Optimised Electric Field Waveforms

• Optimised electric field waveforms are custom-designed signals that maximize process efficiency by balancing power, hardware limits, and nonlinear responses.
• They leverage mathematical optimization, spectral synthesis, and control theory to target applications from oscillator entrainment to biomedical stimulation.
• Experimental and computational methods validate these designs, demonstrating enhanced performance in wireless power transfer, quantum control, and plasma processing.

Optimised electric field waveforms are designed to maximize the efficacy of electric field–driven processes under realistic constraints such as power, hardware nonlinearities, and selectivity criteria. These waveforms have become central across fields ranging from nonlinear oscillator entrainment and strong-field quantum dynamics to wireless power transfer (WPT), plasma processing, and biomedical stimulation. For arXiv readers, a comprehensive treatment requires rigorous mathematical, optimization, and experimental perspectives, reflecting the current state-of-the-art.

1. Mathematical Formulations and Optimization Principles

Optimised waveforms are typically derived by formalizing a maximization (or minimization) problem subject to dynamical and physical constraints. The general approach couples physical models (e.g., phase, kinetic, circuit, hydrodynamic, quantum) to a cost functional and solves for the waveform under power, voltage, or physical process constraints.

For weak oscillator entrainment (Harada et al., 2010), the dynamics are described via a phase model: $\frac{d\psi}{dt} = \omega + Z(\psi) f(\Omega t)$ where $\psi$ is oscillator phase, $\omega$ is its natural frequency, $Z(\psi)$ is the phase response curve (PRC), and $f$ is the forcing waveform. Averaging leads to the interaction function $$\Gamma(\phi) = \langle Z(\theta+\phi) f(\theta)\rangle$$, where brackets denote period average. The optimization seeks $f^*$ maximizing the "locking range" $R$ for fixed power $P$: $$\mathcal{S}[f] = R[f] - \lambda(\langle f^2\rangle - P)$$ The first variation yields the optimal waveform: $$f^*(\theta) = \frac{1}{2\lambda}\big[Z(\theta+\phi_+) - Z(\theta+\phi_-)\big]$$ and the balancing condition involves the PRC derivative.

Strong-field quantum optimization (Schwinger effect) (Kohlfürst et al., 2012) employs quantum kinetic equations for the distribution function $F(q, t)$, and optimizes the field $A(t)$ via optimal control theory: $J[F, A] = -\gamma n[F] + f[A]$ with $f[A]$ imposing constraints (on peak field, pulse energy). Variational methods yield forward–backward propagation of state and adjoint equations, with gradient-based updates on $A(t)$.

Wireless power transfer waveform optimization (Moghadam et al., 2017, Zhang et al., 2022) is formulated in terms of transmit amplitudes/phases and a nonlinear rectifier model. For multisine signals,

$$x(t) = \text{Re}\Big\{\sum_{n=1}^{N}\sqrt{2}s_n e^{j\omega_n t}\Big\}$$

the DC voltage at the receiver is expressed as a nonlinear average over the received signal. Maximization is performed using sequential convex programming (SCP), maximal ratio transmission (MRT), or hybrid approaches, depending on nonlinearity severity.

Pulse shape optimization for pair production (Fillion-Gourdeau et al., 2017) parametrizes the field in Fourier space via B-spline bases, reducing dimensionality. The B-spline expansion allows efficient exploration via gradient-based local search or metaheuristic population methods (differential evolution).

2. Handling Nonlinearities and Constraints

Practical constraints strongly shape the nature of optimal waveforms. Nonlinear device models (rectifiers, power amplifiers), energy and amplitude bounds, and physical process thresholds (e.g., neural activation potential) must be respected.

Wireless power transfer with nonlinear energy harvester and power amplifier (Zhang et al., 2021, Zhang et al., 2022) introduces amplitude and power constraints at both transmitter and receiver. The solid-state power amplifier’s transfer function is nonlinear: $$f_{\text{SSPA}}(x^{in}(t)) = \frac{G x^{in}(t)}{[1 + (G x^{in}(t)/A_s)^{2\beta}]^{1/(2\beta)}}$$ and the harvested DC power is given by

$$z_{DC} = k_2 \mathbb{E}\{ y(t)^2 \} + k_4 \mathbb{E}\{ y(t)^4 \}$$

Optimization shifts from maximally exploiting high–peak-to-average power ratio (PAPR) signals (benefiting the rectenna’s nonlinearity) to single-carrier transmission (when HPA nonlinearity dominates).

In TMS pulse shape optimization (Ma et al., 11 Oct 2025), the cost functional penalizes coil heating $\mathcal{E}(i(t)) = R_{coil}\int i^2(t) dt$, while requiring neuron model activation and imposing coil voltage asymmetry constraints ($r_V = |V_{max}/V_{min}|$). Optimized waveforms feature a slow leading phase and near-rectangular main phase, yielding up to 92% energy loss reduction compared to conventional monophasic pulses.

3. Spectral Synthesis and Temporal Control

Many optimized waveforms employ spectral synthesis—combining several Fourier components with tailored amplitudes and phases—or parametric control via basis functions, to achieve sub-cycle temporal sculpting or frequency-selective effects.

High-harmonic generation (HHG) (Haessler et al., 2013) leverages three-color synthesis, combining a 1030 nm fundamental, its second harmonic (515 nm), and a 1545 nm mid-IR component, with carefully controlled phase delays. This “cycle-shaped” waveform enables attosecond steering of ionization and recollision, boosting XUV flux >100× and extending cutoff energies. Optimization targets quantum trajectories modeled by the Lewenstein saddle-point formalism, which links field waveform directly to recollision dynamics.

Waveform optimization in plasma jets (Vass et al., 11 Jun 2025) synthesizes “Peaks” and “Valleys” waveforms from four harmonics of the base frequency (13.56 MHz) for synergistic spatial–temporal control. Geometrical electrode structuring interacts with voltage waveform tailoring to localize electron power absorption, as confirmed by fluid simulations and optical diagnostics.

4. Experimental Validation, Diagnostics, and Measurement

Optimized waveform shaping is validated experimentally via precise diagnostic techniques and in vivo physiological measurements.

Optical gating of THz field waveforms (Muraviev et al., 2016) uses femtosecond optical pulses to induce ultrafast photoconductivity in AlGaAs/InGaAs HEMTs, enabling subpicosecond time-resolution of THz field traces. The photoconductive response is quenched by the optical gate, allowing direct retrieval of the electric field waveform by scanning the delay between THz and optical pulses.

Single-shot measurement of few-cycle optical waveforms (Liu et al., 2021) uses crossed mid-IR beams on a silicon chip, mapping time-delay onto spatial position for sub-cycle resolution. The process reconstructs carrier-envelope phase–stable optical field waveforms from pixelated detector images, facilitating real-time adaptive optimization.

In TMS (Ma et al., 11 Oct 2025), the “optimised unidirectional rectangular” pulse was validated in human subjects, showing similar motor thresholds but far lower coil heating and statistically significant motor-evoked potential (MEP) latency differences, indicating directional selectivity.

5. Applications and Engineering Implications

The principles of optimised electric field waveform design have broad applicability across scientific and technological domains.

• Oscillator entrainment: Phase model–based variational waveform optimization enables efficient entrainment under power limits (e.g., injection-locked oscillators, biological pacemakers) (Harada et al., 2010).
• Nonlinear strong-field physics: Pulse optimization via control theory and spectral basis expansion enhances electron–positron pair production in QED (Kohlfürst et al., 2012, Fillion-Gourdeau et al., 2017), with systematic approaches for maximum yield under energy constraints.
• Wireless power transfer (WPT): Multisine waveforms and channel-adaptive designs maximize DC output in nonlinear rectenna circuits, with careful management of transmitter and receiver nonlinearities, enabling efficient far-field power delivery (Moghadam et al., 2017, Zhang et al., 2021, Zhang et al., 2022).
• Plasma engineering: Tailoring current or voltage waveforms (e.g., sawtooth, square, multiharmonic) offers control over plasma density, asymmetry, and energy distribution for processing applications (Sharma et al., 2021, Vass et al., 11 Jun 2025).
• Biomedical devices: Asymmetric pulse optimization in TMS achieves precise, energy-efficient, directionally selective neurostimulation (Ma et al., 11 Oct 2025).
• Sensing and metrology: Viewing quantum sensors as LTI systems allows rapid assessment and optimization of sensor response to arbitrary field waveforms (Malvania et al., 30 Apr 2025).

6. Algorithmic, Computational, and Scaling Considerations

Efficient optimization of electric field waveforms benefits from advanced computational strategies. Gradient-based local search, sequential convex programming, mutational metaheuristics (e.g., differential evolution), and spectral basis reduction (e.g., B-splines) enable tractable exploration of large parameter spaces.

• Parametric reductions (e.g., B-spline basis) allow high-dimensional fields (tens of Fourier modes) to be controlled by a small set of parameters, improving computational tractability (Fillion-Gourdeau et al., 2017).
• Diagonal scaling and careful parameter initialization are crucial for convergence and avoidance of poor local optima in multi-parameter landscapes (Kohlfürst et al., 2012).
• Transfer function formalism (LTI system analogy) radically reduces computation time for sensor model simulations, enabling rapid optimization and assessment (Malvania et al., 30 Apr 2025).

7. Future Directions

Multiple lines of further inquiry will shape the evolution of optimised electric field waveform research:

• Incorporating device nonidealities (switching losses, hardware limits) into optimization frameworks for more realistic pulse design (Ma et al., 11 Oct 2025).
• Extending optimal waveform design to MIMO, multi-user, and adaptive schemes in wireless power transfer, addressing hardware impairments and feedback limits (Zhang et al., 2022).
• Real-time active control, stability management, and additional degrees of freedom in optical waveform synthesis for attosecond science and ultrafast processes (Haessler et al., 2013, Liu et al., 2021).
• Application of tailored spatial dispersion models for direct mode-profile engineering in photonic and accelerator devices, bypassing brute-force electromagnetic simulation (Boyd et al., 2018).
• Synergistic control strategies in plasma, combining electrical asymmetry and geometric focusing for radical generation and localized chemistry (Vass et al., 11 Jun 2025).

Optimised electric field waveforms therefore constitute a cross-disciplinary toolbox for enhancing process efficiency, selectivity, and control in electronic, optical, quantum, and biological systems. Their development and application remain strongly guided by mathematical optimization, rigorous experimental diagnostics, and constraints imposed by nonlinearities and device physics.