Enhanced third-harmonic generation and degenerate four-wave mixing in an all-dielectric metasurfaces via Brillouin zone folding-induced bound states in the continuum

Dandan Zhang; Feng Wu; Meibao Qin; Shuyuan Xiao; Tingting Liu

REVIEW 2 major objections 7 minor 59 references

Enhanced third-harmonic generation and degenerate four-wave mixing in an all-dielectric metasurfaces via Brillouin zone folding-induced bound states in the continuum

T0 review · 2 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that Brillouin-zone-folding bound states in a silicon metasurface, created by a periodic gap perturbation, boost third-harmonic generation to 10^-4 W and degenerate four-wave mixing to 10^-2 W at 1 MW/cm^2 input.

desk verdict The dFWM result is internally inconsistent—the reported peaks are pump third harmonics, not the 2ω1+ω2 output claimed—so the paper's headline claim is unsupported, though the THG part and linear analysis are still worth examining. read the letter →

arxiv 2411.12639 v1 pith:2Q4ENAAR submitted 2024-11-19 physics.optics

Meibao Qin , Feng Wu , Tingting Liu , Dandan Zhang , Shuyuan Xiao This is my paper · ORCID

classification physics.optics

keywords boundstatesinthecontinuumBrillouinzonefoldingthird-harmonicgenerationdegeneratefour-wavemixingall-dielectricmetasurfacesiliconphotonicsmultipoledecompositionQfactorengineering

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 sets out to show that bound states in the continuum created by Brillouin-zone folding (BZF-BICs) are a superior platform for third-order nonlinear frequency conversion in an all-dielectric metasurface. By adding a periodic gap perturbation to a silicon membrane with air holes, the authors fold guided modes into the light cone and obtain three types of resonances: ordinary guided resonances, a Γ-point BIC, and a BZF-BIC. They then simulate third-harmonic generation (THG) and degenerate four-wave mixing (dFWM) under x- and y-polarized illumination, reporting that the BZF-BIC produces the largest output powers: $10^{-4}$ W for THG and $10^{-2}$ W for dFWM at an input power density of 1 MW/$cm^{2}$. The significance, if correct, is a simple, fabrication-friendly route to chip-scale nonlinear devices without requiring phase matching.

What carries the argument

The central object is the Brillouin-zone-folding-induced bound state in the continuum (BZF-BIC), a guided mode that becomes radiative when a periodic perturbation doubles the unit cell and folds the band into the light cone. It is realized by changing the gap between air holes in a silicon membrane from L to L−ΔL, with asymmetry parameter α = ΔL/L. Unlike ordinary quasi-BICs whose Q factor drops sharply away from the Γ point, the BZF-BIC keeps a high Q factor over a broad range of in-plane wavevectors and is robust to disorder. The mechanism that carries the argument is resonant local-field enhancement: the BZF-BIC concentrates the pump field (enhancement ~350), and in dFWM the overlap of pump and idler resonances multiplies their Q factors, raising the nonlinear output.

What would settle it

Run the same numerical model with the pump at 1272.08 nm and idler at 1499.24 nm and scan the output spectrum: the claimed dFWM process predicts a peak near 446 nm (from $2\omega_1+\omega_2$), whereas a peak at 424 nm would show the enhancement is simply third-harmonic generation of the pump. Similarly, pump 1648.614 nm and idler 1733.49 nm should give a peak near 559 nm, not 549.5 nm.

Watch

Extended reading notes

Core claim

The paper's central discovery is that a Brillouin-zone-folding-induced bound state in the continuum, realized by doubling the period of a silicon air-hole metasurface through a gap perturbation, delivers stronger third-order nonlinear responses than the other resonances in the same structure. Under an obliquely incident x-polarized pump at 1648.614 nm with asymmetry parameter α = 0.025 and angle 5°, the BZF-BIC gives a local electric-field enhancement of about 350, leading to THG output power above $10^{-4}$ W. When the BZF-BIC is used as the pump and a guided resonance as the idler, the simulated dFWM output near 549.5 nm exceeds $10^{-2}$ W, two orders of magnitude above the BZF-BIC's own THG. The authors attribute this to the overlap of the two modes, giving an effective Q factor proportional to the product of the constituent Q factors.

Load-bearing premise

The reported output powers assume the simulated nonlinear response can be converted into watts without specifying the device area or collection aperture, and the dFWM peaks at 424 nm and 549.5 nm obey the stated relation $\omega_3 = 2\omega_1 + \omega_2$ rather than being the pump's own third harmonics.

Editorial extensions

If this is right

BZF-BICs, being robust in k-space and to disorder, could make nonlinear metasurfaces that work over a range of incidence angles rather than at a single angle.
The same silicon membrane supports THG and dFWM under both x- and y-polarizations, so polarization can be used as a control knob without changing the structure.
Simulated output powers of 10^-4 W (THG) and 10^-2 W (dFWM) at 1 MW/cm^2 suggest moderate pump intensities are enough for useful conversion in an all-dielectric, lossless platform.
The mode-overlap argument implies that pairing high-Q resonances (pump and idler) can multiply the Q factor and further boost nonlinear conversion.

Reading between the lines

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

Editorial: The reported dFWM wavelengths (424 nm and 549.5 nm) are exactly the third harmonics of the pumps at 1272.08 nm and 1648.614 nm. Under the stated relation ω3 = 2ω1 + ω2, the expected dFWM peaks would be near 446 nm and 559 nm, so the simulated 'dFWM' signal may be dominated by the pump's own third-harmonic generation.
Editorial: The conversion of simulated powers to watts is under-specified: without the illuminated area or integration aperture, the values 10^-4 W and 10^-2 W cannot be compared with experimental measurements or other designs.
Editorial: A natural testable extension is to fabricate the gap-perturbed silicon membrane and measure the THG and dFWM spectra at the predicted pump and idler wavelengths, also checking the angle robustness of BZF-BICs by tilting the sample.
Editorial: The mode-overlap idea suggests searching for 'super-BIC' designs where both pump and idler are BICs, potentially pushing the product-Q enhancement beyond what this paper shows.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague,

The short version: the paper's headline dFWM result does not hold up. The reported peaks at 424 nm and 549.5 nm are exactly the third harmonics of the pump beams (1272.08/3 ≈ 424.0; 1648.614/3 ≈ 549.5), not the wavelengths expected from ω3 = 2ω1 + ω2. For the stated pump and idler pairs, Eq. (2) would put the new frequency at ~446.6 nm and ~558.7 nm. The manuscript doesn't show a separate dFWM line and doesn't explain the discrepancy. That is an internal contradiction, not an external interpretation. The dFWM enhancement claim in the abstract and conclusions is unsupported as written.

What's genuinely new: applying BZF-BICs to third-order nonlinear conversion under dual polarization is a legitimate extension of the BIC-metasurface toolkit. The eigenmode analysis, Q-factor trends, and multipole decompositions in Sec. II are careful and internally consistent, and the THG result (BZF-BIC giving 10^-4 W) is plausible even though it depends on the same unit ambiguity.

The soft spots, in order of severity: 1. The dFWM misidentification. If the code actually solved Eq. (2), the output would appear at the sum-frequency wavelengths, not at the pump THG. The authors need to re-examine their simulation setup, show a spectrum with the idler beam present and absent, and report the actual generated wavelength. 2. No defined device area or integration aperture. Output powers in watts mean nothing without knowing the area over which they are computed. This is fixable with a sentence, but it's needed before any comparison to experiment. 3. No convergence or mesh-refinement study. This is routine and should be added.

None of these attack the linear analysis, which is the strongest part of the paper.

The citation pattern is fine; the authors draw on the standard BIC and nonlinear metasurface literature, and self-citation isn't inflated.

Who's this for? Specialists in dielectric metasurface nonlinear optics. They'll find the design route useful if the dFWM issue is resolved, and the THG part is likely salvageable. But as it stands, the paper cannot be trusted on its central quantitative claim.

For peer review: send it out. The flaw is serious but addressable, and the underlying idea is worth testing. A good referee will catch this immediately; the authors should be given the chance to fix the dFWM simulation or remove the claim. If they can't, the paper should be reframed around THG only.

Referee Report

2 major / 7 minor

Summary. The paper studies third-harmonic generation (THG) and degenerate four-wave mixing (dFWM) in a suspended silicon membrane with a rectangular lattice of air holes. By introducing a gap perturbation that doubles the unit cell, the authors fold guided modes into the light cone and identify guided resonances (GRs), a Gamma-BIC, and a Brillouin-zone-folding-induced BIC (BZF-BIC) via eigenfrequency analysis, Q-factor calculations, and Cartesian multipole decomposition. They then use COMSOL frequency-domain simulations with a literature value of silicon chi^(3) to report THG output powers near 10^-4 W and dFWM output powers near 10^-2 W at 1 MW/cm^2, attributing the largest enhancement to the BZF-BIC.

Significance. If the quantitative claims were supported, the demonstration of a BZF-BIC with a more robust Q factor than ordinary BICs would be a useful addition to nonlinear dielectric metasurface design. The linear mode analysis is carried out with standard eigenmode solvers and multipole decomposition, and the nonlinear polarization sources follow Boyd's textbook convention with literature silicon parameters, so the basic machinery is appropriate. However, the absolute output powers are not grounded in a defined device area, and the reported dFWM wavelengths are inconsistent with the stated frequency relation; as a result, the headline enhancement claims are not currently established.

major comments (2)

[Section IV, Eq. (2), Fig. 6] The reported dFWM peaks do not satisfy the stated frequency relation. For the Gamma-BIC pump at lambda_1 = 1272.08 nm and the GR-2 idler at lambda_2 = 1499.24 nm, Eq. (2) with omega_3 = 2 omega_1 + omega_2 gives lambda_3 approximately 446.6 nm, whereas the text reports output near 424 nm, which is exactly lambda_1/3. Similarly, for the BZF-BIC pump at 1648.614 nm and the GR-1 idler at 1733.49 nm, the predicted output is lambda_3 approximately 558.7 nm, while the reported 549.5 nm equals 1648.614/3. The manuscript contains no spectrum showing a distinct line at the predicted dFWM wavelength and no explanation for the coincidence with pump third-harmonic generation. Since the abstract and conclusions headline the 10^-2 W dFWM value, the central dFWM claim is unsupported as written.
[Sections III and IV] The quoted output powers in watts are not tied to a defined device area or integration aperture. The input is specified only as an intensity (1 MW/cm^2), so the absolute output power depends on the lateral extent of the simulated domain and on the surface used to integrate the Poynting flux. Taking the perturbed unit cell used in the simulations (920 nm x 400 nm), the incident power at 1 MW/cm^2 is about 3.7 x 10^-3 W; an output of 10^-2 W would therefore exceed the input power (even if the idler is also incident at the same intensity), which is unphysical for the frequency conversion processes described. If the quoted powers instead refer to a macroscopic illuminated area, that area and the integration aperture must be stated. The absence of a mesh-convergence or refinement study further weakens confidence in these quantitative values.

minor comments (7)

[Title] The title uses "metasurfaces" where "metasurface" is correct.
[Eq. (1)] Equation (1) is written without a degeneracy factor; the authors should state the convention for chi^(3) used in the THG simulation.
[Section IV] The manuscript does not specify whether the idler beam is also incident at 1 MW/cm^2 and how the idler field is included in the COMSOL simulation; this should be stated explicitly.
[Fig. 6 and Section IV] There are typographical errors: "dWFM" should be "dFWM" in Fig. 6 and in the surrounding text, and "outpower" in Section V should be "output power".
[Section II] The phrase "the magnetic vector of Ey-polarized and Ex-polarized plane wave" should be reworded to "the magnetic field vector of an Ey-polarized or Ex-polarized plane wave" for clarity.
[Section IV] The statement that the dFWM enhancement comes from a Q factor proportional to the product of the two mode Q factors is not derived and is physically unclear; the conversion efficiency depends on the local field intensities at the pump and idler frequencies, and this explanation should be revised or removed.
[References] Reference [56] has a malformed author list and should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the simulated THG and dFWM outputs are derived from standard nonlinear polarization equations with literature material parameters and are not fitted to reproduce the claimed output powers.

full rationale

The paper's derivation chain is self-contained rather than circular. The linear eigenmode and multipole analysis is used only to identify excitation conditions for the GRs, Γ-BIC, and BZF-BIC modes; the nonlinear results are then obtained by solving the nonlinear wave equation with the standard THG and dFWM polarization terms in Eqs. (1) and (2), using literature values for silicon's refractive index and χ(3) = 2.45e-19 m2/V2, and a fixed input intensity of 1 MW/cm2. No parameter is fitted to the reported output powers of 10^-4 W (THG) or 10^-2 W (dFWM); the geometry and incident angles are selected from the preceding mode-analysis, not from optimizing the nonlinear outputs. The cited nonlinear polarization formulas appear in Boyd's standard textbook and in prior works including [44] and [46], so the presence of self-citations is not load-bearing: the formulas are independently established. The reported dFWM peak wavelengths in Fig. 6 appear internally inconsistent with Eq. (2) — 424 nm equals 1272.08 nm/3 and 549.5 nm equals 1648.614 nm/3, rather than the stated 2ω1+ω2 combination — but that is a correctness or simulation-validity concern, not a circular reduction of the prediction to an input parameter. Therefore no circularity is identified.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No target quantity is fitted to measured data. The geometry and excitation parameters are chosen by hand, and the material constants are taken from prior literature. The central numerical output is a direct finite-element simulation result, so the free-parameter count is modest. The main unseen support is the validity of the material model and the two-step nonlinear solver approximation.

free parameters (3)

asymmetry parameter alpha = 0.025
Chosen by hand to open radiation channels and tune Q factors; not optimized against measured data and not fitted to the output powers.
oblique incidence angles = 1 degree and 5 degrees
Selected to excite the Gamma-BIC/GR-2 pair and the BZF-BIC/GR-1 pair, respectively; no selection rule or optimization procedure is given.
structural dimensions = h=330 nm, r=130 nm, a1=460 nm (920 nm perturbed), a2=400 nm, L=200 nm
Design parameters chosen by hand; the claimed enhancement depends on these choices but they are conventional rather than fitted.

assumptions (4)

standard math Maxwell's equations solved with the finite element method give the true linear and nonlinear response of the periodic metasurface.
The entire simulation methodology rests on solving Maxwell's equations; this is standard physics, not specific to the paper.
domain assumption Silicon's third-order nonlinearity is described by the scalar susceptibility chi(3)=2.45e-19 m2/V2 with the standard polarization formulas in Eq. (1) and Eq. (2).
The paper imports this material model from the literature and does not test its validity at the high local field enhancements it reports.
domain assumption Symmetry matching between the eigenmode field profiles and the incident plane wave determines which modes can be excited.
The paper uses C2v symmetry and C2/ sigma_y arguments, citing refs. [52,56], to assign excitation polarizations to the Gamma-BIC and BZF-BIC.
domain assumption The nonlinear process can be treated in two steps: linear field computation at the fundamental frequency, then the nonlinear polarization as a source at the harmonic frequency, with no pump depletion or cascaded effects.
This undepleted-pump approximation is standard in the cited BIC nonlinear metasurface papers, but it is not stated explicitly as a limitation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Enhanced third-harmonic generation and degenerate four-wave mixing in an all-dielectric metasurfaces via Brillouin zone folding-induced bound states in the continuum." pith.science (2026). https://pith.science/paper/2Q4ENAAR

@misc{pith2026241112639,
  author       = {Pith},
  title        = {Pith review of: Enhanced third-harmonic generation and degenerate four-wave mixing in an all-dielectric metasurfaces via Brillouin zone folding-induced bound states in the continuum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2Q4ENAAR}},
  note         = {Machine review of arXiv:2411.12639}
}

abstract

Bound states in the continuum (BICs) exhibit significant electric field confinement capabilities and have recently been employed to enhance nonlinear optics response at the nanoscale. In this study, we achieve substantial enhancement of third-harmonic generation (THG) and degenerate four-wave mixing (dFWM) by implementing Brillouin zone folding-induced BICs (BZF-BICs) in an air-hole type nonlinear metasurface. By introducing gap perturbations within the metasurface, guided modes below the light line can be folded into the light cone, resulting in three resonant modes: guided resonances (GRs), $\Gamma$-BICs, and BZF-BICs. Through the eigenvalue analysis and multipole decompositions, we establish their excitation conditions. With their resonantly enhanced local field, we successfully boost both THG and dFWM under $x$- and $y$- polarizations within the same metasurfaces. The simulated results indicate that the BZF-BICs provide the most significant enhancement of third-order nonlinear optical responses, with the output power of THG to 10$^{-4}$ W and dFWM output power of 10$^{-2}$ W under a moderate input power density of 1 MW/cm$^{2}$. These findings demonstrate that the BZF-BICs can offer an effective pathway for chip-scale nonlinear optical applications.

Figures

Figures reproduced from arXiv: 2411.12639 by the authors.

**Figure 2.** FIG. 2. (a) Band diagrams of the transverse electric (TE) pla [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. (a)-(d) Simulated [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Simulated the optical response of the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. Simulated the optical response of the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Enhanced third-order optical nonlinearities throu [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 58 canonical work pages

[1]

0003, 0.003, 0.015, 0.025, and 0.075). (e)-(g) Normalized m ultipole decomposition results of the eigenvalue for the GRs and BICs, with the components |Pα |, |Mα |, and |Tα | representing electric dipole (ED), magnetic dipole (MD), and toroidal di pole (TD), respectively. (h) Normalized radiation powers of the eigenvalue for the GRs and BICs. To more intu...

work page
[2]

R.W.Boyd, Nonlinear Optics (Academic Press, 2020)

work page 2020
[3]

Sheik-Bahae and M

M. Sheik-Bahae and M. P. Hasselbeck, Third-order optica l nonlinearities, Handbook of Optics 4, 16 (2000)

work page 2000
[4]

M. Dinu, F. Quochi, and H. Garcia, Third-order nonlinear ities in silicon at telecom wave- lengths, Applied Physics Letters 82, 2954 (2003)

work page 2003
[5]

G. D. Hutchinson and G. J. Milburn, Nonlinear quantum opt ical computing via measurement, Journal of Modern Optics 51, 1211 (2004)

work page 2004
[6]

L. Xu, K. Zangeneh Kamali, L. Huang, M. Rahmani, A. Smirno v, R. Camacho-Morales, Y. Ma, G. Zhang, M. Woolley, D. Neshev, et al. , Dynamic nonlinear image tuning through magnetic dipole quasi-bic ultrathin resonators, Advanced science 6, 1802119 (2019). 12

work page 2019
[7]

Y. Gao, Y. Fan, Y. Wang, W. Yang, Q. Song, and S. Xiao, Nonli near holographic all-dielectric metasurfaces, Nano letters 18, 8054 (2018)

work page 2018
[8]

P. D. Maker, R. W. Terhune, M. Nisenoﬀ, and C. M. Savage, Eﬀec ts of dispersion and focusing on the production of optical harmonics, Physical Review Let ters 8, 21 (1962)

work page 1962

Show all 59 references

[9]

Lifshitz, A

R. Lifshitz, A. Arie, and A. Bahabad, Photonic quasicrys tals for nonlinear optical frequency conversion, Physical Review Letters 95, 133901 (2005)

2005
[10]

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, Quasi-phase-matched generation of coherent extreme-ultraviolet light, Nature 421, 51 (2003)

2003
[11]

Krasnok, M

A. Krasnok, M. Tymchenko, and A. Al` u, Nonlinear metasu rfaces: a paradigm shift in nonlinear optics, Materials Today 21, 8 (2018)

2018
[12]

Huang, X

T. Huang, X. Zhao, S. Zeng, A. Crunteanu, P. P. Shum, and N . Yu, Planar nonlinear meta- surface optics and their applications, Reports on Progress in Physics 83, 126101 (2020)

2020
[13]

Kivshar, All-dielectric meta-optics and non-linea r nanophotonics, National Science Review 5, 144 (2018)

Y. Kivshar, All-dielectric meta-optics and non-linea r nanophotonics, National Science Review 5, 144 (2018)

2018
[14]

T. Liu, S. Xiao, B. Li, M. Gu, H. Luan, and X. Fang, Third-a nd second-harmonic generation in all-dielectric nanostructures: a mini review, Frontier s in Nanotechnology 4, 891892 (2022)

2022
[15]

Vabishchevich and Y

P. Vabishchevich and Y. Kivshar, Nonlinear photonics w ith metasurfaces, Photonics Research 11, B50 (2023)

2023
[16]

H. Xu, R. Ma, Y. Zhu, A. Dong, H. Jiang, W. Gao, M. Qin, J. Li u, F. Wu, and H. Li, Chirality- controlled second-order nonlinear frequency conversion i n lithium niobate ﬁlm metasurfaces, Optics Letters 49, 6405 (2024)

2024
[17]

J. T. Wang, P. Tonkaev, K. Koshelev, F. Lai, S. Kruk, Q. So ng, Y. Kivshar, and N. C. Panoiu, Resonantly enhanced second-and third-harmonic ge neration in dielectric nonlinear metasurfaces, Opto-Electronic Advances 7, 230186 (2024)

2024
[18]

N. C. Panoiu, W. E. I. Sha, D. Y. Lei, and G.-C. Li, Nonline ar optics in plasmonic nanostruc- tures, Journal of Optics 20, 083001 (2018)

2018
[19]

M. S. Bin-Alam, O. Reshef, Y. Mamchur, M. Z. Alam, G. Carl ow, J. Upham, B. T. Sullivan, J.-M. M´ enard, M. J. Huttunen, R. W. Boyd, and K. Dolgaleva, U ltra-high-q resonances in plasmonic metasurfaces, Nature Communications 12, 974 (2021). 13

2021
[20]

D. Liu, Y. Ren, Y. Huo, Y. Cai, and T. Ning, Second harmoni c generation in plas- monic metasurfaces enhanced by symmetry-protected dual bo und states in the continuum, Optics Express 31, 23127 (2023)

2023
[21]

Y. Yang, W. Wang, A. Boulesbaa, I. I. Kravchenko, D. P. Br iggs, A. Puret- zky, D. Geohegan, and J. Valentine, Nonlinear fano-resonan t dielectric metasurfaces, Nano Letters 15, 7388 (2015)

2015
[22]

Y. Hu, X. Wang, X. Luo, X. Ou, L. Li, Y. Chen, P. Yang, S. Wan g, and H. Duan, All- dielectric metasurfaces for polarization manipulation: p rinciples and emerging applications, Nanophotonics 9, 3755 (2020)

2020
[23]

M. Z. Alam, S. A. Schulz, J. Upham, I. De Leon, and R. W. Boy d, Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material, Nature Photonics 12, 79 (2018)

2018
[24]

E. V. Melik-Gaykazyan, K. L. Koshelev, J.-H. Choi, S. Kr uk, H.-G. Park, A. A. Fedyanin, and Y. S. Kivshar, Enhanced second-harmonic generation wit h structured light in algaas nanoparticles governed by magnetic response, JETP Letters 109, 131 (2019)

2019
[25]

Koshelev, Y

K. Koshelev, Y. Tang, K. Li, D.-Y. Choi, G. Li, and Y. Kivs har, Nonlinear metasurfaces governed by bound states in the continuum, ACS Photonics 6, 1639 (2019)

2019
[26]

X. Qi, J. Wu, F. Wu, M. Ren, Q. Wei, Y. Wang, H. Jiang, Y. Li, Z. Guo, Y. Yang, W. Zheng, Y. Sun, and H. Chen, Steerable merging bound states in the con tinuum on a quasi-ﬂatband of photonic crystal slabs without breaking symmetry, Photo nics Research 11, 1262 (2023)

2023
[27]

G. Sun, Y. Wang, R. Xie, and X. Zhao, Polarization-insen sitive terahertz third-harmonic generation from degenerate pairs of mirror -coupled super-bics, Applied Physics Letters 125, 081702 (2024)

2024
[28]

X. Tu, S. Feng, J. Li, Y. Xing, F. Wu, T. Liu, and S. Xiao, En hanced second- harmonic generation in high-q all-dielectric metasurface s with backward frequency conversion, Physical Review A 109, 063522 (2024)

2024
[29]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Bound states in the continuum, Nature Reviews Materials 1, 16048 (2016)

2016
[30]

Friedrich and D

H. Friedrich and D. Wintgen, Interfering resonances an d bound states in the continuum, Physical Review A 32, 3231 (1985)

1985
[31]

Koshelev, G

K. Koshelev, G. Favraud, A. Bogdanov, Y. Kivshar, and A. Fratalocchi, Nonradiating pho- tonics with resonant dielectric nanostructures, Nanophot onics 8, 725 (2019). 14

2019
[32]

X. Wang, J. Duan, W. Chen, C. Zhou, T. Liu, and S. Xiao, Con trolling light absorption of graphene at critical coupling through magnetic dipole quas i-bound states in the continuum resonance, Physical Review B 102, 155432 (2020)

2020
[33]

F. Wu, M. Luo, J. Wu, C. Fan, X. Qi, Y. Jian, D. Liu, S. Xiao, G. Chen, H. Jiang, Y. Sun, and H. Chen, Dual quasibound states in the continuum i n compound grating waveg- uide structures for large positive and negative goos-h¨ anc hen shifts with perfect reﬂection, Physical Re...

2021
[34]

M. Qin, S. Xiao, W. Liu, M. Ouyang, T. Yu, T. Wang, and Q. Li ao, Strong coupling be- tween excitons and magnetic dipole quasi-bound states in th e continuum in ws2-tio2 hybrid metasurfaces, Optics Express 29, 18026 (2021)

2021
[35]

P. Xie, Z. Liang, T. Jia, D. Li, Y. Chen, P. Chang, H. Zhang , and W. Wang, Strong coupling between excitons in a two-dimensional atom ic crystal and quasibound states in the continuum in a two-dimensional all-dielectri c asymmetric metasurface, Physical Review B 104, 125446 (2021)

2021
[36]

Zeng, G.-D

T.-Y. Zeng, G.-D. Liu, L.-L. Wang, and Q. Lin, Light-mat ter interactions en- hanced by quasi-bound states in the continuum in a graphene- dielectric metasurface, Optics Express 29, 40177 (2021)

2021
[37]

L. Qu, L. Bai, C. Jin, Q. Liu, W. Wu, B. Gao, J. Li, W. Cai, M. Ren, and J. Xu, Giant second harmonic generation from membrane metasurfaces, Nano Lett ers 22, 9652 (2022)

2022
[38]

Q. Liu, L. Qu, Z. Gu, D. Zhang, W. Wu, W. Cai, M. Ren, and J. X u, Boosting second harmonic generation by merging bound states in the continuum, Physic al Review B 107, 245408 (2023)

2023
[39]

Huang, L

L. Huang, L. Xu, D. A. Powell, W. J. Padilla, and A. E. Miro shnichenko, Resonant leaky modes in all-dielectric metasystems: Fundamentals and app lications, Physics Reports 1008, 1 (2023)

2023
[40]

T. Liu, J. Qiu, L. Xu, M. Qin, L. Wan, T. Yu, Q. Liu, L. Huang , and S. Xiao, Edge detection imaging by quasi-bound states in the continuum, Nano Letter s 24, 14466 (2024)

2024
[41]

Zhang, T

D. Zhang, T. Liu, L. Lei, W. Deng, T. Wang, Q. Liao, W. Liu, S. Xiao, and T. Yu, Tailoring intrinsic chirality in a two-dimensional planar waveguide grating via quasibound states in the continuum, Physical Review B 109, 205403 (2024)

2024
[42]

T. Liu, M. Qin, S. Feng, X. Tu, T. Guo, F. Wu, and S. Xiao, Eﬃ cient photon-pair generation empowered by dual quasibound state s in the continuum, 15 Physical Review B 109, 155424 (2024)

2024
[43]

Hajian, X

H. Hajian, X. Zhang, O. McCormack, Y. Zhang, J. Dobie, I. D. Rukhlenko, E. Ozbay, and A. Louise Bradley, Quasi-bound states in the continuum for e lectro magnetic induced trans- parency and strong excitonic coupling, Optics Express 32, 19163 (2024)

2024
[44]

Jiang, K

H. Jiang, K. Sun, Y. Jia, Y. Cai, U. Levy, and Z. Han, Tunab le second harmonic generation with large enhancement in a nonlocal all-dielectric metasu rface over a broad spectral range, Advanced Optical Materials , 2303229 (2024)

2024
[45]

Carletti, S

L. Carletti, S. S. Kruk, A. A. Bogdanov, C. De Angelis, an d Y. Kivshar, High- harmonic generation at the nanoscale boosted by bound state s in the continuum, Physical Review Research 1, 023016 (2019)

2019
[46]

T. Ning, X. Li, Y. Zhao, L. Yin, Y. Huo, L. Zhao, and Q. Yue, Giant enhancement of harmonic generation in all-dielectric resonant waveguide gratings of quasi-bound states in the continuum, Optics Express 28, 34024 (2020)

2020
[47]

S. Xiao, M. Qin, J. Duan, F. Wu, and T. Liu, Polarization- controlled dynamically switchable high-harmonic generation from all-dielectric metasurfac es governed by dual bound states in the continuum, Physical Review B 105, 195440 (2022)

2022
[48]

T. Liu, M. Qin, F. Wu, and S. Xiao, High-eﬃciency optical frequency mixing in an all-dielectric metasurface enabled by multiple bound st ates in the continuum, Physical Review B 107, 075441 (2023)

2023
[49]

W. Tang, Q. Zhao, Z. Wang, Y. Gao, J. He, Y. Zhu, S. Wang, H. Yu, R. Peng, and M. Wang, Realizing high-eﬃciency third harmonic generation via acc idental bound states in the contin- uum, Optics Letters 49, 1169 (2024)

2024
[50]

M. Qin, G. Wei, H. Xu, R. Ma, H. Li, W. Gao, J. Liu, and F. Wu, Polarization-insensitive and polarization-controlled dual-band third-harmonic gener ation in silicon metasurfaces driven by quasi-bound states in the continuum, Applied Physics Lette rs 124, 051703 (2024)

2024
[51]

W. Shi, J. Gu, X. Zhang, Q. Xu, J. Han, Q. Yang, L. Cong, and W. Zhang, Terahertz bound states in the continuum with incident angle robustness indu ced by a dual period metagrating, Photonics research 10, 810 (2022)

2022
[52]

Yan, Y.-J

X.-F. Yan, Y.-J. Xie, S. Chen, and P.-Y. Wu, Brillouin zo ne folding metasurface for near perfect circular dichroism, Advanced Optical Materials 12, 2401027 (2024). 16

2024
[53]

W. Wang, Y. K. Srivastava, T. C. Tan, Z. Wang, and R. Singh , Brillouin zone folding driven bound states in the continuum, Nature Communications 14, 2811 (2023)

2023
[54]

K. Sun, U. Levy, and Z. Han, Exploiting zone-folding ind uced quasi-bound modes to achieve highly coherent thermal emissions, Nano Letters 24, 764 (2024)

2024
[55]

Y. He, G. Guo, T. Feng, Y. Xu, and A. E. Miroshnichenko, To roidal dipole bound states in the continuum, Physical Review B 98, 161112 (2018)

2018
[56]

A. C. Overvig, S. C. Malek, M. J. Carter, S. Shrestha, and N. Yu, Selection rules for quasi- bound states in the continuum, Physical Review B 102, 035434 (2020)

2020
[57]

Lee., S.-L

J. Lee., S.-L. C. Bo Zhen, M. S. Wenjun Qiu, John D. Joanno poulos, and O. Shapira., Ob- servation and diﬀerentiation of unique high-q optical reson ances near zero wave vector in macroscopic photonic crystal slabs, Physical Review Lette rs 109, 067401 (2012)

2012
[58]

Palik, Handbook of Optical Constants of Solids (Academic Press, 2012)

E. Palik, Handbook of Optical Constants of Solids (Academic Press, 2012)

2012
[59]

A. F. Sadreev, Interference traps waves in an open syste m: bound states in the continuum, Reports on Progress in Physics 84, 055901 (2021). 17 Clip A Frame Clip B Frame

2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

0003, 0.003, 0.015, 0.025, and 0.075). (e)-(g) Normalized m ultipole decomposition results of the eigenvalue for the GRs and BICs, with the components |Pα |, |Mα |, and |Tα | representing electric dipole (ED), magnetic dipole (MD), and toroidal di pole (TD), respectively. (h) Normalized radiation powers of the eigenvalue for the GRs and BICs. To more intu...

work page

[2] [2]

R.W.Boyd, Nonlinear Optics (Academic Press, 2020)

work page 2020

[3] [3]

Sheik-Bahae and M

M. Sheik-Bahae and M. P. Hasselbeck, Third-order optica l nonlinearities, Handbook of Optics 4, 16 (2000)

work page 2000

[4] [4]

M. Dinu, F. Quochi, and H. Garcia, Third-order nonlinear ities in silicon at telecom wave- lengths, Applied Physics Letters 82, 2954 (2003)

work page 2003

[5] [5]

G. D. Hutchinson and G. J. Milburn, Nonlinear quantum opt ical computing via measurement, Journal of Modern Optics 51, 1211 (2004)

work page 2004

[6] [6]

L. Xu, K. Zangeneh Kamali, L. Huang, M. Rahmani, A. Smirno v, R. Camacho-Morales, Y. Ma, G. Zhang, M. Woolley, D. Neshev, et al. , Dynamic nonlinear image tuning through magnetic dipole quasi-bic ultrathin resonators, Advanced science 6, 1802119 (2019). 12

work page 2019

[7] [7]

Y. Gao, Y. Fan, Y. Wang, W. Yang, Q. Song, and S. Xiao, Nonli near holographic all-dielectric metasurfaces, Nano letters 18, 8054 (2018)

work page 2018

[8] [8]

P. D. Maker, R. W. Terhune, M. Nisenoﬀ, and C. M. Savage, Eﬀec ts of dispersion and focusing on the production of optical harmonics, Physical Review Let ters 8, 21 (1962)

work page 1962