Stimulated Brillouin scattering (SBS) is one of the most significant nonlinear phenomenon arising from the interaction between optical and acoustic modes through the electrostriction . This nonlinear process has attracted considerable interest and has been gainfully exploited in many important applications and devices such as optical amplifiers , lasers , , optical comb generator , coherent optical storage , slow light generation , , ,  and distributed strain and temperature sensors , , , .
For several decades, distributed optical fiber sensors based on Brillouin scattering have attracted intensive research interest around the world for their ability to replace thousands of point sensors including temperature and strain sensors .
SBS has been well characterized since it was observed in optical fiber in 1972 . Over the past several decades, many researches with various conceptually simple and useful methods tend to maximize the Brillouin gain coefficient (gB) in optical fibers. In fact, in , Mizuno et al. have calculated a gB of 3.09×10-11 m/W in polymer fiber. Using tellurite microstructured optical fiber, Cheng et al.  have reported a Brillouin gain of 8.93×10-11 m/W at 7.588 GHz in single core and 9.55×10-11 m/W at 7.65 GHz in four-core while Cherif et al. have reported a gB = 1.1×10-10 m/W at the acoustic frequency of 8.2GHz . These results are close to that of fused silica fibers. For the first time, in 2005, Abedin has measured a Brillouin gain of 6×10−9 m/W and a Brillouin frequency shift of 7.95 GHz for As2Se3 single mode fiber at 1.55μm . Florea et al. have measured a gB of 3.9×10-9 m/W and 6.75×10−9 m/W in a As2S3 and As2Se3 fiber, respectively .
Later in 2009, a Brillouin gain of 8×10-10 m/W at 8.2 GHz was measured in Gej5Sb20S65 chalcogenide photonic crystal fiber (PCF) by Fatome et al. . In , a large Brillouin gain of 5.91×10−9 m/W is obtained around the acoustic frequency of 8.19 GHz using a small core microstructured chalcogenide fiber. Recently, Beugnot et al. have demonstrated a Brillouin gain coefficient of 6.2×10−10 m/W, 3.72×10−10 m/W, and 2.24×10−10 m/W at the acoustic frequencies 7.37 GHz, 7.53 GHz and 6.87 GHz in hybrid chalcogenide-PMMA microwires . Such studies have focused on characterization of Brillouin in chalcogenide fibers. It is observed from the previous literature that the gain is 150 times larger than in single mode silica fibers.
In order to enhance the SBS, it is desirable to have a medium with a large nonlinear efficiency. Recently, PCF specifically tapered PCF called photonic nanofiber (PnF) made up of As2Se3 chalcogenide glass have shown great promise to increase the nonlinearity , . These fibers have been found to be an excellent candidates for nonlinear applications due to their higher nonlinearity, larger refractive index and higher Brillouin gain coefficient (gB) than silica-based fibers , .
In this paper, we propose a new design of chalcogenide PnF with a wide Brillouin gain spectrum (BGS) having multiple peaks corresponding to higher-order acoustic modes. Brillouin gain and Brillouin shift performances for both single-core and dual-core chalcogenide PnF are numerically investigated. To the best of our knowledge, no such results have been published for SBS characterization in As2Se3 chalcogenide PnF. We show that we can obtain a multi-peaked BGS with a peak value of Brillouin gain coefficients up to 3.324×10-9 m/W. The obtained results will be advantageous for distributed strain and temperature sensors.
This paper is organized as follows: section 2 reports in detail the steps of modeling SBS in optical fibers. In section 3, we present the characteristic of the proposed fibers and we evaluate the variation of Brillouin gain and Brillouin shift frequency for both single-core and dual-core As2Se3 chalcogenide PnF. Finally, section 4 summarizes the paper and draws the conclusion.
SIMULATED BRILLOUIN SCATTERING MODELING IN OPTICAL FIBERS
Numerical procedure based on full-vector finite element (FEM) method with an anisotropic perfectly matched layer (PML) has been used for the modeling of SBS. The propagation equation of the acoustic wave as a function of the total optical field is given by  :
Where is the material density fluctuation around its mean value ρ0. ΓB and VA represent the acoustic damping coefficient and the speed of sound in the fiber. γe = neff4 ε0p12 is the electro-restrictive constant of the material
with p12 is the photo-elastic coefficient. The sum of the fields of pumps and stokes waves result on the total electric field E.
The optical mode field (E) and acoustic mode field (u) in the fiber can be obtained by solving the optical and acoustic wave equations, respectively. The equations can be expressed as two-dimensional (2D) scalar-wave equations ,  :
where λp is the pump wavelength, n is the refractive index of the fiber, neff is the effective index of the fundamental optical mode, ωa is the angular frequency of the acoustic wave, v1 is the longitudinal acoustic velocity, and βa is the propagation constant of the acoustic mode.
The Brillouin frequency shift fB,i that correspond to the ith- order acoustic modes is given by  :
where λp is the optical wavelength in vacuum, neff = βoptλ/2π is the effective index of the optical mode, and veff = ωa/ka is the effective velocity of the longitudinal acoustic mode interacting with the optical mode which depends on temperature and strain.
where E is the field pattern of the interacting optical mode, being the fundamental mode for single-mode fibers. The integral is normalized so that its value does not exceed 1.
The contribution of the ith-order acoustic mode to BGS is SB,I(f), which is evaluated from the overlap between E(x,y) and ui(x,y), and it is given by  :
where ΔfB is the full-width at half maximum (FWHM) defined by ΔfB = 1/πTB = ΓΒ/ π, and gB,i is the Brillouin gain spectrum obtained at the resonance and having the following form:
where p12 is the photo-elastic coefficient and ρ0 is the material density .
The total BGS is the sum of the BGS from each mode, since the acoustic modes are statistically independent:
BRILLOUIN GAIN SPECTRUM ANALYSIS IN As2Se3 CHALCOGENIDE PnF
Fig. 1 (a) and 1 (b) show the cross-section of the designed structure of the single-core chalcogenide PnF and the dual-core chalcogenide PnF, respectively. These two fibers have the same geometric parameters with filling ratio d/Λ=0.5, small core diameter about 0.75 μm and overall fiber diameter of 3.25 μm.
BGS of the proposed As2Se3-based chalcogenide PnF is calculated by adopting the approach of the SBS characterization detailed in the previous section.
The refractive index is determined using the Sellmeier formula expressed by:
Where a=7.56 μm, b=1.03 μm2 and c=0.12 μm4 
Previous works have observed that the birefringence has an important role on the Brillouin spectrum and threshold. It induces a large frequency shift of the whole SBS spectrum. The frequency shift of the Brillouin spectrum increases along the fiber.
Figure (2) illustrates the modal birefringence as a function of the wavelength from 1.45 to 1.65 μm. These curves compare the birefringence of two proposed structures. A small value of birefringence due to the numerical method was found in the single-core PnF, it decreases from 2.17×10-7 to 1.67×10-7. These values increase from 8.76×10-4 to 1.93×10-3 in the dual-core PnF. At λ=1.55μm, the value of the birefringence is 1.95×10-7 for the first nanofiber, while it reaches 1.37 ×10-3 for the second.
As shown in Fig. 3. Aeff of the single-core chalcogenide PnF and the dual-core chalcogenide PnF are calculated in order to compare the magnitude of SBS. It shows that the second is larger than the first. The dual-core chalcogenide PnF behaves like two single mode fibers. Considering the intensities of three coupled waves have a relationship with the effective mode area which plays an important role in the SBS process. The Brillouin gain has a direct relationship with the effective area of the studied mode, it increases When the effective area increases.
We choose the optical wavelength λ=1.55μm and we identified the input parameter: density ρ0= 4640kg/m3, longitudinal photo-elastic coefficient p12= 0.266, FWHM bandwidth Δf= 13.2 MHz and the speed of sound= 2250 m/s of As2Se3 glass .
As shown in Fig. 4, in the single core chalcogenide PnF, there is only one fundamental acoustic mode, that overlaps with the fundamental optical mode corresponding to the peak (gB). Figure 5 (a) plots the fundamental optical mode of the dual-core chalcogenide PnF. This fiber has a fundamental acoustic mode and two high-order acoustic modes. Figure 5 (b) shows the fundamental acoustic mode that overlaps with the fundamental optical mode corresponding to the first peak (gB1). The high-order acoustic modes, presented in Fig. 5 (c) and (d), overlap with the fundamental optical mode corresponding to the second peak (gB2) and the third peak (gB3). The combination of chalcogenide glass and more than one core increases the nonlinear refractive index and create a smaller mode field diameter. As a result, the Brillouin spectrum of the fiber shows multiple peaks with comparable intensities, with a main peak and two sub-resonance peaks due to the guided acoustic modes.
The BGS calculated for the single core chalcogenide PnF and the dual core chalcogenide PnF at λ= 1.55 μm are presented in Figure 6. The first fiber has only one peak at fB= 8.06 GHz with gB= 3.242×10-9 m/W, while the second has three peaks at fB1= 8.11 GHz, fB2= 8.16 GHz, and fB3=8.46 GHz with Brillouin gain about gB1= 3.32× 10-9 m/W, gB2= 2.59× 10-9 m/W, and gB3= 2.42× 10-11 m/W corresponding to these three peaks.
In order to highlight the relevance of each acoustic mode in the Brillouin gain spectrum, we calculate the spatial overlap between the optical mode and acoustic modes defined in equation (5). The only peak of gB for the singlecore chalcogenide PnF is confirmed by the only nonzero value of integral overlap depicted in Fig. 7 (a). While Fig. 7 (b) presents three nonzero value of overlap integral and confirms the presence of three peaks of gB for the dual-core chalcogenide PnF. A set of acoustic modes with resonance frequencies between 8.06 and 8.46 GHz can be seen to have a not negligible overlap to the optical mode. Even if the value of the overlap integral for each of those modes is rather small, the overall effect is the appearance of the spectral peak in the BGS. The values of the overlap integral confirm the existence of multiple acoustic modes with different frequencies and explain the multi-peaked Brillouin gain spectrum.
As the Brillouin frequency shift depends on both the effective refractive index of the fiber mode and the velocity of acoustic waves within the fiber, it changes whenever these quantities change in response to local environmental variations and can be used to deduce the temperature and strain along the fiber. Brillouin frequency is linearly related to the temperature and strain by:
where T is the temperature in C°, CT is the temperature coefficient in MHz/C°, fB0 is the reference Brillouin frequency, ε is the strain, and Cε is the strain coefficient in MHz/με .
The simultaneous measurement of temperature and strain is not directly possible for the usual Brillouin based sensors with a single-core PnF because the Brillouin spectrum has only one peak whose frequency is sensitive to both temperature and strain variations. As a solution to this problem, interest was focused on Brillouin frequency shifts,   because they exhibit dependence on strain and temperature. The use of fiber with multipeak Brillouin spectrum is one of these solutions. This method was proposed by Lee et al.,  using a large- effective-area fiber with a multicomposition fiber core. Lee et al. achieved the simultaneous measurement of temperature and strain with a spatial resolution of 2 m, using the fact that the dependence on frequency shift of the first and second Brillouin peaks is different for temperature but the same for strain, even though those two peaks come from different composition materials of the core. For a PCF, Zou et al. demonstrated that the temperature coefficients were different for two Brillouin peaks that originated from two different doping concentration materials of the core.. They also found the strain coefficients to be different for these two peaks in the PCF described in . So fibers with the presence of multiple peaks are used to separate these two effects, and thus measure both strain and temperature . In this work, the design of the dual-core chalcogenide PnF present three peak Brillouin gain compared to the single-core chalcogenide PnF. This type of fibers has been found to be excellent candidates for sensing applications.
A new way of enhancing SBS in optical fiber is proposed. The highly nonlinear As2Se3 chalcogenide dual core PnF present three peaks of Brillouin gain due to the high degree of overlap between the fundamental optical mode and the acoustic modes. The multi-peaked BGS can be applied in nonlinear optics based on Brillouin effect. This optical fiber is extremely desired for strain/temperature sensing.
This work has been supported by the International Centre for Theoretical Physics ICTP affiliated center at the Optical Society of Tunisia. This work has also been partially supported by the Tunisian-Indian bilateral project “Design, modeling and characterization of highly nonlinear fibers for all-optical high bit-rate networks,” funded by the Ministry of Higher Education and Scientific Research of the Republic of Tunisia and the Department of Science and Technology of India.
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