[en] This paper experimentally and theoretically investigates improving interaction free measurement (IFM) efficiency using a chain of multiple interferometers with transmissive beam splitters of reflectivity less than 50%. The object measured with IFM is present or absent from one of the interferometer paths depending on the polarization of the incident light. The ability to effectively move an object in and out of the system without physically moving it makes implementing a multiple chain interferometer more practical. It also allows verifying the desired interferometer phase while simultaneously making the IFM measurement. A recursive phase model of a chain of interferometers is presented which accounts for both losses and arbitrary phase differences between each pair of arms in the chain. The recursive model predicts the photodiode power and is experimentally validated for a chain of interferometers of lengths one and two. (paper)