1. INTRODUCTION

Since the pioneering work of Birks and Li in 1992 [1], optical microfibers and nanofibers have drawn a widespread interest from both fundamental and applied sciences. These hair-like slivers of silica glass, fabricated by tapering optical fibers down to a few hundred nanometers over several centimeters long, have a number of optical and mechanical properties that make them very attractive for a variety of applications such as optical sensing [2,3], atom trapping and quantum optics [4–7], nonlinear optics [8–10], evanescent coupling [11,12], optical filtering [13], and plasmonics [14,15]. All these applications, however, require a precise measurement of the microfiber dimensions and several methods have been proposed and demonstrated in the past [16–21]. Among these techniques, the most used is the scanning electron microscope (SEM), which can provide an accuracy of 3% of the fiber taper diameter [16]. However, a SEM often requires metallization and careful manipulation of the tapered fiber into the microscope chamber. Other indirect measurement techniques have also been proposed in the literature. For instance, Warken and Giessen used the light scattered from the side of the optical microwires [17] to measure the diameter with an accuracy of 50 nm. Wiedemann et al. have exploited the intermodal phase-matching condition for second-harmonic and third-harmonic generation and deduced the microfiber diameter with an accuracy greater than 2% [18]. Similarly, Sumetsky et al. measured the taper uniformity with a precision of 2% by probing light with a stripped fiber [19].

In addition to providing new optical properties, optical nanofibers also possess unique elastic properties that make them attractive for investigating and exploiting the Brillouin light scattering [22]. For instance, it has recently been shown that, unlike standard silica optical fibers, optical microfibers can support new types of waves, such as surface acoustic waves (SAWs) and bulk hybrid acoustic waves (HAWs), due to the small waveguide boundary conditions [23,24]. It is well known in seismology that longitudinal waves reflect as a superposition of both longitudinal and shear displacements at the surface, the latter being the deadliest for the inhabitants. Similarly, in optical microfibers the surface contribution can no longer be neglected, and near the surface there is a strong coupling between longitudinal and shear elastic components. As a result, this strong coupling leads to the generation of SAWs and HAWs that do not behave as pure axial longitudinal elastic wave, as in standard fibers. It has been specifically shown that both the surface Rayleigh waves and hybrid waves propagate at a lower velocity than pure axial longitudinal waves, giving rise to new Brillouin lines in the backward Brillouin spectrum in a frequency range of 5 GHz–10 GHz [23].

In this paper, we propose to exploit the full elastic properties of tapered optical fibers and demonstrate a simple and accurate measurement technique of their diameter and their uniformity. Our method is based on highly sensitive single-ended heterodyne coherent detection as a tool for Brillouin optical spectral analysis. It enables the mapping of the backward Brillouin spectrum along the optical fiber taper, including both the adiabatic transitions and the uniform section of the optical microwire, by fitting with numerical simulations of the elastodynamics equation. Using this technique and taking advantage of both SAW and HAWs, we achieve a sensitivity as high as a few nanometers for fiber taper diameters ranging from 500 nm to 1.2 μm. The paper is organized as follows. In the next section, we present the theory of Brillouin light scattering in tapered optical fibers based on the elastodynamics equation including electrostriction. Section 3 is devoted to the description of the experimental setup and measurement method. Results are shown and discussed in Section 4. The sensitivity of our method will be discussed and we will show the discrimination of diameter measurements from uniformity.

Before going into details, it is important to note that Brillouin scattering has been widely investigated for remote or distributed measurements of fiber uniformity, strain, and temperature variations [25–27]. Recently, distributed measurements of chalcogenide optical microfibers with a spatial resolution of 9 mm were performed by using the Brillouin optical phase correlation technique [28]. Forward Brillouin scattering, also named guided acoustic wave Brillouin scattering (GAWBS), has been used to estimate the fiber cladding diameter of standard fibers [29], the core diameter of photonic crystal fibers [30], and the diameter of optical fiber tapers [24,31]. In this case, the light scattering does not rely on longitudinal elastic waves but on axial shear (radial and torso-radial) waves that strongly depend on the waveguide boundaries [32]. Although providing a quite good diameter estimate, measurement of these transverse acoustic modes, however, often requires a pump-probe pulsed system and access to the two ends of the fiber taper.

On the other hand, the longitudinal elastic wave involved in Brillouin backscattering in optical fibers is directly related to the opto-acoustic properties and to the effective index of the fiber through the phase-matching condition. For instance, in photonic crystal fibers the variation of the air-hole microstructure along the fiber locally modifies the effective refractive index leading to a broadened Brillouin spectrum [25,33]. In silicon chip waveguides, the core diameter fluctuations can induce a strong reduction of the optomechanical quality factor and Brillouin scattering efficiency [34]. In this work, we specifically exploit those elasto-optic properties to fully characterize silica optical microfibers and nanofibers including the taper transitions.

2. THEORY

Figure 1 schematically shows a typical optical microfiber, including a uniform subwavelength waist section over several centimeters long linked to standard optical fibers by adiabatic transitions. The samples used in this work have been tapered from standard single-mode fibers (SMF-28) used in telecommunication and have a typical uniform microwire length $L=40\mathrm{mm}$ and symmetric transition sections whose length varies from $L_T=87\mathrm{mm}$ to $L_T=108\mathrm{mm}$ depending on the microwire diameter (for details, see Supplement 1).

 

Fig. 1. Schematic of an optical microfiber made using the heat brush technique. $L$ and $L_T$ are the waist and adiabatic transition lengths, respectively. $\varphi$ is the diameter of the microwire.

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Stimulated Brillouin scattering (SBS) in standard optical fibers arises from the interaction between two counterpropagating and frequency-detuned optical waves and a longitudinal elastic wave. This inelastic scattering is usually modeled by three coupled-amplitude equations for the pump, Stokes, and the elastic waves. The latter is generally considered as a plane wave [22]. Unlike single-mode optical fibers, optical microwires have an air cladding and a small core close to both the optical and the acoustical wavelengths. In such tiny waveguides, there is a strong coupling between the longitudinal and shear elastic components because of the waveguide boundaries. The longitudinal elastic wave can no longer be considered as a plane wave but rather as an acoustic mode including both the longitudinal and shear elastic components. To find these new hybrid acoustic modes, we solve the elastodynamics equation driven by the electrostrictive stress [35,36]. This equation can be written in the form

Figure 2 shows the resulting numerical Brillouin spectrum of a silica optical microfiber whose diameter varies from 0.36 μm to 1.2 μm. This diameter range ensures that only the fundamental TE-like optical mode is propagating in the microwire (for details, see Supplement 1). As seen in Fig. 2, there are many Brillouin acoustic resonances and their frequency detuning strongly varies with the taper diameter according to the phase-matching map. The two first modes on the left between 5 and 7 GHz [denoted TR(2,1) and $L(0,1)$] are SAWs whereas higher-order longitudinal modes in the range of 8–11 GHz are bulk HAWs including both a shear and longitudinal components [$L(0,i>2)$], as previously shown in Ref. [23]. In addition, the HAW modes $L(0,2)$ and $L(0,3)$ exhibit a clear-cut avoided crossing near 9 GHz that results from the strong coupling of the axial shear and axial longitudinal elastic components near the waveguide boundaries. This anti-crossing leads to the strong curvature of the phase-matching map shown in Fig. 2. The dotted and dashed black lines show the shear and the longitudinal acoustic branches, respectively. It is important to note that the shear wave branch is mainly given by the boundary conditions while the longitudinal one is driven by the optical effective index variation. Furthermore, the optical dispersion continuously shifts the Brillouin spectrum toward lower frequencies as the taper diameter decreases. The combined effects of optical dispersion and acoustic anti-crossing add to the overall complexity of the phase-matching map.

 

Fig. 2. Numerical 3D Brillouin spectra in silica optical microfibers for a diameter ranging from 0.36 μm to 1.2 μm. The dotted and dashed black lines show the shear and the longitudinal acoustic branches, respectively.

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This curvilinear nature of the phase-matching map supports the idea that each taper diameter provides a different Brillouin spectrum. As a result, in principle, we could find the taper diameter by measuring and fitting the backward Brillouin spectrum with numerical simulations. Two important parameters can be calculated from Fig. 2. First, one can find the sensitivity to the diameter variation that each curve in Fig. 2 exhibits. Second, we can know how each resonance peak participates and improve the diameter discrimination. In Fig. 3, we plot the sensitivity function $S=\partial f_k/\partial d$, which is the derivative of the Brillouin resonances with respect to the microfiber diameter $d$. We thus obtain the frequency shift $\mathrm{\Delta }f=(\partial f_k/\partial d)\mathrm{\Delta }d$ that experiences the Brillouin frequency $f_k$ related to an acoustic mode $K$ for a diameter variation of $\mathrm{\Delta }d$. As expected from Fig. 2, the sensitivity $S$ reaches significant values for small diameters where the elastic modes related to the TR(2,1) and $L(0,2)$ waves undergo large frequency shifts. For instance, a taper diameter below 0.4 μm provides a sensitivity below $-20\mathrm{MHz}·{\mathrm{nm}}^{-1}$. This means that a frequency shift in the Brillouin spectrum of a 20 MHz (typically the Brillouin linewidth) would correspond to a variation of one nanometer in fiber diameter. Figure 3 also shows a green area that provides a sensitivity limit of $1\mathrm{MHz}·{\mathrm{nm}}^{-1}$. In this region, if mode $L(0,4)$ is not considered, the sensitivity falls down to $1\mathrm{MHz}·{\mathrm{nm}}^{-1}$ [$L(0,3)$ value], which results in a minimum diameter variation of 20 nm. This is the case for taper diameters around 0.95 μm where most resonances fall into the green area.

 

Fig. 3. Sensitivity $S=\partial f_k/\partial d$ is computed from the derivative of the Brillouin frequency related to mode $k$ with respect to the diameter $d$. The green area highlights the $\pm 1\mathrm{MHz}/\mathrm{nm}$ sensitivity.

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One of main advantages of our method is the presence of multiple Brillouin resonances that vary in a complex manner with respect to the taper diameter. To evaluate how this improves the diameter measurement, we compute for a diameter $a$ the corresponding Brillouin spectrum attributing to each resonance a linewidth of 30 MHz. For each diameter $a$ spectrum we then calculate the cross-correlation function of all the other spectra corresponding to other diameters $b$. Figure 4 shows the maximum of the computed cross correlation. As expected, the peak values are located on the diagonal where the two diameter spectra match ($a=b$). The correlation peak drops sharply away from the diagonal, meaning that the spectra of two different diameters are strongly uncorrelated. In the case of forward Brillouin scattering, the correlation function does not provide such a good result (see Supplement 1).

 

Fig. 4. Normalized maximum cross correlation between different spectra with respect to diameters.

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3. EXPERIMENTAL SETUP

Figure 5 shows a scheme of the experimental setup for both making and characterizing the optical microfibers using Brillouin spectroscopy. They have been manufactured by tapering standard single-mode fiber (SMF-28) using the heat-brush technique [38]. In short, the fiber is fixed on two translation stages and softened on its central part with a butane flame. The flame is regulated by a mass-flow controller and it is kept motionless while the two translation stages stretch the fiber. The waist and taper transition shapes are fully controlled by the trajectories of the two translation stages.

 

Fig. 5. Schematic of the experimental setup for both making and measuring the optical microwires. PM: power meter. CW: continuous wave, EDFA: erbium-doped fiber amplifier, ESA: electrical spectrum analyzer, FC: fiber coupler.

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To monitor the tapering process, we injected a laser light at 1550 nm into the fiber taper and measured the transmitted power and optical mode beat using a photodiode [39] (for details, see Supplement 1). The measured spectrogram allows us to identify if the optical microwire is optically single mode at the end of the process achieving maximum transmission. In such conditions, the total transmission loss of the optical microfiber at the end of the pulling process is 1.2 dB. The experimental setup for measuring the backward Brillouin spectrum is based on a heterodyne coherent detection (See Fig. 5). In this setup, we used another coherent continuous-wave laser at a wavelength of 1550 nm with a narrow linewidth ($<10\mathrm{KHz}$) that is split into two beams using a 70/30 fiber coupler. One beam is sent to an erbium-doped fiber amplifier (33 dBm output power) and used as the pump and the other beam is used as reference light or local oscillator. The pump light is injected through an optical circulator into the microwire under test. The backscattered Brillouin signal is then mixed with the reference light using another 50/50 coupler to produce an optical beat note that is further detected in the radio-frequency domain by a fast photodiode. The RF signal is then amplified and the resulting Brillouin spectrum is recorded with an electrical spectrum analyzer (ESA). The SBS spectrum is measured on the translation stages once the pulling process is ended and the heat from the flame is dissipated. This technique allows us to avoid any pollution, stress, and temperature effect that could shift the Brillouin lines and alter the measurement results. The measurement could be done in situ during the tapering process. However, the strong temperature increase in the taper waist region during the heating process generates different Brillouin frequency shifts than at room temperature. The heat-brush technique does not only produce a uniform long microwire, it also produces the taper transitions to couple light efficiently from standard fibers. The transition shape results from the mass conservation and have been well predicted by Birks’s model [1]. They are schematically depicted in Fig. 1, where a 40-mm-long microwire is surrounded by adiabatic transitions. The transitions are rather long and cannot be neglected in the Brillouin analysis. Consequently, the whole Brillouin spectrum will result from three contributions: the microwire, the two adiabatic transitions, and the standard fibers present in the experimental setup.

To gain better insight, we perform numerical simulations of the Brillouin spectral dynamics along the fiber taper. Figure 6(a) shows a color plot of the Brillouin spectrum versus fiber elongation including the transitions over 70 mm and the uniform microwire over 20 mm. This elongation corresponds to a taper diameter variation from 10 μm down to 1 μm. The blue line in Fig. 6(b) is a plot of the integrated Brillouin spectrum in the adiabatic transitions. We can see many broad square-shape resonances that result from the superposition of all Brillouin resonances that continuously downshift to lower frequency detuning as the diameter decreases. In the microwire section, we see sharp Brillouin resonances, including both surface and hybrid acoustic modes [Fig. 6(c)], while the total spectrum is the incoherent sum of the transitions and the microwire section, as shown in Fig. 6(d). Note that, in this case, the usual Brillouin resonance from standard fibers has been not considered for better visibility.

 

Fig. 6. (a) Numerical simulation of the Brillouin spectrum along a tapered optical fiber versus elongation from 25 mm to 115 mm associated with a diameter change from 10 μm to 1 μm. (b) The integrated Brillouin spectrum in the transition regions over a 70-mm-length (from 25 mm to 95 mm). (c) The integrated Brillouin spectrum in the uniform section of the microwire with a diameter of 1 μm over 20 mm (from 95 to 115 mm). (d) The total integrated Brillouin spectrum.

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4. EXPERIMENTAL RESULTS

We carried out a series of measurements of the backward Brillouin spectrum in many optical microwires with an increasing diameter from 0.5 μm to 1.2 μm. Figure 7 shows the results of those measurements (white traces) where the amplitude was normalized one by one from electrical tension (μV) of the spectrum analyzer. A 2D numerical mapping of the Brillouin spectra for the microwire diameter range between 0.4 and 1.2 μm was superimposed on the same Fig. 7 for a direct comparison. This colormap is in linear scale and normalized for each taper. As it can be seen, the agreement is excellent. We clearly observe the crossing of the two SAWs and the anti-crossing of hybrid waves. Figure 8 shows a further comparison of the experimental (black) and numerical (green) Brillouin spectra for a microwire diameter of 780 nm (SEM measurement). A microscope image of the microwire is shown in the inset of Fig. 8. We observe two SAW resonances at 5.34 and 5.41 GHz, respectively, and multiple resonances due to HAWs from 8.36 GHz up to 10.5 GHz, with a clear avoided crossing around 9 GHz. The main Brillouin resonance at 8.36 GHz exhibits a linewidth of 26 MHz. Note also that the spectral broadening due to taper transitions is clearly observable. In Fig. 8(a), the green plot depicts the simulated Brillouin spectrum for a microwire diameter of 780 nm. The agreement is quite satisfactory regarding the frequency peaks, except the high-frequency one at 10.8 GHz due to single-mode fibers, which is not accounted for in our model. To go further into details, we have plotted in Figs. 8(b)–8(d) a zoom on the four main acoustic frequency peaks at 5.34 GHz, 5.41 GHz, 8.37 GHz, and 9.7 GHz, respectively. They are compared with a set of three numerical simulations for an optical microwire diameter of 780 nm (dashed green), 800 nm (blue), and 810 nm (red). A slight frequency shift between theory and experiment can be seen and the fit curve corresponds to a wire diameter of 800 nm (blue spectrum).

 

Fig. 7. Direct comparison of the experimental Brillouin spectra (white traces) and 2D numerical mapping in false color as a function of the silica fiber taper diameter including both the transition and microwire sections.

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Fig. 8. (a) Experimental (black) and numerical (green dashed) Brillouin spectra of a silica optical microwire of 780 nm waist. Inset: the SEM image of the microwire. (b), (c), (d) The zooms on the SAW and HAW Brillouin resonances and comparison with numerical simulations for an optical microwire diameter of 780 nm (dashed green), 800 nm (blue), and 810 nm (red).

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As previously shown in Fig. 3, the two first peaks related to $L(0,1)$ and TR(2,1) are not very sensitive for a diameter $d=800\mathrm{nm}$. This is confirmed in Fig. 8(b) where the two peaks do not shift significantly. On the other hand, Brillouin resonances at 8.37 and 9.7 GHz are very sensitive, as shown in Figs. 8(b) and 8(c). The elastic resonance at 8.37 GHz shifts toward higher frequencies while the 9.7 GHz peak shifts to lower ones. The sensitivity curve plotted in Fig. 3 allows the assessment of the fine wire diameter by selecting the best resonance peak to fit. In our case, Figs. 8(d) and 3 show that the $L(0,3)$ HAW mode has the best sensitivity, achieving 25 MHz for a 5 nm diameter step. For a microwire diameter of 800 nm, the sensitivity lies within 1% of the diameter.

This can be generalized to all measurements. Figure 9 shows the experimental Brillouin frequencies extracted from the previous measurements (dots) and compared to the theory (color curves). The vertical lines depict the sensitivity of our measurements that are evaluated as follows. For each measurement, we define the experimental frequency mismatch by $\mathrm{\Delta }f=5\mathrm{MHz}$ and deduced the diameter mismatch $\mathrm{\Delta }d$ by multiplying by $(\partial f_k/\partial d)$ (see Fig. 3). The vertical lines have a height $2\mathrm{\Delta }d$. If the height is too high, the measurement is not sensitive enough. If we consider the previous example at 800 nm, the measurement of the Brillouin peak at $f=5.3\mathrm{GHz}$ shows a very sharp and tall vertical line. It was expected since the TR(2,1) wave has almost a vertical slope at this frequency (see Fig. 2). In contrast, the red and green data points show small vertical lines, meaning that the measurement is sensitive enough for HAW resonances. The maximum sensitivity is typically 20 nm for diameters ranging from 0.36 μm to 1.2 μm. Those results have been compared to SEM images with a very good agreement, as reported in the insets of Figs. 8 and 10. The microwire diameter measurement from microscopic images (see the insets of Figs. 8 and 10) is, however, lower than measurement with Brillouin spectroscopy. This is probably because the Brillouin scattering is sensitive to strain and temperature. For our simulations, we used elastic coefficients for silica at temperature of 300 K without accounting for the strain effect.

 

Fig. 9. Comparison between experimental (dots) and numerical (lines) Brillouin frequency shifts. The vertical lines delimit the measurement sensitivity; longer is less sensitive.

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Fig. 10. Effect of the microwire waist fluctuations on the Brillouin spectrum. (a) The experimental Brillouin spectrum (solid black) and the computed spectrum for a diameter of 1.27 μm (dotted blue). Inset: the SEM image of the microwire. (b) and (c) are zooms on acoustic resonances at 9.54 GHz and 10.15 GHz, respectively, with a linear variation on the waist diameter of 0% (green dotted), 3% (blue), and 5% (red).

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Another important behavior regarding the Brillouin spectrum is the width of each resonance peak that provides additional information about the microwire uniformity. As stated previously, the Brillouin spectrum results from the incoherent superposition of many spectra along the fiber taper. This fact is confirmed by the good agreement between theory and experiment as far as the taper transitions contribute to the Brillouin spectrum. Figure 7 shows light blue vertical shades that correspond to those taper transitions. Those shades are very well modeled using the incoherent superposition assumption. Using the same assumption, we can consider that longitudinal fluctuations of the microwire diameter will lead to an incoherent superposition of slightly shifted frequency peaks, resulting in a spectral broadening of the Brillouin lines. To get more insight, Fig. 10(a) depicts a Brillouin spectrum (black solid line) superposed with a numerical Brillouin spectrum that gives the best fit (blue dotted) using this method. From our estimations, we found that the diameter is $1.27\mathrm{\mu m}\pm 3.0\%$. To assess the microwire uniformity, the Brillouin spectrum was then computed with one period of triangular diameter variations with amplitudes of 0%, 3%, and 5%. We also assume one period of triangular variation along the waist of microwire for not changing the transition profile. Consequently, the maximum diameter variation is in the middle of the waist. The computation included incoherent superposition of many spectra from optical microwires whose diameters are equally distributed within the diameter target variation percentage. Results are shown in Figs. 10(b) and 10(c) by the dotted spectra for the two main Brillouin resonances at 9.54 GHz and 10.15 GHz. One can clearly see the linewidth broadening as the percentage of non-uniformity increases, together with a slight shift of a few MHz. From this simulation, we can estimate the taper diameter fluctuations to about 3%. Note that this estimation relies on the validity of our taper profile that we believe is good for our comparison between theory and experiment.

5. CONCLUSION

To conclude, we have demonstrated a simple and original method that allows for the complete characterization of subwavelength-diameter tapered silica optical fibers. Our technique relies on a highly sensitive direct measurement of the backward Brillouin spectrum using single-ended heterodyne coherent detection. A further comparative analysis of the Brillouin spectra with numerical simulations was then performed. Sensitivity as high as a few nanometers for taper diameters ranging from 500 nm to 1.2 μm was reported, which is comparable to SEM and other destructive methods. This method is valid for all optical wavelengths and for arbitrary glass materials and could help in the design and characterization of micro- and nanoscale photonic chip platforms used in many applications.