4.3 with C = mΔLMI, where m is the modulation index and ΔLMI is the maximum length difference variation. Hence, the total phase of the interference light is:
(4.4)
And the interference intensity is rewritten as:
(4.5)
After being multiplied separately with fundamental and second harmonic carriers cos(ωct) andcos(2ωct), and later with low‐pass filtering, the in‐phase and quadrature components II(t) and IQ(t) are represented as (Dandridge et al., 1982):
(4.6)
where J1(C) and J2(C) are the first‐order and the second‐order Bessel function, respectively, of the first kind. When C is equal to 2.63, it satisfies J1(C) = J2(C). Thus, the phase φ(t) is calculated by:
(4.7)
4.3. EXPERIMENTS AND RESULTS
The PGC‐DAS system setup is illustrated in Figure 4.2. A 1550.15 nm coherent laser with a bandwidth of 3 kHz is modulated by AOM with an extinction ratio of 50 dB to an optical pulse. The pulse width and repetition rate are 50 ns and 8 kHz, respectively. The pulse light travels through an optical isolator (ISO) and is amplified by an erbium‐doped fiber amplifier (EDFA). A fiber Bragg grating is utilized to filter redundancy in amplified spontaneous emission (ASE). The filtered pulse light is launched into the sensing fiber through a circulator. After that, RB light is injected into an unbalanced MI with a one‐way optical path difference of 10 m, i.e., LMI = 10 m. FRMs are used to eliminate the influence of polarization fading. The mixed interference RB light is modulated by a sinusoidal signal with a modulation amplitude of 2.63 rad and arrives at the high‐sensitivity optical detector (PD) with a bandwidth of 80 MHz. After analog‐to‐digital conversion at the analog digital converter (ADC), the obtained RB signal is sampled with a sampling rate of 250 MS/s, corresponding to the minimum sampling interval of 0.4 m. PGC demodulation scheme is implemented on a digital processing unit consisting of field programmable gate array/digital signal processor (FPGA/DSP) circuits and a real‐time controller, which could realize more than 10,000 channels’ real‐time phase calculation. The sensing fiber is a 10 km standard single‐mode fiber, and a fiber stretcher with a 6 m single‐mode fiber wound on a piezoelectric ceramic tube is inserted in the sensing fiber as a unit under test. An isolator is placed at the end of the sensing fiber to remove unwanted end reflection.
Figure 4.2 Setup of PGC‐DAS system.
The time series in Figure 4.3a contains 9,995 data points of Channel #4750. These data points are sampled with a time increment of 0.5 ms, which conceivably allows the time series to contain frequency content up to a Nyquist frequency of 1 kHz (Figure 4.3b). To remove quasi‐static phase drift caused by environmental effects, a high‐pass filter with a cutoff frequency of 2 Hz is adopted in the procedure. Thus, the frequency response range is limited to 2 Hz to 1 kHz.
Figure 4.3 Phase noise of PGC‐DAS system on Channel #4750: (a) Time series and (b) power spectrum.
Under the equation δε = δφ/(2πnLMI/λ), the strain sensitivity is mainly determined by the phase noise δφ and the spatial resolution LMI (defined as the gauge length [Masoudi et al., 2013]). The phase noise is shown in Figure 4.3b, and the average value is around 5 × 10‐4rad/√Hz. With the designed spatial resolution LMI = 10 m, the minimum detected strain of this PGC‐DAS system is as small as 8.5 pε/√Hz.
Figure 4.4a displays a waterfall plot of the magnitude response of each channel in the sensing fiber around the fiber stretcher with a sinusoidal signal of 100 Hz. The y‐axis is proportional to distance along the cable, with a distance increment of 0.4 m, and the color of each cell is proportional to the waveform amplitude. Figure 4.4b shows the superposition result of absolute amplitude of each channel. The signal boundary is defined by the channel of 10% of the absolute peak amplitude. Results show that the sinusoidal signal ranges from Channel #4786 to Channel #4828, and the range is up to 16.8 m. By subtracting the coiled fiber length, the spatial resolution of PGC‐DAS system is about 10.8 m, which is nearly consistent with the optical path difference LMI = 10 m.
Figure 4.4 Intensity map of demodulation magnitude of each channel: (a) Waterfall plot and (b) superposition absolute magnitude.
Figure 4.5 depicts the measurement of frequency response with a linear sweeping frequency signal from 2 Hz to 1 kHz. Each sweeping signal with a constant voltage amplitude of 0.5 Vpp lasts 2 s. Short‐time Fourier transform (STFT) is used to indicate the relative linear and flat frequency response of PGC‐DAS system.
The linearity of PGC‐DAS system is an essential characteristic of quantitative seismic measurement. A sinusoidal strain signal of the fiber stretcher with sweeping voltage from 0.01 Vpp to 1.6 Vpp is used to inspect the amplitude response. The linearity of the strain response is shown in Figure 4.6. From the fitting result, the linear coefficient R2 is 0.99941. An expected linear response capability is presented, and it proves the feasibility of the PGC‐DAS system for the microseismic signal detection.