Differential source-basis encoding for superresolved parameter estimation in a time-reversed Young interferometer

Abstract

We develop a differential source-encoding protocol for local parameter estimation in a time-reversed Young interferometer, where the source plane is used not merely as a scan coordinate but as a programmable measurement basis. Two sequential positive-only source patterns implement an antisymmetric differential probe about a chosen operating point, converting the deterministicc source-coordinate response into a derivative-gradient sensing channel. In the local regime, the differential signal separates naturally into an envelope-gradient term, which is also present in noninterferometric differential sensing, and an interference-gradient term, which is specific to the time-reversed Young fringe law. This decomposition identifies the physical origin of the interferometric advantage and clarifies why it is regime dependent rather than universal. Using a shot-noise-limited Poisson model, we derive the corresponding Fisher information and Cramér--Rao bounds and compare the protocol with raster sampling in the same geometry and with a matched noninterferometric differential baseline. Representative numerical examples show a strong and robust gain over raster sampling, while the additional improvement from the time-reversed Young interference is parameter dependent but can be substantial in favorable regimes. The results establish the time-reversed Young geometry as a practically simple platform for programmable differential interferometric metrology.

Figures (5)

• Figure 1: Schematic of the time-reversed Young interferometer and the differential source-encoding protocol. Panel (a) shows the standard Young geometry. Panel (b) shows the time-reversed geometry with a fixed detector and a programmable source plane. Panel (c) illustrates the positive-only implementation of a signed differential source code.
• Figure 2: Deterministic time-reversed Young response as a function of the source coordinate $x_s$ for several values of the displacement parameter $\theta$. The curves illustrate Eq. (\ref{['eq:mu_general']}) with a localized Gaussian response kernel multiplied by the source-coordinate fringe factor.
• Figure 3: Comparison of a raster sample $S_{\rm raster}(\theta)$, the differential time-reversed Young signal $S_{\rm TRY}(\theta)$, and a noninterferometric differential baseline $S_{\rm SP}(\theta)$. The differential protocol converts the source-coordinate response into a derivative-sensitive channel and steepens the local dependence on the displacement parameter near the operating point.
• Figure 4: Local Fisher information of the differential time-reversed Young protocol as a function of the normalized differential spacing $\delta/\sigma$ and the dimensionless interference parameter $\kappa \sigma$. The marked point indicates the maximum within the plotted parameter range.
• Figure 5: Comparison of the differential time-reversed Young protocol with raster sampling and the noninterferometric differential baseline for the same Gaussian-response model and representative parameter set used in the text. Panel (a) shows the Fisher information, panel (b) the corresponding Cramér--Rao bound, and panel (c) the local sensitivity metric $|\partial_\theta S|/\sqrt{\mathrm{Var}(S)}$, all plotted as functions of the displacement parameter $\theta$. The time-reversed Young differential protocol and the noninterferometric differential baseline both outperform raster sampling over the main operating region, while the relative advantage of the time-reversed Young protocol over the noninterferometric differential case is parameter dependent and must therefore be assessed quantitatively rather than inferred from the signal shapes alone.