Open Access
Add to BibSonomy
Abstract
Stimulated Brillouin scattering (SBS) in solid-core photonic crystal fibers (PCFs) differs significantly from that in standard optical fibers due to the tight confinement of both optical and acoustic fields in their µm-sized fiber cores, as resultantly evident in their Brillouin gain spectra. Despite many theoretical studies based on either simplified models or numerical simulations, the structural dependency of Brillouin gain spectra in small-core PCFs has not been characterized comprehensively using PCFs with elaborated parameter controls. In this work we report a comprehensive characterization on the core-structure dependences of backward SBS effects in solid-core PCFs that are drawn with systematically varied core-diameter, revealing several key trends of the fiber Brillouin spectrum in terms of its gain magnitude, Brillouin shift and multi-peak structure, which have not been reported in detail previously. Our work provides some practical guidance on PCF design for potential applications like Brillouin fiber lasers and Brillouin fiber sensing.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Stimulated Brillouin scattering (SBS) refers to light scattering by acoustic fields driven by optical force through electrostriction and radiation pressure, with photo-elastic effect and moving-boundary effect as backactions that lead to the energy transfer from pump to the Stokes light with a frequency shift determined by the phase-matching condition [1]. In optical fibers [2], due to their waveguide structure the directions of SBS are constrained to be in either forward [3] or backward [4] directions. Backward SBS effects in optical fibers [5] are of particular importance due to the low SBS threshold, which can impose non-trivial limitations on the performances of some fiber-optic systems, such as fiber-optic communication systems [6], high-power fiber lasers [7,8]. Meanwhile, backward SBS has been well exploited as a useful nonlinear effect to achieve single-wavelength lasing [9,10], narrow-band filters [11], distributed sensing [12,13], optical buffering [14,15], etc. As a result, the manipulation of the backward SBS effect in the optical fiber, with the purpose of acquiring on-demanded SBS spectral characteristics (e.g., the gain magnitude, frequency shift, etc.), has been widely regarded as a critical technique in many SBS applications [16–18].
The spectral features of Brillouin gain in the optical fiber mainly depend on two factors. The first one is the phase-matching condition, which can be determined from the dispersion relations of optical and acoustic modes. The second one is the overlapping strength between the optical and acoustic fields which can be retrieved from the mode profiles of these two interacting fields. Both of the two factors depend heavily on the geometric structure of the optical fiber. In the conventional single-mode fiber (SMF), meanwhile the optical mode is guided in its Germanium-doped fiber core, and the confinement of acoustic fields in its core area is very limited [19], leading to a relatively-low SBS gain coefficient of the SMF. The advent of solid-core photonic crystal fibers (PCFs) [20] with microstructured core-region revolutionized the field of SBS in the fiber waveguide, bringing unprecedented flexibility to the light-sound interactions due to the tight confinement of both fields in the PCF core surrounded by many hollow channels [4,21,22]. In the solid-core PCF with µm-scale core diameter and high air-filling ratio, the tight confinement of both the optical and the acoustic modes results in extraordinary overlapping and consequently enhanced gain coefficient of the SBS effect [4] in contrast to the conventional SMF.
The Brillouin gain enhancement in the PCF comes along, however, with a series of complications in some key features. On the one hand, the hollow-channels in the PCF provide higher index contrast than the doped-core as in the conventional SMF, leading to significant changes in mode profile and effective refractive index of the fundamental optical mode in the PCF. On the other hand, the tight confinement of hollow channels also applies to the acoustic field, which leads to non-trivial coupling of longitudinal and shear acoustic waves at the surface of the PCF core, in contrast to the case in the conventional SMF where the acoustic wave can be approximately regarded as pure longitudinal plane-wave along the fiber axis. These complications bring intricate structural dependences of the Brillouin gain spectrum, which have been well explored previously using either simplified analytical models (e.g., an elongated circular strand) [4] or with numerical simulations [23]. In order to gain practical guidance in PCF structure, one can only resort to experimental measurements of PCFs [4] with elaborated structural controls during fabrication. In this work, we report the experimental studies on core-structural dependences of backward SBS effects in solid-core PCFs, with systematically varied core diameters. Our experimental results reveal several key features of the SBS gain spectrum in terms of its magnitude, frequency shift, and multi-peak structure which have not, previously, been demonstrated in detail. These results are expected to provide practical guidance for designing PCF-based Brillouin lasers and sensing systems.
2. Fiber fabrication and experimental set-up
2.1 Fabricated PCF samples
We fabricated a series of solid-core silica PCFs with wavelength-scale core diameters and high air-filling ratios on a customized fiber drawing tower supplied by iFiber Optoelectronics Technology Co. Ltd. By properly designing preform structures and controlling gas-pressure in the hollow-channels during the “stack-and-draw” fabrication procedure, we managed to achieve elaborated core-region control, namely systematically varied core diameters ranging from 1.4 µm to 5 µm. The scanning electron microscopic (SEM) photos of some exemplary PCFs are shown in Fig. 1. Here we define the air-filling ratio of the PCF sample as the following way. Here we define the air-filling ratio S = 1 – W/D concerning only the innermost layer of hollow channels since they dominate the confinement of both optical and acoustic fields, in which W is the waist width of the silica “bridge” between adjacent hollow-channels and D is the averaged diameter of the hollow channels. The air-filling ratios of these PCFs are controlled to be around ∼0.8, although in practice they fluctuated between 0.78 and 0.84.
Fig. 1. SEM photos of the core-regions of the fabricated PCFs with systematically varied core diameters ranging from 1.4 to 5 µm with high air-filling ratios (fluctuation between 0.78 to 0.84).
In order to implement the PCF into the measurement set-up, in the experiment we employed an all-fiber configuration, in which the PCF sample was spliced with the conventional SMF. Due to the relatively large difference in their core diameters, a short length (∼1 cm) of highly nonlinear fiber (HNAF) with an intermediate mode area was used as the transition fiber between the PCF and the SMF [24] so as to mitigate the mode mismatch. The resultant insertion loss of the PCF sample was measured to be 2–5 dB.
2.2 Pump-probe measurement of the fiber SBS gain spectrum
We employed in the experiment a pump-probe set-up [19,25] to measure the backward SBS gain spectrum of the fiber, which is sketched in Fig. 2. We used a single-frequency laser (NKT Koheras, linewidth <100 Hz) at 1550 nm as the seed laser source, and split its output using a 50/50 fiber coupler. One part of the light beam was used as the pump light after amplification in a high-power erbium-doped fiber amplifier (EDFA), reaching 100-mW average power before being launched into the PCF samples. The other part was modulated using an electro-optical modulator (EOM) and a radio-frequency (RF) source, and then the modulated light was filtered using a narrow band-pass filter (BPF) in order to obtain the low-frequency component used as the Stokes (probe) light in the pump-probe measurement. The counter-propagating Stokes light, with an average power of ∼ 2 mW, was amplified in the PCF sample due to the SBS gain, then collected using a fiber-optic circulator. The output power of the amplified Stokes light was measured using a power meter.
Fig. 2. Pump-probe set-up for PCF Brillouin gain spectrum measurement. SFL: single-frequency laser; ISO: isolator; EDFA: erbium-doped fiber amplifier; FPC: fiber polarization controller; EOM: electro-optical modulator; BPF: band-pass filter; IP: in-line polarizer.
By scanning the frequency of the probe light (tuning the modulation frequency of the RF source), we obtained the SBS gain spectra g(ω) of different fiber samples, with a dimension of m-1W-1, expressed below, while the pump depletion was ignored.
3. Experimental results
3.1 SBS gain magnitude and frequency shift versus core-diameter
We measured the SBS gain spectra of the fabricated PCFs with systematically varied core-diameter in the vicinity of phase-matched frequency with maximum Brillouin gain coefficient. Although most of measured spectra exhibited multi-peak profiles, we first concentrated on the dominant peak of each SBS gain spectrum with the maximum gain coefficient, leaving detailed analysis on the multi-peak structures to the latter section. The measured results of dominant SBS gain spectral peaks corresponding to the fabricated PCF samples (listed in Fig. 1) are illustrated in Fig. 3(a) below. For comparison, the SBS gain spectrum of the standard SMF-28 fiber is measured using the pump-probe set-up and the results are also plotted in Fig. 3(a), with a gain coefficient of merely ∼0.17 m-1W-1, which is generally one order of magnitude lower than that of the solid-core PCF.
Fig. 3. (a) Experimentally measured Brillouin gain spectra for fabricated PCF with different core diameters (denoted above respective curves). Only the dominant gain peaks are shown in this plot. (b) The maximum Brillouin gain coefficient (red triangles) and the simulated effective mode area (gray circles) for PCFs with different core diameters. (c) The simulated fundamental optical mode profile (LP01-like mode) for PCF with core size of 1.44 µm and 5.01 µm (the normalized in-plane electric field strength |ET| are shown for the mode profile).
There are a series of interesting trends in the structural dependence of the PCF SBS gain spectrum that can be drawn from these experimental results, particularly concerning the SBS gain magnitude and frequency shift for different PCF samples. (1) Despite that the gain coefficients of the solid-core PCFs are generally one order of magnitude higher than that of the conventional SMF, the measured results exhibit that the SBS gain coefficient of the PCF reached a maximum value of ∼ 4.6 m-1W-1 at a core diameter of ∼1.8 µm, see Fig. 3(a). Below this optimized diameter, the SBS gain coefficient of the PCF declined quickly, despite that the effective mode area of the optical mode continues to become smaller as the PCF core diameter is further decreased (See Fig. 3(b)). (2) Meanwhile, the FWHM bandwidth of the dominant peak decreased from 61 MHz down to 36 MHz as the PCF core diameter decreased from 5.01 µm to 1.84 µm, and then increased again to 66 MHz with PCF core diameter of 1.64 µm. (3) The Brillouin shift of the dominant gain peak was observed to decrease monotonically as we gradually reduced the PCF core diameter (analyzed in the next section).
The existence of this “optimized” PCF-core diameter for reaching the maximum SBS gain coefficient could be interpreted with a few underlying mechanisms as follows. Firstly, although the small-core of PCF can tightly confine both optical and acoustic field, their overlapping ratio does not necessarily increase monotonically as the core size decreases. The optical mode extends outside the core region non-trivially when the core size reaches close to the diffraction limit (see Fig. 3(c)), meanwhile the acoustic field is still confined within the PCF-core, leading to a reduced overlapping [4]. Secondly, since we noticed an evident bandwidth broadening effect at a core diameter of 1.67 µm, we suspected that the acoustic damping becomes stronger when the core size is below the optimized value, leading to a reduced peak coefficient of the SBS gain spectrum. For the case with 1.44-µm core-size, the peak splitting becomes significant, indicating the generation of some hybrid acoustic modes due to the coupling between shear and longitudinal acoustic waves [4,26,27]. Such coupling usually leads to weakened index modulation despite the enhanced optical intensity. Moreover, the moving-boundary effect caused by the gradient forces upon the silica-air boundary might also play a minor role in partial cancellation of the photo-elastic effects caused by electrostrictions [28], although previous theoretical studies [26] show that this moving-boundary effect becomes prominent only if the PCF has a sub-wavelength core size.
3.2 Brillouin frequency shift versus core-diameter
The Brillouin shift of the PCFs as we measured in the experiment decreased monotonically as the PCF core diameter decreased. The measured Brillouin shift values of the PCF sample, as the function of the PCF core diameter, is summarized in Fig. 4 (orange line). In theory, following the phase-matching condition the Brillouin shift can be expressed as Eq. (2).
Fig. 4. Experimentally measured Brillouin frequency shifts for PCFs with different core diameters (orange) and calculated results using ${n_{\textrm{eff}}}$ of LP01-like mode of the PCFs and a constant longitudinal acoustic velocity ${\upsilon _\textrm{a}}$=5940 m/s (blue).
With a fixed pump wavelength (${\lambda _\textrm{P}}$), both the effective refractive index (${n_{\textrm{eff}}}$) and the acoustic velocity along the fiber axis (${\upsilon _\textrm{a}}$) vary as the function of the PCF-core size. We found that ${n_{\textrm{eff}}}$ tends to decrease significantly when the core size is reduced due to the decreased mode confinement. In our simulations, the ${n_{\textrm{eff}}}$ of the fundamental LP01-like mode decreased from 1.4296 to 1.3156 when the PCF core diameter changed from 5 µm to 1.44 µm and the air-filling ratio of the PCF was fixed to 0.8 in simulations. The acoustic velocity would also be varied a bit in principle, due to the hybridization of longitudinal and shear waves, leading to varied acoustic dispersion relation [27]. Nevertheless, we realized that the variation of the ${n_{\textrm{eff}}}$ played the dominant role in the Brillouin shift. The theoretical value of the Brillouin shift, calculated using the simulated ${n_{\textrm{eff}}}$ at different PCF core diameters and a fixed longitudinal acoustic velocity (${\upsilon _\textrm{a}}$=5940 m/s), are plotted as the blue line in Fig. 4, exhibiting actually very good agreements with the measured results, as seen in Fig. 4. The minor discrepancies may be most probably attributed to the slightly-different air-filling ratios of different PCF samples. The decent agreements between measured and calculated results indicate that the acoustic waves involved in the SBS process near the dominant gain peak still follow the acoustic dispersion relation of the longitudinal-wave approximation [26].
3.3 Multi-peak structures of Brillouin gain spectrum
The measured SBS gain spectra of different PCF samples exhibit multi-peak structures due to the existing of multiple phase-matched hybrid acoustic modes, which results from the coupling between longitude and transverse acoustic waves [4,26], especially at the silica-air boundary. In the previous two sections, only the dominant gain peaks are analyzed. In this section, we demonstrated that the multi-peak structure of the PCF gain spectrum also exhibits apparent structure dependency as we observed in our measurements. As illustrated in Fig. 5, we noticed an interesting trend that as the PCF core diameter decreases, the secondary sub-peak in the SBS gain spectrum deviates further away from the dominant peak, together with a decrease of the sub-peak magnitude. The spacing between the two peaks increases from ∼100 MHz to ∼720 MHz when the PCF core diameter decreases from 3.40 µm to 1.67 µm, see Fig. 5. As the PCF core diameter is further reduced to 1.44 µm, the dramatic peak-splitting effect is observed on the SBS gain spectrum, leading to a quite complicated spectral structure. Such complication might result from further splitting of the existing hybrid modes shared by other PCFs with larger cores. The varied structure of multi-peaks in the Brillouin spectrum has been demonstrated numerically in previous work for wavelength-scale glass-strand [4]. Meanwhile, revealing the exact mechanism for the varied multi-peak structures in PCF would demand further theoretical studies. Our measurements can possibly serve as a practical guideline for engineering the Brillouin gain profile of PCFs in various applications (e.g. Brillouin lasers and amplifiers), indicating the existence of a critical PCF-core size below which the Brillouin gain profile may become highly distorted with excessive new gain peaks.
Fig. 5. Multi-peak structure of Brillouin gain spectra for solid-core PCFs with different core diameters (ranging from 1.44 to 3.40 µm), with spacing between dominant peak and secondary sub-peak marked.
3.4 Polarization-dependent Brillouin gain in PCF
The SBS gain of the PCF sample is polarization-dependent as we measured in the experiment. In principle, the Brillouin gain should be maximum when the pump and probe (Stokes) waves are co-polarized, while being trivial when they are orthogonally polarized, given that the optical fiber is polarization-maintaining (See Fig. 6(a)). We adjusted the polarization state of light for both the pump and probe (Stokes) light waves in our experiments using the fiber-based polarization controllers in the set-up (see Fig. 2) to achieve maximum and minimum gain profiles, corresponding to the co-polarized and orthogonally-polarized scheme. The minimum gain profile is almost flat (zero) throughout the measurement span, indicating that for the short lengths of PCF samples used in the experiment, the polarization-maintaining performances of these samples are reasonably good. In fact, the estimated group-birefringence values of the PCF samples used in our experiments are between 10−4 and 10−5 [29].
Fig. 6. (a) Polarization-dependent Brillouin gain, with maximum gain for co-polarized pump and Stokes wave and minimum gain for orthogonally polarized gain. (c) and (d) The Brillouin gain spectra for PCFs with core diameters of 1.84 µm and 5.01 µm, with maximum gain profile (red) measured under co-polarized scheme and minimum gain profile (gray) under orthogonally polarized scheme.
It can be seen that the two SBS gain peaks (corresponding to two acoustic hybrid modes), reached the minimum values at the same polarization state of the Stokes light. This phenomenon verified the understanding that the sub-peak structures of the SBS gain spectrum is mainly originated from the degeneracy of different acoustic modes in the small PCF core [4], influenced little by the property of optical modes in the PCF.
4. Conclusions and discussions
In this study we investigated experimentally multiple aspects of dependences of the SBS gain spectrum on the PCF core structure. Due to the tight confinement of both optical and acoustic waves in the PCF core, the SBS gain spectrum of the PCF sample differs significantly from that in the conventional SMF. Firstly, the Brillouin gain magnitude is generally one order of magnitude higher than that in the conventional SMF, while the peak SBS gain coefficient of the PCF exhibits a maximum value at a core diameter of ∼1.8 µm. Secondly, we found that the Brillouin frequency shift of the PCF decreases monotonically as the PCF core diameter decreased, which is dominantly caused by the decreasing effective refractive index of the fundamental optical mode in the PCF. Thirdly, the Brillouin gain spectra of different PCF samples exhibit multi-peak structures. While the frequency difference between the dominant and secondary spectral peaks was observed to increase as the core diameter decreased, a complicated peak splitting effect was observed at a PCF core diameter of 1.44 µm (sub-wavelength). We also explored the influence of the air-filling ratio of the core structure upon the Brillouin gain spectrum, although the structural control becomes difficult when demanding a constant core diameter simultaneously. We revealed from preliminary measurements using a few non-ideal PCF samples (with air-filling ratio changed from 0.7 to 0.9 while the core-diameter also changed by ∼10%) that the Brillouin gain spectrum only have trivial changes in terms of the Brillouin shift of the dominant peaks as well as the gain magnitudes. Therefore, we can safely say that in our experiments, the core-diameter variation played a dominant role in the changes of Brillouin gain spectra (as shown in Fig. 3(a) and Fig. 5) over the minor fluctuation of air-filling ratio. Previous work suggested that the air-channels could play an important role in Brillouin spectrum broadening due to multiple peaks [30]. In our case the apparent Brillouin spectrum broadening for PCF-core diameters of 1.44 µm and 5.01 µm might be attributed to this effect (see Fig. 3(a)). Nevertheless, the unambiguous dependencies of Brillouin spectrum upon the PCF air-filling ratio would require further experimental studies. In addition, we observed that due to the reasonably-good polarization-maintaining performances of the short PCF samples, the Brillouin gain can be suppressed, in the experiment, to be nearly zero throughout the measurement span when the polarization states of the pump and Stokes light beams were adjusted to be orthogonal. These experimental measurements and analysis with different PCF samples under elaborate structural controls may provide some valuable clues for practical PCF designs, particularly important for fine adjustments of SBS gain magnitude and frequency shift. This capability of fine tuning of SBS effects in the PCF may be useful for Brillouin fiber laser and distributed Brillouin fiber sensing applications.
Further experimental studies can be implemented for the investigation of the following aspects of Brillouin scattering effects in PCFs. Firstly, since the measured Brillouin shift exhibits a clear dependence upon the PCF-core diameter over a wide range, the local derivative of the relation curve can be used for determine non-uniformity of a long piece of PCF from measured Brillouin bandwidth of the PCF [31]. Secondly, Brillouin gain spectra of PCF samples with sub-wavelength core sizes are very interesting and worth detailed investigations [32]. In this sub-wavelength region, the dispersion relations of both optical and acoustic modes in the PCF core would be highly deformed, meanwhile the moving-boundary effect starts to play a significant role in the resultant Brillouin effect [28]. Thirdly, the threshold behavior of the PCF Brillouin effect [2] is worth systematical studies, since the structural dependence of the Brillouin threshold is critical for the PCF-based Brillouin laser and sensing applications. At last, inter-modal [33] or inter-polarization [34,35] Brillouin scattering that involves specific asymmetric acoustic mode is also of great interests for optical non-reciprocity [36], which also demands further studies on their structural dependencies in solid-core PCFs.
Funding
National High-level Talent Youth Project; National Natural Science Foundation of China (62275254); Shanghai Science and Technology Innovation Action Plan (21ZR1482700); National Postdoctoral Program for Innovative Talents (BX2021328); National Natural Science Foundation of China Youth Science Foundation Project (62205353); China Postdoctoral Science Foundation (2021M703325); Zhangjiang Laboratory Construction and Operation Project (20DZ2210300).
Acknowledgment
We thank Prof. Philip St.J. Russell for useful discussions and suggestions on some key descriptions and interpretations of experimental observations.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
The data underlying the results presented in this paper is not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. W. Boyd, Nonlinear Optics, 3 ed. (Academic Press, 2010).
2. E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539–541 (1972). [CrossRef]
3. M. S. Kang, A. Nazarkin, A. Brenn, et al., “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009). [CrossRef]
4. P. Dainese, P. S. J. Russell, N. Joly, et al., “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006). [CrossRef]
5. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010). [CrossRef]
6. D. A. Fishman and J. A. Nagel, “Degradations due to stimulated Brillouin scattering in multigigabit intensity-modulated fiber-optic systems,” J. Lightwave Technol. 11(11), 1721–1728 (1993). [CrossRef]
7. C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013). [CrossRef]
8. R. G. Smith, “Optical Power Handling Capacity of Low Loss Optical Fibers as Determined by Stimulated Raman and Brillouin Scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef]
9. S. Gundavarapu, G. M. Brodnik, M. Puckett, et al., “Sub-hertz fundamental linewidth photonic integrated Brillouin laser,” Nat. Photonics 13(1), 60–67 (2019). [CrossRef]
10. K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “cw Brillouin laser,” Appl. Phys. Lett. 28(10), 608–609 (1976). [CrossRef]
11. D. Marpaung, B. Morrison, R. Pant, et al., “Frequency agile microwave photonic notch filter with anomalously high stopband rejection,” Opt. Lett. 38(21), 4300–4303 (2013). [CrossRef]
12. D. M. Nguyen, B. Stiller, M. W. Lee, et al., “Distributed Brillouin Fiber Sensor With Enhanced Sensitivity Based on Anti-Stokes Single-Sideband Suppressed-Carrier Modulation,” IEEE Photon. Technol. Lett. 25(1), 94–96 (2013). [CrossRef]
13. D. M. Chow, Z. Yang, M. A. Soto, et al., “Distributed forward Brillouin sensor based on local light phase recovery,” Nat. Commun. 9(1), 2990 (2018). [CrossRef]
14. B. Stiller, M. Merklein, C. Wolff, et al., “Coherently refreshing hypersonic phonons for light storage,” Optica 7(5), 492–497 (2020). [CrossRef]
15. Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored Light in an Optical Fiber via Stimulated Brillouin Scattering,” Science 318(5857), 1748–1750 (2007). [CrossRef]
16. K. Shiraki, M. Ohashi, and M. Tateda, “Suppression of stimulated Brillouin scattering in a fibre by changing the core radius,” Electron. Lett. 31(8), 668–669 (1995). [CrossRef]
17. V. I. Kovalev and R. G. Harrison, “Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers,” Opt. Lett. 31(2), 161–163 (2006). [CrossRef]
18. A. Liu, “Suppressing stimulated Brillouin scattering in fiber amplifiers using nonuniform fiber and temperature gradient,” Opt. Express 15(3), 977–984 (2007). [CrossRef]
19. M. Nikles, L. Thevenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997). [CrossRef]
20. P. S. J. Russell, “Photonic Crystal Fibers,” Science 299(5605), 358–362 (2003). [CrossRef]
21. P. Dainese, P. S. J. Russell, G. S. Wiederhecker, et al., “Raman-like light scattering from acoustic phonons in photonic crystal fiber,” Opt. Express 14(9), 4141–4150 (2006). [CrossRef]
22. P. S. J. Russell, E. Marin, A. Díez, et al., “Sonic band gaps in PCF preforms: enhancing the interaction of sound and light,” Opt. Express 11(20), 2555–2560 (2003). [CrossRef]
23. Y. Koyamada, S. Sato, S. Nakamura, et al., “Simulating and Designing Brillouin Gain Spectrum in Single-Mode Fibers,” J. Lightwave Technol. 22(2), 631–639 (2004). [CrossRef]
24. L. Xiao, M. S. Demokan, W. Jin, et al., “Fusion Splicing Photonic Crystal Fibers and Conventional Single-Mode Fibers: Microhole Collapse Effect,” J. Lightwave Technol. 25(11), 3563–3574 (2007). [CrossRef]
25. C. Wolff, M. J. A. Smith, B. Stiller, et al., “Brillouin scattering—theory and experiment: tutorial,” J. Opt. Soc. Am. B 38(4), 1243–1269 (2021). [CrossRef]
26. G. S. Wiederhecker, P. Dainese, and T. P. M. Alegre, “Brillouin optomechanics in nanophotonic structures,” APL Photon. 4(7), 071101 (2019). [CrossRef]
27. B. A. Auld, Acoustic fields and waves in solids, 2nd ed. (Wiley, 1992).
28. O. Florez, P. F. Jarschel, Y. A. V. Espinel, et al., “Brillouin scattering self-cancellation,” Nat. Commun. 7(1), 11759 (2016). [CrossRef]
29. M. Pang, W. He, X. Jiang, et al., “All-optical bit storage in a fibre laser by optomechanically bound states of solitons,” Nat. Photonics 10(7), 454–458 (2016). [CrossRef]
30. J.-C. Beugnot, T. Sylvestre, D. Alasia, et al., “Complete experimental characterization of stimulated Brillouin scattering in photonic crystal fiber,” Opt. Express 15(23), 15517–15522 (2007). [CrossRef]
31. D. M. Chow, J. C. Tchahame, A. Denisov, et al., “Mapping the Uniformity of Optical Microwires Using Phase-Correlation Brillouin Distributed Measurements,” in Frontiers in Optics 2015 (Optica Publishing Group, San Jose, California, 2015), p. FW4F.4.
32. J.-C. Beugnot, S. Lebrun, G. Pauliat, et al., “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5(1), 5242 (2014). [CrossRef]
33. P. F. Jarschel, E. Lamilla, Y. A. V. Espinel, et al., “Intermodal Brillouin scattering in solid-core photonic crystal fibers,” APL Photon. 6(3), 036108 (2021). [CrossRef]
34. H. E. Engan, “Analysis of polarization-mode coupling by acoustic torsional waves in optical fibers,” J. Opt. Soc. Am. A 13(1), 112–118 (1996). [CrossRef]
35. X. Zeng, W. He, M. H. Frosz, et al., “Stimulated Brillouin scattering in chiral photonic crystal fiber,” Photonics Res. 10(3), 711–718 (2022). [CrossRef]
36. M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5(9), 549–553 (2011). [CrossRef]
