Optimal Null-Constrained Source-Basis Sensing in a Time-Reversed Young Interferometer

We develop a general theory of null-constrained parameter estimation in a time-reversed Young (TRY) interferometer, where measurement is performed through programmable source-basis encoding with a fixed detector. We address the fundamental question of how to design source patterns that enforce a true metrological null -- vanishing nominal response at the operating point -- while preserving finite first-order sensitivity to the parameter. Under a general shot-noise-limited channel model, we show that the optimal null-constrained receiver is obtained by projecting the derivative response onto the subspace orthogonal to the nominal background in the inverse-noise metric. This yields a constructive solution in which the optimal source-basis code is given by the inverse-noise-weighted derivative response with its background-parallel component removed. We further derive an exact and universal information-retention law: the locally accessible Fisher information is reduced by a factor $1-\chi^2$, where $\chi$ quantifies the inverse-noise overlap between the nominal and derivative response vectors. This result establishes a precise geometric interpretation of the cost of null enforcement. Numerical examples demonstrate the null-coded TRY receivers can retain nearly the full local information and can be accurately implemented using binary and positive-only source patterns. These findings identify source-basis null engineering as a distinct and practically viable modality for derivative-mode sensing, with implications for superresolution metrology and programmable optical measurement architectures.