Complex-Frequency Waveform Excitation

• Complex-frequency waveform excitation is a technique that applies drives with complex frequency components to embed exponential envelopes into time-domain signals, enabling targeted manipulation of wave responses.
• It compensates for loss and enhances resonator performance by balancing temporal decay or growth with spatial propagation, thereby improving quality factors in various physical systems.
• The method leverages scattering zeros and poles for virtual absorption, selective mode loading, and precise waveform synthesis, with broad applications across optics, acoustics, and molecular sensing.

Complex-frequency waveform excitation denotes the deliberate driving of a physical system with a temporally structured field whose frequency is treated as a complex quantity, so that the usual harmonic factor acquires an exponential envelope in time. In the recent optics, acoustics, resonator, and inverse-problem literature, this is treated not merely as Laplace-domain bookkeeping but as an operational degree of freedom for interrogating analytically continued constitutive laws, scattering functions, and modal responses (Long et al., 13 Oct 2025, Trivedi et al., 2024, Trivedi et al., 2 Jun 2025, Li et al., 15 Jul 2025). Across these works, the method is used to compensate loss without material gain, realize complementary-media behavior in passive systems, probe hidden poles, capture arbitrary waveforms in lossless structures, sharpen effective resonance quality factors, and improve selective addressing in coupled resonator networks (Tian et al., 19 Dec 2025, Guan et al., 2023, Zeng et al., 2023).

1. Definition, scope, and neighboring usages

In the narrow sense now common in electromagnetics and wave physics, complex-frequency excitation means a drive of the form $e^{-i\omega_c t}$ or $e^{j\omega_c t}$ with $\omega_c\in\mathbb{C}$, interpreted as a literal time-domain waveform rather than only as an analytic continuation variable (Long et al., 13 Oct 2025, Huang et al., 13 Mar 2025). If the $e^{-i\omega t}$ convention is used and $\omega=\omega'+i\omega''$, then

$e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$

so $\omega''<0$ gives a temporally decaying waveform and $\omega''>0$ gives a temporally growing waveform (Long et al., 13 Oct 2025). Several papers instead use the $e^{j\omega t}$ convention, in which the same physical envelope appears with the opposite sign assignment for the imaginary part of frequency (Trivedi et al., 2024, Maddi et al., 14 Mar 2025).

A broader waveform-control literature uses related language for deliberately non-sinusoidal or multi-harmonic drives. In symmetric capacitively coupled plasmas, for example, a sawtooth-like current waveform synthesized from $N=50$ harmonics is treated as a tailored excitation that controls plasma asymmetry, dc self-bias, and the ion energy distribution function (Sharma et al., 2021). Circuit-based metasurfaces that distinguish short pulses from continuous waves at the same carrier frequency likewise exploit waveform dependence beyond ordinary frequency selectivity, though they do so through nonlinear rectification and RC/RL memory rather than a single complex carrier frequency (Wakatsuchi et al., 2014). A plausible implication is that “complex-frequency waveform excitation” now names a narrower class within a wider family of temporally engineered excitation strategies.

2. Temporal and spectral foundations

The common mathematical feature is that a nonzero imaginary part of frequency builds an exponential envelope directly into the excitation. In trajectory-based formulations, the analytic input is written as

$e^{j\omega_c t}$0

with instantaneous complex angular frequency

$e^{j\omega_c t}$1

so the real part controls instantaneous oscillation while the imaginary part controls local exponential growth or decay (Krasnok et al., 12 Mar 2026). This extends the pointwise notion of complex frequency into a contour $e^{j\omega_c t}$2 in the complex-frequency plane, enabling finite-energy analytic chirps that probe a device along prescribed trajectories rather than at isolated off-axis points (Krasnok et al., 12 Mar 2026).

A second foundational theme is synthesis. In arbitrary-waveform capture, a lossless scattering system with zeros $e^{j\omega_c t}$3 supports a basis of exponentially rising complex-frequency modes $e^{j\omega_c t}$4, and a waveform can be decomposed as

$e^{j\omega_c t}$5

If each $e^{j\omega_c t}$6 is a scattering zero for the same input eigenchannel, the total scattered field vanishes term-by-term, and the capture efficiency $e^{j\omega_c t}$7 equals the synthesis fidelity $e^{j\omega_c t}$8 (Tian et al., 19 Dec 2025). In the dense-zero limit, the synthesis becomes inverse-Laplace-like rather than purely Fourier-like, which is one of the clearest formal distinctions between real-frequency harmonic analysis and complex-frequency waveform design (Tian et al., 19 Dec 2025).

A third foundation is operator-theoretic. In the generalized virtual wave transform for diffusive systems, analytic continuation maps a thermal spectrum to a virtual wave spectrum evaluated at a complex argument $e^{j\omega_c t}$9. The resulting map is a causal, compact Fredholm operator acting as a nonstationary low-pass filter, and pulse, lock-in, chirped, coded, and PN excitations are interpreted as different projections onto excitation-induced subspaces of the same underlying transform (Zhu et al., 7 Jun 2026). This suggests a general spectral-geometric interpretation: waveform design selects how an analytically continued operator is sampled, weighted, and regularized.

3. Constitutive response, loss compensation, and effective material transformation

The most explicit electromagnetic formulation appears in passive lossy complementary media. For a homogeneous isotropic medium under the $\omega_c\in\mathbb{C}$0 convention, the Maxwell relations are written as

$\omega_c\in\mathbb{C}$1

Because $\omega_c\in\mathbb{C}$2 is complex and $\omega_c\in\mathbb{C}$3, $\omega_c\in\mathbb{C}$4 are evaluated at that complex argument, the relevant propagation conditions depend on $\omega_c\in\mathbb{C}$5 and $\omega_c\in\mathbb{C}$6, not only on the real-frequency signs of $\omega_c\in\mathbb{C}$7 and $\omega_c\in\mathbb{C}$8 (Long et al., 13 Oct 2025). The sufficient conditions

$\omega_c\in\mathbb{C}$9

make the wavevector purely real, so a waveform that decays in time can propagate without decay in space (Long et al., 13 Oct 2025). In that framework, loss ordinarily expressed as spatial attenuation is offset by the temporal envelope of the excitation.

Within a Lorentz–Drude constitutive model,

$e^{-i\omega t}$0

with passive constraints $e^{-i\omega t}$1, $e^{-i\omega t}$2, and $e^{-i\omega t}$3, the paper proves a constructive theorem: for any complex frequency $e^{-i\omega t}$4 with $e^{-i\omega t}$5 and any target $e^{-i\omega t}$6, there exist passive Lorentz–Drude parameters such that $e^{-i\omega t}$7; the same statement is said to apply likewise to $e^{-i\omega t}$8 (Long et al., 13 Oct 2025). This means that at one chosen complex operating point, passive lossy media can realize arbitrary complex-valued effective constitutive parameters. The result is then specialized to complementary media satisfying

$e^{-i\omega t}$9

with $\omega=\omega'+i\omega''$0 and $\omega=\omega'+i\omega''$1, where $\omega=\omega'+i\omega''$2 and $\omega=\omega'+i\omega''$3 are positive real numbers (Long et al., 13 Oct 2025). Under these conditions, passive lossy slabs can emulate “optical antimatter,” ideal lensing, and effective negative-index behavior without material gain.

Cross-domain loss compensation follows the same logic. In phonon polaritons and surface polaritons, a suitably chosen complex excitation frequency can enforce $\omega=\omega'+i\omega''$4, and the response at that complex frequency can be synthesized from measured real-frequency field maps through a coherent sum,

$\omega=\omega'+i\omega''$5

This recovered nearly lossless propagation in hBN and $\omega=\omega'+i\omega''$6-MoO$\omega=\omega'+i\omega''$7, with propagation length increasing from $\omega=\omega'+i\omega''$8 to $\omega=\omega'+i\omega''$9 at the chosen complex frequency and optimal time snapshot (Guan et al., 2023). In acoustics, an experimentally chosen decaying temporal envelope likewise compensates the imaginary part of the duct wavenumber, while a growing temporal envelope restores total absorption in a non-critically coupled Helmholtz resonator by exciting a complex reflection zero (Maddi et al., 14 Mar 2025).

4. Zeros, poles, virtual absorption, and selective loading

Scattering zeros and poles provide the dominant interpretation of complex-frequency excitation in open resonator systems. In lossless multiport scattering, a zero $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$0 of an eigenvalue $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$1 of the scattering matrix satisfies

$e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$2

and the associated exponentially rising waveform $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$3 can produce virtual perfect absorption: there is no outgoing scattering during the excitation window even though the structure itself is lossless (Tian et al., 19 Dec 2025). By engineering a chain of coupled cavities with many such zeros, arbitrary waveforms can be synthesized from zero-associated complex-frequency modes and captured without material absorption. In the reported examples, capture efficiencies for all waveform types approach unity as the number of cavities increases; with $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$4 cavities, all waveforms except exponentially decaying ones exceed $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$5, and with a tailored non-uniform chain of $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$6 cavities the exponentially decaying case reaches $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$7, compared with $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$8 for a uniform chain of the same size (Tian et al., 19 Dec 2025).

The same zero-matching logic underlies selective resonator loading. In a single resonator, complex-frequency driving matched to a reflection zero yields near-unity selected energy storage efficiency $e^{-i\omega t}=e^{-i\omega' t}e^{\omega'' t},$9, whereas optimized Gaussian pulses of the same system reach about $\omega''<0$0 (Trivedi et al., 2 Jun 2025). In a coupled three-resonator system, zero-matched complex-frequency excitation gives $\omega''<0$1–$\omega''<0$2 efficiency together with substantially improved selectivity and crosstalk suppression relative to Gaussian pulses of the same duration (Trivedi et al., 2 Jun 2025). The paper describes this as dynamic critical coupling: the exponentially growing envelope counteracts radiative leakage so that the input port remains effectively matched throughout the loading process.

Complex-frequency driving also reveals poles that are practically invisible to ordinary real-frequency spectroscopy. In RF and optical resonator models, excitation at a complex-conjugate zero can load poles whose imaginary parts dominate their real parts, including nearly non-oscillatory and purely imaginary poles; after truncation of the drive, these hidden poles can re-emit or transfer energy to visible radiative resonances, enabling conversion of a non-oscillating excitation into an oscillating emitted signal (Trivedi et al., 2024). In passive resonators driven at their complex resonance frequencies, the time-domain response acquires additional polynomial factors: for a simple pole, an input proportional to $\omega''<0$3 produces a response approximately proportional to $\omega''<0$4, while a second-order exceptional point yields a $\omega''<0$5 response under simple complex-frequency excitation (Farhi et al., 4 Jun 2025). The practical implication is that complex-frequency excitation does not merely improve matching; it can expose pole multiplicity and non-Hermitian temporal structure directly in time.

5. Implementations and representative domains

A useful organizing distinction is between direct complex-frequency driving, synthesized complex-frequency reconstruction from real-frequency data, and operator- or subspace-based projections. Direct driving appears in passive complementary media, acoustics, coupled resonators, and mechanical resonators; synthesized reconstruction appears in polaritons, molecular sensing, and arbitrary-waveform capture; operator formulations appear in chirped complex-frequency probing and in generalized virtual-wave transforms (Long et al., 13 Oct 2025, Krasnok et al., 12 Mar 2026, Zhu et al., 7 Jun 2026).

Domain	Excitation form	Reported effect
Passive complementary media	Decaying complex-frequency illumination at $\omega''<0$6	Optical antimatter, ideal perfect lensing, superscattering (Long et al., 13 Oct 2025)
Lossless cavity chains	Synthesis of scattering-zero complex frequencies	Capture efficiencies above $\omega''<0$7 for all tested waveforms at $\omega''<0$8 (Tian et al., 19 Dec 2025)
Coupled resonators	Zero-matched exponentially growing waveform	$\omega''<0$9 single-resonator loading; $\omega''>0$0–$\omega''>0$1 in a three-resonator system (Trivedi et al., 2 Jun 2025)
Polaritons	Synthetic reconstruction from multiple real frequencies	Propagation length increase from $\omega''>0$2 to $\omega''>0$3 (Guan et al., 2023)
Molecular sensing	Synthesized complex-frequency spectra from FTIR data	At least order-of-magnitude sensitivity increase; $\omega''>0$4 for a $\omega''>0$5 nm silk layer (Zeng et al., 2023)
Mechanical resonators	$\omega''>0$6	$\omega''>0$7-fold increase in effective quality factor (Li et al., 15 Jul 2025)
Acoustic ducts and resonators	Growing or decaying exponential envelopes	Quasi-total absorption and propagation-loss compensation (Maddi et al., 14 Mar 2025)

In mechanical systems, the core transformation is explicit. For

$\omega''>0$8

the substitution $\omega''>0$9 yields

$e^{j\omega t}$0

so choosing $e^{j\omega t}$1 cancels the transformed damping and makes the amplitude-frequency response approach that of an undamped resonator (Li et al., 15 Jul 2025). Experiments on acrylic cantilever beams showed a $e^{j\omega t}$2-fold increase in effective quality factor under this strategy (Li et al., 15 Jul 2025).

In sensing, complex-frequency synthesis is implemented post-measurement. For graphene-based SEIRA, the thin-film response is reconstructed from real-frequency transmission data via Kramers–Kronig phase retrieval and discrete complex-frequency synthesis, which compensates molecular damping and sharpens weak PIT or Fano features. The method was demonstrated on DON molecules, silk proteins, and BSA in water, and was explicitly said to be universally applicable to molecular sensing in different phases (Zeng et al., 2023). In inverse thermal problems, pulse, lock-in, chirped, coded, and PN excitation schemes are interpreted as sampling strategies over the same generalized analytically continued operator, reinforcing that waveform design and inversion strategy are inseparable in complex-frequency settings (Zhu et al., 7 Jun 2026).

6. Limits, ambiguities, and open questions

Several constraints recur across the literature. First, most constructive results are single-point or single-trajectory statements in the complex plane, not broadband real-axis equalities. The complementary-media theorem guarantees arbitrary target $e^{j\omega t}$3 and $e^{j\omega t}$4 only at one chosen complex operating point and within a local, isotropic Lorentz–Drude effective-medium model (Long et al., 13 Oct 2025). Likewise, the ability to capture arbitrary waveforms without absorption becomes exact only in the continuum limit of sufficiently dense scattering zeros; finite systems capture only the component lying in the span of available complex-frequency modes (Tian et al., 19 Dec 2025).

Second, physical synthesis remains nontrivial. Ideal complex-frequency waveforms often require exponential growth or decay over a finite window, and several papers note finite-time truncation, dynamic-range limits, bandwidth constraints, high energy density near cutoff, and vulnerability to noise as practical restrictions (Trivedi et al., 2024, Krasnok et al., 12 Mar 2026). Trajectory probing with analytic chirps adds another deterministic distortion: finite-duration windowing alters the realized trajectory by an additive term $e^{j\omega t}$5, so only a central time interval can be trusted for comparison with the continued response (Krasnok et al., 12 Mar 2026).

Third, the response object being measured depends on the formal setting. In subsystem non-Hermitian physics, direct complex-frequency excitation (CFE) and complex-frequency synthesis (CFS) probe the exact subsystem complex-frequency Green’s function and generally do not reveal non-Bloch or NHSE behavior, whereas complex-frequency fingerprinting (CFF) probes a distinct double-frequency Green’s function that can reveal NHSE-induced responses (Huang et al., 13 Mar 2025). This is an important corrective to a common misconception: complex-frequency driving is not automatically a universal diagnostic for every complex-plane topology.

Fourth, inverse problems remain fundamentally ill posed. In the generalized virtual wave transform, analytic continuation does not create missing information; it defines a causal, compact Fredholm operator that acts as a nonstationary low-pass filter, so waveform design trades information capacity against conditioning rather than eliminating the intrinsic information loss of diffusion (Zhu et al., 7 Jun 2026). A plausible implication is that many reported gains from complex-frequency waveform excitation should be read as gains in matching, observability, or reconstructibility under structured priors, not as unrestricted reversal of dissipation.

Finally, sign conventions remain a persistent source of ambiguity. Some works use $e^{j\omega t}$6, others $e^{j\omega t}$7, and the sign of the imaginary frequency component therefore flips between “growing” and “decaying” descriptions for the same physical waveform (Long et al., 13 Oct 2025, Maddi et al., 14 Mar 2025). For technically precise comparison, the convention must always be specified.

7. Historical trajectory and conceptual significance

The recent literature shows a shift from viewing complex frequency as a passive modal property to treating it as a user-controlled excitation parameter. Earlier traditions centered on eigenfrequencies, quasinormal modes, or Laplace-domain analysis; the newer works instead emphasize physically synthesized waveforms, scattering-zero matching, and trajectory-based probing in the complex plane (Trivedi et al., 2024, Krasnok et al., 12 Mar 2026). In that sense, complex-frequency waveform excitation converts analytic continuation from a post hoc interpretive tool into a design variable.

Its significance is therefore twofold. On the one hand, it reinterprets loss, radiation leakage, and damping as quantities that can sometimes be balanced by temporal envelope design rather than only by changing material composition or adding gain. On the other hand, it broadens excitation design beyond the real-frequency axis: arbitrary-waveform capture, virtual absorption, loss compensation, trajectory probing, selective mode loading, and effective $e^{j\omega t}$8-factor sharpening all rely on the fact that poles, zeros, constitutive laws, and Green’s functions are intrinsically complex-frequency objects (Tian et al., 19 Dec 2025, Farhi et al., 4 Jun 2025, Li et al., 15 Jul 2025). The resulting field is still heterogeneous in terminology and implementation, but its unifying principle is now clear: complex-frequency waveform excitation uses temporally structured waveforms to operationally access and exploit the analytic continuation of wave systems.