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Source-Prior Engineering for Bayesian Optical Sensing in Time-Reversed Young Interferometry

Jianming Wen

Published Jul 23, 2026• 3 citations

Abstract

Time-reversed Young (TRY) interferometry reconstructs interference from a fixed detector by reading out a programmable source-label distribution. This work formulates the architecture as a source-coded Bayesian response sensor. For a perturbation parameter θ, the detected source-label histogram is a posterior distribution determined by a programmed source prior, an optical likelihood for a fixed-detector click, and an evidence factor equal to the click probability. The key point is not the Bayesian identity itself, but its physical implementation: in TRY the prior is imposed before propagation and can therefore reshape the response ensemble actually sampled by the detector. The normalized posterior is shown to respond through a centered likelihood score, and the detected-event Fisher information is the posterior variance of this score. This identifies posterior-weighted score contrast, rather than local response magnitude alone, as the relevant sensing resource. The framework separates posterior-shape information from evidence information, giving a resource-aware way to judge near-null response enhancement. It also yields practical design rules: a two-label source code converts a weak perturbation into a fixed-detector label imbalance, while the multiparameter score covariance provides a route to nuisance rejection and gives a minimal-label rank condition for sensing multiple perturbations. A passive double-slit implementation with weak one-slit phase and loss perturbations is proposed, requiring only fixed-detector source scans before and after calibrated perturbations. Practical tolerances associated with source-programming error, drift, background, imperfect coherence, and polarization mismatch are analyzed, and extensions to multi-aperture and integrated photonic systems are formulated. The results position TRY as a source-programmable Bayesian sensing architecture complementary to conventional detector-plane Young interferometry.

DOI: 10.3390/s26154698

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Time-reversed Young (TRY) interferometry holds a detector channel fixed while a source coordinate is scanned or programmed. We describe this measurement through a single accepted source-detector probability law obtained from scalar Fresnel propagation, the programmed source prior, and detector acceptance. The conventional fixed-source Young distribution, the source and detector marginals, and the fixed-detector TRY distribution are then different reductions of that common record. Standard information measures can therefore be assigned without ambiguity to the probability law actually measured. For a symmetric double-slit response with matched source and detector windows, the fixed-source and fixed-detector conditional distributions have the same entropy; the physically useful information from detector conditioning is instead expressed by grid-stable entropy deficits and Kullback-Leibler information gains. The canonical example gives a modest prior-to-posterior information gain and establishes a transparent baseline for more selective source programming, detector modes, or operating points. Finite slit width, finite-source element width, finite detector pixels, and finite-count entropy estimation are included to connect the ideal model with experiments. A phase scan then shows that conditional entropy and Fisher information characterize different properties of the same likelihood, while a two-mode coherence model separates record entropy from optical coherence entropy and distinguishes coherent superposition from incoherent mixture. The framework gives a unified statistical interpretation of fixed-detector, source-programmed Young interferometry while keeping optical reciprocity, Bayesian conditioning, local parameter sensitivity, and quantum state disturbance conceptually distinct.

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Time-reversed Young (TRY) interferometry reconstructs interference from a fixed detector by reading out a programmable source-label distribution. This work formulates the architecture as a source-coded Bayesian response sensor. For a perturbation parameter θ, the detected source-label histogram is a posterior distribution determined by a programmed source prior, an optical likelihood for a fixed-detector click, and an evidence factor equal to the click probability. The key point is not the Bayesian identity itself, but its physical implementation: in TRY the prior is imposed before propagation and can therefore reshape the response ensemble actually sampled by the detector. The normalized posterior is shown to respond through a centered likelihood score, and the detected-event Fisher information is the posterior variance of this score. This identifies posterior-weighted score contrast, rather than local response magnitude alone, as the relevant sensing resource. The framework separates posterior-shape information from evidence information, giving a resource-aware way to judge near-null response enhancement. It also yields practical design rules: a two-label source code converts a weak perturbation into a fixed-detector label imbalance, while the multiparameter score covariance provides a route to nuisance rejection and gives a minimal-label rank condition for sensing multiple perturbations. A passive double-slit implementation with weak one-slit phase and loss perturbations is proposed, requiring only fixed-detector source scans before and after calibrated perturbations. Practical tolerances associated with source-programming error, drift, background, imperfect coherence, and polarization mismatch are analyzed, and extensions to multi-aperture and integrated photonic systems are formulated. The results position TRY as a source-programmable Bayesian sensing architecture complementary to conventional detector-plane Young interferometry.

Time-reversed Young (TRY) interferometry holds a detector channel fixed while a source coordinate is scanned or programmed. We describe this measurement through a single accepted source-detector probability law obtained from scalar Fresnel propagation, the programmed source prior, and detector acceptance. The conventional fixed-source Young distribution, the source and detector marginals, and the fixed-detector TRY distribution are then different reductions of that common record. Standard information measures can therefore be assigned without ambiguity to the probability law actually measured. For a symmetric double-slit response with matched source and detector windows, the fixed-source and fixed-detector conditional distributions have the same entropy; the physically useful information from detector conditioning is instead expressed by grid-stable entropy deficits and Kullback-Leibler information gains. The canonical example gives a modest prior-to-posterior information gain and establishes a transparent baseline for more selective source programming, detector modes, or operating points. Finite slit width, finite-source element width, finite detector pixels, and finite-count entropy estimation are included to connect the ideal model with experiments. A phase scan then shows that conditional entropy and Fisher information characterize different properties of the same likelihood, while a two-mode coherence model separates record entropy from optical coherence entropy and distinguishes coherent superposition from incoherent mixture. The framework gives a unified statistical interpretation of fixed-detector, source-programmed Young interferometry while keeping optical reciprocity, Bayesian conditioning, local parameter sensitivity, and quantum state disturbance conceptually distinct.

Young interference is usually read as the gradual statistical accumulation of random detection events. Here we show that a time-reversed Young (TRY) geometry has a different statistical character: the fringe is not a marginal distribution of detector positions, but a conditional response indexed by a programmed source coordinate. With a fixed detector and a scanned source basis, the observable is an operational hybrid correlator between detector signal and source label. The resulting interference is deterministic at the response-function level, while noise enters only through estimation precision. We formulate this distinction using Fisher information, estimator variance, and noise scaling, clarifying why TRY naturally supports calibration, lock-in readout, null-fringe sensing, and source-plane superresolution.

We propose a physically explicit sensing application of a time-reversed Young (TRY) interferometer: simultaneous monitoring of beam tilt and focus drift with a fixed detector. The task is relevant to compact optical relays, free-space links, fiber-coupling stages, and micro-optical alignment modules, where continuous tracking of pointing and focus is needed but downstream wavefront cameras or multiport analyzers are undesirable. Using a finite-width double-slit Fresnel model, we derive the exact local TRY response functions for tilt-like and defocus-like phase perturbations and compute the corresponding optimal upstream source codes numerically. The physical optimal codes are fringe-locked and differ qualitatively from the simple odd/even modes suggested by Gaussian toy models. Two source-coded scalar channels recover essentially all local Fisher information in the full source-resolved TRY record for the physical model considered here. Compared with downstream direct intensity sensing, TRY provides first-order access to the mixed tilt--defocus task with fixed detection; compared with ideal downstream matched-mode sorting, its advantage is architectural rather than fundamental.

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-χ^2

, where

χ

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.

We show that time-reversed Young interferometry reorganizes, rather than reverses, optical entropy. A fixed detector conditions the reciprocal source--detector Green function and produces a source-label probability distribution. Marginal entropies in the standard and time-reversed geometries are generally unequal; the reciprocal invariant is instead the mutual information between source and detector coordinates. Near a destructive response, the conditioned source-label entropy can decrease while Fisher information for small phase, tilt, or defocus perturbations increases. The result identifies time-reversed Young interferometry as a source-space information processor with no analogue in ordinary detector-plane fringe readout.

We develop a compact theory of time-reversed Young (TRY) interference beyond the symmetric two-slit geometry by considering equally spaced three-slit, finite

N

-slit, and infinite periodic slit arrays. In the TRY configuration, a point emitter illuminates the aperture, a position-fixed detector records the signal, and the response is reconstructed in source space by correlating the detector record with the source-coordinate label. We show that the three-slit case already reveals the essential new physics beyond two slits: a quadratic Fresnel phase survives, modifies the reconstructed interference law, and lifts the nominal dark fringes in the generic case. For a general equally spaced

N

-slit array, we identify the exact reconstructed response and show that the familiar textbook grating factor is recovered only when the quadratic phase is negligible, compensated, or reduced to a common phase across the array. In that ideal limit, the reconstructed peaks are source-space analogues of classical grating orders rather than outgoing diffraction beams. For an infinite periodic TRY array, we further show that the same discrete quadratic phase generates full and fractional Talbot-like revivals in source space, governed by a reciprocal-distance condition rather than the conventional Talbot propagation law. These results show that the symmetric two-slit TRY geometry is exceptional, while multi-slit TRY systems naturally combine source-space discrimination with sensitivity to aperture-wide phase structure and periodic-array revival physics.