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Abstract
The design of transparent optical materials with stimulated Brillouin scattering (SBS) suppression is a topic of current interest. We measured two-photon absorption (2PA) cross-section σ2PA and Brillouin gain factor gB of a suspension of hexagonal boron nitride (hBN) in N-methyl-2-pyrrolidone at the second harmonic of a Nd:YAG laser. SBS exhibits a significant quenching with hBN concentration, like previously observed in graphene suspension. The melting of hBN flakes due to a large 2PA and the related changes in the acoustic damping coefficient explain the quenching mechanism.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Up-to-day development of fiber optics and optical communication technologies gives a new significance to stimulated Brillouin scattering (SBS) [1–3]. Both amplification of the effect in optical fibers [4,5] and its suppression [6,7] are currently under study. Two-dimensional nanoparticles are also candidates into the SBS-response modification. In this regard, we have recently shown that a small addition of graphene into liquids dramatically quenches SBS due to its nonlinear optical properties [8].
The electronic properties of hexagonal boron nitride (hBN) are attractive for two- and multiphoton absorption [9,10], photoluminescent [11] and hyperbolic phonon-polariton [12] applications. The material offers perspectives in optical switching and limiting [13], high resolution bio-imaging [14] and hyperlens design for optical microscopy and lithography [15].
The experimental studies of hBN nonlinear optical characteristics seem to be at their initial stage, so studying the interaction with the 2nd harmonic nanosecond Nd:YAG laser beam can help understanding its nonlinear optical, thermo- and acousto-optical properties. Up to now, the two-photon absorption (2PA) of hBN suspensions has only been studied at λ = 1064 nm wavelength [11]. An enormous 2PA cross-section: σ2PA = 57⋅10−46 cm4/(s⋅ph) was found, which is larger than in a solid 5-layer hBN film at 400 nm: σ2PA = 4.5⋅10−46 cm4/(s⋅ph) [9]. As theoretically shown [16], 2PA spectrum of hBN has the maximum at ca. 407 nm and a relatively narrow band shape, making the 2PA efficiency only 4% of its maximum already at 450 nm. Therefore, it should be much smaller at 1064 nm. Its large experimental value was most recently explained by a change of nonlinear absorption character in the NIR from five-photon at low laser fluence to two-photon at fluence >0.6 J/cm2 due to significant photo-induced doping of hBN structure with O and C atoms [10].
Proceeding from the concept of a strong nonlinear absorption of hBN we studied pulsed laser beam propagation through its liquid suspension at 532 nm where the data on its optical nonlinearity is scarce. We applied the method of simultaneous detection of transmitting and backscattering beams previously used for studying diluted suspension of graphene [8]. As a two-dimensional (2D) nanoparticle with a strong linear absorption, graphene sublimates under power laser radiation and quenches stimulated Brillouin scattering (SBS) through density decrease and acoustic absorption increase. In this paper, we report the study of a suspension of the 2D material without linear absorption. We experimentally show that SBS also exhibits a remarkable suppression, and the 2PA cross-section is so large that the BN nanoflakes can absorb enough energy to be heated up to the melting temperature. This produces droplets of melted BN flakes and increases the acoustic absorption resulting in the change of acoustic damping coefficient ΓB, which is also the half line-width of the Brillouin scattering.
2. Materials and methods
An hBN powder (Sigma-Aldrich #255475) was tip-sonicated for 60 minutes in 99.5% pure N-methyl-2-pyrrolidone (NMP) with initial concentration of 5 g/l. The supernatant was taken after 120 min centrifugation at 4000 rpm. The precipitate was dried in a box at ca. 1 Torr pressure and 90°C for 11 days and weighed periodically using Mettler Toledo XS105 analytic balance. After the weight stopped changing, the BN concentration in the suspension was calculated that amounts to ρ = (161 ± 16) mg/l.
BN structure was studied by means of transmission electron microscopy (FEI Tecnai G2 F20 TEM instrument) and micro-Raman analysis (Horiba Jobin-Yvon LabRAM HR800 spectrometer). Figure 1(a) shows a bent flake of several hundred nm size. Pleats on it evidence its thin-sheet form with thickness below 10 nm. The HRTEM study shows a big number of superimposed small flakes of few tens nm size like those we see in Fig. 1(b). FFT in Fig. 1(c) proves their hexagonal structure. From 10 to 20 parallel rows are distinguishable at the edges of the flakes similar to those shown elsewhere [17] that make us suggest the corresponding number of layers. It could give a thickness of 3.5 - 7 nm, which agrees with the pleats observation in Fig. 1(a). A pronounced peak in the Raman spectrum at 1365 cm−1 [see Fig. 1(d)] indicates the E2g phonon mode of a bulk hBN crystal structure [18]. Therefore, we deal with multilayer hBN flakes with a couple dozen layers.
Fig. 1 (a) TEM image, (b) HRTEM image and (c) corresponding FFT pattern, (d) Raman and (e) absorption spectrum of BN nanoflakes. Inset shows concentration dependence of absorbance at 532 nm.
Absorption spectra of the suspensions were measured in 1 cm path cell using PerkinElmer Lambda 750 dual-beam UV/VIS/NIR spectrometer with a reference cell with NMP. The spectrum of the initial suspension is given in Fig. 1(e). The hyperbolic increasing of the absorbance from IR to UV is due to scattering, since the absorption of BN is negligible (the material is white). Six dilutions of the suspension were taken to check the linearity of the absorbance with respect to the BN concentration. The inset in Fig. 1(e) shows that it is quite linear despite it reflects rather scattering than absorption. This allows us to recalculate absorbance to concentration.
Optical limiting (OL) and SBS measurements were performed using the setup and procedure described in our previous work [8] with Continuum Minilite II laser source at λ = 532 nm wavelength, τ = 4 ns pulse duration, fr = 1 Hz repetition rate. The latter is low enough to establish equilibrium in the suspension between the pulses so that measured pulse energies can be considered as the property of a single pulse. A 2 cm-path quartz cell was used in this experiment to avoid possible damage due to a shock wave generated.
The energies of the incident, back-scattered and transmitted beams were measured by silicon detectors Thorlabs PDA100A-EC calibrated by pyroelectric heads: Coherent J10MT-10kHz in the energy region 0.5 - 450 μJ and Coherent J50MB-YAG-1561 from 0.5 to 3.5 mJ.
3. Results of nonlinear optical measurements
3.1 Optical limiting
The transmittance, T, measured as a function of incident pulse energy, E, for different suspension dilutions are presented in Fig. 2(a). This dependence shows a pronounced limiting behavior where the linear declination of the beginning part is due to 2PA:
This expression is obtained with the assumptions that the interaction length is equal to twice the Rayleigh length: L = 2ZRn and that the input beam intensity does not change along L and equal to its average value for the Gaussian beam: I = (4/π)I0, I0 = E/(πw02τ). Beam waist radius w0 = 41 ± 7 μm is obtained from a CCD image analysis. ZR = πw02/λ and L = 2.94 mm. We also neglect the linear absorption of the layer L whose impact in the effective layer Leff is less than 5% in all our measurements. The linear transmittance of the rest of the suspension layer is considered as Tlin. Thus, it is seen from Eq. (1) that the 2PA coefficient, β2PA, can be determined by the linear fitting of experimental points.Fig. 2 (a) Transmittance vs. input energy for different dilutions of BN suspension (legend shows the linear absorption coefficient), and (b) corresponding 2PA coefficients vs. BN concentration. Inset shows Z-scan of the suspension normalized transmittance with ρBN = 161 mg/l in 2 mm cell at E = 4.0 mJ.
The fitting lines are shown in Fig. 2(a) and demonstrate an increase of slopes with BN addition. 2PA coefficients calculated from the slopes using Eq. (1) are plotted in Fig. 2(b) against corresponding BN concentration, ρBN. We also checked the 2PA coefficient in open-aperture Z-scan of the suspension in a 2 mm cell at the same wavelength (w0 = 74 ± 4 μm, Leff = 0.19 cm). This method is poorly sensitive in the case of low-concentration BN suspension.
It requires to increase the laser pulse intensity at which the BN nonlinearity concurs with the cell wall damage threshold. We evaluated β2PA = 0.037 cm/GW from Z-scan for the maximal BN concentration with the error of several tens of percent due to a low signal-to-noise ratio [see inset in Fig. 2(b)]. This value is, however, consistent with the OL data.
Despite relatively large errors the β2PA(ρBN) dependence is fitted by a linear function with a correlation coefficient of 0.97. Its slope provides the 2PA cross-section: σ2PA = (4.6 ± 0.3)⋅10−48 cm4s⋅photon−1⋅molecule−1. This value is two orders of magnitude less than in a BN multilayer film at 400 nm [9], and 33 times less than in a suspension of hydroxylated BN nanoflakes at 532 nm [13].
However, at the input energy E = 1 mJ which is close to the SBS threshold conditions in NMP in our setup, it gives a nonlinear contribution to the absorption cross-section σNL = 1500 cm2/g. This value is of the same order with the linear absorption cross-section in a graphene suspension: 8000 cm2/g [8], hence, local thermal effects known for graphene nanoflakes can be expected. For instance, we can calculate temporal rise of the local temperature of a BN flake following the heat capacity Cp definition:
where is the energy absorbed by a flake of mass for the time since the laser pulse beginning, I(t) is the incident pulse intensity time profile measured by a time-resolved detector [8]; h is the Plank constant; c is the speed of light; MBN is molecular mass of BN. The calculation based on the known value of Cp = 0.79 kJ g−1 K−1 for BN shows that at incident pulse energy E = 1 mJ the melting temperature T = 2967°C is attained after 2.7 ns. Unlike graphene, BN melts at atmospheric pressure [19], and the process obviously starts during the laser pulse. Using BN melting enthalpy value Λ = 81 kJ/mol, we find that at E = 1.25 mJ, full melting of the flake happens during the laser pulse. All the thermodynamic constants of BN are taken from [20].Equation (2) does not include the heat transfer to the environment which in case of a thin-layer particle can be remarkable. It can be evaluated by a factor of d/√(χt) to Eq. (2), where d is the thickness of an instantly heated plate (thickness of the BN layer) and χ is the thermal diffusivity of the solvent [21]. The calculation of the latter for NMP by χ = κ/(ρCp) with thermal conductivity κ = 0.187 W m−1K−1 [22] gives χ = 0.1 mm2/s. The correction is evidently different for different flakes. For instance, at d = 7 nm it shifts the melting temperature achievement time at E = 1 mJ to 3.8 ns and the energy of the full-melting regime to 1.35 mJ. However, it does not change the main conclusion: BN flakes melt at the beginning of the laser pulse at energies slightly exceeding 1 mJ, which is around the SBS threshold Ethr in NMP.
3.2 Scattering
Due to the melting we can consider that the BN suspension transforms to an emulsion of BN droplets for laser pulses with E > Ethr. However, BN molar volumes in solid and in liquid differ only by 9%, and this transition should not remarkably change density nor refraction, and hence should not much enhance light scattering. Indeed, if we compare the Tyndall cone occurring in the suspension at the laser beam propagation with different energies [Figs. 3(a) and 3(b)], no significant changes in size and shape can be noticed, but only in its brightness. However, the scattering coefficient, S = Esc/E, where scattered energy was measured by a Coherent J10MT detector on the side wall of the cell, as shown in Fig. 3(c), reveals some changes above Ethr. Being almost constant at lower energies, the scattering coefficient remarkably increases by 0.5% for the next ca. 1 mJ after Ethr and then tends to saturate.
Fig. 3 Photograph of the cell with 60 mg/l-concentration BN suspension irradiated with pulse energy (a) below and (b) above the SBS threshold; (c) corresponding side-scattering into 1.4 sr solid angle vs. input energy.
Energetic characteristics of the SBS process change the way more dramatically. Figure 4(a) represents measured dependences of the SBS energy, ESBS, as a function of E, in pure NMP and in diluted suspensions of BN. We see a fast decrease in the energies when even small concentrations of BN are added. The sensitivity of the SBS quenching is remarkably greater even than that of the OL performance. Not all the concentrations are studied simultaneously with OL. For the detailed study of the SBS effect, more dilutions were also made.
Fig. 4 (a) SBS energy vs. input energy for different BN concentration (given in legends), and (b) corresponding Brillouin gain factor vs. suspension absorption coefficient: points – experiment, solid curve – simulation with ΔΓ = 14 cm, dashed – simulation with Δn = −5 cm, dashed dot – simulation for graphene suspension with both ΔΓ = 50 cm and Δn = −1.55 cm.
The SBS quenching obviously correlates with the observed side-scattering behavior. Side-scattering in Fig. 3(c) which is spontaneous scattering, shows a pronounced increase when the droplets of melted BN destroy the stimulated scattering conditions (between ca. 1 and 2 mJ), and stops increasing when SBS occurs (above 2 mJ). We thus see where the light energy goes.
In the studied energy range the SBS energy dependences look linear, so that a linear fitting was used to determine Ethr related to SBS intensity gain factor: gB = 25πw02τ/(EthrL), where 25 is the gain value at which SBS can be observed in organic liquids amid spontaneous scattering [23]. The gain factors obtained are shown in Fig. 4(b) for different BN concentrations. To compare the effect with the one earlier obtained in graphene suspensions [8], we plotted all data against the Napierian absorption coefficient obtained from absorption spectra: α = ln(10)Abs/(1 cm). Different icons indicate the corresponding concentration given in the legend.
Like in dispersed graphene, the decrease of gB in BN saturates with concentration growth. It amounts to ΔgB ≈15 cm/GW which is much larger than the β2PA value. Hence 2PA cannot give such effect directly by light extinction. On the other hand, in the case of substance heating and thermal expansion the Brillouin gain factor is determined by a set of electro- and thermo-optical parameters [24]:
i.e. directly proportional to electrostrictive coefficient γe, thermo-optic coupling coefficient γt, squared angular frequency ω0, and inversely proportional to refractive index n, solvent density ρ, acoustic wave velocity vac and ΓB. Last two parameters determine acoustic absorption and scattering: according to the classical approach [25], the sound attenuation coefficient αac = ΓB/(2vac). Here we neglected bulk viscosity and thermal conductivity parts of ΓB considering it to be dependent on the shear viscosity, η, only: ΓB = ωac2η/(ρvac2), which is a good approximation in liquids [26].Light frequency ω0 is constant during the experiment. Based on the analysis we performed earlier [8], we can divide the other parameters in Eq. (3) into two groups: γe, n whose changes are induced by the density change (ρ-dependent), and vac, ΓB (acoustic). For the latter pair no universal relation between them is known. For the case of graphene, we have numerically shown that namely ρ and ΓB are the parameters most responsible for the concentration-induced changes in gB. Density completely determines the ρ-dependent parameters and ΓB contributes in both terms of Eq. (3) unlike vac which is cancelled in the thermo-expansive term, being included in γt in the numerator, and hence becomes less significant.
Similar gB(α) curve simulation for the case of BN shows that the change of acoustic damping coefficient: ΓB = ΓB(0)(1 + ΔΓB⋅α) perfectly fits the data [solid curve in Fig. 4(b)] at the relative decrement value ΔΓB = 14 cm. Meanwhile an analogous decrement of density does not reproduce the dependence correctly, which is shown by the dashed curve in Fig. 4(b), and hence cannot be the reason for the quenching observed. This confirms the fact that rather droplets than bubbles form in the BN suspension under the powerful laser beam.
ΔΓB in BN suspension is 3.6 times less than in graphene that is also understandable in view of the BN droplet formation: the compressibility of liquid is significantly less than gas (carbon vapor bubbles in the case of graphene) stipulating a lower acoustic attenuation. However, we see that it still leads to a remarkable effect on SBS.
Considering the negligible linear absorption of BN nanoparticles, we can anticipate that their small additions into optical fibers is a good means for SBS suppression as compared to the application of special fiber designs [1]. It is worth noting that at moderate intensities BN can reduce the Brillouin gain factor in the fiber geometry due to its 2PA as gB eff = gB – 2β2PA [27]. If we consider, for example, a high-gain polymer with gB = 0.03 cm/GW [28] then the concentration ρBN ≈48 mg/l will be enough to completely suppress SBS in it according to the dependence in Fig. 2(b). In such application BN has an advantage over 2PA dyes due to its high photostability and absence of luminescence. Possible losses due to linear scattering (44 dB/m in our suspension) can be reduced by the improvement of doping technology: selection of finer nanoparticles using ultracentrifugation and tailoring immersion polymer composition. Passing to telecommunication wavelengths will also decrease losses.
4. Conclusion
In this study, we measured the 2PA cross-section of an hBN suspension at 532 nm wavelength and showed that due to the relatively intensive 2PA, BN nanoflakes quench SBS in the liquid almost as intensively as dispersed graphene nanoflakes. Quenching in case of BN should be attributed to the acoustic attenuation increase due to the melting of BN and nanoflakes-into-nanodrops transformation under the laser beam at intensities above the SBS threshold. Since BN dispersed in a liquid with the concentrations under study is highly transparent for low-intensity irradiation, it can be recommended as an appropriate dopant for the design of nonlinear-optical composites for all-optical filters, as well as transparent materials with the SBS suppression. The BN suspension is also an interesting model system for studying nonlinear optical and acousto-optical phenomena.
Funding
National Natural Science Foundation of China (NSFC) (61675217, 61522510); Strategic Priority Research Program of Chinese Academy of Sciences (CAS) (XDB16030700); Key Research Program of Frontier Science of CAS (QYZDB-SSW-JSC041); Program of Shanghai Academic Research Leader (17XD1403900); President’s International Fellowship Initiative (PIFI) of CAS (2017VTA0010, 2017VTB0006, 2018VTB0007).
Acknowledgments
The authors thank the Testing & Analysis Center of Soochou University and the Center for Geo-Environmental Research and Modeling (GEOMODEL) of St. Petersburg State University for the use of their equipment for TEM and Raman studies respectively.
References
1. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010). [CrossRef]
2. B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photonics 5(4), 536–587 (2013). [CrossRef]
3. W. C. Wang, B. Zhou, S. H. Xu, Z. M. Yang, and Q. Y. Zhang, “Recent advances in soft optical glass fiber and fiber lasers,” Prog. Mater. Sci. 101, 90–171 (2019). [CrossRef]
4. J. Zhu, X. Cheng, Y. Liu, R. Wang, M. Jiang, D. Li, B. Lu, and Z. Ren, “Stimulated Brillouin scattering induced all-optical modulation in graphene microfiber,” Photon. Res. 7(1), 8–13 (2019). [CrossRef]
5. E. Giacoumidis, A. Choudhary, E. Magi, D. Marpaung, K. Vu, P. Ma, D.-Y. Choi, S. Madden, B. Corcoran, M. Pelusi, and B. J. Eggleton, “Chip-based Brillouin processing for carrier recovery in self-coherent optical communications,” Optica 5(10), 1191–1199 (2018). [CrossRef]
6. J. Cui, H. Dang, K. Feng, W. Yang, T. Geng, Y. Hu, Y. Zhang, D. Jiang, X. Chen, and J. Tan, “Stimulated Brillouin scattering evolution and suppression in an integrated stimulated thermal Rayleigh scattering-based fiber laser,” Photon. Res. 5(3), 233–238 (2017). [CrossRef]
7. C. Liu, J. Liu, Y. Zhang, Y. Hou, S. Qi, X. Feng, and P. Wang, “Stimulated Brillouin scattering suppression of thulium-doped fiber amplifier with fiber superfluorescent seed source,” Opt. Express 25(9), 9569–9578 (2017). [CrossRef] [PubMed]
8. I. M. Kislyakov, J.-M. Nunzi, X. Zhang, Y. Xie, V. N. Bocharov, and J. Wang, “Stimulated Brillouin scattering in dispersed graphene,” Opt. Express 26(26), 34346–34365 (2018). [CrossRef] [PubMed]
9. Q. Ouyang, K. Zhang, W. Chen, F. Zhou, and W. Ji, “Nonlinear absorption and nonlinear refraction in a chemical vapor deposition-grown, ultrathin hexagonal boron nitride film,” Opt. Lett. 41(7), 1368–1371 (2016). [CrossRef] [PubMed]
10. Y. Dong, P. Parajuli, A. M. Rao, W. Thielemans, S. Eyley, K. R. Sahoo, T. N. Narayanan, and R. Podila, “Intrinsic five-photon non-linear absorption of two-dimensional BN and its conversion to two-photon absorption in the presence of photo-induced defects,” Opt. Mater. 86, 414–420 (2018). [CrossRef]
11. P. Kumbhakar, A. K. Kole, C. S. Tiwary, S. Biswas, S. Vinod, J. Taha-Tijerina, U. Chatterjee, and P. M. Ajayan, “Nonlinear optical properties and temperature-dependent UV–Vis absorption and photoluminescence emission in 2D hexagonal boron nitride nanosheets,” Adv. Opt. Mater. 3(6), 828–835 (2015). [CrossRef]
12. A. J. Giles, S. Dai, I. Vurgaftman, T. Hoffman, S. Liu, L. Lindsay, C. T. Ellis, N. Assefa, I. Chatzakis, T. L. Reinecke, J. G. Tischler, M. M. Fogler, J. H. Edgar, D. N. Basov, and J. D. Caldwell, “Ultralow-loss polaritons in isotopically pure boron nitride,” Nat. Mater. 17(2), 134–139 (2018). [CrossRef] [PubMed]
13. G. Zhao, F. Zhang, Y. Wu, X. Hao, Z. Wang, and X. Xu, “One-step exfoliation and hydroxylation of boron nitride nanosheets with enhanced optical limiting performance,” Adv. Opt. Mater. 4(1), 141–146 (2016). [CrossRef]
14. A. W. Schell, T. T. Tran, H. Takashima, S. Takeuchi, and I. Aharonovich, “Non-linear excitation of quantum emitters in hexagonal boron nitride multiplayers,” APL Photonics 1(9), 091302 (2016). [CrossRef]
15. T. Low, A. Chaves, J. D. Caldwell, A. Kumar, N. X. Fang, P. Avouris, T. F. Heinz, F. Guinea, L. Martin-Moreno, and F. Koppens, “Polaritons in layered two-dimensional materials,” Nat. Mater. 16(2), 182–194 (2017). [CrossRef] [PubMed]
16. C. Attaccalite, M. Grüning, H. Amara, S. Latil, and F. Ducastelle, “Two-photon absorption in two-dimensional materials: The case of hexagonal boron nitride,” Phys. Rev. B 98(16), 165126 (2018). [CrossRef]
17. H. Zhou, J. Zhu, Z. Liu, Z. Yan, X. Fan, J. Lin, G. Wang, Q. Yan, T. Yu, P. M. Ajayan, and J. M. Tour, “High thermal conductivity of suspended few-layer hexagonal boron nitride sheets,” Nano Res. 7(8), 1232–1240 (2014). [CrossRef]
18. R. V. Gorbachev, I. Riaz, R. R. Nair, R. Jalil, L. Britnell, B. D. Belle, E. W. Hill, K. S. Novoselov, K. Watanabe, T. Taniguchi, A. K. Geim, and P. Blake, “Hunting for monolayer boron nitride: optical and Raman signatures,” Small 7(4), 465–468 (2011). [CrossRef] [PubMed]
19. V. L. Solozhenko, V. Z. Turkevich, and W. B. Holzapfel, “Refined phase diagram of boron nitride,” J. Phys. Chem. B 103(15), 2903–2905 (1999). [CrossRef]
20. W. M. Haynes, ed., CRC Handbook of Chemistry and Physics, 97th ed. (CRC, 2017).
21. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena (Dover Publishing, 2002).
22. N. D. Klushin, V. P. Pogodin, and A. F. Vorobev, “Thermal conductivity of solutions of sodium, potassium, and caesium iodides in n-methyl-2-pyrrolidinone in the temperature range 298-348 K,” Zh. Fiz. Khim. 57, 2873–2875 (1983).
23. R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990). [CrossRef] [PubMed]
24. D. Pohl and W. Kaiser, “Time-resolved investigations of stimulated Brillouin scattering in transparent and absorbing media: determination of phonon lifetimes,” Phys. Rev. B 1(1), 31–43 (1970). [CrossRef]
25. T. D. Rossing, ed., Springer Handbook of Acoustics (Springer Science+Business Media, 2007).
26. B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).
27. C. Wolff, P. Gutsche, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Impact of nonlinear loss on stimulated Brillouin scattering,” J. Opt. Soc. Am. B 32(9), 1968–1978 (2015). [CrossRef]
28. Y. Mizuno and K. Nakamura, “Brillouin scattering in polymer optical fibers: fundamental properties and potential use in sensors,” Polymers (Basel) 3(2), 886–898 (2011). [CrossRef]
