Non-Hermitian Wave Funneling

• Non-Hermitian wave funneling is the directed transport and accumulation of wave energy in systems with complex effective Hamiltonians, leveraging asymmetric couplings and non-Bloch modes to concentrate energy at interfaces.
• The methodology spans discrete lattice models exhibiting the non-Hermitian skin effect and continuum frameworks using complex constitutive parameters to quantify skin depths, drift velocities, and energy localization.
• Practical implications include robust energy concentration for photonic guiding, acoustic noise control, and asymmetric channel conversion, with experimental validations highlighting transitions between NHSE funneling and disorder-induced localization.

Non-Hermitian wave funneling is the directed transport and accumulation of wave energy toward a boundary, interface, corner, or internal mode in a system with a complex effective Hamiltonian or complex constitutive parameters. In lattice realizations, it is often the dynamical manifestation of the non-Hermitian skin effect (NHSE), where bulk states become non-Bloch under open boundary conditions and pile up at selected edges. In scattering and continuum settings, the same term also denotes asymmetric external-to-internal coupling, trapping of radiation in guided modes, or steering by locally engineered complex media. Across current realizations, the underlying mechanisms include asymmetric hopping, stochastic temporal fluctuations, mirrored nonreciprocity, asymmetric in/out coupling, local Hilbert-transform design, complex anisotropic density tensors, and reciprocal lattices with gain/loss gradients that support continuum Landau modes (Weidemann et al., 2020, Longhi, 2020, Grineviciute et al., 2024, Ahmed et al., 2020, Tian et al., 20 Aug 2025, Wang et al., 2022).

1. Foundational mechanisms and boundary sensitivity

In the canonical NHSE setting, funneling arises from asymmetric nearest-neighbor couplings. For the discrete-time light walk

$$-i\, a^{(m+1)}_{n}= J_R\, a^{(m)}_{n-1}+J_L\, a^{(m)}_{n+1},$$

the periodic-boundary quasi-energy is

$$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$

and the group or drift velocity at $k=0,\pi$ is

$v_g=\frac{J_R-J_L}{J_R+J_L}.$

Under open boundary conditions, the Brillouin zone deforms to a generalized Brillouin zone with $h=\frac{1}{2}\log(J_R/J_L)$, the eigenvectors acquire the exponential skew $A_n=\sin(nk_l)e^{hn}$, and the deterministic Hatano–Nelson localization length is $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$ (Longhi, 2020).

This boundary sensitivity is the core structural reason why non-Hermitian wave funneling differs from Hermitian edge localization. In a Hermitian SSH interface, only a small number of modes localize; in an NHSE interface, the entire eigenmode spectrum can collapse toward the interface, so arbitrary excitations are pulled there dynamically (Weidemann et al., 2020). In the photonic mesh realization of an anisotropy-flip domain wall, all eigenmodes were exponentially localized at the interface, and the measured intensity flowed toward the interface for excitations launched left of, at, and right of it (Weidemann et al., 2020).

The same logic persists in continuum media, but the mathematical carrier is a complex wavevector rather than a lattice Bloch factor. In a uniform anisotropic acoustic medium with complex density tensor $\boldsymbol{\rho}$, the bulk dispersion

$$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$

admits $$\mathbf{k}=\mathbf{k}_{\mathrm r}+i\boldsymbol{\kappa}$$, so fields acquire factors $$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$0 and accumulate at boundaries or corners selected by the sign of $$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$1 (Tian et al., 20 Aug 2025). This suggests that the central invariant across discrete and continuum formulations is not the lattice itself, but the conversion of ordinary Bloch transport into non-Bloch transport by complex drift.

2. Deterministic lattice funnels and dynamical NHSE

The experimentally demonstrated photonic-mesh funnel based on the NHSE uses direction-dependent amplitude modulation to realize non-reciprocal couplings $$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$2 in a large-scale time-bin lattice. In the corresponding tight-binding picture,

$$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$3

the non-Bloch ansatz $$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$4 yields the generalized-Brillouin-zone condition $$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$5, and the skin depth is $$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$6. Introducing a domain wall where the anisotropy reverses localizes the left and right tails toward the same interface, producing universal funneling rather than a single topological defect mode (Weidemann et al., 2020).

The observed robustness is specific rather than absolute. In the 120-site photonic-mesh experiments, funneling persisted under weak uniformly distributed phase disorder, but under strong disorder Anderson localization dominated and the wave packet localized near the launch site instead of moving to the interface. The data therefore show a continuous crossover from NHSE funneling to Anderson localization rather than an unconditional immunity to disorder (Weidemann et al., 2020).

A complementary formulation starts from a Hermitian equally spaced waveguide array and applies the non-unitary transformation

$$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$7

which yields the effective non-Hermitian Hamiltonian

$$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$8

The field amplitudes obey $$\mu(k)=-\frac{\pi}{2}+i \log \left\{ J_L\, e^{ik}+J_R\, e^{-ik} \right\},$$9, so the Hermitian transport solution is reweighted by an exponential spatial envelope that biases propagation toward one side of the array (Bocanegra et al., 2023). In the infinite-chain dispersion,

$k=0,\pi$0

the real part sets drift while the imaginary part gives direction-dependent amplification and attenuation, providing a band-theoretic picture of funneling without changing the physical spacing of the guides (Bocanegra et al., 2023).

Recent wave-packet analyses clarify that deterministic NHSE funneling is not always a simple constant-velocity drift. In a discrete Hatano–Nelson chain with asymmetric couplings, the center of mass exhibits initial acceleration, then slowdown, then uniform motion, with the late-time drift controlled by the maximally amplifying mode near $k=0,\pi$1. The same work identifies a non-Hermiticity-induced jump, in which the dominant momentum switches abruptly and causes an abrupt spatial shift of the center of mass (Jana et al., 20 Dec 2025). In pseudo-Hermitian lattices, this has been recast as the coexistence of a non-Hermitian front and a Hermitian front, the latter governing reflected and boundary-mediated reorganizations of the former (Beck et al., 17 Dec 2025). A plausible implication is that practical funnel design must account not only for localization lengths, but also for transient front competition.

3. Fluctuation-induced funnels and mirrored interfaces

A central conceptual extension is that static nonreciprocity is not necessary. In the stochastic light walk with symmetric-on-average hopping,

$k=0,\pi$2

with $k=0,\pi$3, the long-time quasi-energy becomes the Lyapunov exponent

$k=0,\pi$4

Although $k=0,\pi$5, the multiplicative logarithmic average breaks detailed balance and generates a finite drift slope in $k=0,\pi$6 (Longhi, 2020).

For uniform noise $k=0,\pi$7, the drift is

$k=0,\pi$8

which vanishes at $k=0,\pi$9 and grows monotonically with $v_g=\frac{J_R-J_L}{J_R+J_L}.$0. Reversing which bond fluctuates reverses the sign of the drift, so the funneling direction is selected by the fluctuating bond rather than by a static asymmetry (Longhi, 2020). At an interface where the right hopping fluctuates for $v_g=\frac{J_R-J_L}{J_R+J_L}.$1 and the left hopping fluctuates for $v_g=\frac{J_R-J_L}{J_R+J_L}.$2, the drift points toward $v_g=\frac{J_R-J_L}{J_R+J_L}.$3 on both sides, and the wavepacket distance

$v_g=\frac{J_R-J_L}{J_R+J_L}.$4

decreases monotonically, evidencing stochastic funneling (Longhi, 2020). Because the system is explicitly time-dependent, the skin effect here is dynamical pile-up rather than a stationary eigenmode construction.

Mirrored nonreciprocity yields a second distinct interface phenomenon: tunneling-like transmission through a nominal NHSE barrier. In a Hatano–Nelson interface $v_g=\frac{J_R-J_L}{J_R+J_L}.$5, each non-Hermitian half individually supports skin accumulation, so the interface appears dark. Yet the imaginary gauge field cancels across the mirrored pair, and under impedance matching the transmission can be unity while the interface remains weakly populated (Jana et al., 2023). The work reports that the tunneling is independent of the interface length, and a perfect transmission can be achieved independently of frequency and nonreciprocity strength (Jana et al., 2023).

The acoustic realization of this mechanism uses an active waveguide with microphones and loudspeakers embedded in the wall. Starting from a controlled mass–spring chain with asymmetric stiffnesses $v_g=\frac{J_R-J_L}{J_R+J_L}.$6, the continuum equation becomes

$v_g=\frac{J_R-J_L}{J_R+J_L}.$7

which contains a convective-like non-Hermitian drift term. The corresponding continuous decay rate is $v_g=\frac{J_R-J_L}{J_R+J_L}.$8, and finite-element simulations and experiments showed a dark tunnel region with strong SPL reduction inside the interface and recovery beyond it (Langfeldt et al., 23 Jul 2025). This broadens the meaning of funneling from accumulation at a boundary to directed transmission through an NHSE-induced barrier.

4. Asymmetric channel conversion and trapped radiation

In scattering systems, non-Hermitian wave funneling often means asymmetric conversion between external and internal channels rather than boundary skin accumulation. In non-Hermitically modulated thin films, the relevant channels are external radiation near normal incidence and planar guided modes inside the film. The complex permittivity modulation is

$v_g=\frac{J_R-J_L}{J_R+J_L}.$9

or equivalently $h=\frac{1}{2}\log(J_R/J_L)$0, with

$h=\frac{1}{2}\log(J_R/J_L)$1

When $h=\frac{1}{2}\log(J_R/J_L)$2 and the quadratures are $h=\frac{1}{2}\log(J_R/J_L)$3-shifted, the modulation is proportional to $h=\frac{1}{2}\log(J_R/J_L)$4, so $h=\frac{1}{2}\log(J_R/J_L)$5 and only one spatial harmonic remains (Grineviciute et al., 2024).

In the temporal coupled-mode theory used there,

$h=\frac{1}{2}\log(J_R/J_L)$6

the coupling coefficients satisfy $h=\frac{1}{2}\log(J_R/J_L)$7 and $h=\frac{1}{2}\log(J_R/J_L)$8. The single-sideband condition therefore realizes $h=\frac{1}{2}\log(J_R/J_L)$9 while $A_n=\sin(nk_l)e^{hn}$0, meaning efficient in-coupling with suppressed outcoupling (Grineviciute et al., 2024). The resulting absorption asymmetry was validated numerically by RCWA and FDTD, and experimentally by spectrophotometry and photothermal measurements on TiO$A_n=\sin(nk_l)e^{hn}$1 thin films with a Ni loss modulation positioned approximately $A_n=\sin(nk_l)e^{hn}$2 shifted relative to the corrugation maxima (Grineviciute et al., 2024).

A common misconception is that this effect violates reciprocity. The thin-film structure contains no magnetic bias or time modulation; it is reciprocal. The asymmetry lies in conversion efficiency and absorption, not in propagation directionality for the same source–detector configuration. Reflection remains nearly symmetric, while transmission and absorption are asymmetric because the internal guided-mode amplitude depends non-commutatively on $A_n=\sin(nk_l)e^{hn}$3 and $A_n=\sin(nk_l)e^{hn}$4 (Grineviciute et al., 2024). The same paper explicitly distinguishes the mechanism from coherent perfect absorption, which requires coherent excitation from multiple ports, and from a bound state in the continuum, because the structure remains leaky and absorptive (Grineviciute et al., 2024).

An earlier minimal scattering realization due to Li, Zhang, Zhang, and Song uses a non-Hermitian triangular ring threaded by Aharonov–Bohm flux. At

$A_n=\sin(nk_l)e^{hn}$5

the scattering amplitudes satisfy

$A_n=\sin(nk_l)e^{hn}$6

so left incidence is perfectly absorbed while right incidence is fully transmitted (Li et al., 2014). This is one-way funneling through a spectral singularity, and it provides an analytically exact precursor of later asymmetric-coupling devices.

5. Continuum design paradigms beyond static asymmetric hopping

Several recent frameworks realize funneling without relying on static asymmetric hoppings. One route uses a local Hilbert-transform relation between the real and imaginary parts of the refractive index,

$A_n=\sin(nk_l)e^{hn}$7

to design locally PT-symmetric landscapes that steer waves into a sink. For a target inward directionality field $A_n=\sin(nk_l)e^{hn}$8, the restricted/local Hilbert transform computes a companion quadrature and then iteratively projects the complex index back to a realizable set $A_n=\sin(nk_l)e^{hn}$9 in the complex plane. The resulting fields, simulated with

$\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$0

concentrate around the center for unrestricted, square-restricted, ring-restricted, and discrete restricted designs (Ahmed et al., 2020). Quantitatively, convergence typically occurs in $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$1 iterations, with $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$2 and $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$3 agreeing to within numerical precision after $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$4 iterations in the square case, differing by $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$5 in the ring case and by $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$6 in the discrete case (Ahmed et al., 2020).

A second route realizes NHSE directly in a homogeneous continuum. In the anisotropic acoustic metamaterial with complex density tensor $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$7, rotated principal densities generate off-diagonal inverse-density components and effective gauge terms

$\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$8

For a strip bounded along $\xi^{-1}=|h|=\frac{1}{2}|\log(J_R/J_L)|$9, the envelope

$\boldsymbol{\rho}$0

localizes on the left or right boundary according to the sign of $\boldsymbol{\rho}$1 (Tian et al., 20 Aug 2025). With hard boundaries on both axes, simultaneous signs of $\boldsymbol{\rho}$2 and $\boldsymbol{\rho}$3 generate second-order NHSE and corner funneling. The reported metamaterial used effective parameters $\boldsymbol{\rho}$4 kg/m$\boldsymbol{\rho}$5, $\boldsymbol{\rho}$6 kg/m$\boldsymbol{\rho}$7, and $\boldsymbol{\rho}$8 kPa, and experiments showed boundary and corner funneling over 6–10 kHz with broadband and wide-angle characteristics (Tian et al., 20 Aug 2025).

A third route dispenses with NHSE altogether. In the reciprocal 1D lattice

$\boldsymbol{\rho}$9

the effective Hamiltonian

$$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$0

supports continuum Landau modes with Gaussian envelopes and centers $$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$1. Under spatially incoherent monochromatic excitation, the steady-state response concentrates near the boundary with larger relative gain over a broad band $$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$2 (Wang et al., 2022). The same work stresses that this funneling requires only reciprocal couplings, shows no spectral winding or point gap, and exhibits OBC and PBC spectra occupying the same region of the complex plane, so it is explicitly distinct from NHSE-based schemes (Wang et al., 2022).

6. Diagnostics, limitations, and applications

The diagnostics of non-Hermitian wave funneling depend on mechanism. In NHSE lattices, the primary signatures are the PBC–OBC spectral dichotomy, generalized-Brillouin-zone deformation, and boundary accumulation. In the stochastic case, the relevant topology is the winding of the Lyapunov spectrum $$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$3 rather than of ordinary eigenvalues (Longhi, 2020). In active acoustic waveguides with periodic electroacoustic feedback, point-gap windings in the complex-$$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$4 plane predict which boundary accumulates the skin modes, with $$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$5 funneling to $$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$6 and $$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$7 to the opposite end (Braghini et al., 2022). In thin films, the observable is asymmetric absorption $$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$8, corroborated by photothermal $$\mathbf{k}^\top \boldsymbol{\rho}^{-1}\mathbf{k}=\frac{\omega^2}{B}$$9 rather than by a skin spectrum (Grineviciute et al., 2024). In local-Hilbert-transform media, correlation coefficients between the target flow and realized flow are the convergence metric (Ahmed et al., 2020).

Several limitations recur. In thin films, the condition $$\mathbf{k}=\mathbf{k}_{\mathrm r}+i\boldsymbol{\kappa}$$0 and exact $$\mathbf{k}=\mathbf{k}_{\mathrm r}+i\boldsymbol{\kappa}$$1 phase shift is critical; deviations produce residual outcoupling and leakage, and stronger funneling narrows the resonance (Grineviciute et al., 2024). In active acoustic tunneling, digital delay, actuator dynamics, and phase lag constrain the achievable nonreciprocity $$\mathbf{k}=\mathbf{k}_{\mathrm r}+i\boldsymbol{\kappa}$$2 and the tunnel depth (Langfeldt et al., 23 Jul 2025). In periodic electroacoustic-feedback waveguides, realistic filters degraded topology and localization, and stable operation required a restricted gain window such as $$\mathbf{k}=\mathbf{k}_{\mathrm r}+i\boldsymbol{\kappa}$$3 for the integral-feedback example (Braghini et al., 2022). In NHSE photonic meshes, strong disorder drives a crossover to Anderson localization (Weidemann et al., 2020).

The application space is correspondingly broad. Stochastic interface funneling concentrates energy at designated locations for sensing and signal routing in photonic platforms (Longhi, 2020). Thin-film asymmetric trapping enables static solar panels, integrated photodetectors, perfect absorbers for specific angles and wavelengths, thermal emitters, and sensors (Grineviciute et al., 2024). Acoustic feedback waveguides are positioned for noise control, wave localization, filtering, and multiplexing (Braghini et al., 2022). Uniform anisotropic media extend funneling to broadband and wide-angle boundary and corner collection without delicate spatial fine-tuning (Tian et al., 20 Aug 2025). A plausible implication is that future classification of the field will be less about a single mechanism than about whether the dominant non-Hermitian resource is asymmetric transport, asymmetric channel conversion, or spatially engineered complex constitutive response.

A final conceptual caution is that “non-Hermitian wave funneling” no longer denotes one narrowly defined phenomenon. In some papers it is literally the NHSE-induced condensation of all bulk modes at a boundary or interface (Weidemann et al., 2020); in others it is fluctuation-induced dynamical pile-up without stationary skin eigenvectors (Longhi, 2020); in others still it is one-way coupling of external radiation into an absorptive guided mode (Grineviciute et al., 2024) or reciprocal gain-gradient concentration without NHSE (Wang et al., 2022). The unifying theme is directed concentration by non-Hermitian means, but the mathematical objects that diagnose it—GBZ radii, Lyapunov exponents, TCMT couplings, local-flow correlations, complex gauge fields, or continuum Landau modes—are mechanism-specific rather than universal.