We present a microcavity design where the overlap between quantum well excitons and the electromagnetic field is much larger than in previously reported structures: quantum wells are located not only in the central cavity layer but also in the Bragg mirrors. We obtain a Rabi splitting of 19 meV. Since the Rabi splitting is larger than the exciton binding energy, exciton excited states have to be taken into account in the reflectivity calculation to describe the spectral shape of the two polarito...