Theory of spin-wave transport in ferromagnet-superconductor heterostructures: Negative refraction, perfect imaging and temperature-controlled spin-wave optics

Merel A. Bouma; Michael Borst; Pim H. Vree; Rembert A. Duine; Thijs van der Meer; Toeno van der Sar; Tomas T. Osterholt

REVIEW 4 major objections 5 minor 95 references

Theory of spin-wave transport in ferromagnet-superconductor heterostructures: Negative refraction, perfect imaging and temperature-controlled spin-wave optics

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Meissner screening of spin-wave stray fields can flatten the spin-wave isofrequency contours to nearly straight lines, making a ferromagnet–superconductor heterostructure a canalizing medium that promises perfect subwavelength spin-wave…

desk verdict The spin-wave Fresnel formalism is solid and new, but the 'perfect imaging' claim outruns the evidence; needs revision before it should be published. read the letter →

arxiv 2608.06476 v1 pith:5MN74LVQ submitted 2026-08-06 cond-mat.mes-hall cond-mat.supr-con

Tomas T. Osterholt , Thijs van der Meer , Pim H. Vree , Merel A. Bouma , Michael Borst , Toeno van der Sar , Rembert A. Duine This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.supr-con

keywords spin-waveopticsMeissnerscreeningferromagnet-superconductorheterostructuresisofrequencycontourscanalizationperfectimagingnegativerefractionmagnonics

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

The paper develops a theory of spin-wave optics in heterostructures made of a ferromagnetic insulator and a superconductor. It claims that Meissner screening of the spin-wave stray fields profoundly changes the spin-wave dispersion, producing negative phase- and group-velocity refraction, non-specular reflection laws, and temperature-dependent Fresnel coefficients. The central finding is that one isofrequency branch becomes nearly straight, a canalizing regime in which the group velocity is nearly independent of wavevector. If correct, this enables perfect subwavelength spin-wave imaging, efficient waveguiding, and temperature-controlled interferometric elements such as phase shifters and beam splitters.

What carries the argument

The central object is the spin-wave dispersion $\omega(\mathbf{k})$ of the FMI-SC heterostructure, derived from the Landau-Lifshitz equation with the superconductor's London response. In the London dipolar limit it reduces to $\omega(\mathbf{k})\approx\gamma B_0\xi+\frac{\gamma\mu_0 M_S}{2}kd\cos(\phi-\psi)[1+\xi\cos(\phi-\psi)]$, where the ellipticity $\xi=\sqrt{1+\mu_0M_S/B_0}$ is the only material parameter controlling the reflection and refraction laws. The almost-straight isofrequency contours make the medium canalizing, $\omega\approx c k_x$; the spin-wave Fresnel coefficients obtained from magnetostatic boundary conditions carry the scattering amplitudes.

What would settle it

Measure the isofrequency contour of an FMI-SC heterostructure at fixed frequency by wavevector-resolved Brillouin light scattering or NV-center magnetometry: if the contour is not straight over a broad range of $k_y$—or if imaging a subwavelength grating through a slab does not reproduce the grating at the image plane—the canalization and perfect-imaging claims fail.

Watch

Extended reading notes

Core claim

The paper claims that the dispersion of dipolar spin waves in a ferromagnetic-insulator–superconductor heterostructure is controlled by Meissner screening, and that in the London dipolar limit the superconductor-response term makes one isofrequency branch nearly straight. This canalizing medium is the key: for a dispersion of the form $\omega = c k_x$, every Fourier component of a spin-wave image acquires the same phase over a fixed distance, so the image is reproduced with subwavelength features. The same framework yields spin-wave analogues of Fresnel equations and reflection laws such as $\sin\phi'\approx a_R(\xi)\sinh(b_R(\xi)\sin\phi)$, with scattering that is tunable through the temperature-dependent London penetration depth $\lambda_L(T)$.

Load-bearing premise

The perfect-imaging claim assumes the dispersion branch is exactly canalizing, $\omega = c k_x$ with $k_x$ independent of $k_y$; the real system only approaches this in the London dipolar limit, so the result holds only as long as residual curvature and the finite range of straight wavevectors can be neglected.

Editorial extensions

If this is right

A grating placed in front of an FMI-SC slab can be imaged with subwavelength features on the far side, because all Fourier components of the object propagate with nearly the same phase.
Varying the temperature across the superconducting transition changes $d_s/\lambda_L$, shifting reflected and refracted angles by up to roughly 30 degrees and opening or closing reflection channels, yielding mirrors and beam splitters that are reconfigurable without changing the device geometry.
Spin-wave beams with different wavevectors self-collimate along a single direction set by the magnetization angle $\psi$, enabling waveguides without geometrical confinement and with enhanced group velocity.
Nonreciprocal phase shifters and beam splitters can be built, with phase accumulation that is direction-dependent and, for appropriate $\psi$, direction-independent on the modified branch, useful for Mach-Zehnder and Michelson magnonic interferometers.
The flattened contours persist as long as dipolar and Zeeman contributions dominate, suggesting the spin-wave optics platform remains robust when additional interactions such as anisotropies or Dzyaloshinskii-Moriya coupling are present.

Reading between the lines

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

The perfect imaging result is idealized: the real dispersion is only nearly canalizing, so the achievable resolution will be limited by residual curvature and by the finite wavevector range over which the contours stay straight; a quantitative estimate of that resolution limit is a natural next step.
The same straight-contour physics could extend to other magnetic systems, such as antiferromagnetic insulators capped with a superconductor, though the dispersion structure would need to be re-derived for those materials.
Beyond imaging, the canalizing medium could serve as a building block for transformation-type spin-wave devices, such as beam steerers and self-collimated routing elements in integrated magnonic circuits.
The temperature-controlled opening and closing of reflection channels suggests a mechanism for thermal switching of magnonic logic elements without moving parts or applied electric fields.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear [Colleague],

The thing to know about this paper is that the scattering framework is the real contribution, while the headline perfect-imaging claim is not backed by the actual dispersion. If you work on spin-wave optics, the Fresnel equations (92)-(95) and the temperature-dependent reflection/refraction analysis give you something to build on. The self-consistent field derivation in Sec. II is careful and, unlike earlier work, computes all components of the response field; the authors are also honest that the dispersion itself comes from Ref. [36]. That is exactly the right way to reuse prior results.

The new material is the spin-wave analogue of Fresnel coefficients at a bare-FMI / FMI-SC interface, plus the demonstration that varying T through d_s/λ_L can tune reflection and refraction angles by up to ~30°. The negative reflection and negative phase/group-velocity refraction analysis is clear and well-illustrated, and the conditions under which they occur are carefully mapped.

Now the soft spots, in proportion. The straight isofrequency contours that motivate the applications are derived from the London dipolar limit, Eq. (55). But inside that limit's validity (kd≪1, kλ_L≪1), the dispersion has the form q + ξ q^2/k = const, which does not give straight contours. The flattening only appears for large perpendicular wavevector, where kd and kλ_L are no longer small and the same approximations break down. So the 'nearly straight' claim is on shaky ground exactly where it matters. The perfect imaging argument in Sec. IV.A then uses an idealized dispersion ω = c kx rather than the actual kx(ky); the authors never show that the exact dispersion canalizes over the spatial-frequency range needed. That is a genuine gap, and the abstract overstates it by saying 'perfect imaging'.

The 'universal' reflection and refraction laws, Eqs. (67) and (76), are empirical fits to numerics, valid only for ξ≥2, with fitting functions that are not derived. They reproduce the numerics well, but calling them 'universal' is too strong. And they don't cover the ξ≈1 regime where the flattening is supposed to occur.

The paper deserves a serious referee: the scattering formalism is a solid contribution, and the experimentalists in this area will want the Fresnel coefficients. But the imaging and straight-contour claims need quantitative support from the exact dispersion, or the title and abstract need to be tempered. I would recommend peer review with the explicit request to fix that gap.

Best, [Your name]

Referee Report

4 major / 5 minor

Summary. The manuscript develops a theoretical framework for dipolar spin waves in ferromagnet–superconductor (FMI-SC) heterostructures. Using a self-consistent solution of Maxwell's and London's equations for the superconducting screening field, the authors re-derive the spin-wave dispersion (Sec. II) and then analyze reflection and refraction at a bare-FMI/FMI-SC interface (Sec. III). They derive spin-wave Fresnel coefficients in the London dipolar limit, present empirical 'universal' reflection/refraction laws, and identify negative reflection, negative phase and group refraction, and temperature-tunable scattering. The most striking claim is that Meissner screening produces nearly straight isofrequency contours, which are argued to enable perfect spin-wave imaging, waveguiding, and interferometric elements (Sec. IV).

Significance. The self-consistent derivation of the response field and the analytic Fresnel coefficients are careful and of technical value, and the mechanism of temperature tuning through the London penetration depth is physically appealing and plausible. If the straight-contour claim could be substantiated quantitatively, the system would be a promising platform for magnonic canalization, going beyond the flat regions available in conventional dipolar films. The paper is clearly written and gives proper credit to the earlier derivation of the dispersion in Ref. [36]. However, at present the headline functional claims—especially 'perfect imaging'—are supported only by an ideal-medium calculation and by qualitative statements about flatness, without a quantitative analysis of the actual k_x(k_y) relation or of the finite-slab transfer function.

major comments (4)

[§IV.A, Eq. (99)] The perfect-imaging demonstration assumes the exact canalization dispersion ω = c k_x, with k_x independent of k_y. The actual London-dipolar dispersion, Eq. (55), gives isofrequency contours q + ξ q^2/k = constant (with q = k_x cosψ + k_y sinψ and k = sqrt(q^2+p^2)); these contours are straight only in the limit |p| >> |q|, where k ≈ |p| and the conditions kd << 1 and kλ_L << 1 underlying Eq. (55) break down. No calculation is presented using the exact dispersion (44) to show that k_x is approximately independent of k_y over the spatial-frequency range of interest, nor to quantify the resulting image fidelity. The 'perfect imaging' claim is therefore not established for the FMI-SC system itself; it is a property of an idealized canalizing medium.
[§III.B.2 and Eq. (83)] The 'nearly straight isofrequency contours' are asserted without a quantitative flatness measure. The text states that straightening occurs for ξ sufficiently close to unity, while the universal reflection/refraction laws of Eqs. (67) and (76) are validated only for ξ ≥ 2 (μ0M_S ≥ 3B_0). These two parameter regimes appear to be different, and the paper does not address whether the straight-contour effect and the universal laws can coexist. A quantitative map of the straightness of the contours (e.g., a flatness tolerance) as a function of ξ and ψ, compared with the bare-FMI contours, is needed to support the canalization claims.
[§III.B.1, Eqs. (67)–(79)] The reflection and refraction laws are empirical fits to numerical solutions, and the paper calls them 'universal' on the basis of their independence of material parameters except ξ. However, no fit-quality metrics (residuals, confidence intervals) are given, and the paper itself notes that the polynomial approximation underlying the fits breaks down at large angles. Since these laws are presented as a central result of the spin-wave optics framework, the authors should either provide a derivation or scaling argument, or quantify the accuracy and domain of validity of the fits.
[§IV.A and IV.C] The imaging and waveguiding applications consider only the phase accumulation through the medium and ignore the amplitude and reflection effects at the boundaries of a finite FMI-SC slab. The Fresnel coefficients (92)–(95) show that transmission amplitudes depend on angle and wavevector, and a finite slab will also produce interface reflections. The perfect-imaging calculation in Eq. (99) would require all Fourier components to acquire the same phase and transmission amplitude; the paper does not compute the transfer function of a finite FMI-SC region. An estimate of the achievable resolution and fidelity is necessary before 'perfect imaging' can be claimed.

minor comments (5)

[General] There are several typos, including 'illusrated' in the Fig. 10 caption and 'Eq. 92-95' which should read 'Eqs. (92)–(95)'.
[§III.B.1 / Eq. (58)] The relation is more commonly called the Gorter-Casimir relation; the paper should be consistent with the naming.
[Figs. 3(d) and 6(d)] The temperature range corresponding to the plotted d_s/λ_L values is not specified; the statement that temperature changes the angle 'by up to 30°' should be tied to a concrete temperature interval.
[§III.B.2, Eq. (83)] Equation (83) is described as a 'nearly straight line' but its right-hand side still contains k_x; it would be clearer to present the asymptotic line q = constant and state the condition under which the quadratic correction term is negligible.
[Abstract and §IV.A] The paper oscillates between 'perfect imaging' and 'nearly straight' contours; the abstract and conclusion say 'perfect', while the body often says 'nearly'. The authors should consistently distinguish the ideal demonstration from the approximate physical realization.

Circularity Check

1 steps flagged · score 2.0 of 10

Central dispersion and Fresnel derivations are self-contained; minor circularity in presenting numerical fits as universal scattering laws.

fitted input called prediction [Section III.A.1, Eqs. (67)-(69) and Section III.B.1, Eqs. (76)-(79)]
"we find that the reflection law for |ϕ|≤90◦ (right-moving incident spin waves) is accurately described by the k-independent equation sinϕ′≈a_R(ξ) sinh(b_R(ξ) sinϕ), where a_R(ξ)≈0.07 + 0.57 ln[0.55 + 0.23ξ] + 0.013ξ, b_R(ξ)≈ 3.16√ξ−0.73 . We have numerically verified the validity of the reflection law for ξ≥2"

The coefficients a_R, b_R (and similarly a_T, b_T, p for refraction) are fitted to the numerical solutions of the exact dispersion relations, Eqs. (44)/(55)-(57), and are then presented as 'universal' and 'independent of the microscopic details'. The reflection and refraction angles are therefore not independently predicted by the claimed laws; the laws are compact parametrizations of the same numerical data used for the fit. This is a mild internal circularity because these fitted laws do not feed back into the derivation of the dispersion or the flat-contour claim, but they are presented as fundamental results rather than as empirical fits.

full rationale

The paper's core derivation is self-contained: the dispersion relation in Section II is obtained from Maxwell's equations, the London equations, and the Landau-Lifshitz equation, with the self-consistent screening field computed explicitly. The self-citation to Ref. [36] is for the same dispersion, but the present paper re-derives it in a more general framework, so the citation is not load-bearing. The Fresnel coefficients in Section III.C are derived from magnetostatic boundary conditions, not imported from prior work. The 'perfect imaging' section uses an idealized dispersion omega = c kx as a demonstrative calculation; the claim that the FMI-SC system approximates this behavior is a separate, under-supported assertion about the actual dispersion, but the imaging formula itself does not presuppose the FMI-SC dispersion. The only identifiable circularity is the fitting of the 'universal' reflection and refraction laws to the paper's own numerics and then presenting those fits as fundamental laws. This is a minor issue and does not compromise the independent content of the dispersion and Fresnel derivations. Overall score 2.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the London-theory description of the superconductor, the quasi-static approximation, the ansatz for the field inside the superconductor, the thin-film and dipolar limits, and the magnetostatic boundary conditions. The fitted functions for the reflection and refraction laws are free parameters tuned to the paper's own numerical solutions.

free parameters (5)

a_R(xi) coefficients = 0.07, 0.57, 0.55, 0.23, 0.013
Empirical coefficients in Eq. (68) fitting the numerically computed reflection law to the functional form of Eq. (67); not derived from the dispersion.
b_R(xi) coefficient = 3.16*sqrt(xi) - 0.73
Empirical fit in Eq. (69) to the numerical reflection angle; determines the nonlinearity of the reflection law.
a_T(xi) = 2.01 + 1.70/(xi - 0.95)
Empirical fit in Eq. (77) for the low-angle refraction ratio n(xi)/n'(xi).
b_T(xi) = 13.0 + 120*exp(-1.70*xi) - 4.21*xi^{-0.18}
Empirical fit in Eq. (78) controlling the angular dependence of the refraction law.
p(xi) = 0.28 + 0.47/(xi + 0.20)
Empirical exponent in Eq. (79) used in the refraction law; fitted to numerical solutions.

assumptions (6)

domain assumption The superconductor is described by the London equations, including the second London equation curl J_S = -B_S/(mu0 lambda_L^2) (Eq. 7).
This is the core model for Meissner screening throughout; it neglects non-local and vortex effects, which the authors acknowledge in the conclusion.
domain assumption Quasi-static approximation, displacement currents ignored (Sec II A 1).
Allows expressing the response field via the Biot-Savart-type integral (Eq. 15); standard for spin-wave frequencies but an approximation.
ad hoc to paper The magnetic field inside the superconductor takes the ansatz f(z) = C1 exp(-zeta z) + C2 exp(zeta z) (Eq. 21) with zeta = sqrt(k^2 + 1/lambda_L^2).
This ansatz is motivated by the infinite-slab solution and verified by substitution; it is the load-bearing form for the response field but is not derived from more basic principles.
domain assumption Exchange boundary conditions (Rado-Weertman) justify the homogeneous spin-wave profile across the film thickness (Sec II B 1).
Needed for the magnetization ansatz Eq. (1); alternative pinning conditions would modify the dispersion.
domain assumption In the London dipolar limit, kd << 1, k lambda_L << 1, d_s >= lambda_L, and exchange is neglected (Sec III).
Defines the regime for the simplified dispersion Eq. (55), the reflection/refraction laws and the flat-contour claims.
domain assumption Magnetostatic boundary conditions are applied to the thickness-averaged normal B and tangential H components only, excluding the z-component (Sec III C and Appendix C).
Needed to derive the Fresnel coefficients; the exclusion of H_z is justified by a scalar-potential argument in Appendix C.

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Cite this review

Pith. "Pith review of Theory of spin-wave transport in ferromagnet-superconductor heterostructures: Negative refraction, perfect imaging and temperature-controlled spin-wave optics." pith.science (2026). https://pith.science/paper/5MN74LVQ

@misc{pith2026260806476,
  author       = {Pith},
  title        = {Pith review of: Theory of spin-wave transport in ferromagnet-superconductor heterostructures: Negative refraction, perfect imaging and temperature-controlled spin-wave optics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MN74LVQ}},
  note         = {Machine review of arXiv:2608.06476}
}

read the original abstract

We investigate spin-wave transport in ferromagnetic insulator-superconductor (FMI-SC) heterostructures and develop a general theoretical framework for spin-wave optics in these hybrid systems. We demonstrate that Meissner screening by the superconductor gives rise to a range of unconventional wave phenomena, including negative phase- and group-velocity refraction, and reflection and refraction laws that differ fundamentally from their optical counterparts. Within this framework, we derive the spin-wave Fresnel equations governing reflection and transmission at FMI-SC interfaces and show that the scattering properties exhibit a pronounced temperature dependence, enabling tunable spin-wave mirrors and refractive elements. Most strikingly, we find that superconducting screening can produce nearly straight isofrequency contours, far flatter than the kinked, intrinsically curved contours attainable in conventional dipolar spin-wave systems. We show that these straight contours enable functionalities such as perfect spin-wave imaging, efficient waveguiding, and interferometric elements, such as phase shifters and beam splitters, with unconventional properties. Our results establish FMI-SC heterostructures as a versatile platform for temperature-tunable spin-wave optics and interferometric magnonic devices.

Figures

Figures reproduced from arXiv: 2608.06476 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

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Reference graph

Works this paper leans on

95 extracted references · 70 canonical work pages

[36]

Qin and S

H. Qin and S. van Dijken, Applied Physics Letters116, 202403 (2020)

work page 2020
[1]

We first consider the case whereψ= 0, i.e., when the applied fieldH 0 is parallel to the interface, for we can then capture the reflection laws by fairly simple formulas

Law of reflection forψ= 0. We first consider the case whereψ= 0, i.e., when the applied fieldH 0 is parallel to the interface, for we can then capture the reflection laws by fairly simple formulas. The reflection law in the bare FMI can be derived di- rectly from either Eq. (47) or Eq. (57), and we have ϕ′ =π−ϕ,(61) k′ =k.(62) We recognize this as the con...
[2]

The first phenomenon we discuss isnegative reflection

Negative reflection and reflection-channel control As briefly mentioned in the previous paragraph, the system exhibits even more intriguing properties when the angleψ, which characterizes the orientation of the satu- ration magnetization, is nonzero. The first phenomenon we discuss isnegative reflection. In optics, the conventional law of reflection dicta...
[3]

Generalization to arbitrary in-plane applied fields So far, we have considered the case in which the ex- ternal magnetic field is oriented along the ˆydirection, B0 =B 0ˆy, and derived the corresponding spin-wave dis- persion and eigenmodes. To analyze spin-wave reflection and refraction at an interface involving the FMI-SC sys- tem, it is necessary to co...
[4]

Law of refraction forψ= 0 Like before, settingψ= 0 and using the approximated dispersion relations of Eq. (55) and (57), we can gain useful insight into the refraction by considering the roots of the polynomialg k(k′ x), given by gk(k′ x) =Q g,k(k′ x) +Rg,k(k′ x),(72) where Qg,k(k′ x) = (ξ2−1)k′4 x + 2Cg,kk′3 x− C2 g,k +k 2 y k′2 x, (73) Rg,k(k′ x) =C g,k...
[5]

The most intriguing phenomenon that occurs exclu- sively whenψ̸= 0 isnegative refractionwith respect to the group velocity

Negative refraction We now consider refraction properties that emerge when the saturation magnetization angleψis nonzero. The most intriguing phenomenon that occurs exclu- sively whenψ̸= 0 isnegative refractionwith respect to the group velocity. Conventional optics dictates that the refracted light ray emerges on the opposite side of the interface normal ...
[6]

Infinite magnetic film We first consider a magnetic film that extends to±∞in both thex- andy-directions

Dipolar field integrals a. Infinite magnetic film We first consider a magnetic film that extends to±∞in both thex- andy-directions. Starting from a magnetization densitym k(r,t) given by mk(r,t) =e i(k·ρ−ωt) θ(z+d/2)−θ(z−d/2)   imx imy mz  ,(A1) we seek to derive the resulting dipolar (magnetizing) fieldH d(r,t). Since no free current density is prese...
[7]

Referring to Eq

Ansatz field integrals We assume the following ansatz for the magnetic-field solution inside the superconducting region Ω S, B(r,t) =e i(k·ρ−ωt)   ikx C1e−ζz +C 2eζz iky C1e−ζz +C 2eζz −k2 ζ C1e−ζz + k2 ζ C2eζz +C 3   ,(A15) wherez∈[d/2,d/2 +d s]. Referring to Eq. (15), we are interested in integrals of the form IΩS(r,t) = Z ΩS d3r′ B(r′,t...

Show all 95 references

[8]

S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln´ ar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science294, 1488 (2001), https://www.science.org/doi/pdf/10.1126/science.1065389

2001 doi
[9]

ˇZuti´ c, J

I. ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.76, 323 (2004)

2004
[10]

Hoffmann and S

A. Hoffmann and S. D. Bader, Phys. Rev. Appl.4, 047001 (2015)

2015
[11]

Hirohata, K

A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu, B. Di´ eny, P. Pirro, and B. Hillebrands, Journal of Magnetism and Magnetic Materials509, 166711 (2020)

2020
[12]

A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Physics11, 453 (2015)

2015
[13]

Pirro, V

P. Pirro, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Reviews Materials6, 1114 (2021)

2021
[14]

H. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, Physics Reports965, 1–74 (2022)

2022
[15]

Csaba, ´A

G. Csaba, ´A. Papp, and W. Porod, Physics Letters A381, 1471 (2017)

2017
[16]

D. R. Candido, G. D. Fuchs, M. Johnston-Halperin, and M. E. Flatt´ e, Materials for Quantum Technology1, 011001 (2020)

2020
[17]

S. Yuan, C. Liu, J. Chen, S. Liu, J. Lan, H. Yu, J. Wu, F. Yan, M.-H. Yung, J. Xiao, L. Jiang, and D. Yu, Physical Review A107, 012434 (2023)

2023
[18]

S. S. Mukherjee, J. H. Kwon, M. Jamali, M. Hayashi, and H. Yang, Phys. Rev. B85, 224408 (2012)

2012
[19]

Rousseau, B

O. Rousseau, B. Rana, R. Anami, M. Yamada, K. Miura, S. Ogawa, and Y. Otani, Scientific Reports5, 9873 (2015)

2015
[20]

Kanazawa, T

N. Kanazawa, T. Goto, K. Sekiguchi, A. B. Granovsky, C. A. Ross, H. Takagi, Y. Nakamura, and M. Inoue, Scientific Reports6, 30268 (2016)

2016
[21]

´A. Papp, W. Porod, and G. Csaba, Nature Communications12, 6422 (2021)

2021
[22]

Loayza, M

N. Loayza, M. B. Jungfleisch, A. Hoffmann, M. Bailleul, and V. Vlaminck, Phys. Rev. B98, 144430 (2018)

2018
[23]

Vlaminck, L

V. Vlaminck, L. Temdie, V. Castel, M. B. Jungfleisch, D. Stoeffler, Y. Henry, and M. Bailleul, Journal of Applied Physics 133, 10.1063/5.0128666 (2023)

2023 doi
[24]

Y. Wang, W. Yan, N. Kuznetsov, L. c. v. Flajˇ sman, H. Qin, and S. van Dijken, Phys. Rev. Appl.22, 014038 (2024)

2024
[25]

Stigloher, M

J. Stigloher, M. Decker, H. S. K¨ orner, K. Tanabe, T. Moriyama, T. Taniguchi, H. Hata, M. Madami, G. Gubbiotti, K. Kobayashi, T. Ono, and C. H. Back, Phys. Rev. Lett.117, 037204 (2016)

2016
[26]

Hioki, R

T. Hioki, R. Tsuboi, T. H. Johansen, Y. Hashimoto, and E. Saitoh, Applied Physics Letters116, 10.1063/1.5141864 (2020)

2020 doi
[27]

Mieszczak, O

S. Mieszczak, O. Busel, P. Gruszecki, A. N. Kuchko, J. W. K los, and M. Krawczyk, Phys. Rev. Appl.13, 054038 (2020)

2020
[28]

Li and R

Y.-H. Li and R. Cheng, Phys. Rev. B102, 094404 (2020)

2020
[29]

H. Wang, M. Madami, J. Chen, H. Jia, Y. Zhang, R. Yuan, Y. Wang, W. He, L. Sheng, Y. Zhang, J. Wang, S. Liu, K. Shen, G. Yu, X. Han, D. Yu, J.-P. Ansermet, G. Gubbiotti, and H. Yu, Phys. Rev. X13, 021016 (2023)

2023
[30]

D. O. Oriekhov, T. T. Osterholt, R. A. Duine, and V. P. Gusynin, Phys. Rev. B112, 045142 (2025)

2025
[31]

T. T. Osterholt, D. O. Oriekhov, L. Eek, C. M. Smith, and R. A. Duine, Twist-modulated magnetic interactions in bilayer van der Waals materials (2025), arXiv:2509.14122 [cond-mat.mes-hall]. 30

2025
[32]

Weiler, L

M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S. Brandt, and S. T. B. Goennenwein, Phys. Rev. Lett.106, 117601 (2011)

2011
[33]

Liu and A

A. Liu and A. M. Finkel’stein, Phys. Rev. B105, L020404 (2022)

2022
[34]

Demokritov, B

S. Demokritov, B. Hillebrands, and A. Slavin, Physics Reports348, 441 (2001)

2001
[35]

Tacchi, P

S. Tacchi, P. Gruszecki, M. Madami, G. Carlotti, J. W. K los, M. Krawczyk, A. Adeyeye, and G. Gubbiotti, Scientific Reports5, 10367 (2015)

2015
[37]

Rovillain, R

P. Rovillain, R. de Sousa, Y. Gallais, A. Sacuto, M. A. M´ easson, D. Colson, A. Forget, M. Bibes, A. Barth´ el´ emy, and M. Cazayous, Nature Materials9, 975 (2010)

2010
[38]

M. Zhu, Z. Zhou, B. Peng, S. Zhao, Y. Zhang, G. Niu, W. Ren, Z. G. Ye, Y. Liu, and M. Liu, Advanced Functional Materials27, 1604088 (2017)

2017
[39]

H. Qin, R. Dreyer, G. Woltersdorf, T. Taniyama, and S. van Dijken, Nature Communications10, 4790 (2019)

2019
[40]

I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, V. V. Bolginov, V. V. Ryazanov, A. A. Golubov, and A. V. Ustinov, Advanced Functional Materials28, 1802375 (2018), https://advanced.onlinelibrary.wiley.com/doi/pdf/10.1002/adfm.201802375

2018 doi
[41]

Golovchanskiy, N

I. Golovchanskiy, N. Abramov, V. Stolyarov, V. Chichkov, M. Silaev, I. Shchetinin, A. Golubov, V. Ryazanov, A. Ustinov, and M. Kupriyanov, Phys. Rev. Appl.14, 024086 (2020)

2020
[42]

Yu and G

T. Yu and G. E. W. Bauer, Phys. Rev. Lett.129, 117201 (2022)

2022
[43]

Borst, P

M. Borst, P. H. Vree, A. Lowther, A. Teepe, S. Kurdi, I. Bertelli, B. G. Simon, Y. M. Blanter, and T. van der Sar, Science 382, 430 (2023), https://www.science.org/doi/pdf/10.1126/science.adj7576

2023 doi
[44]

Ghirri, C

A. Ghirri, C. Bonizzoni, M. Maksutoglu, and M. Affronte, Phys. Rev. Appl.22, 034004 (2024)

2024
[45]

Yu, X.-H

T. Yu, X.-H. Zhou, G. E. Bauer, and I. Bobkova, Physics Reports1151, 1–94 (2026)

2026
[46]

X.-H. Zhou, X. Ye, L. Bai, and T. Yu, Phys. Rev. B110, L020404 (2024)

2024
[47]

Balasubramanian, I

G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, and J. Wrachtrup, Nature455, 648 (2008)

2008
[48]

J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, Nature Physics4, 810 (2008)

2008
[49]

Casola, T

F. Casola, T. van der Sar, and A. Yacoby, Nature Reviews Materials3, 17088 (2018)

2018
[50]

T. T. Osterholt, P. M. Gunnink, and R. A. Duine, Phys. Rev. B110, 134424 (2024)

2024
[51]

One therefore does not have to consider a back-reaction from the superconductor to this applied field

For an infinite rectangular superconducting film in a static and uniform applied fieldB0, the field outside the superconductor will always be equal toB 0 [49]. One therefore does not have to consider a back-reaction from the superconductor to this applied field
[52]

All the effects of the response field on the magnetization of the film are included inH in d,ave, and we therefore haveH in sc,ave = Bin sc,ave/µ0
[53]

Due to the symmetry of the system, a time-independentm y has to be spatially uniform inside the magnetic film
[54]

Rado and J

G. Rado and J. Weertman, Journal of Physics and Chemistry of Solids11, 315 (1959)

1959
[55]

It should be noted that the reflection and refraction behavior depends not only on the ratiod s/λL, but also on the magnitude of the London penetration depthλ L relative to the spin-wave wavelength. The superconducting response is strongest in the regimekλ L ≪1, whereas the in...
[56]

Tinkham,Introduction to Superconductivity, 2nd ed

M. Tinkham,Introduction to Superconductivity, 2nd ed. (Dover Publications, Inc., 1996)

1996
[57]

Zhang and T

C. Zhang and T. J. Cui, Applied Physics Letters91, 194101 (2007)

2007
[58]

´Alvarez P´ erez, J

G. ´Alvarez P´ erez, J. Duan, J. Taboada-Guti´ errez, Q. Ou, E. Nikulina, S. Liu, J. H. Edgar, Q. Bao, V. Giannini, R. Hil- lenbrand, J. Mart´ ın-S´ anchez, A. Y. Nikitin, and P. Alonso-Gonz´ alez, Science Advances8, eabp8486 (2022)

2022
[59]

Lesniewski, Y

N. Lesniewski, Y. Dadoenkova, F. F. L. Bentivegna, and P. Gruszecki, ACS Applied Materials & Interfaces18, 10539 (2026)

2026
[60]

This solution, however, will be located atk≫1/d, and therefore lies far outside the dipolar regime

Strictly speaking, a reflection solution may still exist, as the exchange contribution to the frequency causes the isofrequency contours to close eventually. This solution, however, will be located atk≫1/d, and therefore lies far outside the dipolar regime
[61]

V. G. Veselago, Soviet Physics Uspekhi10, 509 (1968)

1968
[62]

J. B. Pendry, Phys. Rev. Lett.85, 3966 (2000)

2000
[63]

J. B. Pendry, D. Schurig, and D. R. Smith, Science312, 1780 (2006)

2006
[64]

Leonhardt, Science312, 1777 (2006)

U. Leonhardt, Science312, 1777 (2006)

2006
[65]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, Photonics and Nanostructures - Fundamentals and Applications6, 87 (2008), the Seventh International Symposium on Photonic and Electromagnetic Crystal Structures

2008
[66]

H. Chen, C. T. Chan, and P. Sheng, Nature Materials9, 387 (2010)

2010
[67]

S.-K. Kim, S. Choi, K.-S. Lee, D.-S. Han, D.-E. Jung, and Y.-S. Choi, Applied Physics Letters92, 212501 (2008)

2008
[68]

N. W. Ashcroft and N. D. Mermin,Solid State Physics(Saunders College Publishing, 1976)

1976
[69]

Verba, V

R. Verba, V. Tiberkevich, and A. Slavin, Phys. Rev. B101, 144430 (2020)

2020
[70]

Born and E

M. Born and E. Wolf,Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999)

1999
[71]

Hecht,Optics, 5th ed

E. Hecht,Optics, 5th ed. (Pearson, Boston, MA, 2017)

2017
[72]

S. A. Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz, Journal of Modern Optics49, 1747–1762 (2002). 31

2002
[73]

G´ omez-Santos, Phys

G. G´ omez-Santos, Phys. Rev. Lett.90, 077401 (2003)

2003
[74]

P. A. Belov, C. R. Simovski, and P. Ikonen, Phys. Rev. B71, 193105 (2005)

2005
[75]

P. A. Belov, Y. Hao, and S. Sudhakaran, Phys. Rev. B73, 033108 (2006)

2006
[76]

P. A. Belov and Y. Hao, Phys. Rev. B73, 113110 (2006)

2006
[77]

Mansfeld, J

S. Mansfeld, J. Topp, K. Martens, J. N. Toedt, W. Hansen, D. Heitmann, and S. Mendach, Phys. Rev. Lett.108, 10.1103/physrevlett.108.047204 (2012)

2012 doi
[78]

Makartsou, M

U. Makartsou, M. Golebiewski, U. Guzowska, A. Stognij, R. Gieniusz, and M. Krawczyk, Applied Physics Letters124 (2024)

2024
[79]

Toedt, M

J.-N. Toedt, M. Mundkowski, D. Heitmann, S. Mendach, and W. Hansen, Scientific Reports6, 33169 (2016)

2016
[80]

N. J. Whitehead, S. A. R. Horsley, T. G. Philbin, and V. V. Kruglyak, Applied Physics Letters113, 10.1063/1.5049470 (2018)

2018 doi
[81]

Gr¨ afe, P

J. Gr¨ afe, P. Gruszecki, M. Zelent, M. Decker, K. Keskinbora, M. Noske, P. Gawronski, H. Stoll, M. Weigand, M. Krawczyk, C. H. Back, E. J. Goering, and G. Sch¨ utz, Phys. Rev. B102, 024420 (2020)

2020
[82]

W. Bao, Z. Wang, Y. Cao, and P. Yan, Physical Review B102, 10.1103/physrevb.102.014423 (2020)

2020 doi
[83]

H. Dai, Y. Xing, M. Chen, M. Gao, Z. Guo, Y. Zhang, X. Ma, X. Hao, Z. A. Mohamed, H. Zhang, and C. Liu, Journal of Magnetism and Magnetic Materials545, 168743 (2022)

2022
[84]

V. E. Demidov, S. O. Demokritov, D. Birt, B. O’Gorman, M. Tsoi, and X. Li, Phys. Rev. B80, 014429 (2009)

2009
[85]

Wagner, A

K. Wagner, A. K´ akay, K. Schultheiss, A. Henschke, T. Sebastian, and H. Schultheiss, Nature Nanotechnology11, 432 (2016)

2016
[86]

Henry, D

Y. Henry, D. Stoeffler, J.-V. Kim, and M. Bailleul, Phys. Rev. B100, 10.1103/physrevb.100.024416 (2019)

2019 doi
[87]

R. A. Gallardo, P. Alvarado-Seguel, F. Brevis, A. Rold´ an-Molina, K. Lenz, J. Lindner, and P. Landeros, Nanomaterials 12, 10.3390/nano12162785 (2022)

2022 doi
[88]

Schneider, A

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, Phys. Rev. Lett.104, 197203 (2010)

2010
[89]

Abrikosov, Journal of Physics and Chemistry of Solids2, 199 (1957)

A. Abrikosov, Journal of Physics and Chemistry of Solids2, 199 (1957)

1957
[90]

A. A. Bespalov, A. S. Mel’nikov, and A. I. Buzdin, Phys. Rev. B89, 054516 (2014)

2014
[91]

O. V. Dobrovolskiy and A. V. Chumak, Journal of Magnetism and Magnetic Materials543, 168633 (2022)

2022
[92]

Niedzielski, C

B. Niedzielski, C. Jia, and J. Berakdar, Phys. Rev. Appl.19, 024073 (2023)

2023
[93]

D. S. Katkov, S. S. Apostoloff, and I. S. Burmistrov, JETP Letters120, 655 (2024)

2024
[94]

O. V. Dobrovolskiy, Q. Wang, D. Y. Vodolazov, R. Sachser, M. Huth, S. Knauer, and A. I. Buzdin, Nature Nanotechnology 20, 1764 (2025)

2025
[95]

J. D. Jackson,Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 1999)

1999

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [36]

Qin and S

H. Qin and S. van Dijken, Applied Physics Letters116, 202403 (2020)

work page 2020

[2] [1]

We first consider the case whereψ= 0, i.e., when the applied fieldH 0 is parallel to the interface, for we can then capture the reflection laws by fairly simple formulas

Law of reflection forψ= 0. We first consider the case whereψ= 0, i.e., when the applied fieldH 0 is parallel to the interface, for we can then capture the reflection laws by fairly simple formulas. The reflection law in the bare FMI can be derived di- rectly from either Eq. (47) or Eq. (57), and we have ϕ′ =π−ϕ,(61) k′ =k.(62) We recognize this as the con...

[3] [2]

The first phenomenon we discuss isnegative reflection

Negative reflection and reflection-channel control As briefly mentioned in the previous paragraph, the system exhibits even more intriguing properties when the angleψ, which characterizes the orientation of the satu- ration magnetization, is nonzero. The first phenomenon we discuss isnegative reflection. In optics, the conventional law of reflection dicta...

[4] [3]

Generalization to arbitrary in-plane applied fields So far, we have considered the case in which the ex- ternal magnetic field is oriented along the ˆydirection, B0 =B 0ˆy, and derived the corresponding spin-wave dis- persion and eigenmodes. To analyze spin-wave reflection and refraction at an interface involving the FMI-SC sys- tem, it is necessary to co...

[5] [4]

Law of refraction forψ= 0 Like before, settingψ= 0 and using the approximated dispersion relations of Eq. (55) and (57), we can gain useful insight into the refraction by considering the roots of the polynomialg k(k′ x), given by gk(k′ x) =Q g,k(k′ x) +Rg,k(k′ x),(72) where Qg,k(k′ x) = (ξ2−1)k′4 x + 2Cg,kk′3 x− C2 g,k +k 2 y k′2 x, (73) Rg,k(k′ x) =C g,k...

[6] [5]

The most intriguing phenomenon that occurs exclu- sively whenψ̸= 0 isnegative refractionwith respect to the group velocity

Negative refraction We now consider refraction properties that emerge when the saturation magnetization angleψis nonzero. The most intriguing phenomenon that occurs exclu- sively whenψ̸= 0 isnegative refractionwith respect to the group velocity. Conventional optics dictates that the refracted light ray emerges on the opposite side of the interface normal ...

[7] [6]

Infinite magnetic film We first consider a magnetic film that extends to±∞in both thex- andy-directions

Dipolar field integrals a. Infinite magnetic film We first consider a magnetic film that extends to±∞in both thex- andy-directions. Starting from a magnetization densitym k(r,t) given by mk(r,t) =e i(k·ρ−ωt) θ(z+d/2)−θ(z−d/2)   imx imy mz  ,(A1) we seek to derive the resulting dipolar (magnetizing) fieldH d(r,t). Since no free current density is prese...

[8] [7]

Referring to Eq

Ansatz field integrals We assume the following ansatz for the magnetic-field solution inside the superconducting region Ω S, B(r,t) =e i(k·ρ−ωt)   ikx C1e−ζz +C 2eζz iky C1e−ζz +C 2eζz −k2 ζ C1e−ζz + k2 ζ C2eζz +C 3   ,(A15) wherez∈[d/2,d/2 +d s]. Referring to Eq. (15), we are interested in integrals of the form IΩS(r,t) = Z ΩS d3r′ B(r′,t...