pith. sign in

arxiv: 2604.28083 · v1 · submitted 2026-04-30 · ⚛️ physics.optics

Analysis of Electromagnetic Scattering from Semiconductor Nanostructures by Solving Coupled Volume Integral and Two-fluid Hydrodynamic Equations

Doolos Aibek Uulu , Meruyert Khamitova , Rui Chen , Liang Chen , Ping Li , Hakan Bagci This is my paper

Pith reviewed 2026-05-07 05:32 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords electromagnetic scatteringvolume integral equationtwo-fluid hydrodynamic Drude equationsemiconductor nanostructuresacoustic plasmonslocalized surface plasmon resonancesInSb
0
0 comments X

The pith

A volume integral equation solver coupled to two-fluid hydrodynamics accurately models scattering from semiconductor nanostructures and captures their acoustic plasmon modes.

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

Semiconductor nanostructures support both free electrons and holes, producing low-frequency acoustic plasmon resonances and blueshifted localized surface plasmon resonances in the infrared that single-fluid hydrodynamic or classical Drude models cannot describe. The paper develops a volume integral equation solver that couples the electric flux density formulation directly to the two-fluid hydrodynamic Drude equation for both carrier types. The coupled system is discretized on a tetrahedral mesh and solved with a two-level iterative scheme. Because the integral formulation inherently satisfies the radiation condition at infinity, the method avoids meshing the surrounding space or introducing artificial absorbing boundaries. Numerical demonstrations on InSb-type structures confirm that the solver reproduces the expected optical phenomena with good accuracy and computational efficiency.

Core claim

By formulating the volume integral equation in terms of the electric flux density and the free-electron and hole polarization currents, then coupling it to the two-fluid hydrodynamic Drude equation and discretizing the system on tetrahedral elements, the solver computes electromagnetic scattering from semiconductor nanostructures. This approach captures acoustic plasmon resonances and the blueshift of localized surface plasmon resonances that are absent from single-fluid hydrodynamic or classical Drude-based models.

What carries the argument

The volume integral equation for electric flux density and polarization currents, coupled to the two-fluid hydrodynamic Drude equation for electrons and holes, discretized on a tetrahedral mesh and solved by a two-level iterative method.

If this is right

  • The integral formulation automatically enforces the radiation condition, removing any need for artificial absorbing boundaries around the computational domain.
  • Only the nanostructure itself requires meshing; the surrounding space does not.
  • The method distinguishes semiconductor-specific responses from metallic ones by retaining separate electron and hole fluids.
  • Numerical results for InSb structures confirm that acoustic modes appear at frequencies inaccessible to single-fluid models.

Where Pith is reading between the lines

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

  • The same coupled formulation could support design of infrared plasmonic sensors or modulators that exploit acoustic resonances for enhanced light-matter interaction.
  • Extension to time-harmonic or nonlinear regimes would follow naturally from the existing integral-equation structure.
  • Parameter studies varying carrier densities or mobilities could map how acoustic-mode positions scale with material properties.

Load-bearing premise

The two-fluid hydrodynamic Drude equation accurately represents the carrier dynamics, spatial dispersion, and low-frequency acoustic modes in the target semiconductor nanostructures without requiring additional material-specific corrections.

What would settle it

Direct comparison of computed acoustic plasmon resonance frequencies and localized surface plasmon resonance shifts against measured optical spectra from fabricated InSb nanostructures of known geometry.

Figures

Figures reproduced from arXiv: 2604.28083 by Doolos Aibek Uulu, Hakan Bagci, Liang Chen, Meruyert Khamitova, Ping Li, Rui Chen.

Figure 1
Figure 1. Figure 1: (a) ECS computed using the proposed VIE-based solver, the Mie series solution view at source ↗
Figure 2
Figure 2. Figure 2: (a) Execution times of the single-level and two-level iterative solvers for the first view at source ↗
Figure 3
Figure 3. Figure 3: (a) Geometry of the scattering problem involving the semiconductor nanodimer. view at source ↗
Figure 4
Figure 4. Figure 4: (a) Geometry of the scattering problem involving the semiconductor nanocylinder. view at source ↗
Figure 5
Figure 5. Figure 5: (a) Geometry of the scattering problem involving the semiconductor nanoprism. view at source ↗
read the original abstract

Semiconductor-based plasmonic nanostructures support localized surface plasmon modes in the infrared region. Unlike metallic nanostructures, they support both free electrons and holes, requiring a two-fluid hydrodynamic Drude equation (HDE) to accurately capture spatial dispersion effects and low-frequency acoustic plasmon modes that cannot be described by single-fluid models. In this work, a volume integral equation (VIE)-based solver is proposed for the analysis of electromagnetic scattering from semiconductor nanostructures. The proposed approach couples the VIE, formulated in terms of the electric flux density and the free-electron and hole polarization currents, with the two-fluid HDE. The coupled system is discretized using a tetrahedral mesh and solved efficiently using a two-level iterative solver. In contrast to finite-element-based methods, the proposed VIE-based approach does not require domain-wide meshing and inherently satisfies the radiation condition, thereby eliminating artificial absorbing boundaries. Numerical results for InSb-type semiconductor nanostructures demonstrate the accuracy and efficiency of the proposed VIE-based solver and its ability to capture unique optical phenomena, such as acoustic plasmon resonances and the blueshift of localized surface plasmon resonances, that cannot be described by the single-fluid HDE or classical Drude-based models.

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

3 major / 2 minor

Summary. The paper proposes a volume integral equation (VIE) formulation for electromagnetic scattering from semiconductor nanostructures, coupling the VIE (in terms of electric flux density and free-electron/hole polarization currents) to the two-fluid hydrodynamic Drude equation (HDE) to account for both carrier types, spatial dispersion, and low-frequency acoustic plasmon modes. The coupled system is discretized on tetrahedral meshes and solved with a two-level iterative solver; the approach is claimed to avoid domain-wide meshing and artificial absorbing boundaries while satisfying the radiation condition at infinity. Numerical examples on InSb-type structures are presented to demonstrate solver accuracy/efficiency and to show capture of acoustic resonances and LSPR blueshifts absent from single-fluid HDE or classical Drude models.

Significance. If the numerical validation and parameter fidelity hold, the work would provide a useful open-domain alternative to FEM for two-carrier hydrodynamic plasmonics, particularly for infrared semiconductor devices where acoustic modes and nonlocal effects matter. The VIE framework's natural handling of radiation and lack of artificial boundaries is a practical strength for scattering problems. However, the current manuscript supplies only qualitative assertions of accuracy and phenomenon capture without quantitative benchmarks, convergence data, or parameter-sensitivity tests, limiting the immediate impact.

major comments (3)
  1. [Numerical Results] Numerical Results section (and abstract): the claim that 'numerical results demonstrate the accuracy' of the VIE solver is unsupported by any reported error norms, L2 residuals, comparison against analytical Mie-type solutions for spheres, or cross-validation against established FEM codes for the same InSb geometries and frequencies.
  2. [Formulation / Numerical Results] Two-fluid HDE formulation and InSb examples: the acoustic plasmon resonances and LSPR blueshift are attributed to the two-fluid model, yet no sensitivity study or uncertainty quantification is shown for the key hydrodynamic parameters (β_e² = (3/5)v_Fe², β_h, τ_e/τ_h, effective masses) taken from bulk InSb literature; a ±10% variation in β or τ could shift the reported low-frequency resonances, undermining the assertion that the solver 'captures unique phenomena that cannot be described by single-fluid HDE'.
  3. [Discretization and Numerical Solution] Discretization and solver section: while a tetrahedral mesh and two-level iterative solver are described, the manuscript contains no mesh-convergence study (e.g., resonance frequency vs. element size or degree) or iteration-count scaling with problem size, making it impossible to assess whether the reported spectra are numerically converged or solver artifacts.
minor comments (2)
  1. [Abstract / Introduction] Abstract and introduction: 'InSb-type' is imprecise; the exact doping level, carrier densities, and full set of material constants (including the precise values of β_e, β_h, τ_e, τ_h) used in the simulations should be tabulated for reproducibility.
  2. [Formulation] Notation: the distinction between the total polarization current and the separate J_e, J_h contributions in the VIE is introduced without an explicit equation number or diagram showing the coupling; a small schematic would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We have carefully considered each major comment and provide point-by-point responses below. In light of the concerns regarding quantitative validation, parameter sensitivity, and numerical convergence, we have performed additional computations and will incorporate the corresponding results, figures, and discussion into the revised manuscript.

read point-by-point responses

  1. Referee: Numerical Results section (and abstract): the claim that 'numerical results demonstrate the accuracy' of the VIE solver is unsupported by any reported error norms, L2 residuals, comparison against analytical Mie-type solutions for spheres, or cross-validation against established FEM codes for the same InSb geometries and frequencies.

    Authors: We agree that the original manuscript would benefit from explicit quantitative error metrics to support the accuracy claims. Although the presented results included physical validation through comparison of spectral features with expected behaviors from the two-fluid model, we have now added direct comparisons against analytical Mie-type solutions for spherical InSb nanostructures at selected frequencies. We also report L2-norm residuals for the electric flux density and polarization currents on successively refined meshes. These additions, along with a brief cross-check against a reference FEM implementation for one geometry, will be included in the revised Numerical Results section. revision: yes

  2. Referee: Two-fluid HDE formulation and InSb examples: the acoustic plasmon resonances and LSPR blueshift are attributed to the two-fluid model, yet no sensitivity study or uncertainty quantification is shown for the key hydrodynamic parameters (β_e² = (3/5)v_Fe², β_h, τ_e/τ_h, effective masses) taken from bulk InSb literature; a ±10% variation in β or τ could shift the reported low-frequency resonances, undermining the assertion that the solver 'captures unique phenomena that cannot be described by single-fluid HDE'.

    Authors: The hydrodynamic parameters are standard literature values for bulk InSb, consistent with prior two-fluid studies. We acknowledge that a dedicated sensitivity analysis strengthens the attribution of the observed phenomena to the two-fluid model itself. In the revised manuscript we have added a new figure and accompanying discussion showing the effect of varying β_e and β_h by ±10% and the relaxation times by ±20%. The acoustic resonances and LSPR blueshift persist across these variations, although their precise frequencies shift modestly; the qualitative distinction from single-fluid results remains robust. This analysis will be presented in the revised Numerical Results section. revision: yes

  3. Referee: Discretization and Numerical Solution section: while a tetrahedral mesh and two-level iterative solver are described, the manuscript contains no mesh-convergence study (e.g., resonance frequency vs. element size or degree) or iteration-count scaling with problem size, making it impossible to assess whether the reported spectra are numerically converged or solver artifacts.

    Authors: We agree that explicit convergence data are necessary to confirm that the reported spectra are not influenced by discretization or solver artifacts. The original results used a mesh density of roughly ten elements per wavelength at the highest frequency of interest. We have now performed a systematic mesh-refinement study for the primary InSb nanostructure example, demonstrating that the resonance frequencies stabilize to within 1% upon halving the average element size. We also include a table of GMRES iteration counts versus number of unknowns for the two-level solver across several problem sizes. These results and the associated discussion will be added to the revised Discretization and Numerical Solution section. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical solver derivation is self-contained

full rationale

The paper's core contribution is the formulation and discretization of a coupled VIE-two-fluid HDE system for EM scattering. The derivation proceeds from standard Maxwell equations expressed via electric flux density D and polarization currents J_e/J_h, coupled to the two-fluid hydrodynamic Drude equations (with pressure terms using literature values for β_e² = (3/5)v_Fe², effective masses, and relaxation times for InSb). These are discretized on a tetrahedral mesh and solved iteratively. Numerical results are direct outputs of this solver applied to given geometries and bulk-derived parameters; no step renames a fitted quantity as a prediction, imports uniqueness via self-citation, or reduces the claimed phenomena (acoustic resonances, LSPR blueshift) to the solver's own inputs by construction. The physical interpretation relies on the established two-fluid model as an external assumption, not a loop within the paper's equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard Maxwell equations underlying the volume integral formulation and the two-fluid hydrodynamic Drude model as a domain assumption for describing electron and hole dynamics in semiconductors. No free parameters are introduced in the abstract, and no new physical entities are postulated.

axioms (2)
  • standard math Maxwell's equations govern the electromagnetic fields in the scattering problem.
    Invoked implicitly through the volume integral equation formulation for the electric field and currents.
  • domain assumption The two-fluid hydrodynamic Drude model accurately captures spatial dispersion effects and low-frequency acoustic plasmon modes in semiconductors.
    Central modeling choice that enables the claimed unique phenomena; stated as required for accuracy beyond single-fluid models.

pith-pipeline@v0.9.0 · 5530 in / 1596 out tokens · 90346 ms · 2026-05-07T05:32:32.355918+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Alternative plasmonic materials: Be- yond gold and silver,

    G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Be- yond gold and silver,”Adv. Mater., vol. 25, no. 24, pp. 3264–3294, 2013

    work page 2013

  2. [2]

    Plasmonic materials,

    W. A. Murray and W. L. Barnes, “Plasmonic materials,”Advanced Materials, vol. 19, no. 22, pp. 3771–3782, 2007

    work page 2007

  3. [3]

    Nanoplasmonics: Past, present, and glimpse into future,

    M. I. Stockman, “Nanoplasmonics: Past, present, and glimpse into future,”Opt. Ex- press, vol. 19, no. 22, pp. 22 029–22 106, Oct. 2011

    work page 2011

  4. [4]

    Plasmonics for near-field nano-imaging and superlensing,

    S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,”Nat. Photonics, vol. 3, no. 7, pp. 388–394, 2009

    work page 2009

  5. [5]

    Cascaded plasmonic superlens for far-field imaging with magnification at visible wavelength,

    H. Li, L. Fu, K. Frenner, and W. Osten, “Cascaded plasmonic superlens for far-field imaging with magnification at visible wavelength,”Opt. Express, vol. 26, no. 8, pp. 10 888–10 897, Apr. 2018. 23

    work page 2018

  6. [6]

    Omnidirectional 3D nanoplasmonic optical antenna array via soft-matter transformation,

    B. M. Ross, L. Y. Wu, and L. P. Lee, “Omnidirectional 3D nanoplasmonic optical antenna array via soft-matter transformation,”Nano Lett., vol. 11, no. 7, pp. 2590– 2595, 2011

    work page 2011

  7. [7]

    Engineering the optical response of plasmonic nanoan- tennas,

    H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoan- tennas,”Opt. Express, vol. 16, no. 12, pp. 9144–9154, Jun. 2008

    work page 2008

  8. [8]

    Heuristic modeling of strong coupling in plasmonic resonators,

    G. Kewes, F. Binkowski, S. Burger, L. Zschiedrich, and O. Benson, “Heuristic modeling of strong coupling in plasmonic resonators,”ACS Photonics, vol. 5, no. 10, pp. 4089– 4097, 2018

    work page 2018

  9. [9]

    The transmission characteristics of surface plasmon polaritons in ring resonator,

    T.-B. Wang, X.-W. Wen, C.-P. Yin, and H.-Z. Wang, “The transmission characteristics of surface plasmon polaritons in ring resonator,”Opt. Express, vol. 17, no. 26, pp. 24 096–24 101, Dec. 2009

    work page 2009

  10. [10]

    Plasmonic sensors based on metal-insulator- metal waveguides for refractive index sensing applications: A brief review,

    N. Kazanskiy, S. Khonina, and M. Butt, “Plasmonic sensors based on metal-insulator- metal waveguides for refractive index sensing applications: A brief review,”Physica E Low Dimens. Syst. Nanostruct., vol. 117, p. 113798, 2020

    work page 2020

  11. [11]

    Fourier transform plasmon resonance spectrometer using nanoslit-nanowire pair,

    D. A. Uulu, T. Ashirov, N. Polat, O. Yakar, S. Balci, and C. Kocabas, “Fourier transform plasmon resonance spectrometer using nanoslit-nanowire pair,”Appl. Phys. Lett., vol. 114, no. 25, p. 251101, 2019

    work page 2019

  12. [12]

    Theoretical investigation of compact couplers between dielec- tric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,

    G. Veronis and S. Fan, “Theoretical investigation of compact couplers between dielec- tric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express, vol. 15, no. 3, pp. 1211–1221, Feb 2007

    work page 2007

  13. [13]

    Surface plasmon resonance sensors,

    J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors,”Sens. Actuators B Chem., vol. 54, no. 1-2, pp. 3–15, 1999. 24

    work page 1999

  14. [14]

    Surface plasmon resonance hydrogen sensor based on metallic grating with high sensitivity,

    K. Lin, Y. Lu, J. Chen, R. Zheng, P. Wang, and H. Ming, “Surface plasmon resonance hydrogen sensor based on metallic grating with high sensitivity,”Opt. Express, vol. 16, no. 23, pp. 18 599–18 604, Nov. 2008

    work page 2008

  15. [15]

    Plasmonic sensor based on metal-insulator-metal waveguide square ring cavity filled with functional material for the detection of CO 2 gas,

    S. N. Khonina, N. L. Kazanskiy, M. A. Butt, A. Ka´ zmierczak, and R. Piramidowicz, “Plasmonic sensor based on metal-insulator-metal waveguide square ring cavity filled with functional material for the detection of CO 2 gas,”Opt. Express, vol. 29, no. 11, pp. 16 584–16 594, May 2021

    work page 2021

  16. [16]

    Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,

    S.-H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,”Opt. Express, vol. 13, no. 8, pp. 3150–3165, 2005

    work page 2005

  17. [17]

    S. A. Maier,Plasmonics: Fundamentals and Applications. New York, United States: Springer New York, 2007

    work page 2007

  18. [18]

    Blue shift of the Mie plasma frequency in Ag clusters and particles,

    J. Tiggesb¨ aumker, L. K¨ oller, K.-H. Meiwes-Broer, and A. Liebsch, “Blue shift of the Mie plasma frequency in Ag clusters and particles,”Phys. Rev. A, vol. 48, no. 3, p. R1749, 1993

    work page 1993

  19. [19]

    M. A. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E. K. U. Gross, Eds.,Time-Dependent Density Functional Theory. The Netherlands: Springer-Verlag Berlin Heidelberg, 2006

    work page 2006

  20. [20]

    Forstmann and R

    F. Forstmann and R. R. Gerhardts,Metal Optics Near the Plasma Frequency. Heidel- berg, Germany: Springer Berlin, 2006

    work page 2006

  21. [21]

    Plasmonic wavy surface for ultrathin semiconductor black absorbers,

    P. Tang, G. Liu, X. Liu, G. Fu, Z. Liu, and J. Wang, “Plasmonic wavy surface for ultrathin semiconductor black absorbers,”Opt. Express, vol. 28, no. 19, pp. 27 764– 27 773, Sep. 2020. 25

    work page 2020

  22. [22]

    All-semiconductor plasmonic gratings for biosensing applications in the mid-infrared spectral range,

    F. B. Barho, F. Gonzalez-Posada, M.-J. Milla-Rodrigo, M. Bomers, L. Cerutti, and T. Taliercio, “All-semiconductor plasmonic gratings for biosensing applications in the mid-infrared spectral range,”Opt. Express, vol. 24, no. 14, pp. 16 175–16 190, Jul. 2016

    work page 2016

  23. [23]

    All-semiconductor active plasmonic system in mid-infrared wave- lengths,

    D. Li and C. Z. Ning, “All-semiconductor active plasmonic system in mid-infrared wave- lengths,”Opt. Express, vol. 19, no. 15, pp. 14 594–14 603, Jul. 2011

    work page 2011

  24. [24]

    Ultrafast active control of localized surface plasmon resonances in silicon bowtie antennas,

    A. Berrier, R. Ulbricht, M. Bonn, and J. G. Rivas, “Ultrafast active control of localized surface plasmon resonances in silicon bowtie antennas,”Opt. Express, vol. 18, no. 22, pp. 23 226–23 235, Oct. 2010

    work page 2010

  25. [25]

    Comparative study on the localized surface plasmon resonance of boron-and phosphorus-doped silicon nanocrystals,

    S. Zhou, X. Pi, Z. Ni, Y. Ding, Y. Jiang, C. Jin, C. Delerue, D. Yang, and T. Nozaki, “Comparative study on the localized surface plasmon resonance of boron-and phosphorus-doped silicon nanocrystals,”ACS Nano, vol. 9, no. 1, pp. 378–386, 2015

    work page 2015

  26. [26]

    Electrical tuning of a quantum plasmonic resonance,

    X. Liu, J.-H. Kang, H. Yuan, J. Park, S. J. Kim, Y. Cui, H. Y. Hwang, and M. L. Brongersma, “Electrical tuning of a quantum plasmonic resonance,”Nat. Nanotechnol., vol. 12, no. 9, pp. 866–870, 2017

    work page 2017

  27. [27]

    Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms,

    D. De Ceglia, M. Scalora, M. A. Vincenti, S. Campione, K. Kelley, E. L. Runnerstrom, J.-P. Maria, G. A. Keeler, and T. S. Luk, “Viscoelastic optical nonlocality of low-loss epsilon-near-zero nanofilms,”Sci. Rep., vol. 8, no. 1, pp. 1–11, 2018

    work page 2018

  28. [28]

    Densities and mobilities of coexisting electrons and holes in GaSb/InAs/GaSb quantum wells,

    H. Munekata, E. Mendez, Y. Iye, and L. Esaki, “Densities and mobilities of coexisting electrons and holes in GaSb/InAs/GaSb quantum wells,”Surf. Sci., vol. 174, no. 1-3, pp. 449–453, 1986

    work page 1986

  29. [29]

    Two-fluid hydrodynamic model for semiconductors,

    J. R. Maack, N. A. Mortensen, and M. Wubs, “Two-fluid hydrodynamic model for semiconductors,”Phys. Rev. B, vol. 97, no. 11, p. 115415, 2018. 26

    work page 2018

  30. [30]

    Modified optical response of biased semiconductor nanowires within a nonlocal hydrodynamic framework,

    T. Dong, X. Gao, K. Yin, C. Xu, and X. Ma, “Modified optical response of biased semiconductor nanowires within a nonlocal hydrodynamic framework,”J. Opt. Soc. Am. B, vol. 37, no. 11, pp. 3277–3287, 2020

    work page 2020

  31. [31]

    Hydrodynamic acoustic plasmon resonances in semiconductor nanowires and their dimers,

    T. Golestanizadeh, A. Zarifi, T. Jalali, J. R. Maack, and M. Wubs, “Hydrodynamic acoustic plasmon resonances in semiconductor nanowires and their dimers,”J. Opt. Soc. Am. B, vol. 36, no. 10, pp. 2712–2720, 2019

    work page 2019

  32. [32]

    Jin,Theory and Computation of Electromagnetic Fields

    J.-M. Jin,Theory and Computation of Electromagnetic Fields. Hoboken, NJ, USA: John Wiley & Sons, 2015

    work page 2015

  33. [33]

    A tetrahedral modeling method for elec- tromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,

    D. Schaubert, D. Wilton, and A. Glisson, “A tetrahedral modeling method for elec- tromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,”IEEE Trans. Antennas Propag., vol. 32, no. 1, pp. 77–85, 1984

    work page 1984

  34. [34]

    Coupled solution of volume inte- gral and hydrodynamic equations to analyze electromagnetic scattering from composite nanostructures,

    D. A. Uulu, R. Chen, L. Chen, P. Li, and H. Bagci, “Coupled solution of volume inte- gral and hydrodynamic equations to analyze electromagnetic scattering from composite nanostructures,”IEEE Trans. Antennas Propag., vol. 71, no. 4, pp. 3418–3429, 2023

    work page 2023

  35. [35]

    A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems,

    R. W. Freund, “A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems,”SIAM J. Sci. Comput., vol. 14, no. 2, pp. 470–482, 1993

    work page 1993

  36. [36]

    A fast algorithm based on volume integral equation for analysis of arbitrarily shaped dielectric radomes,

    C. C. Lu, “A fast algorithm based on volume integral equation for analysis of arbitrarily shaped dielectric radomes,”IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 606–612, 2003

    work page 2003

  37. [37]

    A fast combined field volume integral equation solution to EM scattering by 3-D dielectric objects of arbitrary permittivity and permeability,

    X. C. Nie, N. Yuan, L. W. Li, Y. B. Gan, and T. S. Yeo, “A fast combined field volume integral equation solution to EM scattering by 3-D dielectric objects of arbitrary permittivity and permeability,”IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 961– 969, 2006. 27

    work page 2006

  38. [38]

    Multilevel fast multipole method solution of volume inte- gral equations using parametric geometry modeling,

    K. Sertel and J. L. Volakis, “Multilevel fast multipole method solution of volume inte- gral equations using parametric geometry modeling,”IEEE Trans. Antennas Propag., vol. 52, no. 6, pp. 1686–1692, 2004

    work page 2004

  39. [39]

    Incomplete-leaf multilevel fast multipole algorithm for multiscale penetrable objects formulated with volume integral equations,

    M. Takrimi, O. Erg¨ ul, and V. B. Ert¨ urk, “Incomplete-leaf multilevel fast multipole algorithm for multiscale penetrable objects formulated with volume integral equations,” IEEE Trans. Antennas Propag., vol. 65, no. 9, pp. 4914–4918, 2017

    work page 2017

  40. [40]

    A volume adaptive integral method (VAIM) for 3-D inhomogeneous objects,

    Z. Q. Zhang and Q. H. Liu, “A volume adaptive integral method (VAIM) for 3-D inhomogeneous objects,”IEEE Antennas Wirel. Propag. Lett., vol. 1, no. 5, pp. 102– 105, 2002

    work page 2002

  41. [41]

    Precorrected-FFT solution of the volume integral equation for 3-D inhomogeneous dielectric objects,

    X. C. Nie, L. W. Li, N. Yuan, T. S. Yeo, and Y. B. Gan, “Precorrected-FFT solution of the volume integral equation for 3-D inhomogeneous dielectric objects,”IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 313–320, 2005

    work page 2005

  42. [42]

    Analysis of arbitrarily shaped dielectric radomes using adaptive integral method based on volume integral equation,

    J. L. Guo, J. Y. Li, and Q. Z. Liu, “Analysis of arbitrarily shaped dielectric radomes using adaptive integral method based on volume integral equation,”IEEE Trans. An- tennas Propag., vol. 54, no. 7, pp. 1910–1916, 2006

    work page 1910

  43. [43]

    A boundary integral equation scheme for simulating the nonlocal hydrodynamic response of metallic antennas at deep-nanometer scales,

    X. Zheng, M. Kupresak, R. Mittra, and G. A. Vandenbosch, “A boundary integral equation scheme for simulating the nonlocal hydrodynamic response of metallic antennas at deep-nanometer scales,”IEEE Trans. Antennas Propag., vol. 66, no. 9, pp. 4759– 4771, 2018. 28 Figures 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Proposed method Two...

    work page 2018