Читать книгу Distributed Acoustic Sensing in Geophysics - Группа авторов - Страница 72

The principle of PGC‐DAS system is shown in Figure 4.1. A coherent input light pulse passes through a circulator into the sensing optical fiber. RB light enters into an unbalanced MI with FRMs at the ends. There is a phase modulator on one arm of MI and an optical delay L_MI on the other arm. RB signal mixes with itself and is detected by one photoelectric detector (PD).

Intensity distribution of RB light is a type of Fourier transform of random permittivity fluctuations (Bao et al., 2016). Assume that the sensing fiber is composed of successive slices with a length of ΔL. Each slice contains M scattering centers, and polarization states between each scattering center are consistent. The interference field of backscattered light at distance L_m = mΔL can be expressed by (Park et al., 1998):

(4.1)

where E₀ is electric field intensity of the incident light; P_m is polarization‐dependent coefficient ranging from 0 to 1; α is optical power attenuation coefficient; r_k and φ_k are scattering coefficient and phase of the kth scattering center, respectively; a_i and φ_i are reflectivity and phase of scattering unit, respectively; and β is propagation constant.

Figure 4.1 Principle of PGC‐DAS system with an unbalanced MI.

Then, scattering light enters into MI, and RB1 and RB2 separated by L_MI interference due to the same optical path. The interference electrical field E(t) is written as:

(4.2)

With simplified coefficients A and B, the interference intensity is given by:

(4.3)

For PGC demodulation algorithm, a sinusoidal signal with a modulation frequency of ω_c is loaded on one arm of MI. Therefore, an additional phase modulation C ⋅ cos (ω_ct) is introduced in Equation 4.3 with C = mΔL_MI, where m is the modulation index and ΔL_MI 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(ω_ct) andcos(2ω_ct), and later with low‐pass filtering, the in‐phase and quadrature components I_I(t) and I_Q(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)