Figure 1 shows the schematic of the proposed SOA-based FRL, which comprises a SOA, an FBG working in reflection mode, an optical isolator, and an optical circulator. In the device, the SOA acts as a gain medium [34] and an in-loop pulse modulator in the cavity. It is switched by a programmable pulse generator (PPG). Thus, the SOA yields a pulsed output with a repetition rate equal to the applied modulation frequency. The addition of the FBG modifies the intra-cavity loss profile. The spectral shift of the reflected light from the FBG changes as a result of exchange changes, which can be dynamically observed in the output light intensity of the filter. The FBG is also a wavelength-selective filter [30], which filters the broadband pulsed light emitted from the SOA within the bandwidth of the FBG, as shown in Figure 2. Most reflected light circulates continuously within the cavity to ensure a stable coupled output. Optical isolators are used to maintain the unidirectional nature of the cavity and to protect the SOA, which is connected to a 10:90 single mode (SM) coupler and an erbium-doped fiber amplifier (EDFA). The EDFA amplifies the output pulsed light from the SM coupler.

The SOA-based FRL operates as a switchable pulsed mode-locked laser [35], which works fundamentally differently than traditional continuous wave (CW) or mode-locked lasers. The cavity dynamics are much more complicated than either CW or mode-locked lasers because of the multiple physical effects involved. To date, a great deal of theoretical modeling and investigation of the laser dynamics in the cavities of the devices has been demonstrated [36–38]. The basic principle of the device is resonating frequency is related to the transmitted light’s round-trip time, which is given by: f c a v = ν g L 0 + 2 L F B G (1) where ν g is the group velocity, and L 0 is the fixed fiber length corresponding to the length of the ring. L F B G is the fiber length between the circulator and the FBG.

It is worth noting that using the SOA and FBG in the ring cavity aims to achieve a low side-mode suppression ratio (SMSR) laser [35], which shows a high stability in theory. Two main factors affect the stability of lasers: one is relaxation oscillation, and the other is mode competition [39]. Since the SOA’s carrier recovery time is significantly shorter than the photon rise time in the cavity [40], the small fluctuations in optical power at the stimulated wavelength caused by external environmental disturbances decay rapidly with time. Thus, no relaxation oscillation is generated. The only factor that significantly affects the stability of the device is mode competition. As the fiber medium in fiber lasers cannot effectively compress the side modes [41], the side mode oscillation will compete with the main mode. When the mode hopping occurs, the SNR of the fiber laser output decreases due to interference between the two modes [42]. However, in the case of a fiber laser as a gain medium, the side-mode oscillation is suppressed by the gain saturation effect as long as it is in the gain saturation state [43]. Then, the laser output power becomes flat, and the SNR is improved. The side mode suppression effect is due to the nonlinear effect of the SOA. The nonlinear compression effect in the proposed SOA-based FRL is derived in detail below. The main mode and one of the side modes of the FRL are assumed to have the same small signal gain. At the input of the main mode, the main mode is E d = E ∙ cos ω 0 t , the side mode is written as E s = δ i ∙ E ∙ cos ω 0 + Δ ω t + ϕ , and the SMSR is δ i ≪ 1 . Then, the SOA’s optical input power is obtained as follows: P i = 1 2 1 + δ i 2 E 2 + δ i E 2 cos Δ ω t + ϕ = P i ¯ + Δ P i t (2)

The relative change in input power is defined as follows: R i = Δ P i max P i ¯ = 2 δ i 1 + δ i 2 (3)

Similarly, the relative changes of the side mode extinction ratio and power of SOA output can be derived from the small signal dynamic equation. The integral gain of SOA can be expressed as follows [44]: h t = ∫ 0 l g z , t d z = h ¯ + Δ h t (4) where Δ h t is the time-varying term due to the input optical power, which is determined by the following equation: τ c ∙ d Δ h t d t + ( 1 + P i ¯ ∙ e h ¯ P s a t ) Δ h t = − Δ P t t P s a t e h ¯ − 1 (5) where τ c is the carrier lifetime. P s a t is saturated power of the SOA. Define the saturation coefficient to characterize the depth of saturation as follows: S = P i ¯ ∙ e h ¯ P s a t (6)

Then Δ h t can be obtained as follows: Δ h t = − S R i ( 1 − e h ¯ ) 1 + S 2 + τ c Δ ω 2 ∙ 1 + S ∙ cos Δ ω t + ϕ + τ c ∙ sin Δ ω t + ϕ (7)

The output power of the SOA can be written as follows: P o = P i ∙ e h ≈ P i ¯ ∙ e h ¯ + Δ P i t ∙ e h ¯ + P i ¯ ∙ e h ¯ ∙ Δ h t (8)

Ignoring the higher-order small signals in Eq. 8, the expression for the output power can be derived as follows: Δ P o t ≈ Δ P o t ∙ e h ¯ + P i ¯ ∙ e h ¯ ∙ Δ h t = P i ¯ R i e h ¯ ( 1 − 1 + S S ( 1 − e h ¯ ) 1 + S 2 + τ c Δ ω 2 ) cos Δ ω t + ϕ − ( 1 − S ( 1 − e h ¯ ) τ c Δ ω 1 + S 2 + τ c Δ ω 2 ) sin Δ ω t + ϕ (9)

The relative change in output power is R o = Δ P i max P i ¯ = 1 − 1 + S 2 A 2 − A 1 + S 2 + τ c Δ ω 2 ∙ R i (10) where A = S 1 − e h ¯ 1 + S for 0 < A < 1 , then it can be derived that R o < R i . When δ < 1 , R decreases monotonically as δ decreases, so δ 0 < δ i . It indicates that the side mode has a smaller effective gain than the main mode. Although the side and main modes have the same small signal gain, the oscillations are suppressed by the smaller effective gain of the side mode.

Considering δ i ≪ 1 , then R o R i = δ o δ i ∙ 1 + δ i 2 1 + δ o 2 ≈ δ o δ i (11)

The ratio of the effective gain of the side mode to the main mode can be obtained from the above equation as follows: η Δ ω = G s G d = δ o δ i 2 ≈ R o R i 2 = 1 − 1 + S 2 A 2 − A 1 + S 2 + τ c Δ ω 2 (12)

The equation above suggests that η decreases as Δ ω decreases. Therefore, the closer the side mode is to the main mode, the stronger the suppression effect is. The side mode farther away from the main mode has a weaker non-linear compression effect. However, the small signal gain of it is also relatively lower. Additionally, the FBG, as a frequency-selective element, makes the side mode farther away a significant loss. Thus, these frequencies far from the main mode can also be suppressed by the non-linear effect. When Δ ω is fixed, if the average input power is high, then S and A are also relatively high, then η is low. In other words, the side modes are effectively suppressed when the SOA is in deep saturation. Thus, the device is of high stability in theory due to low SMSR output.

The working principle of Φ-OTDRs is the interference effect of Rayleigh backscattering (RBS) light in optical fibers [30]. When subjected to external disturbances, the scattered light changes during its transmission through the fiber (intensity, phase, etc.). By detecting changes in these properties, scattered light can be identified. Observation of changes in the scattered light provides information about external disturbances.

In contrast to a conventional Φ-OTDR system, the probe pulses used in the proposed method are a pair of pulses. The principle of operation based on dual pulse heterodyne detection is shown in Figure 3. The receiver is disturbed by two RBS from the pair of probe pulses. This causes a heterodyne signal to be generated at the receiver. The acoustic signal can be obtained with a high SNR using a suitable demodulation method with OTDR. The waveform in the time domain and the signal’s frequency spectrum can be effectively recovered.

Discrete the optical fiber intervals of ∆ L for each probe pulse [45]at the point Z m = m ∆ L . The RBS within a pulse width can be expressed as follows: E Z m = E 0 ∑ k = m m + N − 1 γ k σ k e j θ k e j φ k e − α k Δ L (13) where E 0 is the amplitude of the incident light, and α is the fiber attenuation coefficient. γ k , σ k and θ k denote the polarization attenuation coefficient, Rayleigh scattering cross section and phase delay of the pulse through the point, respectively. φ k is the phase change caused by the acoustic signal at the point. N is the total number of slices through which a single pulse passes. All slices within the pulse width have the same properties. Then, γ k , σ k and θ k can be considered as constants, and the vibration point spacing is related to ∆ L . The above equation can be simplified as: E Z m = S m ∙ e j ϕ m (14) where S m = E 0 e − α m Δ L γ m σ m e j θ m , and ϕ m denotes the total phase change caused by the vibration at that point. As long as the two pulse widths are long enough, the total expression of the detected RBS can be expressed as: E Z m = S m ∙ e j ϕ m ∙ e j 2 π f 1 t + φ 1 ∙ S m − N d ∙ e j ϕ m − N d ∙ e j 2 π f 2 t + φ 2 (15) where φ 1 and φ 2 are the initial phase of the dual pulses. Then the AC component of the interferce signal can be expressed as: I s Z m = S m S m − N d cos 2 π Δ f t + Φ t + Δ φ 0 (16) where Δ f is the heterodyne frequency mentioned above. Φ t = ϕ m − ϕ m − N d is phase change caused by the acoustic signal, and Δ φ 0 = φ 1 − φ 2 .

Then, the output of the system based on the equation above can be simplified as follows: I s t = A cos 2 π Δ f t + Φ t + Δ φ 0 (17) where A = S m S m − N d is the intensity amplitude of the interferometric light, and 2 π Δ f t is the heterodyne carrier term. Φ t is the phase change term caused by the external vibration; Δ φ 0 is the initial phase noise of the dual pulses.

Φ (t) can be demodulated by an in-phase/quadrature (IQ) phase demodulation algorithm [46], as shown in Figure 4. The output signal needs to be mixed with cos 2 π Δ f t and sin 2 π Δ f t respectively. Then, the mixed frequency terms will be low-pass filtered, giving: I t = A 2 cos Φ t + Δ φ 0 (18) Q t = A 2 sin Φ t + Δ φ 0 (19)

The intensity and phase information of RBS can be obtained by solving the summation and arctangent of the above two signals as follows: A = I 2 t + Q 2 t (20) Φ t = tan − 1 Q t I t − Δ φ 0 (21)

As the phase noise Δ φ 0 is a slow variable, which can be filtered using a high-pass filter. Thus, the phase variation Φ t due to external acoustic signals can be obtained. The range of values of the arctangent method is − π 2 , π 2 , and can be extended by the use of the unwrapping algorithm [46].

Furthermore, as the pulses propagate forward in the fiber, the phase information from the previous point is carried back. As a result, the RBS at the location behind the interference point has the interference information. This RBS cannot complete the positioning of the interference signal. Separating the phase differences at a certain distance can eliminate the problem of phase accumulation. It is worth noting that the RBS results from coherence within a half pulse. Therefore, the length of the phase difference should be greater than half a pulse width. This should be greater than the spatial resolution in this system. By the phase difference algorithm, the phase change of the corresponding part of the fiber can be demodulated as follows: Δ Φ t = Φ z 1 t − Φ z 2 t (22)

For the demodulated amplitudes, the same is true. Thus, using the different algorithms on the demodulation results can achieve localization of the acoustic signal.

To verify the effectiveness of the SOA-based FRL, an experimental setup is constructed, as shown in Figure 5. The modulated signals are generated from a PPG. The driver circuit of the SOA amplifies them. The amplified modulated signals are applied to the SOA. The driver circuit integrates the temperature control system and the current control system of the SOA optical chip. This mainly aims to prevent damage to the optical chip during operation and improve the laser’s power stability. The bandwidth of the FBG is <0.3 nm, and the reflectance index is >90%. The light from the FRL is connected to a photodetector after passing through a 20 dB attenuator (ATT). The bandwidth of the PD is 200 MHz. The signal from the PD is acquired by a high-speed data acquisition card (DAQ) with a sampling rate of 250 MHz. The output results are displayed on a personal computer (PC). The output results of applying different pulse duration and repetition frequency modulations to the SOA are shown in Figure 6. The amplitude of the SOA-based FRL is large enough for DAS. The short rise/fall SOA results also show good switching performance.

As shown in Figure 7, an unbalanced Mach-Zehnder interferometer (MZI) converts the single pulse from the SOA-based FRL into periodic double pulses separated by ∆ T . The single pulse is split into the MZI by a 30:70 SM coupler. The split pulses are propagated along two different paths. The short path propagates through an acoustic-optic modulator (AOM) with a frequency shift of ∆ f = 80 M H z , and the long path is through an attenuator and a delay. The AOM is used only as a frequency shifter, driven by a DC source and an RF driver. The center wavelength of the SOA-based FRL’s output is f c a v . Then, f 1 = f c a v and f 2 = f c a v + ∆ f . Thus, ∆ f , the heterodyne frequency, also separates the dual pulses in the frequency domain. The difference in length L d of the two arms of the MZI is 50 m, which corresponds to the pulse interval Δ T = n L d c ≈ 245 n s . The pulse repetition frequency is 10 kHz. The PD with a bandwidth of 200 MHz detects the pulse traces from the MZI. Figure 8 shows the detected dual pulse with ten traces. The results show that when the pulse width is 240 ns, the dual pulses can also be separated in the time domain. However, when the pulse width is 250 ns, the dual pulses overlap. The experimental results show that the rise/fall time of the SOA-based FRL is less than 10 ns, which has significant advantages over the commonly used AOM [47].

An experimental system is shown in Figure 9 to verify the ability of the proposed method to detect acoustic signals. The length of the FUT is 3 km long single-mode fiber (SMF). An arbitrary function generator (AFG) controls a piezoelectric ceramic transducer (PZT). The PZT simulates acoustic signals applied to 2 km of the FUT. In the experiments, the AOM shifted the light by 80 MHz. The PPG modulates the SOA so that the SOA-based FRL outputs an optical pulse with τ = 240 n s width and f r = 10 k H z repetition frequency. For each measurement, M = 2000 traces of RBS signals are collected. In this case, the theoretical minimum detectable frequency is f r M = 5 H z , while the maximum is 5 kHz. And the maximum detection range is up to about L s = c 2 n f r M ≈ 10.2 k m . Then the given spatial resolution is w τ + L d 2 = 11.8 m , where w τ is the pulse width in length. The interference signals can be reconstructed by extracting specific points in each trace. The heterodyne demodulation technique algorithm is used to retrieve the signal’s phase information. The demodulation process is conducted in the PC.

Figure 10 shows the demodulation results when the modulation signal frequency with a peak amplitude of 1 V applied to the PZT is 1 kHz. A power spectral density (PSD) analysis is conducted with the demodulation results. The results are in the time domain and frequency domain.

A differential algorithm can be applied to the demodulated signal, allowing the acoustic signal to be located. Figure 11 shows the amplitude and phase variance results for a 1 Vpp and 1 kHz signal. The results agree well with the location of the PZT. The results imply that the system can localize the acoustic signal accurately.

The modulation signals applied to the PZT are varied with a fixed amplitude to demonstrate the system’s response to other frequencies. The sinusoidal signal applied to the PZT was fixed at a constant peak voltage of 1 V. The frequency was gradually changed for the measurement and demodulated, as shown in Figures 12, 13. As can be seen from the demodulation results, the system can accurately detect the acoustic signals at the fixed point of the fiber with a complete signal waveform. The spectrum also shows that the system can achieve a high SNR of 42.51 dB and accurately demodulate the signal frequencies.

The modulation signals applied to the PZT are varied in amplitude at a fixed frequency to demonstrate the system’s dynamic response. The sinusoidal signal applied to the PZT has been set at 3 kHz; the amplitude is 100 mV and 12 V, respectively, corresponding to 0.19 rad and 22.8 rad. The results of the demodulation are shown in Figures 14, 15. The demodulation results show that the system can accurately detect acoustic signals with different amplitudes.

The experimental results illustrate that the proposed system can detect acoustic signals with different frequencies and amplitudes. The demodulation and positioning results are accurate, fitting the modulation signals applied to the PZT well. For a 3 kHz acoustic signal, the SNR can achieve 42.51 dB. Above 100 Hz, the SNR of the demodulation results remained at 27.38 dB and above.