Inverse Scattering for Dirac Equations Arising in Waveguide Arrays

Technical Summary: Inverse Scattering for Dirac Equations Arising in Waveguide Arrays

Problem Statement
This paper investigates inverse scattering problems for two specific Dirac equations that serve as continuum models for waveguide arrays. The authors consider two distinct formulations:

1. The Chiral Model: A hyperbolic equation of the form $-i \partial_x \psi + i \alpha \partial_y \psi + k(\beta + V)\psi = 0$. Here, $x$ acts as a time-like variable, and the inverse problem involves reconstructing the scattering potential $V(x,y)$ from measurements of the scattered field at a final time $x=L_x$.
2. The Antichiral Model: An elliptic equation of the form $i \beta \partial_x \psi + i \alpha \partial_y \psi + k(V - 1)\psi = 0$. The inverse problem here seeks to recover $V$ from boundary measurements of the scattered field on $\partial \Omega$.

The primary objective is to determine the scattering potential $V$ from suitable measurements of the spinor field $\psi$. The authors note that while elliptic inverse problems are typically more ill-posed than hyperbolic ones, the specific measurement data available (full boundary data vs. final-time data) significantly influences the reconstruction behavior.

Methodology
The core approach employs the Inverse Born Series (IBS) and the Reduced Inverse Born Series (RIBS).

• Forward Problem: The authors first establish the well-posedness of both models. For the chiral model, they treat the equation as an evolution problem and prove existence and uniqueness using semigroup theory and the Banach fixed-point theorem. For the antichiral model, they utilize the Lippmann–Schwinger equation and Green's function representations, proving uniqueness under a smallness condition on the potential $V$.
• Series Expansion: The scattered field is expressed as a Born series, a multilinear expansion of the potential. Inverting this expansion yields the IBS, which reconstructs $V$ as a multilinear series of the measured scattered field.
• Reduced Series: The authors also investigate the RIBS, which retains only a distinguished subset of terms from the full IBS. This reduction is motivated by cancellation phenomena among higher-order terms, aiming to lower computational costs.
• Numerical Implementation:
  • For the chiral model, a finite-difference scheme discretizes the space-like variable $y$, while a Crank–Nicolson marching scheme propagates the solution in the time-like variable $x$.
  • For the antichiral model, the problem is reformulated as an integral equation for an auxiliary field $\sigma$, solved using the Generalized Minimal Residual (GMRES) method with Fast Fourier Transform (FFT) acceleration for the convolution operator.
  • The inverse solvers utilize a conjugate-gradient method with Tikhonov regularization to compute the pseudoinverse of the leading-order operator, followed by the evaluation of IBS or RIBS terms.

Key Contributions

1. Well-Posedness Analysis: The paper provides rigorous proofs for the well-posedness of the forward problems for both chiral and antichiral Dirac equations in the context of waveguide arrays.
2. Convergence and Error Analysis: The authors derive convergence criteria and rigorous error estimates for both the IBS and RIBS. They establish specific conditions on the norm of the potential $V$ required for the series to converge.
3. Algorithm Development: Detailed numerical algorithms are presented for solving both the forward and inverse problems, including specific discretization strategies for the hyperbolic and elliptic cases.
4. Comparative Numerical Study: Extensive numerical experiments are conducted to validate the algorithms. The study compares the performance of the full IBS against the RIBS across various scattering scenarios (low, medium, and high contrast) for both models.

Results
The numerical experiments demonstrate the following:

• Contrast Dependence: In the low-contrast regime, one or two terms of the IBS are sufficient for accurate reconstruction. As contrast increases to moderate levels, more terms are required for convergence. In the high-contrast regime, the reconstruction fails, consistent with theoretical limitations of the Born series approach.
• Model Comparison: Contrary to the intuition that elliptic problems are more ill-posed, the antichiral model yields better reconstruction quality than the chiral model. The authors attribute this not to the ellipticity itself, but to the availability of richer measurement data (full boundary data in the antichiral case versus one-sided final-time data in the chiral case).
• RIBS Effectiveness: The Reduced Inverse Born Series (RIBS) performs comparably to, and in some cases better than, the full IBS. This suggests that the RIBS can significantly reduce computational costs without sacrificing accuracy, likely due to the cancellation of higher-order terms observed in the experiments.

Significance
The paper claims to contribute to the relatively limited literature on inverse scattering for Dirac equations by providing a comprehensive framework that includes well-posedness analysis, convergence theory, and practical numerical algorithms. The work highlights that the effectiveness of inverse scattering methods depends critically on the type of available measurement data. Furthermore, the demonstration that the RIBS is effective for these specific waveguide array models suggests a viable pathway for efficient reconstruction in quantum optical settings, offering a balance between computational efficiency and reconstruction accuracy. The authors note that a deeper theoretical understanding of the cancellation phenomena observed in the RIBS remains a subject for future investigation.