Theory of phase transmission fibre-optic deformation sensing

Abstract

Summary We present a theory and conceptual examples for fibre-optic deformation sensing based on phase changes of transmitted light. As a first result, we establish an exact relation between observable phase changes and the deformation tensor along the fibre. This relation is non-linear and includes effects related to both local changes in fibre length and deformation-induced changes of the local refractive index. In cases where the norm of the deformation tensor is much smaller than 1, a useful first-order relation can be derived. It connects phase changes to an integral over in-line strain along the fibre times the local refractive index. When spatial variations of the refractive index are negligible, this permits the calculation of phase change measurements from distributed strain measurements, for instance, from Distributed Acoustic Sensing (DAS). An alternative form of the first-order relation reveals that a directional sensitivity determines the ability of a point along the fibre to measure deformation. This directional sensitivity is proportional to fibre curvature and spatial variability of the refractive index. In a series of simple conceptual examples, we illustrate how a seismic wavefield is represented in a phase change time series and what the role of higher-order effects may be. Specifically, we demonstrate that variable curvature along the fibre may lead to a multiplication of seismic waves, meaning that a single seismic wave appears multiple times in a recording of optical phase changes. Furthermore, we show that higher-order effects may be observable in specific scenarios, including deformation exactly perpendicular to the fibre orientation. Though higher-order effects may be realised in controlled laboratory settings, they are unlikely to occur in seismic experiments where fibre geometries are irregular and waves asymptotically propagate in all directions with all possible polarisations as a consequence of 3-D heterogeneity. Our results provide the mathematical foundation for the analysis of emerging transmission-based fibre-optic sensing data, and their later use in seismic event characterisation and studies of Earth structure.

Sensitive seismic sensors based on microwave frequency fiber interferometry in commercially deployed cables

“…The spatial resolution depends on the integration time and sampling rate at each side 51 and could be in the order of hundreds of meters or even less which is adequate for earthquake detection as wavelengths related to earthquakes are on the order of several hundred meters or several kilometers. Beyond that straightforward approach, continuum mechanics analysis of the relation between optical phase changes and the strain tensor reveal that the sensitivity of a fiber segment to deformation is proportional to the local fiber curvature 52 . This implies that strongly curved segments, such as tight loops, effectively act as individual sensors that contribute large phase measurements ϕ(t) at distinct times when a wavefront reaches the segment.…”

Linking Distributed and Integrated Fiber‐Optic Sensing

“…However, as shown in more detail in Fichtner, Bogris, Nikas, et al. (2022), points of fiber curvature are crucial for observing strains since it is only at such locations where the fibers see differing contributions from the wavefield. This means that interpreting the signal from such a system is both complex and offers opportunity for new analysis since the variable sensitivities along a fiber can be exploited (Fichtner, Bogris, Bowden, et al., 2022).…”

“…A more detailed version of the following paragraphs can be found in Fichtner, Bogris, Nikas, et al. (2022). The only assumption is that the traveltime of the pulse is much smaller than the characteristic time scales of deformation, meaning that the fiber does not deform significantly while a pulse is propagating.…”

“…Before trying to find a quantitative relation between DAS and phase transmission systems such as MFFI, we consider the dependence of the optical phase ϕ(t) on the deformation tensor F(x, t) along the fiber. A more detailed version of the following paragraphs can be found in Fichtner, Bogris, Nikas, et al (2022). The only assumption is that the traveltime of the pulse is much smaller than the characteristic time scales of deformation, meaning that the fiber does not deform significantly while a pulse is propagating.…”

“…In a sense, that hypothetical situation may be considered analogous to the gauge-length effects known to the DAS community (Lindsey & Martin, 2021) where a given segment of DAS is assumed to be straight. However, as shown in more detail in Fichtner, Bogris, Nikas, et al (2022), points of fiber curvature are crucial for observing strains since it is only at such locations where the fibers see differing contributions from the wavefield. This means that interpreting the signal from such a system is both complex and offers opportunity for new analysis since the variable sensitivities along a fiber can be exploited .…”