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Fiber Brillouin amplifier based on orthogonal double pumps
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Fiber Brillouin amplifier based on orthogonal double pumps Kuanlin Mua, Jianming Shangb, Zhengkang Wanga, Lihua Tanga, Song Yub, Yaojun Qiaoa,*
T
a School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, No. 10, Xitucheng Road, Haidian District, Beijing, 100876, China b Institute of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, No. 10, Xitucheng Road, Haidian District, Beijing, 100876, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Stimulated Brillouin scattering Fiber Brillouin amplifier State of polarization Birefringence
We introduce a fiber Brillouin amplifier (FBA) based on the use of orthogonal double pumps to overcome the fiber random birefringence effect which will reduces the gain and introduces the amplified signal power fluctuation in FBA. The signal with arbitrary state of polarization (SOP) can obtain the highest amplification gain without polarization control. By using the proposed scheme, compared with the traditional FBA which based on just single pump, the measured maximum gain increases by 3.74dB and the gain gap of the signal with different SOPs reduces to 2.7dB. Theoretically, the gain difference and fluctuation for signal with arbitrary SOPs can be further diminished to zero by the proposed double orthogonal pumps mechanism.
1. Introduction Stimulated Brillouin scattering (SBS) is one of the most important nonlinear effects in single-mode fiber [1,2]. And SBS plays an important role in various applications, including slow light [3], fiber lasers [4], fiber-optic sensors [5] and fiber amplification [6–9]. In a traditional fiber Billouin amplifier (FBA), in order to amplify the weak signal, another strong pump of an appropriate wavelength is fed in the counter-propagating direction of the fiber. The SBS process mediates power transfer from the pump to the weak signal along the fiber, causing energy attenuation for the pump and amplification for the signal. The pump frequency is higher than the signal frequency. And the interaction is efficient only when the difference between the pump and probe signal is very close to the Brillouin shift vB , which is of the order of 10−11 GHz in silica fiber at room temperature and at telecommunication wavelengths [10,11]. The strength of the SBS interacting between the pump and signal is often quantified in terms of an gain coefficient γ0 . And the gain coefficient depends on the relative state of polarization (SOP) of the pump and signal: maximum for parallel and zero for orthogonal SOP [12]. In order to obtain the maximum gain efficient, polarization controllers must be used to control the input polarizations of the counter-propagating pump and signal [13]. But because of the random birefringence effect [14] exists in the real fiber while the length of fiber which be used as the amplifying medium in the FBA is in the order of kilometers. The initial input SOPs of pump and signal can’t be neither maintained nor completely scrambled. So the strength of the SBS may vary at different positions along the fiber and the overall gain of the FBA will reduce. In addition to simply analyzing the gain magnitude of the FBA, preceding studies have claimed that as the instability of birefringence, the power detected at the output of FBA varies along with the unstable birefringence [15,16]. So it’s a key challenge faced by the FBA to overcome the birefringence problem to achieve the stable highest amplification
⁎
Corresponding author. E-mail address: [email protected] (Y. Qiao).
https://doi.org/10.1016/j.ijleo.2020.164206 Received 23 September 2019; Received in revised form 26 December 2019; Accepted 7 January 2020 0030-4026/ © 2020 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
K. Mu, et al.
gain. In this paper, vector propagation equations for FBA based on orthogonal double pumps, incorporating SBS and random birefringence effect, were given in Stokes space. Then, the commonly used concatenated random wave-plate model was used to simulate the amplification process, based on single pump and orthogonal double pumps respectively, in ordinary fiber with random birefringence property. We also proved the effectiveness of the technique in the experiment, and higher gain with smaller gap than the FBA based on single pump is obtained for signals with different SOP. In a word, the paper shows the orthogonal characteristic guarantee the two SBS interactions between the double pumps and the signal can complement each other at every position of the fiber. Therefore, the double orthogonal pumps mechanism has the advantages of increasing gain, reducing the power fluctuation and omitting the steps of adjusting the relative SOP of the pump and signal in FBA. 2. Theory When considering the special case of continuous wave operation in the long single mode fiber, the SBS usually can be described as nonlinear physical phenomenon interacting among a pump wave, a Stokes wave and an acoustic wave. Vector formulations were used to analyze the polarization properties of SBS amplification [17]. And when considering the random birefringence effect in the optical fibers, Eq. (12) in reference [18] indicates the local birefringence can be regarded as the driving force which will change the orientation of the light SOP. The power and SOP propagation of the pump and signal along the fiber are governed as below:
−
γ dIs (z ) = 0 (1 + sˆp (z )⋅sˆs (z )) Ip (z ) Is (z ) − αIs (z ) dz 2
(1a)
dsˆs (z ) → = β (z ) × sˆs (z ) dz dIp (z ) dz
dsˆp (z ) dz
=−
(1b)
γ0 (1 + sˆp (z )⋅sˆs (z )) Is (z ) Ip (z ) − αIp (z ) 2
(2a)
= β˜ (z ) × sˆp (z )
(2b)
Eqs. (1a) and (2a) represent the propagation of the signal and pump power along the fiber, respectively. Eqs. (1b) and (2b) specify the birefringence-induced evolution of the signal and pump SOP along the fiber in the opposite direction. Here Ip and Is denote the pump and signal power, respectively. sˆs = [s1, s s2, s s3, s ]T and similarly sˆp = [s1, p s2, p s3, p ]T are 3 × 1 normalized Stokes vectors (s1,2 s / p + s2,2 s / p + s3,2 s / p = 1), describing the evolution of the polarizations of the counter-propagating signal and pump. γ0 is the gain → coefficient and α represents the fiber attenuation. The three-dimensional vectors β = [β β β ]T and β˜ = [β β − β ]T describe the 1
2
3
1
2
3
fiber birefringence in Stokes space When we use orthogonal double pumps, just as the schematic diagram shown in Fig. 1, the signal will be amplified by the two pumps simultaneously. Due to the unitary nature of the fiber, the two pumps will keep their relative orthogonal polarization states, even though the states themselves are continuously changing along the fiber [19]. The propagation equations for the signal and two orthogonal pumps should be updated as Eqs. (3)–(5). Eq. (3a) still represents the propagation of the signal power along the fiber, but it includes three parts. The first two parts denote the interacting between the signal and the two orthogonal pumps. The third part represents the signal attenuation along the fiber. Eq. (3b) is used to analyze the evolution of the signal SOP along the fiber. The Eqs (4a) and (4b) and Eqs. (5a) and (5b) represent the power propagation and the evolution of the SOPs along the fiber of the orthogonal double pumps.
−
γ Ip _ ↕ (z ) dIs (z ) = 0 (1 + sˆp _ ↕ (z )⋅sˆs (z )) Is (z ) dz 2 γ0 Ip _ ↔ (z ) (1 + sˆp _ ↔(z )⋅sˆs (z )) Is (z ) 2 −αIs (z )
(3a)
dsˆs (z ) → = β (z ) × sˆs (z ) dz dIp _ ↕ (z ) dz
=−
(3b)
γ0 (1 + sˆp _ ↕ (z )⋅sˆs (z )) Is (z ) Ip _ ↕ (z ) − αIp _ ↕ (z ) 2
(4a)
Fig. 1. The schematic diagram of FBA based on orthogonal double pumps. The double orthogonal pumps and signal input from different ends of the → fiber, and βi denotes variable birefringence. 2
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
K. Mu, et al.
dsˆp _ ↕ (z ) dz dIp _ ↔ (z ) dz
dsˆp _ ↔(z ) dz
= β˜ (z ) × sˆp _ ↕ (z ) =−
(4b)
γ0 (1 + sˆp _ ↔(z )⋅sˆs (z )) Is (z ) Ip _ ↔ (z ) − αIp _ ↔ (z ) 2
(5a)
= β˜ (z ) × sˆp _ ↔(z )
(5b)
The double pumps propagate in the + z direction, whereas the signal travels in the –z direction. In the FBA based on orthogonal double pumps, the normalized Stokes vectors describing the SOPs of orthogonal double pumps can be given by sˆp _ ↕ (z ) = [s1, p s2, p s3, p ]T and sˆp _ ↔(z ) = [−s1, p − s2, p − s3, p ]T . So the polarization dependence of the two SBS gain efficiencies caused by the double pumps can be described as:
γ↕ =
γ0 (1 + s1, p s1, s + s2, p s2, s + s3, p s3, s ) 2
(6)
γ γ↔ = 0 (1 − s1, p s1, s − s2, p s2, s − s3, p s3, s ) 2
(7)
Therefore, by simply adding γ↕ and γ↔ , it is found that the two part interactions will be complementary: γ↕ + γ↔ = γ0 . So we believe, theoretically, no matter what input SOP of the signal is, the FBA based on orthogonal double pumps can achieve fixed and high gain. The same high gain can also be achieved in FBA just based on one pump only when the SOPs of the pump and signal are always aligned along the fiber. But limited by the birefringence effect of the long single mode fiber, this ideal state is impossible to maintain to achieve the maximum amplification efficiency. 3. Simulation The gain of FBA based on single pump and orthogonal double pumps are numerically examined and shown in Figs. 2 and 3, using Eqs. (1) and (2) and Eqs. (3)–(5), respectively. Each point on the Poincare sphere represents an input SOP of the signal, and the variable colors correspond to different gains. Simulations are based on the commonly used concatenated random wave-plate model, → with the three components of β (z ) and β˜ (z ) be determined utilizing the random modulus model (RMM), which has been experi→ mentally verified [20]. The average beat length LB = 2π 〈 |β | 〉 equals 14.75 m. Results below are obtained for a fiber length of L = 5km comprising 8000 plates. The initial SOP of the single pump is chosen as [0 1 0]T , while the input Stokes vectors of the two orthogonal pumps are set as [0 1 0]T and [0 − 1 0]T , respectively. Then we can simulate the SOPs of the pumps in each wave→ plate by formula β˜ (z ) × sˆp (z ) . Similarly, the variation of the signal SOP can be obtained by β (z ) × sˆs (z ) when a particular Stokes vector sˆs (L) is selected. With the gain efficiency (γ0 2)(1 + sˆp⋅sˆs ) be determined, the power of amplified signal with different input SOP can be iterated by shooting method. Here, the SBS gain coefficient γ0 = 0.086[W ⋅m]−1 and the fiber attenuation coefficient α is set as 0.2dB km . The input signal power is Is (L) = 0.01mW and all the input pumps power are set as equal, Ip (0) = 30mW , Ip _ ↕ (0) = 30mW , Ip _ ↔ (0) = 30mW . Fig. 2 shows the simulation result of FBA gains based on just one pump as a function of different input SOP of signal. We see the maximum gain reaches 27.7 dB when the minimum gain is just 14.1 dB. The gain disparity is as high as 13.6 dB when every kind of signal SOPs are examined. From Fig. 3, when two orthogonal pumps are used, we can see the gain difference is reduced to 2.3 dB as we calculate out the maximum gain of 33.1 dB while the minimum gain is 30.8 dB. By comparison the results of Figs. 2 and 3, the orthogonal double pumps method can not only overcome the influence of the random birefringence effect of the fibers to increase the gain of the FBA, but also maintain the high gain for signal with arbitrarily SOPs.
Fig. 2. The FBA gain based on single pump. 3
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
K. Mu, et al.
Fig. 3. The FBA gain based on orthogonal double pumps.
More simulations have been performed to determine the average gain 〈G〉 = 〈Is (0) 〉 Is (L) and the relative gain standard deviation (STD) σs = 〈Is2 (0) 〉 〈Is (0) 〉2 − 1 , valuing the quantity of signal fluctuation. Here, we chose a specific input signal Stokes vector sˆp (L) = [− 2 2 0 − 2 2]T , So the SOP of single pump which parallel or orthogonal to that of the signal is [− 2 2 0 − 2 2]T 2 2]T , respectively. And the input Stokes vectors of the two orthogonal pumps are still set as [0 1 0]T and and [ 2 2 0 [0 − 1 0]T . In Fig. 4, the average gains of FBA with varied beat length are provided. The blue and pink line show the average gains based on conventional single pump while the input relative SOPs of the signal and pump are orthogonal and parallel, respectively. And the red line represents the average gain based on the improved orthogonal double pumps. The results suggest that as the random birefringence lead to the randomness of polarization, the average gains of FBA which based on just one pump are closely related to the initial SOPs. When the beat length is small, the random birefringence results in the loss of original orthogonality and parallelism. Consequently, there almost is no discrepancy between the blue and pink line. As the beat length becomes larger, namely the fiber can be treated as isotropic, the parallel SOPs result in maximum average gain while the orthogonal ones contribute to minimum value. When the orthogonal double pumps structure is used, the average gain can be basically unchanged regardless of the beat length, as shown in the red line in Fig. 4. Fig. 5 shows the standard deviation of signal fluctuation at the output of FBA as function of the beat length for single pump and orthogonal double pumps. It suggests, compare with the single pump, that the signal fluctuation of the FBA which based on orthogonal double pumps is really small. And we believe the tiny fluctuation represents by the red line is due to the calculation error, so the gain instability caused by random birefringence can be completely eliminated by the orthogonal double pumps structure, theoretically. 4. Experiment The experimental setup shown in Fig. 6 was assembled in order to demonstrate the effect of FBA based on orthogonal double pumps. In the left side, pump emitted from a tunable laser source was split by a 50:50 coupler and both the lower and upper pump lights were amplified by Erbium-doped fiber amplifier (EDFA). Before integrated together, the SOPs of the two pumps were modulated to be orthogonal to each other by two polarization controllers (PCs). Then the orthogonal double pumps were directed into the fiber via a circulator. The length of the fiber under test was 5 km, and its Brillouin frequency shift was about 11 GHz. In the
Fig. 4. Average gain as function of the beat length. 4
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
K. Mu, et al.
Fig. 5. Signal fluctuation as function of the beat length.
Fig. 6. Experimental setup for FBA based on orthogonal double pumps.
right side, the signal was controlled by a variable optical attenuator (VOA) and a PC to adjust the attenuation and SOP, respectively. For the lack of polarization analyzer, the probe signal was also split by a 50:50 coupler before amplified in the fiber, and a polarization beam splitter (PBS) was used to infer the adjusted SOP angle of the signal. Following the port 3 of the circulator, another PC and PBS was combined to filter the amplified signal from the amplified spontaneous Brillouin scattering (ABS) noise. Fig. 7 shows the gain of FBA as function of the adjusted SOP angle of the input signal. The downward-triangle dashed line shows the gain based on conventional single pump while the upward-triangle dashed line represents the gain based on improved orthogonal double pumps. The power of the orthogonal double pumps and the power of the single pump were both modulated to 30 mW. We can see that the net gain gap of the FBA based on the orthogonal double pumps reduces to 2.7 dB while the gain difference based on single pump still reaches 9 dB, and the maximum gain has increased 3.74 dB. As obtained in the simulations, the experimental results show that the scheme based on orthogonal double pumps does reduce the gain gap of signals with different input SOPs. In addition, the gain based on the orthogonal double pumps is higher than the traditional one single pump. The small gain difference in the simulation for the orthogonal double pumps scheme should be caused by the approximate value
Fig. 7. The measurement FBA gain based on single pump and orthogonal double pumps. 5
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
K. Mu, et al.
for different SOPs in the iterative process. The gain gap in the experimental can be blamed to the incomplete orthogonal of the double pumps and fail to adjust their powers to fully consistency. So the effect of the scheme can be improved by optimizing the system parameters. As we know, the gain spectrum of SBS is Lorentz-shaped and its bandwidth is narrow (only about 10 MHz). Therefore, in the experiment, the gain coefficient of FBA can’t be maintained at a fixed value due to the jitter of the central wavelength of the laser source that we used. So we failed to verify the improvement of signal fluctuation by the orthogonal double pumps structure because of the serious influence of the light central wavelength shift on the gain stability. 5. Conclusion In summary, a new FBA based on orthogonal double pumps has been put forward in this paper. The improved structure can not only omit the steps of adjusting the relative SOP of the pump and signal, but also prevent the gain reduction and signal fluctuation which caused by the random birefringence effect of the fiber in traditional single pump FBA. And the validity of the structure is proved by theory, simulation and experiment. Funding This work was supported by the National Natural Science Foundation of China (NSFC) [61531003, 61690195, 61701040, 61427813]; Youth Research and Innovation Program of BUPT [2017RC13]; Open Funds of IPOC [IPOC2017ZT14]. References [1] A. Kobyakov, M. Sauer, D. Chowdhury, Stimulated Brillouin scattering in optical fibers, Adv. Opt. Photonics 2 (2010) 1–59. [2] W. Chen, Z. Meng, H.J. Zhou, Phono lifetime measurement by stimulated Brillouin scattering slow light technique in optical fiber, Chin. Phys. Lett. 30 (2013) 074209-1–074209-4. [3] A. Ghosh, D. Venkitesh, R. Vijaya, Study of Brillouin amplifier characteristics toward optimized conditions for slow light generation, Appl. Opt. 48 (2009) G48–G52. [4] G.S. Qin, H. Sotobayashi, M. Tsuchiya, A. Mori, T. Suzuki, Y. Ohishi, Stimulated Brillouin scattering in a single-mode tellurite fiber for amplification, lasing, and slow light generation, J. Lightwave Technol. 26 (2008) 492–498. [5] Z.H. Ou, J.F. Li, L.X. Zhang, Z.Y. Dai, Y.Z. Liu, An approximate analytic solution of the steady state Brillouin scattering in single mode optical fiber without neglecting the attenuation coefficient, J. Lightwave Technol. 282 (2009) 3812–3816. [6] L.W. Sheng, D.X. Ba, Z.W. Lu, Low-noise and high-gain of stimulated Brillouin amplification via orbital angular momentum mode division filtering, Appl. Opt. 58 (2019) 147–151. [7] H. Yuan, Y.L. Wang, Z.W. Lü, Z.X. Zheng, Small-scale self-focusing of 200 ps laser pulses in Brillouin amplification, Chin. Phys. B 24 (2015) 094210-1–094210-4. [8] O. Terra, G. Grosche, H. Schnatz, Brillouin amplification in phase coherent transfer of optical frequencies over 480 km fiber, Opt. Express 18 (2010) 16102–16111. [9] S.M.F. Raupach, A. Koczwara, G. Grosche, Optical frequency transfer via a 660 km underground fiber link using a remote Brillouin amplifier, Opt. Express 22 (2014) 26537–26547. [10] L. Chen, X. Bao, Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber, Opt. Commun. 152 (1998) 65–70. [11] M. Niklès, L. Thévenaz, P.A. Robert, Brillouin gain spectrum characterization in single-mode optical fibers, J. Lightwave Technol. 15 (1997) 1842–1851. [12] M. Oskar van Deventer, André J. Boot, Polarization properties of stimulated Brillouin scattering in single-mode fibers, J. Lightwave Technol. 12 (1994) 585–590. [13] O. Shlomovits, M. Tur, Vector analysis of depleted stimulated Brillouin scattering amplification in standard single-mode fibers with nonzero birefringence, Opt. Lett. 38 (2013) 836–838. [14] A. Galtarossa, L. Palmieri, M. Schiano, T. Tambosso, Measurement of birefringence correlation length in long, single-mode fibers, Opt. Lett. 26 (2001) 962–964. [15] A. Galtarossa, L. Palmieri, M. Santagiustina, L. Schenato, L. Ursini, Polarized Brillouin amplification in randomly birefringent and unidirectionally spun fibers, IEEE Photonics Technol. Lett. 20 (2008) 1420–1422. [16] S. Cao, M. Zhang, Polarized and Birefringence-dependent stimulated Brillouin scattering in single mode fiber, Optik 131 (2017) 374–382. [17] K.L. Mu, J.M. Shang, L.H. Tang, Z.K. Wang, S. Yu, Y.J. Qiao, Polarization dependence of gain and amplified spontaneous Brillouin scattering noise analysis for fiber Brillouin amplifier, Chin. Phys. B 28 (2019) 094216-1–094216-5. [18] P.K.A. Wai, C.R. Menyuk, Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence, J. Lightwave Technol. 14 (1996) 148–157. [19] J. Urricelqui, F. López-Fernandino, M. Sagues, A. Loayssa, Polarization diversity scheme for BOTDA sensors based on a double orthogonal pump interaction, J. Lightwave Technol. 33 (2015) 2633–1637.. [20] A. Galtarossa, L. Palmieri, M. Schiano, T. Tambosso, Statistical characterization of fiber random birefringence, Opt. Lett. 25 (2000) 1322–1324.
6
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Fiber Brillouin amplifier based on orthogonal double pumps Kuanlin Mua, Jianming Shangb, Zhengkang Wanga, Lihua Tanga, Song Yub, Yaojun Qiaoa,*
T
a School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, No. 10, Xitucheng Road, Haidian District, Beijing, 100876, China b Institute of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, No. 10, Xitucheng Road, Haidian District, Beijing, 100876, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Stimulated Brillouin scattering Fiber Brillouin amplifier State of polarization Birefringence
We introduce a fiber Brillouin amplifier (FBA) based on the use of orthogonal double pumps to overcome the fiber random birefringence effect which will reduces the gain and introduces the amplified signal power fluctuation in FBA. The signal with arbitrary state of polarization (SOP) can obtain the highest amplification gain without polarization control. By using the proposed scheme, compared with the traditional FBA which based on just single pump, the measured maximum gain increases by 3.74dB and the gain gap of the signal with different SOPs reduces to 2.7dB. Theoretically, the gain difference and fluctuation for signal with arbitrary SOPs can be further diminished to zero by the proposed double orthogonal pumps mechanism.
1. Introduction Stimulated Brillouin scattering (SBS) is one of the most important nonlinear effects in single-mode fiber [1,2]. And SBS plays an important role in various applications, including slow light [3], fiber lasers [4], fiber-optic sensors [5] and fiber amplification [6–9]. In a traditional fiber Billouin amplifier (FBA), in order to amplify the weak signal, another strong pump of an appropriate wavelength is fed in the counter-propagating direction of the fiber. The SBS process mediates power transfer from the pump to the weak signal along the fiber, causing energy attenuation for the pump and amplification for the signal. The pump frequency is higher than the signal frequency. And the interaction is efficient only when the difference between the pump and probe signal is very close to the Brillouin shift vB , which is of the order of 10−11 GHz in silica fiber at room temperature and at telecommunication wavelengths [10,11]. The strength of the SBS interacting between the pump and signal is often quantified in terms of an gain coefficient γ0 . And the gain coefficient depends on the relative state of polarization (SOP) of the pump and signal: maximum for parallel and zero for orthogonal SOP [12]. In order to obtain the maximum gain efficient, polarization controllers must be used to control the input polarizations of the counter-propagating pump and signal [13]. But because of the random birefringence effect [14] exists in the real fiber while the length of fiber which be used as the amplifying medium in the FBA is in the order of kilometers. The initial input SOPs of pump and signal can’t be neither maintained nor completely scrambled. So the strength of the SBS may vary at different positions along the fiber and the overall gain of the FBA will reduce. In addition to simply analyzing the gain magnitude of the FBA, preceding studies have claimed that as the instability of birefringence, the power detected at the output of FBA varies along with the unstable birefringence [15,16]. So it’s a key challenge faced by the FBA to overcome the birefringence problem to achieve the stable highest amplification
⁎
Corresponding author. E-mail address: [email protected] (Y. Qiao).
https://doi.org/10.1016/j.ijleo.2020.164206 Received 23 September 2019; Received in revised form 26 December 2019; Accepted 7 January 2020 0030-4026/ © 2020 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
K. Mu, et al.
gain. In this paper, vector propagation equations for FBA based on orthogonal double pumps, incorporating SBS and random birefringence effect, were given in Stokes space. Then, the commonly used concatenated random wave-plate model was used to simulate the amplification process, based on single pump and orthogonal double pumps respectively, in ordinary fiber with random birefringence property. We also proved the effectiveness of the technique in the experiment, and higher gain with smaller gap than the FBA based on single pump is obtained for signals with different SOP. In a word, the paper shows the orthogonal characteristic guarantee the two SBS interactions between the double pumps and the signal can complement each other at every position of the fiber. Therefore, the double orthogonal pumps mechanism has the advantages of increasing gain, reducing the power fluctuation and omitting the steps of adjusting the relative SOP of the pump and signal in FBA. 2. Theory When considering the special case of continuous wave operation in the long single mode fiber, the SBS usually can be described as nonlinear physical phenomenon interacting among a pump wave, a Stokes wave and an acoustic wave. Vector formulations were used to analyze the polarization properties of SBS amplification [17]. And when considering the random birefringence effect in the optical fibers, Eq. (12) in reference [18] indicates the local birefringence can be regarded as the driving force which will change the orientation of the light SOP. The power and SOP propagation of the pump and signal along the fiber are governed as below:
−
γ dIs (z ) = 0 (1 + sˆp (z )⋅sˆs (z )) Ip (z ) Is (z ) − αIs (z ) dz 2
(1a)
dsˆs (z ) → = β (z ) × sˆs (z ) dz dIp (z ) dz
dsˆp (z ) dz
=−
(1b)
γ0 (1 + sˆp (z )⋅sˆs (z )) Is (z ) Ip (z ) − αIp (z ) 2
(2a)
= β˜ (z ) × sˆp (z )
(2b)
Eqs. (1a) and (2a) represent the propagation of the signal and pump power along the fiber, respectively. Eqs. (1b) and (2b) specify the birefringence-induced evolution of the signal and pump SOP along the fiber in the opposite direction. Here Ip and Is denote the pump and signal power, respectively. sˆs = [s1, s s2, s s3, s ]T and similarly sˆp = [s1, p s2, p s3, p ]T are 3 × 1 normalized Stokes vectors (s1,2 s / p + s2,2 s / p + s3,2 s / p = 1), describing the evolution of the polarizations of the counter-propagating signal and pump. γ0 is the gain → coefficient and α represents the fiber attenuation. The three-dimensional vectors β = [β β β ]T and β˜ = [β β − β ]T describe the 1
2
3
1
2
3
fiber birefringence in Stokes space When we use orthogonal double pumps, just as the schematic diagram shown in Fig. 1, the signal will be amplified by the two pumps simultaneously. Due to the unitary nature of the fiber, the two pumps will keep their relative orthogonal polarization states, even though the states themselves are continuously changing along the fiber [19]. The propagation equations for the signal and two orthogonal pumps should be updated as Eqs. (3)–(5). Eq. (3a) still represents the propagation of the signal power along the fiber, but it includes three parts. The first two parts denote the interacting between the signal and the two orthogonal pumps. The third part represents the signal attenuation along the fiber. Eq. (3b) is used to analyze the evolution of the signal SOP along the fiber. The Eqs (4a) and (4b) and Eqs. (5a) and (5b) represent the power propagation and the evolution of the SOPs along the fiber of the orthogonal double pumps.
−
γ Ip _ ↕ (z ) dIs (z ) = 0 (1 + sˆp _ ↕ (z )⋅sˆs (z )) Is (z ) dz 2 γ0 Ip _ ↔ (z ) (1 + sˆp _ ↔(z )⋅sˆs (z )) Is (z ) 2 −αIs (z )
(3a)
dsˆs (z ) → = β (z ) × sˆs (z ) dz dIp _ ↕ (z ) dz
=−
(3b)
γ0 (1 + sˆp _ ↕ (z )⋅sˆs (z )) Is (z ) Ip _ ↕ (z ) − αIp _ ↕ (z ) 2
(4a)
Fig. 1. The schematic diagram of FBA based on orthogonal double pumps. The double orthogonal pumps and signal input from different ends of the → fiber, and βi denotes variable birefringence. 2
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
K. Mu, et al.
dsˆp _ ↕ (z ) dz dIp _ ↔ (z ) dz
dsˆp _ ↔(z ) dz
= β˜ (z ) × sˆp _ ↕ (z ) =−
(4b)
γ0 (1 + sˆp _ ↔(z )⋅sˆs (z )) Is (z ) Ip _ ↔ (z ) − αIp _ ↔ (z ) 2
(5a)
= β˜ (z ) × sˆp _ ↔(z )
(5b)
The double pumps propagate in the + z direction, whereas the signal travels in the –z direction. In the FBA based on orthogonal double pumps, the normalized Stokes vectors describing the SOPs of orthogonal double pumps can be given by sˆp _ ↕ (z ) = [s1, p s2, p s3, p ]T and sˆp _ ↔(z ) = [−s1, p − s2, p − s3, p ]T . So the polarization dependence of the two SBS gain efficiencies caused by the double pumps can be described as:
γ↕ =
γ0 (1 + s1, p s1, s + s2, p s2, s + s3, p s3, s ) 2
(6)
γ γ↔ = 0 (1 − s1, p s1, s − s2, p s2, s − s3, p s3, s ) 2
(7)
Therefore, by simply adding γ↕ and γ↔ , it is found that the two part interactions will be complementary: γ↕ + γ↔ = γ0 . So we believe, theoretically, no matter what input SOP of the signal is, the FBA based on orthogonal double pumps can achieve fixed and high gain. The same high gain can also be achieved in FBA just based on one pump only when the SOPs of the pump and signal are always aligned along the fiber. But limited by the birefringence effect of the long single mode fiber, this ideal state is impossible to maintain to achieve the maximum amplification efficiency. 3. Simulation The gain of FBA based on single pump and orthogonal double pumps are numerically examined and shown in Figs. 2 and 3, using Eqs. (1) and (2) and Eqs. (3)–(5), respectively. Each point on the Poincare sphere represents an input SOP of the signal, and the variable colors correspond to different gains. Simulations are based on the commonly used concatenated random wave-plate model, → with the three components of β (z ) and β˜ (z ) be determined utilizing the random modulus model (RMM), which has been experi→ mentally verified [20]. The average beat length LB = 2π 〈 |β | 〉 equals 14.75 m. Results below are obtained for a fiber length of L = 5km comprising 8000 plates. The initial SOP of the single pump is chosen as [0 1 0]T , while the input Stokes vectors of the two orthogonal pumps are set as [0 1 0]T and [0 − 1 0]T , respectively. Then we can simulate the SOPs of the pumps in each wave→ plate by formula β˜ (z ) × sˆp (z ) . Similarly, the variation of the signal SOP can be obtained by β (z ) × sˆs (z ) when a particular Stokes vector sˆs (L) is selected. With the gain efficiency (γ0 2)(1 + sˆp⋅sˆs ) be determined, the power of amplified signal with different input SOP can be iterated by shooting method. Here, the SBS gain coefficient γ0 = 0.086[W ⋅m]−1 and the fiber attenuation coefficient α is set as 0.2dB km . The input signal power is Is (L) = 0.01mW and all the input pumps power are set as equal, Ip (0) = 30mW , Ip _ ↕ (0) = 30mW , Ip _ ↔ (0) = 30mW . Fig. 2 shows the simulation result of FBA gains based on just one pump as a function of different input SOP of signal. We see the maximum gain reaches 27.7 dB when the minimum gain is just 14.1 dB. The gain disparity is as high as 13.6 dB when every kind of signal SOPs are examined. From Fig. 3, when two orthogonal pumps are used, we can see the gain difference is reduced to 2.3 dB as we calculate out the maximum gain of 33.1 dB while the minimum gain is 30.8 dB. By comparison the results of Figs. 2 and 3, the orthogonal double pumps method can not only overcome the influence of the random birefringence effect of the fibers to increase the gain of the FBA, but also maintain the high gain for signal with arbitrarily SOPs.
Fig. 2. The FBA gain based on single pump. 3
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Fig. 3. The FBA gain based on orthogonal double pumps.
More simulations have been performed to determine the average gain 〈G〉 = 〈Is (0) 〉 Is (L) and the relative gain standard deviation (STD) σs = 〈Is2 (0) 〉 〈Is (0) 〉2 − 1 , valuing the quantity of signal fluctuation. Here, we chose a specific input signal Stokes vector sˆp (L) = [− 2 2 0 − 2 2]T , So the SOP of single pump which parallel or orthogonal to that of the signal is [− 2 2 0 − 2 2]T 2 2]T , respectively. And the input Stokes vectors of the two orthogonal pumps are still set as [0 1 0]T and and [ 2 2 0 [0 − 1 0]T . In Fig. 4, the average gains of FBA with varied beat length are provided. The blue and pink line show the average gains based on conventional single pump while the input relative SOPs of the signal and pump are orthogonal and parallel, respectively. And the red line represents the average gain based on the improved orthogonal double pumps. The results suggest that as the random birefringence lead to the randomness of polarization, the average gains of FBA which based on just one pump are closely related to the initial SOPs. When the beat length is small, the random birefringence results in the loss of original orthogonality and parallelism. Consequently, there almost is no discrepancy between the blue and pink line. As the beat length becomes larger, namely the fiber can be treated as isotropic, the parallel SOPs result in maximum average gain while the orthogonal ones contribute to minimum value. When the orthogonal double pumps structure is used, the average gain can be basically unchanged regardless of the beat length, as shown in the red line in Fig. 4. Fig. 5 shows the standard deviation of signal fluctuation at the output of FBA as function of the beat length for single pump and orthogonal double pumps. It suggests, compare with the single pump, that the signal fluctuation of the FBA which based on orthogonal double pumps is really small. And we believe the tiny fluctuation represents by the red line is due to the calculation error, so the gain instability caused by random birefringence can be completely eliminated by the orthogonal double pumps structure, theoretically. 4. Experiment The experimental setup shown in Fig. 6 was assembled in order to demonstrate the effect of FBA based on orthogonal double pumps. In the left side, pump emitted from a tunable laser source was split by a 50:50 coupler and both the lower and upper pump lights were amplified by Erbium-doped fiber amplifier (EDFA). Before integrated together, the SOPs of the two pumps were modulated to be orthogonal to each other by two polarization controllers (PCs). Then the orthogonal double pumps were directed into the fiber via a circulator. The length of the fiber under test was 5 km, and its Brillouin frequency shift was about 11 GHz. In the
Fig. 4. Average gain as function of the beat length. 4
Optik - International Journal for Light and Electron Optics 204 (2020) 164206
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Fig. 5. Signal fluctuation as function of the beat length.
Fig. 6. Experimental setup for FBA based on orthogonal double pumps.
right side, the signal was controlled by a variable optical attenuator (VOA) and a PC to adjust the attenuation and SOP, respectively. For the lack of polarization analyzer, the probe signal was also split by a 50:50 coupler before amplified in the fiber, and a polarization beam splitter (PBS) was used to infer the adjusted SOP angle of the signal. Following the port 3 of the circulator, another PC and PBS was combined to filter the amplified signal from the amplified spontaneous Brillouin scattering (ABS) noise. Fig. 7 shows the gain of FBA as function of the adjusted SOP angle of the input signal. The downward-triangle dashed line shows the gain based on conventional single pump while the upward-triangle dashed line represents the gain based on improved orthogonal double pumps. The power of the orthogonal double pumps and the power of the single pump were both modulated to 30 mW. We can see that the net gain gap of the FBA based on the orthogonal double pumps reduces to 2.7 dB while the gain difference based on single pump still reaches 9 dB, and the maximum gain has increased 3.74 dB. As obtained in the simulations, the experimental results show that the scheme based on orthogonal double pumps does reduce the gain gap of signals with different input SOPs. In addition, the gain based on the orthogonal double pumps is higher than the traditional one single pump. The small gain difference in the simulation for the orthogonal double pumps scheme should be caused by the approximate value
Fig. 7. The measurement FBA gain based on single pump and orthogonal double pumps. 5
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for different SOPs in the iterative process. The gain gap in the experimental can be blamed to the incomplete orthogonal of the double pumps and fail to adjust their powers to fully consistency. So the effect of the scheme can be improved by optimizing the system parameters. As we know, the gain spectrum of SBS is Lorentz-shaped and its bandwidth is narrow (only about 10 MHz). Therefore, in the experiment, the gain coefficient of FBA can’t be maintained at a fixed value due to the jitter of the central wavelength of the laser source that we used. So we failed to verify the improvement of signal fluctuation by the orthogonal double pumps structure because of the serious influence of the light central wavelength shift on the gain stability. 5. Conclusion In summary, a new FBA based on orthogonal double pumps has been put forward in this paper. The improved structure can not only omit the steps of adjusting the relative SOP of the pump and signal, but also prevent the gain reduction and signal fluctuation which caused by the random birefringence effect of the fiber in traditional single pump FBA. And the validity of the structure is proved by theory, simulation and experiment. Funding This work was supported by the National Natural Science Foundation of China (NSFC) [61531003, 61690195, 61701040, 61427813]; Youth Research and Innovation Program of BUPT [2017RC13]; Open Funds of IPOC [IPOC2017ZT14]. References [1] A. Kobyakov, M. Sauer, D. Chowdhury, Stimulated Brillouin scattering in optical fibers, Adv. Opt. Photonics 2 (2010) 1–59. [2] W. Chen, Z. Meng, H.J. Zhou, Phono lifetime measurement by stimulated Brillouin scattering slow light technique in optical fiber, Chin. Phys. Lett. 30 (2013) 074209-1–074209-4. [3] A. Ghosh, D. Venkitesh, R. 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