Synthetic Reflectionless Modes (SRM)

• Synthetic Reflectionless Modes (SRM) are engineered scattering solutions that yield zero reflection in designated input channels while allowing transmission or conversion.
• SRM are realized through auxiliary non-selfadjoint operators and filtered scattering matrices, with synthetic frequency dimensions shifting R-zeros onto the real axis.
• These modes enable precise control in photonic, microwave, and quantum systems, facilitating exceptional point degeneracy and tailored modal responses.

Synthetic Reflectionless Modes (SRM) are engineered reflectionless scattering states for which an incident field produces zero reflection in a specified input channel or input-channel set while remaining a genuine scattering solution rather than a bound state. In the most explicit recent usage, SRM are solutions of an auxiliary wave operator defined in a synthetic frequency dimension and describe incoming reflectionless waves onto a Floquet-driven cavity (Tuxbury et al., 5 Aug 2025). Closely related literature formulates the same phenomenon as reflectionless modes in obstacle-loaded waveguides (Dhia et al., 2018) or as real-frequency reflectionless scattering modes (RSMs), obtained when complex-frequency reflection zeros, or R-zeros, are tuned to the real axis (Sweeney et al., 2019, Stone et al., 2020). Across these formulations, reflectionlessness is treated as a spectral property of non-selfadjoint operators, filtered scattering matrices, or auxiliary synthetic lattices.

1. Definition and Terminological Scope

In the waveguide setting, a reflectionless mode is defined at a wavenumber $k$ for which there exists a nonzero incident modal vector $(a_n)_{n=0}^N$ such that all reflected propagating coefficients vanish, equivalently

$$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$

The corresponding total field is ingoing on the input side, outgoing on the output side, and on the back side only evanescent terms remain. This is the operational meaning of perfect transmission through an obstacle in the presence of propagating and evanescent channels (Dhia et al., 2018).

In the more general scattering-theoretic formulation, one chooses a subset $F$ of asymptotic channels as inputs and defines a filtered reflection matrix

$${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$

An R-zero satisfies

$${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$

and an RSM is the corresponding real-frequency state obtained when such a zero lies on the real axis. Except in single-channel systems, the reflectionless state is tied to a specific coherent input wavefront given by the zero-eigenvalue eigenvector of ${\bf R}_{\rm in}$ (Sweeney et al., 2019, Stone et al., 2020).

The terminology is not uniform across the literature. Several papers explicitly use reflectionless modes, reflectionless scattering modes, and R-zeros, while the exact label Synthetic Reflectionless Modes is central in the Floquet synthetic-frequency work (Tuxbury et al., 5 Aug 2025). Other papers state explicitly that the term SRM does not appear in their own text and that the closest internal terminology is RSM or related reflectionless states (Stone et al., 2020).

2. Waveguide Spectral Formulation and the Link to Trapped Modes

A foundational formulation appears for the 2D waveguide

$\Omega=\{(x,y)\in\mathbb{R}^2: 0<y<1\},$

with Helmholtz problem

$$\Delta u + k^2 \gamma u = 0 \quad \text{in }\Omega, \qquad \partial_y u = 0 \quad \text{on } y=0,1,$$

where $\gamma=1$ outside a compact set $(a_n)_{n=0}^N$0. For $(a_n)_{n=0}^N$1, the propagating indices are $(a_n)_{n=0}^N$2, with

$(a_n)_{n=0}^N$3

and $(a_n)_{n=0}^N$4, $(a_n)_{n=0}^N$5 (Dhia et al., 2018).

The central spectral construction introduces a conjugated complex scaling that selects ingoing waves on the left and outgoing waves on the right,

$(a_n)_{n=0}^N$6

leading to the non-selfadjoint operator

$(a_n)_{n=0}^N$7

with

$(a_n)_{n=0}^N$8

Its essential spectrum is

$(a_n)_{n=0}^N$9

and the full spectrum lies in

$$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$0

The key theorem is that if

$$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$1

then $$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$2 is real if and only if it belongs to the union of trapped-mode frequencies $$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$3 and reflectionless-mode frequencies $$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$4. The same non-selfadjoint spectral problem therefore contains both trapped modes and reflectionless modes. Complex eigenvalues are also informative: the paper remarks numerically that complex eigenvalues near the real axis correlate with minima of the reflection coefficient and hence with weakly reflectionless behavior (Dhia et al., 2018).

A real eigenpair $$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$5 can be classified by

$$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$6

If $$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$7, the mode is trapped; if $$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$8, it is reflectionless. The incident amplitudes are

$$\ker R(k)\neq\{0\}, \qquad R(k):=(s_{np}^-)_{0\le n,p\le N}.$$9

When $F$0, the operator is $F$1-symmetric,

$F$2

which explains why isolated eigenvalues close to the real axis are often forced onto the real axis in symmetric geometries (Dhia et al., 2018).

3. R-Zeros, RSMs, and the General Scattering Theory

The general theory extends reflectionless states beyond 1D left-right scattering to arbitrary finite photonic structures in any dimension, as well as to acoustic and quantum scattering (Sweeney et al., 2019). In this framework, R-zeros form a countably infinite discrete set of complex frequencies. They are distinct from resonances: resonances impose purely outgoing boundary conditions, whereas R-zeros impose zero reflection back into a chosen input set while allowing scattering into the complementary output set.

A central result is that steady-state RSMs are obtained by moving an R-zero onto the real-frequency axis. The literature identifies two broad routes. One is index or geometric tuning in flux-conserving systems; the other is gain-loss tuning in non-flux-conserving systems. In a single-resonance approximation, the complex R-zero frequency is

$F$3

with $F$4 and $F$5 the effective radiative couplings into the chosen input and output channels, and $F$6 the nonradiative loss or gain. Reflectionlessness at real frequency is therefore a balance condition among input-channel coupling, output-channel coupling, and internal loss or gain (Stone et al., 2020).

The same theory specifies how much tuning is generically required. In a generic finite structure with no special symmetry, one continuous structural parameter is typically enough to move an R-zero onto the real axis. In the absence of geometric symmetries, the tuning of at least one structural parameter is necessary to achieve reflectionless excitation. By contrast, in structures with parity and time-reversal symmetry or with parity-time symmetry, a subset of R-zeros is real, so reflectionless states can exist without structural tuning (Stone et al., 2020).

Symmetry also controls directionality. For passive flux-conserving cavities, RSMs are bidirectional in the left-right language. For non-flux-conserving cavities they are generically unidirectional. In $F$7-symmetric systems, unidirectional R-zeros appear in complex-conjugate pairs, and reflectionless states arise naturally at real frequencies for small gain-loss parameter before leaving the real axis after a spontaneous $F$8-breaking transition (Sweeney et al., 2019).

4. SRM in Synthetic Frequency Dimensions

The explicit SRM construction introduced for Floquet-driven systems begins from a driven two-mode cavity described in coupled-mode theory by time-dependent equations with modulation

$F$9

and resonator detuning chosen so that

$${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$0

Because the system is time periodic, the analysis is recast in an infinite-dimensional synthetic lattice, or Floquet ladder, whose sites are harmonic replicas of the original cavity and whose inter-replica couplings are induced by the modulation (Tuxbury et al., 5 Aug 2025).

In that enlarged space, the Floquet scattering matrix is

$${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$1

after truncation of the ladder. The SRM construction then introduces an auxiliary operator $${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$2 by imposing reflectionless scattering boundary conditions at the input channel. Physically, the effective loss at the driven input site is replaced by an effective gain in the auxiliary problem. The scattering problem is thereby reduced to the eigenvalue problem

$${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$3

and synthetic reflectionless solutions are the eigenstates of this operator. The criterion for an SRM is the existence of a real eigenfrequency of $${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$4 at which the corresponding scattering state has zero reflection in the specified input channel (Tuxbury et al., 5 Aug 2025).

For a single input channel, the paper states that if the auxiliary operator has a second-order zero on the real axis at $${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$5, separated from poles of the physical system, then

$${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$6

This quartic reflectance scaling is the signature of the SRM exceptional point degeneracy, or SRM-EPD. In the regime $${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$7, distant Floquet replicas can be decimated, leaving an effective two-site auxiliary system that becomes a $${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$8-symmetric dimer. The degeneracy occurs when

$${\bf R}_{\rm in}(\omega) = {\bf F}\,{\bf S}(\omega)\,{\bf F}^\dagger .$$9

and the supplementary derivation gives the reduced auxiliary Hamiltonian as

$${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$0

up to the exact conventions used in the paper’s representation (Tuxbury et al., 5 Aug 2025).

This auxiliary-lattice formulation is not limited to suppressing reflection. It is also presented as a route to targeted up-conversion and down-conversion. SRM guarantee zero reflection into the input harmonic, while the modulation frequency and synthetic connectivity determine whether the wave exits at the $${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$1 or $${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$2 harmonic. The paper validates the theory numerically and through driven RF resonator simulations, using seven Floquet replicas to ensure convergence (Tuxbury et al., 5 Aug 2025).

5. Symmetry, Exceptional Degeneracies, and Flattened Line Shapes

A recurring theme is that reflectionless spectra are especially structured in $${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$3-symmetric or parity-symmetric settings. The general RSM theory describes a spontaneous symmetry-breaking transition in which two real RSMs meet, coalesce, and then leave the real axis as a complex-conjugate pair of R-zeros. At the coalescence, the reflection and transmission resonance line shape becomes quartically flat (Stone et al., 2020).

A direct optical realization is provided by two coupled optical cavities with a $${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$4-symmetric spectrum of reflectionless modes implemented as a three-mirror resonator of alternating ZnS and cryolite layers (Riboli et al., 2023). In the symmetric coupled-mode description,

$${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$5

and the eigenvalues of the auxiliary operator are

$${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$6

The exact $${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$7-symmetric phase corresponds to $${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$8, the broken phase to $${\bf R}_{\rm in}(\omega_{\rm RZ})\,{\boldsymbol \alpha}_{\rm in} = {\bf 0}, \qquad \det {\bf R}_{\rm in}(\omega_{\rm RZ}) = 0,$$9, and the exceptional point of degeneracy occurs at

${\bf R}_{\rm in}$0

At this point the transmittance is

${\bf R}_{\rm in}$1

which is the reported quartic flat-top passband (Riboli et al., 2023).

The same multilayer platform also demonstrates how reflectionless-mode degeneracy can be destroyed. At fluences below ${\bf R}_{\rm in}$2, the structure exhibits a flat-top passband at ${\bf R}_{\rm in}$3. At higher fluences, thermo-optic detuning in ZnS breaks the ${\bf R}_{\rm in}$4 symmetry of the reflectionless-mode spectrum, the real reflection zeros disappear, and the multilayer becomes highly reflective, functioning as an optical limiter (Riboli et al., 2023).

The Floquet SRM-EPD is closely analogous in spectral structure: an emergent local ${\bf R}_{\rm in}$5 symmetry in synthetic frequency space produces a second-order reflection zero, quartic reflectance scaling, and a flattened transmission spectrum (Tuxbury et al., 5 Aug 2025). This suggests that exceptional degeneracy is not incidental to SRM theory but one of its most robust organizing principles.

6. Related Realizations, Applications, and Conceptual Boundaries

The SRM idea appears in several adjacent literatures, often under different names but with closely related reflectionless constructions.

Domain	SRM-related object	Salient result
SNAP optical fibers	Reflectionless axial potential for whispering gallery modes (Suchkov et al., 2015)	A matched ${\bf R}_{\rm in}$6 radius modulation supports reflectionless propagation, local transmission/delay control, and close packing without cross-talk.
Transformation optics	Impedance-tunable reflectionless elements (Cao et al., 2013)	Compressors, expanders, bends, shifters, and splitters are designed by tuning impedance without changing refractive index.
Passive lossless metasurfaces	Auxiliary-field synthesis for reflectionless beam splitting (Epstein et al., 2016)	Auxiliary evanescent fields enforce local power conservation and permit exact reactive solutions.
Graded-index media	One-way reflectionless and nearly non-transmitting profiles (King et al., 2017)	Spatial Kramers-Kronig media remain reflectionless from one side while transmission can be made arbitrarily small.
Nonlinear soliton scattering	Reflectionless potentials matched to flat-top and thin-top solitons (Sakkaf et al., 2022)	Potentials built from the stationary soliton density produce sharp resonances between full transmission and full quantum reflection.
Multiterminal Josephson junctions	Zero-energy reflectionless scattering modes (Ohnmacht et al., 13 Mar 2025)	Zero-RSMs of the normal-state scattering matrix generate topological phase boundaries and Weyl nodes.
Ultracompact objects and wormholes	Quasi-RSMs, RSMs, and echo modes (Rosato et al., 27 Jan 2025, Qian et al., 1 Nov 2025)	High-frequency quasi-reflectionless scattering modes govern echoes; symmetric cavities admit exact reflectionless modes, and asymmetry shifts them into the complex plane.
Microwave networks	Reflectionless filter structures (Morgan et al., 2014)	Symmetry, even/odd-mode duality, and matched internal subnetworks give filters with identically zero reflection coefficient at all frequencies.

Several conceptual boundaries are emphasized repeatedly in this literature. Reflectionlessness does not mean absence of scattering: the defining condition is zero reflection into the chosen input channels, while substantial transmission, conversion, or rerouting into complementary channels may remain (Sweeney et al., 2019). Reflectionlessness also does not by itself specify the output channel; in the Floquet SRM setting, the modulation frequency and synthetic connectivity decide whether the wave exits in the ${\bf R}_{\rm in}$7 or ${\bf R}_{\rm in}$8 harmonic (Tuxbury et al., 5 Aug 2025). In ultracompact-object scattering, high-frequency quasi-reflectionless scattering modes rather than low-frequency resonances are identified as the direct source of time-domain echoes (Rosato et al., 27 Jan 2025).

The nomenclature is likewise nonuniform. Some works explicitly speak of SRM, some of reflectionless modes, some of RSMs and R-zeros, and some use the language of reflectionless potentials or reflectionless filters. This suggests that SRM is best understood as a cross-platform category of engineered reflectionless states whose mathematical realization may be a non-selfadjoint eigenproblem, a filtered scattering zero, or an auxiliary operator in a synthetic dimension.