Reflectionless Scattering Modes (RSMs)

• RSMs are scattering states defined by zero reflection in selected input channels using a tailored scattering matrix framework.
• The theory employs complex-frequency R-zeros and coherent wavefront tuning, distinguishing RSMs from conventional resonances and coherent perfect absorption.
• Experimental implementations in photonic and microwave systems demonstrate index or gain–loss tuning to achieve near-complete reflection suppression and effective wave routing.

Reflectionless scattering modes (RSMs) are scattering states defined relative to a chosen partition of asymptotic channels into inputs and complementary outputs, such that there is zero reflection back into the selected input channels. In the modern scattering-theoretic formulation, the generic object is the complex-frequency “R-zero,” and an RSM is the special case in which that reflection-zero condition is tuned onto the real-frequency axis, producing a physically excitable steady state (Sweeney et al., 2019). Across wave physics, the term now covers a family of related phenomena: discrete reflection zeros in multichannel photonic and microwave systems, exact reflectionless frequencies in symmetric cavity problems, and analytically continued reflection zeros in ultracompact-object scattering, where they are closely connected to echo phenomenology (Stone et al., 2020, Rosato et al., 27 Jan 2025, Qian et al., 1 Nov 2025).

1. Definition and core scattering formalism

The modern definition starts from the scattering matrix relation

$$\boldsymbol{\beta}=\mathbf S(\omega)\boldsymbol{\alpha},$$

where $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ are incoming and outgoing channel amplitudes. One then chooses a subset of channels as inputs and defines the corresponding reflection subblock $\mathbf R_{\rm in}$. The reflectionless condition is

$$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$

equivalently

$\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$

A solution at generally complex frequency is an R-zero; when $\omega_{\rm RZ}\in\mathbb R$, it is an RSM (Sweeney et al., 2019, Stone et al., 2020).

This definition is broader than the familiar one-dimensional statement “zero reflected wave.” It applies to arbitrary finite photonic structures, in any dimension, with any number of channels, and with arbitrary choices of input and output sets (Stone et al., 2020). In multichannel systems the reflectionless state is not specified by frequency alone: the required excitation is a particular coherent input wavefront, namely the zero-eigenvector of the relevant reflection matrix. The four-mode waveguide mode-converter example makes this explicit: at the RSM frequency only the special superposition

$$\alpha_0(\omega_{\rm RSM})=[0.7982,\,-0.5642+0.2158i,\,0,\,0]$$

is reflectionless, whereas individual mode inputs are not (Stone et al., 2020).

RSMs are distinct from ordinary resonances and from coherent perfect absorption (CPA). A resonance is a pole of the scattering matrix and corresponds to purely outgoing boundary conditions; an RSM instead imposes mixed boundary conditions, outgoing into the complementary outputs and reflectionless into the chosen inputs (Sweeney et al., 2019). CPA is the special case in which the full scattering matrix has a zero eigenvalue at a real frequency and all outgoing amplitudes vanish, whereas an RSM only requires zero reflection into the selected inputs and may route energy into other output channels (Sol et al., 2022). In the absence of geometric symmetries, at least one structural parameter must generically be tuned to move an R-zero onto the real axis (Stone et al., 2020).

2. Symmetry, directionality, and exceptional-point structure

The directionality of RSMs is controlled by flux conservation and symmetry. In flux-conserving passive cavities, steady-state RSMs are bidirectional: if one partition supports a real-frequency reflectionless state, the complementary partition does as well (Sweeney et al., 2019). In non-flux-conserving systems with gain and/or absorption, RSMs are generically unidirectional (Sweeney et al., 2019).

Parity, time-reversal, and $\mathcal{PT}$ symmetry organize the complex-frequency R-zero spectrum. In $\mathcal{PT}$-symmetric structures, R-zeros are either real or occur in complex-conjugate pairs; real-frequency reflectionless states can therefore exist without structural tuning in the symmetry-unbroken regime, but move off the real axis after a spontaneous $\boldsymbol{\alpha}$0-breaking transition (Sweeney et al., 2019, Stone et al., 2020). In ordinary Fabry–Perot-type resonators with both $\boldsymbol{\alpha}$1 and $\boldsymbol{\alpha}$2, the R-zeros are real and correspond to bidirectional RSMs (Stone et al., 2020).

A further development is the exceptional point formed by coalescing reflectionless states. In the $\boldsymbol{\alpha}$3-symmetric reflectionless-operator framework,

$\boldsymbol{\alpha}$4

the reflectionless eigenvalues are

$\boldsymbol{\alpha}$5

The reflectionless exceptional point occurs at

$\boldsymbol{\alpha}$6

that is, when the spacing between the central frequencies of two natural resonances equals the decay rate into incoming and outgoing channels (Ferise et al., 2021). At that point the reflection minimum becomes quartically flat,

$\boldsymbol{\alpha}$7

rather than quadratically sharp (Ferise et al., 2021).

Detuning enriches this picture. In coupled-cavity non-Hermitian scattering, bi-directional RSMs and RSM exceptional points are distinct objects: a bi-directional RSM requires $\boldsymbol{\alpha}$8, whereas an RSM EP is the coalescence of two reflection zeros. Detuning can either destroy an existing RSM by pushing its zero off the real axis or create a new one by bringing a zero onto it (Xu et al., 2024). The same work identifies transmission exceptional points, defined by coalescing transmission peaks rather than reflection zeros, showing that non-Hermitian scattering singularities extend beyond the Hamiltonian-EP paradigm (Xu et al., 2024).

3. Ultracompact objects, greybody factors, and echo physics

In gravitational-wave scattering by ultracompact horizonless objects, RSMs arise from cavity interference between effective barriers. For axial perturbations of a static, spherically symmetric ultracompact object,

$\boldsymbol{\alpha}$9

the reflection amplitude is

$\boldsymbol{\beta}$0

and on the real axis the reflectivity and greybody factor satisfy

$\boldsymbol{\beta}$1

Real-axis RSMs are therefore the frequencies for which

$\boldsymbol{\beta}$2

(Rosato et al., 27 Jan 2025).

That real-axis formulation yields an important physical claim: in ultracompact horizonless objects, the time-domain echo train is controlled by the high-frequency quasi-reflectionless sector rather than by low-frequency trapped-mode resonances. Fourier transforming the reflection amplitude,

$\boldsymbol{\beta}$3

shows that retaining the high-frequency oscillatory structure reproduces the echo pattern, whereas retaining only low-frequency resonances does not (Rosato et al., 27 Jan 2025). In symmetric double-barrier systems such as idealized wormholes, the reflectivity vanishes at discrete real frequencies and the oscillation scale is

$\boldsymbol{\beta}$4

with echo delay of order $\boldsymbol{\beta}$5 (Rosato et al., 27 Jan 2025).

The asymmetric Damour–Solodukhin wormhole analysis extends this notion from quasi-RSMs on the real axis to exact reflectionless modes in the complex frequency plane. The wormhole potential is written as

$\boldsymbol{\beta}$6

with throat at $\boldsymbol{\beta}$7 and large separation $\boldsymbol{\beta}$8 (Qian et al., 1 Nov 2025). The RSM boundary condition is

$\boldsymbol{\beta}$9

implemented in transfer-matrix language by

$\mathbf R_{\rm in}$0

The resulting quantization condition is

$\mathbf R_{\rm in}$1

and the mode spacing is approximately

$\mathbf R_{\rm in}$2

so the spectrum forms a ladder parallel to the real axis; in the time domain this gives

$\mathbf R_{\rm in}$3

for the echo period (Qian et al., 1 Nov 2025).

The key asymptotic result is that echo modes and RSMs share the same real-part spacing, and in the regime

$\mathbf R_{\rm in}$4

the real parts of echo modes coincide with those of the reflectionless modes (Qian et al., 1 Nov 2025). In symmetric Damour–Solodukhin wormholes the RSMs lie exactly on the real axis,

$\mathbf R_{\rm in}$5

whereas asymmetry pushes them into the lower-half plane, with $\mathbf R_{\rm in}$6 interpreted as a measure of asymmetry (Qian et al., 1 Nov 2025). The same work explicitly warns that once the problem is analytically continued to complex $\mathbf R_{\rm in}$7, the real-axis greybody-factor criterion no longer survives: for complex frequency one no longer has $\mathbf R_{\rm in}$8, so the correct definition is vanishing reflection amplitude in the complex plane, not unit transmission (Qian et al., 1 Nov 2025).

4. Photonic and microwave realizations

RSMs and closely related reflectionless constructions have been realized experimentally in photonics and microwaves. In stratified optics, reflectionless potentials were implemented as Al$\mathbf R_{\rm in}$9O$$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$0/TiO$$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$1 heterolayers approximating a designed reflectionless refractive-index profile. The measured response showed < 1% reflection over 350 nm to 2500 nm, for incidence angles 0° to 50°, with polarization-independent transmission, together with a negative delay = 0.031 ± 0.01 ps for two successive structures and discernible pulse narrowing (Thekkekara et al., 2013). That work is not phrased in the later multichannel RSM language, but it is a direct realization of one-dimensional reflectionless scattering (Thekkekara et al., 2013).

The general Maxwell theory of reflectionless excitation places these experiments in a wider setting. It formulates R-zeros and RSMs for arbitrary finite photonic structures, in any dimension, and shows that reflectionless behavior can be obtained either by index tuning in flux-conserving systems or by gain–loss tuning in non-flux-conserving ones (Stone et al., 2020). The central determinant relation,

$$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$2

shows that R-zeros are discrete complex solutions distinct from ordinary resonance poles (Stone et al., 2020).

Microwave experiments have used this framework for functionalized scattering control. Reflectionless programmable signal routing was demonstrated in an irregular cavity with strong modal overlap and a 304-element programmable metasurface. The experiments showed in-situ routing functionalities such as wavelength demultiplexing, including cases in which multi-channel excitation required adapted coherent input wavefronts (Sol et al., 2022). In the reported proof-of-principle cavity, the achieved performance included reflection suppression at least about $$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$3 dB, undesired transmission suppression by at least $$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$4 dB, and desired transmission attenuated by at most $$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$5 dB (Sol et al., 2022). An important conceptual point of that work is that the simple critical-coupling picture fails in such a highly overdamped multi-resonance system; instead, strong absorption broadens the distribution of R-zero damping rates so that some weakly damped zeros remain tunable toward functional RSMs (Sol et al., 2022).

5. Topological and statistical formulations

A recent non-Hermitian viewpoint treats RSMs as one of the fundamental scattering singularities. In that framework, an RSM is the zero of an individual reflection matrix element,

$$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$6

with the physical consequence of zero reflected power at channel $$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$7 (Shaibe et al., 18 Jul 2025). Unlike CPA, which is a zero of $$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$8, or a transmissionless scattering mode, which is a zero of $$\mathbf R_{\rm in}(\omega)\,\boldsymbol{\alpha}_{\rm in}=0,$$9, an RSM is a single-channel reflection zero (Shaibe et al., 18 Jul 2025).

Because $\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$0 is a complex scalar field over a two-dimensional parameter space, its zeros behave as topologically protected vortices. They are stable under small perturbations, can move in parameter space, and can only disappear via pairwise annihilation with opposite winding (Shaibe et al., 18 Jul 2025). The same work reports a universal statistical law: any quantity diverging only at such a point singularity has a probability distribution with a $\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$1 power-law tail. For RSM-related observables this includes

$\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$2

and normalized inverse-amplitude quantities such as

$\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$3

whose PDFs scale as $\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$4 in the tail (Shaibe et al., 18 Jul 2025). Homogeneous dissipation $\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$5 is identified there as the most important control parameter for singularity density: increasing uniform loss reduces the abundance of RSMs and other fundamental singularities (Shaibe et al., 18 Jul 2025). By contrast, higher-order coincident objects such as RSM-2 events are not topologically protected and do not exhibit universal statistics (Shaibe et al., 18 Jul 2025).

RSMs also enter topological condensed-matter settings. In multiterminal Josephson junctions, zero-energy reflectionless scattering modes of the normal scattering matrix generate topological phase boundaries. Writing the normal scattering matrix in block form,

$\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$6

a zero-energy RSM is present when

$\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$7

This implies a transparent channel in the normal region, and the corresponding zero-RSM is shown to be the source of topological phase boundaries and Weyl nodes in the Andreev bound-state spectrum (Ohnmacht et al., 13 Mar 2025). The same analysis emphasizes that many, though not all, topological boundaries in symmetric multiterminal junctions can be traced back to such reflectionless conditions in an appropriate reflection subblock (Ohnmacht et al., 13 Mar 2025).

6. Related reflectionless structures and conceptual boundaries

The modern RSM concept sits within a larger reflectionless-scattering landscape. Classical reflectionless Schrödinger operators on the line are defined by vanishing reflection coefficients,

$\det \mathbf R_{\rm in}(\omega_{\rm RZ})=0.$8

and can be reconstructed from discrete eigenvalues and norming constants by inverse scattering (Hryniv et al., 2020). In discrete spectral theory, Jacobi matrices and CMV matrices are reflectionless exactly when the associated two-channel scattering matrices are off-diagonal, so that the diagonal reflection amplitudes vanish and transmission has unit modulus on the absolutely continuous spectrum (Jaksic et al., 2013, Chu et al., 2014). These are not multichannel RSMs in the later filtered-subblock sense, but they provide a mathematically precise antecedent for “reflectionless” as “no backscattering.”

Integrable and nonlinear wave systems furnish another layer of related phenomena. Linearizations of several integrable PDEs around stationary solitons yield operators that are reflectionless at all energies, typically through intertwining relations with free operators; the examples include attractive NLS/Gross–Pitaevskii, sine-Gordon, KdV, and Liouville equations (Koller et al., 2013). Nonlinear analogues include reflectionless potentials for flat-top and thin-top solitons in the cubic–quintic NLSE (Sakkaf et al., 2022), nonreflecting wavepackets built from modified Pöschl–Teller eigenstates (Mousavi, 2015), and scattering of one-dimensional quantum droplets by a Pöschl–Teller reflectionless potential, where a sharp threshold separates full reflection from full transmission and the critical trapped mode becomes spatially asymmetric for large flat-top droplets (Hu et al., 2023). Modulated reflectionless potential wells have also been used to generate soliton ejection, although that regime deliberately breaks the exact reflectionless condition and uses the trapped mode as a nonlinear bound state rather than as the outgoing reflectionless mode itself (Uthayakumar et al., 2021).

Several recurrent misconceptions are resolved by the recent literature. First, RSMs are not identical to ordinary quasinormal modes or resonances; reflection zeros and scattering poles are different singular structures (Sweeney et al., 2019, Ferise et al., 2021). Second, except in single-channel systems, an RSM is a frequency–wavefront pair rather than a frequency alone (Stone et al., 2020). Third, the equality “reflectionless” = “unit transmission” is valid on the real axis in the one-dimensional greybody-factor normalization used for symmetric cavity problems, but it fails after analytic continuation to complex frequency, where vanishing reflection amplitude is the correct criterion (Rosato et al., 27 Jan 2025, Qian et al., 1 Nov 2025). This suggests a useful taxonomy: classical reflectionless potentials describe all-energy transparent operators; nonlinear reflectionless transport describes matched solitary-wave or droplet dynamics; modern RSM theory isolates discrete real- or complex-frequency reflection zeros associated with a chosen input subspace.