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We study the wave functions of exciton polaritons trapped in the elliptical traps of a patterned microcavity. A homodyne detection setup with numerical off-axis filtering allows us to retrieve the amplitude and the phase of the wave functions. Doublet states are observed as the result of the ellipticity of the confinement potential and are successfully compared to even and odd solutions of Mathieu equations. We also show how superpositions of odd and even states can be used to produce "donut" and "eight-shape" states which can be interpreted as polariton vortices.