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REVIEW 3 major objections 4 minor 54 references

Application of Madelung Hydrodynamics to Plasmonics and Nonlinear Optics in Two-Dimensional Materials

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Nonlinearity renormalizes surface-plasmon dispersion in 2D electron gases and acts as an extra nonlocal pressure.

desk verdict A useful hydrodynamic review with a couple of questionable new results; the core framework is sound but the self-modulation claim needs more work. read the letter →

arxiv 2412.07903 v3 pith:Y7JKY2N6 submitted 2024-12-10 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.quant-gas

Simão S. Cardoso , A. J. Chaves , N. Asger Mortensen , N. M. R. Peres This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-scicond-mat.quant-gas PACS 73.20.Mf42.65.Ky
keywords Madelungtransformationquantumhydrodynamicssurfaceplasmonpolaritonsnonlocalplasmonicssecond-harmonicgenerationnonlineardispersionrenormalizationtwo-dimensionalelectrongasmagneto-opticalconductivity
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 establishes that nonlinearity renormalizes the dispersion of surface-plasmon polaritons (coupled light–charge oscillations at an interface) in a two-dimensional electron gas, acting as an extra nonlocal contribution alongside the Fermi pressure. Using the Madelung hydrodynamic picture, the authors derive an analytic expression, Eq. (82), that ties the nonlinear velocity amplitude $v_\omega$ to a positive frequency shift of the plasmon spectrum. The same framework reproduces known second-harmonic-generation and magneto-optical conductivity results, and shows that plasmon excitation resonantly boosts second-harmonic emission. The payoff is a simple analytic route to nonlinear nonlocal optical response in 2D materials, where kinetic-equation approaches are heavier.

What carries the argument

The central object is the Madelung transformation, $\Psi = \sqrt{n}\,e^{iS}$, which converts the Schrödinger equation into continuity and Euler-type equations for the electron density $n$ and velocity field $v$. Supplemented by the Fermi pressure and Poisson's equation, this hydrodynamic system is the machinery that carries every result: harmonic expansions of $n$ and $v$ at $\omega$ and $2\omega$ produce algebraic equations whose solution yields the magnetoplasmon spectrum, the nonlocal conductivity tensor, the second-harmonic current, and finally the renormalized dispersion relation in Eq. (82).

What would settle it

Measure the surface-plasmon dispersion of a gated two-dimensional electron gas by near-field or attenuated-total-reflection spectroscopy while increasing the driving field intensity at fixed wavevector; if the resonance frequency does not shift upward according to Eq. (82), or shifts downward, the central prediction is wrong. A complementary check is a fully nonlinear numerical solution of the hydrodynamic equations without truncating at second order, which should reproduce the analytic spectrum only where the perturbative assumption holds.

Watch

Extended reading notes

Core claim

The central claim is that solving the Madelung equations to second order in the electron velocity field, together with Poisson's equation, gives a self-modulated surface-plasmon-polariton dispersion of the form $\omega^2 = q^2 n_0 k/(2m\epsilon_0) + \beta^2 k^2 + k v_\omega^2 F(k,\omega)$, so that as the velocity amplitude $|v_\omega|$ grows the plasmon frequency increases. The paper argues that nonlinear effects play a role analogous to the Fermi pressure, supplying another nonlocal contribution to the optical response. For the same framework, the paper derives nonlocal magnetoplasmon spectra and magneto-optical conductivities, and shows that second-harmonic generation is resonantly enhanced at the fundamental and second-harmonic plasmon poles, with the hydrodynamic results matching earlier Boltzmann kinetic calculations in the appropriate limits.

Load-bearing premise

The load-bearing premise is that the nonlinear velocity amplitude stays small enough that the first and second harmonics are only weak corrections; the paper itself notes that once $|v_\omega|$ approaches about $40 v_F$, the density fluctuation reaches the full background density and the perturbative expansion is no longer valid.

Editorial extensions

If this is right

  • At high driving amplitudes the surface-plasmon frequency rises with $|v_\omega|$, so intense plasmon fields are predicted to blue-shift the resonant response of 2D electron gases.
  • Because the nonlinear term enters multiplied by the wavevector $k$ in Eq. (82), nonlinearity is a nonlocal correction and grows in importance at shorter wavelengths.
  • Second-harmonic generation from a 2D electron gas is resonantly enhanced when either $\omega$ or $2\omega$ matches a plasmon pole of the nonlocal dielectric function, and vanishes at normal incidence by symmetry.
  • The hydrodynamic results for the nonlinear current and the magneto-optical conductivity coincide with previous Boltzmann kinetic calculations in the collisionless limit, so the simpler analytic expressions can be used for modeling.

Reading between the lines

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

  • The intensity-dependent blue shift predicted by Eq. (82) suggests a concrete experimental signature: in graphene, tuning the pump intensity should shift the near-field plasmon resonance peak, and the shift should grow with wavevector $k$.
  • Because the perturbative expansion breaks down near $|v_\omega| \approx 40 v_F$, strongly driven systems likely require retaining higher harmonics or a fully nonlinear treatment, a regime the paper leaves open.
  • The Madelung framework could be extended to include relaxation and retardation, predicting how the nonlinear blue shift competes with damping in realistic surface-plasmon waveguides; the paper treats the renormalized spectrum in the lossless case.
  • If the renormalized dispersion is measured, the slope of the frequency shift versus intensity would provide a direct estimate of the nonlinear velocity amplitude $v_\omega$ in the two-dimensional electron gas.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper applies the Madelung (quantum) hydrodynamic formalism to several problems in two-dimensional plasmonics and nonlinear optics. It derives the magnetoplasmon spectrum and magneto-optical conductivity with nonlocal corrections from the Fermi pressure, computes second-harmonic generation both in the bare and in the plasmon-assisted self-consistent case, and presents a new self-modulation analysis of surface-plasmon polaritons. The central new claim is that nonlinearity renormalizes the SPP dispersion and acts as an additional source of nonlocal behavior, expressed analytically in Eq. (82) and illustrated in Fig. 4. The manuscript also reviews the derivation of the Madelung equations from the Wigner function and verifies several results against independent Boltzmann kinetic calculations.

Significance. If the central claim holds, the paper offers a simple hydrodynamic derivation of nonlinear nonlocal corrections in 2D materials, with a compact analytic formula for the renormalized SPP dispersion. The cross-checks against Boltzmann kinetic results in Refs. 41, 44, and 48, and the independent derivation of the nonlinear current via the polarization formalism in Sec. IIIC.1, are clear strengths. The self-modulation result is the main novelty, but its demonstration relies on a perturbative regime whose validity is not established for the parameter values used to display the effect.

major comments (3)
  1. [Sec. IIIA, Eq. (14)] The matrix in Eq. (14) is inconsistent with the preceding equations. Substituting the continuity solution (12a) and the self-consistent potential (13) into the linearized Euler equation (12b) produces a matrix that contains an additional contribution proportional to q^2 n0/(2 m epsilon0 |k|) in the entries, reflecting the Coulomb interaction. The matrix as printed contains only the Fermi-pressure term and the cyclotron term; its determinant gives omega^2 = beta^2 k^2 + omega_c^2, without the ak term that appears in Eq. (15). The derivation should be corrected so that the matrix reproduces the claimed dispersion relation.
  2. [Sec. IIIE, Eqs. (81)-(83) and Fig. 4] The perturbative expansion underlying Eq. (82) is not controlled in the regime where the self-modulation effect is claimed to be significant. The paper acknowledges at the end of Sec. IIIE that for |v_omega| = 40 v_F the density fluctuation approaches n0 and the approach is no longer valid. However, Fig. 4B shows that already for |v_omega| = 20 v_F the density fluctuation is a sizable fraction of n0 at moderate wavevectors, so the neglect of higher harmonics and of higher-order nonlinearities is not justified. No estimate of the small parameter or of the magnitude of the neglected 3-omega terms is provided. Consequently, Eq. (82) is not validated in the regime that supports the headline conclusion that nonlinearity renormalizes the SPP dispersion; the claim should either be restricted to the |v_omega| << v_F regime (where the effect is negligible) or be supported by a controlled expansion with a quantitative error bound.
  3. [Sec. IIIE, Eqs. (74a)-(74d)] The assumption that the harmonic amplitudes n_omega, n_2omega, v_omega, and v_2omega are real is introduced without justification. For a propagating SPP described by a factor exp(i k x - i omega t), the complex amplitudes carry phase information; setting them real imposes a specific phase relation that is not generally valid. This assumption is essential for reducing Eqs. (73) to (74), and the authors should either state the physical conditions under which it holds or redo the harmonic balance with complex amplitudes. Without this, the derivation of Eq. (82) rests on an unverified ansatz.
minor comments (4)
  1. [Sec. IIIC, Eq. (29a)] The continuity equation (29a) mixes perturbation orders: it combines the first-order terms partial_t n1 + n0 div v1 with the second-order terms div(n1 v1) + n0 div v2. It would be clearer to write separate equations at each order and then collect harmonics.
  2. [Sec. IIIE, Eq. (72)] Eq. (72) contains a typographical error: E_{l omega}^2 should likely be E_{l omega} (or a different notation), since the electric field is not squared.
  3. [Secs. IIIC and IIIE] The notation for the density harmonics is inconsistent: n1 is used for the total first-order density (containing both omega and 2omega components), while n2 in Eq. (36) denotes the second-order density at 2omega. Standardizing the notation would improve readability.
  4. [Sec. IIIE, Eq. (82)] The paper would benefit from a brief discussion of the physical meaning of the amplitude v_omega and how it relates to an external drive or to the intensity of the SPP, since Eq. (82) is expressed in terms of this input parameter.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all central formulas are derived from the hydrodynamic equations and cross-checked against independent kinetic-theory benchmarks.

full rationale

I find no circular step. The dispersion relations in Secs. IIIA and IIID follow from solving the linearized continuity/Euler equations together with Poisson's equation; Eq. (16) is obtained from a zero-determinant condition, and the SHG results in Sec. IIIC are obtained by explicit harmonic expansion, with the same nonlinear current re-derived independently via the polarization method in Sec. IIIC.1. The zero-damping limits are compared with Boltzmann kinetic calculations in Refs. 44 and 48, which are independent external benchmarks rather than self-citations. Equation (82) in Sec. IIIE is produced by closing the harmonic hierarchy and eliminating n_omega, n_2omega, and v_2omega algebraically; the only free input is the velocity amplitude v_omega, which is a stated state amplitude rather than a fitted parameter, and setting v_omega = 0 correctly recovers Eq. (16), as the paper itself notes. The acknowledged breakdown of the perturbative expansion at |v_omega| = 40 v_F is a validity limitation of the derivation, not a circular reduction of the result to its input. No load-bearing claim is justified solely by a self-citation, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The formalism adds no new particles or fields; the only ad hoc elements are the phenomenological damping gamma and the hand-set velocity amplitude in the self-modulation section.

free parameters (3)
  • velocity amplitude |v_omega| = 0, 20 vF, 40 vF (chosen by hand)
    Enters Eq. (82) as the nonlinearity parameter; the paper does not compute it from the driving field or intensity, so the renormalized dispersion is a parametric family rather than a closed prediction.
  • damping rate gamma = hbar gamma = 4 meV (Fig. 2), 0.25 meV (Fig. 3)
    Phenomenological relaxation rate added to the Euler equation; values chosen for illustration, not fitted to experiment.
  • sheet density n0 = 1e12 cm^-2 (Fig. 3), 1e14 cm^-2 (Fig. 4)
    Input material parameter for the 2D electron gas; chosen for illustration in the figures.
assumptions (5)
  • domain assumption Fermi pressure of a degenerate parabolic-band 2D electron gas is pF = pi hbar^2 n^2/(2m)
    Used in Secs. IIIA-IIID to introduce nonlocal corrections; does not apply to Dirac (graphene) dispersion without modification.
  • domain assumption Quasi-static approximation replaces Maxwell's equations by Poisson's equation
    Justified in Appendix B for SPPs with k >> omega/c; used for optical-frequency SHG, limiting validity to large in-plane wavevectors.
  • domain assumption Fields are expanded perturbatively to second order in the external field
    The authors note at the end of Sec. IIIE that the perturbation breaks down when n_omega approaches n0.
  • domain assumption Weak magnetic field does not modify the zero-field Fermi pressure expression
    Assumed before Eq. (9) in Sec. IIIA; no derivation for the magnetized 2DEG is given.
  • domain assumption In the lossless limit the velocity field components are real and parallel to k
    Used in Sec. IIIE to reduce Eq. (73) to Eq. (74) for the self-modulation calculation.

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Pith. "Pith review of Application of Madelung Hydrodynamics to Plasmonics and Nonlinear Optics in Two-Dimensional Materials." pith.science (2026). https://pith.science/paper/Y7JKY2N6

@misc{pith2026241207903,
  author       = {Pith},
  title        = {Pith review of: Application of Madelung Hydrodynamics to Plasmonics and Nonlinear Optics in Two-Dimensional Materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y7JKY2N6}},
  note         = {Machine review of arXiv:2412.07903}
}
read the original abstract

This paper explores the application of Madelung hydrodynamic models to study two-dimensional electron gases, with a focus on nonlocal plasmonics and nonlinear optics. We begin by reviewing the derivation of the Madelung equations. Using the Madelung equations in conjunction with Poisson's equation, we calculate the spectrum of magnetoplasmons and the magneto-optical conductivity in the electrostatic regime, incorporating nonlocal corrections due to the Fermi pressure. In the absence of a magnetic field, we analyze nonlinear and nonlocal second-harmonic generation, demonstrating how plasmon excitation enhances this process. We further discuss the emergence of self-modulation phenomena driven by nonlinearity, leading to the renormalization of the plasmon dispersion. Notably, we show that nonlinearity amplifies nonlocal effects and, leveraging the hydrodynamic formalism, derive a simple analytic expression for the renormalized spectra.

Figures

Figures reproduced from arXiv: 2412.07903 by the authors.

Figure 1
Figure 1. FIG. 1: Second-harmonic generation. A graphene [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Nonlinear optical response for finite [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Second-harmonic generation assisted by [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: A) Plasmon-polaritons dispersion relation [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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Works this paper leans on

54 extracted references · 53 canonical work pages

  1. [1]

    2: Nonlinear optical response for finite ℏγ = 4 meV according to Boltzmann kinetic equation [44] and hydrodynamic approach (our results)

    Derivation of the nonlinear current using the polarization To assess the correctness of our previous approach, we derive the same results using a more common method [45–47], where the quantum hydrodynamic 6 FIG. 2: Nonlinear optical response for finite ℏγ = 4 meV according to Boltzmann kinetic equation [44] and hydrodynamic approach (our results). equatio...

    work page

  2. [2]

    1 2ω + iγ 1 ω + iγ − 1 ω (E1 · ∇)E1 + 1 ω 1 ω + iγ (∇ · E1)E1 + 1 2ω 1 2ω + iγ ∇(E1 · E1) # e−i2ωt − i q3n0 m2

    = 0, (71a) − i2ωn2 + ∇ · (n1v1) + n0∇ · v2 = 0, (71b) − iωv1 + (v∗ 1 · ∇) v2 + (v2 · ∇) v∗ 1 = q m E1 − b∇n1, (71c) − i2ωv2 + (v1 · ∇) v1 = q m E2 − β2 n0 ∇n2, (71d) where b = πℏ2/m2. Referringtotheelectricpotentials given in Eqs. (62)–(63), the electric fields can be determined by taking their gradients, yielding Elω 2 = −ik q 2ε0k nlω, (72) where l = 1,...

    work page 2020

  3. [3]

    (C1) 12 Appendix D: Derivation of Eq

    − 1 ω 1 ω − iγ 1 2ω + iγ (∇ · E∗ 1)E2 + 1 2ω 1 ω − iγ 1 2ω + iγ (∇ · E2)E∗ 1 # e−iωt + c.c. (C1) 12 Appendix D: Derivation of Eq. A4 We start with the time derivative of Eq. (A1), that results in ∂ ∂t f (r, p, t) = 1 (2π)2ℏ2 X α Pα Z dr′ ∂ ∂t ψ∗ α(r + r′/2, t) ψα(r − r′/2, t)eip·r′/ℏ, + 1 (2π)2ℏ2 X α Pα Z dr′ψ∗ α(r + r′/2, t) ∂ ∂t ψα(r − r′/2, t)eip·r′/ℏ,...

    work page

  4. [4]

    Eguiluz and J

    A. Eguiluz and J. J. Quinn, Phys. Rev. B14, 1347 (1976)

    work page 1976

  5. [5]

    Ciraci, J

    C. Ciraci, J. B. Pendry, and D. R. Smith, Chem. Phys. Chem. 14, 1109 (2013). 13

    work page 2013

  6. [6]

    Yan, Phys

    W. Yan, Phys. Rev. B91, 115416 (2015)

    work page 2015

  7. [7]

    Ciracì and F

    C. Ciracì and F. Della Sala, Phys. Rev. B93, 205405 (2016)

    work page 2016

  8. [8]

    H. M. Baghramyan, F. Della Sala, and C. Ciracì, Phys. Rev. X11, 011049 (2021)

    work page 2021

Show all 54 references
  1. [9]

    Corvi and W

    M. Corvi and W. L. Schaich, Phys. Rev. B33, 3688 (1986)

    1986

  2. [10]

    Brevet, Surface Second Harmonic Generation , Cahiers de chimie (Presses polytechniques et universitaires romandes, 1997)

    P. Brevet, Surface Second Harmonic Generation , Cahiers de chimie (Presses polytechniques et universitaires romandes, 1997)

    1997

  3. [11]

    Ginzburg, A

    P. Ginzburg, A. V. Krasavin, G. A. Wurtz, and A. V. Zayats, ACS Photonics2, 8 (2015)

    2015

  4. [12]

    A. V. Krasavin, P. Ginzburg, and A. V. Zayats, Laser Photon. Rev. 12, 1700082 (2018)

    2018

  5. [13]

    Crabb, X

    J. Crabb, X. Cantos-Roman, J. M. Jornet, and G. R. Aizin, Phys. Rev. B104, 155440 (2021)

    2021

  6. [14]

    Müller, J

    M. Müller, J. Schmalian, and L. Fritz, Phys. Rev. Lett. 103, 025301 (2009)

    2009

  7. [15]

    A. J. Chaves, N. M. R. Peres, G. Smirnov, and N. A. Mortensen, Phys. Rev. B96, 195438 (2017)

    2017

  8. [16]

    Lucas and K

    A. Lucas and K. C. Fong, J. Phys.: Cond. Matter30, 053001 (2018)

    2018

  9. [17]

    Pongsangangan, T

    K. Pongsangangan, T. Ludwig, H. T. C. Stoof, and L. Fritz, Phys. Rev. B106, 205126 (2022)

    2022

  10. [18]

    J. R. Maack, N. A. Mortensen, and M. Wubs, Phys. Rev. B 97, 115415 (2018)

    2018

  11. [19]

    Born, Zeitschrift für Physik37, 863 (1926)

    M. Born, Zeitschrift für Physik37, 863 (1926)

    1926

  12. [20]

    Pais, Science218, 1193 (1982)

    A. Pais, Science218, 1193 (1982)

    1982

  13. [21]

    Bernstein, Am

    J. Bernstein, Am. J. Phys.73, 999 (2005)

    2005

  14. [22]

    Madelung, Z

    E. Madelung, Z. Physik40, 322 (1927)

    1927

  15. [23]

    Z. A. Moldabekov, M. Bonitz, and T. S. Ramazanov, Phys. Plasmas 25, 031903 (2018)

    2018

  16. [24]

    Ding and C

    K. Ding and C. T. Chan, Phys. Rev. B96, 125134 (2017)

    2017

  17. [25]

    Manfredi and F

    G. Manfredi and F. Haas, Phys. Rev. B64, 075316 (2001)

    2001

  18. [26]

    R. A. Alves, A. Guerreiro, and M. Navarro-Cía, Phys. Rev. B 101, 235412 (2020)

    2020

  19. [27]

    R. A. Alves, V. Pacheco-Peña, and M. Navarro-Cía, J. Phys. Chem. C126, 14758 (2022)

    2022

  20. [28]

    A. L. Fetter, Phys. Rev. B32, 7676 (1985)

    1985

  21. [29]

    Fessatidis and H

    V. Fessatidis and H. L. Cui, Phys. Rev. B43, 11725 (1991)

    1991

  22. [30]

    Dempsey and B

    J. Dempsey and B. I. Halperin, Phys. Rev. B45, 1719 (1992)

    1992

  23. [31]

    Dulikravich and S

    G. Dulikravich and S. Lynn, Int. J. Non-Linear Mech. 32, 913 (1997)

    1997

  24. [32]

    R. Sano, R. Toshio, and N. Kawakami, Phys. Rev. B 104, L241106 (2021)

    2021

  25. [33]

    W. L. Barnes, Am. J. Phys.84, 593 (2016)

    2016

  26. [34]

    Locarno and D

    M. Locarno and D. Brinks, Am. J. Phys. 91, 538 (2023)

    2023

  27. [35]

    Quantum hydrodynamics,

    S. A. Khan and M. Bonitz, “Quantum hydrodynamics,” in Complex Plasmas: Scientific Challenges and Technological Opportunities, edited by M. Bonitz, J. Lopez, K. Becker, and H. Thomsen (Springer International Publishing, Cham, 2014) pp. 103–152

    2014

  28. [36]

    27,35(1969)

    L.Janossy,ActaPhys.Acad.Sci.Hung. 27,35(1969)

    1969

  29. [37]

    Akbari-Moghanjoughi, Phys

    M. Akbari-Moghanjoughi, Phys. Plasmas20, 102115 (2013)

    2013

  30. [38]

    Mazzarella, L

    G. Mazzarella, L. Salasnich, and F. Toigo, Phys. Rev. A 79, 023615 (2009)

    2009

  31. [39]

    Halevi, Phys

    P. Halevi, Phys. Rev. B51, 7497 (1995)

    1995

  32. [40]

    del Castillo-Mussot, G

    M. del Castillo-Mussot, G. J. V. Fonseca, and G. H. Cocoletzi, Rev. Mex. Fis.49, 231 (2003)

    2003

  33. [41]

    Wegner, D.-N

    G. Wegner, D.-N. Huynh, N. A. Mortensen, F. Intravaia, and K. Busch, Phys. Rev. B107, 115425 (2023)

    2023

  34. [42]

    Lindhard, Mat.-Fys

    J. Lindhard, Mat.-Fys. Medd. Dan. Vid. Selsk.28, 1 (1954)

    1954

  35. [43]

    N. D. Mermin, Phys. Rev. B1, 2362 (1970)

    1970

  36. [44]

    Ferreira, J

    A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. B84, 235410 (2011)

    2011

  37. [45]

    Andreeva, D

    V. Andreeva, D. A. Bandurin, M. Luskin, and D. Margetis, Phys. Rev. B102, 205411 (2020)

    2020

  38. [46]

    M. T. Manzoni, I. Silveiro, F. J. G. de Abajo, and D. E. Chang, New J. Phys.17, 083031 (2015)

    2015

  39. [47]

    J. D. Cox, R. Yu, and F. J. García de Abajo, Phys. Rev. B 96, 045442 (2017)

    2017

  40. [48]

    J. E. Sipe, V. C. Y. So, M. Fukui, and G.I. Stegeman, Phys. Rev. B21, 4389 (1980)

    1980

  41. [49]

    Scalora, M

    M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, Phys. Rev. A 82, 043828 (2010)

    2010

  42. [50]

    Ciracì, E

    C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, Phys. Rev. B86, 115451 (2012)

    2012

  43. [51]

    S. A. Mikhailov, Phys. Rev. B84, 045432 (2011)

    2011

  44. [52]

    De Leon, J

    I. De Leon, J. E. Sipe, and R. W. Boyd, Phys. Rev. A 89, 013855 (2014)

    2014

  45. [53]

    Otto, Zeitschrift für Physik A Hadrons and nuclei 216 (1968)

    A. Otto, Zeitschrift für Physik A Hadrons and nuclei 216 (1968)

    1968

  46. [54]

    J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. De Abajo, R. Hillenbrand, and F. H. L. Koppens, Nature 487 (2012)

    2012

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