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arxiv: 2604.03149 · v1 · submitted 2026-04-03 · 🧮 math-ph · math.MP· physics.optics

Scattering of TE and TM waves by inhomogeneities of a 2D material, low-frequency behavior of the scattering amplitude, and low-frequency invisibility

Farhang Loran , Ali Mostafazadeh This is my paper

Pith reviewed 2026-05-13 18:49 UTC · model grok-4.3

classification 🧮 math-ph math.MPphysics.optics
keywords TE TM scatteringlow-frequency expansiontransfer matrixDyson series2D cloakingBergmann equationscattering amplitudeinvisibility
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The pith

A Dyson series for the fundamental transfer matrix yields the leading two terms of the low-frequency scattering amplitude for TE and TM waves in a thin 2D layer, enabling an explicit cloaking construction.

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

The paper treats stationary scattering of TE and TM waves governed by Bergmann's acoustic equation in an effectively two-dimensional isotropic medium. It introduces the fundamental transfer matrix as an integral operator that obeys a Dyson series generated by a non-Hermitian Hamiltonian. When the material inhomogeneities are confined to a slab of thickness ℓ, this series produces an explicit power-series expansion of the scattering amplitude in the small parameter kℓ. The first two coefficients are obtained in closed form, verified on solvable models, and then used to derive a low-frequency cloaking condition that works simultaneously for both polarizations.

Core claim

The stationary scattering problem for TE and TM waves is recast in terms of a fundamental transfer matrix operator M-hat whose Dyson series in the confined-layer geometry supplies analytic expressions for the O(kℓ) and O((kℓ)²) contributions to the scattering amplitude; these expressions directly furnish a low-frequency cloaking scheme valid for both wave types.

What carries the argument

The fundamental transfer matrix operator M-hat, an integral operator whose Dyson series encodes the full scattering data and permits systematic expansion in kℓ.

If this is right

  • The scattering amplitude is given to second order by two explicit integral expressions involving the inhomogeneity profile.
  • A cloaking condition follows at once by setting the leading two coefficients to zero, and this condition is identical for TE and TM waves.
  • The same expansion applies verbatim to acoustic waves in 2D fluids because they obey Bergmann's equation.
  • The Dyson series supplies a practical computational route that avoids full numerical solution of the wave equation at long wavelengths.

Where Pith is reading between the lines

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

  • The same operator construction may be adapted to other first-order wave systems whose Green's function admits a similar Dyson expansion.
  • Low-frequency cloaking achieved this way requires no active elements and may therefore be realizable in passive 2D metamaterial sheets.
  • Because the next-to-leading term involves a non-Hermitian Hamiltonian, it may encode directional asymmetry or gain-loss effects that are invisible in the leading-order term.

Load-bearing premise

All material inhomogeneities are confined inside a layer whose thickness ℓ satisfies kℓ ≪ 1.

What would settle it

For a concrete piecewise-constant or exactly solvable profile confined to thickness ℓ, compute the exact scattering amplitude at a chosen small kℓ and verify that the difference from the two-term analytic approximation scales as O((kℓ)³).

Figures

Figures reproduced from arXiv: 2604.03149 by Ali Mostafazadeh, Farhang Loran.

Figure 1
Figure 1. Figure 1: Schematic views of the scattering setup for the scatter [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic views of the scattering of a TM wave with wavenu [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graph of θ1± as a function of k/K for z1 and z2 given by (113). The two curves meet at k/K = κ0/K = 0.510. a function of k/K. In [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graphs of the real and imaginary parts of [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Graphs of |τ1±| 2×104 as functions of kℓ for z1 and z2 given by (113), ℓ = 100 nm, and K = π µm−1 . The yellow region corresponds to κ0 < k ≤ K. The dashed and dotted curves respectively correspond to the outcome of the first- and second-order low-frequency approximations, while the solid curve corresponds to the exact calculation of τ0−. The media fulfilling this condition do not scatter TE and TM waves o… view at source ↗
Figure 6
Figure 6. Figure 6: Schematic view of the slab S⋆ and its coatings S− and S+ respectively colored in shades of blue, green, and red. The coloring patterns represent the dependence of the inhomogeneities of the slab on both x and y, and dependence of the inhomogeneities of the coatings on y. Distances are measured in units of ℓ. If E−(y) = E+(y) = 1, we can satisfy (130) by setting ˆℓ±(y) = 0. In this case, ℓ±(y) = 0, ℓS(y) = … view at source ↗
Figure 7
Figure 7. Figure 7: Schematic view of the coated slab on the left and plots of th [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
read the original abstract

The propagation of the transverse electric (TE) and transverse magnetic (TM) waves in an effectively two-dimensional (2D) isotropic medium is described by Bergmann's equation of acoustics. We develop a dynamical formulation of the stationary scattering of these waves and explore its application in the study of the low-frequency behavior of the scattering data. Specifically, we introduce a suitable notion of fundamental transfer matrix for TE and TM waves in 2D. This is an integral operator $\widehat{\mathbf{M}}$ that carries the information about the scattering properties of the medium and admits a Dyson series expansion involving a non-Hermitian Hamiltonian operator. For situations where the inhomogeneities of the medium are confined to a layer of thickness $\ell$, we use the Dyson series for $\widehat{\mathbf{M}}$ to construct the series expansion of the scattering amplitude in powers of $k\ell$, where $k$ is the incident wavenumber. We derive analytic expressions for the leading- and next-to-leading-order terms of this series, verify the effectiveness of their application to a class of exactly solvable models, and use them to study low-frequency invisibility. In particular, we develop a low-frequency cloaking scheme which is applicable for both TE and TM waves. Our results have immediate applications in the study of low-frequency scattering of acoustic waves in a 2D fluid as these waves are also described by Bergmann's equation.

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 dynamical formulation for stationary scattering of TE and TM waves in 2D isotropic media governed by Bergmann's equation. It introduces a fundamental transfer matrix as an integral operator admitting a Dyson series expansion with a non-Hermitian Hamiltonian. For inhomogeneities confined to a layer of thickness ℓ, analytic expressions are derived for the leading- and next-to-leading-order terms in the low-frequency expansion of the scattering amplitude in powers of kℓ. These are verified on exactly solvable models and used to construct a low-frequency cloaking scheme applicable to both TE and TM waves, with applications noted for 2D acoustic scattering.

Significance. If the derivations hold, the work supplies explicit analytic tools for low-frequency scattering and invisibility in 2D media, extending directly to acoustics via Bergmann's equation. The verification against exactly solvable models and the parameter-free character of the leading-order terms constitute concrete strengths that support the cloaking construction without numerical fitting.

major comments (1)
  1. [Abstract and the section deriving the kℓ expansion from the Dyson series] The conversion of the Dyson series for the transfer matrix into an explicit power series in kℓ (and the subsequent cancellation of O(kℓ) and O((kℓ)²) coefficients in the cloaking scheme) is performed only after restricting the support of the inhomogeneity to a slab of width ℓ. The manuscript verifies the resulting formulae on models obeying this restriction but supplies neither an a priori radius-of-convergence estimate nor error bounds that remain valid when the slab condition is relaxed even modestly. This assumption is load-bearing for the stated applicability of the low-frequency invisibility results.
minor comments (2)
  1. [Section introducing the fundamental transfer matrix] The definition and domain of the non-Hermitian Hamiltonian appearing in the Dyson kernel should be stated more explicitly, including its action on the relevant function space, to aid independent reproduction of the series.
  2. [Verification section] In the verification against solvable models, list the precise functional forms, boundary conditions, and numerical values of parameters used so that the agreement with the analytic expressions can be checked directly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. We address the single major comment below, clarifying the scope of our assumptions while maintaining the manuscript's focus.

read point-by-point responses

  1. Referee: [Abstract and the section deriving the kℓ expansion from the Dyson series] The conversion of the Dyson series for the transfer matrix into an explicit power series in kℓ (and the subsequent cancellation of O(kℓ) and O((kℓ)²) coefficients in the cloaking scheme) is performed only after restricting the support of the inhomogeneity to a slab of width ℓ. The manuscript verifies the resulting formulae on models obeying this restriction but supplies neither an a priori radius-of-convergence estimate nor error bounds that remain valid when the slab condition is relaxed even modestly. This assumption is load-bearing for the stated applicability of the low-frequency invisibility results.

    Authors: We appreciate the referee's observation. The restriction of inhomogeneities to a slab of thickness ℓ is stated explicitly in the abstract and is the geometric setting used throughout the derivation in Section 3, where the Dyson series for the fundamental transfer matrix is converted to a power series in kℓ. This layered geometry is a standard and physically relevant assumption for the low-frequency cloaking construction we propose, which applies to both TE and TM waves (and hence to 2D acoustics via Bergmann's equation). The exact solvability of the verification models is likewise tied to this slab support. We do not supply a priori radius-of-convergence estimates or uniform error bounds outside the slab setting because the manuscript's scope is confined to that case; the underlying Dyson series converges for sufficiently small kℓ (controlled by the operator norm of the potential), but explicit bounds are not derived. We are prepared to insert a short clarifying sentence in the introduction reiterating that the analytic expansions and cloaking scheme are derived under the slab restriction, thereby making the domain of applicability fully transparent. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation from Bergmann equation via Dyson series under explicit slab assumption

full rationale

The paper begins from the standard Bergmann equation for TE/TM waves, introduces the integral operator M̂ as the fundamental transfer matrix, and applies the standard Dyson series to it. The power-series expansion in kℓ is obtained only after imposing the modeling assumption that inhomogeneities are confined to a slab of thickness ℓ; this is stated explicitly as a prerequisite rather than derived. Analytic expressions for the leading and next-to-leading coefficients are obtained by direct term-by-term integration of the series, then checked against independent exactly solvable models that satisfy the same slab condition. No parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a uniqueness theorem, and no ansatz is imported from prior work by the same authors. The cloaking construction follows directly from setting the derived O(kℓ) and O((kℓ)²) coefficients to zero. The derivation chain is therefore self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard Bergmann equation as domain assumption and introduces one new operator without free parameters or fitted values.

axioms (1)
  • domain assumption Propagation of TE and TM waves in an effectively 2D isotropic medium is described by Bergmann's equation of acoustics.
    This equation is invoked as the starting point for the entire scattering formulation.
invented entities (1)
  • fundamental transfer matrix M-hat no independent evidence
    purpose: Integral operator carrying all scattering information for the 2D medium and admitting Dyson series expansion
    Newly defined in the paper for TE and TM waves; no independent evidence outside this work is provided.

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Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    L. L. S´ anchez-Soto, J. J. Monz´ ona, A. G. Barriuso, and J. F . Cari˜ nena, The transfer matrix: A geometrical perspective, Phys. Rep. 513, 191 (2012)

    work page 2012

  2. [2]

    Mostafazadeh, Transfer matrix in scattering theory: A sur vey of basic properties and recent developments, Turkish J

    A. Mostafazadeh, Transfer matrix in scattering theory: A sur vey of basic properties and recent developments, Turkish J. Phys. 44, 472-527 (2020)

    work page 2020

  3. [3]

    R. C. Jones, A new calculus for the treatment of optical system s I. Description and discussion of the Calculus, J. Opt. Soc. Am. 31, 488-493 (1941)

    work page 1941

  4. [4]

    Abel` es, Recherches sur la propagation des ondes ´ electro magn´ etiques sinuso ¨ ıdales dans les milieux stratifı´ es Application aux couches minces, Ann

    F. Abel` es, Recherches sur la propagation des ondes ´ electro magn´ etiques sinuso ¨ ıdales dans les milieux stratifı´ es Application aux couches minces, Ann. Phys. (Paris) 12, 596-640 (1950)

    work page 1950

  5. [5]

    W. T. Thompson, Transmission of elastic waves through a stratifi ed solid medium, J. Appl. Phys. 21, 89-93 (1950)

    work page 1950

  6. [6]

    Optics in stratified and anisotropic media: 4 × 4-matrix formulation,

    D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502-510 (1972)

    work page 1972

  7. [7]

    A. S. McLean and J. B. Pendry, A polarized transfer matrix for e lectromagnetic waves in structured media, J. Mod. Opt. 41, 1781-1802 (1994). 27

    work page 1994

  8. [8]

    Transfer matrix techniques for elec tromagnetic waves,

    J. B. Pendry and P. M. Bell, “Transfer matrix techniques for elec tromagnetic waves,” in Photonic Band Gap Materials , pp 203-228, edited by Soukoulis C. M., NATO ASI Series, vol. 315 (Springer, Dordrecht, 1996)

    work page 1996

  9. [9]

    P. Yeh, A. Yariv, and C.-S. Hong, Electromagnetic propagation in periodic stratified media. I. General theory, J. Opt. Soc. Am. 67, 423-438 (1977)

    work page 1977

  10. [10]

    Pereyra, Resonant tunneling and band mixing in multichannel s uperlattices, Phys

    P. Pereyra, Resonant tunneling and band mixing in multichannel s uperlattices, Phys. Rev. Lett. 80, 2677-2680 (1998)

    work page 1998

  11. [11]

    D. J. Griffiths and C. A. Steinke, Waves in locally periodic media, Am. J. Phys. 69, 137-154 (2001)

    work page 2001

  12. [12]

    Mostafazadeh, A dynamical formulation of one-dimensional scattering theory and its ap- plications in optics, Ann

    A. Mostafazadeh, A dynamical formulation of one-dimensional scattering theory and its ap- plications in optics, Ann. Phys. (NY) 341, 77 (2014)

    work page 2014

  13. [13]

    Loran and A

    F. Loran and A. Mostafazadeh, Fundamental transfer matr ix and dynamical formulation of stationary scattering in two and three dimensions, Phys. Rev. A 104, 032222 (2021)

    work page 2021

  14. [14]

    Loran and A

    F. Loran and A. Mostafazadeh, Fundamental transfer matr ix for electromagnetic waves, scat- tering by a planar collection of point scatterers, and anti-PT -symm etry, Phys. Rev. A 107, 012203 (2023)

    work page 2023

  15. [15]

    Loran and A

    F. Loran and A. Mostafazadeh, Renormalization of multi-delta- function point scatterers in two and three dimensions, the coincidence-limit problem, and its reso lution, Ann. Phys. (NY) 443, 168966 (2022)

    work page 2022

  16. [16]

    Loran and A

    F. Loran and A. Mostafazadeh, Exactness of the first Born a pproximation in electromagnetic scattering, Prog. Theor. Exp. Phys. 2024, 023A03 (2024)

    work page 2024

  17. [17]

    Loran and A

    F. Loran and A. Mostafazadeh, Can N-th order Born approxim ation be exact? J. Phys. A: Math. Theor. 57, 335205 (2024)

    work page 2024

  18. [18]

    Loran and A

    F. Loran and A. Mostafazadeh, Dynamical formulation of low-e nergy scattering in one di- mension, J. Math. Phys. 62, 042103 (2021)

    work page 2021

  19. [19]

    Loran and A

    F. Loran and A. Mostafazadeh, Low-frequency scattering d efined by the Helmholtz equation in one dimension, J. Phys. A: Math. Theor. 54, 315204 (2021)

    work page 2021

  20. [20]

    Loran and A

    F. Loran and A. Mostafazadeh, Dynamical formulation of low-f requency scattering in two and three dimensions, Phys. Rev. A 111, 032215 (2025)

    work page 2025

  21. [21]

    N. P. Zhuck and A. G. Yarovoy, Two-dimensional scattering fr om an inhomogeneous dielectric cylinder embedded in a stratified medium: Case of TM polarization, IEE E Trans. Antennas Propag. 42, 16-21 (1994)

    work page 1994

  22. [22]

    Yu and N

    J.-W. Yu and N. H. Myung, Scattering by a dielectric-loaded nonp lanar slit–TM case, IEEE Trans. Antennas Propag. 46, 598-600 (1998). 28

    work page 1998

  23. [23]

    Kashihara and J

    A. Kashihara and J. Nakayama, Scattering of TM plane wave fro m a finite periodic surface, Electron. Commun. Jpn. 89, 493-501 (2006)

    work page 2006

  24. [24]

    Lechleiter and D.-L

    A. Lechleiter and D.-L. Nguyen, A trigonometric Galerkin method for volume integral equa- tions arising in TM grating scattering, Adv. Comput. Math. 40, 1-25 (2014)

    work page 2014

  25. [25]

    Loran and A

    F. Loran and A. Mostafazadeh, Scattering of TE and TM waves and quantum dynamics generated by non-Hermitian Hamiltonians, Prog. Theor. Exp. Phys . 2024, 123A01 (2024)

    work page 2024

  26. [26]

    Loran, A

    F. Loran, A. Mostafazadeh, and C. Yeti¸ smi¸ so˘ glu, Low-Frequency Scattering of TE and TM Waves by an Inhomogeneous Medium with Planar Symmetry, J. Phys. A: Math. Theor. 58, 395202 (2025)

    work page 2025

  27. [27]

    Born and E

    M. Born and E. Wolf, Principles of Optics (Cambridge University Pr ess, Cambridge, 1999)

    work page 1999

  28. [28]

    Taflove, Computational Electrodynamics: The Finite-Difference Tim e-Domain Method (Artech House, Boston, 1995)

    A. Taflove, Computational Electrodynamics: The Finite-Difference Tim e-Domain Method (Artech House, Boston, 1995)

    work page 1995

  29. [29]

    P. G. Bergmann, The wave equation in a medium with a variable index of refraction, J. Acoust. Soc. Am. 17, 329-333 (1946)

    work page 1946

  30. [30]

    P. A. Martin, Acoustic scattering by inhomogeneous obstacles , SIAM J. Appl. 64, 297-308 (2003)

    work page 2003

  31. [31]

    S. Y. Reutskiya and B. Tirozzib, A meshless boundary method fo r 2D problems of elec- tromagnetic scattering from inhomogeneous bodies; H-polarized w aves, J. Quant. Spectrosc. Radiat. Transf. 83, 313-320 (2004)

    work page 2004

  32. [32]

    D. R. Yafaev, Mathematical Scattering Theory (AMS, Providence, 2010)

    work page 2010

  33. [33]

    S. K. Adhikari, Quantum scattering in two dimensions, Amer. J. P hys. 54, 362 (1986)

    work page 1986

  34. [34]

    A. F. Stevenson, Solution of electromagnetic scattering prob lems as power series in the ratio (dimension of scatterer)/wavelength, J. Appl. Phys. 24, 1134-1142 (1953)

    work page 1953

  35. [35]

    G. A. Kriegsmann and E. L. Reiss, Low frequency scattering by local inhomogeneities, SIAM J. App. Math. 43, 923-934 (1983)

    work page 1983

  36. [36]

    Dassios and R

    G. Dassios and R. Kleinman, Low Frequency Scatterings (Oxford University Press, Oxford, 2000)

    work page 2000

  37. [37]

    Ammari and J.-C

    A. Ammari and J.-C. N´ ed´ elec, Low-frequency electromagne tic scattering, SIAM J. Math. Anal. 31, 836-861 (2000)

    work page 2000

  38. [38]

    Loran and A

    F. Loran and A. Mostafazadeh, Class of exactly solvable scatt ering potentials in two dimen- sions, entangled-state pair generation, and a grazing-angle reso nance effect, Phys. Rev. A 96, 063837 (2017)

    work page 2017

  39. [39]

    M. V. Berry, Lop-sided diffraction by absorbing crystals, J. Ph ys. A: Math. Gen. 31, 3493 (1998). 29

    work page 1998

  40. [40]

    Supplementary Material: Exact solution of the scattering prob lem for the scattering of TM waves by a grating

    work page

  41. [41]

    Mostafazadeh, Invisibility and PT-symmetry, Phys

    A. Mostafazadeh, Invisibility and PT-symmetry, Phys. Rev. A 87, 012103 (2013)

    work page 2013

  42. [42]

    W. T. Silfvast, Laser Fundamentals (Cambridge University Press, 1996)

    work page 1996

  43. [43]

    R. G. Newton, Scattering Theory of Waves and Particles , 2nd Ed., (Dover, New York, 2013)

    work page 2013

  44. [44]

    J. R. Taylor, Scattering Theory (Dover, New York, 2000)

    work page 2000

  45. [45]

    J. B. Pendry, A transfer matrix approach to localisation in 3D, J . Phys. C: Solid State Phys. 17, 5317-5336 (1984)

    work page 1984

  46. [46]

    J. B. Pendry, Transfer matrices and conductivity in two- and t hree-dimensional systems. I. Formalism, J. Phys.: Condens. Matter 2, 3273-3286 (1990)

    work page 1990

  47. [47]

    J. B. Pendry, Photonic band structures, J. Mod. Opt. 41, 209-229 (1994). 30

    work page 1994