Auto-calibrating Universal Programmable Photonic Circuits: Hardware Error-Correction and Defect Resilience

TL;DR

The paper addresses how to realize arbitrary $U \in U(N)$ unitaries on photonic chips with robust auto-calibration. It proposes an interlaced factorization $U = F P_M F \cdots P_1 F$ where $F$ is the Discrete Fractional Fourier Transform realized by a $Jx$ lattice and $P_m$ are diagonal phase matrices, achieving universality for $M = N+1$ with $N(N+1)$ tunable phases; a Levenberg–Marquardt optimization minimizes $L = \frac{1}{N^2}\|U - U_t\|^2$ over Haar-random targets. The study shows that perturbations in $F$ can be countered by a second optimization over phase shifters, restoring the target unitary to numerical-noise accuracy, and that the architecture exhibits defect resilience when at most one faulty phase shifter resides per layer, owing to its over-determined nature. These findings support post-fabrication calibration and scalable, defect-tolerant programmable photonic circuits for optical information processing, with implications for large-scale photonic computation and communication systems.

Abstract

It is recently shown that discrete $N\times N$ linear unitary operators can be represented by interlacing $N+1$ phase shift layers with a fixed intervening operator such as Discrete Fractional Fourier Transform (DFrFT). Here, we show that introducing perturbations to the intervening operations does not compromise the universality of this architecture. Furthermore, we show that this architecture is resilient to defects in the phase shifters as long as no more than one faulty phase shifter is present in each layer. These properties enable post-fabrication auto-calibration of such universal photonic circuits, effectively compensating for fabrication errors and defects in phase components.

Figures (4)

• Figure 1: (a) The proposed architecture of the N-port system, consisting of alternating layers of Discrete Fractional Fourier Transforms (DFrFT) and diagonal phase shifts layers (PS${}_{j}$). (b) The photonic realization for $N = 6$ using photonic waveguide lattices and phase shifters (red squares). (c) The mean-squared error norms Eq. (\ref{['eq_L']}) of the optimization for $N=4$ and $N=8$, versus the number of phase layers $M$ (for $N=4$, we considered $M=3,4,5,6$ while for $N=8$, we considered $M=7,8,9,10$ phase layers).
• Figure 2: The auto-calibration property of the proposed architecture. By perturbing the DFrFT matrices, the error norm jumps to large values, but after a second optimization, new phases are found so that error norms reduce to numerical noise levels. This analysis is done for $(N = 8, M=9)$, and by considering $100$ random target matrices. Here, the perturbation magnitude parameter $\sigma_k$ has been chosen such that the relative error $\Delta F$ is $0.76\%$ (upper row), $2.28\%$ (middle row), and $4.55\%$ (lower row).
• Figure 3: Means $\mu_{\Delta x}$ and standard deviations $\sigma_{\Delta x}$ of the difference vector of the original and re-calibrated phases. Each point represents a single run of truncated LMA with 50 iterations for a single perturbed structure and target, with the color representing the norm log$_{10}(L)$. (a) Initial vector chosen within 10% of the unperturbed vector. (b) Initial vector randomly chosen.
• Figure 4: (Left column) Skecth of faulty phase shifters denoted in green. (Right column) The corresponding loss function for 100 randomly generated Haar target matrices for the faulty combinations in the horizontal label. A device with $N=4$ ports has been considered for $k=1$ (first row), $k=2$ (second row), $k=3$ (third row), and $k=4$ (fourth row) faulty phase shifters.