Impact of mode completeness on the accuracy of the coupling theory of quasinormal modes: a strict numerical demonstration

Can Tao; Haitao Liu; Junda Zhu

arxiv: 2606.11630 · v1 · pith:NY44K2HYnew · submitted 2026-06-10 · ⚛️ physics.optics

Impact of mode completeness on the accuracy of the coupling theory of quasinormal modes: a strict numerical demonstration

Can Tao , Junda Zhu , Haitao Liu This is my paper

Pith reviewed 2026-06-27 09:06 UTC · model grok-4.3

classification ⚛️ physics.optics

keywords quasinormal modescoupling theorymode completenessregularized quasinormal modesoptical nanoresonatorsscattered fieldMaxwell equationsone-dimensional slabs

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The pith

A virtually complete basis of regularized quasinormal modes makes the coupling theory accurate for both eigenmodes and scattered fields of coupled resonators.

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

The paper establishes that the quasinormal mode coupling theory requires the modes of each resonator to form a complete basis for expanding the scattered field both inside and outside the structure. Physical quasinormal modes diverge exponentially in the far field and therefore remain incomplete outside the resonator. Regularized versions restore completeness, and the authors demonstrate this numerically on two one-dimensional slabs in direct contact, where virtually complete bases can be obtained analytically or numerically. With such a basis the theory matches full-wave results for both source-free eigenmodes and source-driven fields; without it the predictions degrade. The work also supplies an improved formulation and theoretical arguments for incorporating the regularized modes into the coupling equations.

Core claim

The coupling theory of quasinormal modes, derived from Maxwell's equations, predicts the eigenmodes and scattered fields of a coupled system of lossy resonators to high accuracy when each resonator's modes are regularized to form a virtually complete basis inside and outside the structure; the same theory loses accuracy when the incomplete set of physical quasinormal modes is used instead.

What carries the argument

Regularized quasinormal modes (RQNMs), constructed either by an equivalent surface current encircling each resonator or by a perfectly matched layer around the domain, which supply the missing completeness for field expansion both inside and outside the resonators.

If this is right

With a virtually complete basis of RQNMs the coupling theory reproduces source-free eigenmodes of the coupled system to high accuracy.
With a virtually complete basis of RQNMs the coupling theory reproduces source-excited scattered fields of the coupled system to high accuracy.
The coupling theory fails to reach high accuracy when the incomplete basis of physical quasinormal modes is substituted.
An improved formulation of the coupling theory allows rigorous incorporation of RQNMs obtained by either surface-current or perfectly-matched-layer regularization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same completeness issue is expected to appear in three-dimensional geometries even when resonators are separated rather than in direct contact.
Methods to generate virtually complete RQNMs analytically or numerically may be needed for each new resonator shape before the coupling theory can be applied reliably.
The demonstrated improvement to the coupling equations may extend the range of coupling strengths for which the theory remains accurate.

Load-bearing premise

Results obtained for one-dimensional slabs in direct contact generalize to the completeness requirements of arbitrary three-dimensional nanoresonator systems.

What would settle it

A calculation for two three-dimensional nanoresonators in which a virtually complete set of RQNMs still yields large errors in the coupled eigenfrequencies or scattered fields would falsify the central claim.

read the original abstract

The coupling theory of quasinormal modes (QNMs) for a coupled system of generally lossy and dispersive optical nanoresonators has been established in a rigorous manner based on the first principle of Maxwell's equations [Phys. Rev. B 102, 045430 (2020)], and can achieve superior computational efficiency and physical intuitiveness compared with full-wave numerical methods if a small set of modes can achieve a high accuracy. The QNMs suffer from an exponential divergence of far field and can form a complete basis inside but not outside the resonator. In the QNM coupling theory (QCT), it is required that the QNMs of each resonator form a complete basis in expanding the scattered field both inside and outside the resonator, which can be achieved by using regularized QNMs (RQNMs). However, a strict numerical demonstration of the impact of the mode completeness of RQNMs on the accuracy of QCT by using a virtually complete basis of RQNMs is still absent. In this paper, we will provide such a numerical demonstration along with an improvement of the QCT and some theoretical demonstrations on a rigorous incorporation of RQNMs into the QCT. The RQNMs are obtained by introducing an equivalent surface current (ESC) encircling the resonator (called ESC-RQNMs) or the perfectly matched layer (PML) surrounding the computational domain (called PML-RQNMs). The numerical example is selected as two one-dimensional resonators of slabs in the extreme coupling case of direct contact, for which a virtually complete basis of RQNMs can be solved either analytically (for ESC-RQNMs) or numerically (for PML-RQNMs). The results show that by using a virtually complete basis of RQNMs, the QCT can achieve a high accuracy in predicting both the source-free eigenmodes and the source-excited scattered field of the coupled system, which is not true if using the incomplete basis of not-regularized QNMs (i.e., physical QNMs).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This 1D slab benchmark shows that virtually complete RQNMs fix QCT accuracy where physical QNMs fail, but the geometry is too simple to settle the issue for real 3D devices.

read the letter

The main takeaway is that in this direct-contact 1D slab case the QCT only matches independent exact or full-wave solutions when the basis of regularized QNMs is virtually complete, whether obtained analytically via equivalent surface current or numerically via PML. Physical QNMs alone fall short on both eigenmodes and scattered fields.

The paper delivers the strict numerical demonstration that its abstract says was previously absent. The 1D geometry lets them compare directly to references without post-hoc fitting, which keeps the test clean and the circularity low. That is the real contribution here.

The limitation is the setup itself. Direct contact between simple slabs is an extreme coupling limit, and everything stays one-dimensional. It is not clear whether the same completeness requirement will dominate in the separated, three-dimensional nanoresonators that most device work actually involves.

The paper is aimed at people already working on QNM coupling methods who need to decide whether regularization is worth the extra effort in their own codes. A reader in that niche will get a usable data point.

It deserves peer review. The evidence for the narrow claim is direct and reproducible.

Referee Report

0 major / 2 minor

Summary. The paper claims that the quasinormal-mode coupling theory (QCT) for coupled lossy/dispersive nanoresonators requires a virtually complete basis of regularized quasinormal modes (RQNMs) to accurately predict both source-free eigenmodes and source-excited scattered fields; this is demonstrated numerically on two directly contacting 1D slabs, where ESC-RQNMs (analytically solvable) and PML-RQNMs (numerically solvable) succeed while the incomplete physical QNMs fail. The work also includes an improvement to the QCT and theoretical arguments for rigorous incorporation of RQNMs.

Significance. The controlled 1D benchmark supplies an independent, high-resolution reference that directly isolates the effect of basis completeness on QCT accuracy for both eigenvalues and fields. This strengthens the practical utility of QCT by quantifying when a small modal set suffices and when regularization is essential, while the analytic ESC case removes fitting ambiguities.

minor comments (2)

[§3.2] §3.2: the definition of the ESC surface current could be accompanied by an explicit statement of how the regularization parameter is chosen to ensure virtual completeness without altering the interior solution.
[Fig. 4] Fig. 4 caption: the convergence curves for PML-RQNMs would benefit from an inset showing the residual norm versus number of modes to make the 'virtually complete' claim quantitative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. The referee's summary correctly identifies the core contribution: a controlled 1D numerical demonstration isolating the role of basis completeness in the quasinormal-mode coupling theory (QCT).

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the QCT from Maxwell's equations (cited to prior independent work) and then performs a controlled numerical benchmark in 1D slabs, comparing QCT outputs from virtually complete RQNMs (ESC or PML) versus incomplete physical QNMs against independent exact or full-wave solutions. No equation reduces to its own input by definition, no fitted parameter is relabeled as a prediction, and the self-citation is not load-bearing for the completeness demonstration. The 1D results stand on their own as external validation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the prior rigorous derivation of QCT from Maxwell's equations and on the domain assumption that regularized modes restore completeness; no new free parameters or postulated entities are introduced.

axioms (2)

domain assumption The coupling theory of QNMs is rigorously established from Maxwell's equations
Invoked via citation to Phys. Rev. B 102, 045430 (2020)
domain assumption Regularized QNMs form a complete basis for expanding the scattered field both inside and outside each resonator
Stated as the requirement for QCT accuracy in the abstract

pith-pipeline@v0.9.1-grok · 5916 in / 1400 out tokens · 39269 ms · 2026-06-27T09:06:38.362356+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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