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REVIEW 4 major objections 3 minor 48 references

The paper claims that the Brewster angle in a thin film is not a single electric-dipole effect: its familiar zero in p-polarized reflection survives only because magnetic-dipole and electric-quadrupole terms cancel each other with a π phase

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 19:22 UTC pith:QQIXZGOZ

load-bearing objection Solid multipole model of thin-film reflection, but the headline Brewster-angle claims are undercut by equation inconsistencies and an untested origin dependence in the decomposition. the 4 major comments →

arxiv 2603.02377 v2 pith:QQIXZGOZ submitted 2026-03-02 physics.optics

Physics of Dipole and Quadrupole Brewster Angles in Thin Films

Edward H.Krock , Mughaid Ali , Haizhong Weng , C. M. Smith , John F.Donegan This is my paper
classification physics.optics PACS 42.25.Gy42.25.Bs78.20.Ci
keywords Brewster anglemultipole decompositionthin film reflectionsilicon nitridemagnetic dipoleelectric quadrupolepolarizationFabry-Perot modes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that the Brewster angle, usually taught as a single-interface effect set by the refractive index, has a richer multipole structure in thin films. Using a multipole expansion of the reflected field for a free-standing 455 nm silicon-nitride film, the authors re-derive the standard Brewster angle from the electric-dipole term and derive new Brewster-angle equations for the magnetic dipole, electric quadrupole, and magnetic quadrupole. The central new claim is that the observed zero at the Brewster angle requires both electric-dipole alignment and destructive interference between the magnetic-dipole and electric-quadrupole terms, which become equal in magnitude and out of phase by π. The model reproduces measured reflection from 750 to 2200 nm and 0 to 70 degrees, with the paper noting that the description begins to break down at larger angles and shorter wavelengths. If correct, zero-reflection angles in thin films are a multipole interference effect that can be understood and engineered rather than a simple index-matching condition.

Core claim

From a multipole decomposition of the complex reflection coefficient of a thin film, the paper derives per-multipole Brewster conditions: tan(θ_B,ED) = p_x/p_z for the electric dipole, with analogous equations for the magnetic dipole and for p- and s-polarized electric and magnetic quadrupoles. Each condition requires the complex multipole angle to become real, meaning both the real part aligns with the angle of incidence and the imaginary part is zero. Applied to a 455 nm SiN film, the well-known Brewster zero at roughly 64 degrees is shown to occur when the electric-dipole reflection coefficient is zero and, simultaneously, the magnetic-dipole and electric-quadrupole reflection coefficient

What carries the argument

The load-bearing object is the multipole expansion of the reflected field, equation (2), which expresses the complex reflection coefficient as a sum over electric-dipole, magnetic-dipole, electric-quadrupole, magnetic-quadrupole, and octupole contributions, each multiplied by an angular vector obtained from cross products with the reflection direction. Each multipole term can be set to zero to produce its own Brewster condition, and the complex-angle representation θ = θ_real + i θ_imag converts the alignment requirement into the double condition θ_real = θ_incidence and θ_imag = 0. The decisive relation for the classic Brewster angle is the magnetic-dipole/electric-quadrupole cancellation:

Load-bearing premise

The central claim rests on assuming that multipole moments computed by volume integrals over one cubic unit cell of the film capture the full angle- and wavelength-dependent reflection of the infinite thin film, an assumption the paper itself concedes begins to break down at larger angles and shorter wavelengths.

What would settle it

Measure p- and s-polarized reflection from a free-standing 455 nm SiN film at the nominal Brewster angle while scanning wavelength around 850 nm, and separately extract the magnetic-dipole and electric-quadrupole reflection coefficients from the multipole model; if any wavelength shows a true zero in total reflection where |r_MD| and |r_EQ| are not equal or their phase difference is not π, the claimed MD-EQ cancellation is not necessary. Alternatively, in simulation, set the MD and EQ contributions to zero at the Brewster angle and check whether the zero in total reflection disappears.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The standard Brewster angle in thin films is shown to involve a hidden multipole cancellation, not only electric-dipole alignment, so the intuitive single-dipole picture is incomplete.
  • Each higher multipole has its own Brewster angle, some p-polarized and some s-polarized, and these occur only over narrow wavelength ranges, in contrast to the electric-dipole Brewster angle.
  • Brewster-angle zeros and Fabry-Perot zeros can be told apart by their multipole interference structure: Brewster arises from MD-EQ destructive interference, while Fabry-Perot arises from destructive interference between ED and the MD/EQ pair.
  • The derived equations apply to metamaterials with matched substrate and superstrate, offering a route to design dual-polarization or wavelength-selective Brewster angles without exotic structures.
  • Loss-induced pseudo-Brewster minima are explained as a failure of the imaginary-angle condition and incomplete MD-EQ cancellation, giving a quantitative account of a known but often separate phenomenon.

Where Pith is reading between the lines

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

  • A testable extension beyond the paper: in other film thicknesses and refractive-index contrasts, the depth and sharpness of the classic Brewster zero should track how closely the magnetic-dipole and electric-quadrupole amplitudes match at that angle; deliberately detuning thickness should turn the zero into a shallow minimum.
  • The wavelength-narrowness of the magnetic-dipole and quadrupole Brewster angles suggests a spectroscopic fingerprint: sweeping wavelength through the narrow condition should make these zeros appear abruptly, which could be used to identify multipole contributions in thin-film reflectance maps.
  • An implication the authors leave implicit: if the multipole expansion is truncated at octupoles and already shows hints of octupole Brewster angles near 15, 40, and 64 degrees, then at shorter wavelengths where the model breaks down, higher-order multipoles likely dominate and could be characterized directly by the same complex-angle condition.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper analyzes angle-dependent reflection from a 455 nm free-standing SiN thin film using a multipole decomposition of the fields inside one cubic unit cell. It derives Brewster-angle conditions for the electric dipole, magnetic dipole, electric quadrupole, and magnetic quadrupole terms, and claims that the usual thin-film Brewster zero requires destructive interference between the magnetic dipole and electric quadrupole contributions. The total-reflection predictions are compared with S-parameter simulations and transmission-based reflection measurements over 750–2200 nm and 0–70°, with good agreement reported. The paper also identifies wavelength-selective 'Brewster angles' for higher multipoles and discusses pseudo-Brewster and Fabry-Perot effects.

Significance. If the central decomposition claim is correct, the work would add a new mechanistic layer to the textbook Brewster effect: not only electric-dipole alignment, but also magnetic-dipole/electric-quadrupole cancellation, with higher multipoles having their own wavelength-selective zero conditions. The experimental and S-parameter validation of the total reflection is a genuine strength, as is the detailed supplementary derivation. However, the central claim is weakened by internal equation inconsistencies and by the acknowledged origin dependence of the higher multipole moments, which is not tested. The new multipole Brewster angles are also zeros of individual decomposition terms, not observable reflection zeros, so their physical status is presently unclear.

major comments (4)
  1. [Eqs. (9)–(12) vs Supplementary S.32, S.34, S.37, S.39] The main-text Brewster equations are internally inconsistent with the supplementary derivations. Eq. (10) gives tan θ_B,MQ,p = −MQ_zy/MQ_xy, while S.37 gives −MQ_xy/MQ_zy (reciprocal and different component assignment). Eq. (11) gives tan θ_B,EQ,s = EQ_zy/EQ_xy, while S.34 gives EQ_xy/EQ_zy. Eq. (9) has denominator EQ_zz−EQ_xx; S.32 has EQ_xx−EQ_zz (opposite sign). Eq. (12) has denominator MQ_zz−MQ_xx; S.39 has MQ_xx−MQ_zz (opposite sign). These are not equivalent definitions; a reader cannot reproduce the plotted Brewster angles from the stated equations. The main text and supplement must be harmonized and the figures regenerated from the corrected formulas.
  2. [Eq. (13) and Supplementary S2.5–S2.6] Eq. (13) asserts the chain p_x/p_z = m_x/m_z = EQ_zy/EQ_xy = n. The ED equality is supported by the Snell-law argument in S2.1. For the MD, S2.5 invokes Snell's law and θ_T = θ_MD − 90° without deriving why the magnetic-dipole angle must obey that relation for this film. For the EQ, S2.6 establishes m = Q only for s-polarized light with J_y-only and a negligible toroidal term; the main-text claim of MD/EQ cancellation at 64° is made for p-polarized light (Fig. 3(g)), where J has both x and z components and the toroidal suppression is not demonstrated. Thus the chain to n is asserted rather than derived for the cases used in the central claim. Please provide a derivation from the multipole integrals or explicitly label Eq. (13) as a numerical observation.
  3. [Methods, Multipole Decomposition; Fig. 3(g)–(h)] The Methods state that 'the magnetic dipole moment, and all following multipoles have an origin dependence.' The central new claim — that the thin-film Brewster zero requires destructive interference between MD and EQ — relies on the magnitudes and phases of these origin-dependent moments. No test is given of how the decomposition changes when the origin is shifted within the unit cell. Because total reflection is origin-independent, agreement with the S-parameter model and experiment validates only the total, not the MD/EQ split. I request a concrete numerical test: repeat the decomposition with the origin shifted by, e.g., ±100 nm along z and report r_ED, r_MD, r_EQ and their phases at the 64° Brewster condition. If the r_MD + r_EQ = 0 cancellation does not persist, the claimed interference effect is an artifact of the chosen origin.
  4. [Section 3 and Methods; Fig. 4(d)–(j)] The multipole Brewster-angle maps cover a range where the authors state the model 'begins to break down at larger angles and shorter wavelengths.' The unit cell is a cube of side equal to the film thickness (455 nm); for a homogeneous film this lateral size is a modeling choice, not a physical period. The robustness of the derived Brewster conditions should be tested against (a) the lateral unit-cell size (e.g., 455 nm vs 1 μm) and (b) the truncation order (e.g., including octupoles where they are currently neglected). Without such convergence tests, the wavelength-selective new Brewster angles may reflect the specific unit-cell and truncation choices rather than intrinsic properties of the film.
minor comments (3)
  1. [Supplementary S.12–S.13] The p-polarized expression in S.13 includes 'O_e x' (electric octupole) where the magnetic quadrupole term M_x should appear, based on the main-text Eq. (2). This appears to be a typo and should be corrected.
  2. [Section 2 / Supplementary S2.5] The main text refers to 'supplemental information section S1.3' for the magnetic-dipole Brewster angle, but the relevant derivation is in S2.5. The cross-reference should be updated.
  3. [Throughout] There are several typographical issues: 'quadruple' should be 'quadrupole' in several places; Fig. 2 caption contains stray labels ('aa b c'); and the Methods equation numbering (Eq. 18 vs 'equations 13-16') is inconsistent.

Circularity Check

2 steps flagged

New multipole Brewster angles are definitional; the claimed MD–EQ cancellation is an origin-dependent decomposition artifact.

specific steps
  1. self definitional [Section 2, Multipole Theory of Brewster Angles, after Eq. (2), pp. 3–4]
    "Each multipole will have its own Brewster angle which we denote for electric dipole θB,ED . In addition, each multipole has it's own angle, θED , θMD, θEQ, θMQ that are in general complex. ... As with the electric dipole derivation, we presume that only a single multipole is contributing to reflection."

    A 'multipole Brewster angle' is introduced as the angle at which that multipole's term in Eq. (2) vanishes. Eqs. (5), (8)–(12) are then obtained by setting the relevant p- or s-component of that single term to zero, and Fig. 2 'demonstrates' them by evaluating the same moment ratios in the same COMSOL model. The numerical angles (e.g., 45° at 1800 nm for EQ) are therefore not predictions of an external observable; they are restatements of the defining zero condition, so the 'derivation' reduces to the definition.

  2. renaming known result [Section 3, discussion of Fig. 3(g)–(i); Methods, Multipole Decomposition]
    "We note the new result that the MD and EQ terms are large and equal but according to Figure 3 (g) are out of phase by π so that |rMD, p−pol|+|rEQ, p−pol|=0, maintaining the well-known zero in reflection at this angle. ... The magnetic dipole moment, and all following multipoles have an origin dependence, and the moments of these multipoles appear to depend on selection of a reference point, we set our origin to the x, y, z co-ordinates 0, 0, 0..."

    The zero in reflection at Brewster's angle is the known phenomenon being explained, and the explanation is extracted from the multipole decomposition of the same simulated fields (Eq. 2). Because the paper itself states that MD and all higher multipole moments are origin-dependent, the equality and π phase difference between r_MD and r_EQ are a property of the chosen unit-cell origin, not an invariant of the film. An origin shift would redistribute the moments and change the claimed cancellation, so 'destructive interference between magnetic dipole and electric quadrupole terms' is a representation-dependent renaming of the known Fresnel zero rather than a derived physical requirement.

full rationale

The paper's total-reflection model is tested against S-parameter simulation and measured reflection, so the overall model has independent support. The algebraic derivation of individual-multipole zero conditions is internally consistent. However, the two central claims that go beyond the standard picture—the new multipole Brewster angles and the MD–EQ destructive-interference mechanism—reduce to the definitions and to origin-dependent moments. No load-bearing self-citation appears. Because the central 'discovery' is not invariant and the new angles are definitional, but part of the paper is independently grounded, score 6.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The central claim rests on the validity of the multipole expansion for a homogeneous film, the specular-direction ansatz, and the extrapolated refractive index; no new physical entities are introduced. The fitted Cauchy coefficients enter every quantitative prediction, including the wavelengths of the new Brewster angles.

free parameters (2)
  • Cauchy ellipsometry coefficients A-F for SiN refractive index = A=2.032, B=2.671e-2, C=5.626e-5; k set to 0 (measurement 2, Table S1)
    The multipole and S-parameter models use this fitted dispersion; the predicted new Brewster-angle wavelengths (1700-1800 nm) lie beyond the 200-900 nm measurement range over which the fit was made.
  • Surface roughness layer thicknesses = 13 nm and 5 nm mixed air/SiN layers
    Used in measurement 2 of Table S1 to extract the bulk refractive index; the bulk model in the paper assumes k=0.
axioms (6)
  • domain assumption Multipole expansion of eq. (2) (Evlyukhin & Chichkov) describes reflection of a homogeneous thin film when moments are computed by volume integrals over a cubic unit cell of side 455 nm with Floquet periodic boundary conditions.
    Methods section; the central BA conditions are derived from this expansion. Truncation at octupoles is stated to fail at large angles/short wavelengths.
  • domain assumption Scattering direction is the specular direction n = (-sinθI, 0, -cosθI), with θ_scattering = θI - 180°.
    Supplementary S1, eq. S.1; all subsequent multipole angle dependencies follow from this geometry.
  • ad hoc to paper For s-polarized light with negligible toroidal EQ term, the MD and EQ moments coincide: m_x=Q_xy and m_z=Q_zy.
    Supplementary S2.6 imposes J_y-only and negligible toroidal term to force equality; this is not shown to hold generally in the film.
  • domain assumption Imaginary part of the SiN refractive index is zero from 750 to 2200 nm, so R=1-T.
    Methods and Supplementary S8 state 'assuming that absorption is zero'; this affects all measured reflection values.
  • ad hoc to paper Cauchy dispersion fitted at 200-900 nm extrapolates to 2200 nm.
    Supplementary S5: n measured only to 900 nm and extrapolated; the EQ/MQ BA wavelengths (1700-1800 nm) fall in this region.
  • standard math Standard mathematical identities and Snell's law used in S2.1 to identify p_x/p_z with n.
    Used to connect the multipole condition to tanθB=n; this is the place where the derivation leans on the known result rather than deriving it independently.

pith-pipeline@v1.3.0-alltime-deepseek · 23322 in / 17558 out tokens · 159373 ms · 2026-08-02T19:22:10.082950+00:00 · methodology

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read the original abstract

The Brewster angle is a well-known-phenomenon that describes the angle at which the intensity of reflection of p-polarized light is zero for a single dielectric interface. We investigate the angle dependent reflection in a simple SiN thin film, with thickness in the hundreds of nanometres, a common thickness used in optical waveguides, Fabry-Perot resonators, sensors and lasers. We describe the reflection of our SiN thin film in terms of electric and magnetic multipoles through a multipole expansion of the fields inside our film. Previous theoretical studies on Fabry Perot modes in GaP films have only considered the reflection of unpolarized light at normal incidence. Our investigation expands on this work to s- and p-polarization and angle dependent effects permitting the study of both Fabry Perot and Brewster angle effects together. Our approach allows us to re-derive the well-known Brewster angle equation from the electric dipole term. We then derive several new Brewster angle equations associated with the magnetic dipole and electric/magnetic quadrupoles in our model. Our model is then validated by obtaining good agreement between the predicted reflection from our multipoles to the measured reflection of the same thin film. The distinction between the standard electric dipole Brewster angle and our newly discovered Brewster angles is the destructive interference between remaining multipoles. It is this destructive interference which produces the zero in measured and modelled reflection, associated with the Brewster angle. In addition, the Brewster condition of the magnetic dipole and quadrupoles are only satisfied at specific wavelengths. This multipole model brings additional understanding of how light interacts with thin film dielectric materials.

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Works this paper leans on

48 extracted references · 27 canonical work pages

  1. [1]

    write newline

    " write newline " cite write " FUNCTION editor.postfix editor num.names #1 > "( )" "( )" if FUNCTION editor.trans.postfix editor num.names #1 > "( )" "( )" if FUNCTION trans.postfix translator num.names #1 > "( )" "( )" if FUNCTION authors.editors.reflist.apa5 'field := 'dot := field num.names 'numnames := numnames 'format.num.names := format.num.names na...

  2. [2]

    sn-aps.bst

    FUNCTION identify.aps.version "sn-aps.bst" " [2024/07/19 v1.1 APS bibliography style]" * top ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key keywords month note number organization pages publisher school series title type url volume year eprint archive archivePrefix primaryClass adsurl adsnote version lab...

    2024

  3. [3]

    write newline

    " write newline "" before.all 'output.state := FUNCTION if.digit duplicate "0" = swap duplicate "1" = swap duplicate "2" = swap duplicate "3" = swap duplicate "4" = swap duplicate "5" = swap duplicate "6" = swap duplicate "7" = swap duplicate "8" = swap "9" = or or or or or or or or or FUNCTION n.separate 't := "" #0 'numnames := t empty not t #-1 #1 subs...

  4. [4]

    barticle Brewster , D. : X. On the laws which regulate the polarisation of light by reflexion from transparent bodies. By David Brewster, LL. D. F. R. S. Edin. and F. S. A. Edin. In a letter addressed to Right Hon. Sir Joseph Banks, Bart. K. B. P. R. S. . Philosophical Transactions of the Royal Society of London 105 , 125 -- 159 ( 1815 ) 10.1098/rstl.1815...

    arXiv

  5. [6]

    , ElKabbash , M

    barticle Sreekanth , K.V. , ElKabbash , M. , Medwal , R. , Zhang , J. , Letsou , T. , Strangi , G. , Hinczewski , M. , Rawat , R.S. , Guo , C. , Singh , R. : generalized Brewster Angle Effect in Thin-Film Optical Absorbers and Its Application for Graphene Hydrogen Sensing . ACS Photonics 6 ( 7 ), 1610 -- 1617 ( 2019 ) 10.1021/acsphotonics.9b00564 . barticle

    work page doi:10.1021/acsphotonics.9b00564 2019

  6. [7]

    , Yu , Y.F

    botherref Paniagua-Domínguez , R. , Yu , Y.F. , Miroshnichenko , A.E. , Krivitsky , L.A. , Fu , Y.H. , Valuckas , V. , Gonzaga , L. , Toh , Y.T. , Kay , A.Y.S. , Lukyanchuk , B. , Kuznetsov , A.I. Generalized Brewster effect in dielectric metasurfaces. Nat Commun 7 (2016) 10.1038/ncomms10362 botherref

    work page doi:10.1038/ncomms10362 2016

  7. [8]

    , Che , Z

    barticle Zhang , Z. , Che , Z. , Liang , X. , Chu , J. , Zeng , J. , Huang , H. , Guan , F. , Shi , L. , Liu , X. , Zi , J. : Realizing Generalized Brewster Effect by Generalized Kerker Effect. . Phys. Rev. Appl. 16 , 054017 ( 2021 ) 10.1103/PhysRevApplied.16.054017 barticle

    work page doi:10.1103/physrevapplied.16.054017 2021

  8. [9]

    , Martin , O.J.F

    barticle Tiukuvaara , V. , Martin , O.J.F. , Achouri , K. : Quadrupolar susceptibility modeling of substrated metasurfaces with application to the generalized Brewster effect . Opt. Express 31 ( 14 ), 22982 -- 22996 ( 2023 ) 10.1364/OE.488529 barticle

    work page doi:10.1364/oe.488529 2023

  9. [10]

    , Novikov , S.M

    botherref Evlyukhin , A.B. , Novikov , S.M. , Zywietz , U. , Eriksen , R.L. , Reinhardt , C. , Bozhevolnyi , S.I. , Chichkov , B.N. : Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region. Nano Lett 12 (2012) 10.1021/nl301594s botherref

    work page doi:10.1021/nl301594s 2012

  10. [11]

    , Patoux , A

    barticle Majorel , C. , Patoux , A. , Estrada-Real , A. , Urbaszek , B. , Girard , C. , Arbouet , A. , Wiecha , P.R. : Generalizing the exact multipole expansion: Density of multipole modes in complex photonic nanostructures . Nanophotonics 11 , 3663 -- 3678 ( 2022 ) 10.1515/nanoph-2022-0308 barticle

    work page doi:10.1515/nanoph-2022-0308 2022

  11. [12]

    , Alù , A

    botherref Monticone , F. , Alù , A. : The quest for optical magnetism: From split-ring resonators to plasmonic nanoparticles and nanoclusters. Journal of Materials Chemistry C 2 (2014) 10.1039/c4tc01406e botherref

    work page doi:10.1039/c4tc01406e 2014

  12. [13]

    , Fedotov , V.A

    barticle Kaelberer , T. , Fedotov , V.A. , Papasimakis , N. , Tsai , D.P. , Zheludev , N.I. : Toroidal Dipolar Response in a Metamaterial . Science 330 , 1510 -- 2 ( 2010 ) 10.1126/science.1197172 barticle

    work page doi:10.1126/science.1197172 2010

  13. [14]

    , Papasimakis , N

    barticle Savinov , V. , Papasimakis , N. , Tsai , D.P. , Zheludev , N. : Optical anapoles . Commun Phys 2 , 69 ( 2019 ) 10.1038/s42005-019-0167-z barticle

    doi:10.1038/s42005-019-0167-z 2019

  14. [15]

    , Smirnova , D.A

    botherref Baryshnikova , K.V. , Smirnova , D.A. , Luk'yanchuk , B.S. , Kivshar , Y.S. : Optical anapoles: Concepts and applications. Advanced Optical Materials 7 (2019) 10.1002/adom.201801350 botherref

    work page doi:10.1002/adom.201801350 2019

  15. [16]

    , Evlyukhin , A.B

    botherref Miroshnichenko , A.E. , Evlyukhin , A.B. , Yu , Y.F. , Bakker , R.M. , Chipouline , A. , Kuznetsov , A.I. , Luk'yanchuk , B. , Chichkov , B.N. , Kivshar , Y.S. : Nonradiating anapole modes in dielectric nanoparticles. Nature Communications 6 (2015) 10.1038/ncomms9069 botherref

    doi:10.1038/ncomms9069 2015

  16. [17]

    , Babicheva , V.E

    botherref Terekhov , P.D. , Babicheva , V.E. , Baryshnikova , K.V. , Shalin , A.S. , Karabchevsky , A. , Evlyukhin , A.B. : Multipole analysis of dielectric metasurfaces composed of nonspherical nanoparticles and lattice invisibility effect. Physical Review B 99 (2019) 10.1103/PhysRevB.99.045424 botherref

    work page doi:10.1103/physrevb.99.045424 2019

  17. [18]

    , Safari , A

    barticle Alaee , R. , Safari , A. , Sandoghdar , V. , Boyd , R.W. : Kerker effect, superscattering, and scattering dark states in atomic antennas . Phys. Rev. Res. 2 , 043409 ( 2020 ) 10.1103/PhysRevResearch.2.043409 barticle

    doi:10.1103/physrevresearch.2.043409 2020

  18. [19]

    , Pecherkin , V

    botherref Bukharin , M. , Pecherkin , V. , Ospanova , A. , Il’in , V. , Vasilyak , L. , Basharin , A. , Lukiyanchuk , B. Transverse kerker effect in all-dielectric spheroidal particles Sci Rep,12, 7997 (2022) 10.21203/rs.3.rs-947900/v1 botherref

    work page doi:10.21203/rs.3.rs-947900/v1 2022

  19. [20]

    Metamaterials and the landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism

    botherref Merlin , R. Metamaterials and the landau-Lifshitz permeability argument: Large permittivity begets high-frequency magnetism. Proceedings of the National Academy of Sciences of the United States of America 106 (2009) 10.1073/pnas.0808478106 botherref

    work page doi:10.1073/pnas.0808478106 2009

  20. [21]

    , Monticone , F

    botherref Shafiei , F. , Monticone , F. , Le , K.Q. , Liu , X.X. , Hartsfield , T. , Alù , A. , Li , X. A subwavelength plasmonic metamolecule exhibiting magnetic- based optical fano resonance. Nature Nanotechnology 8 (2013) 10.1038/nnano.2012.249 botherref

    work page doi:10.1038/nnano.2012.249 2013

  21. [22]

    : Negative refraction

    barticle Pendry , J.B. : Negative refraction . Contemporary Physics 45(3) , 191 -- 202 ( 2004 ) 10.1080/00107510410001667434 barticle

    doi:10.1080/00107510410001667434 2004

  22. [23]

    , Fleischman , D

    botherref Papadakis , G.T. , Fleischman , D. , Davoyan , A. , Yeh , P. , Atwater , H.A. Optical magnetism in planar metamaterial heterostructures. Nature Communications 9 (2018) 10.1038/s41467-017-02589-8 botherref

    work page doi:10.1038/s41467-017-02589-8 2018

  23. [24]

    , Papasimakis , N

    botherref Li , J. , Papasimakis , N. , MacDonald , K.F. , Zheludev , N.I. Optical magnetic response without metamaterials. APL Photonics 6 (2021) 10.1063/5.0054752 botherref

    work page doi:10.1063/5.0054752 2021

  24. [25]

    42 (2025) 10.1364/OPN.36.11.000042 botherref

    botherref Enjavi, Mohammad , Javadizadeh, Saeed Adibi, Ali .Silicon Nitride: From Workhorse to Superstar Optics and Photonics News 36. 42 (2025) 10.1364/OPN.36.11.000042 botherref

    work page doi:10.1364/opn.36.11.000042 2025

  25. [26]

    , Webb , K.J

    bchapter Man , M. , Webb , K.J. : On the Microscopic Origins of Optical Magnetism — Insight from Mie Theory . Frontiers in Optics 2016, OSA Technical Digest ( 2016 ). 10.1364/FIO.2016.JW4A.5 bchapter

    work page doi:10.1364/fio.2016.jw4a.5 2016

  26. [27]

    , Alu , A

    botherref Monticone , F. , Alu , A. Metamaterial, plasmonic and nanophotonic devices. Reports on Progress in Physics 80 (2017) 10.1088/1361-6633/aa518f botherref

    work page doi:10.1088/1361-6633/aa518f 2017

  27. [28]

    , Lifshitz , E.M

    botherref Landau , L.D. , Lifshitz , E.M. , Chapter XII - Spatial Dispersion, In Course of Theoretical Physics,Electrodynamics of continuous media. (Second Edition) 29 (1984) 10.1016/B978-0-08-030275-1.50018-7 botherref

    work page doi:10.1016/b978-0-08-030275-1.50018-7 1984

  28. [29]

    , Gartstein , Y.N

    botherref Agranovich , V.M. , Gartstein , Y.N. Spatial dispersion and negative refraction of light. Uspekhi Fizicheskih Nauk 176 (2006) 10.3367/ufnr.0176.200610c.1051 botherref

    arXiv 2006

  29. [30]

    , Chichkov , B.N

    botherref Evlyukhin , A.B. , Chichkov , B.N. Multipole decompositions for directional light scattering. Physical Review B 100 (2019) 10.1103/PhysRevB.100.125415 botherref

    work page doi:10.1103/physrevb.100.125415 2019

  30. [31]

    https://silson.com/product/silicon-nitride-membranes/

    botherref Silicon nitride membranes - dielectric substrate materials. https://silson.com/product/silicon-nitride-membranes/. Accessed: 2024-01-27 botherref

    2024

  31. [32]

    sn-basic.bst

    FUNCTION identify.basic.version "sn-basic.bst" " [2024/07/19 v1.1 bibliography style]" * top ENTRY address archive author booktitle chapter doi edition editor eid eprint howpublished institution journal key keywords month note number organization pages publisher school series title type url volume year archivePrefix primaryClass adsurl adsnote version lab...

    2024

  32. [33]

    write newline

    " write newline "" before.all 'output.state := FUNCTION add.period duplicate empty 'skip "." * add.blank if FUNCTION if.digit duplicate "0" = swap duplicate "1" = swap duplicate "2" = swap duplicate "3" = swap duplicate "4" = swap duplicate "5" = swap duplicate "6" = swap duplicate "7" = swap duplicate "8" = swap "9" = or or or or or or or or or FUNCTION ...

  33. [34]

    write newline

    " write newline "" before.all 'output.state := FUNCTION output.doi doi empty skip "doi:" doi * "" * output if FUNCTION format.archive archivePrefix empty "" archivePrefix ":" * if FUNCTION format.primaryClass primaryClass empty "" " [" primaryClass * "] " * if FUNCTION format.eprint eprint empty "" archive empty " https://arxiv.org/abs/" eprint * " " * " ...

  34. [35]

    write newline

    " write newline "" before.all 'output.state := FUNCTION string.to.integer 't := t text.length 'k := #1 'char.num := t char.num #1 substring 's := s is.num s "." = or char.num k = not and char.num #1 + 'char.num := while char.num #1 - 'char.num := t #1 char.num substring FUNCTION find.integer 't := #0 'int := int not t empty not and t #1 #1 substring 's :=...

  35. [36]

    write newline

    " write newline "" before.all 'output.state := FUNCTION string.to.integer 't := t text.length 'k := #1 'char.num := t char.num #1 substring 's := s is.num s "." = or char.num k = not and char.num #1 + 'char.num := while char.num #1 - 'char.num := t #1 char.num substring FUNCTION find.integer 't := #0 'int := int not t empty not and t #1 #1 substring 's :=...

  36. [37]

    sn-nature.bst

    FUNCTION identify.nature.version "sn-nature.bst" " [2024/07/19 v1.1 bibliography style]" * top ENTRY address archive author booktitle chapter edition editor eprint howpublished institution journal key keywords month note number organization pages publisher school series title type url doi volume year archivePrefix primaryClass eid adsurl adsnote version l...

    2024

  37. [38]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  38. [39]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize ":" * " " *...

  39. [40]

    sn-vancouver-num.bst

    FUNCTION identify.vancouver.version "sn-vancouver-num.bst" " [2024/07/19 v1.1 Vancouver bibliography style]" * top ENTRY address assignee author booktitle chapter cartographer day edition editor howpublished institution inventor journal key keywords month note number organization pages part publisher school series title type volume word year eprint doi ur...

    2024

  40. [41]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize ":" * " " *...

  41. [42]

    , Hanafusa , H

    barticle Gan , M. , Hanafusa , H. , Kadoya , Y. : Experimental investigation on generalized Brewster effect for s-polarized terahertz waves at planar si-air interfaces with a thin doped layer . Optics Express 33 , 9306 -- 9316 ( 2025 ) 10.1364/OE.550968 barticle

    work page doi:10.1364/oe.550968 2025

  42. [43]

    , Qiu , T

    barticle Ding , L. , Qiu , T. , Zhang , J. , Wen , X. : Generalized Brewster effect tuned optically in a graphene/substrate system . Journal of Optics 21 ( 12 ), 125602 ( 2019 ) 10.1088/2040-8986/ab4fa1 barticle

    work page doi:10.1088/2040-8986/ab4fa1 2019

  43. [44]

    : Would brewster recognize today's brewster angle? Optics News 15 ( 6 ), 14 -- 18 ( 1989 ) 10.1364/ON.15.6.000014 barticle

    barticle Lakhtakia , A. : Would brewster recognize today's brewster angle? Optics News 15 ( 6 ), 14 -- 18 ( 1989 ) 10.1364/ON.15.6.000014 barticle

    work page doi:10.1364/on.15.6.000014 1989

  44. [45]

    : Fresnel's interface reflection coefficients for the parallel and perpendicular polarizations: global properties and facts not found in your textbook

    bchapter Azzam , R.M.A. : Fresnel's interface reflection coefficients for the parallel and perpendicular polarizations: global properties and facts not found in your textbook . Polarization Analysis and Measurement II , vol. 2265 ( 1994 ). 10.1117/12.186660 bchapter

    work page doi:10.1117/12.186660 1994

  45. [46]

    , Thonn , T.F

    barticle Azzam , R.M.A. , Thonn , T.F. : Pseudo-Brewster and second-Brewster angles of an absorbing substrate coated by a transparent thin film . Appl. Opt. 22 ( 24 ), 4155 -- 4165 ( 1983 ) 10.1364/AO.22.004155 barticle

    work page doi:10.1364/ao.22.004155 1983

  46. [47]

    : Optical Properties of Solids vol

    bbook Fox , M. : Optical Properties of Solids vol. 70 , p. ( 2001 ). 10.1119/1.1691372 bbook

    work page doi:10.1119/1.1691372 2001

  47. [48]

    https://uk.mathworks.com/matlabcentral/fileexchange/102234-driver-for-thorlabs-bbd302-prm1z8-k10cr1-motorized-stages

    botherref Driver for Thorlabs BBD302/PRM1Z8/K10CR1 motorized stages. https://uk.mathworks.com/matlabcentral/fileexchange/102234-driver-for-thorlabs-bbd302-prm1z8-k10cr1-motorized-stages. Accessed: 2025-08-18 botherref

    2025

  48. [49]

    bchapter Bahaa E. A. Saleh , M.C.T. : 9 . Resonator Optics , pp. 310 -- 341 . John Wiley & Sons, Ltd , ( 1991 ). 10.1002/0471213748.ch9 . https://onlinelibrary.wiley.com/doi/abs/10.1002/0471213748.ch9 bchapter

    work page doi:10.1002/0471213748.ch9 1991