Transmissionless Scattering Modes (TSMs)

• Transmissionless Scattering Modes (TSMs) are defined by exact zero-transmission conditions, such as Sxy=0 or J-zeros, that precisely control energy flow in scattering systems.
• They encompass multiple formulations including channel-resolved zeros, multiport block zeros, and near-zero eigenchannels produced by destructive interference among modal contributions.
• Experimental realizations in programmable metasurfaces, waveguides, and nonlinear devices demonstrate TSMs’ applications in filtering, routing, and manipulating scattering responses.

Searching arXiv for papers directly relevant to Transmissionless Scattering Modes and closely adjacent scattering-zero frameworks. Searching arXiv for "Transmissionless Scattering Modes scattering zeros reflectionless scattering modes". Transmissionless Scattering Modes (TSMs) are scattering states or scattering singularities characterized by vanishing transmission into a specified channel set. Across the literature, the term is used in more than one non-equivalent sense. In one line of work, a TSM is the complex zero of an off-diagonal element of the full scattering matrix, $S_{xy}=0$, so that the transmitted power from input channel $y$ to output channel $x$ vanishes (Shaibe et al., 18 Jul 2025). In multiport filter theory, it is a $J$-zero, namely a zero of the determinant of an off-diagonal square block of $S$; in a 3-port device this reduces to a zero of an individual transmission coefficient $S_{ij}$ on the real-frequency axis (Faul et al., 2024). In single-mode waveguides, the same physical condition appears as exact zero transmission, $s_{12}=0$, implying complete reflection (Chesnel et al., 2018). A looser but physically important usage identifies “near-TSMs” with very low-transmission eigenchannels of the transmission matrix $t$, whose small transmission arises from off-resonant and destructively phased modal superpositions rather than from an explicit zero of the full $S$-matrix (Shi et al., 2014).

1. Definitions and conceptual scope

The defining object depends on the formulation. In the full scattering-matrix language, $S$ maps incoming channel amplitudes to outgoing channel amplitudes, and a channel-resolved TSM is the condition

$y$0

In that sense, the transmitted amplitude from input channel $y$1 to output channel $y$2 is exactly zero, and therefore the transmitted power $y$3 also vanishes (Shaibe et al., 18 Jul 2025). This definition is explicitly channel-resolved and matrix-element-based.

In programmable multiport filtering, the relevant quantity is not necessarily a single matrix element but a transmission block. There, TSMs are defined as $y$4-zeros, i.e. zeros of the determinant of an off-diagonal square block of the scattering matrix located on the real frequency axis. For a 3-port system, the off-diagonal transmission blocks are $y$5, so the condition again reduces to $y$6 (Faul et al., 2024).

In one-propagating-mode waveguides, the notion simplifies further. The scattering matrix is

$y$7

and the transmissionless condition is

$y$8

Because the scattering matrix is unitary in that setting, $y$9, so the incident propagating mode is completely reflected (Chesnel et al., 2018).

A distinct but related notion comes from transmission eigenchannels. For a multichannel disordered waveguide with transmission matrix $x$0, singular-value decomposition gives

$x$1

The $x$2-th transmission eigenchannel is the input singular vector $x$3, and low-$x$4 channels are physically close to transmissionless states. However, they are eigenchannels of $x$5, not eigenstates of the full scattering matrix $x$6, so they are not strict TSMs in the channel-resolved $x$7 sense (Shi et al., 2014).

Formulation	Vanishing object	Physical meaning
Channel-resolved scattering singularity	$x$8	Zero transmitted power from $x$9 to $J$0
Multiport $J$1-zero	$J$2 for off-diagonal block	Zero transmission through a selected transmission block
One-mode waveguide	$J$3	Complete reflection, no transmitted propagating mode
Near-TSM transmission eigenchannel	$J$4	Strongly suppressed transmission, not necessarily $J$5-matrix zero

A central misconception is that all these definitions are interchangeable. They are not. A matrix-element zero, a transmission-block zero, a vanishing singular value of $J$6, and a zero eigenvalue of the full $J$7-matrix are different conditions.

2. Scattering-matrix structure and exact TSM conditions

For a two-sided scattering system,

$J$8

so the transmissionless condition can be formulated either as a zero of a specific matrix element, a zero of a transmission block, or the existence of an input in the null space of a transmission map. The literature surveyed here treats all three, but not under a single universal convention (Shi et al., 2014).

The most explicit scalar definition is

$J$9

with

$S$0

This formulation regards a TSM as a fundamental scattering singularity of the complex scalar field $S$1 (Shaibe et al., 18 Jul 2025).

The multiport generalization used in programmable filtering is

$S$2

where $S$3 is an off-diagonal square block of $S$4. In the experimentally realized 3-port case, this becomes simply

$S$5

at a real frequency. The corresponding optimization target is

$S$6

which is minimized over metasurface configurations to impose a TSM at the desired target frequency $S$7 (Faul et al., 2024).

For single-mode waveguides, the exact transmissionless condition is again scalar: $S$8 The analysis of a 2D waveguide with a finite side branch shows that, under the coupling condition

$S$9

there exists an infinite sequence of branch lengths $S_{ij}$0 such that

$S_{ij}$1

The proof does not require geometric symmetry; it uses an asymptotic formula for $S_{ij}$2, the fact that its asymptotic trajectory is a circle in the complex plane, and the unitarity of the scattering matrix (Chesnel et al., 2018).

A more abstract projected-scattering formulation is developed for reflectionless states. There, one chooses an input channel subset and studies a filtered block $S_{ij}$3 of $S_{ij}$4, with reflectionless states defined by

$S_{ij}$5

A plausible extension to TSMs is to replace $S_{ij}$6 by an appropriate filtered transmission block. The paper does not derive a full TSM theory, but it explicitly notes that, for suitable background-normalized operators, the resulting “reflectionless” mode can in some cases be a transmissionless mode of the original $S_{ij}$7 (Sweeney et al., 2019).

3. Modal mechanisms: destructive interference, off-resonant coupling, and zero formation

In disordered media, the transmission matrix at frequency $S_{ij}$8 can be decomposed into modal contributions,

$S_{ij}$9

Each quasi-normal mode $s_{12}=0$0 contributes a Lorentzian-like complex response determined by its center frequency $s_{12}=0$1, linewidth $s_{12}=0$2, and modal transmission matrix $s_{12}=0$3 (Shi et al., 2014).

With the empirical rank-1 approximation

$s_{12}=0$4

the singular value of the $s_{12}=0$5-th transmission eigenchannel can be written as

$s_{12}=0$6

Hence

$s_{12}=0$7

This gives a direct modal interpretation of suppressed transmission: if the $s_{12}=0$8 are individually weak and destructively phased, the resultant transmission becomes small (Shi et al., 2014).

High-transmission channels are dominated by near-resonant modes. Low-transmission channels are built from many spectrally remote modes whose off-resonant amplitudes are weak,

$s_{12}=0$9

and whose phases reduce the net sum by destructive interference. This is the clearest modal account of near-TSMs in random media. It is explicitly a theory of suppressed transmission through $t$0, not a theorem about exact zeros of the full $t$1-matrix (Shi et al., 2014).

A complementary radiative-mode picture appears in the theory of material-independent modes for electromagnetic scattering. There, the scattered field is expanded as

$t$2

Any channel-projected observable becomes a rational function of $t$3, schematically

$t$4

Suppression of a selected scattering channel is therefore a modal cancellation condition,

$t$5

produced by destructive interference among radiative contributions with geometry-controlled poles $t$6 (Forestiere et al., 2016).

An analogous cancellation mechanism appears in nonlinear second-harmonic metasurfaces. For forward-pumped second-harmonic generation, the transmitted second-harmonic field is governed by the nonlinear scattering tensor $t$7, and the transmissionless condition is

$t$8

The paper gives explicit tensor conditions under which electric-type and magnetic-type nonlinear source contributions cancel in the forward channel. This is a transmissionless state in a generated harmonic channel rather than a passive linear eigenmode (Achouri et al., 2018).

4. Topological and statistical theory of TSM singularities

When a TSM is defined by

$t$9

it becomes a zero of a complex scalar field over parameter space,

$S$0

In a two-dimensional parameter space $S$1, a TSM occurs where

$S$2

so the intersections of the two nodal curves are isolated points (Shaibe et al., 18 Jul 2025).

These points are vortex singularities. Their topological charge is the phase winding

$S$3

Because they are fundamental singularities of a complex scalar field, they persist under small perturbations and can only be removed by pairwise annihilation with opposite winding number. This topological stability is one of the sharpest distinctions between exact TSMs and merely low-transmission states (Shaibe et al., 18 Jul 2025).

The associated divergent observables include the transmission time delay

$S$4

and, more generally, parameter-derivative observables such as

$S$5

These diverge exactly at the TSMs of the corresponding scalar field (Shaibe et al., 18 Jul 2025).

For point singularities in 2D parameter space, the probability distribution of an observable that diverges only at the singularity has a universal heavy tail

$S$6

For TSMs, an explicit example is

$S$7

whose divergence probes the density of transmissionless singularities (Shaibe et al., 18 Jul 2025).

Symmetry alters the singularity structure. In non-reciprocal systems, $S$8 and $S$9 can vanish independently, so a TSM can be unidirectional. In reciprocal systems, $S$0, and the paper states that reciprocal systems have fundamental TSM-2s and can only have TSM-Ps of even order $S$1. In a reciprocal two-port model at zero loss, the TSM-2 condition can form a curve rather than isolated points; with added loss, those lines break into points (Shaibe et al., 18 Jul 2025).

The programmable chaotic-cavity platform adds an experimentally observed higher-order phenomenon: transmissionless exceptional points. There, two complex $S$2-zeros approach and anti-cross just below the real-frequency axis near $S$3 GHz and $S$4 GHz. In the ideal lossless $S$5-symmetric limit, they would coalesce on the real axis into a true transmissionless EP before becoming a complex-conjugate pair (Faul et al., 2024).

5. Experimental realizations and platform-specific manifestations

A mechanically programmable, chaotic-cavity-backed non-local metasurface gives a direct experimental realization of channel-resolved TSMs as programmable transfer-function zeros. The device is a 3-port quasi-2D low-loss chaotic cavity with 14 continuously tunable meta-elements. Because the off-diagonal transmission blocks are scalar, TSM synthesis reduces to minimizing

$S$6

The platform demonstrates TSMs imposed at chosen frequencies such as $S$7 GHz, $S$8 GHz, and $S$9 GHz, with undesired output powers well below $y$00 dB (Faul et al., 2024).

The same platform shows that TSMs are practical building blocks for routing and filtering. In reflectionless routing demonstrations, the TSM is the suppression of the undesired transmission path, with measured undesired transmission suppressions of $y$01 dB, $y$02 dB, $y$03 dB, $y$04 dB, $y$05 dB, and $y$06 dB, while desired transmission attenuation is about $y$07 dB. The broader filter demonstrations extend the single-frequency TSM concept to finite-band reject bands and multi-band programmable filters (Faul et al., 2024).

A mathematically rigorous single-mode realization arises in a 2D waveguide with a finite side branch. For $y$08, only one propagating mode exists in each horizontal lead. The transmission coefficient satisfies the asymptotic relation

$y$09

so as the branch length $y$10 varies, $y$11 asymptotically moves on a circle in the complex plane. Unitarity then forces exact zeros, yielding infinitely many lengths $y$12 for which $y$13. These transmissionless states serve as precursor configurations for constructing trapped modes in a truncated half-waveguide, where the augmented-scattering condition $y$14 yields an embedded eigenvalue (Chesnel et al., 2018).

Nonlinear metasurfaces provide a distinct realization in frequency-converting channels. In second-harmonic generation, the transmissionless state is defined not at the pump frequency but at $y$15, via

$y$16

The resulting channel extinction is engineered by balancing nonlinear electric and magnetic source contributions dressed by the linear response at $y$17 (Achouri et al., 2018).

6. Related concepts, reciprocity constraints, and common confusions

TSMs are not identical to reflectionless scattering modes (RSMs), coherent perfect absorption (CPA), scattering invariant modes (SIMs), transmission eigenchannels, or bound states in the continuum. An RSM is a zero of a reflection block or, in the channel-resolved scalar formulation, a zero of $y$18. A CPA state is a zero eigenvalue of the full $y$19-matrix; in the two-channel case the enabling condition is $y$20. A TSM, by contrast, concerns only transmission extinction into a selected channel or block (Shaibe et al., 18 Jul 2025).

The theory of RSMs is nonetheless structurally important. It treats special scattering states as zeros of projected channel maps: $y$21 This projected-scattering viewpoint strongly suggests an analogous TSM formulation based on a filtered transmission block, but that extension is not explicitly derived in the paper (Sweeney et al., 2019).

Reciprocity constrains the channel-space structure of any TSM analysis. For reciprocal scatterers with arbitrary waveguide modes, including degenerate, evanescent, and complex modes, the general theorem is

$y$22

where

$y$23

is the generalized modal orthogonality matrix. Only in special orthogonal normalizations does this reduce to $y$24. Therefore, channel-space transmission zeros must be interpreted in the correct modal metric, especially when evanescent channels are retained (Svendsen et al., 2013).

PT- and RT-symmetric multimode waveguides provide conservation laws that constrain transmission suppression, but they do not by themselves define TSMs. In the multimode notation, a natural transmissionless state would satisfy

$y$25

yet the cited PT/RT paper does not present a dedicated TSM theory; it supplies the scattering formalism and matrix conservation laws needed to formulate one (Ge et al., 2015).

Several adjacent frameworks are useful but not direct TSM constructions. The TE/TM planar-scattering dynamical formalism gives exact transfer-matrix and Riccati equations, but it does not provide generic finite-slab $y$26 states; in the regular finite-slab setting,

$y$27

so exact zero transmission is not produced by regular finite solutions of the Riccati problem (Loran et al., 2024). The low-frequency Bergmann-equation analysis similarly shows that regular slab scattering naturally gives

$y$28

or, under stronger conditions, $y$29, making it more useful for excluding TSMs in that regime than for constructing them (Loran et al., 12 Sep 2025). SIMs are also not TSMs: they are transmitting states defined by output-field invariance relative to ballistic propagation,

$y$30

and are therefore conceptually closer to reference-relative open channels than to transmission-suppressed states (Pai et al., 2020).

The main conceptual boundary is therefore clear. Exact TSMs are scattering zeros of selected transmission channels or transmission blocks. Near-TSMs are strongly suppressed transmission states, often produced by off-resonant modal tails and destructive interference. Both are physically significant, but only the former are strict scattering singularities.