  This manuscript is concerned with phase changes of signals transmitted
through deforming optical fibres. As a first result, it establishes an exact
relation between observable phase changes and the deformation tensor along the
fibre. This relation is non-linear, and it includes effects related to both
local changes in fibre length and deformation-induced changes of the local
speed of light or refractive index. In seismic applications, where the norm of
the earthquake-induced deformation tensor is orders of magnitude smaller than
1, a useful first-order relation can be derived. It simply connects phase
changes to an integral over in-line strain along the fibre times the local
refractive index. Under the assumption that spatial variations of the
refractive index are fast compared to the seismic wavelength, this permits a
direct synthesis of phase change measurements from distributed strain
measurements, for instance, from Distributed Acoustic Sensing (DAS). A more
detailed analysis of the first-order relation reveals that a perfectly straight
fibre would produce zero phase change measurements, unless deformation was
sufficiently widespread to affect the starting and end points of the fibre. If
this condition is not met, non-zero measurements can only result from curvature
of the fibre or from a heterogeneous distribution of the refractive index.
Segments of the fibre that are strongly curved generally make larger
contributions to the observed phase change.
