Forward Brillouin scattering fiber sensors can detect and analyze media outside the cladding of standard fibers, where guided light does not reach. Nearly all such sensors reported to-date have relied on the radially symmetric guided acoustic modes of the fiber. Wave motion in these modes is strictly dilatational. However, forward Brillouin scattering also takes place through torsional–radial guided acoustic modes of the fiber. Torsional–radial modes exhibit more complex tensor characteristics, and they consist of both dilatational and shear wave contributions. In this work, we show that forward Brillouin sensing through torsional–radial acoustic modes is qualitatively different from processes based on the radial ones. While dilatational wave components may dissipate toward liquids outside the fiber cladding, shear waves do not. Consequently, the effect of outside liquids varies among torsional–radial modes. Those modes that are dominated by their dilatational components undergo faster decay rates, whereas other modes with large shear contributions decay at much slower rates in the same liquid. The difference in decay rates may reach a factor of seven. Experimental observations are well supported by the analysis. The differences among modes are also found with liquid outside specific coating layers. Large changes in decay rates are observed when a phase transition between solid and liquid occurs outside the cladding boundary. The monitoring of multiple mode categories provides more complete assessment of outside media and enhances the capabilities of forward Brillouin scattering fiber sensors.

Standard optical fibers are an excellent sensing platform. The propagation of light in the fiber can be affected by many metrics of interest, such as temperature, static strain and pressure, sound and vibrations, electrical and magnetic fields, chemical and biological reagents, and radiation fields.1–4 Fiber sensors may cover hundreds of km, can be installed within many structures with little disruption of functionality, are comparatively immune to electromagnetic interference, and are suitable for hazardous environments.1–4 Optical fibers can support spatially distributed mapping, in which each fiber segment serves as an independent node of a sensory network.5 Optical fiber sensors, therefore, support a thriving research community for 50 years, as well as a rapidly growing industry.

The use of standard single-mode fibers toward sensing applications is, nevertheless, facing an inherent challenge. Light is confined to the inner core of the fiber; hence, measurements require that the condition of interest should prevail within the core. This requirement is easily met, for example, in the monitoring of temperature or axial strain, but it is more difficult to accommodate in the sensing of chemicals outside the cladding boundary.6–8 Optical fiber sensors of chemicals often rely on indirect transduction mechanisms, which convert a signal of interest to changes in temperature or strain within the core.9,10 Alternatively, sensors can be based on structural modifications of standard fibers11–14 or on non-standard cross section geometries and materials.15–20 While these sensing schemes may be clever, sensitive, and specific, they are often restricted to point measurements in a discrete, pre-selected set of locations.9 In addition, the deviations from standard, unmodified single-mode fibers take away from the merit and applicability of the proposed sensors.

Starting in 2016, a new concept for the sensing of liquids outside the cladding boundary of standard, unmodified, single-mode fibers has been proposed and demonstrated by several groups.21–32 The measurements rely on the process of forward stimulated Brillouin scattering (F-SBS):33–35 the opto-mechanical interaction among co-propagating guided optical and acoustic wave components in the common fiber medium. Optical pump pulses may stimulate the oscillations of a guided acoustic mode.33–35 The acoustic wave, in turn, may scatter and modulate optical probe light.33–35 The effect was first proposed and demonstrated by Shelby, Levenson, and Bayer in 1985.33 The guided acoustic modes involved in F-SBS span the entire cross section of the fiber cladding and reach its outer boundary. The presence of media outside the cladding modifies the boundary conditions that define the acoustic mode and, therefore, manifests in the characteristics of the entire F-SBS process.21 

The guided acoustic modes that can be stimulated by forward Brillouin scattering in standard fibers must maintain specific symmetries. In one category of relevant acoustic modes, the transverse profile of the material displacement vector is radially symmetric.33–35 The optical stimulation of the radially symmetric modes is the most efficient,33–35 and that category of modes has served in nearly all demonstrations of F-SBS sensing to-date.21,23–32 The radially symmetric guided acoustic modes taking part in F-SBS are essentially one-dimensional and dilatational: Both the material displacement vector and the wave vector are almost entirely aligned with the transverse radial direction.33–35 When a bare fiber is immersed in a liquid of finite mechanical impedance, the dilatational, radially symmetric acoustic waves partially dissipate outward into the liquid.21 The loss of acoustic energy at the boundary appears as faster decay rates of the stimulated acoustic modes. The decay rate is the same for all radially symmetric modes with a given liquid outside the cladding.21 This property has formed the basis for the quantitative sensing of liquids outside the cladding of standard fibers, where guided light in the single core mode cannot reach.21,23–32 The principle has successfully been extended to point-measurements,30 spatially distributed analysis,23,25,26,29,31,32 and the monitoring of liquids outside certain types of polymer coating layers.24,26

Guided acoustic waves of the fiber cross section, however, generally exhibit more complicated tensor attributes than one-dimensional, dilatational wave motion in the radial direction only. F-SBS in standard fibers may also take place through another class of guided acoustic modes, in which the transverse profiles of the material displacement vectors maintain twofold azimuthal symmetry. In these so-called torsional–radial (TR) modes, the displacement vector comprises both radial and azimuthal components, and it combines dilatational as well as shear wave motion terms. F-SBS in bare fibers through the TR modes was proposed and demonstrated already in the initial work of Shelby et al. in 1985.33 However, to the best of our knowledge, the analysis of the process has not yet been extended to fibers immersed in liquids.

The boundary conditions for the TR modes are more complex, and they present additional opportunities for the sensing of outside media. Yet only few studies have attempted F-SBS sensing via the TR modes. In one report by Hayashi and co-workers,22 the decay rate of one specific TR mode was shown to increase as a function of the mechanical impedance of a liquid outside the cladding of a bare fiber. However, the observed decay rate was an order of magnitude slower than that of the radially symmetric modes under the same conditions.22 This large difference was not discussed, and follow-up studies continued to rely exclusively on the radially symmetric modes. The study of the tensor properties of F-SBS with liquids outside the fiber cladding, and the potential use of these properties toward sensing applications, remained to be completed.

In this work, we extend the study of F-SBS to include TR modes with liquids outside the cladding of bare fibers, in coated fibers, and for liquids outside the coated fibers. The relevant boundary conditions are formulated and solved. The analysis and experiment indicate that unlike the radially symmetric modes, the decay rates of the TR modes with liquids outside the cladding are mode-specific. Following the recent work of Sánchez and co-workers,36–39 we distinguish between two types of TR modes: Ones that are primarily dilatational, and others that are dominated by their shear terms. The presence of liquids outside the cladding or coating does not modify the boundary condition for the shear wave component. Accordingly, the decay rates of shear-dominated modes exhibit only a little increase when the fiber under test is put in liquids, whereas modes of dilatational characteristics decay much faster. The results also resolve the difference between the earlier report of sensing through a TR mode22 and studies based on the radially symmetric modes.21,23–32 The distinction between mode categories is also found with liquids outside fibers with certain types of coating.

The differences between mode types become pronounced when the medium in contact with the fiber cladding changes between solid and liquid. This property is illustrated in monitoring the penetration of acetone through a coating layer of fluoroacrylate polymer. When the liquid reaches the fiber cladding, the decay rates of stimulated shear-type TR modes become slower by an order of magnitude. The observed changes are reversible. The results demonstrate that the tensor analysis of F-SBS can give a strong indication of phase transitions at the fiber boundary. The observation of multiple acoustic mode types may provide more complete assessment of the elastic properties of outside media. The preliminary results have been presented in a recent conference paper.40 

We regard a standard single-mode bare fiber as a uniform cylindrical rod of silica with radius a and neglect small scale differences between the elastic properties of core and cladding regions. The fiber supports discrete sets of guided acoustic modes that propagate in the axial direction ẑ. In this work, we focus on TR modes of twofold azimuthal symmetry, referred to as TR2m modes, where m is an integer. It is the only set of guided acoustic modes of standard fibers, other than the purely radial ones, which may be stimulated by pump pulses through F-SBS processes.33,34 Each mode is characterized by a cutoff acoustic angular frequency Ω2m, below which it may not propagate in the axial direction. Optical stimulation is effective at acoustic frequencies close to cutoff.33,34 At that limit, both the wave vector and the displacement vector are almost entirely confined to the transverse plane.33,34

The normalized displacement vectors of the torsional–radial TR2m modes consist of dilatational and shear wave terms. The dilatational term may be expressed as follows (Ref. 41, see the supplementary material):
u2mLr,ϕ=A2mΩ2mvLJ2Ω2mvLrcos2ϕsin2ϕr̂+2rJ2Ω2mvLrsin2ϕcos2ϕϕ̂.
(1)
The units of u2mL are m−1. In Eq. (1), r is the transverse radial coordinate and ϕ is the azimuthal coordinate with respect to the x̂ axis. r̂ and ϕ̂ denote the unit vectors in the radial and azimuthal directions, respectively. vL is the velocity of dilatational acoustic waves in silica, and J2 is the second-order Bessel function of the first kind. The prime superscript signifies the derivative of a function with respect to its argument. The terms within the curled brackets represent two spatially orthogonal azimuthal dependence functions. The two solutions share the same cutoff frequency and radial dependence, and they are identical except for a π/4 rotation. The unitless magnitude constant A2m is found through the boundary conditions (see below).
The normalized displacement vector also includes a shear wave component (Ref. 41, see the supplementary material),
u2mSr,ϕ=C2m2rJ2Ω2mvSrcos2ϕsin2ϕr̂+Ω2mvSJ2Ω2mvSrsin2ϕcos2ϕϕ̂,
(2)
where vS is the velocity of acoustic shear waves in silica and C2m is a constant magnitude. The overall normalized displacement vector is given by the sum of the two contributions,
u2mr,ϕ=u2mLr,ϕ+u2mSr,ϕ.
(3)
The sum vector u2mr,ϕ is normalized by a common factor so that 02π0au2mu2mrdrdϕ=1. We define the radial and azimuthal components of the combined normalized displacement vector as follows:
u2mr,ϕ=u2m,rr,ϕr̂+u2m,ϕr,ϕϕ̂,
(4)
u2m,rr,ϕ=A2mΩ2mvLJ2Ω2mvLr+C2m2rJ2Ω2mvSrcos2ϕsin2ϕ,
(5)
u2m,ϕr,ϕ=A2m2rJ2Ω2mvLr+C2mΩ2mvSJ2Ω2mvSrsin2ϕcos2ϕ.
(6)
The boundary conditions at the edge of the cladding at r = a require that both the radial stress and the shear stress equal zero (Ref. 41 see the supplementary material). The boundary conditions may be brought to the form of the following coefficient matrix MΩ:41 
J2ΩvLa122Θ2ΩvLaΩvSa2J2ΩvSa12+4Θ2ΩvSaJ2ΩvLa12+4Θ2ΩvLaJ2ΩvSa122Θ2ΩvSaΩvSa2A2C2=0,
(7)
where Θ2ξ=ξJ1ξ/J2ξ and J1 is the first-order Bessel function of the first kind. The solutions to detMΩ=0 represent the cutoff frequencies Ω2m of the torsional–radial TR2m modes. The corresponding eigenvectors are proportional to A2mC2mT; hence, they express the relative magnitudes of dilatational and shear wave contributions to a given TR2m mode. Supplementary videos 1 and 2 present the calculation of TR2m mode wave-packets in a transverse cross-section of the fiber. The dilatational components of the torsional-radial modes propagate at the velocity vL, whereas shear wave components propagate at the velocity vS.

Figure 1 presents the ratios C2m2/C2m2+A2m2 and A2m2/C2m2+A2m2, as functions of the modal cutoff frequencies Ω2m/2π. Large values of the former ratio describe modes that are primarily of shear characteristics, whereas high values of the latter suggest modes that are predominantly dilatational. The distinction between the two types follows that of Refs. 36–39, where the classification is based on approximations to the boundary conditions equation. Note that all torsional–radial TR2m modes contain nonzero components of both dilatational and shear wave motions. As explained below, the distinction between the two types of TR2m modes is significant in F-SBS sensing. The modal oscillations are also characterized by a decay rate (or linewidth) Γ2m. With air outside the bare fiber, the decay rates are given by acoustic dissipation in silica, and they range between 2π × 30 and 2π × 200 kHz.

FIG. 1.

Calculated relative magnitudes squared of the shear terms C2m2/C2m2+A2m2 (top) and dilatational terms A2m2/C2m2+A2m2 (center) of TR2m modes, as functions of their cutoff frequencies Ω2m/2π. Certain modes are dominated by their shear terms, whereas others are primarily of dilatational characteristics. Bottom: Calculated decay rates Γ2m/2π of the TR2m modes with water outside the cladding of a bare fiber, as a function of the cutoff frequencies. The decay rates of modes that are dominated by their dilatational wave components are consistently faster than those of modes that are predominantly of shear characteristics.

FIG. 1.

Calculated relative magnitudes squared of the shear terms C2m2/C2m2+A2m2 (top) and dilatational terms A2m2/C2m2+A2m2 (center) of TR2m modes, as functions of their cutoff frequencies Ω2m/2π. Certain modes are dominated by their shear terms, whereas others are primarily of dilatational characteristics. Bottom: Calculated decay rates Γ2m/2π of the TR2m modes with water outside the cladding of a bare fiber, as a function of the cutoff frequencies. The decay rates of modes that are dominated by their dilatational wave components are consistently faster than those of modes that are predominantly of shear characteristics.

Close modal

The acoustic oscillations are modified by the presence of liquid outside the bare fiber cladding. The radial stress term now equals the pressure applied by the liquid.42,43 Due to the finite impedance of the surrounding medium, dilatational acoustic waves partially dissipate outward, away from the fiber.21,41 The decay rates of the dilatational waves increase accordingly. The radially symmetric guided acoustic modes of the fiber, R0m, are strictly dilatational.42,43 The decay rates Γ0m of these modes are simply related to the impedance Zout of the liquid medium outside, and the relation is the same for all modal orders m.21 The measurement of decay rates Γ0m of radially symmetric modes forms the basis of nearly all F-SBS sensing demonstrations reported to-date.21,23–32

An outside liquid also modifies the boundary conditions of the torsional–radial TR2m modes; however, its effect is more complex. Just like the case of the radial R0m modes, the radial stress term of the TR2m modes at the cladding boundary should equal the liquid pressure.42,43 By contrast, non-viscous liquids do not support shear wave motion; hence, the shear stress at the boundary remains zero just like the case of bare fiber in air. While the dilatational components of the torsional–radial TR2m modes are affected by the presence and properties of the surrounding liquids, the corresponding shear wave components are not. Unlike the radial R0m modes, we expect the decay rates Γ2m to vary among the set of TR2m modes. We anticipate that modes dominated by their dilatational components (see Fig. 1) would experience a stronger dissipation and a faster decay than other modes that are predominantly of shear characteristics. This prediction is tested in both calculations and experiments.

The coefficients of the boundary condition matrix MΩ for the torsional–radial TR2m modes with liquids outside the cladding are modified as follows (Refs. 42 and 43 see the supplementary material):
M11Ω=ρ0vL2ΩvL2J2ΩvLaρ0vL22vS21aΩvLJ2ΩvLaρ0vL22vS22a2J2ΩvLa+ZoutΩH2ΩvLoutaH2ΩvLoutaΩvLJ2ΩvLa,
(8)
M12Ω=2ρ0vS22aΩvSJ2ΩvSa2ρ0vL2vS22a2J2ΩvSa+ZoutΩH2ΩvLoutaH2ΩvLouta2aJ2ΩvSa,
(9)
M21Ω=ρ0vS22aΩvLJ2ΩvLaρ0vS22a2J2ΩvLa,
(10)
M22Ω=ρ0vS22a2J2ΩvSa+12ρ0vS2ΩvS2J2ΩvSa12ρ0vS21aΩvSJ2ΩvSa,
(11)
where vLout is the velocity of dilatational acoustic waves in the surrounding liquid, ρ0 is the density of silica, H2 is the second-order Hankel function of the first kind, and the double prime sucperscript denotes the second derivative of a function with respect to its argument. The solutions of the equation detMΩ=0 are now complex-valued. Their real parts represent the cutoff frequencies Ω2m of the TR2m modes with liquid outside the cladding, and their imaginary parts equal half the modal decay rates, Γ2m/2. As shown in Fig. 1, the decay rates of modes dominated by their dilatational wave terms are, indeed, considerably faster, in agreement with expectations. Unlike the radial R0m modes, which are strictly dilatational, we cannot assign a single decay rate to all torsional–radial TR2m modes of a fiber immersed in a given liquid.
In this work and many other studies, the F-SBS processes are experimentally characterized through the photoelastic scattering and modulation of optical probe waves, induced by the stimulated acoustic modes.21,33–35,41,44,45 The processes are quantified by modal opto-mechanical coefficients, in units of W−1 m−1,44,
γpmΩγpmΩpm11+2jΩpmΩ/Γpm.
(12)
Here, p equals either 0 or 2 for the R0m or the TR2m modes, respectively, and Ω0m is the cutoff frequency of a radial mode R0m. The opto-mechanical coefficient of F-SBS through each mode takes up its maximum magnitude γpmΩpm at its cutoff frequency. This magnitude is determined by spatial overlap considerations, the cutoff frequency, and the decay rate (see the supplementary material Ref. 41).

Figure 2 shows the numerical calculations of the normalized magnitude γ2mΩ of the opto-mechanical coefficients, with air or water outside the cladding boundary. The spectra of torsional–radial TR2m modes of dilatational characteristics broaden considerably with water outside the cladding due to the faster decay rates of these modes. For example, the linewidth of the mode at 2π × 223.5 MHz is 2π × 2.5 MHz. The shear-dominated modes remain much narrower (linewidth of only 2π × 250 kHz for the mode at 2π × 140 MHz).

FIG. 2.

Calculated normalized opto-mechanical coefficients γ2mΩ of forward stimulated Brillouin scattering through torsional–radial TR2m modes as functions of acoustic frequency Ω/2π. The bare fiber was kept in air (dashed lines) or immersed in water (solid lines). The modal decay rates Γ2m with air outside the cladding were taken from the experimental data, whereas those with water outside were obtained through solutions of the boundary conditions equations (see the text). Immersion in water considerably broadens the spectra of some TR2m modes but has much lesser effect on others. Modes that are dominated by their dilatational components undergo more pronounced broadening than those of predominantly shear wave characteristics (see Fig. 1).

FIG. 2.

Calculated normalized opto-mechanical coefficients γ2mΩ of forward stimulated Brillouin scattering through torsional–radial TR2m modes as functions of acoustic frequency Ω/2π. The bare fiber was kept in air (dashed lines) or immersed in water (solid lines). The modal decay rates Γ2m with air outside the cladding were taken from the experimental data, whereas those with water outside were obtained through solutions of the boundary conditions equations (see the text). Immersion in water considerably broadens the spectra of some TR2m modes but has much lesser effect on others. Modes that are dominated by their dilatational components undergo more pronounced broadening than those of predominantly shear wave characteristics (see Fig. 1).

Close modal
The radial R0m acoustic modes induce phase modulation of optical probe waves that co-propagate with pump pulses, at frequency Ω.21,41,45 The phase modulation is scalar: it is independent of the probe wave polarization and does not modify it.46 The magnitude of the phase modulation at the output of a fiber of length L is given by41,
Δφ̃0mΩ,L=γ0mΩP̃ΩL,
(13)
where P̃Ω (W rad−1 Hz−1) is the Fourier-transform of the instantaneous power Pt of pump pulses, in which t stands for time. Phase modulation also modifies the instantaneous frequency of the output probe wave, which oscillates at frequency Ω as well. The magnitude of frequency modulation equals
Δω̃0mΩ,L=jΩΔφ̃0mΩ,L=jΩγ0mΩP̃ΩL.
(14)
The frequency modulation of probe waves is used later in this work toward the characterization of F-SBS through the radial R0m modes and in sensing applications of the same process.46,47
The scattering of probe waves by the stimulated torsional–radial TR2m modes is polarization dependent.22,46 The acoustic waves induce birefringence between the state of polarization of pump pulses, denoted without loss of generality as x̂, and the orthogonal ŷ axis.46 The magnitudes of phase modulation acquired by probe waves polarized along the two axes are given by41,46
Δφ̃2mx,yΩ,L=±γ2mΩP̃ΩL.
(15)
The acoustically induced birefringence may rotate the instantaneous state of polarization of properly aligned probe waves, at frequency Ω.46 The extent of polarization rotation scales with Δφ̃2mx,y. The radial acoustic modes do not imprint such a polarization rotation. The monitoring of probe wave polarization, therefore, serves for the selective measurements of F-SBS through the torsional–radial TR2m modes.46 

Figure 3 shows a schematic illustration of the experimental setup. Pump light from a laser diode of 1550 nm wavelength was intensity modulated to short and isolated repeating pulses of 2 ns duration and 5 µs period, respectively. The pulses were amplified by an erbium-doped fiber amplifier to an average power of 2 W and launched into a fiber under test. The temporal shape of the pump pulses Pt was recorded, and its Fourier transform P̃Ω was calculated offline. A tunable optical bandpass filter blocked the pump pulses at the output end of the fiber. The pulses stimulated a packet of acoustic modes via F-SBS. A continuous optical probe wave was drawn from a second laser diode of 1544 nm wavelength and 16 mW power. The probe wave co-propagated in the fiber section under test, alongside the pump pulses. The optical bandpass filter at the fiber output was tuned to transmit the probe wavelength.

FIG. 3.

Experimental setup.46,47 EDFA: erbium-doped fiber amplifier; BPF: optical bandpass filter; EOM: electro-optic amplitude modulator; and FBG: fiber Bragg grating.

FIG. 3.

Experimental setup.46,47 EDFA: erbium-doped fiber amplifier; BPF: optical bandpass filter; EOM: electro-optic amplitude modulator; and FBG: fiber Bragg grating.

Close modal
The output probe was split into two detection channels. In the first channel, the probe passed through an in-line fiber polarizer, aligned for 50% power transmission. Photoelastic scattering by the stimulated torsional–radial TR2m modes resulted in polarization rotation of the probe wave as discussed above, which was converted to intensity modulation at the polarizer output.22,46 The polarization of the pump pulses was adjusted for maximum probe wave rotation.22,46 Since the purely radial R0m acoustic modes do not rotate the polarization of the probe, the intensity modulation at the output of the polarizer represented contributions of the TR2m modes only.46 The probe light at the polarizer output was detected by a photoreceiver of 1.6 GHz bandwidth. The detected voltage Vpolt was sampled by a digitizing oscilloscope at 4 giga-samples per second and stored for further offline processing. The acquired traces were averaged over 132 000 repeating pump pulses. The normalized opto-mechanical coefficient of the TR2m modes was retrieved through the Fourier transform of the detected voltage, ṼpolΩ [see Eq. (16)],
γ2mΩṼpolΩP̃Ω.
(16)
The output probe light at the second detection branch was reflected from a fiber Bragg grating (FBG). The probe wavelength was aligned with a slope of the FBG reflectivity spectrum. Following reflection from the FBG, the instantaneous frequency oscillations of the probe wave due to the radial R0m modes [see Eq. (14)] were converted to intensity modulation. Polarization scrambling of the pump pulses removed the contributions of the torsional–radial TR2m modes to the probe frequency modulation.46 The reflected probe wave was detected by a second photoreceiver of the same bandwidth, and the detected voltage VFBGt was sampled by the digitizing oscilloscope. The normalized opto-mechanical coefficient of the radial R0m modes was estimated through the Fourier transform ṼFBGΩ46,47 [Eq. (14)],
γ0mΩṼFBGΩΩP̃Ω.
(17)
Pump-induced modulation and polarization rotation of the probe wave take place through the stimulated acoustic waves and the Kerr effect combined.48 However, the contribution of the instantaneous Kerr effect ends when the pump pulse is through, whereas oscillations of the stimulated acoustic modes continue for hundreds of nanoseconds or longer. Time gating of the Vpol,FBGt traces effectively removed the Kerr effect from the subsequent data analysis.21 

Figure 4(a) shows the measured VFBGt traces for a 15 m long section of bare fiber, kept in air or in water (an impedance of 1.48 kg mm−2 s−1). The modulation of the probe wave consists of a series of impulses, separated by the group delay of dilatational acoustic waves from the fiber axis to the edge of the cladding and back: τL = 2a/vL ∼ 21.8 ns. The decay of the impulse series is much faster with the fiber immersed in liquid, as expected.21, Figure 4(b) presents the corresponding normalized spectra ṼFBGΩ. The observed peaks match the cutoff frequencies of the radial R0m modes.33 Immersion in liquid leads to similar broadening of all R0m stimulated modes, with full widths at half maximum (FWHMs) of 2π × 6 MHz.

FIG. 4.

(a) Measurements of the modulated probe output power following reflection from a fiber Bragg grating. A 15 m long section of standard, bare fiber was kept in air (black lines) or immersed in water (blue lines). A series of modulation impulses are separated by 21.8 ns, the group delay of dilatational acoustic waves from the fiber axis to the edge of the cladding and back. Immersion in liquids leads to faster decay of the impulse series. (b) Normalized frequency-domain magnitude of the opto-mechanical interaction, γ0mΩ. Multiple peaks correspond to the cutoff frequencies of the purely radial guided acoustic modes R0m. All modes undergo similar broadening to full-widths-at-half-maximum linewidths of 2π × 6 MHz when the fiber is immersed in liquid. (c) Measurements of the modulated probe output power following transmission through a polarizer, for the same fiber section in air and water. The series of impulses is separated by 33.3 ns, the two-way group delay of acoustic shear waves from the fiber axis to the edge of the cladding and back. Compared with panel (a), immersion in liquid has lesser effect on the decay rate of the series. (d) Normalized frequency-domain magnitude of the opto-mechanical interaction, γ2mΩ. Multiple peaks correspond to the cutoff frequencies of the TR2m modes. Spectral widths following immersion of the fiber in water are uneven. Modes with dominant dilatational components broaden considerably, whereas those of primarily shear characteristics are less affected. (e) Magnified view of panel (d). Peaks are labeled as dilatational (L) or shear (S), according to calculations of relative magnitudes of the two components (see Fig. 1). The measured spectrum with water outside the cladding agrees very well with calculations [panels (f) and (g)].

FIG. 4.

(a) Measurements of the modulated probe output power following reflection from a fiber Bragg grating. A 15 m long section of standard, bare fiber was kept in air (black lines) or immersed in water (blue lines). A series of modulation impulses are separated by 21.8 ns, the group delay of dilatational acoustic waves from the fiber axis to the edge of the cladding and back. Immersion in liquids leads to faster decay of the impulse series. (b) Normalized frequency-domain magnitude of the opto-mechanical interaction, γ0mΩ. Multiple peaks correspond to the cutoff frequencies of the purely radial guided acoustic modes R0m. All modes undergo similar broadening to full-widths-at-half-maximum linewidths of 2π × 6 MHz when the fiber is immersed in liquid. (c) Measurements of the modulated probe output power following transmission through a polarizer, for the same fiber section in air and water. The series of impulses is separated by 33.3 ns, the two-way group delay of acoustic shear waves from the fiber axis to the edge of the cladding and back. Compared with panel (a), immersion in liquid has lesser effect on the decay rate of the series. (d) Normalized frequency-domain magnitude of the opto-mechanical interaction, γ2mΩ. Multiple peaks correspond to the cutoff frequencies of the TR2m modes. Spectral widths following immersion of the fiber in water are uneven. Modes with dominant dilatational components broaden considerably, whereas those of primarily shear characteristics are less affected. (e) Magnified view of panel (d). Peaks are labeled as dilatational (L) or shear (S), according to calculations of relative magnitudes of the two components (see Fig. 1). The measured spectrum with water outside the cladding agrees very well with calculations [panels (f) and (g)].

Close modal

Figure 4(c) shows the measured Vpolt traces for the same fiber, in air and water. The primary impulses in the traces are separated by the group delay of acoustic shear waves: τS = 2a/vS ∼ 33.3 ns. The results suggest that the torsional–radial TR2m modes of the fiber are dominated by their shear wave components. Compared with Fig. 4(a), immersion in the liquid has much lesser effect on the decay rate of the impulse series. The spectra ṼpolΩ are shown in Fig. 4(d), and a magnified view is shown in Fig. 4(e). The spectral broadening of the TR2m modes is not uniform, as expected. Some modes become much broader than others. Referring to Fig. 1, those TR2m modes that are dominated by their dilatational wave components become much broader than other modes that are predominantly of shear characteristics. For example, the FWHM of the mode at 2π × 176 MHz with the fiber in water was broadened to 2π × 6.5 MHz, whereas that of the mode at 2π × 108 MHz remained only 2π × 600 kHz. The results agree well with the calculation of γ2mΩ with water outside the cladding [see Figs. 4(f) and 4(g)]. The 2π × 108 MHz mode was used by Hayashi et al. in the sensing of liquids outside the cladding in 2017.22 They too observed only modest broadening, to 2π × 320 kHz widths.22 This observation is now placed in a broader context and accounted for.

Figure 5(a) presents the measured normalized γ2mΩ for a 15 m long fiber with polyimide coating (outer diameter of 141 µm). The section was kept in air (blue lines) or immersed in water (red lines). Here too, dilatational TR2m modes exhibit larger broadening with liquid outside the fiber. The FWHM of the mode at 2π × 176 MHz with the fiber in water was 2π × 5.4 MHz, whereas that of the mode at 2π × 168.5 MHz was only 2π × 900 kHz.

FIG. 5.

Measured frequency-domain magnitude of the opto-mechanical interaction γ2mΩ. (a) A 15 m long polyimide coated fiber section was kept in air (blue) or immersed in water (red). Both traces were normalized to a maximum of unity. Immersion resulted in significant broadening of the predominantly dilatational modes only. (b) A 10 m-long fluoroacrylate coated fiber. Red: minutes after immersion in acetone. Black: 24 h following immersion. Both traces were normalized to equal magnitudes of the Kerr effect cross-phase modulation of the probe wave.48 Significant enhancement and narrowing of the shear-dominated modes are seen, as acetone penetrated the coating and reached the cladding boundary. The trace taken after 24 h matches that of a bare fiber in acetone (not drawn). In both panels, modes are labeled as dilatational (L) or shear (S) according to the analysis of Fig. 1.

FIG. 5.

Measured frequency-domain magnitude of the opto-mechanical interaction γ2mΩ. (a) A 15 m long polyimide coated fiber section was kept in air (blue) or immersed in water (red). Both traces were normalized to a maximum of unity. Immersion resulted in significant broadening of the predominantly dilatational modes only. (b) A 10 m-long fluoroacrylate coated fiber. Red: minutes after immersion in acetone. Black: 24 h following immersion. Both traces were normalized to equal magnitudes of the Kerr effect cross-phase modulation of the probe wave.48 Significant enhancement and narrowing of the shear-dominated modes are seen, as acetone penetrated the coating and reached the cladding boundary. The trace taken after 24 h matches that of a bare fiber in acetone (not drawn). In both panels, modes are labeled as dilatational (L) or shear (S) according to the analysis of Fig. 1.

Close modal

The tensor properties of the boundary conditions for the torsional–radial TR2m modes may generate large signals in response to phase transitions between solid and liquid media outside the cladding. To illustrate this prospect, we have monitored the F-SBS spectrum of a 10 m long fiber section coated with a fluoroacrylate layer (outer diameter 136 µm) for 24 h following immersion in acetone (an impedance of 1.07 kg mm−2 s−1). Previous studies have shown that acetone penetrates this coating layer and reaches the cladding boundary within hours.49, Figure 5(b) shows γ2mΩ immediately following immersion (red) and after 24 h in the liquid (black). Both traces were normalized for equal magnitudes of the Kerr effect cross-phase modulation of the probe wave.48 This term is unaffected by conditions outside the core. Significant enhancement and narrowing are observed for the shear-dominated modes, as the liquid acetone replaced the solid coating in contact with the fiber cladding. For example, the FWHM of the mode at 2π × 108 MHz was reduced from 2π × 3.3 MHz to only 2π × 450 kHz after 24 h in acetone. Penetration of acetone modified the dilatational mode spectra to a much lesser extent. The trace obtained after 24 h is very similar to that of a bare fiber in acetone (not drawn for better clarity). The F-SBS spectrum returned to its initial state following the evaporation of acetone (not shown).

In this work, we have extended the analysis of F-SBS fiber sensors of liquids outside the cladding to include the torsional–radial TR2m modes. Unlike the purely radial R0m modes of standard fibers, which are strictly dilatational, the TR2m modes consist of both dilatational and shear wave components. Non-viscous liquids outside the cladding of the fiber do not support the propagation of shear waves. Therefore, while the dilatational wave components of TR2m modes may radiate outward into the surrounding media, the shear wave components do not. Consequently, the effect of liquids outside the cladding on the F-SBS processes varies among the different TR2m modes. The analysis and experiment show that torsional–radial modes dominated by their dilatational wave components decay much faster than modes that are predominantly of shear wave characteristics. The decay rates may differ by a factor of seven. The analysis accounts for the earlier observation of modest linewidth broadening of a specific TR2m mode with liquid outside the cladding.22 The measurements are in very good agreement with calculations.

The distinction among the spectral broadening of different torsional–radial TR2m modes was also found with liquids outside fibers with polyimide coating. Note, however, that a solid coating layer does modify the boundary conditions of the shear wave components at the edge of the cladding. Both shear-type and dilatational-type TR2m modes may be broadened by coating layers, and the linewidths of specific modes depend on interference effects within the coating. The linewidths are also affected by the elastic properties of the particular coating.

The pronounced differences in linewidths among modes may produce large signals when phase transitions take place outside the fiber cladding. This prospect was illustrated in the monitoring of a fluoroacrylate coated fiber section immersed in acetone. When the liquid penetrated the coating layer and replaced the solid coating layer immediately in contact with the cladding, the linewidths of the shear-dominated modes decreased by an order of magnitude and their peaks were enhanced accordingly. The penetration of the liquid was identified by the difference between the linewidths of shear-type and dilatational-type modes. The monitoring of both types of modes allows for a more complete and accurate assessment of the elastic properties of media outside the fiber and enhances the capabilities of F-SBS fiber sensing. The linewidths of the dilatational-type torsional–radial TR2m modes may still provide quantitative estimates of the impedance of liquids, similar to the purely radial R0m modes.

This report does not include the calculations of torsional–radial TR2m modes in fibers with liquids outside coating layers of finite thickness, and the measurements reported in Fig. 5 are not accompanied by quantitative predictions. The modal profiles for the TR2m modes in the three layers: fiber, coating, and liquid, as well as the boundary conditions at the two interfaces, were formulated. However, a sufficient quantitative agreement with experiment could not be obtained. We suspect that residual ellipticity, non-concentricity, and thickness variations along the coating layer within production tolerances restrict the fitting of model parameters. Nevertheless, the detection and analysis of liquids outside the coated fibers through the TR2m modes are still possible with proper pre-calibration. Similar challenges were also encountered and overcome by F-SBS sensors based on the radial R0m modes of coated fibers.24,26

In conclusion, the phenomenon of F-SBS in standard optical fibers is of more complex tensor characteristics than those of one-dimensional radial acoustic modes. The tensor analysis of F-SBS with liquids outside the cladding not only improves the understanding of the mechanism but also promotes its application as a new and promising fiber sensing modality.

See the supplementary material for additional analysis and videos of numerical calculations results.

Pazy Foundation, Israel Agency for Atomic Energy and Israeli Universities Planning and Budgeting Committee, Grant ID113-2020.

The authors have no conflicts to disclose.

Alon Bernstein and Elad Zehavi contributed equally to this work.

Alon Bernstein: Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – review & editing (equal). Elad Zehavi: Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – review & editing (equal). Yosef London: Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Supervision (supporting); Writing – review & editing (supporting). Mirit Hen: Investigation (supporting); Supervision (supporting); Writing – review & editing (supporting). Rafael Suna: Investigation (supporting); Methodology (supporting); Writing – review & editing (supporting). Shai Ben-Ami: Investigation (supporting); Methodology (supporting); Writing – review & editing (supporting). Avi Zadok: Conceptualization (lead); Formal analysis (supporting); Funding acquisition (lead); Methodology (supporting); Project administration (lead); Resources (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material