Surface polaritons (SPs) are electromagnetic waves bound to a surface through their interaction with charge carriers in the surface material. Hyperbolic SPs can be supported by optically anisotropic materials where the in-plane and out-of-plane permittivies have opposite signs. Here we report what we believe to be the first experimental study of hyperbolic surface exciton polaritons (HSEPs). We study the intensity and phase response of HSEPs in the J-aggregate TDBC (a type-II natural hyperbolic material). HSEPs can be used to generate phase singularities; the behaviour of these phase singularities is a consequence of the hyperbolic nature of TDBC. The combined intensity and phase response of non-hyperbolic and hyperbolic SPs suggests that they are topologically distinct. We predict analogous effects for hyperbolic surface phonon polaritons in hexagonal boron nitride. Our work suggests that organic materials can provide a new platform for the exploration of hyperbolic surface polaritonics at visible frequencies.

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