J. Tchahame, J. Beugnot, A. Kudlinski, and T. Sylvestre, "Multimode Brillouin spectrum in a long tapered birefringent photonic crystal fiber," Opt. Lett. 40, 4281-4284 (2015). doi: 10.1364/OL.40.004281
W. W. Ke, X. J. Wang and X. Tang, "Stimulated Brillouin Scattering Model in Multi-Mode Fiber Lasers," in IEEE Journal of Selected Topics in Quantum Electronics, vol. 20, no. 5, pp. 305-314, Sept.-Oct. 2014. doi: 10.1109/JSTQE.2014.2303256.
Optical fibers are commonly isolated from the external perturbations by means of cabling techniques and used as communication channels where light waves propagate along haul distances. However, by enhancing the environment influences on the properties of the light that travels into the waveguide, the fibers can be used to detect, to monitor, and even to measure external perturbations in an integral or distributed format . Applications of optical fiber sensors have been growing rapidly in the last years . In particular, distributed fiber-optic sensors (DFS) are attracting a lot of attention for industrial applications, environmental and structural-health monitoring, offering a unique and truly distributed sensing mechanism for long-range applications. Brillouin scattering is a nonlinear optical effect that results from the interaction between light and acoustic waves   and the most widely exploited effects in distributed optical sensing. The Brillouin frequency shift has a linear dependence (for values of strain and temperature within its tolerance ranges) on the applied strain ε and the temperature variation ΔT (at a reference temperature T0) that can be written as .
where Cε is the strain coefficient (MHz / µε), CT is the temperature coefficient (MHz / °C) and reference strain ε0. These values are mostly determined by the fiber composition, pump wavelength, fiber coatings, and jackets. Cε and CT are believed to be constants, and was calibrated as 1.10±0.02 MHz/K, 0.0483±0.0004MHz/µε at 1.55µm . Brillouin scattering was proposed for the first time to measure temperature in 1989 , and, currently, it is widely used for distributed temperature and strain sensing because the Stokes side-lobe is temperature and strain dependent. Based on the linear correlation between Brillouin Frequency and temperature (or strain), the temperature (or strain) distribution along the fiber can be extracted. In this context, SBS has recently been the subject of a renewed interest in multimode (MM) sub-wavelength optical waveguides such as MM-tapered photonic crystal fibers (TPCF). The use of multimode optical fibers as nonlinear media for SBS generation provides a long interaction length which decreases the SBS power threshold. Some efforts have devoted to study the SBS phenomenon in multimode fiber, with both experimental and theoretical methods. In  and , the authors reported the strong SBS between different modes by characterizing the SBS in some two-mode fibers. Recently, Song et al.  measured the profile and peak value of Brillouin gain spectrum (BGS) for each optical mode pair in a four-mode fiber carefully. It was found that the BGS is of multi-peak structure and the intramodel SBS is stronger than the corresponding intermodel SBS. On the other hand, Tei et al.  suggested to describe SBS by a set of special coupling equations and derived a new threshold formula in order to develop a theoretical model. The model employed some assumptions in which both the acoustic and optical modes are treated as the plane waves. In general it is valid in bulk material only. The beam cleanup phenomenon observed in fibers  implies that the acoustic waveguide properties of fiber should be considered. And it is clearly that the fibers with special doping to suppress SBS  cannot be described by a plane acoustic wave model. A complete theoretical model on the stimulated Brillouin scattering (SBS) for MM-fibers is developed by solving the optical-acoustic coupling wave equations. It shows that all optical modes of the signal and of SBS are coupling each other through proper acoustic waves. The Brillouin gain spectrum of each optical mode pair is obtained by certain three-wave couplings (optical-acoustic-optical).
In this work, we evaluate the variation of Brillouin gain and Brillouin shift frequency (BFS) for highly birefringent chalcogenide PCF. We obtain a wide Brillouin gain spectrum (BGS) having certain numbers of peaks corresponding to higher-order acoustic modes with different frequencies. Brillouin gain and Brillouin shift performances for both tapered and untapered chalcogenide PCF are numerically investigated. To the best of our knowledge, no such results have been published for SBS characterization in As2Se3 chalcogenide tapered photonic crystal fiber. We show that we can obtain a multi-peaked BGS with a peak value of Brillouin gain coefficients up to 6.41×10-9 m/W by tapering the fiber. The obtained results will be advantageous for distributed strain and temperature sensors.
EVALUATION OF STIMULATED BRILLOUIN SCATTERING FOR TPCF
Figure 1 depicts the cross section of the untapered and tapered PCF. Two bigger holes were inserted around the core to make it elliptical, therefore enhancing its form-induced birefringence. The core size of the untapered PCF is about 1.36µm and decreases down to 0.6µm. From the microstructure cross section, we then computed the effective refractive index of the fundamental mode at 1.55 µm as a function of the core diameter. The results are shown in Fig. 2 for the slow and fast axis of the birefringent TPCF.
From the acoustic point of view, the longitudinal variations of effective indices theoretically would yield BFS of almost 1 GHz, between 7.5 and 8.5 GHz. This range can be readily calculated from the standard formula νB = 2neff VL / λ, where νB is the Brillouin frequency shift; neff is the effective index; VL= 2250 m/s is the longitudinal acoustic wave velocity of chalcogenide glass and λ = 1.55 µm is the optical wavelength . We identified the input parameter: density ρ0= 4640kg/m3, longitudinal photo-elastic coefficient p12= 0.266, and FWHM bandwidth Δf= 13.2 MHz .
where Aeff is the core area of the fiber, Leff its effective SBS interaction length.
The Brillouin frequency shift fB,i that correspond to the ith-order acoustic modes is given by :
Figure 3 and 4 show the calculated Brillouin spectra for the two tapered and untapered PCFs and for two input states of polarization. As can be seen, figure 3 exhibits a narrow Brillouin spectrum with low Brillouin gain. In TPCF with small-core, light strongly interacts with the air hole microstructure and the waveguide boundaries, leading to the generation of several hybrid elastic modes with different BFS -. For comparison, the Brillouin spectrum of untapered PCF has almost three peaks at 8.20 GHz, 8.23 GHz and 8.26 GHz with a linewidth of 20 MHz (FWHM) each one. In addition, the red and blue Brillouin spectra in Figs 3(a) and (b) also highlight the influence of fiber birefringence on Brillouin gain spectra. This results in a large frequency shift due the refractive index change between the two orthogonal polarizations, as shown in Fig. 2.
For the tapered PCF, the birefringence-induced frequency shift between the multiple peaks significantly increases when the BFS decreases. This behavior confirms that the birefringence increases with the tapering. While a broad Brillouin spectrum spanning almost 400 MHz and including several resonance peaks is presented in Fig. 4 for tapered PCF. Specifically, we can and in both polarizations 4 frequency-detuned acoustic peaks with different weight and linewidth. A further comparison with the Brillouin spectra of the untapered clearly shows the role of the taper to decreases the BFS and broadens the spectrum along the fiber.
The fiber supports several tens of optical and acoustic modes. As each pair of optical modes interacts through a different set of acoustic modes and with a different efficiency, the BGS depends on the input pump and probe fields. We show that new Brillouin dynamics also occurs in the chalcogenide TPCF where the core size shrinks from 8.88µm down to 3.88µm. Our simulations show in particular the generation of a broadband and multi-peaked Brillouin spectrum in comparison with an untapered PCF, while increasing the SBS threshold. Furthermore, we observe that the birefringence induces a large frequency shift of the whole SBS spectrum in both fibers. The Brillouin spectrum broadens and becomes multimode as the fiber core decreases.
The BGS calculated for the chalcogenide TPCF at λ = 1.55 µm for two input states of polarization are presented in Fig. 4. We report the outcome of the numerical computation for a fiber radius of 1.94 µm when field propagates as the fundamental mode. The fiber has four peaks with different values for both polarizations. The peaks with brillouin gain gB1= 5.51×10-11 m/W, gB2= 8.47×10-9 m/W, gB3= 1.48×10-9 m/W and gB4= 7.68×10-13 m/W at fB1= 8.15 GHz, fB2= 8.24 GHz, fB3= 8.32 GHz and fB4= 8.45 GHz, respectively, for the slow polarization axis become gB1= 8.84×10-12 m/W, gB2= 4.07×10-9 m/W, gB3= 1.85×10-9 m/W and gB4= 5.76×10-13 m/W at fB1= 8.09 GHz, fB2= 8.18 GHz, fB3= 8.26 GHz and fB4= 8.36 GHz, respectively, for the slow polarization axis.
Figure (5) shows the simulated plots of the acoustic mode distributions of the multimode chalcogenide TPCF. This fiber has a fundamental acoustic mode and three high-order acoustic modes. Figure 5 (a) is the fundamental optical mode (LP01) that overlaps with the fundamental acoustic mode corresponding to the first peak (gB1). The high-order acoustic modes, presented in Fig. 5 (b), (c) and (d), overlap with the fundamental optical mode corresponding to the second peak (gB2), the third peak (gB3) and the fourth peak (gB4). As a result, the Brillouin spectrum of the fiber shows multiple peaks with comparable intensities, with a main peak and three sub-resonance peaks due to guided acoustic modes.
In order to highlight the relevance of each acoustic mode in the Brillouin gain spectrum, we calculate the spatial overlap integral (Ii) between the optical mode and acoustic modes defined as:
The four peaks of gB for the chalcogenide MM-TPCF in the slow polarization axis are confirmed by the four nonzero values of integral overlap depicted in Fig. 6. A set of acoustic modes with resonance frequencies between 8.15 and 8.45 GHz can be seen to have a not negligible overlap to the optical mode. The multi-peaked Brillouin gain spectrum has been explained by the different values of the overlap integral and confirms the existence of multiple acoustic modes in different frequencies.
Figure 7 (a) presents the Brillouin gain spectrum for slow polarization axis for the intermodal SBS in the tapered PCF with optical mode LP11. Two peaks are shown with brillouin gain gB1= 5.66×10-9 m/W and gB2= 2.03×10-9 m/W at fB1= 8.10 GHz and fB2= 8.15 GHz, respectively. These two peaks are confirmed by the calculated overlap integral which presents two nonzero values at 8.10 and 8.15 GHz corresponding to the overlap of the higher acoustic modes (Fig. 5 (g-h)) with the optical mode LP11.
The Brillouin power (PB) was shown to follow the relation:
where A is a constant which can be evaluated by measuring the Brillouin power and frequency at a position in the fiber where the temperature and strain are known .
By measuring the Brillouin power and frequency at all points in the fiber, it is possible to calculate the temperature distribution from (5). The temperature distribution can then be used in (1) to determine the strain distribution. Thus, by measuring the distributions of the Brillouin power and frequency shift, we can derive the temperature and strain distributions.
In conclusion, we have investigated the stimulated Brillouin scattering in birefringent untapered and tapered As2Se3 chalcogenide photonic crystal fiber at 1.55 µm and we have observed a strong increase of the SBS threshold associated with a broad and multimode Brillouin gain spectrum. We have numerically identified this multi-peaked dynamics as the result of fiber tapering. This type of small core multimode fiber opens another way for controlling and shaping the Brillouin gain spectrum and it can find potential applications 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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