Inverse-designed metamaterial blocks enclosed in a feedback network can solve linear Fredholm integral equations of the second kind at the speed of wave propagation, with the solution emerging as the steady-state complex electromagnetic field. The 2019 microwave proof-of-concept left a clearly stated set of generalization goals: solving nonlinear integral equations by introducing controllable nonlinearity into the metamaterial kernel or feedback path, estimating eigenvalues and spectral properties of an integral kernel within the wave platform, implementing asymmetric kernels in a compact reflective layout, porting the solver to integrated silicon-photonics platforms at optical frequencies, and quantitatively characterizing robustness to noise, fabrication variability, imperfect couplers, and nonzero electrical lengths. Reconfigurable/programmable linear equation solving and free-space optical-frequency operation have since been demonstrated, while the nonlinear-integral-equation and kernel-eigenvalue capabilities remain undemonstrated in any wave-based analog platform, and the nonreciprocal, on-chip-integrated, and quantitatively robust variants remain open.

Background

Wave-based analog computing uses the physics of wave propagation, rather than digital logic, to evaluate mathematical operations: an input waveform encodes a function, the medium transforms it, and the output waveform encodes the result. Early 'computing metasurfaces' (Silva et al., Science 2014) implemented fixed mathematical operations such as spatial differentiation by engineering the nonlocal Green's-function response of thin metamaterial layers — engineered materials whose subwavelength building blocks produce optical responses unavailable in natural media. Such spatial metasurface processors, however, were restricted to a fixed menu of translationally invariant operations. Mohammadi Estakhri, Edwards and Engheta (Science 2019) introduced a fundamentally more general platform: a metamaterial 'metastructure' whose dielectric permittivity distribution is found by inverse design (adjoint-based numerical optimization that discovers the geometry realizing a target scattering response), so that its scattered output field implements a prescribed general integral operator with a complex, translationally variant kernel K(u,v)K(u,v). When this kernel block is embedded in a feedback loop with couplers and phase shifters, the circulating field iterates the Neumann series — the fixed-point expansion g(n+1)(u)=c(u)+∫K(u,v)g(n)(v) dvg^{(n+1)}(u)=c(u)+\int K(u,v)g^{(n)}(v)\,dv — and at steady state the output field equals the solution g(u)g(u) of the Fredholm integral equation of the second kind g(u)=c(u)+∫abK(u,v)g(v) dvg(u)=c(u)+\int_a^b K(u,v)g(v)\,dv, provided the spectral radius of the kernel operator (the largest modulus of its eigenvalues) is below one. The platform was demonstrated experimentally at microwave frequencies in guided-wave setups, treating both symmetric and non-symmetric kernels (the non-symmetric case in a two-sided, transmissive layout), with the input function c(u)c(u) and solution g(u)g(u) carried as complex monochromatic fields. The authors identified a set of unresolved generalization directions for this platform, confirmed and restated in their group's later publications: (i) solving nonlinear integral equations by introducing and controlling nonlinearity in the metamaterial kernel or feedback path; (ii) estimating eigenvalues and spectral properties of a (possibly unknown) kernel operator within the wave platform, which would require active elements such as controllable amplifiers, attenuators, and phase shifters, or gain media, in the feedback path, and amounts to solving the associated homogeneous Fredholm equation; (iii) reconfigurable, programmable, or self-configuring implementations that switch kernels or transfer functions without fabricating new permittivity distributions; (iv) asymmetric (non-symmetric) kernels in a compact reflective layout, which requires breaking Lorentz reciprocity — the scattering symmetry of any linear, time-invariant medium without magnetic bias or temporal modulation, which forces the reflection matrix of a single reflective interface to be symmetric, whereas a two-sided transmissive guided-wave layout can realize an asymmetric kernel; (v) scaling from the microwave proof-of-concept to integrated silicon-photonic platforms at optical frequencies, addressing fabrication tolerances, losses, and coupler de-embedding (removing, from measured scattering parameters, the response of the coupling structures so that the kernel block alone is characterized); and (vi) quantitative characterization of noise tolerance, fabrication variability, imperfect couplers, and nonzero electrical lengths (the finite phase accumulation of spatially extended real components, as opposed to idealized point-like couplers), and their effect on solution fidelity. Subsequent work has closed parts of this outlook: fully reconfigurable and programmable linear-equation solvers have been demonstrated at microwave frequencies (Tzarouchis et al. 2022, 2025; Yang et al. 2023, 2024; Fu et al. 2024), and ultrathin inverse-designed silicon metagratings with a semitransparent feedback mirror have solved Fredholm integral equations of the second kind with free-space visible radiation (Cordaro et al. 2023). What no later work has demonstrated is the nonlinear and spectral core of the original outlook: no wave-based analog platform solves nonlinear integral equations, and none estimates kernel eigenvalues within the feedback-loop architecture.

Problem Statement

Generalize the wave-based analog equation-solving metastructure platform of Mohammadi Estakhri, Edwards and Engheta (Science 2019) beyond linear, convergent-kernel, microwave, proof-of-concept implementations. Concretely: (i) design and demonstrate a metamaterial-based solver for nonlinear integral equations, in which nonlinearity is introduced and controlled in the metamaterial kernel or in the feedback path (for example, Hammerstein-type equations g(u)=c(u)+∫abK(u,v) N(g(v)) dvg(u)=c(u)+\int_a^b K(u,v)\,N(g(v))\,dv, in which the nonlinear operand NN acts on the unknown inside the integral), together with the convergence and stability theory for the resulting nonlinear iterative wave feedback; (ii) estimate the eigenvalues and spectral properties of an integral kernel operator within the wave platform, and identify which eigenpairs are extracted from the steady-state fields; (iii) implement asymmetric (non-symmetric) kernels in a compact reflective layout, which requires nonreciprocal physical implementations; (iv) realize the solver on integrated silicon-photonic platforms at optical frequencies, addressing fabrication tolerances, losses, and coupler de-embedding; and (v) quantitatively characterize how noise, fabrication variability, imperfect couplers, and nonzero electrical lengths degrade solution fidelity, beyond the initial supplementary analyses. Reconfigurable and programmable solving of linear equations, and free-space optical-frequency operation of linear integral-equation solvers, have already been demonstrated and are excluded from the open target; the least-developed and central open cores are (i) and (ii), while (iii)-(v) are still-open supporting capabilities on the same platform.

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