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where e(z) is a coherent optical pulse and Ω(z) ∝ v(z) is the Doppler shifted angular frequency, which is proportional to the local acoustic speed—see Figure 1.25.
The scattering coefficient for engineered fiber can be represented by a spatially periodic function (Farhadiroushan et al., 2021), meaning a reflection coefficient r(z) can be represented by a set of defined scatter center zones separated by sampling distance LS.
where comb(z ) is the Dirac comb function, or sampling operator. If the gauge length is s times larger than sampling distance, L0 = s LS, s = 1, 2…, then r(z) = r(z − L0), and the reflectivity function r(z)can be taken out of the brackets:
Figure 1.25 Optical fiber with defined scatter center zones and the corresponding Doppler shifted angular frequency sampled between the zones. The length occupied by optical pulse is less than the distance between the zones. The gray line corresponds to spatially integrated DAS output, following a linear spline approximation.
(1.36)
To prevent cross‐interference and fading, the spatial length of the optical pulse should be smaller or equal to the distance between scatter center zones, so the spatial sampling of the optical field (Equation 1.35) can be represented by a train of pulses:
(1.37)
The optical pulses from each zone are separated (see Figure 1.25), so the maximum signal intensity and maximum SNR can be delivered if the pulsewidth is equal to the sampling distance, or τ(z) = θ(z + LS) − θ(z), where θ(z) is the Heaviside step function whose value is 0 for negative argument and 1 for positive argument. In this case, intensity can be calculated from the interference between pulses with the same index j, and, for each pulse, an acoustic signal A(z) = F · ∂Φ/∂t, where Φ = ΔΩ(z)t can be recovered from Equation 1.34 using A0 from Equation 1.14 as:
where A0 = 115nm. Equation 1.38 can also be represented in convolution as:
The main parameter for spatial resolution is still the gauge length L0, and the sampling distance can be chosen to have two points per gauge length LS = L0/2. We are considering here the physical spatial sampling, which is defined by the optical configuration, keeping in mind that the photocurrent sampling can have a higher rate. The difference from conventional fiber is an absence of averaging, as the detected signal is deterministic for engineered fiber, and excessive noise from non‐averaged components will hence disappear. Also, the generated optical field can be significantly larger than with conventional Rayleigh backscattering, so the shot noise limitation can be reduced significantly.
The velocity field can be recovered by spatial integration starting from a motionless point as:
(1.40)
So Equation 1.39 can be transformed to:
Formally, the engineered fiber DAS signal expression
