Sensitivity Distribution Properties of a Phase-Shifted Fiber Bragg Grating Sensor to Ultrasonic Waves

3.1. Bragg Wavelength Shift by Strains in Three Orthogonal Axes

A FBG is manufactured by writing a periodic variation of refractive index along a certain length of single-mode optical fiber via ultraviolet light. When a π phase shift is inserted into the middle of the grating area, a PS-FBG can be manufactured, as shown in Figure 2a. Although the equations to describe the spectra of PS-FBG and normal FBG differ, the Bragg wavelength expression for both gratings can be described simply by Equation (1) [9]:

Firstly, we consider an ideal case; i.e., the fiber is in a free space. Furthermore, the strain from ultrasonic wave perfectly couples to the fiber, and the wavelength of the ultrasonic wave is much longer than the grating length. In this case, the n and Λ change according to Equations (2) and (3), respectively [10]:

Herein, we consider two different conditions. In the first case, the ultrasonic wave only propagates along the Z-axis. Because the fiber approximates isotropic material, the lateral strains are related to the axial strain by a standard Poisson ratio ν. Thus, in this case, the strains in the three axes have the relations expressed in Equation (4):

By substituting Equation (4) into Equations (1)–(3), the Bragg wavelength shift Δ λ_B is expressed in Equation (5):

Alternately, when the fiber only experiences lateral strain along the X-axis or Y-axis, the stains in the three orthogonal axes have the relations shown in Equation (6). These relations result because the cylindrical fiber is rotationally symmetric in the X- and Y-axes, and the fiber material is quasi-isotropic:

By substituting Equation (6) into Equations (1)–(3), the Bragg wavelength shift Δ λ_B is expressed in Equation (7):

In the first case, the Bragg wavelength shift is proportional to the stain in the Z-axis, according to Equation (5), and in the second case, the shift is proportional to the strain in the X- or Y-axis, according to Equation (7). The absolute value of C_z is larger than the absolute value of C_x or C_y. However, the negative sign in C_x and C_y means that the same strain will cause the Bragg wavelength to shift in the opposite direction, reflected as an opposite initial phase in detected waveforms.

3.2. Influence from Surface Attachment Method

The above analysis is based on the assumption that the ultrasonic wave perfectively couples to the optical fiber. However, in both our experimental condition and other practical conditions where the fiber is attached to the surface of a plate, the effective ultrasonic coupling performance between fiber and plate should be considered. Thus, we add the influence from the surface attachment method. The strain from the Z-axis has the best coupling performance, while the strain from the X-axis is difficult to couple to the fiber due to the limited effective contact area. Although no strain exists from the Y-axis because of the fiber's surface attachment method, acceleration in the Y-axis will cause dynamic strain to influence the fiber. Thus, in the actual case, the parameters C_x, C_y, and C_z will deteriorate to C_x, C_y, and C_z, the absolute values of which have relations |C_z| > |C_x| and |C_z| > |C_y|. The exact actual values differ depending largely on coupling performance; however, the negative sign in C_x and C_y will always exist.

3.3. Bragg Wavelength Shift by Directional Strain

A more common case in which the ultrasonic wave propagates from a direction other than the three orthogonal axes is considered. When the PZT actuator and the PS-FBG sensor have an angle of α, as shown in Figure 2b, the actual strain applied to the fiber can be written as the composite of the strains from the Z-axis and the strain from the X- or Y-axis. For example, when the stain is in the X-Z plane, the Bragg wavelength shift under strain ε can be written as:

Due to the negative sign in C_x, there should exist an angle α_min where the Bragg wavelength shift is minimal. Because the Bragg wavelength shift is proportional to the output voltage, α_min can be predicted according to Equation (9), where V|ε_z and V|ε_x are the amplitudes of the detected waveforms measured when the stain exists only in the Z-axis and the X-axis, respectively:

3.4. Frequency Responses to Different Angles

In this part of analysis, we add consideration of the influence from the wavelength of ultrasonic waves because ultrasonic wave changes the grating periodically rather than uniformly as quasi-static strain. As demonstrated in previous research [11,12], the fiber's frequency response to ultrasonic waves depends on the effective sensing length. In general, as the effective sensing length lengthens, the high-frequency response worsens. When an ultrasonic wave propagates along the Z-axis, the effective sensing length of PS-FBG is the effective grating length with about a few hundred micrometers [6]. When an ultrasonic wave propagates along the X- or Y-axis, the effective sensing length is the fiber's 150-micrometer diameter, which is shorter than the effective grating length. Thus, without considering the influence from distance or the surface attachment method, the PS-FBG sensor will have a higher response to the ultrasonic wave that propagates in the X- or Y-axis. Furthermore, considering the actual case, the sensitivity decrement of PS-FBG to high frequencies is slower than the decrement to low frequencies when the angle between sensor and actuator changes from 0° to 90°. Paralleling this effect, the detected signal cannot be completely removed at α_min because the exact α_min differs depending on frequency.

4.1. Sensitivity Distribution on an Aluminum Plate

Typically, the sensitivity is defined by the signal amplitude caused by a given level of input (ultrasound signal in this case). In this study, we use signal-to-noise ratio to reflect the sensitivity distribution of the sensor because the ultrasonic signal has the same original input voltage. Because the noise level, determined by the demodulation system, was constant in this experiment, the amplitudes of the detected waveforms can be used to directly evaluate the sensitivity of PS-FBG. A Hilbert transform was used to obtain the envelopes of the detected waveforms, and the amplitudes were then obtained by averaging the thrice-measured data. Figure 3 shows the sensitivity distribution properties of the PS-FBG on an aluminum plate in logarithmic scale. According to Figure 3, the detected waves' amplitudes decrease with the increase in distance between the PS-FBG sensor and the ultrasonic source. However, when the distance is smaller than 4 cm, the amplitudes are relatively large. Figure 4 was obtained by normalizing the amplitudes at 4 cm for different angles. In Figure 4, there is a clear corner at 4 cm. After 4 cm, the slopes of all curves are similar, due to attenuation of the ultrasonic wave. However, before 4 cm, the curves' slopes change greatly, especially in data from large angles, such as 75° and 90°. When the distance between sensor and actuator is small (corresponding to area B), the dynamic strain from the Y-axis caused by acceleration is relatively large and effectively shifts the Bragg wavelength, which leads to relatively large amplitudes.

According to Figure 3, unlike a traditional PZT sensor (which is omnidirectionally sensitive), the sensitivity of the PS-FBG has an obvious directional dependence. Extracting the amplitudes at every distance in Figure 3, the sensitivity distribution at different angles is shown in Figure 5. As this figure depicts, the amplitudes generally decrease with the angle's increment. However, odd phenomena are observed from 75° to 90° after 4 cm, as amplified in the inset of Figure 5. These phenomena result because the waveforms with smallest amplitudes to the angle change are located around 84° rather than 90°, corresponding to area C in Figure 1b.

4.2. Phenomena in Area A

The sensitivity distribution in area A is simple because the effect from X-axis strain can be omitted due to the small value of C_x, while the effect from Y-axis dynamic strain can also be omitted due to the plate's geometric dimensions. The open marks in Figure 6 are the data from area A. The black curve in Figure 6 is obtained by normalizing the data to the value in 0° according to different distances, and this figure shows the cosine relation between sensitivity and angle. Furthermore, in area A, the detected waveforms and the waves' initial phases are always constant for different distances.

4.3. Phenomena in Area B

Figure 7a shows the detected waveforms at dots $D_{\mathit{2}}^{\mathit{0}}$, $D_{\mathit{2}}^{\mathit{75}}$ and $D_{\mathit{2}}^{\mathit{90}}$. The figure indicates that the amplitude of the wave at $D_{\mathit{2}}^{\mathit{90}}$ further decreases from $D_{\mathit{2}}^{\mathit{75}}$ and $D_{\mathit{2}}^{\mathit{0}}$. Additionally, the initial phase of the waveform from $D_{\mathit{2}}^{\mathit{90}}$ is opposite to the initial phase of $D_{\mathit{2}}^{\mathit{0}}$ or $D_{\mathit{2}}^{\mathit{75}}$, as shown in the inset in Figure 7a. This phenomenon matches the description in Section 3.1, in which lateral strain is dominant for $D_{\mathit{2}}^{\mathit{90}}$. Furthermore, the waveforms for $D_{\mathit{2}}^{\mathit{90}}$ and ${\mathit{D}}_{\mathit{2}}^{75}$ differ, and this effect results from different responses to certain frequencies when the ultrasonic wave couples to the fiber from different angles, as explained in Section 3.4. Figure 7b was obtained by performing a Fast Fourier Transform to corresponding waveforms. Although the corresponding spectra of $D_{\mathit{2}}^{\mathit{0}}$, $D_{\mathit{2}}^{\mathit{75}}$ and $D_{\mathit{2}}^{\mathit{90}}$ have a similar frequency range and shape, the responses to different frequencies are dissimilar. For example, the difference in amplitudes at frequency 0.32 MHz and 0.71 MHz is 4.16 dB at $D_{\mathit{2}}^{\mathit{0}}$. However, the amplitude difference decreases to 3.44 dB at $D_{\mathit{2}}^{\mathit{75}}$, while this difference further decreases to 2.16 dB at $D_{\mathit{2}}^{\mathit{90}}$. This decrease is a common phenomenon, which we can observe in almost all cases when the angle changes from 90° to 0°.

Moreover, no α_min is observed. The absence of α_min occurs because, in area B, the strains from all three axes are present, and the distance between PS-FBG and the ultrasonic source is very short; therefore, in this range, relatively large waveforms can always be detected.

4.4. Phenomena in Area C

Figure 8 shows the input waveform and the detected waveforms for $D_2^0$, $D_{10}^{81}$ and $D_{10}^{90}$, respectively. The α_min is around 84°, as shown in the inset of Figure 5, and this value can be explained by the formula for angle α_min, as discussed in Section 3.3. After undergoing a Hilbert transform, the amplitudes of $D_{10}^0$ and $D_{10}^{90}$ are 0.0978 V and 0.0045 V, respectively, as shown in Figure 8. Substituting these values into Equation (9), the predicted angle where the minimal amplitude occurs is 87°, which is very close to the observed result.

Furthermore, in Figure 8, the waveform at $D_{10}^{90}$ is similar to the waveform at $D_{10}^0$ but has opposite phase, the reason for which is explained in Section 3.1. However, at the other observational dots between 75° and 90°, the waveforms are not constant and instead show complex waveform changes, such as the waveform for $D_{10}^{81}$. Continuous wavelet transforms (CWT) are used to further analyze the detected waveforms, as shown in Figure 9. In the CWT result for the input waveform, the frequency range approaches 1 MHz, and the input time is about 20 μs. The CWT results at $D_{10}^0$ and $D_{10}^{90}$ are very similar, and these results present a clear S₀ mode and A₀ mode. The S₀ mode and A₀ mode were determined by subtracting the input signal's arrival time and comparing with the theoretical dispersion curve for this aluminum plate. However, the A₀ mode detected at $D_{10}^{90}$ contains slightly more relative energy in the high-frequency range. Also, the S₀ mode for $D_{10}^{90}$ is more apparent than the S₀ mode at $D_{10}^0$. These phenomena also can be explained as different responses to different frequencies, as discussed in Section 3.4. For example, S₀ mode contains relative high frequency about 0.8 MHz, i.e., it has relative short wavelength. When this component of wave propagates in Z axis, it does not effectively shift the Bragg wavelength compared to the case of static strain. However, when this component of wave propagates in the X or Y axis, it shifts the Bragg wavelength as the behavior of static strain because it is under quasi-uniform strain due to the short diameter of the fiber. Alternately, although the complex CWT result for $D_{10}^{81}$ contains a major A₀ mode and S₀ mode, there also exist several vague areas. These vague areas will confuse researchers and lead to analysis mistakes. Thus, we call these areas “fake modes” in Figure 9. The fake modes occur when the strain from Z axis and X/Y axis are approximately equivalent.