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Abstract
The Brillouin gain spectrum (BGS) provides key information for stimulated Brillouin scattering (SBS), such as the Brillouin frequency shift (BFS), Brillouin spontaneous linewidth, and Brillouin gain coefficient. In this study, we theoretically investigate the field distributions and BGS characterization of Ge-doped, Al-doped, and Al/Ge co-doped fibers. Additionally, we analyzed and compared the relationship between the BGS and acoustic refractive index. In particular, we demonstrate the crucial role played by acoustic modes in anti-waveguide structures. The simulation results show that the Brillouin gain coefficient decreases with a decreasing acoustic index in the fiber core region. Furthermore, we experimentally measure the SBS threshold and BGS of the Al/Ge co-doped fiber to examine the validity of the numerical model. The simulated and experimental results are consistent.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fiber lasers have various applications in the fields of industrial processing, optical communication, medical cosmetology, national defense, and scientific research owing to their high conversion efficiency, high beam quality, compactness, and reliability [1–3]. In particular, high-power narrow linewidth lasers have attracted increasing attention owing to their high coherence and high beam quality [4–7]. Unfortunately, a series of physical factors such as nonlinear optical effects, optical surface damage, and thermal loads can limit the light intensity [8]. In particular, stimulated Brillouin scattering (SBS) is a critical factor. SBS can be divided into two types: one is forward scattering, which degraded the performance of quantum key distribution optical communication [9–11], the other is backscattering, which limits the performance of single-fiber lasers with the increasing requirements of transmission distance and input power for high-power narrow linewidth fiber lasers owing to their low threshold characteristics [12,13]. In the following, SBS refers to the backward Brillouin scattering type. Once the SBS threshold is reached, SBS converts most of the input power to Stokes light, which is not conducive to improving the output power of the system and may even damage the optical components. A high SBS threshold and small Brillouin gain coefficient are necessary for high-power fiber applications. Thus, there is an urgent need to overcome the constraints of SBS. In this context, multichannel laser beam combination technology has become an effective method for overcoming the power and brightness limitations of single-fiber lasers and passive transmission fibers have become key components [14,15]. To the best of our knowledge, the most popular SBS suppression method involves optimizing the radial refractive index profile to change the optical and acoustic field distributions. Generally, the dopants in silica glass can control the refractive index profiles for both optical and acoustic waves (as shown in Table 1) [16,17]. Here, the acoustic refractive index is defined as the ratio of the acoustic velocity in silica to that in the core: N(r) = VSiO2/V(r); thus, the acoustic refractive index is inversely proportional to the acoustic velocity [18].
Table 1. Variation trend of acoustic and optical refractive index of different dopants in silica.
The addition of certain dopants, such as Ge, P, Ti, and Al can increase the optical refractive index of pure silica fiber. However, other dopants, such as F and B can decrease the optical refractive index. Nevertheless, all of the dopants increase the acoustic index except for Al, which decreases it. Therefore, the usage of Al2O3 anti-guides acoustic waves and reduces the spatial overlap between the optical and acoustic fields. Consequently, ramp-like acoustic index profiles and a negative acoustic lens-like structure have been designed [16]. Because Al2O3 is linearly ramped down from the center of the fiber core to the edge of the core, whereas GeO2 is linearly ramped up, the energy of the acoustic modes is localized primarily in the multi-mode region of the pedestal. In 2007 and 2008, strong SBS suppression was achieved using Yb/Al/Ge-doped “anti-guide” acoustic structures and the SBS thresholds reached 6.5 dB and 11.2 dB higher in comparison with those obtained using the GeO2-doped fiber with a step acoustic index [16,19]. An alternative design used the F-doping of the cladding of a fiber to structure an acoustic waveguide, and the optical mode was scattered only by acoustic cladding modes that decreased the acoustic velocity in the cladding instead of the core [20]. In general, because F reduces the optical refractive index of silica but increases its acoustic refractive index, the acoustic guiding property should be reversed. Several experimental studies have demonstrated that the F-doped fibers can achieve 3–8 dB of SBS suppression more than that of conventional uniformly doped large mode area fibers [20–23].
Although previous studies have yielded good results, SBS research for many fibers has focused on the Brillouin gain spectrum (BGS) [21,24]. The BGS contains important information, including the Brillouin frequency shift (BFS), Brillouin spontaneous linewidth, and Brillouin gain coefficient. The Brillouin lines in bulk silica were first observed in 1950 [25], and different BGS testing techniques have been extensively studied [26–29]. Nevertheless, there are less studies have reported on the BGS performance of Ge/Al-doped fibers at a wavelength of 1064 nm. As such, this study comprises two main parts. First, we present a numerical study to comprehensively discuss and investigate the BGS characteristics by considering the contributions from the dominant acoustic modes of Al- or Ge-doped fibers. Second, we conduct experiments on the Brillouin threshold and BGS measurements to verify the accuracy of the simulation model. A comparison of the theoretical and experimental results indicated that the actual fiber exhibits a significant improvement in the SBS threshold and Brillouin gain.
2. Simulated results
SBS involves the transfer of optical and acoustic waveguides to Stokes scattered light through electrostriction. This generates a large amount of Stokes scattered light in the laser, and the excited scattered light strengthens the acoustic waveguide. Hence, the interaction produces strong scattering [12]. To understand the fiber parameters affecting SBS, the threshold formula of SBS can be approximated using Eq. (1) as follows [29]:
In the above, Pth is the threshold of SBS. Aeff is the optical effective mode area and can be expressed as shown in Eq. (2), where the angular brackets represent the integration of the fiber cross-section, f(r) denotes the radial profiles of the fundamental optical modes of the fiber, and gB is the peak Brillouin gain for the dominant acoustic mode. Leff is the effective length of the fiber and can be expressed as shown in Eq. (3), where α is a coefficient of the propagation loss, and L is the actual length of the fiber. Evidently, Eq. (1) indicates that increasing the optical effective mode area Aeff, reducing the effective length of the fiber Leff, and reducing the Brillouin gain coefficient gB are the most direct means of improving the SBS threshold. However, on the premise of remaining single-mode operation, bending loss and modal discrimination will occur when the core radius is increased or the numerical aperture (NA) is decreased [30,31]. Hence, reducing the Brillouin gain coefficient gB may be the most effective approach to increasing the SBS threshold. Thus, the BGS can be described by a multi-Lorentzian form using Eq. (4) as follows [32]:
We can numerically simulate the BGS using Eq. (4), where P12 is an elasto-optic coefficient (considered as 0.26 [33]), c is the velocity of light in vacuum, λ is the pump wavelength (considered as 1064 nm), and ρ0 is the material density (considered as 2200 kg/m3 for pure silica).In addition, fB,i is the frequency of the acoustic mode of order i and can be expresses as shown in Eq. (5), where Va is the acoustic velocity within the fiber, n is the effective refractive index of the fiber mode. ΔfB is the BGS line width (Full width at half maximum) [28]. Ii is the normalized overlap integral between the electric and acoustic fields and can be expressed as follows [29,34]:
In the above, ρm(r) denotes the radial profiles of the mth acoustic modes of the fiber. In particular, Aao,m is the acousto-optic effective area and differs from the Aeff expected when the optical refractive index profile retains a step structure. The parameter Aao,m determines the strength of the acousto-optic interaction [12].
We numerically simulated the BGS of three types of fibers, and the simulation model could be employed for any type of fiber. We focused on the Al- or Ge-doped fiber structures acting as anti-waveguides for the acoustic field owing to their higher SBS thresholds [16]. Their contributions to the longitudinal acoustic velocity v and refractive index n of pure silica are listed in Table 2 [35]. Figure 1 illustrates the optical and acoustic refractive index profiles of these fibers: Fiber-1 with a GeO2-doped core, which was chosen as the reference fiber; Fiber-2 with an Al2O3 core, which can guide the optical wave but anti-guides the acoustic wave; and Fiber-3 with both GeO2 and Al2O3 co-existing in the core, such that the inner and outer core regions are doped with Al2O3 and GeO2, respectively. The diameters of the core and cladding of all the fibers were 10 μm and 100 μm, respectively, and the NA of the cores were 0.07. In the calculations, only the dopant content was regulated and the optical index profile maintained a step structure, as shown in Fig. 1(a).
Fig. 1. (a) The optical index profiles and the acoustic index profiles of (b) Fiber-1 (c) Fiber-2 (d) Fiber-3.
Table 2. Fractional variation of longitudinal acoustic velocity v and refractive index n with dopant concentration w% relative to pure silica [35].
Figure 2 shows the field distributions of the fundamental optical field LP01 and acoustic field of Fiber-1. The doping of GeO2 can increase both the acoustic and optical indices, as shown in Fig. 1(a) and (c). Consequently, acoustic and optical waveguides are simultaneously created. At a frequency of 16.1166 GHz, the fundamental optical mode LP01 and lowest order acoustic mode overlap almost perfectly with an overlap integral of 0.9568, i.e., very close to 1. Thus, this fiber can be used as a reference to measure the SBS threshold improvement.
Figure 3 presents the field distributions of the fundamental optical field LP01 and the three lowest-order acoustic modes of Fiber-2. Acoustic phonons play a major role in SBS. The doping of Al2O3 can reduce the interaction between the acoustic and optical fields in the fiber core region. Usually, an optical fiber supports many acoustic modes, which indicates that for the same wave vector of an optical wave, multiple wave vectors of acoustic wave modes of different frequencies will satisfy the SBS phase-match condition for the acoustic and optical propagation constants βm = 2β0 [16,36], where β0 is the propagation constant of the optical field and βm is the propagation constant of the acoustic mode. The leaky acoustic core modes with the greatest overlap with the LP01 optical modes appear at 16.403 GHz, where the overlap integral and acousto-optic effective area are 0.6157 and 181.3391 μm2, respectively. Hence, the SBS threshold improves by 1.92 dB relative to that of Fiber-1 with the same refractive index profile.
Figure 4 shows the field distributions of the fundamental optical mode LP01 and those of the three lowest-order acoustic modes of Fiber-3. The inner and outer cores were doped with Al and Ge, respectively. The core possesses the same optical index profile as the reference Fiber-1; however, the acoustic index profile is no longer a simple step in the core, as shown in Fig. 1(d). The Al2O3 dopant decreases the acoustic index and separates the acoustic and optical fields in the core. The GeO2 dopant increases the acoustic index, creating an acoustic waveguide at the edge of the outer core region. This significantly reduces the overlap between the optical and acoustic fields and increases the acousto-optic effective area Aao. As shown in Fig. 4, leaky acoustic core modes and acoustic ring modes are simulated when the phase-matching condition is fulfilled. The optical fundamental mode is fully confined in the inner core, whereas the lowest order of the acoustic ring modes is localized in the outer core. The highest overlap integral exhibits a value of 0.3597 at 16.1468 GHz. In comparison with that of Fiber-1, the SBS threshold increases by 4.25 dB and the acousto-optic effective area reaches 310.3711 μm2. Table 3 lists the values of the acousto-optic effective area and overlap integral for these fibers at different resonance frequencies.
Table 3. The simulated results of three types of fiber.
The simulated BGS of Fiber-1, Fiber-2, and Fiber-3 are plotted in Fig. 5. We considered only the acoustic modes with the largest overlap integrals and their contributions to the BGS. In addition, numerous higher-order modes with very small values of the overlap integral are observed owing to the oscillation, but contribute little to the BGS. For example, in Fiber-1 and Fiber-2, these Brillouin gain peaks are dense and can be ignored; therefore, the Brillouin gain spectrum exhibits a single peak structure centered at 16.1166 GHz and 16.403 GHz, respectively. However, the simulated BGS of Fiber-3 exhibits three apparent gain bands owing to the main acoustic modes centered at 16.4178, 16.2426, and 16.1468 GHz. The peak values of the Brillouin gain coefficient for Fiber-1, Fiber-2, and Fiber-3 are 9.0965 × 10−11, 5.3027 × 10−11, and 3.4134 × 10−11, respectively. Notably, the Brillouin gain coefficient can be lowered by doping with Al2O3 because it reduces the acoustic index and anti-guides the acoustic wave. The peak value of the Brillouin gain coefficient is proportional to the overlap integral and inversely proportional to the acousto-optic effective area. It is independent of the optical mode distribution because we only consider the fundamental optical mode, and the acoustic waveguide modes play an important role in SBS suppression.
3. Experimental results
3.1 Characterization of the fibers
Several measurements were performed on the preform to verify our simulation model. First, the refractive index profile of the preform was measured using a preform analyzer (PK26000) as shown in Fig. 6, and the corresponding NA was approximately 0.07. Two symmetric bulges were observed in the center and the actual index profile differed slightly from the theoretical design profile owing to the fabrication preparation of the preform and fiber drawing. Next, the distributions of the preform and fiber were detected using JXA-8230 electron probe microscopy analysis (EPMA). The Si and O ions exhibited homogeneous distributions. The EPMA line scans of Al and Ge in the fiber preform and fiber are shown in Fig. 7, where the scatter plots and solid lines represent the test and fitted data, respectively. The doping concentration in the fiber is almost the same as that in the preform. Because the core is only 10 μm and the accuracy of the testing machine is not enough, so the signal-to-noise ratio is relatively poor. The measured result looks not as smooth as that of the preform, but the basic spectrum type is the same. The optical loss was 2.3 dB/km, which is only slightly higher than that of the conventional single-mode fiber owing to the dopants. Finally, the SBS threshold and BGS of the Al/Ge-doped fibers were measured.
Fig. 6. Radial refractive index profile of the preform. (The inset is the photo of the preform and micrograph of cross section.
3.2 SBS threshold and BGS measurements
The backward SBS threshold was measured using the setup shown in Fig. 8, where the laser power, backward power, and backward spectrum could be monitored simultaneously. The seed, with a wavelength of 1067 nm, was amplified to 10 W by 2 stages pre-amplifier and entered the circulator through port 1. The circulator was used to protect the system from damage by the backward Stokes light generated in the fiber under test (FUT), and the spectrum and power of the backward Stokes light were obtained by port 3 of the circulator. The output light from port 2 of the circulator was connected to a master oscillator power amplifier through a mode field adapter. The amplified light entered the FUT, and the output power was monitored using power meter 2. Simultaneously, the Brillouin signal generated in the FUT was transmitted backwards into port 2 and out of port 3 of the circulator. The optical spectrum and average power of the backward light were monitored continuously during the experiments to observe the onset of SBS.
The power and spectral characteristics of the backward light are plotted in Fig. 9. Figure 9(a) shows that a sudden increase in the backward power occurred at approximately 2.5 W, indicating that the SBS threshold had reached, whereas the backward light spectrum was detected by the spectrometer, as shown in Fig. 9(b). When the output power was 2.5 W, the Stokes peak was almost equal in magnitude to the Rayleigh scattered peak. This indicated that the relative level of Brillouin-scattered light had significantly increased and approached the SBS threshold. The SBS threshold of the 100-m G652D fiber was measured at 0.292 W using the same method. Evidently, the threshold of the Al/Ge co-doped fiber was higher than that of the conventional single-mode fiber without considering the optical loss. In contrast, the SBS threshold of the Al/Ge co-doped fiber was 9.44 dB higher than that of the commercial G652D fiber. Moreover, the Brillouin gain coefficient of the fiber was calculated according to Eq. (1) as 1.041 × 10−11 m/W.
Fig. 9. (a) Backward propagating power measured as a function of forward output power. (b) Spectra of backward propagating light for output powers.
A schematic of the BGS measurement is shown in Fig. 10. The seed source was a commercial single-frequency fiber laser with a linewidth of < 10 KHz and center wavelength of approximately 1067 nm. The output power from this source was 50 mW and was divided into two arms by the 80:20 coupler 1, where approximately 9 mW of light acted as the reference light for the next 80:20 coupler 3. The power was approximately 1.7 mW at the coupler 3 output. In the other arm, approximately 38 mW of light entered a Yb-doped fiber amplifier as SBS pumped light through a circulator and then connected to the input end of the FUT. The generated backward Stokes light in the fiber was coupled to the 50:50 coupler 2 through port 3 of the circulator. The Stokes light power was reduced to 1.7 mW by adjusting the fiber attenuator, and the beat signals of the two light waves were received and converted into electrical signals by a photodetector connected to the signal analyzer. The FUT was wound around a disk with a radius of 10 cm and cleaved with 8° to prevent feedback.
The BGS of the Al/Ge co-doped fiber was measured using the optical beat frequency, as shown in Fig. 11. The BGS is a double-peaked structure because the peak structure of the BGS is determined primarily by the characteristics of the acoustic field and is independent of the optical field mode. Two main acoustic modes are involved in the Brillouin scattering effect. When the wave vector matches with the optical wave vector, these two acoustic wave modes have different characteristic frequencies. Therefore, when the light was modulated by these two acoustic modes, the two Stokes lights generated by the scattering have different frequencies, and the gain spectrum exhibited two peaks. The intensities of the two peaks were unequal, and the gap between them was 43 MHz. The BFS and Brillouin bandwidth of the main peak were 16.22 GHz and 17.72 MHz, respectively.
Considering the errors in the preform production process, we imported the real concentration data for Al and Ge in the actual fiber core into the COMSOL simulation software and re-simulated them. The normalized optical intensity and acoustic density fluctuation modes are shown in Fig. 12 (a). Figure 12(b) shows the simulated (linear) and measured (dotted) BGS of the Al/Ge co-doped fiber. The BGS exhibited two relatively high peaks at approximately 16.1712GHz and 16.2214 GHz, which differed by 0.0058 GHz and 0.0014 GHz, respectively, from the measured frequencies shown in Fig. 10 and correspond to differences of 0.036% and 0.009%, respectively, which are within the experimental error limits of the measurement. The simulated SBS gain coefficient is 6.4269 × 10−12 m/W. The simulated and measured BGS values are in good agreement, thus suggesting that our numerical study is reliable. The simulation results are presented in Table 4.
Fig. 12. (a) The fundamental optical mode and lowest order longitudinal acoustic modes of the Al/Ge co-doped fiber. (b) The simulated and measured BGS of the Al/Ge co-doped fiber.
Table 4. The simulated results of Al/Ge co-doped fiber.
4. Conclusions
In conclusion, we theoretically analyzed the BGS from three types of fibers with different acoustic refractive profiles. The Brillouin gain coefficient was observed to be proportional to the acoustic velocity and overlap integral and inversely proportional to the acousto-optic effective area. Repeated numerical simulations and experimental measurements confirmed the accuracy of the simulation model. Finally, we experimentally demonstrated that the SBS threshold of an Al/Ge co-doped fiber was 9.4 dB higher than that of a conventional single-mode fiber G.652D and that the Brillouin gain coefficient was 1.041 × 10−11 m/W. Thus, the acoustic waveguide modes play an essential role in SBS suppression, and the simulated and measured BGS values are consistent. These results indicate that fibers with a higher SBS threshold can be applied in the field of high-power lasers after further optimizing the production process.
Funding
National Natural Science Foundation of China (51972317, 61875052); Special project for industrialization of high-tech science and technology between Jilin Province and the Chinese Academy of Sciences (2021SYHZ0029); Natural Science Foundation of Shanghai (No.22ZR1470700).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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