Characterising optical fibre transmission matrices using metasurface reflector stacks for lensless imaging without distal access

The ability to form images through hair-thin optical fibres promises to open up new applications from biomedical imaging to industrial inspection. Unfortunately, deployment has been limited because small changes in mechanical deformation (e.g. bending) and temperature can completely scramble optical information, distorting images. Since such changes are dynamic, correcting them requires measurement of the fibre transmission matrix (TM) in situ immediately before imaging. TM calibration typically requires access to both the proximal and distal facets of the fibre simultaneously, which is not feasible during most realistic usage scenarios without compromising the thin form factor with bulky distal optics. Here, we introduce a new approach to determine the TM of multi-mode fibre (MMF) or multi-core fibre (MCF) in a reflection-mode configuration without access to the distal facet. A thin stack of structured metasurface reflectors is used at the distal facet to introduce wavelength-dependent, spatially heterogeneous reflectance profiles. We derive a first-order fibre model that compensates these wavelength-dependent changes in the TM and show that, consequently, the reflected data at 3 wavelengths can be used to unambiguously reconstruct the full TM by an iterative optimisation algorithm. We then present a method for sample illumination and imaging following TM reconstruction. Unlike previous approaches, our method does not require the TM to be unitary making it applicable to physically realistic fibre systems. We demonstrate TM reconstruction and imaging first using simulated non-unitary fibres and noisy reflection matrices, then using much larger experimentally-measured TMs of a densely-packed MCF, and finally on an experimentally-measured multi-wavelength set of TMs recorded from a MMF. Our findings pave the way for online transmission matrix calibration in situ in hair-thin optical fibres