pump, high-gain, an saturation regimes alike. 208 Optical Society of America OCIS coes: ( ) Scattering, stimulate Brillouin; ( ) Fiber optics amplifiers an oscillators; ( ) Nonlinear optics, fibers. INTRODUCTION Since the first emonstration of light amplification in optical fibers with stimulate Brillouin scattering (SBS), the Brillouin fiber amplifier (BFA) configuration [] has been implemente in a wie range of applications, such as microwave photonics [2], shape-ajustable narrowban optical filtering [3], raioover-fiber technology [4], generation of millimeter-wave signals [5], exploiting as tunable slow-light elay buffers [6] an fiber optic sensing [7]. To initiate SBS, a small amount of optical power at the Stokes frequency is launche from the opposite en of the fiber. In the BFA configuration, SBS can be use for efficient narrowban amplification when the see Stokes wave is input from the rear (opposite to the pump) en of the fiber. Interaction between the pump an the Stokes wave ue to SBS is escribe by a system of orinary ifferential equations (ODEs) [8]. The system of ODEs for BFAs has well-efine bounary conitions: P p 0 P 0 an P S L P see. Such a mathematical problem is known as the two-point bounary value problem. Since the bounary value of the P p L an/or P s 0, with L being the fiber length, is unetermine, such systems of nonlinear ODEs is typically aresse numerically. The exact analytical solution to the system of ODEs is known only for lossless meia [9], with the exception of an analytical solution of integration constant C. However, the contribution of loss is a very important effect in optical fibers because the pump Stokes interaction occurs over long istances, an both the pump an the Stokes waves can be attenuate by orers of magnitue [8]. Several approximations have been propose for the spatial evolution of guie optical powers of the pump P p an Stokes P S wave by solving a set of couple ODEs with the consieration of fiber loss. In [0], the authors moifie the ODEs an propose a system of two transcenental equations to reuce the computation time of numerical solution. In [], assuming low-attenuation fiber, it was propose to reverse the sign of the loss term in one of the ODEs to make the approximate set of equations integrable. However, this results in a system of two transcenental equations, which inclues the Stokes power at z 0. Since this power is not known a priori, it can be solve only numerically, an no close-form solution is possible. Below critical pump powers, the uneplete pump approximation (UPA) can be use. However, in the high-gain regime, since the pump becomes eplete above the SBS threshol, the UPA solution strongly overestimates the real Brillouin gain an loses its valiity, so it cannot be use in the above-mentione applications, which typically require pump powers above the SBS threshol. In orer to improve the accuracy of the UPA, one can apply the UPA solution to the Stokes wave an substitute the obtaine Stokes wave to the pump wave, again. One can show, however, that for SBS the correction to the UPA contains an exponential integral function that quickly iverges with increasing pump power [8,2]. In another approach [3], a perturbation technique is successfully applie to obtain erivation of pump an Stokes power evolution for a lossy fiber using the solution obtaine for the lossless fiber. This solution has some rawbacks, X/8/ Journal 208 Optical Society of America

2 608 Vol. 57, No. 4 / February 208 / Applie Optics Research Article however; at the borer of the high-gain an saturation regions, the conserve quantity of lossless fiber either unerestimates an/or prouces imaginary values (i) when the Stokes pump power is aroun one milliwatt (aroun the beginning of the saturation region), an (ii) when the fiber length is increase beyon 20 km, an the accuracy ecreases. On the other han, consiering only the small perturbations, the authors consierably simplifie the ρ parameter, which ecreases the accuracy, an also, the approximation is vali only in the high-gain region an is not operational in the saturation region. In [4], an approximate analytical solution of the steaystate couple equations has been propose. However, similar to Ref. [], these equations also require Stokes power at the fiber input (at z 0), which is not known a priori. In Ref. [5], analytical solutions were obtaine in fiber lengths of up to a few kilometers by neglecting the attenuating terms ue to the short fiber lengths inherent in the moel, which limits its applicability where the fiber length can be 50 to 00 kilometers. In Ref. [6], the authors investigate the linear response of the BFA to a weak amplitue moulation of the injecte signal. However, it is not practical in reality, since response of the BFA is highly nonlinear, especially for the strong pump an Stokes powers, which rive the BFA working in the saturation region. Here, we moifie the metho that is applie in Ref. [3] an extensively improve the accuracy of both the pump an Stokes wave evolutions. The analytical solution is ivie into three regions, epening on the input pump power: the first region is below the critical pump power, where the UPA solution can be use. The secon part is the high-gain region, where pump power excees the critical power, an the epletion is moerate, an when the Stokes power is aroun 2 mw, epening on the critical pump power. The thir part escribes the saturation region, where high Stokes powers (> 2 mw) are use an epletion is high, i.e., when low-gain amplification occurs. Following this formulism, we erive an approximation for the Brillouin gain of the BFA analytically. The pump conition between the regions where high- an low-gain amplification occurs is erive. The propose solution is valiate both experimentally an numerically, where we consier a full integration of couple ODEs. Lastly, we iscuss the valiity ranges of the foun solution. 2. THEORETICAL MODEL The couple ODEs for the evolution of the intensities of pump I p an Stokes I S can be written as [9,0,7] I p g B I p I s αi p I s g B I p I s αi s ; () where 0 z L is the propagation istance along the optical fiber of the total length L, α is the fiber loss coefficient, an g B is the Brillouin gain coefficient. BFA requires that a Stokes wave be injecte at the opposite en of the fiber [,8]. The two counter-propagating light waves are couple through electrostriction, which leas to the amplification of the Stokes at the expense of the pump. Here, we assume a Stokes wave launche from the rear en of the fiber. Then the known values of the input pump intensity I p0 an the input Stokes intensity I sl are the bounary values. In our approach, we represent the unknowns through perturbations to the lossless systems, i.e., as I p;2 I pp ρ ;2 ; I s;2 I ss μ ;2 ; (2) where I pp an I ss are the solutions of the lossless system, i.e., I pp g B I pp I ss ; I ss g B I pp I ss ; (3) with the same bounary values I pp 0 I p0, I ss L I sl,as in Eq. (). The solution of Eq. (3) is[9] I pp zc ;2 I p0 I p0 c ;2 I p0 exp c ;2 g B z I ss zc ;2 I p0 c ;2 I p0 expc ;2 g B zc ;2 I p0 ; (4) where c ;2 I pp I ss is the conserve quantity of Eq. (3). To explicitly fin the value of parameter c ;2, we approximately solve the equation: I sl c ;2 I p0 c ;2 I p0 expc ;2 g B Lc ;2 I p0 ; (5) using bounary conition I sl. The solution of c ;2 iffers for high-gain an saturation regions. We have obtaine c ;2 by eliminating ifferent parameters of secon-orer ifferential equations. A. Solution for High-Gain Region In this region, the solution of c ;2 becomes 0 c k B Λ ln Λ Λ k ln Λ ke Λ e Λ C A I p0 ; (6) where ε I sl I p0, k g B I p0 L, an Λ lnεk. From Eqs. () (3), we then obtain a system of ODEs for the perturbation ρ an μ: ρ g B I ss μ α ρ μ g B I pp ρ α μ ; (7) with the bounary values ρ 0 0 an μ L 0. In this region, c is use in the calculation of I pp an I ss, an the solution of Eq. (7) leas to g B I p ρ α g B I p α g B I s ρ z e αzc g B α c g B α 2 ρ ρ 2 ρ ρ g2 B I pi s g B I s α ρ g B I s α ρ I p0αe c g B z Φ c I p0 c g B α c c I p0 g B α ; (8a)! 2α Φc g B α ; (8b) where Φ 0. log 0 P sl.7 is the curve-fitting parameter, which is substitute instea of the combine effects of nonlinear terms 2 ρ 2 an ρ 2, which arise from the solution of Eq. (8a). The combine effect of these two terms is ominative on the transition from one regime into

3 Research Article Vol. 57, No. 4 / February 208 / Applie Optics 609 another, an their exact values epen on the ρ z 0, which is not known a priori. Moreover, they are highly nonlinear ue to the component of ρ 2. The solution of Eq. (7) also leas to g B I s μ α g B I s α 2 μ μ 2 g B I p μ μ g2 B I pi s g B I p α μ g B I p α μ ; (9a) μ z 0.9Λe c g z B e.zc g α B e.αl e.αl z I p0 c 0. ; (9b) where 0.9Λ is another curve-fitting parameter that is substitute instea of the combine effects of nonlinear terms 2 μ 2 an μ 2, which arise from the solution of Eq. (9a). Curve-fitting parameters are vali only in the highgain regime. In the saturation regime, another formulation is use. The evolution of pump an Stokes waves for the high-gain region is I phigh-gain I pp ρ ; (0a) I shigh-gain I ss μ : The UPA solution for the I p z an I s z is I p z I p0 e αz (0b) I s z I sl e I p0 g B α e αz e αl z αl z : (a) The BFA gain is efine as the ratio of the amplifie Stokes power an the launche Stokes power G BFA P S 0 P S L: G BFA e I e p0g αl B α αl : (b) The parameters in Fig. are chosen to moel a 0 km long BFA, base on a stanar single-moe fiber (α 0.25 B km) with a pump power of either 0 mw (k 4) or50mw (k 70). As can be seen from Fig., approximation (0a) accurately escribes epletion of the transmitte pump not only for moerate pump powers that are the SBS threshol [Fig. (c)], but also for strong input pump an Stokes powers extening the saturation region [Fig. ()]. The preiction of the analytical formula for the amplifie Stokes wave (0b) is quite accurate, even for high pump powers [Fig. (b)]. Note that a iscrepancy in the initial value of the Stokes wave I s z L [Fig. (b)] is ue to the approximation taken in Eq. (6). The BFA gain can be calculate from the approximation in (0b), Eq. (4), an expresse in terms of the physical parameters as G BFA A eff Pp0 c P sl A eff 0 B ln PsL g B L 0.9 C Pp0 0. A: (2) c Fig.. Stokes wave P s (a), (b) an pump wave P p (c), () as a function of fiber istance, z. In [(a), (c)], P p0 0 mw, P sl 0 μw, (k 4); an in [(b), ()], P p0 50 mw, P sl 50 μw, (k 70). Thick soli curves stan for approximate analytical solution (0a) an (0b); ashe curves stan for full numerical solution (). In both figures, L 0 km, 80 μm 2, g B.09 0, α 0.25 B km, an ε 0 3. In our experimental setup [Fig. 2], similar to Ref. [2], a tunable laser with λ p nm was use together with an erbium-ope fiber amplifier (EDFA) to generate up to 80 mw of pump power. Stokes an pump sources are erbiumope fiber lasers, whose linewiths are smaller than 00 khz. Their frequency ifference is controlle with a phase-locke loop an is locke to the Brillouin frequency, ν B, of the stanar single-moe optical fiber uner test. Power meters were use to monitor the input pump power P p0, the transmitte pump power P pl, the launche Stokes power P sl, an the amplifie Stokes power P s0. Fiber with a length of 0 km was experimentally stuie. The experimental results are plotte in [Fig. 3] together with theoretical preictions. We fin a high level of agreement for the transmitte pump for all input pump levels above critical pump power [Fig. 3(a)]. As for the amplifier gain G BFA, we fin an excellent agreement between preictions from our analytical formula (2) an the measure gain [Fig. 3(b)]. We note that Eq. (2) is applicable only when the pump power excees the Fig. 2. meter. Experimental setup for BFA measurements. PWM: power

4 60 Vol. 57, No. 4 / February 208 / Applie Optics Research Article where Lambert W is a function that satisfies Lambert W x explambert W x x. In this region Eq. (7) is moifie as ρ 2 g B I ss μ 2 α ρ 2 μ 2 g B I pp ρ 2 α μ 2 ; (4) with new ρ 2 ; μ 2 ;I ss, an I pp functions with moifie c 2 value. The solution of Eq. (4) leas to a moifie solution of Eqs. (8a) an (9a) with the new replacements of 2 μ 2 2 an μ 2 2. The moifie functions of ρ 2 ; μ 2 will be as follows: μ 2 z c 2 I ss ρ 2 z e e αz L 2α 2α ln Issz L 2 IsszL ln c 2 g B c 2 g B α I ss z I ss z L 2 ; (5) z c 2 g B α e αl e αz : (6) In this case, new c 2 is inserte into Eq. (4) to get new values of I pp an I ss intensities. The evolution of power an Stokes waves for the saturation region is I psaturation I pp ρ 2 I ssaturation I ss μ 2 : (7) The experimental results are plotte in Fig. 4 together with theoretical preictions of Eq. (7). The same fiber parameters is use as in Fig. with a pump powers of either 40 mw (with P sl =20μW) or 20 mw (with P sl =40μW). As can be seen from Fig. 4, approximation (7) accurately escribes the epletion of the transmitte pump for the strong input Stokes powers. The preiction of the analytical formula for Fig. 3. (a) Transmitte pump power P p L an (b) BFA gain G BFA versus the input pump power P p 0. Open circles, experimental ata; thick soli curves, preictions of the analytical formula (2); soli circle curves, calculations base on UPA; ashe curves, numerical results. In both figures, L 0 km, P SL.55 μw, 80 μm 2, g B.09 0, an α 0.25 B km. critical value, P p0 >P cr Λ 0.08 g B L Λ. For the weakpump region, the UPA-base estimation for the Brillouin gain can be use [Fig. 3(b)]. The accuracy of Eq. (2) remains high for any fiber length, even if it extens more than 00 km. The tren in [Fig. 3(b)] remains the same for any fiber length. B. Solution for Saturation Region If Λ < 0, the saturation region ominates, showing high nonlinearity. In this region, the solution of Eq. (6) can be analytically efine as c P sl g B :L Lambert W P sl g A B L e 0.99 P sl eff g B L ; (3) Fig. 4. Stokes wave P s (a), (b) an pump wave P p (c), () versus z. In [(a), (c)], P p0 40 mw, P sl 20 μw, (k 55); an in [(b), ()], P p0 20 mw, P sl 40 μw, (k 28). Thick soli curves, approximate analytical solution (0a) an (0b); ashe curves, full numerical solution of Eq. (). In both figures, L 0 km, 80 μm 2, g B.09 0, an α 0.25 B km.

5 Research Article Vol. 57, No. 4 / February 208 / Applie Optics 6 of the three separate conitions are etermine, an the respective BFA gains to these regions are accurately efine. Especially, the gain approximation of BFA for the saturation region, without any power an/or fiber length limitation, is introuce for the first time, to our best knowlege. The results obtaine can be use to optimize performance of Brillouin fiber amplifiers, which are wiely use in microwave photonics, raio-over-fiber technology, an sensing applications, which especially employ the high-gain region an spatial resolution of more than m. Fig. 5. BFA gain versus the input pump power P p 0 for saturation region. Open circles stan for experimental ata, thick soli curves stan for preictions of the analytical formula (8), an ashe curves stan for numerical results. P SL 4mW, L 0 km, 80 μm 2, g B.09 0, an α 0.25 B km. Table. Brief Usage of the Equations Criteria Region Equation P p0 <P cr Λ 0.08 g B L Λ UPA Eqs. () P p0 >P cr Λ 0.08 g B L Λ High-gain region Eqs. (0) Λ < 0 Saturation region Eqs. (7) the amplifie Stokes wave (7) is quite accurate, even for Stokes powers bigger than pump powers [Figs. 4(b) 4()]. The BFA gain in saturation can be calculate from Eq. (7) an expresse in terms of the physical parameters as 0 e αlc 2 g g P p0 B B G P sl g B 2α ln A e c 2 g B L 2 P p0 eff A c 2 eff c 2 Pp0 A: (8) The experimental results of the amplifier gain, G BFA, an the theoretical preictions show a high level of agreement (iscrepancy 2%) with our analytical formula (8) [Fig. 5]. We note that Eq. (8) is applicable only when Λ < 0. Table summarizes the usage of the equations escribe. 3. CONCLUSIONS We have presente an approximate analytical solution to the system of SBS equations in a lossy meium that accurately take into account pump epletion in three ifferent regimes, namely, UPA, high-gain, an saturation regions. The limits REFERENCES. N. A. Olsson an J. P. Van er Ziel, Cancellation of fiber loss by semiconuctor laser pumpe Brillouin amplification at.5 μm, Appl. Phys. Lett. 48, (986). 2. A. Loayssa, D. Benito, an M. J. Gare, Applications of optical carrier Brillouin processing to microwave photonics, Opt. Fiber Technol. 8, (2002). 3. T. Tanemura, Y. Takushima, an K. Kikuchi, Narrowban optical filter, with a variable transmission spectrum, using stimulate Brillouin scattering in optical fiber, Opt. Lett. 27, (2002). 4. M. J. LaGasse, W. Charczenko, M. C. Hamilton, an S. Thaniyavarn, Optical carrier filtering for high ynamic range fiber optic links, Electron. Lett. 30, (994). 5. T. Schneier, M. Junker, an D. Hannover, Generation of millimetrewave signals by stimulate Brillouin scattering for raio over fiber systems, Electron. Lett. 40, (2004). 6. L. Xing, L. Zhan, S. Luo, an Y. Xia, High-power low-noise fiber Brillouin amplifier for tunable slow-light elay buffer, IEEE J. Quantum Electron. 44, (2008). 7. D. Culverhouse, F. Frahi, C. N. Pannell, an D. A. Jackson, Potential of stimulate Brillouin scattering as sensing mechanism for istribute temperature sensor, Electron. Lett. 25, (989). 8. A. Kobyakov, M. Sauer, an D. Chowhury, Stimulate Brillouin scattering in optical fibers, Av. Opt. Photon. 2, 59 (200). 9. R. W. Boy, Nonlinear Optics, 3r e. (Acaemic, 2007), Chap L. Chen an X. Bao, Analytical an numerical solutions for steay state stimulate Brillouin scattering in a single-moe fiber, Opt. Commun. 52, (998).. S. Le Floch an P. Cambon, Theoretical evaluation of the Brillouin threshol an the steay-state Brillouin equations in stanar single-moe optical fibers, J. Opt. Soc. Am. A 20, (2003). 2. F. S. Gokhan, Moerate-gain Brillouin amplification: an analytical solution below pump threshol, Opt. Commun. 284, (20). 3. A. Kobyakov, S. Darmanyan, M. Sauer, an D. Chowhury, Highgain Brillouin amplification: an analytical approach, Opt. Lett. 3, (2006). 4. Z. Ou, J. Li, L. Zhang, Z. Dai, an Y. Liu, An approximate analytic solution of the steay state Brillouin scattering in single moe optical fiber without neglecting the attenuation coefficient, Opt. Commun. 282, (2009). 5. D. Williams, X. Bao, an L. Chen, Characterization of high nonlinearity in Brillouin amplification in optical fibers with applications in fiber sensing an photonic logic, Photon. Res. 2, 9 (204). 6. L. Stépien, S. Ranoux, an J. Zemmouri, Origin of spectral hole burning in Brillouin fiber amplifiers an generators, Phys. Rev. A 65, (2002). 7. G. P. Agrawal, Nonlinear Fiber Optics, 3r e. (Acaemic, 20). 8. R. W. Tkach, A. R. Chraplyvy, an R. M. Derosier, Performance of a WDM network base on stimulate Brillouin scattering, IEEE Photon. Technol. Lett., 3 (989).