Resonance modes in microstructured photonic waveguides: Efficient and accurate computation based on AAA rational approximation

Felix Binkowski; Fridtjof Betz; Lin Zschiedrich; Martin Hammerschmidt; Sven Burger

REVIEW 4 major objections 4 minor 24 references

Resonance modes in microstructured photonic waveguides: Efficient and accurate computation based on AAA rational approximation

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper shows that applying AAA rational approximation to the scattered field of a single centrally placed line source recovers the fundamental resonance mode of a microstructured waveguide while skipping the cladding and higher-order…

desk verdict A credible, well-documented numerical demonstration of AAA-based resonance-mode computation for a hollow-core fiber; the method is incremental, and the two real soft spots are the unquantified efficiency claim and the informal pole-selection step. read the letter →

arxiv 2412.13826 v1 pith:HZZMLPYX submitted 2024-12-18 physics.comp-ph cs.NAmath.NAphysics.optics

Felix Binkowski , Fridtjof Betz , Martin Hammerschmidt , Lin Zschiedrich , Sven Burger This is my paper · ORCID

classification physics.comp-phcs.NAmath.NAphysics.optics

keywords AAAalgorithmrationalapproximationresonancemodesphotonicwaveguidehollow-corecrystalfibereigenvaluecomputationsourcecouplingMaxwellequations

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

This paper claims that the fundamental resonance mode of a microstructured photonic waveguide can be computed directly, without computing the cladding and higher-order modes that crowd the spectrum. The method applies AAA rational approximation to the scattered field produced by a line source placed at the waveguide center, so that the poles of the rational approximation correspond to modes the source actually excites. On a hollow-core photonic crystal fiber from the literature, the approach returns the fundamental mode's effective index with real-part errors below $10^{-14}$ and imaginary-part errors below $10^{-5}$ relative to reference eigensolver computations, and mode-field errors below $10^{-5}$. If this holds, it removes the need to solve a large eigenproblem and filter irrelevant modes, and it makes sensitivity information available at negligible extra cost.

What carries the argument

The load-bearing object is the AAA (adaptive Antoulas-Anderson) rational approximation in barycentric form, $r(z) = n(z)/d(z)$, whose poles $z_{\mathrm{pole},n}$ and residues $a_n$ are read off directly. The paper reuses the scalar weights and poles in a vector-valued residue formula to assemble the mode field from the finite-element coefficient vectors of the scattered field. The second mechanism is source selection: a line source at the center of the hollow core couples strongly to the fundamental mode and negligibly to cladding and higher-order modes, so the rational approximation is dominated by the wanted eigenpair.

What would settle it

Run the same AAA workflow with a line source moved into the cladding or shaped like a higher-order mode profile, and compare the returned poles against the full reference spectrum: a significant pole that matches a cladding mode but misses the fundamental mode would confirm the output is source-selected, while a significant pole with no nearby reference eigenvalue would refute the claim that significant poles are eigenpairs.

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Extended reading notes

Core claim

The central claim is that one scalar projection of scattering data is enough to isolate a wanted resonance mode, provided the illuminating source couples strongly to that mode. Concretely, the paper solves the time-harmonic Maxwell scattering problem for an $x$-polarized line source at the center of the fiber at 40 sampling values of the effective index, forms the scalar function $y^T E_x$ with a random vector $y$, and fits it with AAA rational approximation. The dominant poles of that fit are interpreted as eigenvalues, and the vector-valued residue formula reconstructs the mode field. For the hollow-core photonic crystal fiber, this yields $n_{\mathrm{eff}1} = 0.9993596784939 + 0.000000003376i$ and $n_{\mathrm{eff}2} = 0.996754264645 + 0.00000190093i$, matching reference eigenvalues to $10^{-14}$ in the real part and to $10^{-5}$--$10^{-8}$ in the imaginary part. The identifying property of the fundamental mode is its central-core localization, which gives the on-axis source a dominant coupling to it.

Load-bearing premise

The method assumes that whenever a pole has a significant influence on the rational approximation, its vector residue is an accurate eigenvector, because the source couples strongly to that mode; modes the source does not excite are assumed to stay out of the approximation.

Editorial extensions

If this is right

Only the relevant modes are computed: in the demonstration, two core-localized modes emerge from 40 scattering solves instead of the 512 eigenvalues the reference eigensolver produces.
No mode-filtering post-processing is needed, because undesired cladding and higher-order modes never enter the rational approximation.
Because the approach relies on scattering solves, sensitivities of the eigenvalues with respect to geometry or material parameters come at negligible extra cost via algorithmic differentiation.
Accuracy can be pushed further by choosing complex sampling points near the physical eigenvalues, as the paper notes.
Other source types, such as multiple line sources, a fundamental-mode field of a single-mode fiber, or a Gaussian beam, can be used in the same framework.

Reading between the lines

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

The same source-coupling principle should let one target any specific mode, not just the fundamental one, by engineering an incident field whose spatial profile overlaps that mode and is mostly orthogonal to the others.
A direct test of the method's generality is to replace the line source with a Gaussian beam or a measured higher-order-mode profile and check whether the returned pole tracks the intended mode's eigenvalue.
In inverse design, the low-cost sensitivities could make this procedure a practical objective-function evaluator, optimizing the waveguide geometry to place a selected mode's eigenvalue at a target value without re-solving the full spectrum.

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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

Colleague,

Short version: this is a credible incremental numerical paper, and I would send it out. It takes the AAA-with-special-source idea from the authors' own Ref [19] and demonstrates it on a hollow-core photonic crystal fiber, getting neff1 = 0.9993596784939 + 3.376e-9 i with real-part errors below 1e-14 and mode errors below 1e-5 against Arnoldi. That is a concrete, reproducible result, and the Zenodo deposit backs it up.

What the paper does well: the accuracy validation is honest. The AAA eigenvalues are outputs of a rational fit to scattering data, not fitted to the Arnoldi values; the reference is used only for comparison. The convergence study over number of sampling points is a real check. The source-placement argument is intuitive and works in the example. The authors are also candid that the core computation uses the approach of Ref [19]; self-citation here is legitimate because that is where the algorithm is defined.

Where it is soft: the headline word "efficient" has no numbers behind it. We never learn runtime or memory for AAA versus computing 512 Arnoldi modes, so the practical selling point is unquantified. The pole-selection step is inspection: "two significant peaks" is not a defined criterion, and details on the other eigenvalues are pushed to the data publication. That matters because the framework's generality rests on the assumption that significant poles correspond to strongly coupled modes. The stress-test concern about high-Q weak-coupling modes is correct in principle: the example has the fundamental mode with Im(neff) = 3.4e-9, so it naturally dominates the rational approximation. That does not invalidate the demonstrated result, but it does mean the general claim that only relevant modes are computed should be stated as a heuristic that holds when the source is well chosen. Using the same FEM solver for both scattering data and Arnoldi reference is a mild shared-discretization circularity, not a fatal one.

Who should read it: people in computational photonics who want targeted resonance-mode calculations and are deciding between eigensolvers and scattering-based rational approximation. They will find a useful data point, but should not expect a general-purpose algorithm with formal guarantees.

Recommendation: send to peer review. Ask the authors to benchmark against a standard eigensolver, to specify an automatic peak-selection rule, and ideally to add a second waveguide where the spectrum is denser or the desired mode is not the lowest-loss one. The paper is solid enough to warrant that revision.

Referee Report

4 major / 4 minor

Summary. The paper proposes a framework for computing selected resonance modes of microstructured photonic waveguides by combining a specially placed line source with AAA rational approximation of the resulting scalar scattering data. For the hollow-core photonic crystal fiber example, the AAA-derived eigenvalues converge to Arnoldi reference values with real-part relative errors below 10^-14, imaginary-part errors below 10^-5 (and 10^-8 for the second mode), and mode-field errors below 10^-5. The authors argue that the source coupling avoids the computation of cladding and higher-order modes and thus eliminates post-processing mode filtering.

Significance. If the central claim is sustained, the approach would be practically useful because it targets relevant modes directly while using only black-box scattering solves, and its reproducibility assets are strong: source code and simulation data are deposited in an open data publication, and the numerical convergence studies are clearly reported. The main limitation is that the example itself cannot establish the general coupling assumption, since the demonstrated fundamental mode has an extremely small imaginary part and therefore dominates the rational approximation for spectral reasons, not necessarily because of superior source coupling.

major comments (4)

[Abstract and Section I] The term "efficient" is load-bearing in the title and abstract, but the manuscript reports no runtime, no number of scattering solves, and no comparison with the Arnoldi computation. Reporting 40 sampling points is not sufficient; a wall-clock time or solver-cost comparison with the Arnoldi reference computation would be needed to support the efficiency claim.
[Section II, paragraph after Eq. (4)] The assumption that a pole with significant influence on r(z) implies an eigenpair with significant source coupling is explicitly stated but never validated. The HC-PCF example has Im(neff1)=3.376e-9, so the fundamental mode's pole dominates the rational approximation because of its very small imaginary part, not necessarily because of exceptional source coupling. To support the general claim, the authors should either prove or numerically test the selection rule for cases with weak coupling and high quality factor, or with strong coupling and low quality factor.
[Section III B, Fig. 3] The selection of "two significant peaks" is made by visual inspection, and no objective criterion is given for distinguishing relevant poles from spurious poles, background continuum, or poles outside the sampled interval. Since the central claim is that only relevant modes are computed, a threshold on residue magnitude, pole proximity, or another well-defined measure is required, together with a demonstration that the selected set is stable with respect to that threshold.
[Section III A and III B, Figs. 3 and 4] The reference Arnoldi solutions are obtained with the same FEM solver, material model, and computational domain as the scattering data used in the AAA approximation. Therefore the reported convergence demonstrates consistency between two methods sharing a common discretization, not absolute accuracy. An independent check, such as FEM mesh refinement or comparison with an analytic waveguide benchmark, is needed to substantiate the claim of accurate computation.

minor comments (4)

[Section III C, Fig. 4 caption] The caption says the mode E2 corresponds to neff1; this should read neff2.
[Section II, Eq. (4)] The residue formula as typeset appears to list two factors separated by a comma in the denominator; the intended expression is likely a single quotient, and the notation should be corrected for clarity.
[Section III B] The random projection vector y is drawn from a uniform distribution, but the manuscript does not report the random seed or the variation of the results over different draws of y; this information should be included so that the numerical experiment is reproducible.
[Section II] The term "special light sources" is used informally in the introduction and conclusion; since the actual source is a singular line source, the terminology should be defined consistently at first use.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: eigenvalues are outputs of a rational approximation to scattering data, with Arnoldi used only for independent validation.

full rationale

The paper's derivation chain is self-contained with respect to the Arnoldi reference: the reported eigenvalues neff1 and neff2 are poles of the AAA rational approximant to the scalar observable y^T Ex obtained from FEM scattering solutions, and the Arnoldi reference solutions enter only in the convergence and error plots (Figs. 3(c,d) and 4(c)). No parameter of the AAA computation (sampling points, weights, poles) is fitted to the Arnoldi values. The only step that could superficially look circular is the source-coupling assumption in Section II ('When a pole z_pole has a significant influence on the rational approximation r(z), then we assume ...'), but the paper explicitly labels this as an assumption and validates it independently against the source-free eigenproblem solved by Arnoldi; an unproven assumption is a correctness risk, not a circular reduction. The sampling interval [0.995, 1] is chosen in the vicinity of the expected fundamental mode, which is a practical heuristic, but the eigenvalues are not prescribed by the interval. Refs [19] and [23] are self-citations, but they are used to attribute the method's origin and the sensitivity extension, not to justify the numerical result; the comparison with the Arnoldi algorithm provides independent support. The informal selection of 'two significant peaks' is an implementation detail that does not make the output equal to the input. Therefore, no specific circular step can be exhibited.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central method introduces no new physical entities. Its free parameters are algorithmic and modeling choices: the sampling grid, the source position and polarization, the random projection, and the pole-selection threshold. The two ad hoc assumptions about pole-to-eigenpair correspondence and source-mode coupling are the real cost of the approach, since they are not proven and may fail for other waveguide geometries.

free parameters (5)

Number of sampling points M = 40
Chosen by hand for the demonstration; the convergence study varies it to confirm stability, so it is not fitted to the target eigenvalue, but it is a user-selected algorithmic parameter.
Sampling interval in effective index = [0.995, 1]
Chosen to bracket the expected fundamental mode. If the interval excluded the mode, the mode would not be found, so this hand-selected range is a free parameter of the numerical experiment.
Source position and polarization = center of core, x-polarized
The special light source is placed at the center and polarized along x to maximize coupling with the fundamental mode. This modeling choice determines which modes are visible to the rational approximation.
Random projection vector y = uniform in (-1,1)
The scalar function y^T E_x is used for the rational fit. The random draw is a stochastic choice that could in principle suppress a mode, though the probability is negligible.
Pole selection threshold = not specified
The two reported eigenvalues are selected as the significant peaks of the rational approximation, but no quantitative significance criterion is given. This is a hidden user choice that affects which modes are reported.

assumptions (5)

standard math Time-harmonic Maxwell equations with open boundary conditions describe waveguide resonance modes.
Used in Section II and III as the governing equation for both scattering problems and the source-free eigenproblem.
domain assumption The fiber is modeled as infinite in z with harmonic dependence exp(i k_z z), reducing the problem to a 2D cross-section.
Section III: 'we model the HC-PCF with an infinite length in z-direction and we assume a harmonic dependence...' This is a standard approximation for long waveguides.
domain assumption Perfectly matched layers accurately model open boundaries.
Section III: 'open boundaries realized by perfectly matched layers are applied'; numerical convergence with FEM parameters is asserted but not shown in detail.
ad hoc to paper A pole with significant influence on the rational approximation corresponds to an eigenpair of the nonlinear eigenproblem with significant source coupling.
Section II: 'When a pole has a significant influence on the rational approximation, then we assume that a_n and z_pole are a good approximation to an eigenpair.' This is the central unproven premise of the method.
ad hoc to paper The chosen line source couples significantly to the wanted mode and negligibly to unwanted modes.
Section III: the source is placed at the center because the fundamental mode is localized there. The conclusion that only relevant modes appear depends on this coupling separation, which is not guaranteed in general.

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Pith. "Pith review of Resonance modes in microstructured photonic waveguides: Efficient and accurate computation based on AAA rational approximation." pith.science (2026). https://pith.science/paper/HZZMLPYX

@misc{pith2026241213826,
  author       = {Pith},
  title        = {Pith review of: Resonance modes in microstructured photonic waveguides: Efficient and accurate computation based on AAA rational approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZZMLPYX}},
  note         = {Machine review of arXiv:2412.13826}
}

read the original abstract

We present a framework for the efficient and accurate computation of resonance modes in photonic waveguides. The framework is based on AAA rational approximation with the application of special light sources. It allows one to calculate only relevant modes, such as the fundamental resonance modes localized in the central core of the waveguides. We demonstrate the framework using an example from the literature, a hollow-core photonic crystal fiber. This waveguide supports many other modes, such as cladding modes and higher-order modes. These nonrelevant modes are not calculated, so that challenging post-processing with mode filtering is not required.

Figures

Figures reproduced from arXiv: 2412.13826 by the authors.

**Figure 1.** FIG. 1. Computation of the fundamental resonance mode of a microstructured waveguide. The waveguide is illuminated with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. Sketch of the HC-PCF presented in Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Illumination of the HC-PCF sketched in Fig. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Computation of resonance modes of the HC-PCF sketched in Fig. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [19]

O. P. Bruno, M. A. Santana, and L. N. Trefethen, preprint arXiv:2405.19582v2 10.48550/arXiv.2405.19582 (2024)

work page Pith review arXiv doi:10.48550/arxiv.2405.19582 2024

[2] [1]

Note that, to compute the solutionf (z) of Eq

This means that the resonance modean corresponding to the eigenvaluezpole n has a significant coupling with the source term s(z) [19]. Note that, to compute the solutionf (z) of Eq. (3), any black-box solver can be used. For the proposed approach, only access to the solution is required, i.e., the way in which the solution is calculated is not taken into ...

work page

[3] [2]

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K. Kao and G. Hockham, Proc. Inst. Electr. Eng.113, 1151 (1966)

work page 1966

[4] [3]

P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, Nat. Photonics8, 278 (2014)

work page 2014

[5] [4]

Ritari, J

T. Ritari, J. Tuominen, H. Ludvigsen, J. C. Petersen, T. Sørensen, T. P. Hansen, and H. R. Simonsen, Opt. Express 12, 4080 (2004)

work page 2004

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Z. Wen, Z. Dong, Q. Deng, C. Pang, C. F. Kaminski, X. Xu, H. Yan, L. Wang, S. Liu, J. Tang, W. Chen, X. Liu, and Q. Yang, Nat. Photonics17, 679 (2023)

work page 2023

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Lalanne, W

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, Laser Photonics Rev.12, 1700113 (2018)

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Lalanne, W

P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugo- nin, M. Besbes, G. Demésy, M. D. Truong, B. Gralak, F. Zolla, A. Nicolet, F. Binkowski, L. Zschiedrich, S. Burger, J. Zimmerling, R. Remis, P. Urbach, H. T. Liu, and T. Weiss, J. Opt. Soc. Am. A36, 686 (2019)

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