Fig. 7: Remote characterisation of the input probe.

Experimental results (bars) and corresponding best theoretical fits (red-dashed lines) showing the output probe power fraction coupled to mode M₁ versus BCB peak power in a 0.4-m long bimodal fibre (DCF, see Supplementary Information 1). Panels (a–c) correspond to different input probe mode states and BCB mode distributions, measured experimentally and reported on the top of each panel. The best theoretical fit is calculated from Eq. (4), assuming the same input probe and BCB relative powers and optimising the input probe relative phase to minimise the least squares difference with experimental data. Note that in all the 3 cases the estimated optimal least-squares value \(\Delta {\widetilde{{{\rm{\phi }}}}}_{{in},12}\) (0.06 rad, 5.72 rad, 1.26 rad in panels (a–c) respectively) is close to the measured \(\Delta {{{\rm{\phi }}}}_{{in},12}\) (0.3 rad, 5.7 rad, 1.2 rad in panels (a–c) respectively). This demonstrates our ability to detect from remote the relative phase of the input probe modes by analysing the output probe response to the BCB. Note that the larger error in panel (a) is due to the large power imbalance among the two input probe modes (92% and 8%, respectively).