Broadband Control of Light through Complex Media via Automatic Self-Referencing Transmission Matrix Characterisation
Light propagation through complex media underpins critical optical technologies, from imaging distant stars and ground-to-space communication to imaging inside biological tissue with hair-thin multimode fibre endoscopes. Central to controlling light through such disordered media lies the transmission matrix. Inaccessible to accurate modelling, the transmission matrix must be measured experimentally$-$yet this conventionally relies on techniques involving an external phase reference. For low-coherence or broadband sources, the stringent coherence, mode-matching, temporal-overlap, and stability requirements of that external reference can make such characterisation prohibitively difficult or fundamentally infeasible. Alternatively, existing self-referencing techniques use algorithmically fragile global optimisation methods, relying on fixed preselected internal reference(s), whose incomplete overlap with the transmitted field can create measurement blind spots. Here, we introduce an automatic self-referencing measurement technique based on spatial state tomography that circumvents these coherence and algorithmic limitations. Rather than relying on a preselected reference or complex phase retrieval, our approach systematically leverages the local interference among all propagating modes as distributed phase references without prior assumptions. We demonstrate this framework experimentally for a multimode fibre across coherent, low-coherence, and broadband regimes, recovering its complete optical transmission matrix and performing high-fidelity spatial and polarisation beam shaping in each case. By enabling robust, source-matched, self-referencing transmission-matrix measurement, our method extends light control through complex media into broadband illumination regimes relevant to biomedical imaging, optical communications, and high-power laser applications.