On the Fiber Geometry Dependence of Forward Stimulated Brillouin Scattering in Optical Fiber

Nowadays, forward Brillouin scattering in optical fibers has attracted massive research interests worldwide, owing to its great potential for applications in sensing, filtering, lasing, and all‐optical signal processing, etc. The manipulation of spectrum properties of forward stimulated Brillouin scattering (FSBS) turns out to be particularly important for various application scenarios. However, the manipulation approaches are still very limited by now. Herein, for the first time to the best of one's knowledge, a thorough investigation on the fiber geometry dependence of FSBS effect in optical fibers is presented, where the dependencies of FSBS on fiber core diameter and cladding diameter are investigated, revealing the characteristics of FSBS in terms of linewidth, frequency interval, and nonlinear coefficient, etc. It is indicated in the result that changing the size of the fiber core/cladding might give rise to considerable modification of the FSBS spectrum. The investigation paves the way to engineer FSBS in optical fibers, where acousto‐optic interaction can be manipulated and the FSBS spectrum can be tailored by adjusting the geometric size of the fiber for various applications.


Introduction
In the past decades, Brillouin scattering as an intriguing nonlinear effect involving the acousto-optic interaction has been investigated intensively. [1][2][3][4] It has found applications in sensing, filtering, and lasing, etc., [5][6][7] among which Brillouin backscattering has been used to develop distributed fiber sensors to measure position-resolved temperature and strain. [8,9] Recently, forward stimulated Brillouin scattering (FSBS) in optical fibers has also gained a massive research interest owing to its capability of measuring new parameters for fiber sensors. FSBS is a kind of acousto-optic interaction that are caused by radial acoustic modes (R 0,m ) and torsional-radial acoustic modes (TR 2,m ), both of which are transverse acoustic waves on the cross section of fibers. To date, a variety of investigations on FSBS have been carried out, such as the linewidth dependencies on temperature, [10,11] frequency shift dependencies on temperature and acoustic impedance, [12,13] and the influence of polarization maintaining fiber structure on FSBS spectrum. [14] However, most of the investigations are focused on the impact of the surrounding environment of the fiber or the impact of a special fiber structure on the FSBS effect, and the fiber geometry dependence of FSBS has not been investigated until now.
In this work, we present the investigation on the fiber geometry dependence of FSBS effect in optical fibers, where the dependencies of FSBS on fiber core diameter and cladding diameter were investigated thoroughly, revealing the characteristics of FSBS in terms of linewidth, frequency interval, and nonlinear coefficient. The result shows that a gain improvement of 35.9% can be achieved by varying the diameter of core from 9 to 6 μm of a 125 μm cladding fiber with only a 0.19 MHz increment of linewidth and a 0.2 MHz decrement of frequency interval. Meanwhile, the 80 μm fiber has a frequency interval increment of 27.4 MHz and linewidth increment of 2.44 MHz, while maintaining the same level of FSBS gain, compared with the 125 μm fiber.

Simulation Process
The simulation was performed on step-index fiber in air, based on the optomechanical model in single-mode fiber (SMF). [15] The mth resonant frequencies Ω m of FSBS were given by the solutions to the boundary condition equation corresponding to the free fiber surface. [16] The linewidth Γ m of the mth FSBS resonant frequency that was related to the R 0,m acoustic mode was given by Γ m ¼ Γ m,int þ Γ mir , where Γ m,int was an acoustic modedependent inherent parameter that was related to the fiber cladding inhomogeneity and acoustic dissipation, and might be assumed as Γ m,int ¼ Ω m =1000. [17] In addition, Γ mir was boundary related and independent of the choice of m, which was govern by [15,18] DOI: 10.1002/adpr.202200298 Nowadays, forward Brillouin scattering in optical fibers has attracted massive research interests worldwide, owing to its great potential for applications in sensing, filtering, lasing, and all-optical signal processing, etc. The manipulation of spectrum properties of forward stimulated Brillouin scattering (FSBS) turns out to be particularly important for various application scenarios. However, the manipulation approaches are still very limited by now. Herein, for the first time to the best of one's knowledge, a thorough investigation on the fiber geometry dependence of FSBS effect in optical fibers is presented, where the dependencies of FSBS on fiber core diameter and cladding diameter are investigated, revealing the characteristics of FSBS in terms of linewidth, frequency interval, and nonlinear coefficient, etc. It is indicated in the result that changing the size of the fiber core/cladding might give rise to considerable modification of the FSBS spectrum. The investigation paves the way to engineer FSBS in optical fibers, where acousto-optic interaction can be manipulated and the FSBS spectrum can be tailored by adjusting the geometric size of the fiber for various applications.
where Z f is the mechanical impedance of the fiber and Z is the acoustic impedance of the environment, while t r is the acoustic propagation delay across the fiber diameter. The nonlinear optomechanical coefficient of FSBS gain induced by each acoustic mode could be given by where the maximum value γ ðmÞ 0 is obtained on the resonant frequency of R 0,m acoustic mode, and Ω is the frequency shift of incident light induced by FSBS. The maximum value could be calculated by where n denotes the refractive index, k 0 denotes the wave number of the incident light, ρ 0 denotes the mass density of the fiber, and c refers to the speed of light in the vacuum.
Q ðmÞ ES is the electrostrictive overlap integral, and Q ðmÞ PE is the photoelastic overlap integral, both of which were affected by the distribution of optical wave and acoustic mode. [15] It could be inferred that the nonlinear coefficient and the linewidth of FSBS would differ as the fiber geometric size changes, since Q ðmÞ ES , Q ðmÞ PE , and Γ m were all radial parameters dependent. For the excitation of multiple acoustic modes, the total FSBS gain can be expressed as [17] gðΩÞ ¼ In the simulations, the step index fiber structure was set based on G.652 fiber, assuming that the core and the cladding of the fiber had the same acoustic properties, and the fiber coating was not considered. The material parameters used in the simulations are listed in Table 1, [18,19] unless otherwise stated. Meanwhile, the acoustic impedance for silica was 13.19e6 kg m À2 s À1 . [18] For both the incident and scattered waves, only the fundamental mode (HE 11 ) was considered in the calculation. As for the acoustic wave, only the R 0,m acoustic modes were considered, as the TR 2,m acoustic modes were negligible in the contribution to the nonlinear coefficients compared with R 0,m acoustic modes. [15,20] Therefore, according to Equations (3) and (4), the nonlinear coefficient could be determined with separately calculated optical mode and acoustic wave. [15]

FSBS Measurement
The experimental setup shown in Figure 1 was used to experimentally investigate the dependence of fiber geometric size on R 0,m acoustic mode-induced FSBS spectrum in SMFs. [10] Here, we chose the Sagnac loop-based FSBS measurement approach mainly for its simplicity and convenience for setting up the implementation. The laser source worked at 1550 nm. After passing through an optical isolator (ISO), the incident light with a power of 7.6 dBm was launched into an optical fiber Sagnac loop, which consisted of a 50:50 coupler, a fiber under test (FUT), and a polarization controller (PC). We measured two pairs of SMFs. The pair of 125 μm cladding SMFs used for analyzing the core diameter dependence were about 183 m in length and had different core diameters of 9 and 6 μm (YOFC, China), respectively. In contrast, the pair of 9 μm core SMFs utilized for analyzing the cladding diameter dependence were about 2 km in length and had different cladding diameters of 125 and 80 μm (YOFC, China), respectively. It's worth mentioning that the TR 2,m acoustic mode-induced FSBS needed to be suppressed as much as possible by adjusting the PC, to avoid undesired phase modulation upon the signal. Then, in the Sagnac loop, the R 0,m acoustic mode-induced phase modulation could be converted to intensity modulation upon the injected light wave. [10] The output of the Sagnac loop was connected to a variable optical attenuator (VOA), and the attenuated optical signal was then detected by a 1.6 GHz photodetector (PD). Eventually, an electrical spectrum analyzer (ESA) was used for data acquisition. The frequency resolution bandwidth (RBW) of ESA was set at 10 kHz, and the FSBS spectrums were averaged by 1000 times.

Investigation of the Fiber Core Diameter Dependence on FSBS
First, we present the investigation on the dependence of core diameter on FSBS. Two kinds of available fibers with different core diameters have been employed for characterization, that is, 125 μm cladding fibers with 9 and 6 μm core, respectively.  www.advancedsciencenews.com www.adpr-journal.com Simulation investigation has been carried out, and the simulated normalized FSBS spectrums for the two fibers are shown in Figure 2, where the normalization is carried out by dividing the spectrums with the intensity of the highest peak of the spectrums. It can be found that, first, when the diameter of fiber core decreases, the resonant frequency and linewidth of each peak remain unchanged as the acoustic property difference between the core and the cladding is negligible. [21] Second, the frequency interval between adjacent resonant peaks of both fibers remains the same. But, the intensity of each resonant peak is generally increased, and the intensities of resonance peaks of the high frequencies have more increment than that of the low frequencies.
To investigate the principle of the core diameter dependence on FSBS, the transverse profiles for the optical field and R 0,7 acoustic mode between the 9 and 6 μm core fiber with 125 μm cladding have been compared, as shown in Figure 3, where the R 0,7 acoustic mode is usually the strongest acoustic mode in commercial 125 μm SMF. [18] It can be found from Figure 3a-c that the optical field has different distribution on the cross section of fibers with varied core diameter. While, the material displacement induced by the R 0,7 acoustic mode remains unchanged because the core and cladding have very similar acoustic property, as shown in Figure 3d-f. According to Equation (3), the nonlinear coefficient changes as the overlap  www.advancedsciencenews.com www.adpr-journal.com integrals Q ðmÞ ES and Q ðmÞ PE vary due to the variations in optical and acoustic field with different core diameter. A stronger acoustooptic interaction can be achieved with more positive overlap for the integral between the optical field and acoustic waves, which leads to a higher FSBS gain. This helps improve the signalto-noise ratio (SNR) of sensing schemes, as well as facilitating all-optical-signal-processing applications, where FSBS gain is usually weak and requires low-SNR detection method. [15,22] For further investigation on the core diameter dependence of FSBS spectrum, simulations have been carried out by changing the core diameter from 9 to 6 μm, while keeping a 125 μm cladding diameter and the same refractive index of 1.4505 for convenience. [20] The spectrums and the intensity variations of the FSBS peaks with different core diameters are shown in Figure 4. It can be found from Figure 4a that the nonlinear coefficient varies with the decrement of core diameter, when the refractive index is kept the same. To show the variation of nonlinear coefficient, the intensity of each peak is divided by the sum of the intensity of peaks for normalization, and the normalized intensity variation of resonant peaks, presenting the energy percentage taken up of each peak in each fiber, with respect to the peaks of the 9 μm fiber is shown in Figure 4b, that is, the result of the 9 μm fiber is used as a reference, and the results of other fibers with various core diameters are compared with it for characterization. It can be inferred that, as the core diameter decreases, the intensity percentage of resonant peaks at the high-frequency region will be increased, while the intensity percentage at the low-frequency region will be relatively reduced, where more positive overlap between the optical field and acoustic field for a thinner core will lead to a relatively higher intensity. Therefore, it's realized that the enhancement and the weakening of FSBS effect result from a thinner core diameter may be utilized for some practical use. As a thinner core will lead to the enhancement of high-frequency peaks, it could be useful to customize the core diameter for tailoring the FSBS spectrum depending upon application scenarios.
For better observation and the convenience of analysis, the FSBS peaks induced by the R 0,7 acoustic mode in different fibers with core diameter ranging from 9 to 6 μm are selected for further analyzing, as shown in Figure 5. It can be found from Figure 5a that the linewidth of the R 0,7 acoustic mode-induced FSBS peak of different optical fibers remains the same, while the intensity first increases and then decreases as the core diameter decreases. The normalized intensity of the FSBS peak induced by the R 0,7 acoustic mode is shown in Figure 5b, indicating that a gain optimization could be achieved by varying the diameter of   core. It is worth mentioning that the variation trend cannot be observed in Figure 4b because the normalization in Figure 4b is performed within each fiber to facilitate the comparison of intensity distribution between different fibers. In addition, the results presented in Figures 4 and 5 also indicate that different FSBS gain property can be generated in optical fiber by adjusting the core diameter of fiber upon application. Specifically, the spectrum of FSBS may also be tailored by designing the geometric structure of fiber. Experiments have also been carried out to characterize the FSBS spectrums of the 125 μm fibers with different core diameters, including the fibers with 9 and 6 μm core, respectively. The measured FSBS spectrums of the two fibers are shown in Figure 6a. Figure 6b presents the enlarged view of a specific FSBS peak that is generated by the R 0,7 acoustic mode, which contributes to the strongest resonant peak in the FSBS spectrum. It can be found from Figure 6a that the frequency intervals for the two fibers are 47.7 and 47.5 MHz, respectively, indicating that the frequency interval is nearly not changed with different core diameter. In addition, it should be mentioned that the noise around the first resonant peak originates from the coherent beating noise. In contrast, Figure 6b indicates that the linewidth of FSBS peak for the 6 μm core fiber is 4.63 MHz, which is slightly increased with respect to the 9 μm core fiber with 4.44 MHz linewidth. In addition, the peak intensity of the 6 μm core fiber also shows about 35.9% increment in comparison with that of the 9 μm core fiber. Here, it's worth mentioning that the measured linewidth shows difference from the simulated result, which is mainly due to the reason that linewidth broadening caused by fiber coating is not considered in the simulation. As a matter of fact, the acoustic properties between the core and the cladding are different, which will also lead to the change of the line widths of the resonant peaks with the core diameter, while the variation in the linewidth induced by the distinct acoustic properties is still negligible when compared to the impact of cladding diameter. The result verifies again that a thinner fiber core can help to achieve a profitable gain improvement for FSBS.
In contrast, it needs to be pointed out that the minor peaks appeared in Figure 6a may be induced by insufficiently suppressed TR 2,m acoustic modes. [10] To compare with the simulated result, the FSBS peaks in Figure 6a that are generated by the R 0,m acoustic modes have been fitted with Lorentzian curves, and the comparison between the Lorentzian-fitted peak spectrums and the simulated spectrums of the two fibers are shown in Figure 7a,b, respectively. It can be found that the experimental result is in good agreement with simulation in terms of the resonant frequencies and intensities of peaks for both fibers.

Investigation of the Fiber Cladding Diameter Dependence on FSBS
In addition to the diameter of fiber core, fiber cladding diameter may also have an impact on the FSBS; therefore, the investigation of cladding diameter dependence on FSBS has also been implemented. Two available fibers with 9 μm core but respectively 125 and 80 μm cladding diameters have been used for simulation and experiment. The simulated intensity-normalized FSBS spectrums for the two fibers are shown in Figure 8. The result reveals that as the diameter of fiber cladding decreases, the resonant frequencies of the corresponding acoustic modes will right shift to higher frequencies accordingly, and the resonant frequency interval has increased by 27.1 MHz with the decrement of fiber cladding from 125 to 80 μm, whose frequency intervals are 48.1 and 75.2 MHz, respectively. Meanwhile, the peak intensity of each resonant peak is also increased considerably, owing to the decrease of cladding diameter. The comparison of the transverse profiles for optical field and R 0,7 acoustic mode between the 125 and 80 μm cladding fibers has also been conducted, as shown in Figure 9. It can be found from Figure 9a-c that the two fibers have the same distribution of optical field on the cross section of fibers. However, the material displacement induced by R 0,7 acoustic mode has changed significantly due to the variation of cladding diameter, as shown in Figure 9d-f. The variations in optical and acoustic fields with different cladding diameter will lead to the change of nonlinear coefficient as the overlap integrals Q ðmÞ ES and Q ðmÞ PE are changed. As a result, a stronger acousto-optic interaction can be achieved with more positive overlap for the integral between the optical field and acoustic waves for the 80 μm cladding fiber. Therefore, the result reveals that a thinner cladding diameter will lead to a higher FSBS gain.
To further investigate the dependence of FSBS on cladding diameter, simulations have been carried out by changing the cladding diameter from 125 to 80 μm, while keeping the 9 μm fiber core, and the results are shown in Figure 10. It can be found  from Figure 10a,b that the intensity of FSBS peaks increases gradually as the cladding diameter decreases, while the distribution of peak intensity with respect to frequency has almost no variation. The frequency interval profile for different cladding diameters, calculated from the resonant peaks that are generated by the R 0,7 and R 0,6 acoustic modes, respectively, is shown in Figure 10c, where gradual frequency interval widening can be observed as the cladding diameter decreases. Therefore, the result reveals that the peak intensity and frequency interval of FSBS spectrum can be manipulated by changing the cladding diameter, which offers another flexible way to engineer the FSBS spectrum in addition to adjusting the fiber core.
In addition, the peak intensities of the strongest peak and the specific FSBS peak that is generated by the R 0,7 acoustic mode in fibers with different cladding diameters have also been compared, as shown in Figure 10d. It reveals that a specific acoustic mode-induced (e.g., R 0,7 ) FSBS resonant peak has its optimized cladding diameter for intensity, while the intensity of the strongest peak increases as the cladding diameter decreases. Note that the strongest peak may be generated by different acoustic mode as the cladding diameter varies. Meanwhile, the linewidth of Γ mir , which is a part of the resonant peak linewidth, becomes larger with the decrement of cladding diameter toward the same acoustic impedance of environment outside the fiber, as shown in Figure 10e. Since the broader the linewidth, the smaller the gain, [15,22] and a larger linewidth resulted from reducing the size of fiber cladding, when the fiber is not placed in air or is coated, may partly cancel out the enhancement of nonlinear coefficient. Therefore, the result reveals that the cladding diameter dependence of FSBS could be utilized for programming desired resonant peaks for applications which require specific frequency profiles.
The FSBS spectrums of the two fibers with 9 μm core but 125 and 80 μm cladding have also been experimentally characterized. The comparison of the measured FSBS spectrums of the 125 and 80 μm cladding fiber is shown in Figure 11. It can be found from Figure 11a that for each fiber the adjacent resonant frequencies of FSBS that are generated by R 0,m acoustic modes still have an equal frequency interval. The peaks of the measured original FSBS spectrum are then fitted with Lorentzian curve, and the extracted peak frequencies with normalized intensities of the two fibers have been presented in Figure 11b. The result reveals that the 80 μm cladding fiber has an increased frequency interval of 75.2 MHz, while the standard 125 μm cladding fiber only has a Figure 10. Simulated FSBS spectrum for the 9 μm core fiber with different cladding diameters. a) FSBS spectrums for the 9 μm core fibers with cladding diameter ranging from 125 to 80 μm, and b) the spectrums are plotted together in. c) The frequency interval of fibers with different cladding diameters as shown in (a). d) Normalized peak intensity profiles for the strongest peak and the peak generated by R 0,7 acoustic mode of the 9 μm core fibers with cladding diameter ranging from 125 to 80 μm. e) Profiles for linewidth of Γ mir toward different environmental acoustic impedance using the same fibers as in (d).
www.advancedsciencenews.com www.adpr-journal.com frequency interval of 47.8 MHz. Meanwhile, it is realized that both of the frequency intervals of the fibers are well matched with the simulation. Figure 11c shows the FSBS spectrums around 275 MHz for both fibers, which is generated by R 0,4 acoustic mode in the 80 μm fiber and R 0,6 acoustic mode in the 125 μm fiber, respectively. The linewidth in the 80 μm fiber is 7.08 MHz, which is 2.44 MHz larger than the linewidth of 4.64 MHz in the 125 μm fiber. However, the nonlinear coefficient is not increased significantly as revealed by simulation, which may be caused by the fiber coating induced further broadening of the linewidth for a thinner cladded fiber, as the broader the linewidth, the smaller the gain. [15] Note that the fiber coating is not considered in the simulation. Therefore, the result indicates that the thinner fiber cladding, the larger linewidth of FSBS spectrum will be when keeping the same level of FSBS gain for coated fibers. By customizing the core diameter and cladding diameter, a desired linewidth and nonlinear coefficient of FSBS effect could be generated. Meanwhile, the measured FSBS spectrums of the two fibers are compared with the simulation results, as shown in Figure 12a,b, respectively. It can be found that the measured spectrum is in good agreement with simulation in terms of the resonant frequencies and intensity for both fibers, although the measured resonant peaks frequencies in Figure 12b are slightly different from simulation, which possibly results from a difference of cladding diameter from 80 μm for the used fiber. [18]

Conclusion
In conclusion, we have reported for the first time to our knowledge the investigation on fiber geometry dependence of FSBS in optical fibers by simulation and experiments. Compared with the 125 μm cladding fiber with 9 μm core, the fiber with 6 μm core has an improvement of 35.9% on FSBS gain, while the linewidth and frequency interval remain almost the same. In contrast, compared with the 125 μm fiber, the 80 μm fiber with the same core diameter has a frequency interval increment of 27.4 MHz and a linewidth increment of 2.44 MHz, while maintaining the same level of FSBS gain. The experimental results are well matched with the simulations. The result indicates that targeted linewidth and nonlinear coefficient of FSBS Figure 11. The measured FSBS spectrums for the 9 μm core fiber with different cladding diameters. a) Measured FSBS spectrums for the 125 μm fiber and 80 μm fiber. b) Extracted peak frequencies of the two fibers after Lorentzian fitting with normalized intensity. c) Enlarged view of two FSBS peaks from the 125 and 80 μm fibers, respectively. www.advancedsciencenews.com www.adpr-journal.com peak may be achieved by adjusting the fiber geometry, which could be utilized for designing special fibers that satisfy different requirements toward fiber sensing, signal processing, etc. In the future, these findings could provide guidance to engineering fiber FSBS spectrum for various applications.