Can a nonradiating mode be externally excited? Nonscattering states vs. embedded eigenstates

Alex Krasnok; Andrea Al\`u; Dimitrios Sounas; Francesco Monticone

REVIEW 2 major objections 4 minor 43 references

Can a nonradiating mode be externally excited? Nonscattering states vs. embedded eigenstates

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Ideal nonradiating eigenmodes of open cavities have zero modal amplitude under any external causal illumination, because Lorentz reciprocity forces the overlap between the nonradiating polarization current and the incident field to vanish.

desk verdict A clean conceptual clarification—anapoles as scattering zeros, embedded eigenstates as unexcitable poles—but the zero-excitation proof is proved only for strictly confined eigenmodes, so the abstract's general BIC claim needs qualifying. read the letter →

arxiv 1908.00956 v1 pith:IZKBHXZI submitted 2019-08-02 physics.optics

Francesco Monticone , Dimitrios Sounas , Alex Krasnok , Andrea Alù This is my paper · ORCID

classification physics.optics

keywords nonradiatingsourcesanapolemodesembeddedeigenstatesboundstatesinthecontinuumLorentzreciprocityMiescatteringzerosopencavitiesquasinormal

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Nonradiating field distributions come in two kinds, and only one kind can be switched on from outside. The paper argues that anapole states, scattering zeros where a particle's dipolar response vanishes, are not eigenmodes of the open cavity at all: they exist only while an incident field is present, and they begin radiating as soon as the excitation is switched off. In contrast, an ideal embedded eigenstate, also called a bound state in the continuum, is a self-sustained nonradiating eigenmode of an open cavity, and Lorentz reciprocity forces its modal amplitude under external illumination to be exactly zero. In other words, an exact nonradiating eigenmode of a linear reciprocal open system cannot be excited by external light, although arbitrarily close approximations with high quality factors can be. The distinction matters because it separates 'invisible under illumination' from 'trapped light that needs no illumination'.

What carries the argument

The load-bearing mechanism is the nonradiating-source orthogonality theorem combined with the quasinormal-mode expansion of fields inside open cavities. The theorem states that any continuous nonradiating current distribution confined to a finite volume is orthogonal, under a volume integral, to any field in that volume that satisfies Maxwell's equations; applied to the induced polarization current $(\varepsilon(\mathbf r)-\varepsilon_0)\mathbf E_{\mathrm{int}}(\mathbf r)$ with the incident field as the test field, it makes the overlap integral that determines modal excitation vanish. The companion piece is the complex-frequency pole picture: eigenmodes of open cavities sit at poles of Mie coefficients in the upper half of the complex-frequency plane, whereas an embedded eigenstate has a purely real eigenfrequency, and an anapole is a zero, not a pole, along the real axis. The time-domain expansion of the internal field into these eigenmodes converts the vanishing overlap into zero modal amplitude.

What would settle it

Construct an open reciprocal cavity whose purported embedded eigenstate is computed numerically, illuminate it with a time-domain plane-wave pulse tuned to the real eigenfrequency, and measure the field energy inside the cavity after the pulse has left: if the energy is nonzero and persists longer than the pulse, the zero-excitation theorem is violated. Alternatively, compute the overlap integral $\int_{V_s} (\varepsilon-\varepsilon_0)\mathbf E_{\mathrm{int}}\cdot \mathbf E_{\mathrm{inc}}\,dV$ for a mode with a finite exterior tail and show it is nonzero while scattering remains zero.

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Extended reading notes

Core claim

The paper's central claim is that radiationless field distributions split into two classes that behave differently under external excitation. A zero of a Mie scattering coefficient, such as the anapole of a dielectric sphere, is an induced polarization pattern that happens not to radiate: it is forced by the incident wave, does not satisfy boundary conditions on its own, and therefore radiates away the moment the incident field is removed. An embedded eigenstate, by contrast, is a genuine eigenmode with real eigenfrequency despite the open nature of the cavity; its internal field satisfies the continuity conditions with zero external field. The paper proves, via Lorentz reciprocity and a time-domain modal expansion, that for such an ideal embedded eigenstate the overlap integral between the nonradiating polarization current and any causal incident field vanishes, so the modal amplitude is $a_n=0$ for every excitation frequency, including at resonance. Thus an exact nonradiating eigenmode of a linear reciprocal open cavity cannot be excited from outside, and the energy stored in it under external causal illumination is identically zero.

Load-bearing premise

The proof assumes the embedded eigenmode has a strictly confined induced polarization current inside a finite volume and exactly zero exterior field, so if any realistic candidate state has an exterior tail or surface current the orthogonality integral no longer vanishes and external excitation may become possible.

Editorial extensions

If this is right

An anapole-based scatterer is invisible only under steady monochromatic illumination; when the incident field is switched off, the stored energy is released as radiation on a time scale set by the nearby complex eigenfrequencies.
At an exact embedded eigenstate, the incident field cannot deposit any energy into the mode; the modal amplitude $a_n$ and the stored modal energy are identically zero.
Near, but not at, an ideal embedded eigenstate, external excitation can produce very large internal fields with quality factors that diverge as the eigenstate is approached.
A laser based on the anapole cannot operate on the anapole itself, since lasing requires a pole of the scattering matrix; gain must instead bring one of the complex poles near the anapole onto the real axis.
The same classification applies to acoustic, elastic, and matter-wave scattering, since the argument uses only linear scattering theory and reciprocity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the orthogonality argument holds, reported 'excitations' of bound states in the continuum in open reciprocal systems are probably excitations of nearby high-Q states, not of the exact eigenstate; a direct test is to measure the dark time after switching off illumination.
The reciprocity argument suggests a design strategy: in nonreciprocal or time-modulated systems, the vanishing-overlap obstruction may be bypassed, potentially allowing direct external excitation of exact embedded eigenstates.
The same orthogonality integral could be used as an inverse-design objective, searching over permittivity profiles for field distributions whose overlap with all possible incident fields vanishes, creating cavities with arbitrarily long confinement without mirrors.
One could classify any proposed nonradiating mode operationally by its transient response: anapole-type states radiate a burst after the source turns off, whereas true eigenstates remain dark; this avoids relying on steady-state scattering measurements that cannot distinguish the two.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: worth reading and worth sending to referees, but the central claim is proved for a narrower class than the abstract promises. The paper makes a genuinely useful distinction between radiationless scattering states (anapoles) and nonradiating eigenmodes (embedded eigenstates/BICs), and shows via Lorentz reciprocity that an ideal nonradiating eigenmode with strictly confined fields cannot be excited by external illumination. The anapole-as-scattering-zero story is classic Mie theory repackaged, but the framing is helpful, and the transient demonstration that an anapole radiates when the pump is switched off is a nice concrete illustration.

The reciprocity argument itself is sound: if the induced polarization current of the eigenmode is nonradiating in the strict Wolf sense—field identically zero outside the object—then Eq. (3) forces the overlap integral in the modal expansion to vanish, so the excitation coefficient a_n = 0. That is a clean result. The problem is the jump from this strict confinement to the general category of embedded eigenstates and BICs. Many BICs in photonic crystals and waveguide arrays have non-zero evanescent fields outside the structure; they are nonradiating in power-flux terms but do not have zero exterior fields. For those, the Wolf orthogonality argument does not directly apply, and the paper does not prove a_n = 0. The specific example in Fig. 2(b) (zero-permittivity shell) satisfies the strict condition, so the theorem holds there, but the abstract and introduction overclaim generality.

Also worth noting: the key modal expansion Eq. (5) is deferred to Supporting Information not included in the arXiv posting, and the printed formula is garbled by OCR. This is addressable, but it means the main text's central equation can't be checked from the arXiv alone. There's also a physically implausible '2 µs' decay time for a 100-nm sphere (likely a typo for ps or ns).

None of this overturns the paper. The taxonomy is important for people working in nonradiating photonics, and the reciprocity proof is a useful formal anchor. A thoughtful referee can fix the overclaim and the typos. My recommendation: send it to peer review, but ask the authors to state clearly that the zero-excitation theorem applies to strictly confined embedded eigenstates, and to clarify which BIC classes satisfy that condition.

Referee Report

2 major / 4 minor

Summary. The paper addresses a controversy in nanophotonics about whether radiationless states can be excited by external illumination. It distinguishes two classes: anapole states, which are scattering zeros of an open cavity and therefore require an incident field to exist, and embedded eigenstates (bound states in the continuum), which are true nonradiating eigenmodes. The central claim is that an ideal embedded eigenstate has identically zero excitation amplitude under any external causal illumination, because its induced polarization current is a nonradiating source, so Lorentz reciprocity (Eqs. 1-3) forces the spatial overlap integral in the modal expansion (Eq. 5) to vanish. The paper supports this with Mie-theory plots, field distributions, and transient simulations, and concludes that anapole-based lasing is not directly possible.

Significance. If the central claim holds, the paper provides a useful conceptual clarification in a literature where the terms 'anapole mode' and 'embedded eigenstate' are often conflated. The reciprocity argument is elegant, parameter-free, and grounded in Wolf's classical nonradiating-source theorem, and the distinction between a scattering zero and a nonradiating eigenmode is likely to be influential. The paper also gives a clear physical picture for why anapoles radiate after the incident field is switched off. However, the significance is tempered by the fact that the proof, as written, applies to strictly confined embedded eigenstates (fields vanishing exactly outside the scatterer) rather than to the broad class of open-cavity BICs with evanescent tails that the introduction claims.

major comments (2)

[Role of Lorentz Reciprocity, Eq. (5)] The central conclusion that a_n = 0 for an embedded eigenstate rests on the modal-expansion coefficient formula (5), but the printed equation is corrupted by OCR artifacts (e.g., the garbled '2( ) ) ( () ( ), np iT it n nn nnV n ic e e d iat' fragment) and its derivation is relegated to a Supporting Information document that is not included in the arXiv posting. This makes the load-bearing step unverifiable from the main text. The authors should provide the full derivation of Eq. (5) and a clean, legible equation in the paper or in an accessible supplement.
[Eqs. (1)-(3), Fig. 2(b)] The orthogonality argument requires that the nonradiating eigenmode's field vanish identically in the exterior volume V\V_s, so that the induced polarization current is a nonradiating source in the strict Wolf sense. Many bound states in the continuum, such as photonic-crystal slab modes or waveguide-array modes, have evanescent tails in the exterior cladding and are nonradiating only in the sense of zero power flux at infinity, not zero field. For such modes, the right-hand side of Eq. (1) includes a contribution from the exterior field, and Eq. (3) need not vanish. The abstract and introduction claim the conclusion for embedded eigenstates and BICs in open cavities generally, but the proof as written establishes it only for strictly confined embedded eigenstates such as the zero-permittivity-shell example in Fig. 2(b). Please either restrict the claim to strictly confined eigenstates or extend the proof to modes with nonzero exterior tails.

minor comments (4)

[Fig. 4 caption and surrounding text] The transient simulation in Fig. 4 is described only qualitatively; please provide simulation parameters such as pulse duration T_p, polarization, mesh resolution, and boundary conditions so that the radiative-decay timescale can be reproduced.
[Eq. (3) and general notation] The sign and time-harmonic convention for the induced polarization current J_1 = -iω(ε(r)-ε0)E_int are not defined in Eq. (1)-(3); the convention is mentioned only in the Fig. 1 caption. Please state the convention explicitly where J is first introduced.
[Throughout] The text contains numerous typesetting and OCR artifacts (e.g., '1 1TMc' for the Mie coefficient, 'ite ω−' in the Fig. 1 caption, and the missing closing parenthesis in reference [39]). A careful copyediting pass is needed.
[Section 'Anapoles vs. Embedded Eigenstates', final paragraph] The statement that an anapole state 'cannot be used, directly, to achieve lasing' is presented as a firm conclusion; consider softening this to a conjecture or supporting it with a more detailed pole-zero analysis, since the proposal in Ref. [15] may involve gain and nonlinearities not covered by the present linear scattering argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the zero-excitation result is a direct application of Lorentz reciprocity and Wolf's nonradiating-source orthogonality, with no fitted parameters or self-citation chain.

full rationale

The paper's central claim—that an ideal nonradiating eigenmode cannot be excited by external causal illumination—is derived from two independent external results: the Lorentz reciprocity theorem (Eq. 1) and Wolf's orthogonality theorem for nonradiating sources (Ref. [28], Eq. 3.6). The key step (Eq. 3) is that for a strictly nonradiating induced polarization current, E1 = 0 in V\V_s, so the reciprocity overlap integral ∫(ε−ε0)E_int·E_inc must vanish (Eq. 2). This is then carried into the modal expansion (Eq. 5), whose spatial integral is the same overlap, yielding a_n = 0. This is not circular: the nonradiating condition (zero exterior field) is an assumption about the ideal eigenmode, and the conclusion (zero modal amplitude under external illumination) is a distinct consequence obtained via an external theorem. The anapole case is distinguished precisely because its nonradiating current is not an eigenmode, so the same orthogonality integral can vanish without implying zero internal field. No parameters are fitted to data, no prediction reduces to an input, and the authors' prior work (Refs. [20–22]) is cited only as examples, not as the load-bearing justification. The printed Eq. (5) is garbled and its SI derivation is absent, which is a completeness concern but not a circularity; similarly, the restriction to eigenmodes with exactly zero exterior field (rather than evanescent tails) is a scope or correctness caveat, not a circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters, fitted constants, or new physical entities. Its argument is built on standard axioms: Lorentz reciprocity, completeness of quasinormal modes for spherical cavities, and a confinement assumption for nonradiating eigenmodes. The completeness assumption is cited to prior literature; the confinement assumption is the main physical idealization.

assumptions (4)

domain assumption Lorentz reciprocity theorem holds for the reciprocal, passive, isotropic media considered.
Used as the basis for Eqs. (1)-(3); standard in electromagnetics, but it is an assumption about the media (no magneto-optic or nonlinear response).
standard math The eigenmodes of the open spherical cavity form a complete set inside the scatterer.
Cited to Refs. 31-38; used to justify the modal expansion (4) and the expression (5) for modal amplitudes.
domain assumption A nonradiating eigenmode has exactly zero external field and its polarization current is confined to the scatterer volume V_s.
Defines the class of embedded eigenstates treated here; the proof of orthogonality (Eq. 3) depends on the current being confined and nonradiating.
standard math The incident field is a source-free solution of Maxwell's equations in the background medium inside V_s, representable as the field of an external current.
Needed to apply Lorentz reciprocity to a plane-wave illumination in Eq. (3).

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Cite this review

Pith. "Pith review of Can a nonradiating mode be externally excited? Nonscattering states vs. embedded eigenstates." pith.science (2026). https://pith.science/paper/IZKBHXZI

@misc{pith2026190800956,
  author       = {Pith},
  title        = {Pith review of: Can a nonradiating mode be externally excited? Nonscattering states vs. embedded eigenstates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IZKBHXZI}},
  note         = {Machine review of arXiv:1908.00956}
}

read the original abstract

In this Letter, we discuss the general problem of exciting radiationless field distributions in open cavities, with the goal of clarifying recent findings on this topic. We point out that the radiationless scattering states, like anapoles, considered in several recent studies, are not eigenmodes of an open cavity; therefore, their external excitation is neither surprising nor challenging (similar to the excitation of nonzero internal fields in a transparent, or cloaked, object). Even more, the radiationless anapole field distribution cannot be sustained without the actual presence of external incident fields. Conversely, we prove that the Lorentz reciprocity theorem prevents the external excitation of radiationless optical eigenmodes, as in the case of embedded eigenstates and bound states in the continuum in open cavities. Our discussion clarifies the analogies and differences between invisible bodies, nonradiating sources, anapole scatterers and emitters, and embedded eigenstates, especially in relation to their external excitation.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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F. Monticone, C. Argyropoulos, and A. Alù, "Multilayered Plasmonic Covers for Comblike Scattering Response and Optical Tagging," Phys. Rev. Lett. 110, 113901 (2013)

work page 2013