## Abstract

Recently published experimental works on remotely bonded fiber Bragg grating (FBG) ultrasound (US) sensors show that they display some unique characteristics that are not observed with directly bonded FBG sensors. These studies suggest that the bonding of the optical fiber strongly influences how the ultrasound waves are coupled from the structure to the FBG sensor. In this paper, the analytical model of the structure-adhesive-optical fiber section, treated as an ultrasound coupler, is derived and analyzed to explain the observed experimental phenomena. The resulting dispersion curve shows that the ultrasound coupler possesses a cutoff frequency, above which a dispersive longitudinal mode exists. The low propagation speed of the dispersive longitudinal mode leads to multiple resonances at and above the cutoff frequency. To characterize the resonant characteristics of the ultrasound coupler, a semi-analytical model is implemented and the scattering parameters (S-parameters) are introduced for broadband time-frequency analysis. The simulation was able to reproduce the experiment observations reported by other researchers. Furthuremore, the behaviors of the remotely bonded FBG sensors can be explained based on its resonant characteristics.

## 1 Introduction

Structural health monitoring (SHM) technology has been under intensive studies in the past decades because it has the potential to shift maintenance of infrastructures from safe life practice or schedule-based schemes to condition based maintenance [14]. Since detectable damage could take a long time to develop and its location is typically unknown, an effective SHM system should be able to detect damage over a large area without incurring significant cost or weight penalty. Due to this requirement, ultrasound (US)-based detection and optical fiber sensors are two of the most common sensing schemes for SHM systems. Ultrasound-based techniques detect the abnormalities in the ultrasound or guided waves propagating in the structures and infer the health condition of the structures from these abnormalities. Since ultrasound waves can propagate over a long distance in plates, tubes, cylinders, etc., one ultrasound transducer can cover an area that is much larger than its physical size [5,6]. Optical fiber sensors, on the other hand, detect damage based on the characteristics of light propagating inside the fiber core. They are attractive for SHM primarily due to their light weight, compact size, low cost, and immunity to electromagnetic interferences, etc. [79]. Among various optical fiber sensors, fiber Bragg grating (FBG) based sensors are the most widely accepted sensors [1012]. Typically, FBG sensors are bonded directly on the structure to ensure that the FBG experiences the same displacement, and thus the strain, as the hosting structure. The displacement changes the FBG periods, leading to a shift in the FBG reflectance frequency. Compared with other optical fiber sensors, one unique advantage of the FBG sensors is that the FBG is directly inscribed into a conventional optical fiber. As such, the interface between the sensing element (i.e., the FBG section) and the optical fiber for signal transmission is seamless. Incorporating FBG sensors in an optical fiber therefore does not require labor-intensive integration. In addition, the physical measurands extracted from the spectral parameter of the FBG render the measurements more reliable, more robust, and more sensitive to minute changes. Since the reflectance spectrum of an FBG can have a very narrow bandwidth of a fraction of nanometers, multiple FBG sensors can be implemented in a single strand of optical fiber based on the principle of wavelength division multiplexing [13]. This unique feature enables deploying a large number of FBG sensors without incurring substantial cost or weight penalties.

While optical fibers are mainly used as optical waveguides, studies have been carried out in the past to investigate optical fibers as ultrasound waveguides [1416]. Dubbed “acoustic fiber,” optical fibers were considered as a means for long-distance data and energy transfer as well as delay lines [17]. A focus of these studies was on designing the mechanical properties of the fiber cladding and core to confine the ultrasound wave within the fiber core. However, analysis done by Mbamou et al. [15] concluded that “the usual glass fibers are not as good for acoustical as for optical applications.” A different strategy was developed by the SHM community in exploiting the optical fiber as ultrasound waveguide sensors [1822]. In these applications, the ultrasound wavelength of interest is much larger than the fiber diameter. As such, the optical fiber can be treated as being homogenous and the differences in the material properties of the fiber core, cladding, and coating are neglected. Based on similar principles, fibers made of different materials, such as copper [23,24], aluminum [25], steel [26,27], etc., were also studied as ultrasound waveguides for environmental monitoring or epoxy curing. Compared with other SHM sensors, however, the ultrasound waveguide sensors received rather limited attention.

In this paper, we present an analytical model for studying ultrasound wave coupling between two ultrasound waveguides, e.g., a structure and an optical fiber, through an adhesive layer. Treating the structure-adhesive-fiber section as an ultrasound coupler having four ports, the concept of scattering parameters is introduced to characterize its resonant characteristics. The response of the ultrasound coupler to a narrowband tone-burst input is simulated numerically by varying the parameters of the adhesive layer. These parametric studies reproduce the experimental observations reported in the literature and provide physical explanation to these observations.

## 2 Analytical and Numerical Simulation Model

The physical model of an optical fiber bonded to a structure is shown in Fig. 1(a). In finite element simulation models [30,35,36], the optical fiber is fully or partially encapsulated in the top portion of the adhesive layer. Assuming the ultrasound wave originates at the left side of the structure and propagates toward the bonded section, upon encountering the bonded section, it is coupled to the optical fiber in both forward (i.e., to the right) and backward (i.e., to the left) propagating directions. The physical model is idealized as the simplified model shown in Fig. 1(b), in which the top portion of the adhesive with the embedded optical fiber is homogenized as a superstrate with material properties differing from the rest of the adhesive layer. The optical fibers leading to and from the bonded section are assumed to be connected to the superstrate at the edges. Since the optical fiber only supports the longitudinal wave [30], the simplified model shown in Fig. 1(b) can be represented by the one-dimensional (1D) extensional bar model shown in Fig. 1(c). Considering that the optical fiber is very light and has a very low attenuation, the forward and backward propagating ultrasound waves in the optical fibers are expected to have the same amplitudes as the displacements at the left and right edges of the superstrate, respectively. Therefore, including the optical fibers in the 1D simulation model is not necessary. Ultrasound waves are generated by applying a time-varying force at the left edge of the substrate. The response of the system to this time-varying force is calculated in the frequency domain, following a procedure described in Refs. [37,38]. Two absorption sections were added to the left and right ends of the substrate (i.e., the structure) to eliminate any reflections that may cause numerical aliasing. To minimize the refection at the absorber–substrate interface, the material properties of the absorption sections are identical to those of the substrate except that they have a very small mechanical loss coefficient and a very large length (e.g., 100 m). By implementing the model semi-analytically without dividing the absorbers into small elements, the large lengths do not introduce any additional computation burden.

Fig. 1
Fig. 1
Close modal
The 1D simulation model is sectioned along the interfaces where the cross-sectional area changes, i.e., at the edges of the ultrasound coupler and the absorber–substrate interfaces. As such, the model can be divided into two types of homogenous section, i.e., the absorber/ substrate section and the ultrasound coupler section. For the absorber/substrate sections, the extensional bar model is adopted to simulate the longitudinal ultrasound modes. For the ultrasound coupler, its governing equation can be derived assuming the displacements of the substrate and superstrate are coupled through the shear deformation of the adhesive layer [37,38] (see Fig. 2). As such, the shear stress τ of the adhesive layer can be expressed as
$τ(x,t)=Gaγa=Ga[ub(x,t)−up(x,t)ha]$
(1)
where γa is the shear strain of the adhesive and Ga is the adhesive shear modulus. The subscribes b, p, a represent the substrate, the superstrate, and the adhesive, respectively. u represents the displacement and h represents the thickness.
Fig. 2
Fig. 2
Close modal
The governing equations for the longitudinal deformations of the substrate and superstrate are [37,39,40]
$∂2ub∂x2−ρbEb∂2ub∂t2=−τbEbhb$
(2a)
and
$∂2up∂x2−ρpEp∂2up∂t2=τpEphp$
(2b)
in which, ρ and E stand for the density and the Young's modulus. τb = τ and τp = ατ for an adhesive having a shear transfer ratio of α. Combining Eqs. (1) and (2) results in an analytical governing equation for the ultrasound coupler, i.e.
$∂4u¯p∂x4+A∂2u¯p∂x2+Bu¯p=0$
(3)
whose solution is
$u¯p(x)=aieβix+die−βix,i=1,2$
(4)
in which βi are the roots of
$βi4+Aβi2+B=0$
(5)
The two constants A and B are functions of the geometrical and mechanical properties of the substrate, superstrate, and adhesive layer as well as the angular frequency ω, i.e.
$A=C2C1+ρbEb(ω2−αρbhbGaha)$
(6a)
$B=C2C1ρbEb(ω2−αρbhbGaha)+αC1ρbEb1ρbhbGaha$
(6b)
$C1=−(ρphp)(Epρp)(haGa)$
(6c)
and
$C2=1−ω2(ρphp)(haGa)$
(6d)

The design parameters of the ultrasound coupler, therefore, include the Young's modulus-density ratio Eb/ρb as well as the density-thickness product ρbhb of the substrate and superstrate, and two adhesive parameters, i.e., the shear modulus-thickness ratio Ga/ha and the shear transfer ratio α.

## 3 Propagation Modes and Dispersion Curve of Ultrasound Coupler—Analytical Solution

The resonant characteristics of the ultrasound coupler can be explained based on the governing equation given in Eq. (5). The characteristic roots of the Eq. (5) can be expressed as

$β1=−A−A2−4B2andβ2=−A+A2−4B2$
(7)
To support wave propagation, at least one of the roots βi, i = 1, 2 must be complex. Since
$Δ=A2−4B=[C2C1−1Eb(ρbω2−αGahbha)]2+4αGaEbhbha×GaEphpha>0$
(8)
whether βi is complex or not depends on the signs of A and B. As tabulated in Table 1, the ultrasound coupler supports only one mode if B < 0 or B = 0 $∧$A > 0 and it supports two modes if B > 0 $∧$A ≥ 0. Consequently, the cutoff frequency for the second propagation mode can be analytically solved by setting B = 0, i.e.
$fc=ωc2π=12πGaha(αhbρb+1hpρp)$
(9)
Table 1

Relationship between the signs of A and B and the characteristic roots of ultrasound coupler's governing equation

A < 0A = 0A > 0
β1β2β1β2β1β2
B < 0CRCRCR
B = 0RR00CR
B > 0RRCCCC
A < 0A = 0A > 0
β1β2β1β2β1β2
B < 0CRCRCR
B = 0RR00CR
B > 0RRCCCC

Clearly, fc is dependent of the adhesive property Ga/ha as well as the substrate and superstrate mass parameters, hbρb, and hpρp. On the other hand, it is independent of the Young's moduli of the substrate or superstrate.

The dispersion curve of the ultrasound coupler, which represents the relationship between the group velocities of the two modes and the frequency, is calculated from the characteristic roots βi and shown in Fig. 3. The substrate is an aluminum alloy with mechanical properties as the followings: Young's modulus E = 71 GPa, density ρ = 2770 kg/m3, and Poisson's ratio υ = 0.33. The superstrate is assumed to have the same properties as the optical fiber, i.e., E = 66 GPa, ρ = 2170 kg/m3, and υ = 0.15 (see tabe 1 in Wee et al. [36]). The mechanical properties of the adhesive are typically unknown, and their values provided in the publications can vary widely [41,42]. To study the adhesive effects, it is common to vary the adhesive properties in a selected range [36,37,43]. The adhesive for this study is initially assumed to have a Young's modulus of 2.5 GPa and a Poisson ratio of υ = 0.39. For an adhesive thickness of 185 µm, a substrate thickness of 0.8 mm, and a superstrate thickness of 125 µm, the cutoff frequency of the coupler is calculated from Eq. (9) as 713 kHz. Below the cutoff frequency, there is only one propagation mode with a group velocity identical to that of the substrate. Above the cutoff frequency, one propagation mode has a group velocity that reduces at a very gradual rate with the increasing frequencies. The group velocity of the second propagation mode, however, increases rapidly from zero at fc and approaches that of the substrate at high frequencies. In other words, the ultrasound coupler supports a dispersive wave above the cutoff frequency. This is different from conventional extensional bars, which only have one nondispersive mode [44].

Fig. 3
Fig. 3
Close modal

## 4 Resonant Characteristics of Ultrasound Coupler—S-parameter Representation

Once the governing equation for the ultrasound coupler is established, the numerical simulation of the 1D model shown in Fig. 1(c) can be implemented by adopting the reverberation matrix method (RMM) described in Refs. [45,46] and applying the boundary and continuity conditions [37,38]. For more detailed descriptions of the RMM and the simulation method, the readers should refer to the cited Refs. [37,38,45,46]. Since the constants A and B are functions of the angular frequency ω, it is expected that the behavior of the ultrasound coupler is frequency dependent. Therefore, a broadband analysis of the ultrasound coupler is necessary, which can be facilitated using the S-parameters [47], a concept that is commonly used in the microwave community for representing a linear-time-invariant network. As shown in Fig. 4, an ultrasound coupler can be considered as a 4-port network; port 1 and 2 represent the left and right edges of the substrate while port 3 and 4 represent the left and right edges of the superstrate, respectively. The transmission S-parameter Sj1 is the frequency spectrum of the output uj(x, ωi) at port j (j = 2, 3, and 4), when the ultrasound is generated at port 1 using an impulse force F(ωi) = 1. Port 1 is selected to be at several wavelengths away from the left edge of the bonding section to eliminate the edge resonance effect (see discussions in Sec. 5.1). Once the S-parameters are available, the time-frequency response of the ultrasound coupler can be calculated using the procedure described in Refs. [47,48].

Fig. 4
Fig. 4
Close modal

Figure 5 shows the S-parameters for three different adhesive lengths La. When La is small, i.e., La = 1 mm, only one resonant peak is observed at the cutoff-frequency fc, due to the very small group velocity of the dispersive ultrasound mode. As La increases to 10 mm, four additional resonant peaks appear above fc. Below fc, the S41 curve only has a slight “bulge” at around 100 kHz, as highlighted by the circle. However, it is difficult to discern whether it is a resonant peak or not. At La = 50 mm, the number of resonance peaks increased dramatically above fc. In addition, there are clear resonant peaks below fc, e.g., at 83, 130, 178 kHz, etc. It is interesting to note that the resonant peak at fc exists regardless of the adhesive length La while the other resonant peaks change locations with La and the number of resonant peaks increases with La. We suspect that the resonance peaks are related to the ultrasound waves being bounced back and forth between the two free edges of the superstrate. If this hypothesis is true, the resonance frequencies would be functions of the propagation speed and the bonding length. This explains why a resonance peak exists at the cutoff frequency fc with any bonding length because of the low propagation speed at fc. Verifying such a hypothesis, however, would require more extensive investigations and will be a subject of future study. The S21 curve displays a few notches at high frequencies. These notches represent the antiresonances, which is similar to the ultrasound spectrum generated using a surface bonded piezoelectric wafer active transducer [5,49]. Notice also that these notches have very narrow bandwidths. In order to observe these notches experimentally, broadband frequency-domain measurements, such as laser ultrasonics, may be needed and will be a subject of future study.

Fig. 5
Fig. 5
Close modal

## 5 Explanations of Experimental Observations

Taking advantage of the computation efficiency and time-frequency analysis capability of the simulation model, we were able to perform comprehensive parametric studies on the bonding condition of remotely bonded FBG sensors. These studies provide the theoretical explanations to the experimental observations reported in published works [29,3133], as discussed below.

### 5.1 Why Does Remotely Bonded Fiber Bragg Gratings Display Enhanced Sensor Responses?.

Wee et al. reported that the response of a remotely bonded FBG could be five times larger than if the FBG is directly bonded [29]. When an FBG is bonded directly on a structure, the adhesive typically covers the entire length of the FBG and even the optical fiber leading to and from the FBG. Therefore, the displacement measured by a directly bonded FBG can be approximated as the displacements at the center of the superstrate. In contrast, when the FBG is bonded on the structure remotely, the displacements at the edges of the superstrate is coupled to the optical fiber. The spectra of the displacements at three locations of the superstrate, i.e., at the left edge, the center, and the right edge, for an adhesive length of 10 mm, are shown in Fig. 6(a). Below the cutoff frequency fc, the displacements at the right edge of the superstrate are consistently larger than the displacements at the center or at the left edge. Above fc, however, both edges experience the same displacements while the center has a slightly lower displacement, except at the resonant peaks. The maximum displacements along the length of the substrate and superstrate are shown in Fig. 6(b), generated using a 300 kHz 5.5 cycle tone-burst excitation. Near the left edge of the ultrasound coupler (i.e., at x = 0.2 m), the maximum displacement of the substrate fluctuates along the length and the displacement at the left edge of the superstrate is substantially smaller than the rest of the superstrate. In contrast, the right edge of the superstrate displays a substantially larger maximum displacement than other locations. Away from the edges, the substrate and superstrate of the ultrasound coupler have almost identical displacements. The differences in the maximum displacements at the edges and at the center of the ultrasound coupler is due to the edge resonance effect. Edge resonant effect refers to the generation of large displacements in the near-field of scattering sources, such as free edges [50,51], step cross-sectional changes [52,53], cracks [54,55], wedges [56], etc. At a scattering source, waves with propagation constants different from that of the incident waves are excited in order to satisfy the boundary condition. The interference of the incident and scattered waves leads to wave enhancements in the immediate vicinity of the scattering source [54,57,58]. This effect is more obvious when the adhesive length increases to 50 mm, as shown in Fig. 6(c). In this case, the large displacements are seen at the locations of the substrate close to the left edge of the coupler and at the right edge of the superstrate. The displacements decay rapidly with the distance near the edges and remains constants at locations that are more than about one wavelength away from the edges. It worth noting that the substrate also displays some edge effects near the edges of the bonded section, albeit the amplitude is much smaller than the superstrate. This is because the substrate is continuous while the superstrate has two free edges. In other words, the substrate sections to the left and right side of the bonded section limit the displacement of the substrate under the superstrate.

Fig. 6
Fig. 6
Close modal

While the present work is focused on the remotely bonded FBG ultrasound sensor, some insights can be also drawn with respect to the directly bonded FBG sensors. For the directly bonded FBG sensors, the assumption is that the displacement experienced by the FBG sensor is the same as that of the structure. This is true only when the FBG sensor is more than one or two wavelengths away from the edges of the adhesive, as Figs. 6(b) and 6(c) indicate. Otherwise, the edge effect will have an impact on the response of the directly bonded FBG sensor as well. In addition, when the adhesive length is short, as in the case of Fig. 6(b), the maximum displacements of the FBG sensor may vary along its length. In other words, different portion of the FBG may experience different displacement amplitudes. This could lead to the broadening of the FBG spectrum. Therefore, the adhesive length should be sufficiently long to ensure uniform displacement amplitude along the FBG length, as in the case of Fig. 6(c).

Fig. 7
Fig. 7
Close modal

### 5.3 Why Does the Coupling Efficiency Increase With the Bonding Length?.

Wee et al. discovered that the FBG response increases with the adhesive length up to a certain distance [31]. To investigate the effect of the bonding length La on the coupling efficiency of the ultrasound coupler, we performed a parametric study on La. The maximum displacements at the right edge of the superstrate with different La are normalized with the displacement of La = 1 mm and are plotted in Fig. 8(a). Again, the excitation was selected to be a 300 kHz 5.5 cycle tone-burst signal. The Young's modulus of the adhesive was 500 MPa and the thickness was 22 µm [33]. Since the shear transfer ratio of the adhesive is typically unknown, two shear transfer ratios, i.e., α = 1 and 0.5, were studied. For α = 1, the normalized maximum displacement increased initially with La, reaching a maximum value of 1.35 at La = 6 mm and then decreases with La until La = 10 mm. Its value then fluctuates slightly with La. Changing α to 0.5 reduces this fluctuation, making the trend agree better with the experimental results. The S41 parameters for these two cases are shown in Figs. 8(b) and 8(c). For α = 1, the S41 parameters do not have any resonance peak below the cutoff frequency fc for any bonding length less than 6 mm. A small resonance peak starts to appear when La = 7 mm. This resonance peak becomes more prominent as La increases. In addition, the resonance frequency shifts toward the left and additional resonance peaks, albeit small, appear at higher frequencies as La increases. Since the tone-burst frequency was fixed at 300 kHz, the shift of the resonance peaks resulted in the amplitude fluctuation of the tone-burst response. In comparison, the resonance peaks are less prominent, i.e., have lower amplitudes and broader bandwidths, for α = 0.5 and thus the fluctuation of the tone-burst response is less significant. For smaller La, however, the amplitude of the tone-burst response is not affected by the shear transfer ratio α. These results suggest that the bonding length La should be optimized to achieve the maximum tone-burst response, especially when the shear transfer ratio is large. Unfortunately, the shear transfer ratio of the adhesive is typically unknown. Measuring the shear transfer ratio from the resonance characteristics of the ultrasound coupler could be a subject of future study.

Fig. 8
Fig. 8
Close modal

## 6 Conclusions

The analytical model of an ultrasound coupler, coupling longitudinal waves from one waveguide to the other via the adhesive layer, is developed. We discovered that the ultrasound coupler possesses a cutoff frequency, above which a dispersive longitudinal mode can propagate. Treating the ultrasound coupler as a 4-port network, a semi-analytical model was implemented to calculate its broadband S-parameters. Parametric studies show that the ultrasound coupler displays very different resonant behaviors at frequencies below and above the cutoff frequency and the adhesive properties have strong influences on these behaviors. We also discovered that the unique behaviors of remotely bonded FBG ultrasound sensors are contributed by the resonance of the ultrasound coupler. In the future, more detailed investigations of the resonant characteristics of ultrasound couplers will be carried out using noncontact ultrasound sensing technique with the aim of inversely determining the adhesive properties from the measured resonances. In addition, the source of the resonances and the relationship between the resonance frequency, the wave speed, and the bonding length, will need more detailed investigations.

## Acknowledgment

This work is supported by the Office of Naval Research (Grant No. N00014-19-1-2098). The supports and suggestions of the program manager, Dr. Ignacio Perez, are greatly appreciated. Professor Huang also thanks Drs. Wee and Peters at North Carolina State University for stimulating discussions.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. Data provided by a third party listed in Acknowledgment.

### Appendix

#### Remotely Bonded Fiber Bragg Grating Ultrasound Sensors

As shown in Fig. 9(a), an FBG is a periodic modulation of the refractive index inscribed in the core of a single mode fiber [13]. There are two different ways to interrogate an FBG, i.e., based on the spectrum or the intensity. For spectrum-based interrogation, the FBG is onnected to the broadband source of an optical spectrum analyzer (OSA) through an optical circulator, as shown in Fig. 9(b). The broadband light, guided inside the fiber core, is first routed toward the FBG by the circulator. When it encounters the FBG, a portion of the light is reflected at the interfaces with a refractive index change. Since the light reflected at different interfaces have different phases, the superposition of the light results in a reflection with a narrow wavelength λB, which is governed by the effective refractive index of the optical fiber neff and the grating period Λ, i.e., λB = 2 neffΛ. The reflected light is then re-directed by the circulator to the input of the OSA. The OSA outputs the spectrum of the reflected light, based on which the FBG wavelength λB can be determined. Ultrasound sensing, however, requires a much higher sampling rate than that of an OSA. To track the high-speed variation of the ultrasound wave, intensity-based interrogation schemes, such as the one shown in Fig. 9(c), was developed [59]. The interrogation light, emitted by a laser diode with a narrow wavelength λI, is tuned to the midpoint of the slope of the FBG reflection spectrum. The FBG spectrum shifts in response to the ultrasound wave, causing the intensity of the reflected light to fluctuate. This fluctuation can be measured using a photodiode to achieve the required high sampling rate. An FBG ultrasound sensor is typically bonded directly on a structure using adhesive [60] (see Fig. 9(e)). The deformation of the structure is transferred to the FBG via the adhesive layer. As such, the grating period and in turn the FBG reflectance spectrum change with the deformation of the structure. Recently, researchers experimented bonding the optical fiber at a location away from the FBG [2833]. In these works, the ultrasound wave propagating in the structure is coupled to the optical fiber through the adhesive layer and then propagates along the optical fiber to reach the FBG sensor, as shown in Fig. 9(d). As such, the FBG sensor does not measure the deformation of the structure directly. Rather, it measures the displacement of the optical fiber that is coupled from the structure by the adhesive. In other words, “The FBG-inscribed optical fiber was used not only as an optical transmission line but also as an ultrasonic transmission line” [28].

Fig. 9
Fig. 9
Close modal

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