Adrien Godet

Jacques Chretien

Kien Phan

Jean-Charles Beugnot
email: jc.beugnot@femto-st.fr

Micronewton nanofiber force sensor using Brillouin scattering

We present a new class of force sensor based on Brillouin scattering in an optical nanofiber. The sensor is a silica nanofiber of few centimeters with submicron transverse dimension. This extreme form factor enables one to measure forces ranging from 10 𝜇N to 0.2N. The linearity of the sensor can be ensured using the multimode character of the Brillouin spectrum in optical nanofibers. We also demonstrated non-static operation and a competitive signal to noise ratio as compared to commercial resistive force sensor.

Introduction

Force measurement is a major technological challenge as new applications are identified in air/spacecraft industries, electric power facilities or micro and nanotechnologies for biology [START_REF] Wei | An overview of micro-force sensing techniques[END_REF][START_REF] Liu | Miniature fiber-optic force sensor based on low-coherence fabry-pérot interferometry for vitreoretinal microsurgery[END_REF][START_REF] Polygerinos | Novel miniature mricompatible fiber-optic force sensor for cardiac catheterization procedures[END_REF][START_REF] Song | Large-capacity high-resolution optomechanical mass sensing based on free-space optical cavity[END_REF][START_REF] Puangmali | Miniature 3-axis distal force sensor for minimally invasive surgical palpation[END_REF]. Force measurement systems are typically based on resistive, piezo-resistive, capacitive, piezoelectric, magnetic, inductive, or optic measurement methods [START_REF] Eichenfield | A picogram-and nanometre-scale photonic-crystal optomechanical cavity[END_REF][START_REF] Pernice | Analysis of short range forces in opto-mechanical devices with a nanogap[END_REF][START_REF] Yu | Sensing and exploiting static femto-newton optical forces by a nanofiber with white-light interferometry[END_REF]. While each has benefits and drawbacks, fiber optical sensors are uniquely suited while electromagnetic immunity and high resolution are required, or chemically aggressive and conductive liquids environments are present [START_REF] Liang | Multi-component fbg-based force sensing systems by comparison with other sensing technologies: a review[END_REF]. The resolution and dynamic range of different force measurement systems are reported in Fig. 1. Notice that for every technology there is a compromise between resolution and dynamic range. Most optical fiber sensors, as shown in green and blue in Fig. 1, are using either fiber Bragg gratings (FBG), Fabry-Perot or another interferometric scheme. They offer both a broad detection range and a high sensitivity that make them very attractive for applications such as earthquake monitoring or automation systems [START_REF] Huang | Broadband fbg resonator seismometer: principle, key technique, self-noise, and seismic response analysis[END_REF]. Concurrently, Brillouin fiber sensors were developed for distributed sensing. They were rarely used as force measurements. However, they offer a major benefit which is the absence of grating or interferometer manufacturing, because they use the interaction between light and an acoustic wave that behaves as a moving Bragg reflector. As shown in blue in Fig. 1, their use is limited to milli-Newton force measurements.

With the development of MEMS a new class of high sensitive sensors have come of age, as they benefit from smaller dimensions, their sensitivity can range from micro to tens of nano Newtons, as shown in grey and purple in Fig. 1. MEMS sensors dynamic range covers tens of milli-Newton to dozen Newtons with performances no single mode fiber (SMF) based Fiber could ever reach. Achieving the higher sensitivity requires 3-D scale reduction. In the case of SBS fiber based force sensors, it limits the efficiency since long distance of interaction are needed to provide strong Brillouin scattered signal. Due to the small transverse dimension combined with the large longitudinal dimension, the optical nanofiber (ONF) possesses the requisite sensitivity at scale thanks to the reduced transverse dimension while providing strong Brillouin scattered signal thanks to the longer longitudinal dimension. Consequently, it's possible to increase force sensitivity by several orders of magnitude compared to fiber force sensor as Bragg grating in conventionally sized fibers. [START_REF] Tang | A highly sensitive force sensor based on two identical fiber bragg gratings[END_REF][START_REF] Kou | Microfiber-based bragg gratings for sensing applications: a review[END_REF][START_REF] Chung | Highly sensitive compact force sensor based on microfiber bragg grating[END_REF][START_REF] Li | High-sensitivity microfiber strain and force sensors[END_REF].

In our previous work, we demonstrated high sensitivity longitudinal stress measurements using Brillouin scattering in ONFs [START_REF] Godet | Nonlinear elasticity of silica nanofiber[END_REF]. Knowing the nanoscale cross-section of nanofibers (the waist part of the tapered fiber), it paves the way for a new class of highly sensitive force sensors whose sensitivity scales inversely with the cross-section. In Fig. 1, we report few of our experimental results, described below, that demonstrate that this new class of sensors covers a large area of measurement sensitivities and dynamic range. [START_REF] Liang | Multi-component fbg-based force sensing systems by comparison with other sensing technologies: a review[END_REF][START_REF] Gao | High-sensitivity temperature and strain measurement in dual-core hybrid tapers[END_REF]), not limited to fiber based sensor (Load cell MEMS [START_REF]F329 and f332 Deci-Newton Loadcell[END_REF] and femtotools [START_REF]anoindenter, the ft-nmt04[END_REF]) and for previous results obtain with tapered optical fiber [START_REF] Chen | A miniature reflective micro-force sensor based on a microfiber coupler[END_REF][START_REF] Luo | Ultra-highly sensitive surface-corrugated microfiber bragg grating force sensor[END_REF][START_REF] Wang | Thermal and mechanical properties of tapered single mode fiber measured by ofdr and its application for high-sensitivity force measurement[END_REF][START_REF] Wieduwilt | High force measurement sensitivity with fiber bragg gratings fabricated in uniform-waist fiber tapers[END_REF]. The black line represent the limit resolutions equal to dynamic. In comparison with other nanofiber platforms, our solution with Brillouin scattering show higher resolution and much lower losses at the same time. And the inline measurement without functionalisation of fiber open new possibility in the region (1um resolution, 1mN dynamic range) in complementary to fiber tip sensor [START_REF] Pevec | Miniature all-fiber force sensor[END_REF][START_REF] Zou | Fiber-tip polymer clamped-beam probe for high-sensitivity nanoforce measurements[END_REF].

The article is organized as follows. We first give the theoretical background of the nanofiber Brillouin sensor. We explain how high sensitivity is obtained and describe the experimental setup. We then present the measurements obtained with this setup, and how the sensitivity evolves with the diameter of the nanofiber and the operating range of our sensors. In the third section, we demonstrate how to ensure the linearity of the sensor response through several examples. Finally, we show that our sensors are not limited to the static regime and compare the sensor signal-to-noise ratio with a commercial strain gauge sensor.

Optical force sensor

The force sensor is built around an optical nanofiber (ONF) obtained by drawing a standard optical fiber using the well-known heat-brush technique until its diameter reaches the nanoscale [START_REF] Birks | The shape of fiber tapers[END_REF][START_REF] Tong | Micro/nanofibre optical sensors: challenges and prospects[END_REF] as shown in Fig. 1(a). When a pump laser light propagates in the ONF (in blue), it interacts with the acoustic waves supported by the ONF. Those acoustic waves behave as a moving mirror and reflect light (pink) via Brillouin scattering. This backscattered Stokes light is shifted in frequency by the Doppler effect. This shift is therefore related to the elastic waves velocity. If a force is applied to one end of the fiber (purple arrow), the nanowire under stress will be strained. The yellow to red color gradient shows how the stress and strain are distributed along the tapered fiber. Because of the strong tapering ratio (125 µm to a few hundreds of a nanometer) the strain is much stronger in the center part call the optical nanofiber [START_REF] Holleis | Experimental stress-strain analysis of tapered silica optical fibers with nanofiber waist[END_REF].

Since the strain alters the acoustic wave's velocity, and causes a variation of the Brillouin frequency shift, we can deduce the applied force by measuring this frequency shift. In our setup shown in Fig. 2, the optical nanofiber is held securely with clamps. A commericial resistive force sensor (ME-systeme KD-78) allows us to measure the longitudinal force that we apply due to motorized translation stages (Newport XML210). The frequency shift is measured through a heterodyne setup. A part of the pump beam is mixed with the backscattered beam to produce an optical beat measured with an radiofrequency electric spectrum analyser. The frequency shifts for each acoustic mode are retrieved with respect to the measured force using a lorentzian fit [START_REF] Godet | Brillouin spectroscopy of optical microfibers and nanofibers[END_REF].

𝜎 𝑧𝑧 = 𝐹 𝑧𝑧 𝑆 , (1)
where 𝐹 𝑧𝑧 is a longitudinal force and 𝑆 is the cross-sectional area over which the force is applied [START_REF] Royer | Elastic waves in solids I: Free and guided propagation[END_REF]. This expression shows how a very small force 𝐹 𝑧𝑧 can result in relatively strong stress 𝜎 𝑧𝑧 providing that the cross-sectional area 𝑆 is small. Additionally, the approximated 1-D Hooke's law tells us that the longitudinal stress represented as

𝜎 = 𝐸𝜖, (2)
where 𝐸 is Young modulus (73 GPa in silica) and 𝜖 𝑧𝑧 is the longitudinal tensile strain. In this case, for a sufficiently small cross-sectional area, a small longitudinal force can lead to a measurable tensile strain 𝜖 𝑧𝑧 . This is schematically illustrated by the figure 2(a), where a tapered optical fiber is depicted. A longitudinal force is applied at one end. Most of the strain appears in the central region highlighted in red where the cross sectional area is the smallest [START_REF] Godet | Nonlinear elasticity of silica nanofiber[END_REF]. In a standard optical fiber, the Brillouin frequency shift varies linearly with an applied tensile strain 𝜖 𝑧𝑧 inferior to 2% and a normalized Brillouin strain coefficient that is given by

𝐶 𝜖 = 1 𝜈 𝐵 𝜕𝜈 𝐵 𝜕𝜖 𝑧𝑧 , (3)
where 𝜈 𝐵 , Brillouin frequency shift. For a SMF-28 it is typically 4.25 [START_REF] Galindez-Jamioy | Brillouin distributed fiber sensors: an overview and applications[END_REF]. Combining eq 1, 2 and 3 we can compute the sensitivity S that gives the Brillouin Frequency shift for a given applied force,

𝑆 = 𝜕𝜈 𝐵 𝜕𝐹 𝑧𝑧 = 4 𝜈 𝐵 𝐶 𝜀 𝜋𝐸 𝑑 2 , ( 4
)
where d is the diameter of the silica nanofiber. This expression scales inversely with the square of the nanofiber diameter which means that reducing an optical fiber diameter from 125𝜇m to few hundreds of a nanometer could increase the sensitivity by four or more orders of magnitude.

Although the benefit from reducing the diameter of the fiber is obvious, it also raises questions. In order to make a usable sensor, it is necessary to discriminate the light scattered in the ONF from that scattered in the standard fibers of the optical setup. Also the scattered light must be strong enough to allow detection and measurement of the Brillouin frequency shift. This is only possible if the acousto-optic interaction is strong which requires low optical losses, long (few centimeters) ONF and strong light confinement. We will see in the next paragraph, through the experimental results that an ONF can fulfill all those requirements.

Sensitivity enhancement

Brillouin spectra were measured with a ONF of 630 nm diameter with different applied force going from 0 to 611 𝜇N as shown by Fig. 3(a). The ONF is 40 mm long and the losses are below 0.2 dB at telecommunication wavelength. We observe many acoustic resonances, such as hybrid acoustic waves (HAW) in the range (8 GHz-9 GHz), the resonances due to the taper transitions (9 GHz to 10.8 GHz) and the untapered optical fibers (10.8 bGHz). The presence of many resonances is due to the fact that in an ONF the acoustic waves behave as modes, each having its own velocity and therefore Brillouin frequency shift. It enables to discriminate easily the contribution of the ONF to the Brillouin spectrum and thus eliminates the need for active or passive stabilization of the fibers belonging to the rest of the heterodyne circuit. Those resonances experience a blue frequency shift that increases with the applied force. The main resonance just below 11 GHz comes from the single mode fibers contribution to the Brillouin spectrum (See Fig. 2). As those fibers belong to the heterodyne setup, no force is applied to them. It results in the absence of frequency shift for the 11 GHz resonance. Similar measurements were performed for 10 different ONF diameters ranging from 18 µm to 630nm. The geometric parameters of tapered optical fibers as nanofiber diameter and homogeneity are precisely estimated by Brillouin spectroscopy [START_REF] Godet | Brillouin spectroscopy of optical microfibers and nanofibers[END_REF]. For each diameter, the sensitivity is measured for strain smaller than 2% for 9-10 GHz Hybrid Acoustic Waves (HAW) resonances [START_REF] Godet | Nonlinear elasticity of silica nanofiber[END_REF] and reported in Fig. 3(b). The experiment data were fitted using equation 4. A force sensitivity around 1.9 MHz/𝜇N is reached for an ONF of diameter 630 nm. This means that force sensitivity is higher by more than five orders of magnitude compared to an untapered fiber [START_REF] Galindez-Jamioy | Brillouin distributed fiber sensors: an overview and applications[END_REF]. The analytical model is in good agreement with the experimental measurements and the small differences can be explained by small temperature variations.

Range

In order to explore the potential operating range of a family of force sensors based on optical nanofibers, we performed measurements with ONF of diameters ranging from 580 nm to 18 microns. All the ONFs were 40 mm long with losses less than 0.2 dB. The results are shown in Fig. 4, where the Brillouin frequency shifts are plotted as a function of applied force (the total nanofiber elongation is shorter to 2% to maintain the linear regime). Note that logarithmic scales were used to better represent the wide operating range of our sensor family. This shows that we could in principle make measurements of few tens of micrometers and even reach values lower than the reliability limit of our commercial strain gauge (blue area). We also note that for a diameter of 18 𝜇m, we can measure forces approaching 0.2 N. This limit can be pushed back since nanofibers of diameters going up to hundreds of micron could be developed with further manufacturing effort. The improvement in sensitivity for smaller diameters is again confirmed by Fig. 4, as the measurements from small ONF diameters correspond to small forces and measurements associated with large diameters to large forces. However, a number of details raise questions. The measurements for the 580 and 630 nm diameters appear to correspond to curves that would intersect. The same is true for the 1.07 and 1.25 𝜇m diameters. Also, the ONF points of diameter 9 𝜇m seem to show an inflection. We will explain these observations through the following three examples in the following.

Linearization

In this paragraph, we will focus on measurements obtained for ONF with diameters 610 nm, 855 nm and 1570 nm. The results are shown in Fig. 5(a-c), where the Brillouin spectrum are plotted in false color with respect of applied force. In those cases, the ONF were stretch until 6% showing clear sensitivity variation from one resonance to the other as well as nonlinear behaviour. Fig. 5(a) for example, shows the surface acoustic waves (SAW) between 5 and 6.5 GHz contribution almost crossing each other for higher applied force. The HAW resonance starting here at 10 GHz in Fig. 5(c) is also a great example of nonlinearity. It can be seen that the 10 GHz and 11 GHz resonances move closer together as the applied force increases, then move apart for forces exceeding 5 mN. The nonlinear behavior results from both the third order elastic constant and from the acoustic modal dispersion [START_REF] Godet | Nonlinear elasticity of silica nanofiber[END_REF]. A common point of Fig. 5(a-c) is the presence of three resonances corresponding to three acoustic modes. For each resonance from Fig. 5(a-c), we interpolated the Brillouin frequency variation as a function of the applied force by a third order polynomial. The three obtained polynomials form a 3x3 system

Δ𝜈 0 Δ𝜈 1 Δ𝜈 2 = 𝑚 11 𝑚 12 𝑚 13 𝑚 21 𝑚 22 𝑚 23 𝑚 31 𝑚 32 𝑚 33 𝐹 𝐹 2 𝐹 3 , (5)
with the coefficient (𝑚 𝑖 𝑗 ) {1,2,3} 2 forming the matrix M. Thanks to the acoustic mode dispersion, each Brillouin frequency shift varies differently enough for equation 5 to be invertible. The matrix M inversion allows us to obtain 𝐹 𝑙𝑙𝑖𝑛 as a linear combination of the Brillouin frequency shift of each resonance.

𝐹 lin = 𝑎 01 Δ𝜈 0 + 𝑎 02 Δ𝜈 1 + 𝑎 03 Δ𝜈 2 , (6)
where the weights 𝑎 01 , 𝑎 02 and 𝑎 03 are the coefficient of the first line of the inverted matrix 𝑀 -1 . We plot in Fig. 6(a), the measured 𝐹 lin with respect to the applied force measured with the strain gauge for the three measurements shown Fig. 5. The measurements from the blue, orange and green dots coming from the 1570 nm, 855 nm and 610 nm diameters ONFs respectively show excellent linearity (gray line). We thus demonstrate that the quadratic and cubic contribution of the nonlinearity are well compensated by the contibution of the second and third resonance frequency shifts. We also show that interpolating the measured frequency shifts with cubic polynomials, neglecting the fourth order coefficient is a valid approximation. The same results, are also shown in log scale in Fig. 6(b), to better illustrate the range covered by the three experiments. Measurements ranging from 20 𝜇N to 7 mN confirm that the linearization does not affect the range while maintaining a good sensitivity.

Non-static regime and Signal-to-Noise Ratio

One of functionalities for a force sensor is the ability to measure in a non-static regime. This could be usefull if a control loop mechanism employing feedback is used to dynamically maintain or alter the applied force. Also for this kind of application a good signal-to-noise ratio is required.

To test the ability of our sensor to perform these kind of measurements, we fabricated an ONF with a diameter of 885 nm with an homogeneous section of 40 mm length. We isolated a Brillouin resonances at 8 GHz corresponding to hybrid elastic mode (L(0,2)) which is one of the most sensitive to longitudinal strain. On the top of Fig 7(a), we represent the stretching pattern applied to the ONF as a function of time. At each cycle, the ONF is stretched by a distance increased of 100 µm and then released to its inital state. We sequentially launch 15 cycles manually until a maximum stretch of 1500 µm is reached. As shown in Fig. 7(b) The force applied by the fiber stretching is measured by both the resistive force sensor (blue) and Brillouin scattering (red). The red curve showing the response of the optical sensor clearly presents a better signal-to-noise ratio. This is particularly evident for the first cycle, for which the commercial strain gauge has difficulty in producing a clean measurement. We can also guess from the slopes that the optical sensor has a better frequency response. To evaluate the noise performance, we have plotted the distribution of normalized measurements around the mean value for each of the sensors in Fig. 8(a). The distribution produced by the resistive commercial sensor in blue shows a broad Gaussian distribution (standard deviation of 0.0167) while the normalized distribution obtained with the optical sensor is mostly concentrated in the central bin. By reducing the size of the bins by a factor of 90, the normalized distribution of the optical measurements can be represented again and show a Gaussian of standard deviation 2.3 × 10 -4 . This confirms the better signal-to-noise performance of our sensor by nearly two orders of magnitude. In order to confirm has been considered within a cycle of increasing and decreasing applied force with the same step. The force measurement with Brillouin scattering, depicted in fig 9, demonstrate the absence of hysteresis. The better SNR performance and dynamic performance makes the ONF force sensor a better candidate for non-static measurements requested for control loop PID based force actuator application.

Conclusion

In conclusion, we have presented a new family of all-optical force sensors based on the measurement of Brillouin scattering in optical nano fibers. The unusual properties of these ONFs, such as low losses, 1:80,000 form factors and modal guidance of both optical and acoustic waves, give a whole new dimension to the traditional fiber Brillouin sensors. Thus we report measurements ranging from the tens of micro Newton to 0.2 N which allows this family of sensors to cover a whole new measurement range. The reduction of the diameter of the ONF to the nanoscale over a length of a few centimeters makes it possible to reach sensitivities of the order of ten micro Newtons while preserving a significant measurement range. The linearity of the sensor, essential for a real-world application, can be obtained by taking advantage of the presence of several acoustic modes in the nanowire and their dispersion and without any specific signal processing. Finally, we have compared our sensor with a commercial resistive force sensor showing its superiority in non-static operation and showing an increase of signal-to-noise ratio close to two order of magnitude. Future improvements include packaging and surface coating in order to protect the ONF from environmental contamination.

Fig. 2 .

 2 Fig. 2. (a) Schematic of the nanofiber force sensor. (b) Experimental setup for measuring the longitudinal force as a function of the Brillouin spectrum. Inset picture: ME-systeme KD-78 resistive force sensor completed with an amplifier GSV-2TSD-DI One of the most challenging specification to meet for an optical force sensor is high sensitivity. ONF uses the expression of the longitudinal normal strain to improve sensitivity

Fig. 3 .

 3 Fig. 3. (a) Experimental backscattering Brillouin spectrum in a ONF with a diameter of 630nm a length of 40 mm for 5 different applied forces. (b) frequency sensitivity normalized to 𝜇N as a funtion of the nanofiber diameter, theory (red), experiment (blue point).

Fig. 4 .

 4 Fig. 4. Measurements of Brillouin frequency shift as a function of applied force for different diameter of tapered optical fibres ranging from 580 nm to 18 𝜇m. Force and frequency are plotted in log scale. Blue area for measurements inferior to 70 𝜇N is where our measurements are less reliable due to the strain gauge limitation. The pink area represents the limits of our ability to measure Brillouin frequency shifts with Lorentzian fits.

Fig. 5 .

 5 Fig. 5. (a), (b) and (c) In false color, experimental Brillouin spectrum with respect of the applied Force measured with the strain gauge for a 1570 nm, 855 nm and 610 nm diameter ONF respectively. The Circle represent the theoretical model from [15].

Force measured with 3

 3 Brillouin lines (mN)

Fig. 6 .

 6 Fig. 6. (a) In false color, Brillouin spectrum computed with Finite-Element Method for different ONF diameters. Black line: 855 nm diameter. (b) Brillouin frequencies for the TR21 (red), L01, L02 and L03 (blue) acoustic mode resonances for a 855 nam diameter ONF with respect to strain. In gray, same resonances without modal dispersion.

Fig. 8 .FrequencyFig. 9 .

 89 Fig. 8. (a) Measurements distribution around the mean force reported in fig 7(a) for commericial resistive sensor (blue) and optical sensor (light brown). (b) Optical force measurements distribution around the mean force reported in fig 7(a) with proper scale.

Acknowledgement

Authors thank Maxime Zerbib for cycling force measurement.

Funding

The authors would like to acknowledge the financial support of Agence Nationale de la Recherche (ANR) via EQUIPEX+ SMARTLIGHT (ANR-21-ESRE-0040) and EIPHI Graduate School (ANR-17-EURE-0002) and Bourgogne-Franche-Comté Region.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Disclosures

The authors declare no conflicts of interest.


Fan Yang
email: fanyang808@gmail.com

Flavien Gyger

Adrien Godet

Jacques Chrétien

Li Zhang

Meng Pang

Jean-Charles Beugnot
email: jean-charles.beugnot@femto-st.fr

Luc Thévenaz

Large evanescently-induced Brillouin scattering at the surrounding of a nanofibre

Brillouin scattering has been widely exploited for advanced photonics functionalities such as microwave photonics, signal processing, sensing, lasing, and more recently in micro-and nano-photonic waveguides. So far, all the works have focused on the opto-acoustic interaction driven from the core region of micro-and nano-waveguides. Here we observe, for the first time, an efficient Brillouin scattering generated by an evanescent field nearby a sub-wavelength waveguide embedded in a pressurised gas cell, with a maximum gain coefficient of 18.90±0.17 m -1 W -1 . This gain is 11 times larger than the highest Brillouin gain obtained in a hollow-core fibre and 79 times larger than in a standard single-mode fibre. The realisation of strong free-space Brillouin scattering from a waveguide benefits from the flexibility of confined light while providing a direct access to the opto-acoustic interaction, as required in free-space optoacoustics such as Brillouin spectroscopy and microscopy. Therefore, our work creates an important bridge between Brillouin scattering in waveguides, Brillouin spectroscopy and microscopy, and opens new avenues in light-sound interactions, optomechanics, sensing, lasing and imaging.

Brillouin scattering involves light-sound interactions and has been used in nonlinear optics [1][2][3][4], microwave photonics [5], slow and fast light [6], lasing [7], sensing [8] and imaging [START_REF] Prevedel | Antonacci[END_REF]10]. It has been observed in various platforms, including optical fibres [8,[11][12][13][14], whisperinggallery-mode resonators [15][16][17][18] and integrated waveguides [19][20][21][22][23][24][25].

The advent of micro-and nano-photonic waveguides has recently driven a renewed interest for Brillouin scattering as a tool to perform nonlinear optics and optical signal processing in waveguides. In 2006, a photonic crystal fibre was used to greatly intensify the Brillouin interactions by confining both acoustic and optical fields to a 1 µm microstructured core [12]. Another interesting work proposed that Brillouin interactions could be drastically enhanced by radiation pressure on subwavelengthscale waveguide boundaries [26], and triggered intense research interest into Brillouin scattering in micro-and nano-photonic waveguides. Since the gain of the interaction is proportional to the acousto-optic overlap integral as well as to the pump power, two different approaches have been addressed to increase the Brillouin interactions: the first one focused on the design of waveguides with good optical and acoustic fields overlap [12,13,[19][20][21], while the second approach used ultra-low-loss waveguides with high-Q resonant enhancement [15-18, 22, 24]. Generally, all prior works were devoted to generating efficient Brillouin scattering in the core region of the waveguide. So far, no strong Brillouin scattering has been gen-erated from an optical waveguide into the surrounding medium located in the waveguide's vicinity.

Free-space Brillouin scattering has found applications in microscopy: in the past decade, Brillouin microscopy has been used for analysing the mechanical properties of biological samples [27] and hydrogel materials [28]. It provides label-free, non-contact, 3D imaging capabilities at typical optical resolution. So far, Brillouin microscopy has exclusively used free-space illumination, which requires careful optical alignment and has limited throughput [29]. Fluorescence microscopy based on total internal reflection in photonic waveguides [30] and structured illumination microscopy [31] solve these issues for the case of fluorescence microscopy; however, there is currently no alternative for Brillouin microscopy.

Here, we use a nanofibre waveguide and report the first observation of strong Brillouin scattering generated by the evanescent field of a guided lightwave in pressurised carbon dioxide gas located in the immediate vicinity of the nanowaveguide. We obtain a 11-times higher peak Brillouin gain coefficient in the nanofibre gas cell compared to the highest Brillouin gain realised in a hollowcore fibre gas cell [8] and 79-times higher compared to that in a standard single-mode fibre. The higher gain results from a tighter optical mode confinement as well as a large overlap between the evanescent optical field and the acoustic mode in the surrounding medium. Our results pave the way for many potential applications in sensing, lasing, signal processing, etc and suggest a way to realise a novel waveguide-based Brillouin spectroscopy and microscopy that no longer requires free-space illumination.

Results

Operation scheme. Brillouin scattering is an inelastic process within a medium involving a net energy transfer from the optical fields to the medium or vice versa. Spontaneous Brillouin scattering occurs when light scatters from thermally-excited sound waves, giving rise to frequency-shifted Stokes (i.e. phonon generation) and anti-Stokes waves (phonon annihilation). A conceptual illustration of Brillouin scattering around a nanowaveguide is shown in Fig. 1(a). In a waveguide, only photons scattered in the backward and forward directions are guided and hence detected. Here we focus on backward Brillouin scattering. The backward scattering efficiency is maximised when phase matching between all interacting waves is satisfied. For the Stokes process, an incident pump photon of frequency ω P is converted to a lower frequency Stokes photon of frequency ω S through the scattering from the acoustic wave and thereby creating a phonon of angular frequency Ω. The phase matching condition requires ω P = ω S + Ω and k P = k S + q, where k P , k S and q are the wave vectors of the pump, Stokes, and phonon modes, respectively. In this case, the Brillouin frequency shift ν B , which is the frequency difference between pump and Stokes waves under phase matching condition, is given by [START_REF] Boyd | Nonlinear Optics -3rd Edition[END_REF]:

ν B = 2n eff v a /λ, (1)
where n eff is the effective refractive index of the optical mode, v a is the acoustic velocity of the medium -the surrounding gas in our case -and λ is the pump wavelength in vacuum.

The small nanofibre dimensions (740 nm in this work) compared to the optical wavelength results in 58% of the light field intensity propagating outside the nanofibre, which can interact with the surrounding gas or other fluid material. This evanescent field is visible in the finiteelement simulation of the power distribution of the fun-damental optical mode illustrated in Fig. 1(b). The gas molecular displacement of the acoustic mode along the fibre axis direction is shown in Fig. 1(c) (see Methods for the simulation details). Nanofibre gas cell. The diameter of the nanofibre is elaborately chosen in order to provide the maximum Brillouin backscattering gain generated from the elastic wave in the gas surrounding the silica nanofibre. The large evanescent field creates a strong electrostrictive force spatially distributed around the nanofibre. The elastic wave in the gas generated via electrostriction is confined by the evanescent optical field. Figure 2 shows a numerical calculation giving the map of the Brillouin spectrum in 40 bar CO 2 surrounding a nanofibre with diameters ranging The backward Brillouin scattering of the nanofibre is measured with a set-up we developed in 2017 [START_REF] Godet | [END_REF]. The Brillouin signal is normalised to the of the electrical spectrum analyser when there is no pump. The Brillouin spectra from the surface acoustic modes and the hybrid acoustic modes of the silica nanofibre during fabrication are calculated by the model in [START_REF] Godet | [END_REF]. The blue line shows the measured Brillouin spectrum of the nanofibre on the stages after tapering. The dotted red line shows the simulated Brillouin spectrum of the nanofibre with a waist diameter of 740 nm. The dotted purple line shows the measured Brillouin spectrum of the nanofibre after packaging the nanofibre in a metallic tube with a 2.15% strain. Note that the signal from SMF pigtail is not taken into account in the simulation and any response from interactions in the gas is not visible in the frequency range covered by the set-up. Brillouin scattering from the standard single-mode fibre pigtail of the nanofibre is also observed at a frequency of 10.86 GHz. The inset shows the fabricated nanofibre gas cell with the untapered SMF segments exiting the gas cell at each end. Gases can be pressurised in or vacuum pumped out through the gas inlet/outlet port.

from 0.5 to 1.5 µm. This calculation indicates the geometrical sizes of the nanofibre maximising the light-sound interaction in the surrounding gas, which is a trade-off between the optical mode's effective overlap and the optical intensity in the evanescent field. The optimal diameter of the nanofibre maximising Brillouin scattering by the evanescent field is calculated to be 740 nm.

The detailed fabrication process of the nanofibre gas cell is described in Methods. We infer the waist diameter of the nanofibre with a few nanometre sensitivity by mapping the backscattered Brillouin spectrum along the optical fibre taper and fitting with numerical simulations of the elastodynamic equations (a technique we developed in 2017 [START_REF] Godet | [END_REF]). The fabricated nanofibre sample presented in this work shows a waist diameter 740 nm ± 5 nm, a waist length of 10 cm, two adiabatic transitions of 79.4 mm ± 5 mm of total length and an insertion loss of 0.14 dB. The backscattered Brillouin signal of the nanofibre is shown in Fig. 3. The blue and dotted red lines in Fig. 3 are respectively the experimental and numerical simulation results in the nanofibre after the tapering process. The experimental Brillouin spectrum, including the sur-face acoustic modes and hybrid acoustic modes, matches perfectly with the numerical simulation for a nanofibre with a diameter of 740 nm ± 5 nm. After fabrication, the nanofibre is packaged in a simple metallic tube as a gas cell while applying a controlled strain. The measured Brillouin spectrum after applying the strain is shown in the dotted purple line of Fig. 3. As a result of the strain, the resonances due to surface acoustic modes and hybrid acoustic modes both shift to higher frequencies, perfectly matching with our previous analysis [34].

Evanescently-induced Brillouin scattering in the nanofibre gas cell. The Brillouin gain coefficient and spectrum can be obtained from a spontaneous Brillouin scattering measurement [35]. The detailed calculation and calibration of the Brillouin gain coefficient from the spontaneous Brillouin measurements are described in Supplementary Section 1. All the experiments were performed at an environmental temperature of 24 ± 1 • C. The output of a continuous-wave external-cavity diode laser (ECDL) is split: one branch is amplified by an erbium-doped fibre amplifier (EDFA) and used to pump the nanofibre as well as the SMF. The other branch is up-shifted in frequency (+110 MHz) by an acousto-optic modulator (AOM) and combined with the backward spontaneous Brillouin scattering for heterodyne mixing. The total length of the nanofibre (including the adiabatic tapering transition and the nanofibre waist) is 179.4 mm ± 5 mm with a 10 cm long 740 nm ± 5 nm diameter nanofibre. Note that the total insertion loss of the nanofibre is 1 dB including the 0.14 dB tapering-induced loss, the splicing of the nanofibre to two fibre pigtails at each end, as well as the loss of the two angled-physical contact connectors. The heterodyne spectrum is measured by a photodetector combined with a radio-frequency electrical spectrum analyser (ESA). The AOM is used to separate the Stokes and anti-Stokes Brillouin scatterings. The polarisations of the pump light and the local oscillator are adjusted using two polarisation controllers (PCs) to achieve the highest heterodyne response. (b) Heterodyne beating spectra of the 10 cm nanofibre gas cell and the appended 53.5 m SMF. The Brillouin frequency shifts of the 40 bar CO2 in the 10 cm nanofibre gas cell, the silica contribution in the 10 cm nanofibre gas cell as well as the 53.5 m SMF are 0.34 GHz, 9 GHz and 10.86 GHz respectively. The red and blue lines in the inset of (b) are the backward Stokes Brillouin gain spectra for the 40 bar CO2 nanofibre gas cell and for the SMF, respectively. The horizontal scales span over the same frequency width for a good comparison. nanofibre gas cell. A 53.5 m single-mode fibre (SMF) is appended to the nanofibre and provides a direct comparative response for the Brillouin gain. The backward spontaneous Brillouin scattering signal is measured using heterodyne beating with a frequency-upshifted fraction of the pump laser (as a local oscillator) and detected by a radio-frequency electrical spectrum analyser via a photodetector. The beating spectra from the 40 bar CO 2 contribution in the 10 cm nanofibre gas cell, from the silica contribution in the 10 cm nanofibre as well as from the 53.5 m SMF are shown in Fig. 4(b). The Brillouin frequency shift for the 40 bar CO 2 contribution, silica contribution and SMF contribution are 340 MHz, 9 GHz and 10.86 GHz respectively. Note that the 9 GHz signal is from the silica nanofibre hybrid acoustic mode which has been analysed in Fig. 3. The +110 MHz frequencyshifted local oscillator splits the Stokes and anti-Stokes components. The right peaks identify the Stokes scattering while the left peaks originate from the anti-Stokes scattering. Remarkably, we can observe that the peak (c) Measured Brillouin linewidth defined as twice the value of the left half-width measured spectrum using a Lorentzian fitting on the experimental curves in (a). Each red data point shows the estimated mean and the RMS error bar a total of 10 measurements at each pressure. The blue line shows the theoretical Brillouin linewidth of free-space CO2 gas at different pressures.

Brillouin signal for 10 cm nanofibre filled with 40 bar CO 2 is ∼ 10 times higher than that of the 53.5 m SMF.

The red and blue lines in the inset of Fig. 4 (b) show the Brillouin gain spectra of the 40 bar CO 2 nanofibre gas cell and the SMF respectively, when the polarisations are tuned to maximise the specific Brillouin signal. The peak Brillouin gain coefficient for the nanofibre gas cell filled with 40 bar CO 2 is 8.20±0.08 m -1 W -1 which is 34 times higher than that of the SMF (0.24 m -1 W -1 ). The error bar is calculated from the standard deviation of the peak gain over 10 measurements. Note that the peak Brillouin gain coefficient for the SMF obtained from our spontaneous scattering measurement is in good agreement with the standard value for this type of fibre [36], which confirms the solidity of our test bench and calibrations based on spontaneous scattering. The shape of the Brillouin gain spectrum of the standard SMF is a fully symmetric Lorentzian distribution while the shape of the Brillouin gain spectrum of the 40 bar CO 2 nanofibre gas cell is evidently asymmetric, as explained in the next section.

Different gas pressures. Figure 5(a) shows the Brillouin gain coefficient as a function of the CO 2 pressure in the nanofibre gas cell. The peak gain coefficient increases with the pressure while the linewidth is simultaneously reduced, in agreement with the model presented in [8]. The peak gain coefficient is 18.90 ± 0.17 m -1 W -1 at 57 bar which is 79 times stronger than that in a SMF. The error bar is calculated from the standard deviation of the peak gain for 10 measurements.

It is interesting to see in Fig. 5(a) that the shape of the Brillouin spectra is skewed. This asymmetry results from the contribution of the tapered transition region, since this segment shows a position-varying evanescent optical field that keeps interacting with the sound wave in the gas. Let us take 40 bar CO 2 as an example and analyse its spectrum in Fig. 5(b). The blue line is the theoretical Brillouin spectrum in 40 bar CO 2 (along the homogeneous nanofibre section) and the red line is the theoretical Brillouin spectrum including the uniform nanofibre section as well as the tapered region (see Supplementary Section 2 for the detailed analysis). The tapered region only contributes to the higher frequency region because its gradually larger diameter increases the effective refractive index, leading to a larger Brillouin frequency shift, as indicated by the phase-matching condition given in Eq. 1. So the tapered region presents a higher Brillouin frequency shift and makes the Brillouin spectrum of the nanofibre slightly skewed. The theoretical Brillouin spectrum taking into account this asymmetry matches well with our experimental results.

Figure 5(c) shows the measured Brillouin linewidth for the nanofibre gas cell filled with CO 2 at different pressures as well as the theoretical estimation of the freespace gas Brillouin linewidth. Here, to exclude the contribution from the tapered regions, the Brillouin linewidth for the nanofibre gas cell is defined as twice the value of the left half-width-half-maximum of the spectrum fitted with a Lorentzian (i.e. considering only the left half of the spectrum). The measured Brillouin linewidth decreases with pressure in the range from 5 to 50 bar. It then increases from 50 to 57 bar, as a result of the close vicinity of the gas-liquid phase transition at room temperature, since the acoustic damping increases when the gas phase is approaching the liquid transition. It should be Measured Brillouin gain spectra for 2 bar SF6, 5 bar CO2 and 10 bar N2. For the CO2 and N2 measurements, the noise equivalent bandwidth of the electrical spectrum analyser (ESA) is 1 kHz, while for the SF6 measurement, the noise equivalent bandwidth of the ESA is 100 Hz.

mentioned that the measured Brillouin linewidth for the nanofibre at a specific pressure is ∼ 10 MHz larger than the free space gas Brillouin linewidth. This linewidth broadening is thought to originate from the coupling of the light field with a continuum of free-space modes [37], unlike in our simulation in which we only considered one acoustic mode propagating longitudinally.

Different types of gas. We then carried out the study of Brillouin scattering in a nanofibre gas cell filled with different types of gas, namely CO 2 , sulfur hexafluoride (SF 6 ) and nitrogen (N 2 ). The Brillouin gain spectra for these three gases at specific pressures are shown in Fig. 6. This result shows the possibilities of the evanescent Brillouin scattering of our nanofibre to be used for Brillouin spectroscopy and gas analysis as well as the flexibility of our platform in tailoring the Brillouin gain at will.

Discussion

In this work, we have demonstrated the generation of a strong Brillouin scattering driven by the evanescent field of a nanofibre placed in a gas-filled cell. The nanofibre is obtained by fused-tapering of a single mode fibre, so that the mode size transition to a submicron dimension can be adiabatically realised with very low added loss: this is a crucial advantage for the implementation of such a gain stage in photonic systems.

The tighter light confinement in nanofibre compared with hollow-core fibre results in a measured Brillouin gain coefficient in our 40 bar CO 2 gas filled nanofibre gas cell 5-times larger than in a hollow-core fibre under the same pressure [8], but it is expected that higher gains can be achieved by using noble gases (e.g. xenon) [38]. Using our nanofibre, we can increase the pressure close to the gas-liquid phase transition of CO 2 (64 bar at ambient temperature) with negligible absorption loss because of the relatively shorter length of the nanofibre and smaller percentage of light interact with the gas molecules, in contrast with hollow core fibres where light molecular absorption turns out to limit the maximum pressure. In these conditions, we have achieved a peak Brillouin gain coefficient of 18.90 ± 0.17 m -1 W -1 at 57 bar, which is 11 times higher than that obtained in a hollow-core fibre and is 79 times higher than that in a SMF. Furthermore, unlike in hollow-core fibres [8] where both optical and acoustic modes are confined in the hollow core, this work has demonstrated the first possibility to probe the acoustic wave in the surrounding medium of a waveguide using an all-optical method thanks to the strong Brillouin scattering generated by the evanescent field of the waveguide. This feature offers the attractive possibility to realise compact optical amplification stages, since a gain similar to that generated over several tens of meter of standard fibre can be obtained over a few centimetres along the nanofibre, with a very minor insertion loss. The concept may be extended to any planar waveguiding structure showing a substantial evanescent field and can be the essence of gain blocks in photonic integrated circuits. For instance, a suspended [21] or slot nano-waveguide [39] could be used for strong light-sound interactions in gas. Various functional optical devices and sensors could be generated based on this platform.

Our nanofibre gas cell can also be used for novel approaches in pressure and temperature sensing. For in-stance, the measured Brillouin frequency shift along the nanofibre gas cell filled with CO 2 at different pressures (shown in Fig. S2 in Supplementary Section 3) shows a pressure sensitivity of ∼ -1 MHz/bar. This large sensitivity enables our nanofibre gas cell to be used for pressure measurement over a range from 5 to 57 bar. Temperature variations also change the gas acoustic velocity [8] and hence the Brillouin frequency shift in the nanofibre gas cell.

In order to estimate the capability of our nanofibre waveguide for microscopy and spectroscopy of biological samples, we have simulated the Brillouin gain spectrum of a nanofibre immersed in water. Since the acoustic velocities in the surrounding medium (e.g. ∼ 300 m/s in gas and 1500 m/s in water) and that of the solid silica nanofibre (6000 m/s) are very different, the respective Brillouin frequency shifts in the surrounding medium and the nanofibre are well separated. The peak Brillouin gain is calculated to be 1 m -1 W -1 for a 450 nm diameter nanofibre immersed in water, which is 8 times larger than that in 2 bar of SF 6 shown in Fig. 6. The detailed simulation is described in Fig. S3 in Supplementary Section 4. The accessibility of the acoustic wave, present in the surrounding medium and thus external to the waveguide, and the high Brillouin gain of a nanofibre immersed in water opens new perspectives and creates an important bridge between Brillouin scattering in waveguide and Brillouin microscopy and spectroscopy. So far, Brillouin microscopy and spectroscopy have exclusively relied on free-space light excitation. For Brillouin spectroscopy [28], our platform provides an efficient way for light illumination and Brillouin scattering signal collection. For Brillouin microscopy, our work suggests a novel imaging namely waveguide-illumination Brillouin microscopy, which inherits many benefits of on-chip based microscopy [30,31], such as separation between the illumination and detection light paths, facilitated alignment and high throughput [29].

Methods

Simulations. The transverse dimension of the silica nanofibre is close to the acoustic wavelength. Therefore, in such a waveguide, the boundaries conditions induce a strong coupling between longitudinal, shear and surface elastic components. The generation of a confined elastic wave in the gas-silica rod assembling by the two components of optical field (guided and evanescent) is calculated by using the elasto-dynamic equation driven by the electrostrictive stress [40,41]. All calculations for CO 2 at 40 bar are realised using: an optical wavelength of 1550 nm, a refractive index of 1.01804 [8], an acoustic velocity of 250 m/s, and a density of 71.35 kg/m 3 . To exclude the contribution from the tapered regions, the Brillouin linewidth used in the simulation for 40 bar CO 2 is 14.2 MHz which is defined as twice the value of the left half-width-half-maximum of the spectrum fitted with a Lorentzian (i.e. considering only the left half of the spectrum). Fabrication of the nanofibre gas cell. The nanofibre is fabricated by tapering a standard single-mode fibre (SMF) using the heat-brush technique. To package the nanofibre into the gas cell, we applied the following procedure: first, we move the nanofibre sample from the tapering translation stage to a fixed-length stage with two fibre clamps on both ends of the SMF pigtails. Second, we orient the stage perpendicular to the ground, release the lower fibre clamp, insert the lower end of the nanofibre sample into a metallic tube, to finally hermetically glue the lower end of the tube. Third, to avoid any contact of the nanofibre with the tube wall, we apply a 2.15% strain on the nanofibre. We then apply glue on the other side while precisely controlling the strain and monitoring the backward Brillouin spectrum.

  density, a.u. Long. displacement, a.u.

Figure 1 .

 1 Figure 1. Brillouin scattering in a nanofibre gas cell. (a) Conceptual view of the Brillouin scattering in a nanofibre gas cell. Pump (in green) goes from left to right and the backscattered light includes Rayleigh scattering from the same frequency and frequency down-shifted Stokes (in red) and frequency up-shifted anti-Stokes (in blue) Brillouin scattering. (b) Computed spatial optical power distribution of the fundamental optical mode at a wavelength of 1550 nm for a 740 nm diameter nanofibre surrounded with 40 bar CO2 gas. The effective mode refractive index is 1.0829. (c) Computed longitudinal displacement of the elastic wave at 350 MHz as a result of backward Brillouin scattering in the surrounding gas. The black circles in (b) and (c) show the silica nanofibre boundary. See Methods for the simulation details.

Figure 2 .

 2 Figure 2. 3D calculation of Brillouin spectra in silica nanofibres surrounded by 40 bar CO2 for a diameter ranging from 500 nm to 1.5 µm.

FrequencyFigure 3 .

 3 Figure3. Backward Brillouin scattering in the silica material of the nanofibre during fabrication process. The backward Brillouin scattering of the nanofibre is measured with a set-up we developed in 2017[START_REF] Godet | [END_REF]. The Brillouin signal is normalised to the of the electrical spectrum analyser when there is no pump. The Brillouin spectra from the surface acoustic modes and the hybrid acoustic modes of the silica nanofibre during fabrication are calculated by the model in[START_REF] Godet | [END_REF]. The blue line shows the measured Brillouin spectrum of the nanofibre on the stages after tapering. The dotted red line shows the simulated Brillouin spectrum of the nanofibre with a waist diameter of 740 nm. The dotted purple line shows the measured Brillouin spectrum of the nanofibre after packaging the nanofibre in a metallic tube with a 2.15% strain. Note that the signal from SMF pigtail is not taken into account in the simulation and any response from interactions in the gas is not visible in the frequency range covered by the set-up. Brillouin scattering from the standard single-mode fibre pigtail of the nanofibre is also observed at a frequency of 10.86 GHz. The inset shows the fabricated nanofibre gas cell with the untapered SMF segments exiting the gas cell at each end. Gases can be pressurised in or vacuum pumped out through the gas inlet/outlet port.

Figure 4 (

 4 PhotodetectorLocal oscillator PC

Figure 4 .

 4 Figure 4. Backward Brillouin scattering in the nanofibre gas cell. (a) Heterodyne experimental set-up. To make a comparison between peak Brillouin gains, a 53.5 m single-mode fibre (SMF) is connected to the far end of the nanofibre.The output of a continuous-wave external-cavity diode laser (ECDL) is split: one branch is amplified by an erbium-doped fibre amplifier (EDFA) and used to pump the nanofibre as well as the SMF. The other branch is up-shifted in frequency (+110 MHz) by an acousto-optic modulator (AOM) and combined with the backward spontaneous Brillouin scattering for heterodyne mixing. The total length of the nanofibre (including the adiabatic tapering transition and the nanofibre waist) is 179.4 mm ± 5 mm with a 10 cm long 740 nm ± 5 nm diameter nanofibre. Note that the total insertion loss of the nanofibre is 1 dB including the 0.14 dB tapering-induced loss, the splicing of the nanofibre to two fibre pigtails at each end, as well as the loss of the two angled-physical contact connectors. The heterodyne spectrum is measured by a photodetector combined with a radio-frequency electrical spectrum analyser (ESA). The AOM is used to separate the Stokes and anti-Stokes Brillouin scatterings. The polarisations of the pump light and the local oscillator are adjusted using two polarisation controllers (PCs) to achieve the highest heterodyne response. (b) Heterodyne beating spectra of the 10 cm nanofibre gas cell and the appended 53.5 m SMF. The Brillouin frequency shifts of the 40 bar CO2 in the 10 cm nanofibre gas cell, the silica contribution in the 10 cm nanofibre gas cell as well as the 53.5 m SMF are 0.34 GHz, 9 GHz and 10.86 GHz respectively. The red and blue lines in the inset of (b) are the backward Stokes Brillouin gain spectra for the 40 bar CO2 nanofibre gas cell and for the SMF, respectively. The horizontal scales span over the same frequency width for a good comparison.

FrequencyFigure 5 .

 5 Figure 5. Experimental Brillouin gain spectra in the nanofibre placed in a gas cell filled with CO2. (a) Measured Brillouin gain spectra along the nanofibre gas cell filled with CO2 at different pressures. (b) Normalised Brillouin gain spectra in 40 bar CO2 obtained in a nanofibre with a waist of 740 nm. The blue line shows the theoretical Brillouin spectrum at the nanofibre waist; the red line shows the compound theoretical Brillouin spectrum along the nanofibre waist and the tapered region. The black line shows the experimental Brillouin spectrum along the nanofibre in the gas cell filled with 40 bar CO2.(c) Measured Brillouin linewidth defined as twice the value of the left half-width measured spectrum using a Lorentzian fitting on the experimental curves in (a). Each red data point shows the estimated mean and the RMS error bar a total of 10 measurements at each pressure. The blue line shows the theoretical Brillouin linewidth of free-space CO2 gas at different pressures.

2 Figure 6 .

 26 Figure 6. Experimental Brillouin gain spectra from the nanofibre gas cell filled with different types of gas. Measured Brillouin gain spectra for 2 bar SF6, 5 bar CO2 and 10 bar N2. For the CO2 and N2 measurements, the noise equivalent bandwidth of the electrical spectrum analyser (ESA) is 1 kHz, while for the SF6 measurement, the noise equivalent bandwidth of the ESA is 100 Hz.

Acknowledgments. The authors would like to acknowledge the support by the Swiss National Science Foundation (SNSF) under grant No. 159897 and 178895, and the financial support of Agence Nationale de la Recherche (ANR-16-CE24-0010-03), EIPHI Graduate School (contract ANR-17-EURE-0002) and Bourgogne-Franche-Comté Region.

Data Availability Statement: The code and data used to produce the plots within this work will be released on the repository Zenodo upon publication of this preprint.