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Abstract
We present a new class of force sensor based on Brillouin scattering in an optical nanofiber. The sensor is a silica nanofiber of a few centimeters with a 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 force sensor resistor.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. 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 [1–5]. Force measurement systems are typically based on resistive, piezo-resistive, capacitive, piezoelectric, magnetic, inductive, or optic measurement methods [6–8]. 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 [9]. 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 [10]. 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.
Fig. 1. Comparison of key force sensor figure of merit (resolution and dynamical range) for this work, for state of the art optical fiber force sensor (SBS and FBG in single mode fiber [9,16]), not limited to fiber based sensor (Load cell MEMS [17] and femtotools [18]) and for previous results obtain with tapered optical fiber [19–22]. 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, 1 mN dynamic range) in complementary to fiber tip sensor [23,24].
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. [11–14].
In our previous work, we demonstrated high sensitivity longitudinal stress measurements using Brillouin scattering in ONFs [15]. 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.
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.
2. 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 [25,26] 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 ${\mathrm \mu }$m to a few hundreds of a nanometer) the strain is much stronger in the center part call the optical nanofiber [27].
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 commercial force sensing resistor (ME-system 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 radio-frequency electric spectrum analyser. The frequency shifts for each acoustic mode are retrieved with respect to the measured force using a Lorentzian fit [28].
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-system KD-78 force sensor resistor 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
where $F_{zz}$ is a longitudinal force and $S$ is the cross-sectional area over which the force is applied [29]. This expression shows how a very small force $F_{zz}$ can result in relatively strong stress $\sigma _{zz}$ 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 where $E$ is Young modulus (73 GPa in silica) and $\epsilon _{zz}$ 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 $\epsilon _{zz}$. This is schematically illustrated by the Fig. 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 [15]. In a standard optical fiber, the Brillouin frequency shift varies linearly with an applied tensile strain $\epsilon _{zz}$ inferior to 2% and a normalized Brillouin strain coefficient that is given by where $\nu _{B}$, Brillouin frequency shift. For a SMF-28 it is typically 4.25 [30]. Combining Eq. (1), 2 and 3 we can compute the sensitivity S that gives the Brillouin Frequency shift for a given applied force, 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 125$\mu$m 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.
3. Sensitivity enhancement
Brillouin spectra were measured with a ONF of 630 nm diameter with different applied force going from 0 to 611 $\mu$N 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 GHz). 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 ${\mathrm \mu }$m to 630nm. The geometric parameters of tapered optical fibers as nanofiber diameter and homogeneity are precisely estimated by Brillouin spectroscopy [28]. For each diameter, the sensitivity is measured for strain smaller than 2% for 9-10 GHz Hybrid Acoustic Waves (HAW) resonances [15] and reported in Fig. 3(b). The experiment data were fitted using Eq. (4). A force sensitivity around 1.9 MHz/$\mu$N 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 [30]. The analytical model is in good agreement with the experimental measurements and the small differences can be explained by small temperature variations.
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 $\mu$N as a function of the nanofiber diameter, theory (red), experiment (blue point).
4. 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 $\mu$m, 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 $\mu$m diameters. Also, the ONF points of diameter 9 $\mu$m seem to show an inflection. We will explain these observations through the following three examples in the following.
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 $\mu$m. Force and frequency are plotted in log scale. Blue area for measurements inferior to 70 $\mu$N 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.
5. 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. Figure 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 non-linearity. 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 [15]. 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
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].
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 nm diameter ONF with respect to strain. In gray, same resonances without modal dispersion.
6. 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 useful 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 ${\mathrm \mu }$m and then released to its initial state. We sequentially launch 15 cycles manually until a maximum stretch of 1500 ${\mathrm \mu }$m is reached. As shown in Fig. 7(b) The force applied by the fiber stretching is measured by both the force sensor resistor (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 commercial force 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 $2.3\times 10^{-4}$. This confirms the better signal-to-noise performance of our sensor by nearly two orders of magnitude. In order to confirm the good behavior of the optical sensor in dynamic regime compared to the commercial sensor, we have computed the Fourier transforms of the measurements from Fig. 7(b) and plotted the spectrum of the optical sensor measurement in blue and the one from the commercial strain gauge in light brown as shown in Fig. 9. Although the measurement cycle were performed manually at a low frequency rate, the commercial strain gauge signal spectrum (in light brown) shows clear limitations with a cut-off around 1 kHz. Above this frequency, only noise is to be found whereas the optical sensor reveals rich harmonic content (in blue). The hysteresis effect 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.
Fig. 7. (a) Stretching pattern applied to the ONF for non-static measurement. (b) Experimental force measurement from commercial strain gauge in blue and the ONF of diameter of 885 nm optical force measurement in red.
Fig. 8. (a) Measurements distribution around the mean force reported in Fig. 7(a) for commercial sensor (blue) and optical sensor (light brown). (b) Optical force measurements distribution around the mean force reported in Fig. 7(a) with proper scale.
Fig. 9. Spectrum computed from dynamic measurements shown in Fig. 7(b). The optical sensor spectrum is in blue and the commercial force sensor in light brown. (b) Measured force as a function of strain for increasing (blue circle) and decreasing force (red point).
7. 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 force sensor resistor 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.
Funding
Conseil régional de Bourgogne-Franche-Comté; Agence Nationale de la Recherche (ANR-17-EURE-0002, ANR-21-ESRE-0040).
Acknowledgement
Authors thank Maxime Zerbib for cycling force measurement.
Disclosures
The authors declare no conflicts of interest.
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.
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