Compact 1D Angular Correlation of Annihilation Radiation using Hadamard-Inspired Filters

Angular correlation of annihilation radiation (ACAR) is a powerful technique for probing the momentum distribution of electrons in solids, but its implementation is often limited by the need for long flight paths or confinement to 2-dimensional measurements. We present a theoretical framework for solving the inverse problem in 1-dimensional ACAR experiments using large, position-insensitive detectors. By combining such detectors with specially designed transmission filters, we show that a series of filtered measurements can be used to reconstruct the angular photon distribution without position-sensitive detection. Using Hadamard matrices and null space analysis, we determine the optimal filter set under specific geometric constraints. To test reconstruction, we simulate an ACAR experiment using random numbers drawn from an assumed emission probability function. The results demonstrate the feasibility of this technique for reconstructing the angular distribution across a range of emission profiles. This work lays the groundwork for a compact ACAR apparatus with improved angular and energy resolution.