Eigenpulses in Time-Varying Media

• Eigenpulses are incident waveforms that retain their spectral envelope during scattering by time-varying media, defined as eigenfunctions of scattering operators or reflection matrices.
• They are analyzed using operator theory, exact moving-impedance solutions, and experimental reflection matrix measurements, revealing methods for broadband, distortion-free pulse transmission.
• Experimental studies demonstrate that eigenpulses enable optimized spectral focusing, minimal reflection, and advanced modal control for applications in secure communications and dynamic filtering.

Eigenpulses are incident waveforms whose spectra are preserved under scattering by a time-varying medium, up to multiplication by a scalar reflection or transmission factor. In the literature on dispersive and ultrafast time-varying media, the term denotes eigenfunctions of scattering operators or of spectrally resolved reflection matrices, so that the output retains the input spectral shape even though the medium exchanges energy with the wave (Horsley et al., 2022, Hooper et al., 18 Aug 2025). In exact treatments of moving impedance profiles, the same term is used for specially shaped broadband pulses that enter a finite modulated section without reflection, providing a pulse-level analogue of a transmission resonance (Li et al., 2024). Across these settings, eigenpulses generalize the role of monochromatic modes in static media to temporally modulated systems.

1. Definition and core eigenvalue problems

For a single-mode line terminated by a time-varying impedance, the input–output relation can be written in the time domain as

$$\bar v(t)=\int_{-\infty}^{t} {\rm d}t'\;\bar R\bigl(t,t-t'\bigr)\,\bar u(t').$$

After Fourier transformation, the reflected spectrum obeys

$$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$

For periodic time modulation with period $T$, this becomes the discrete matrix equation

$v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$

where the sampled frequencies are separated by $2\pi/T$. The eigenpulses are then defined by the spectral integral equation

$$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$

or, in the discrete basis,

$\sum_n R_{mn}\,a_n=\lambda\,a_m.$

When $R(\omega,\omega')$ is square-integrable, the kernel is Hilbert–Schmidt, so one obtains a countable set of eigenvalues $\lambda_k$ and orthonormal eigenfunctions $a_k(\omega)$; after discretization over $$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$0 frequency points, this reduces to a standard $$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$1 complex matrix eigenproblem (Hooper et al., 18 Aug 2025).

The associated time-domain waveform is

$$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$2

By construction, injecting $$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$3 yields

$$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$4

so the reflection is spectrally distortion-free: the pulse shape, including its relative spectral phases, is unchanged apart from an overall scale factor (Hooper et al., 18 Aug 2025).

A parallel formulation appears in operator-based scattering theory for dispersive time-varying media. There, a reflection eigenpulse satisfies

$$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$5

which is equivalent to

$$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$6

with normalization

$$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$7

Any such $$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$8 is an eigenpulse because its reflected spectrum is $$v(\omega)=\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\,R(\omega,\omega')\,u(\omega').$$9; the same eigenproblem also determines transmission eigenpulses, with transmitted amplitude $T$0 (Horsley et al., 2022).

2. Operator, matrix, and exact-solution frameworks

The three main formulations appearing in the current literature are complementary rather than competing. One is operator-theoretic, one is exact for a specific class of space–time-varying impedances, and one is directly experimental through measured reflection matrices.

Framework	Setting	Eigenpulse characterization
Operator scattering	Dispersive time-varying media	Eigenvectors of $T$1 or $T$2
Exact moving-impedance solution	Constant $T$3, impedance $T$4	Periodic condition $T$5
Measured reflection matrix	Ultrafast periodically modulated termination	Eigenvectors of $T$6

In the operator formulation, the medium response is described by a two-time conductivity kernel,

$T$7

which becomes an operator-valued conductivity $T$8 in the frequency domain. Maxwell’s equations reduce to an operator Helmholtz equation, and continuity at an interface yields operator-valued Fresnel relations such as

$T$9

The transverse-magnetic case has the same form with $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$0 in place of $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$1. For a finite slab, one similarly obtains explicit operator expressions for $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$2 and $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$3, after which diagonalization or singular-value decomposition identifies reflection eigenpulses and maximal-transmission pulses (Horsley et al., 2022).

In the exact moving-impedance treatment, the refractive index is constant while the impedance depends on the co-moving coordinate $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$4. Introducing

$v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$5

reduces Maxwell’s equations to a form admitting closed-form solutions in terms of arbitrary functions $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$6 and $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$7. For a finite slab $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$8, matching to vacuum at both interfaces completely fixes the reflected and transmitted waves in terms of the incident pulse (Li et al., 2024).

In the experimental matrix formulation, the medium is characterized by a measured spectrally resolved reflection matrix $v_m=\sum_n R_{mn}u_n,\qquad v=Ru,$9, and eigenpulses are obtained numerically from the matrix eigenproblem $2\pi/T$0 using standard linear-algebra routines such as MATLAB eig or Python numpy.linalg.eig (Hooper et al., 18 Aug 2025).

3. Broadband reflectionless pulses and the moving-impedance picture

For a finite section of a medium with constant refractive index and impedance profile $2\pi/T$1, Li and Horsley identified exact broadband reflectionless eigenpulses. The key condition is that the auxiliary function $2\pi/T$2 be periodic with period $2\pi/T$3,

$2\pi/T$4

Under this condition, the net reflection from the two interfaces cancels exactly, yielding eigenpulses that enter without reflection. The paper characterizes this as an analogue of a transmission resonance, but at the level of pulse shaping rather than discrete monochromatic frequencies (Li et al., 2024).

This pulse-level resonance generalizes the familiar Fabry–Pérot condition. In a uniform slab, zero reflection occurs at discrete frequencies satisfying $2\pi/T$5. In the moving-impedance problem, the analogous condition is not a single frequency constraint but the broadband pulse-shaping condition $2\pi/T$6, which makes the field on the front face at time $2\pi/T$7 identical to that on the back face at the delayed time $2\pi/T$8. The two partial reflections therefore cancel for all time (Li et al., 2024).

Any periodic profile $2\pi/T$9 with period $$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$0 admits the Fourier series

$$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$1

so the incident eigenpulse acquires a comb-like spectrum with lines at

$$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$2

weighted by the Fourier transform of the impedance-envelope prefactor. By contrast, a non-eigen incident pulse excites both co- and counter-propagating waves, and the rapidly varying impedance mixes frequencies in the reflected field. For weak, rapid modulations of the form

$$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$3

the leading reflected amplitude carries a factor $$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$4, corresponding to an up-shift of the spectrum by $$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$5 (Li et al., 2024).

A common misconception is to equate eigenpulses with ordinary continuous waves. The moving-impedance results make clear that broadband pulse trains can be the exact reflectionless solutions, and that their defining property is a spatiotemporal matching condition rather than monochromaticity (Li et al., 2024).

4. Experimental characterization in ultrafast time-varying media

An experimental realization of eigenpulses in an ultrafast time-varying medium was reported using a ring resonator on a PCB with two varactor diodes in opposite gaps, inductively coupled to a coaxial transmission line via a small loop antenna. Reverse-biasing the diodes sets a static resonance, while fast modulation of that bias produces a response function $$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$6 with strong coupling between many frequencies (Hooper et al., 18 Aug 2025).

Two classes of control-voltage modulation were used: a sinusoidal modulation at $$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$7, and a “random” modulation formed as a sum of 100 tones from $$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$8 with random phases and amplitudes. All modulations were periodic with $$\int R(\omega,\omega')\,a(\omega')\,d\omega'=\lambda\,a(\omega),$$9, giving a frequency grid $\sum_n R_{mn}\,a_n=\lambda\,a_m.$0 (Hooper et al., 18 Aug 2025).

The reflection matrix was measured sequentially. Continuous-wave tones were injected at $\sum_n R_{mn}\,a_n=\lambda\,a_m.$1 in $\sum_n R_{mn}\,a_n=\lambda\,a_m.$2 steps using an AWG. For each injected tone, the time trace of the reflected signal was recorded over one period $\sum_n R_{mn}\,a_n=\lambda\,a_m.$3 using a fast oscilloscope. The resulting matrix $\sum_n R_{mn}\,a_n=\lambda\,a_m.$4 was corrected by subtracting a $\sum_n R_{mn}\,a_n=\lambda\,a_m.$5-phase-shifted measurement to remove spurious bias-modulation leakage, and a Fourier transform along the time axis produced the spectrally resolved matrix $\sum_n R_{mn}\,a_n=\lambda\,a_m.$6 (Hooper et al., 18 Aug 2025).

Under the $\sum_n R_{mn}\,a_n=\lambda\,a_m.$7 sinusoidal modulation, the measured $\sum_n R_{mn}\,a_n=\lambda\,a_m.$8 displayed a near-diagonal “ladder” of sidebands corresponding to first-, second-, and higher-order scattering. The eigenvalues formed a quasi-continuous band, and each eigenvector $\sum_n R_{mn}\,a_n=\lambda\,a_m.$9 spanned several sidebands. When three representative eigenpulses were injected, the measured reflected spectra perfectly overlapped $R(\omega,\omega')$0, and the time-domain waveforms $R(\omega,\omega')$1 and $R(\omega,\omega')$2 were identical up to scale. Under the “random” modulation, $R(\omega,\omega')$3 became a fully dense matrix, yet two sample eigenpulses still reflected without spectral distortion (Hooper et al., 18 Aug 2025).

These measurements establish that the eigenpulse concept does not depend on a sparse sideband structure. A plausible implication is that the relevant object is the full frequency-coupling operator itself, whether it is near-diagonal or fully dense.

5. Power extremization, absorption, and spectral focusing

The measured reflection matrix also supports optimization problems beyond distortion-free reflection. To find pulses that extremize the total reflected power, one forms $R(\omega,\omega')$4 and diagonalizes it. Its right singular vectors are the relevant incident waveforms, with the top singular vector maximizing total reflected power and the bottom singular vector minimizing it. In the reported experiment, the minimally reflected pulse yielded broadband absorption greater than $R(\omega,\omega')$5 (Hooper et al., 18 Aug 2025).

Selective spectral concentration is obtained by partitioning the reflected spectrum into masked bands $R(\omega,\omega')$6 and $R(\omega,\omega')$7, defining submatrices $R(\omega,\omega')$8 and $R(\omega,\omega')$9, and constructing the contrast operator

$\lambda_k$0

The top eigenvector of $\lambda_k$1 injects a pulse whose reflected energy is concentrated into the user-defined bands $\lambda_k$2; swapping $\lambda_k$3 and $\lambda_k$4 concentrates the reflection into $\lambda_k$5 instead. The reported contrast exceeded $\lambda_k$6 (Hooper et al., 18 Aug 2025).

Within the terminology of the paper, these optimized inputs are frequency eigenchannels of the ultrafast time-varying medium. The eigenpulse problem identifies waveforms that are invariant in spectral shape, while the singular-vector and contrast-operator problems identify waveforms that extremize global reflected power or redistribute that power into selected spectral windows. This suggests that eigenpulse analysis is part of a broader linear-algebraic control framework for dynamic scattering (Hooper et al., 18 Aug 2025).

6. Physical interpretation, bound states, and prospective applications

The physical intuition offered across the literature is that eigenpulses are “temporal wavefronts.” In a static medium, the natural distortion-free modes are monochromatic continuous waves. In a time-varying medium, by contrast, the incident pulse spectrum must be pre-shaped so that temporal modulation and dispersion cancel. This is explicitly compared to spatial wavefront shaping and eigenchannels in multimode fibres (Hooper et al., 18 Aug 2025).

The corresponding eigenvalues encode qualitatively distinct regimes. In the experimental reflection-matrix picture, $\lambda_k$7 when there is loss, interpreted as absorption into unguided channels. The same framework also identifies waveforms for which $\lambda_k$8, associated with parametric amplification, and waveforms with $\lambda_k$9, associated with broadband absorption (Hooper et al., 18 Aug 2025). In the operator theory, reflection and transmission amplitudes are set by the impedance eigenvalue through $a_k(\omega)$0 and $a_k(\omega)$1, so eigenpulse behavior is directly linked to the spectral properties of $a_k(\omega)$2 (Horsley et al., 2022).

Operator poles introduce a distinct but related phenomenon. In dispersive time-varying media, the poles of the reflection or transmission operators represent non-time-harmonic bound states. Since

$a_k(\omega)$3

poles occur when

$a_k(\omega)$4

equivalently when there exists a nontrivial solution of

$a_k(\omega)$5

These $a_k(\omega)$6 are described as surface-plasmon-like eigenpulses confined to the interface and decaying evanescently away from it (Horsley et al., 2022).

Potential applications listed in the experimental work include distortion-free communications through dynamic channels by encoding in the eigenbasis, dynamic spectral filtering and focusing, temporal dispersion compensation in optics and microwaves, secure key-exchange keyed by knowledge of the time-varying reflection matrix, and inverse design of time-varying media to realize a target set of eigenpulses (Hooper et al., 18 Aug 2025). In the moving-impedance setting, the strong asymmetry between co-propagating and counter-propagating waves further suggests one-way, spectrum-reshaping mirrors, while the exact zero-reflection solutions provide a broadband analogue of resonance-based impedance matching (Li et al., 2024).

Taken together, these results place eigenpulses at the intersection of operator scattering theory, exact space–time electromagnetics, and experimentally measured frequency-eigenchannel control. They are not merely pulses that survive propagation unchanged in a trivial sense; rather, they are the specific spectral states selected by a time-varying scattering operator, a moving-impedance boundary-value problem, or a measured ultrafast reflection matrix, and they therefore provide a natural modal language for dynamic wave–matter interactions.