Theory of phase transmission fibre-optic deformation sensing

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 nonlinear 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 realized 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 polarizations 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 characterization and studies of Earth structure.