Exciton-polaritons in a microcavity are composite two-dimensional bosonic quasiparticles, arising from the strong coupling between confined light modes in a resonant planar optical cavity and excitonic transitions, typically using excitons in semiconductor quantum wells (QWs) placed at the antinodes of the same cavity. Quantum phenomena such as Bose-Einstein condensation (BEC), quantized vortices, and macroscopic quantum states have been reported at temperatures from tens of Kelvin up to room temperatures, and polaritonic devices such as spin switches cite{Amo2010} and optical transistors have also been reported. Many of these effects of exciton-polaritons depend crucially on the polariton-polariton interaction strength. Despite the importance of this parameter, it has been difficult to make an accurate experimental measurement, mostly because of the difficulty of determining the absolute densities of polaritons and bare excitons. Here we report the direct measurement of the polariton-polariton interaction strength in a very high-Q microcavity structure. By allowing polaritons to propagate over 40 $mu$m to the center of a laser-generated annular trap, we are able to separate the polariton-polariton interactions from polariton-exciton interactions. The interaction strength is deduced from the energy renormalization of the polariton dispersion as the polariton density is increased, using the polariton condensation as a benchmark for the density. We find that the interaction strength is about two orders of magnitude larger than previous theoretical estimates, putting polaritons squarely into the strongly-interacting regime. When there is a condensate, we see a sharp transition to a different dependence of the renormalization on the density, which is evidence of many-body effects.
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