Tuning the probability detection of OAM entangled photons in Helical Ince-Gauss modes

In this work we provide a detailed theoretical and experimental analysis of the two-photon Orbital Angular Momentum (OAM) entangled state, generated by a type-I Spontaneous Parametric Down Conversion (SPDC) process, when decomposed in terms of the Helical Ince-Gauss (HIG) modes basis. By exploiting the unique characteristics of this modal basis we show how the probability detection of the photon-pair can be tuned with the ellipticity parameter of the modes. We also show that, on the HIG basis the SPDC state has the contribution of two different symmetric Bell states, and it is possible to maximize the probability of each HIG symmetric Bell state separately, also by tuning the elipticity of the projected basis. The observed properties are confirmed experimentally by implementing measurements of the HIG modal joint probability of the SPDC two-photon state and Bell-type inequality violation experiments.