Bidirectional Nonlinear Optical Tomography: Unbiased Characterization of Off- and On-Chip Coupling Efficiencies

TL;DR

The paper tackles bias in evaluating nonlinear photonic integrated circuits caused by relying on linear calibration that only yields the product of input/output couplings $η_1η_2$. It introduces bidirectional nonlinear optical tomography (BNOT), a direction-aware metrology that uses forward and backward pumping to break this degeneracy and estimate the individual couplings $η_1^{(2ω)}$ and $η_2^{(ω)}$ via a joint constrained optimization with an observation model that includes pump fluctuations and detector noise. Monte Carlo results demonstrate unbiased convergence of the estimates to ground truth with reduced variance, enabling accurate reconstruction of on-chip squeezing $\\mathcal{S}_{ON}$ and SHG efficiency $\\mathcal{E}_{ON}$ from off-chip measurements. The method is hardware-compatible and platform-agnostic, offering coupling-resolved benchmarking across nonlinear processes and enabling reproducible, scalable performance assessment for quantum optics, frequency conversion, and precision metrology.

Abstract

Accurate evaluation of nonlinear photonic integrated circuits requires separating input and output coupling efficiencies (i.e., $η_1$ and $η_2$), yet the conventional linear-transmission calibration method recovers only their product (i.e., $η_1\,η_2$) and therefore introduces systematic bias when inferring on-chip performance from off-chip data. We present bidirectional nonlinear optical tomography (BNOT), a direction-aware metrology that uses forward and backward pumping of complementary nonlinear probes, with process-appropriate detection, to break the ``degeneracy'' of $η_1\,η_2$ and estimate individual interface efficiencies with tight confidence intervals. The method links off-chip measurements to on-chip quantities through a compact observation model that explicitly incorporates pump fluctuations and detector noise, and it frames efficiency extraction as a joint constrained optimization. Monte Carlo studies show unbiased convergence of the estimated efficiencies to ground truth with low error across realistic operating regimes. Using these efficiency estimates to reconstruct on-chip nonlinear figures of merit yields distributions centered on the true values with reduced variance, whereas conventional ``degenerate'' calibration is biased and can substantially misestimate on-chip performance. BNOT is hardware-compatible and platform-agnostic, and provides unbiased characterization of off- and on-chip coupling efficiencies across nonlinear processes, enabling reproducible, coupling-resolved benchmarking for scalable systems in quantum optics, frequency conversion, and precision metrology.

Figures (5)

• Figure 1: (a) Examples of integrated structures and nonlinear optical processes. (b) Calibration protocols: (i) linear transmission, yielding a response $\propto \eta_{1}\eta_{2}$, and (ii--iii) nonlinear processes with forward (left-to-right direction) and backward (right-to-left) pumping, described by the asymmetric functions $f_{AS}(\eta_{1},\eta_{2})$ and $f_{AS}(\eta_{2},\eta_{1})$. The right panel compares SHG scaling with input power for each scheme. (c) Bidirectional nonlinear optical tomography on a PPLN waveguide. Pumps at $\omega$ and $2\omega$ generate SHG and squeezing under forward pumping ($\eta_\text{in}=\eta^{(2\omega)}_1$, $\eta_\text{out}=\eta^{(\omega)}_2$) and backward pumping ($\eta_\text{in}=\eta^{(\omega)}_2$, $\eta_\text{out}=\eta^{(2\omega)}_1$), with detection by direct detection (DD) and balanced homodyne detection (BHD) using local oscillator (LO).
• Figure 2: Histograms of Monte Carlo (MC) simulations with the ground-truth interface efficiencies in Eq. \ref{['eq:ground_truth']} and parameters in Tab. \ref{['tab:parameters']}. (a) Simulated off-chip squeezing and (b) SHG efficiency. (c) BNOT estimates of $\hat{\eta}_{1,\,\text{B}}$ (blue) and $\hat{\eta}_{2,\,\text{B}}$ (red), with shaded regions showing 95 % confidence intervals (2.5–97.5 percentiles).
• Figure 3: (a) Total mean-square error (MSE) map of efficiency estimation using BNOT, $e_1^2 + e_2^2$, where $e^2_1$ and $e^2_2$ are the estimator's MSEs of $\eta^{(2\omega)}_1$ and $\eta^{(\omega)}_2$, respectively. The red dashed contour marks the $-20$ dB boundary. Colored stars indicate the parameter sets used for Monte Carlo simulations. (b) MSE of $\hat{\eta}_{1,\,\text{B}}$ and (c) MSE of $\hat{\eta}_{2,\,\text{B}}$ as functions of the number of independent trials, with colors matching the parameter sets in (a).
• Figure 4: MC simulations of squeezing and SHG efficiency. Histograms of (a) off-chip squeezing and (b) SHG efficiency. The estimated (c) on-chip squeezing and (d) SHG efficiency are shown. Black dashed lines mark the ground-truth of on-chip squeezing and SHG efficiency: $\mathcal{S}^{(\varepsilon_1,\,\Delta k)}_\text{ON}$ and $\mathcal{E}^{(\varepsilon_2,\,\Delta k)}_\text{ON}$. In (c) and (d), the brown histograms represent conventional calibration assuming "degenerate" coupling efficiencies, while the green histograms show the estimates of BNOT.
• Figure 5: (a) On-chip squeezing versus $\eta_1^{(2\omega)}$. (b) Off-chip squeezing versus $\eta_1^{(2\omega)}$ and $\eta_2^{(\omega)}$. The orange horizontal line in (a) and curve in (b) denote the on-chip and off-chip $15$-dB squeezing threshold.