pith. sign in

arxiv: 2605.15527 · v1 · pith:PUUDPGYInew · submitted 2026-05-15 · 🧮 math.NA · cs.NA· physics.comp-ph

Method of Fundamental Solutions for Maxwell's Equations in Bi-Periodic Multilayered Media

Jared Weed , Bowei Wu , Jingfang Huang , Min Hyung Cho This is my paper

Pith reviewed 2026-05-19 14:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph
keywords Method of Fundamental SolutionsMaxwell equationsbi-periodic mediamultilayered structuresperiodizationquasi-periodic scatteringelectromagnetic wavesnumerical method
0
0 comments X

The pith

A periodization scheme using proxy sources on spheres makes the Method of Fundamental Solutions accurate for Maxwell's equations in bi-periodic multilayered media.

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

The paper develops a numerical method to solve time-harmonic Maxwell's equations for electromagnetic waves in structures that repeat periodically in two directions and consist of many stacked layers. Fields in each layer are written as a combination of direct interactions with nearby periodic copies plus an approximation of all distant copies through a handful of proxy source points placed on a sphere around the unit cell. Continuity conditions at layer interfaces, quasi-periodicity on the sides of the cell, and a radiation condition expressed via Rayleigh-Bloch series then produce a linear system that is solved with a Schur complement and a stable direct solver. This construction is verified to converge exponentially to near machine precision even when dozens of interfaces are present, which matters for simulating realistic devices in optics and electromagnetics that contain many thin layers.

Core claim

The authors show that the electric and magnetic fields can be represented using the Method of Fundamental Solutions with a split into near and distant interactions, where distant periodic effects are captured by proxy sources on enclosing spheres. Enforcement of tangential field continuity across interfaces, quasi-periodicity on vertical walls, and the radiation condition at infinity yields a square system for the unknown coefficients. The resulting scheme exhibits exponential convergence reaching 10^{-14} for both single-interface and multi-interface test cases, including a demonstration on a stack of 39 layers.

What carries the argument

The periodization scheme that approximates distant interactions by proxy source points placed on spheres surrounding the unit cell.

If this is right

  • The solver maintains exponential convergence close to machine precision for configurations with both single and multiple interfaces.
  • An example computation with 39 interfaces demonstrates practical performance for thick layered structures.
  • The approach satisfies all interface and boundary conditions exactly within the chosen basis and produces a backward-stable linear system.
  • Results indicate the method can be extended to a fast boundary integral equation solver for applications involving large numbers of layers in electromagnetics and optics.

Where Pith is reading between the lines

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

  • If the number of proxy points remains fixed as the number of layers grows, the method could simulate arbitrarily thick periodic media at modest cost.
  • The same near-far split might be applied to acoustic or elastic wave problems in similar bi-periodic geometries.
  • Accuracy for lossy or dispersive materials would need separate verification, since the examples assume lossless media.

Load-bearing premise

The distant periodic interactions can be replaced by a small fixed set of proxy sources on surrounding spheres without introducing errors that grow with the number of layers or destroy the exponential convergence.

What would settle it

A comparison of the computed fields against an independent high-accuracy reference solution for a 50-layer bi-periodic stack, where the relative error fails to stay below 10^{-10} or the convergence rate drops below exponential.

Figures

Figures reproduced from arXiv: 2605.15527 by Bowei Wu, Jared Weed, Jingfang Huang, Min Hyung Cho.

Figure 1
Figure 1. Figure 1: Free Space Problem Setup wavevector k = (kx, ky, kz) scattered by Ω can be found by solving the time-harmonic Maxwell’s equations ∇ × El = iωµlHl , (1) ∇ × Hl = −iωϵlEl , (2) ∇ · El = 0, (3) ∇ · Hl = 0, (4) with continuity of tangential fields at the boundary of Ω, n × E1 = n × E0 + n × E inc and n × H1 = n × H0, (5) and a Silver-Müller radiation condition, where l = 0, 1 and n is the outward unit normal v… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Unit cell and artificial source points above ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Two interfaces and MFS source points above and below each interface. (b) The unit cell and surrounding [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: One flat interface. Left: Real part of Ey. Center: Pointwise convergence, flux error, and divergence at (0, 0, 0.25) (+ marker) and (0, 0, −0.25) (o marker) as a function of MFS source points against the exact solution. Right: Pointwise convergence, flux error, and divergence at (0, 0, 0.25) (+ marker) and (0, 0, −0.25) (o marker) as a function of proxy source points against the exact solution. 5. Numerica… view at source ↗
Figure 5
Figure 5. Figure 5: One interface with f(x, y) = 0.1 sin(2πx). Left: Real part of Ey. Center: Pointwise convergence, flux error, and divergence at (0, 0, 0.25) (+ marker) and (0, 0, −0.25) (o marker) as a function of MFS source points against the reference solution. Right: Pointwise convergence, flux error, and divergence at (0, 0, 0.25) (+ marker) and (0, 0, −0.25) (o marker)as a function of proxy source points against the r… view at source ↗
Figure 6
Figure 6. Figure 6: One interface with f(x, y) = 0.1 sin(2πx) cos(2πy). Left: Real part of Ex, Ey, and Ez . Center: Pointwise convergence at (0, 0, ±0.25), flux error, and divergence as a function of MFS source points. Right: Pointwise convergence at (0, 0, ±0.25) , flux error, and divergence as a function of proxy source points. cell is shown, along with pointwise convergence in Ey, flux error, and divergence are presented i… view at source ↗
Figure 7
Figure 7. Figure 7: Flux Error of MFS solutions for a single interface defined as [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Five interfaces with fl(x, y) = −l + 0.1 sin(2πx) cos(2πy), l = 0, . . . , 4. Left: Real part of Ey for the entire cell. Center: Pointwise convergence at (0, 0, 0.25) and (0, 0, −4.25), flux error, and divergence as a function of MFS source points with P fixed. Right: Pointwise convergence at (0, 0, 0.25) and (0, 0, −4.25), flux error, and divergence as a function of proxy source points with N fixed [PITH… view at source ↗
Figure 9
Figure 9. Figure 9: Five interfaces with fl(x, y) = −l + 0.1 sin(2πx) cos(2πy), l = 0, . . . , 4, where ω = 10. Left: Real part of Ex for the entire cell. Center: Real part of Ey for the entire cell. Right: Real part of Ez for the entire cell. 6. Conclusion The combination of MFS and periodization scheme yields accurate numerical solutions of the time-harmonic Maxwell’s equations for multiple layers near machine precision. We… view at source ↗
Figure 10
Figure 10. Figure 10: A portion of the field solutions between layers 22 and 27 for 40 bi-periodic interfaces defined as [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Reflectance and transmittance plots for a frequency band of [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
read the original abstract

In this paper, we present an accurate numerical method for the time-harmonic Maxwell's equations for bi-periodic multilayered media with quasi-periodic incident waves using the Method of Fundamental Solutions in conjunction with a periodization scheme. Following an approach used in acoustic scattering problems, the electric and magnetic fields in each layer are expressed as a sum of near and distant interactions. The near interaction comprises interactions between the unit cell and its nearest neighboring copies, while the distant interaction is approximated by proxy source points placed on spheres surrounding the unit cell. Imposing continuity of tangential components at the layer interface, quasi-periodicity conditions on the walls of the unit cell, and Rayleigh-Bloch expansion for the radiation condition yields a system of equations for the unknown coefficients, which can be solved by Schur complement and a backward-stable solver. The scheme is verified with known solutions and exhibits exponential convergence close to $10^{-14}$ for both single and multiple interfaces. An example with 39 interfaces is presented to demonstrate the solver's performance. The paper provides promising results for extending this method to a fast and accurate boundary integral equation solver for many cutting-edge applications involving a large number of layers in electromagnetics and optics.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a Method of Fundamental Solutions for the time-harmonic Maxwell equations in bi-periodic multilayered media with quasi-periodic incidence. Fields in each layer are split into explicit near-neighbor interactions and distant interactions approximated by a finite set of proxy sources placed on spheres enclosing the unit cell. Interface continuity, quasi-periodicity, and Rayleigh-Bloch radiation conditions are enforced to obtain a linear system solved via Schur complement; the scheme is reported to achieve exponential convergence to approximately 10^{-14} and is demonstrated on an example with 39 interfaces.

Significance. If the proxy approximation remains accurate and non-accumulating, the method supplies a high-order, mesh-free solver for electromagnetic scattering in periodic multilayer stacks that extends prior acoustic work. The explicit numerical verification against known solutions together with the 39-interface demonstration constitute concrete evidence of practical performance for applications in optics and photonics.

major comments (1)
  1. [Method section (proxy approximation) and Numerical results (39-interface example)] The central claim of sustained 10^{-14} accuracy at 39 interfaces rests on the distant-interaction proxy representation (described after the near/distant decomposition in the method section) introducing an error smaller than the target tolerance and independent of layer count. No a-priori truncation bound for the finite proxy sources on the enclosing spheres is supplied, nor is stability of the fixed proxy configuration demonstrated for varying contrasts or still larger stacks; the Schur-complement solve can in principle propagate any residual mismatch across the multilayer system.
minor comments (2)
  1. [Abstract] The abstract states exponential convergence 'close to 10^{-14}' but does not specify the norm (e.g., L^2 on the interfaces or maximum norm on coefficients) in which this rate is measured.
  2. [Numerical results section] Notation for the proxy-point count and sphere radii is introduced without a dedicated table or explicit parameter list, making it difficult to reproduce the exact configuration used for the 39-interface test.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the detailed and constructive feedback on our manuscript. The positive evaluation of the method's potential for applications in optics and photonics is encouraging. Below, we provide a point-by-point response to the major comment and outline the revisions we plan to make to the manuscript.

read point-by-point responses

  1. Referee: [Method section (proxy approximation) and Numerical results (39-interface example)] The central claim of sustained 10^{-14} accuracy at 39 interfaces rests on the distant-interaction proxy representation (described after the near/distant decomposition in the method section) introducing an error smaller than the target tolerance and independent of layer count. No a-priori truncation bound for the finite proxy sources on the enclosing spheres is supplied, nor is stability of the fixed proxy configuration demonstrated for varying contrasts or still larger stacks; the Schur-complement solve can in principle propagate any residual mismatch across the multilayer system.

    Authors: We thank the referee for highlighting this important aspect of the proxy approximation. The manuscript demonstrates the method's performance through extensive numerical verification, including exponential convergence to near machine precision (~10^{-14}) for both single and multiple interfaces, and a specific example with 39 interfaces where the accuracy is maintained. The proxy sources are placed on spheres enclosing the unit cell, with the number chosen to achieve the target accuracy based on the acoustic precedent and empirical testing. While an analytical a-priori truncation error bound for the proxy representation is not derived in the current work (as the focus is on the numerical scheme and its practical performance), the numerical results indicate that the approximation error does not accumulate across layers, as evidenced by the sustained high accuracy in the 39-layer case. The Schur complement solver is backward stable, minimizing propagation of errors. To address the concern, we will add a new subsection in the numerical results discussing the choice of proxy parameters and include additional experiments showing stability under varying contrasts (e.g., high-contrast dielectrics) and for stacks with up to 100 interfaces. This will provide further empirical evidence of the method's robustness without altering the core claims. revision: partial

Circularity Check

0 steps flagged

Numerical construction is self-contained; no load-bearing step reduces to fitted input or self-definition

full rationale

The paper constructs a direct numerical scheme for Maxwell's equations via the Method of Fundamental Solutions in bi-periodic multilayered media. Fields in each layer are expressed as sums of explicit near-neighbor interactions plus proxy-source approximations for distant periodic contributions; interface continuity, quasi-periodicity, and Rayleigh-Bloch radiation conditions are imposed to obtain a linear system solved by Schur complement. Verification against known solutions and reported exponential convergence to 10^{-14} constitute empirical testing of the assembled discretization, not a derivation in which any claimed accuracy or result is forced by construction from the same fitted quantities or prior self-citations. The approach follows standard periodization techniques from scattering literature without importing a uniqueness theorem or ansatz that itself depends on the target result. Consequently the central claims remain independent of the inputs they are tested against.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical properties of fundamental solutions for Maxwell's equations and the validity of the Rayleigh-Bloch expansion for the radiation condition in periodic media. No free parameters or invented entities are explicitly introduced in the abstract; the proxy source approximation is a modeling choice whose accuracy is asserted via numerical tests.

axioms (2)
  • standard math Fundamental solutions satisfy Maxwell's equations away from source points and the chosen radiation condition holds at infinity.
    Invoked when expressing fields as sums of near and distant interactions.
  • domain assumption Quasi-periodicity and continuity of tangential components are sufficient to determine the unknown coefficients uniquely.
    Used when imposing interface and cell-wall conditions to form the linear system.

pith-pipeline@v0.9.0 · 5755 in / 1502 out tokens · 37980 ms · 2026-05-19T14:37:08.846865+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    B. T. DeBoi, A. N. Lemmon, B. McPherson, B. Passmore, Improved methodology for parasitic analysis of high-performance silicon carbide power modules, IEEE Transactions on Power Electronics 37 (10) (2022) 12415–12425

    work page 2022

  2. [2]

    B. Kim, H. Jeon, D. Park, G. Kim, N.-H. Cho, J. Khim, Emi shielding leadless package solution for automotive, Journal of Advanced Joining Processes 5 (2022) 100102. 19

    work page 2022

  3. [3]

    H. A. Atwater, A. Polman, Plasmonics for improved photovoltaic devices, Materials for sustainable energy: a collection of peer-reviewed research and review articles from Nature Publishing Group (2011) 1–11

    work page 2011

  4. [4]

    M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J. M. Spurgeon, R. M. Briggs, N. S. Lewis, H. A. Atwater, Enhanced absorp- tion and carrier collection in si wire arrays for photovoltaic applications, Nature materials 9 (3) (2010) 239–244

    work page 2010

  5. [5]

    Perry, R

    M. Perry, R. Boyd, J. Britten, D. Decker, B. Shore, C. Shannon, E. Shults, High-efficiency multilayer dielectric diffraction gratings, Optics letters 20 (8) (1995) 940–942

    work page 1995

  6. [6]

    Barty, M

    C. Barty, M. Key, J. Britten, et al., An overview of llnl high-energy short-pulse technology for advanced radiography of laser fusion experiments, Nuclear Fusion 44 (12) (2004) S266

    work page 2004

  7. [7]

    Monk, et al., Finite element methods for Maxwell’s equations, Oxford University Press, 2003

    P. Monk, et al., Finite element methods for Maxwell’s equations, Oxford University Press, 2003

    work page 2003

  8. [8]

    Bao, Finite element approximation of time harmonic waves in periodic structures, SIAM Journal on Numerical Analysis 32 (4) (1995) 1155–1169

    G. Bao, Finite element approximation of time harmonic waves in periodic structures, SIAM Journal on Numerical Analysis 32 (4) (1995) 1155–1169

    work page 1995

  9. [9]

    Taflove, S

    A. Taflove, S. C. Hagness, M. Piket-May, Computational electromagnetics: the finite- difference time-domain method, The Electrical Engineering Handbook 3 (2005) 629–670

    work page 2005

  10. [10]

    Yee, Numerical solution of initial boundary value problems involving maxwell’s equa- tions in isotropic media, IEEE Transactions on antennas and propagation 14 (3) (1966) 302–307

    K. Yee, Numerical solution of initial boundary value problems involving maxwell’s equa- tions in isotropic media, IEEE Transactions on antennas and propagation 14 (3) (1966) 302–307

    work page 1966

  11. [11]

    R. F. Harrington, Field computation by moment methods, Wiley-IEEE Press, 1993

    work page 1993

  12. [12]

    Moharam, T

    M. Moharam, T. Gaylord, Rigorous coupled-wave analysis of planar-grating diffraction, JOSA 71 (7) (1981) 811–818

    work page 1981

  13. [13]

    Li, Use of fourier series in the analysis of discontinuous periodic structures, JOSA A 13 (9) (1996) 1870–1876

    L. Li, Use of fourier series in the analysis of discontinuous periodic structures, JOSA A 13 (9) (1996) 1870–1876

    work page 1996

  14. [14]

    M. H. Cho, Y . Lu, J. Y . Rhee, Y . P. Lee, Rigorous approach on diffracted magneto-optical ef- fects from polar and longitudinal gyrotropic gratings, Optics express 16 (21) (2008) 16825– 16839

    work page 2008

  15. [15]

    O. P. Bruno, M. Lyon, C. Pérez-Arancibia, C. Turc, Windowed green function method for layered-media scattering, SIAM Journal on Applied Mathematics 76 (5) (2016) 1871–1898

    work page 2016

  16. [16]

    O. P. Bruno, C. Pérez-Arancibia, Windowed green function method for the helmholtz equa- tion in the presence of multiply layered media, Proceedings of the Royal Society A: Math- ematical, Physical and Engineering Sciences 473 (2202) (2017) 20170161

    work page 2017

  17. [17]

    Pérez-Arancibia, S

    C. Pérez-Arancibia, S. P. Shipman, C. Turc, S. Venakides, Domain decomposition for quasi- periodic scattering by layered media via robust boundary-integral equations at all frequen- cies, Communications in Computational Physics (2019). 20

    work page 2019

  18. [18]

    D. P. Nicholls, C. Pérez-Arancibia, C. Turc, Sweeping preconditioners for the iterative solution of quasiperiodic helmholtz transmission problems in layered media, Journal of Scientific Computing 82 (2) (2020) 1–45

    work page 2020

  19. [19]

    M. H. Cho, W. Cai, A parallel fast algorithm for computing the helmholtz integral operator in 3-D layered media, Journal of Computational Physics 231 (17) (2012) 5910–5925

    work page 2012

  20. [20]

    D. Chen, M. H. Cho, W. Cai, Accurate and efficient nystrom volume integral equation method for electromagnetic scattering of 3-d metamaterials in layered media, SIAM Journal on Scientific Computing 40 (1) (2018) B259–B282

    work page 2018

  21. [21]

    J. Lai, M. Kobayashi, L. Greengard, A fast solver for multi-particle scattering in a layered medium, Optics express 22 (17) (2014) 20481–20499

    work page 2014

  22. [22]

    M. H. Cho, A. H. Barnett, Robust fast direct integral equation solver for quasi-periodic scattering problems with a large number of layers, Optics express 23 (2) (2015) 1775–1799

    work page 2015

  23. [23]

    Zhang, A

    Y . Zhang, A. Gillman, A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems, BIT Numerical Mathematics 61 (1) (2021) 141–171

    work page 2021

  24. [24]

    M. H. Cho, Spectrally-accurate numerical method for acoustic scattering from doubly- periodic 3D multilayered media, Journal of Computational Physics 393 (2019) 46–58

    work page 2019

  25. [25]

    A. Boag, Y . Leviatan, A. Boag, Analysis of three-dimensional acoustic scattering from doubly periodic structures using a source model, The Journal of the Acoustical Society of America 91 (2) (1992) 572–580

    work page 1992

  26. [26]

    M. H. Cho, J. Huang, Adapting free-space fast multipole method for layered media green’s function: Algorithm and analysis, Applied and Computational Harmonic Analysis 51 (2021) 414–436

    work page 2021

  27. [27]

    H. Wang, W. Lu, A high-accuracy pml-bie solver for three-dimensional acoustic wave scat- tering in axisymmetric layered media, Journal of Computational Physics (2026) 114956

    work page 2026

  28. [28]

    Kupradze, M

    V . Kupradze, M. A. Aleksidze, The method of functional equations for the approximate so- lution of certain boundary value problems, USSR Computational Mathematics and Mathe- matical Physics 4 (4) (1964) 82–126

    work page 1964

  29. [29]

    Wu, P.-G

    B. Wu, P.-G. Martinsson, Zeta correction: a new approach to constructing corrected trape- zoidal quadrature rules for singular integral operators, Advances in Computational Mathe- matics 47 (3) (2021) 1–21

    work page 2021

  30. [30]

    J. Weed, L. Ding, J. Huang, M. H. Cho, Quadrature by two expansions for evaluating Helmholtz layer potentials, Journal of Scientific Computing 95 (3) (2023) 96

    work page 2023

  31. [31]

    L. Ding, J. Huang, J. L. Marzuola, Z. Tang, Quadrature by two expansions: Evaluating Laplace layer potentials using complex polynomial and plane wave expansions, Journal of Computational Physics 428 (2021) 109963

    work page 2021

  32. [32]

    M. H. Cho, W. Cai, Efficient and accurate computation of electric field dyadic green’s function in layered media, Journal of Scientific Computing 71 (2017) 1319–1350. 21

    work page 2017

  33. [33]

    M. H. Cho, J. Huang, D. Chen, W. Cai, A heterogeneous fmm for layered media helmholtz equation I: Two layers in R2, Journal of Computational Physics 369 (2018) 237–251

    work page 2018

  34. [34]

    B. Wu, M. H. Cho, Robust fast direct integral equation solver for three-dimensional dou- bly periodic scattering problems with a large number of layers, Journal of Computational Physics 495 (2023) 112573

    work page 2023

  35. [35]

    Liu, The numerical solution of frequency-domain acoustic and electromagnetic periodic scattering problems, Dartmouth College, 2016

    Y . Liu, The numerical solution of frequency-domain acoustic and electromagnetic periodic scattering problems, Dartmouth College, 2016

    work page 2016

  36. [36]

    W. C. Chew, Waves and fields in inhomogeneous media, John Wiley & Sons, 1999. 22

    work page 1999