Fano lineshapes and Rabi splittings: Can they be artificially generated or obscured by the numerical aperture?

Fano resonances and Rabi splittings are routinely reported in the scientific literature. Asymmetric resonance lineshapes are usually associated with Fano resonances, and two split peaks in the spectrum are often attributed to a Rabi splitting. True Fano resonances and Rabi splittings are unequivocal signatures of coherent coupling between subsystems. However, can the same spectral lineshapes characterizing Fano resonances and Rabi splittings arise from a purely incoherent sum of intensities? Here we answer this question through experiments with a tunable Fabry-Pérot cavity containing a CsPbBr\textsubscript{3} perovskite crystal. By measuring the transmission and photoluminescence of this system using microscope objectives with different numerical aperture ($NA$), we find that even a modest $NA=0.4$ can artificially generate Fano resonances and Rabi splittings. We furthermore show that this modest $NA$ can obscure the anti-crossing of a bona fide strongly coupled light-matter system. Through transfer matrix calculations we confirm that these spectral artefacts are due to the incoherent sum of transmitted intensities at different angles captured by the $NA$. Our results are relevant to the wide nanophotonics community, characterizing dispersive optical systems with high numerical aperture microscope objectives. We conclude with general guidelines to avoid pitfalls in the characterization of such optical systems.

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