Analog Wave-Domain Computing

• Analog wave-domain computing is a paradigm that performs mathematical operations directly on propagating waves, exploiting continuity and spectral richness for real-time processing.
• Implementations using photonic circuits, plasmonic devices, and metasurfaces achieve high bandwidths, miniaturization, and energy efficiency beyond conventional digital hardware.
• Programmable and dynamic wave networks enable complex tasks such as matrix inversion and machine learning by harnessing passive physics and tunable wave-matter interactions.

Analog wave-domain computing encompasses a diverse family of physical architectures and methodologies for executing mathematical operations directly on propagating waves—optical, microwave, acoustic, or other—leveraging their inherent continuity, parallelism, and spectral richness. The core principle is to engineer wave-matter interactions so that the transformation between input and output waveforms implements a prescribed mathematical operator, often in real time and across broad bandwidths. These approaches exploit passive and programmable devices (photonic chips, metasurfaces, waveguide networks, random scattering media, etc.), often far exceeding the energy and speed limitations of conventional digital hardware.

1. Mathematical Foundations and Operator Realization

Analog wave-domain computation is mathematically rooted in the signal and systems theory of linear, time-invariant (LTI) filters and operators. Any desired operation—such as differentiation, integration, or solving ordinary/partial differential equations (ODEs/PDEs)—is encapsulated by a transfer function $H(\omega)$ or $H(k_x)$ in the frequency or spatial-frequency domain. For example, a first-order differentiator implements $H_{\text{diff}}(\omega) = j\omega$, while an ODE solver for $dy/dt + k y = x(t)$ realizes $H_{\text{ODE}}(\omega) = 1/(j\omega + k)$ (Dong et al., 2014).

In wave-optical platforms, engineering the physical response—such as the interference phase in a Mach–Zehnder interferometer (MZI) or the resonance of a microring resonator—results in the desired spectral transfer. Such tailoring extends to higher-order or fractional operators via cascading devices or programming additional loss/dispersion elements.

2. Photonic and Plasmonic Circuit Implementations

Photonic integrated circuits (PICs) on silicon or lithium niobate often serve as wave-domain analog processors. Spectral engineering in passive devices enables differentiators, integrators, and ODE solvers with distinct advantages: bandwidths in excess of 100–200 GHz (MZI), footprints of $<0.015$ mm$^2$, and insertion losses $<3$ dB per stage. Tunability (via carrier injection or heaters) further enables programmable operation, including fractional differentiation and cascaded higher-order functions (Dong et al., 2014). Experimental results demonstrate accurate temporal differentiation and ODE solutions on ps-scale pulse trains.

Graphene-based platforms exploit the tunable surface impedance and strong plasmon confinement, achieving ultra-compact transmit-arrays ("metalines") suitable for spatial Fourier operations such as differentiation and integration. A lens-mapped spatial-frequency ($k_x$) profile is imprinted via gate-controlled chemical potential profiles, achieving accurate analog computation with sub-micron device lengths, $\approx$60-fold reduction compared to bulk lens-based architectures (Abdollahramezani et al., 2015).

3. Metasurface- and Metagrating-Based Architectures

Gradient metasurfaces and metagratings extend analog wave-domain computing to planar and compact geometries. Engineered nanobricks (such as high-index silicon with overlapping electric/magnetic dipole resonances) locally modify amplitude and phase, mapping spatial location to transfer-function value, thus enabling broadband operators like $i k_x$ (differentiator) and $1/(i k_x)$ (integrator) at telecommunication wavelengths (Chizari et al., 2016).

Programmable time-modulated metasurfaces employ digital meta-atom coding sequences ($2$-bit states) rapidly cycled via electronic bias, creating effective analog transfer functions $H(k_x)$ in selected harmonics. This technique supports flexible real-time implementation of calculus operations—including 1st/2nd-order differentiation, integration, and integro-differential equation solving—as well as continuous image-processing (edge detection), all within a passive, subwavelength structure (Rajabalipanah et al., 2020). High-order Floquet harmonics in compound gratings enable multiple parallel channel operations (e.g., odd/even transfer functions for differentiation) on a single surface (Rajabalipanah et al., 2021).

4. Programmable and Dynamical Wave Networks

Beyond static analog operations, programmable metastructures realize general-purpose linear algebra, matrix inversion, and iterative algorithmic solutions. Architectures such as the Direct Complex-Matrix (DCM) machine consist of cross-bar waveguide networks embedded with independently adjustable phase and amplitude multipliers, supporting true $n\times n$ programmable matrix multiplication and inversion at MHz to GHz rates. Reconfiguration (via embedded microcontrollers) enables stationary (matrix inversion) and non-stationary (Newton’s method, constrained optimization) algorithms, with vector-to-vector processing and error floors in the $10^{-3}$ regime (Tzarouchis et al., 2022).

Wave-dynamical networks implement field-programmable problem-solving via interconnected nodes and edges capable of frequency mixing, time-delay, filtering, and phase shifting. NP-hard combinatorial problems (number partitioning, knapsack, traveling salesman) are encoded directly into spectral and temporal signatures: the interference of exponentially many solution paths yields measurable features (spectral lines, arrival times) corresponding to optimal solutions, all in one analog wave propagation event (Liu et al., 7 Aug 2025). Such architectures exploit intrinsic space, time, and frequency parallelism limited only by bandwidth and signal-to-noise ratios.

5. Reservoir and Extreme Learning with Wave Physics

Analog wave propagation has been mapped onto recurrent neural network (RNN) models for machine learning. Reservoir computing (RC) systems use multimode waveguides, modal mixing, and optical speckle as random high-dimensional projectors, with node nonlinearity arising from photodetection and electronic feedback. Chip-scale photonic reservoir computers perform time-domain classification tasks, such as multivariate audio (vowel recognition), at GHz rates and with scalability to thousands of optical "neurons" (Paudel et al., 2019).

Nonlinear time-Floquet media enable tunable non-linear entanglement of input features, overcoming the typically weak nonlinearities of physical wave materials. By phase-modulating dielectric slabs at low-frequency ($\omega_m$), the device causes high-order frequency mixing amongst many Floquet harmonics, supporting efficient extreme learning and reservoir computing for regression, classification, chaotic time-series forecasting, and parallel multi-task image analysis. Only a trivial readout layer requires training; the rest is achieved natively via physics—even for highly nonlinear tasks (Momeni et al., 2021).

The inhomogeneous wave equation (with space-dependent $c(x,y)$) also functions as an RNN, with trainable "weights" encoded via dielectric patterns. Inverse design and adjoint backpropagation enable high-accuracy classification, with inference bandwidths orders-of-magnitude above digital approaches (Hughes et al., 2019).

6. Analog Network Algorithms in Microwave and Random Media

Microwave linear analog computers (MiLACs) and random scattering media realize wave-based computation as multiport network problems. Reconfigurable microwave networks of tunable admittance elements ($P^2$ components for $P$ ports) can be programmed to perform any linear algebra operation: $y = Ax$, LMMSE estimation, matrix inversion. All multiplications and inversions occur in a single analog clock cycle, with complexity reduction from $O(P^3)$ to $O(P^2)$ for matrix inversion, and energy advantages over digital DSP (Nerini et al., 9 Apr 2025, Nerini et al., 10 Apr 2025).

Random or chaotic media likewise serve as analog computing substrates. Calibrating the impact/scattering matrix of a random cavity (e.g., a room with reconfigurable wall metasurfaces) allows encoding arbitrary linear operations (including DFT) on propagating Wi-Fi signals, with per-operation energy costs of $\sim$1 pJ and $10^6$ operations/sec throughput (Hougne et al., 2018).

7. Physical Limits, Integration, and Power Consumption

A distinguishing feature of analog wave-domain computing is its ability to operate at real-time speeds set by wave propagation—GHz to THz—without active amplification, and with near-zero static power (in passive designs, floating-gate graphene, or photonic circuits). CMOS-compatible fabrication (e.g., metatronic circuits in indium-tin-oxide) supports board-scale integration, enabling direct solutions of Laplace, Poisson, and wave PDEs via mesh-wide nonlocal Green’s functions, programmable at femtojoule energies via carrier injection (Miscuglio et al., 2020).

Integrated electro-optic digital-to-analog links (EO-DiALs) for photonic computing combine high-fidelity DAC and EO modulation directly in a single thin-film device, supporting ultrabroadband analog waveform generation at 186 Gb/s and energy consumption of 0.058 pJ/bit. Parallel architectures enable the extension to convolutional and matrix multiplication accelerators in analog photonic computing (Song et al., 2024).

8. Limitations, Scalability, and Outlook

Bandwidth, signal dispersion, dynamic range, fabrication tolerances, and calibration all constrain wave-domain computing. Scaling node and edge counts increases the overhead of programmable components and demands intricate anti-collision coding for unique spectral signatures in combinatorial algorithms. Passive platforms are typically limited to moderate-precision analog operations; digital or active refinement may be needed for high-accuracy tasks (Tzarouchis et al., 2022, Liu et al., 7 Aug 2025).

Nevertheless, analog wave-domain computing constitutes a robust and versatile framework for real-time, parallel, energy-efficient mathematical transformations—including but not limited to calculus, matrix algebra, differential equation solving, and machine learning—on a diversity of physical platforms. Progress in integrated fabrication, tunable materials, metasurface design, and hybrid photonic-microwave architectures continues to expand the practical scope and impact of these technologies.