Complex Frequency Excitations

• Complex Frequency Excitations are waveforms with a complex-valued frequency that combine oscillatory phase evolution with exponential growth or decay.
• They are realized through analytic waveforms and exponentially weighted sinusoids, directly probing off-axis poles and zeros in resonant and scattering systems.
• Applications span resonator control, mechanical sensing, and photonics, offering enhanced parameter estimation, effective-Q improvement, and dynamic critical coupling.

Searching arXiv for papers on complex frequency excitations and related methods. Complex frequency excitations are drives or probes whose frequency parameter is extended from the real axis into the complex plane, so that oscillation and exponential growth or decay are controlled simultaneously. Across the literature, this notion appears in several closely related forms: as analytic waveforms with instantaneous complex frequency $\tilde{\omega}(t)=\omega_r(t)+i\omega_i(t)$ for contour probing of continued scattering responses (Krasnok et al., 12 Mar 2026); as exponentially weighted sinusoids $e^{(\alpha+i\omega)t}$ or $e^{- \alpha t}\cos(\omega t)$ for resonant control, sensing, and effective-$Q$ enhancement (Li et al., 15 Jul 2025, Li et al., 13 May 2026); and as a geometric or system-theoretic generalization of eigenvalues, in which the real part encodes amplitude variation and the imaginary part encodes rotation (Sofos et al., 20 Mar 2026). In all of these settings, complex frequency excitations provide access to off-axis analytic structure—poles, zeros, modal decay rates, and phase winding—that conventional real-frequency probing samples only indirectly through model fitting (Krasnok et al., 12 Mar 2026, Trivedi et al., 2024).

1. Definitions and mathematical representations

Complex frequency excitation most directly denotes a waveform whose temporal dependence is governed by a complex-valued frequency. In the contour-probing formulation, the incident analytic signal at a calibrated reference plane is written

$$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$

with instantaneous complex angular frequency

$$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$

Here $\omega_r(t)$ sets the instantaneous phase slope, whereas $\omega_i(t)$ sets the local exponential growth or decay rate of the analytic envelope (Krasnok et al., 12 Mar 2026).

A prescribed trajectory $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$ yields the ideal core waveform

$$s_{\mathrm{core}}(t)=s_0\exp\!\left(\int_0^t\omega_i(\tau)\,d\tau\right)\exp\!\left(-i\int_0^t\omega_r(\tau)\,d\tau\right),$$

which is then windowed to enforce finite energy:

$e^{(\alpha+i\omega)t}$0

The realized trajectory is deterministically deformed by the window according to

$e^{(\alpha+i\omega)t}$1

so only a central interval where the window slope is small can be used for faithful comparison between desired and realized complex-frequency paths (Krasnok et al., 12 Mar 2026).

In a more classical modal form, complex frequency excitations are sums of damped or growing exponentials,

$e^{(\alpha+i\omega)t}$2

or, in continuous time,

$e^{(\alpha+i\omega)t}$3

In this representation, $e^{(\alpha+i\omega)t}$4 is the exponential damping or growth rate and $e^{(\alpha+i\omega)t}$5 is the angular frequency (Andersson et al., 2011).

A related geometric representation defines, for a complex signal $e^{(\alpha+i\omega)t}$6, the instantaneous complex frequency

$e^{(\alpha+i\omega)t}$7

so that $e^{(\alpha+i\omega)t}$8 is the instantaneous logarithmic amplitude rate and $e^{(\alpha+i\omega)t}$9 is the instantaneous angular frequency (Sofos et al., 20 Mar 2026). This system-theoretic perspective connects complex frequency directly to eigenvalue structure in linear time-invariant systems.

2. Relation to poles, zeros, and continued responses

The principal motivation for complex frequency excitations is that poles and zeros governing resonant scattering lie away from the real-frequency axis. Standard measurements sample only the real axis and infer off-axis structure from fitted models, with the greatest uncertainty near singularities (Krasnok et al., 12 Mar 2026). Complex-frequency chirps instead probe the analytically continued response $e^{- \alpha t}\cos(\omega t)$0 directly along prescribed open or closed contours in the complex plane (Krasnok et al., 12 Mar 2026).

For a stable LTI device with transfer function

$e^{- \alpha t}\cos(\omega t)$1

the output approximately obeys

$e^{- \alpha t}\cos(\omega t)$2

when the envelope and $e^{- \alpha t}\cos(\omega t)$3 vary slowly relative to internal relaxation (Krasnok et al., 12 Mar 2026). This gives a direct operational meaning to complex-frequency excitation: the device is interrogated at the continued response evaluated at a non-real argument.

In one-port temporal coupled-mode theory, the reflected response of a resonator is

$e^{- \alpha t}\cos(\omega t)$4

with pole

$e^{- \alpha t}\cos(\omega t)$5

and zero

$e^{- \alpha t}\cos(\omega t)$6

Complex-frequency trajectories can be chosen to encircle the zero while avoiding the pole, permitting direct validation of the continued response and argument-principle checks based on phase winding (Krasnok et al., 12 Mar 2026).

A complementary resonator-control literature emphasizes excitation at complex reflection zeros. For a single resonator with reflection coefficient extended to the Laplace variable $e^{- \alpha t}\cos(\omega t)$7,

$e^{- \alpha t}\cos(\omega t)$8

the reflection zero is

$e^{- \alpha t}\cos(\omega t)$9

Driving with

$Q$0

dynamically cancels reflections and realizes dynamic critical coupling without hardware reconfiguration (Trivedi et al., 2 Jun 2025). This is a zero-matching formulation of complex-frequency excitation rather than a contour-probing formulation, but both rely on analytic continuation away from the real axis.

The same off-axis logic appears in scattering theory for “invisible” poles: poles with large imaginary parts may contribute weakly on the real axis yet become directly excitable by complex-frequency drives whose growth or decay matches their location in the complex plane (Trivedi et al., 2024). A different line of work shows that complex-frequency excitation and synthesis in frequency-dependent non-Hermitian subsystems measure the exact complex-frequency Green’s function, whereas a “complex-frequency fingerprint” using a double-frequency Green’s function can expose non-Hermitian skin-effect signatures that complex-frequency excitation alone does not reveal (Huang et al., 13 Mar 2025).

3. Geometric and system-theoretic interpretation

A system-theoretic account identifies complex frequency as a generalization of eigenvalues for diagonalizable LTI systems (Sofos et al., 20 Mar 2026). For

$Q$1

there exists a real, invertible, generally non-isometric transformation $Q$2 such that in transformed coordinates $Q$3 the system decomposes into $Q$4 real blocks and $Q$5 real blocks

$Q$6

The complex frequency computed on each decoupled $Q$7D block equals the corresponding eigenvalue $Q$8 (Sofos et al., 20 Mar 2026).

This interpretation separates symmetric and skew-symmetric local dynamics. If

$Q$9

then

$$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$0

The symmetric part governs amplitude scaling, while the skew-symmetric part generates rotation (Sofos et al., 20 Mar 2026). In two dimensions, the instantaneous complex frequency

$$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$1

therefore combines radial and angular dynamics into a single quantity.

This geometric formulation extends beyond LTI systems. For nonlinear flows $$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$2, the equivalence “complex frequency = eigenvalues” generally fails because a global time-invariant decoupling transformation does not exist, but the geometric frequency remains well defined as an instantaneous descriptor of normalized radial and rotational rates (Sofos et al., 20 Mar 2026). This suggests that “complex frequency excitation” can be interpreted either as an externally imposed waveform design or as an intrinsic description of modal or trajectory dynamics, depending on context.

An analogous conceptual expansion appears in power systems, where the complex frequency of a bus voltage

$$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$3

is defined as

$$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$4

so that amplitude dynamics and frequency deviation are treated on equal footing (Ponce et al., 2023, Milano, 2021). Although this usage concerns state evolution rather than externally engineered waveforms, it preserves the same decomposition of exponential-amplitude variation and oscillatory phase evolution.

4. Excitation design, synthesis, and measurement workflows

The contour-probing framework gives an explicit synthesis rule suitable for arbitrary waveform generation and I/Q modulation (Krasnok et al., 12 Mar 2026). For a carrier $$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$5 and detuning $$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$6, the baseband signal is

$$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$7

with DAC outputs

$$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$8

where

$$s_{\mathrm{in}}(t)=A(t)e^{-i\phi(t)},\qquad A(t)>0,$$9

This enables standard AWG–I/Q-modulator–coherent-receiver implementations (Krasnok et al., 12 Mar 2026).

Windowing is not incidental but structurally important. Finite-duration excitation deforms the realized trajectory through the term $$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$0, so analysis is restricted to a plateau interval satisfying

$$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$1

For closed contours, the condition

$$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$2

returns the unwindowed envelope to its initial level after one traversal (Krasnok et al., 12 Mar 2026).

Response extraction is performed with a time-local least-squares estimator

$$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$3

reported only when the denominator exceeds a noise threshold (Krasnok et al., 12 Mar 2026). Under quasi-steady tracking, $$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$4 follows the continued response $$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$5.

The same paper provides a nine-step design workflow: prescribe a contour, choose carrier and traversal speed, construct $$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$6 and $$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$7, choose a flat-plateau window, generate I/Q, monitor input and output coherently, estimate the realized $$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$8 from short-window fits to phase and log-amplitude, compute $$\tilde{\omega}(t)\equiv i\frac{d}{dt}\ln s_{\mathrm{in}}(t)=\dot{\phi}(t)+i\frac{d}{dt}\ln A(t)\equiv \omega_r(t)+i\omega_i(t).$$9, and iterate speed, contour placement, and pre-distortion until trajectory and response agree over the analysis interval (Krasnok et al., 12 Mar 2026).

In resonator-zero matching, the design procedure is more direct: determine or fit the reflection transfer function $\omega_r(t)$0, compute the relevant zero $\omega_r(t)$1, synthesize

$\omega_r(t)$2

and verify near-zero reflection and stored-energy efficiency (Trivedi et al., 2 Jun 2025).

Post-detection synthesis constitutes a third workflow. In systems where direct real-time complex-frequency generation is difficult, one may reconstruct the complex-frequency response numerically from real-frequency data. However, recent analysis shows that this synthesized CF response is noise-limited and, in realistic conditions, often inferior to physical real-time complex-frequency excitation; in low-noise regimes, simpler post-detection filtering may perform as well as or better than synthesized CF reconstruction (Khurgin et al., 2 Jul 2026).

5. Applications across resonant, mechanical, and wave systems

Complex frequency excitations have been used for at least four distinct operational goals: direct off-axis characterization, selective resonator control, effective-$\omega_r(t)$3 enhancement, and improved parameter estimation.

In coupled resonator systems, driving at a complex reflection zero can achieve near-unity selected energy storage efficiency in a single resonator and substantial crosstalk suppression in multi-resonator networks (Trivedi et al., 2 Jun 2025). The reported single-resonator theory gives

$\omega_r(t)$4

with practical excitation approaching unity within approximately $\omega_r(t)$5 for the example parameters, compared with optimized Gaussian pulses peaking near $\omega_r(t)$6 under equal-duration constraints (Trivedi et al., 2 Jun 2025). In a three-resonator system, complex-frequency pulses yielded target efficiencies of approximately $\omega_r(t)$7–$\omega_r(t)$8 with near-zero reflection and much higher selectivity than Gaussian pulses of the same duration (Trivedi et al., 2 Jun 2025).

In mechanical resonators, exponentially decaying harmonic drives

$\omega_r(t)$9

or equivalently $\omega_i(t)$0 define a complex-frequency excitation (Li et al., 15 Jul 2025, Li et al., 13 May 2026). Under the transform

$\omega_i(t)$1

the physical response becomes a steady-state harmonic response in transformed coordinates with effective damping

$\omega_i(t)$2

and effective stiffness

$\omega_i(t)$3

Choosing

$\omega_i(t)$4

cancels damping in the transformed coordinate, so the effective response approaches that of an undamped resonator (Li et al., 15 Jul 2025). Experiments with acrylic cantilever beams reported a $\omega_i(t)$5-fold increase in effective quality factor under such excitation (Li et al., 15 Jul 2025).

The same mechanical setting also supports a Fisher-information analysis for natural-frequency estimation. In a single-degree-of-freedom underdamped system, applying

$\omega_i(t)$6

and mapping the measured displacement via $\omega_i(t)$7 shifts the poles of the transfer function to sharpen the mapped response when $\omega_i(t)$8 (Li et al., 13 May 2026). Monte Carlo and experimental results showed markedly improved estimation accuracy under strong noise, including a reduction in absolute frequency-estimation error from $\omega_i(t)$9 to $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$0 and variance from $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$1 to $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$2 at $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$3 in a cantilever-beam experiment (Li et al., 13 May 2026).

In photonics and scattering, complex-frequency excitation has been used to reveal sharp spectral features hidden by loss (Khurgin et al., 2 Jul 2026), to exceed conventional real-frequency scattering bounds over finite windows (Kim et al., 2022), to emulate anisotropic gain in passive structures via “anisotropic virtual gain” (Zouros et al., 2023), to reveal poles with predominantly imaginary components (Trivedi et al., 2024), and to improve flat-superlens resolution by probing transfer functions at complex frequencies selected to maximize evanescent transmission (Lalanne et al., 14 Aug 2025). A common mechanism in these works is partial compensation of intrinsic damping by the excitation envelope.

A further extension maps diffusive systems to virtual wave fields by analytic continuation in the complex-frequency plane, with pulse, lock-in, chirped, and coded excitations interpreted as different projections of a single operator-valued transformation (Zhu et al., 7 Jun 2026). This suggests that “complex frequency excitations” can also be understood as a design language for spectral sampling, not merely as a class of exponentially weighted sinusoids.

6. Limits, controversies, and methodological trade-offs

Several limitations recur across the literature. The contour-probing framework presumes linear, time-invariant behavior over the burst, with errors growing at higher traversal speeds and near resonant poles (Krasnok et al., 12 Mar 2026). An adiabatic estimate gives

$$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$4

where

$$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$5

This makes explicit that proximity to poles and fast contour traversal degrade the quasi-steady mapping to $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$6 (Krasnok et al., 12 Mar 2026).

Windowing creates a deterministic trajectory bias rather than random error, so failure to verify the realized input trajectory can invalidate interpretation (Krasnok et al., 12 Mar 2026). Estimation of instantaneous complex frequency is unstable near amplitude nulls, and sufficient dynamic range is required because the extrema of $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$7 determine the necessary amplitude excursion (Krasnok et al., 12 Mar 2026).

In physical real-time complex-frequency excitation, waveform generation can be technologically challenging when the required decay times are very short. Optical implementations may demand decay times in the range of tens of femtoseconds, making real-time synthesis difficult (Khurgin et al., 2 Jul 2026). Post-detection synthetic CF methods avoid this generation challenge but are often constrained by shot noise, relative intensity noise, detector nonuniformities, and the ill-conditioning of exponential time-domain reweighting (Khurgin et al., 2 Jul 2026).

A significant methodological controversy concerns what complex-frequency measurements actually reveal in non-Hermitian or open systems. In frequency-dependent non-Hermitian subsystems, direct complex-frequency excitation and its real-frequency synthesis measure the exact continued Green’s function and are therefore incompatible with constant non-Hermitian approximations used to diagnose the non-Hermitian skin effect; only the double-frequency “complex-frequency fingerprint” recovers the frozen-frequency non-Hermitian topology relevant to that phenomenon (Huang et al., 13 Mar 2025). This distinguishes exact analytic continuation from approximate effective-Hamiltonian diagnostics.

There is also a conceptual tension between physical real-time excitation and post hoc synthesis. Recent work argues that physical CF excitation robustly sharpens spectral features in noisy settings because the detector collects charge predominantly from the late-time, narrow response, whereas numerical synthesis after detection merely reweights already-acquired noisy data (Khurgin et al., 2 Jul 2026). A plausible implication is that complex-frequency excitation is not just a different representation of real-frequency information once measurement noise and detector physics are included.

Finally, some interpretations rely on “virtual gain,” where a passive medium under complex-frequency illumination behaves as if it were active (Zouros et al., 2023, Lalanne et al., 14 Aug 2025). These studies do not claim literal gain media but rather an exact or approximate equivalence at the level of analytically continued constitutive response or transfer function. This suggests that complex-frequency excitation can operationally emulate active behavior without changing material parameters, but only within finite-time, causal, and stability-constrained protocols.

7. Broader significance and open directions

Complex frequency excitations have developed into a unifying concept spanning waveform engineering, modal analysis, analytic continuation, and non-Hermitian response theory. One line of development emphasizes direct measurement along contours in the complex plane to characterize resonant scatterers without relying exclusively on fitted models (Krasnok et al., 12 Mar 2026). Another uses tailored decaying or growing envelopes to match zeros or poles for selective energy delivery, crosstalk suppression, and dynamic critical coupling (Trivedi et al., 2 Jun 2025). A third uses exponential envelopes to reshape effective damping, thereby sharpening resonance peaks, boosting effective quality factors, and improving estimation accuracy in noisy systems (Li et al., 15 Jul 2025, Li et al., 13 May 2026). A fourth treats complex frequency as a generalized eigenvalue or geometric invariant, linking instantaneous amplitude and rotation in a single quantity (Sofos et al., 20 Mar 2026).

Several directions remain open. Multiport generalizations of contour probing are explicitly proposed through matrix least-squares estimates of $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$8 and invariants such as $$\tilde{\omega}_{\mathrm{des}}(t)=\omega_r(t)+i\omega_i(t)$$9 over closed contours (Krasnok et al., 12 Mar 2026). Closed-loop pre-distortion based on monitored trajectory estimates is also identified as a natural extension (Krasnok et al., 12 Mar 2026). In resonator networks, adaptive complex driving to track drifting or nonlinear zeros is proposed for lossy or Kerr-shifted systems (Trivedi et al., 2 Jun 2025). In mechanical sensing, the single-mode Fisher-information theory points toward multi-parameter extensions for multimode systems and joint estimation of damping and resonance (Li et al., 13 May 2026). In vibro-acoustic simulation, rational approximation of complex-frequency-dependent finite-element operators provides a route from Laplace-domain constitutive models to time-domain auxiliary-state realizations (Deckers et al., 2022).

A broader synthesis is that complex frequency excitations convert hidden analytic structure into measurable time-domain behavior. Depending on the protocol, they can turn off-axis continuation into a directly observed trajectory, make a lossy passive structure emulate a gain medium, convert diffusive evolution into a virtual wave description, or transform a damped resonator into an effectively undamped one over a finite window (Krasnok et al., 12 Mar 2026, Zouros et al., 2023, Zhu et al., 7 Jun 2026, Li et al., 15 Jul 2025). This does not eliminate causality, passivity, or finite-energy constraints; rather, it exploits them through analytic continuation, tailored envelopes, and carefully delimited time windows.