1. Introduction

Fiber-optic interferometric sensors have been extensively studied for measuring a variety of parameters including temperature, strain, pressure, vibration and acoustics due to their high detection sensitivity and small size [1–3]. Compared with electronic sensors, they have additional advantages of immunity to electromagnetic interference, resistance to harsh environments, and capability to have many sensors multiplexed on the same fiber, which make them attractive for many applications. The information of the measurand is typically encoded in the phase of the spectral fringes of the sensors. For sensors based on Michelson interferometers, Mach-Zehnder interferometers, and low-finesse Fabry-Perot interferometers (FPI), their spectral fringes are approximately sinusoidal [1,3], and phase generated carrier (PGC) demodulation, originally proposed and demonstrated almost three decades ago [4], has been widely used for extracting the phase information of the fringes and consequently the information of the measurand [4–7]. In this scheme, a carrier with a frequency at least twice the highest frequency of the measurand is produced in the interferometer. The laser reflected from the sensor is measured by a photodetector (PD), and the output of the PD is the sum of infinite series of frequency components at the carrier frequency and its harmonics. The amplitude of a frequency component is proportional to either the cosine or sine of the phase of sensor fringes. Heterodyne detection is performed on 2 appropriate frequency components to extract the cosine and the sine terms by mixing the PD output and an oscillation signal of appropriate frequency and phase. The phase of the fringes, which contains the information of the measurand, can be reconstructed by a series of operations on the 2 quadrature terms that involves differentiation, cross-multiplication, summing, and integration. The PGC demodulation scheme overcomes the signal fading due to environmental perturbation and has the advantages of large dynamic range and good linearity. However, it may lack the sensitivity for detecting small dynamic signals such as ultrasound that causes only minute changes in the phase [5]. In current heterodyne demodulation scheme, after the mixer, DC coupled amplifiers could be used to amplify the quadrature components, and the DC components usually limit the maximum gain that can be used. Moreover, the carrier in PGC demodulation is produced by either modulating the sensor (e.g., through a piezoelectric transducer) [8] or modulate the laser frequency (e.g., through injection current modulation) [9,10]. The former method could be difficult to implement in fiber-optic sensors; while the latter is often accompanied with the unwanted laser-intensity variations that cause degradation of the demodulation performance [11–13].

In this paper, we propose and demonstrate a modified PGC demodulation scheme optimized for the detection of small dynamic signals whose frequency is much higher than the environmental perturbation such as ultrasound. Using small-signal approximation, for each of the 2 quadrature terms, we separate the contributions from the environmental perturbations and from the signal-of-interest. Therefore, AC-coupled amplifiers with large gains in the signal bandwidth can be applied to the signal components to achieve the optimal signal-to-noise ratio. The signal processing in the modified PGC scheme only involves simple algebraic operations such as multiplication and summing to construct the ultrasound signal with good linearity. Moreover, we produce the carrier signal by modulating the laser phase using an electro-optic phase modulator, while previously high-speed phase modulators were employed to modulate one of the arms in interferometric sensors in PGC schemes [14]. Modulating the laser source through a high-speed phase modulator improves the modulation bandwidth, does not require tunable lasers, and can be conveniently implemented in a fiber-optic sensor system. Unlike laser frequency modulation through current injection modulation, the phase modulator avoids undesirable intensity modulations. In addition, a single-phase modulator can be used for multiple lasers in a multiplexed sensor configuration. Recently, we have demonstrated a polarization-insensitive, omnidirectional fiber coil sensor system for detecting ultrasound on a solid surface [15]. The demodulation system employs a phase modulator generate 2 quadrature terms of phase using the dc component and the first-harmonic component that can be amplified using AC-coupled amplifiers. However, a drawback of the demodulation technique presented in Ref. [15] is that the 2 quadrature terms cannot be used to construct a signal that is linearly proportional to the intensity of the ultrasound. Therefore, the output of the demodulation system lack the information on the intensity of the ultrasound, which can greatly limit the application of the sensor system.

2. Theory

The system is schematically shown in Fig. 1. The sensor used for the demonstration of the demodulation method is made from a pair of weak FBGs that form a low-finesse FPI on a coiled single mode fiber for ultrasound detection on a solid surface [15]. The length of the fiber coil is controlled in such a way that a round-trip phase difference of 2Nπ (where, N = 0, 1. . .) between the fast and slow axes is generated by the bend-induced birefringence of the fiber. Therefore, the fringes at the 2 principal polarization states of the fiber overlap and thus made the sensor operation insensitive to the variations of laser polarization. The long cavity length results in dense spectral fringes for improved detection sensitivity. The circular symmetry of the sensor structure also renders omnidirectional response to ultrasound. Note that this sensor is an example of many types of sensors that the proposed modified PCG demodulation scheme can be applied to.

 

Fig. 1. Schematic of signal demodulation for fiber coil sensor: an illustration of concept description. PD: photodetector, AMP: amplifier, DAQ: data acquisition, LPF: low pass filter, BPF: band pass filter.

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A monochromatic laser is phase modulated with a frequency much larger than that of the signal of interest (ultrasound) before it is injected into a low-finesse FPI sensor. The electric field of the phase-modulated laser can be written as:

Considering that the signal of interest (ultrasound) as well as other factors such as laser wavelength drift and the environmental effects modulate the optical length of the FPI, leading to changes of $\tau $ with time, denoted by $\Delta \tau (t )$, the output of the PD given in Eq. (3) can be re-written as follows:

It is seen that the quadrature components, $\cos [{{\phi_0} + \Delta \phi (t )} ]$ and $\sin [{{\phi_0} + \Delta \phi (t )} ]$, which contain the information of the signal of interest, are modulated by carrier signals with frequencies of the even and odd multiples of $\mathrm{\Omega }$, respectively. Note that, the phase of the carrier signals is also modulated by the environmental perturbations and the signal of interest. However, because $\mathrm{\Omega }\Delta \tau (t )\ll 1,$ $\cos \mathrm{\Omega }\Delta \tau (t )\approx 1$ and $\cos 2\mathrm{\Omega }\Delta \tau (t )\approx 1.$ Using these approximations, the terms at carrier frequencies $\mathrm{\Omega }$ and $2\mathrm{\Omega }$, or $- 2B{J_1}(C )\sin [{{\phi_0} + \Delta \phi (t )} ]$ and $- 2B{J_2}(C )\cos [{{\phi_0} + \Delta \phi (t )} ]$, can be obtained by mixing the output from PD with proper signals at the same frequency and applying a low pass filter with a bandwidth covering the highest frequency of signal-of-interest. For small signals, the following approximation is valid:

The quasi-DC terms, $\sin {\phi _0}$ and $\cos {\phi _0}$, can be separated from the ultrasound signal using low-pass filters with a cut-off frequency below the lower end of the ultrasound bandwidth. The AC terms corresponding to the ultrasound signal, $\cos {\phi _0}\Delta \phi (t )$ and $- \sin {\phi _0}\Delta \phi (t )$, can also be separated by bandpass filters that match the bandwidth of the ultrasound and further amplified. Four signals are obtained, as shown in Fig. 1, which can be used to extract the ultrasound signal that does not fade with the environmental perturbations:

It has been assumed that the 2 amplifiers for the ultrasonic signal have identical gains. It is worth noting that, in practice, there might be discrepancies in the gains. The amplifier gains can be calibrated, and Eq. (8) can be easily adapted to account for the discrepancies. For example, assuming ${G_1}$ and ${G_2}$ are the gains of the amplifiers for the ${I_{1a}}$ and ${I_{2a}}$ channels, respectively, and replacing the corresponding gain in Eqs. (7b) and (7d) with them, $\Delta \phi (t )$ can be obtained by performing the following operation:

Equations (8) and (9) indicate that the ultrasonic signal can be obtained from simple algebraic operations on the four terms in Eq. (7) involving multiplying and subtracting. Note that the system output given by Eq. (9) is in the unit of V². A calibration is necessary to obtain the absolute intensity of the ultrasound. For detection of ultrasound on a solid surface, a laser Doppler vibrometer can be used to find the conversion coefficient that converts the output from the fiber-coil sensor system to the displacement or velocity of the motion of the surface.

3. Experimental setup

The demodulation scheme was demonstrated and studied using a setup schematically shown in Fig. 2(a). The low-finesse FPI sensor was made from two 5-mm-long FBGs each having a reflectivity of ∼20% centered at 1550 nm fabricated in-house on a 39 cm coiled bend-insensitive fiber (F-SBC, Newport), corresponding to a round-trip time delay of ${{\boldsymbol \tau }_0} = $ ∼3.8 ns and resulting in approximately sinusoidal reflection fringes with a free-spectral range of ∼2 pm. The inner and outer loop diameter of the fiber coil sensor were 8 and 10 mm, respectively. Care was taken to reduce the sensitivity of spectral fringes to light polarizations using bend-induced birefringence. The sensor was glued on an aluminum plate (91 cm × 66 cm × 1 mm) for the detection of ultrasound on the plate surface. A picture of the fiber-coil sensor glued on the plate is shown in Fig. 2(b). The reflection spectrum of the fiber-coil sensor measured by a wavelength-scanning laser is shown in Fig. 2(c). The fabrication process, sensor spectral characteristics, response to laser polarization variations, and response to ultrasound of the sensor have been described in detail elsewhere [15].

 

Fig. 2. (a) Schematic of experimental setup. (b) An image of the installed fiber coil sensor; (c) Reflection spectrum of the fiber-coil sensor measured by a wavelength-scanning laser.

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The light source (Model: 6328-H, New Focus) was a narrow-linewidth wavelength-tunable external-cavity semiconductor laser and the phase modulator (Model: 2942, Covega) was a high-speed lithium niobate electro-optic modulator. The wavelength tuning capability of the laser provides a convenient way to change the operating point of the sensor to study the sensor’s operation and performance. The polarization of laser was controlled to match the phase modulator through a polarization controller (PC) to maximize the modulation efficiency. The laser after phase modulation was injected to the sensor via an optical circulator. The phase modulator was driven by a 130 MHz sinusoidal waveform from a function generator along with a RF amplifier. The choice of the modulation frequency, $\mathrm{\Omega } = 130\;\textrm{MHz}$, yields $\sin ({\mathrm{\Omega }{\mathrm{\tau }_0}/2} )\approx 1$ that maximizes C for a given modulation depth, M, for the phase modulator. The power of the RF source was adjusted to maximize ${J_1}(C ){J_2}(C )$ in Eq. (8). Note that C is insensitive to environmental perturbations. The returned optical signal from the sensor was directed to a PD via the circulator. The PD has an AC-coupled amplifier with a 3-dB bandwidth of 5-270 MHz. The sine and cosine terms were obtained by mixing the PD output with sinusoidal waves at the fundamental and first harmonic of the carrier with appropriate phases that match the phases of the corresponding carrier signals (or differ by an integer multiple of π) to maximize the output after passing through a 500 kHz low pass filter (LPF). The phase matching for the first harmonic of the carrier was achieved through the integrated phase shifter in the function generator. A dedicated phase shifter was used after the frequency doubler for phase matching of the second harmonic of the carrier. After the 500 kHz low-pass filter, the signal of interest, which was ultrasound, was separated and amplified by a 40 dB amplifier in the 50-500 kHz frequency range. Another 25 kHz LPF was applied to obtain the quasi-DC terms. Then we have four signals (2 quasi-DC and 2 AC) represented by Eqs. (7a)–(7d) and recorded by an oscilloscope or a data acquisition (DAQ) system as shown in Fig. 2(a), which were used to extract the ultrasound signal using Eq. (9).

Two difference methods were used to generate the ultrasonic signal used for testing. The first method was to use a piezo transducer glued on the aluminum plate about 2 cm away from the center of the fiber coil sensor. It was driven by 150 kHz, 10 V peak-to-peak, 4 cycle burst (100 ms period) sinusoidal signal from a function generator. Such a method can produce narrow, synchronized, and identical ultrasonic pulses needed for the study of the sensors. The second method was to use a Hsu-Nielsen pencil lead break (PLB) source [16] where a pencil lead is broken by pressing it against the plate surface. PLB is a standard method for generating reproducible, broadband ultrasonic signals for characterizing acoustic emission sensors [17–19]. A reference piezoelectric sensor was placed close to the fiber-coil sensor to monitor the ultrasonic signal.

4. Results and discussion

First, we verified that the setup could generate the sine and cosine terms [Eqs. (7a) and (7c)]. For this purpose, we scanned the laser wavelength using a triangular wave and recorded 2 quasi-DC terms, ${I_{1d}}(t )$ and ${I_{2d}}(t )$. Figure 3(a) shows the recorded signal during the linear range of the scanning signal. It is seen that, as expected, the system produced 2 sinusoidal waveforms with 0 off-sets and quadrature phase relationship. The difference in the amplitudes of the 2 waves is attributed to differences in the values of the Bessel functions in Eqs. (7a) and (7c) determined by the amplitude of the phase modulation ($C$) as well as in the power of the local oscillator signals to the mixers and the efficiency of the mixers. Then we normalized the 2 waveforms to have the same unit amplitude and used the normalized waveforms to generate the Lissajous curve, as shown in Fig. 3(b). The circular shape of the Lissajous indicates a 90° phase difference between the 2 waveforms.

 

Fig. 3. (a) Two DC channel outputs as the laser wavelength was scanned linearly with time; (b) Lissajous curve of the signals shown in (a) normalized to unit amplitude.

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Next, we studied the effect of the operating point on the sensor operation. In this case, the ultrasonic pulses generated by the piezo transducer were applied to the sensor and the laser wavelength was tuned to different operating points on the sine and cosine terms. Figure 4(a) shows the eight different operating points as depicted by the different relative positions of the laser line on the spectral fringes. At each operating point, the four terms depicted in Eq. (7) as well as the signal obtained by performing the operation in Eq. (9) from the four terms are shown.

 

Fig. 4. (a) Positions of the eight operating points; (b)-(d), (e)-(g), (h)-(j), and (k)-(m) are the results for operating points 1-4, respectively. For each operating point, the results show, from left to right, the 2 DC channel outputs, the 2 AC channel outputs (offsets were applied for clarity), and the calculated ultrasonic signal ultrasound response. (n) and (o) show the calculated ultrasound response for all eight operating points and from the reference piezoelectric sensor. See Visualization 1 for recorded signal evolution as the laser wavelength was scanned.

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Figures 4(b)–4(d) show the results for operating point 1, including the 2 DC channel outputs [Fig. 4(b)], the 2 AC channel outputs [Fig. 4(c)], and the calculated ultrasonic signal [Fig. 4(d)]. In this case, the laser line was at the peak of the fringes of DC channel 2 with 0 slope and least ultrasonic sensitivity, and the corresponding AC channel 2 did not record any ultrasound signal. However, at this operating point, the DC channel 1 was at the middle point (around 0 DC values) of the fringes with the maximum slope and highest sensitivity to ultrasound, and the corresponding AC channel 1 recorded the ultrasound signal with the largest amplitude. At operating point 2, the laser line was close to the intersection of the 2 fringes where the 2 DC outputs show similar amplitudes [Fig. 4(e)] and similar slope but with opposite slope directions. As a result, the 2 AC channels recorded ultrasonic response of similar amplitudes [Fig. 4(f)], but the recorded ultrasound waves show a phase difference of 180°. The calculated ultrasonic signal [Fig. 4(g)] is similar to that obtained at operating point 1. For operating point 3 [Figs. 4(h)–4(j)], the laser line was on the peak of fringes of DC channel 1 and the middle point of fringes of DC channel 2, then the response from AC channel 1 vanishes and the response from AC channel 2 reached to the maximum. At operating point 4 [Figs. 4(k)–4(m)], the laser line was on the fringes of the 2 DC channels with similar slope and opposite values. Therefore, as expected, the responses to the ultrasound from the 2 AC channels show similar amplitudes and the same phase. For all the four operating points, the calculated ultrasonic signals [Figs. 4(d), 4(g), 4(j), and 4(m)] were similar. The details of the four channel outputs and the calculated ultrasound signal at other operating points (from 5 to 8) can be found in the Fig. S1. For comparison, the calculated ultrasound responses of the fiber coil sensor at the eight different operating points of the laser are plotted in Figs. 4(n) and 4(o) and they are approximately identical. Therefore, the ultrasound signal detected by the fiber coil sensor through the proposed method is not affected by the spectral shifts occurred due to the laser wavelength drift or the environmental perturbations. As a reference, the ultrasound signal detected by a commercial piezoelectric sensor is also shown [top curves in Figs. 4(n) and 4(o)]. The continuous data recording at different operating points of the laser can be seen in Visualization 1. Note that the discrepancy in the waveforms detected by the fiber-coil sensor and the piezo-sensor is attributed to the fact that they are responsive to different strain components of the ultrasound field on the plate. Specifically, fiber-coil sensor is more sensitive to the in-plane strain components, while the piezo-sensor is more sensitive to the out-plane strain components.

We tested the sensor system for detecting the ultrasound signal generated by a PLB source. In this experiment, the laser was set at fixed wavelength. Figures 5(a) and 5(b) show, respectively, the outputs of 2 DC channels (normalized to its possible maximum amplitude) and the 2 AC channels. Figure 5(c) is the Lissajous curve for the 2 normalized DC signals shown in Fig. 5(a). It is seen that pressing the pencil lead on the aluminum plate to break the pencil lead caused deformation of the plate that resulted in operating points changes over the range of more than one FSR. The circular shape of the Lissajous curve indicates that the quadrature relationship of the 2 DC channels were maintained during the test process. Figure 5(d) shows the calculated ultrasound signal using Eq. (9). The experiment was repeated several times, and despite the different initial operating points, the sensor showed similar responses to the signals generated from the PLB source.

 

Fig. 5. (a) Two DC channel outputs normalized by their maximum possible amplitudes and (b) 2 AC channel outputs (offsets were applied for clarity) in response to the signal generated from PLB source. (c) Lissajous curve of the signals shown in (a). (d) Calculated ultrasound signal using Eq. (9).

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We studied the detection limit of the system in terms of noise-equivalent wavelength shift of the spectral fringes of the fiber-coil sensor. For simplicity, we consider the case where the operating point is at the position where one of quasi-DC terms is 0. In this case, the system output including the noise is primarily contributed by the AC channel corresponding to the 0 quasi-DC channel according to Eq. (9). Figure 6(a) shows the 2 quasi-DC terms when the laser wavelength was scanned. As an example, we consider the operating point indicated by the vertical dashed line in Fig. 6(a) where the slope vanishes for DC channel 1 and reaches maximum for DC channel 2, which has an amplitude of 0.056 V. Considering the fiber-coil sensor has an FSR of 2 pm, the slope of DC channel 2 in terms of output voltage change per unit wavelength shift is found to be 0.176 V/pm. Figure 6(b) shows the 2 DC channel outputs and the 2 AC channel outputs in response to the ultrasound generated by the piezo-actuator. As expected, AC channel 1 shows little response and AC channel 2 has a large response. Then we measured the noise of the AC channels by turning off the piezo-actuator and recording the system outputs. The result is shown in Fig. 6(c). The noise of AC channel 2 has a standard deviation of 7.4 mV. Considering that a 40 dB amplifier was applied in the AC channels, a sensitivity of 17.6 V/pm for AC channel 2 can be inferred from the slope of DC channel 2 at the operating point. Therefore, the noise level of 7.4 mV is equivalent to a wavelength shift 4.2 × 10⁻⁴ pm of the fringe wavelength shift, which is considered as the detection limit of the system.

 

Fig. 6. (a) Two DC channel outputs as the laser wavelength is scanned. Two AC channel outputs (offsets were applied for clarity) and 2 DC channel outputs in response to the ultrasound signal from the piezo-actuator (b) and when no ultrasound was applied to the sensor (c). Results shown in (b) and (c) were obtained for the operating point indicated by the dashed line in (a).

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The system can be divided into 3 modules including the laser source, the PD (with integrated transimpedance amplifier), and the RF circuit, all of which can contribute to the system noise. We performed experiment to identify the main noise sources. Figure 7(a) shows the typical noise in the 2 AC channels when the system is under normal operation, which is the overall noise of the system including all three modules. Then we turned off the laser to eliminate the noise contribution from the laser, and the result is shown in Fig. 7(b). It is seen that, compared with Fig. 7(a), the noise level (standard deviation) drops from 5.6 mV to 2.9 mV for AC channel 1 and from 4.1 mV to 3.5 mV for AC channel 2, indicating that the laser is a significant noise source. Note that laser noise consists of frequency noise and relative intensity noise and their contributions are dependent on the operating point, which is believed to be responsible for the discrepancy in the changes of the noise level when the laser was turned off. Finally, we disconnected the PD from the RF circuit, in which case the noise is mainly contributed from the RF circuit, and the result is shown in Fig. 7(c). A significant noise reduction to 0.74 mV for both AC channels is observed. The experimental results show that the laser and the PD are main contributors to the noises compared with the RF circuit. We note that, in our experiment, the PD has a saturation optical power of >5 mW. The optical power injected into the PD in our experiment was estimated to be <0.2 mW. Therefore, we expect a laser source with high power and lower frequency/intensity noise can greatly improve the noise performance of the system.

 

Fig. 7. Noise of the 2 AC channel outputs when the system was under normal operation (a), when the laser was turned off (b), and when the PD was disconnected from the RF circuit (c).

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5. Conclusion

In summary, we proposed and demonstrated a modified PGC demodulation scheme optimized for small dynamic signal such as ultrasound detection from interferometric sensors with large optical path length difference. Carrier signal was generated by modulating the phase of the laser source using a high-speed electro-optic phase modulator to obtain the sine and the cosine terms of the phase. Using small signal approximation for each of the 2 quadrature terms, a quasi-DC component corresponding to environmental effect and an AC component corresponding to the ultrasound signal are separated. The AC component is amplified by an amplifier with a 40 dB gain. The ultrasound signal was obtained by multiplying and subtraction. The potential of higher order terms for passive quadrature demodulation have been studied to detect the ultrasound signal that does not fade with environmental perturbation. The modified PGC method was demonstrated using a sensor made from a pair of identical and weak FBGs at the 2 ends of a coiled bending insensitive fiber that forms a low-finesse FPI with sinusoidal spectral fringes. The cavity length of the FPI was ∼39 cm, resulting a FSR of 2 pm. The performance of the system was studied for the detection of ultrasound signal generated from a piezo transducer and a PLB source. This is a robust and simple demodulation method with high sensitivity for fiber-optic interferometric ultrasound sensors and is applicable to other types of interferometric configurations with sinusoidal spectral fringes.