Twisted bilayers of α-MoO3 support phonon polaritons whose dispersion undergoes a twist-controlled topological transition, and at the transition (photonic magic) angle the flattened dispersion yields low-loss, diffractionless canalization of nanolight. The 2020 discovery paper established that only the integer counting the anti-crossing points of the two layers' dispersions, NACPN_{\text{ACP}}, is topologically robust, while the quality of canalization itself — propagation length, inverse damping ratio, angular collimation of the group velocity, and operating bandwidth — has no protection. The quantitative limit of magic-angle canalization remains open: how realistic disorder (twist-angle inhomogeneity, interfacial roughness and contamination, thickness variation, impurities) and intrinsic anharmonic phonon damping degrade canalization, and which confinement/propagation-length/bandwidth/tunability tradeoffs are simultaneously achievable in bilayers and in the stack designs later proposed to overcome them.

Background

Phonon polaritons (PhPs) are hybrid excitations of infrared light and lattice vibrations in polar crystals; in slabs of the van der Waals (vdW) semiconductor α-MoO3 (α-phase molybdenum trioxide) they are strongly subwavelength and, within the mid-infrared Reststrahlen band (the frequency interval between a polar crystal's transverse and longitudinal optical-phonon resonances, in which the permittivity is negative along the corresponding axis) from 818 to 974 cm^-1, in-plane hyperbolic: the permittivity components along the [100] and [001] crystal axes have opposite signs, so the isofrequency contour (IFC) — the set of in-plane wavevectors k⃗\vec{k} at fixed frequency — is an open hyperbola rather than a closed ellipse, and energy preferentially flows along the asymptote directions. Hu, Ou, Si, Wu, Wu, Dai, Krasnok, Mazor, Zhang, Bao, Qiu and Alù (Nature 582, 209–213, 2020) stacked two such slabs with a relative twist angle Δθ\Delta\theta and showed by scattering-type scanning near-field optical microscopy (s-SNOM, a technique in which a metallized atomic-force-microscope tip both launches polaritons and scatters their evanescent fields to the far field for detection) that the two layers' hyperbolic branches hybridize and anti-cross, and that the number of anti-crossing points, NACPN_{\text{ACP}}, fixes whether the bilayer IFC is open (hyperbolic, NACP=2N_{\text{ACP}}=2) or closed (elliptical, NACP=4N_{\text{ACP}}=4). Changing Δθ\Delta\theta therefore drives a topological transition of the IFC, the photonic analogue of the Lifshitz transition of a Fermi surface in metals (a change of the Fermi surface's topology); the precursor concept of moiré-coupled hyperbolic metasurfaces and the NACPN_{\text{ACP}} rule was given by Hu, Krasnok, Mazor, Qiu and Alù (Nano Lett. 20, 3217–3224, 2020), and two companion experiments (Chen et al., Nat. Mater. 19, 1307–1311, 2020; Duan et al., Nano Lett. 20, 5323–5329, 2020) reported the same transition independently. At the transition angle — termed the photonic magic angle, in analogy to magic-angle flat bands in twisted bilayer graphene — the IFC flattens, all wavevectors share one group-velocity direction, and the paper demonstrated canalization: diffractionless, unidirectional propagation with sub-diffractional resolution better than λ0/40\lambda_0/40 (λ0\lambda_0 the free-space wavelength). The canalization quality is conventionally quantified by the propagation length versus the polariton wavelength (their ratio is the inverse damping ratio) and by the transverse full width at half maximum of the launched beam as a function of propagation distance. The authors also stated a robustness claim that is central to the present problem: the open/closed nature of the dispersion is strongly robust to disorder as long as the integer NACPN_{\text{ACP}} does not change. This is a statement about a topological invariant of the clean dispersion; it does not protect the canalization regime itself, which depends on continuously variable quantities — loss, group-velocity collimation, and the frequency window over which the band stays flat — all of which were measured only in clean, uniformly twisted samples. Reviews of vdW polaritonics (Zhang et al., Nature 597, 187–195, 2021) present magic-angle canalization as an established capability, while the loss and bandwidth sides of the problem have since been sharpened: the microscopic damping of α-MoO3 phonon polaritons was traced to specific anharmonic phonon channels by Taboada-Gutiérrez et al. (ACS Photonics 11, 3570–3577, 2024), and Voronin et al. (arXiv:2508.15993, 2025) showed that in lossy anisotropic slabs the directions of propagation and of decay are strongly misaligned, complicating the extraction of polariton parameters from near-field images. No published source provides the quantitative link between realistic imperfections, intrinsic loss, finite bandwidth, and the achievable canalization performance.

Problem Statement

Establish, quantitatively and with verification against existing experiments, the robustness and performance limits of low-loss diffractionless polariton canalization at photonic magic angles in twisted α-MoO3 bilayers — recognizing that the demonstrated topological protection covers only the integer NACPN_{\text{ACP}} (the open/closed nature of the dispersion), not the canalization regime itself. Concretely: (1) determine how realistic imperfections — twist-angle inhomogeneity across the stack, interfacial contamination and roughness between the transferred flakes, flake thickness variation, impurities and substrate disorder — degrade canalization, as measured by the polariton propagation length and inverse damping ratio (propagation length divided by polariton wavelength), by the residual angular spread of the group velocity about the canalization direction near the flattened band including the small-|k⃗\vec{k}| modes, and by mode mixing between the two hybridized dispersion branches; (2) determine the tradeoffs that intrinsic anharmonic phonon damping of α-MoO3 and the single-magic-angle-per-frequency condition of a bilayer impose on the simultaneously achievable field confinement, propagation length, operating bandwidth, and twist- or gate-based tunability, and establish how far these limits carry over to, or are lifted by, the alternative platforms now available (reconfigurable trilayers, asymmetric stacks with unidirectional ray polaritons, single slabs on negative-permittivity substrates, and other biaxial van der Waals crystals). Parts (1) and (2) are jointly posed and together describe one objective, assessed against existing published measurements rather than new experiments: a predictive account of what diffractionless transport in twisted polaritonic slabs can deliver — the achievable propagation length and bandwidth as functions of disorder level and material choice. The imperfection and platform lists above name the class of admissible cases rather than an exhaustive checklist.

References

  1. Guangwei Hu, Qingdong Ou, Guangyuan Si, Yingjie Wu, Jing Wu, Zhigao Dai, Alex Krasnok, Yarden Mazor, Qing Zhang, Qiaoliang Bao, Cheng-Wei Qiu, Andrea Alù. Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers. Nature 582, 209–213 (2020). DOI 10.1038/s41586-020-2359-9, https://doi.org/10.1038/s41586-020-2359-9
  2. Guangwei Hu, Alex Krasnok, Yarden Mazor, Cheng-Wei Qiu, Andrea Alù. Moiré Hyperbolic Metasurfaces. Nano Letters 20, 3217–3224 (2020). DOI 10.1021/acs.nanolett.9b05319, https://doi.org/10.1021/acs.nanolett.9b05319
  3. Mingyuan Chen, Xiao Lin, Thao H. Dinh, Zhiren Zheng, Jialiang Shen, Qiong Ma, Hongsheng Chen, Pablo Jarillo-Herrero, Siyuan Dai. Configurable phonon polaritons in twisted α-MoO3. Nature Materials 19, 1307–1311 (2020). DOI 10.1038/s41563-020-0732-6, https://doi.org/10.1038/s41563-020-0732-6
  4. Jiahua Duan, Nathaniel Capote-Robayna, Javier Taboada-Gutiérrez, Gonzalo Álvarez-Pérez, Iván Prieto, Javier Martín-Sánchez, Alexey Y. Nikitin, Pablo Alonso-González. Twisted Nano-Optics: Manipulating Light at the Nanoscale with Twisted Phonon Polaritonic Slabs. Nano Letters 20, 5323–5329 (2020). DOI 10.1021/acs.nanolett.0c01673, https://doi.org/10.1021/acs.nanolett.0c01673 (arXiv:2004.14599, https://arxiv.org/abs/2004.14599)
  5. Zebo Zheng, Fengsheng Sun, Wuchao Huang, Jingyao Jiang, Runze Zhan, Yanlin Ke, Huanjun Chen, Shaozhi Deng. Phonon Polaritons in Twisted Double-Layers of Hyperbolic van der Waals Crystals. Nano Letters 20, 5301–5308 (2020). DOI 10.1021/acs.nanolett.0c01627, https://doi.org/10.1021/acs.nanolett.0c01627
  6. Qing Zhang, Guangwei Hu, Weiliang Ma, Peining Li, Alex Krasnok, Rainer Hillenbrand, Andrea Alù, Cheng-Wei Qiu. Interface nano-optics with van der Waals polaritons. Nature 597, 187–195 (2021). DOI 10.1038/s41586-021-03581-5, https://doi.org/10.1038/s41586-021-03581-5
  7. Hai Hu, Na Chen, Hanchao Teng, Renwen Yu, Yunpeng Qu, Jianzhe Sun, Mengfei Xue, Debo Hu, Bin Wu, Chi Li, Jianing Chen, Mengkun Liu, Zhipei Sun, Yunqi Liu, Peining Li, Shanhui Fan, F. Javier García de Abajo, Qing Dai. Doping-driven topological polaritons in graphene/α-MoO3 heterostructures. Nature Nanotechnology 17, 940–946 (2022). DOI 10.1038/s41565-022-01185-2, https://doi.org/10.1038/s41565-022-01185-2 (arXiv:2201.00930, https://arxiv.org/abs/2201.00930)
  8. J. Duan, G. Álvarez-Pérez, C. Lanza, K. Voronin, A. I. F. Tresguerres-Mata, N. Capote-Robayna, J. Álvarez-Cuervo, A. Tarazaga Martín-Luengo, J. Martín-Sánchez, V. S. Volkov, A. Y. Nikitin, P. Alonso-González. Multiple and spectrally robust photonic magic angles in reconfigurable α-MoO3 trilayers. Nature Materials 22, 867–872 (2023). DOI 10.1038/s41563-023-01582-5, https://doi.org/10.1038/s41563-023-01582-5 (arXiv:2311.04173, https://arxiv.org/abs/2311.04173)
  9. Maximilian Obst, Tobias Nörenberg, Gonzalo Álvarez-Pérez, Thales V. A. G. de Oliveira, Javier Taboada-Gutiérrez, Flávio H. Feres, Felix G. Kaps, Osama Hatem, Andrei Luferau, Alexey Y. Nikitin, J. Michael Klopf, Pablo Alonso-González, Susanne C. Kehr, Lukas M. Eng. Terahertz Twistoptics–Engineering Canalized Phonon Polaritons. ACS Nano 17, 19313–19322 (2023). DOI 10.1021/acsnano.3c06477, https://doi.org/10.1021/acsnano.3c06477 (arXiv:2308.03343, https://arxiv.org/abs/2308.03343)
  10. J. Álvarez-Cuervo, M. Obst, S. Dixit, G. Carini, A. I. F. Tresguerres-Mata, C. Lanza, E. Terán-García, G. Álvarez-Pérez, L. F. Álvarez-Tomillo, K. Diaz-Granados, R. Kowalski, A. S. Senerath, N. S. Mueller, L. Herrer, J. M. De Teresa, S. Wasserroth, J. M. Klopf, T. Beechem, M. Wolf, L. M. Eng, T. G. Folland, A. Tarazaga Martín-Luengo, J. Martín-Sánchez, S. C. Kehr, A. Y. Nikitin, J. D. Caldwell, P. Alonso-González, A. Paarmann. Unidirectional ray polaritons in twisted asymmetric stacks. Nature Communications 15, 9042 (2024). DOI 10.1038/s41467-024-52750-3, https://doi.org/10.1038/s41467-024-52750-3 (arXiv:2403.18657, https://arxiv.org/abs/2403.18657)
  11. Javier Taboada-Gutiérrez, Yixi Zhou, Ana I. Tresguerres-Mata, Christian Lanza, Abel Martínez-Suárez, Gonzalo Álvarez-Pérez, Jiahua Duan, José Ignacio Martín, María Vélez, Iván Prieto, Adrien Bercher, Jérémie Teyssier, Ion Errea, Alexey Y. Nikitin, Javier Martín-Sánchez, Alexey B. Kuzmenko, Pablo Alonso-González. Unveiling the Mechanism of Phonon-Polariton Damping in α-MoO3. ACS Photonics 11, 3570–3577 (2024). DOI 10.1021/acsphotonics.4c00485, https://doi.org/10.1021/acsphotonics.4c00485
  12. Jiahua Duan, Aitana Tarazaga Martín-Luengo, Christian Lanza, Stefan Partel, Kirill Voronin, Ana Isabel F. Tresguerres-Mata, Gonzalo Álvarez-Pérez, Alexey Y. Nikitin, Javier Martín-Sánchez, Pablo Alonso-González. Canalization-based super-resolution imaging using an individual van der Waals thin layer. Science Advances 11, eads0569 (2025). DOI 10.1126/sciadv.ads0569, https://doi.org/10.1126/sciadv.ads0569 (arXiv:2404.14876, https://arxiv.org/abs/2404.14876)
  13. Kirill V. Voronin, Gonzalo Álvarez-Pérez, Aitana Tarazaga Martín-Luengo, Pablo Alonso-González, Alexey Y. Nikitin. Misalignment between the Directions of Propagation and Decay of Nanoscale-confined Polaritons (2025). arXiv:2508.15993, https://arxiv.org/abs/2508.15993
  14. Qingdong Ou, Shuwen Xue, Weiliang Ma, Jiong Yang, Guangyuan Si, Lu Liu, Gang Zhong, Jingying Liu, Zongyuan Xie, Ying Xiao, Tian Sun, Ding Yuan, Kourosh Kalantar-Zadeh, Peining Li, Zhigao Dai, Huanyang Chen, Qiaoliang Bao. Natural van der Waals Canalization Lens for Non-Destructive Nanoelectronic Circuit Imaging and Inspection. Advanced Materials 37, 2504526 (2025). DOI 10.1002/adma.202504526, https://doi.org/10.1002/adma.202504526 (arXiv:2502.09308, https://arxiv.org/abs/2502.09308)
  15. Farid Aghashirinov, Andrea Mancini, Lin Nan, Giacomo Venturi, Bettina Frank, Harald Giessen, Antonio Ambrosio. Intrinsic plasmon canalization in the biaxial van der Waals crystal MoOCl2. Nature Nanotechnology 21, 1238–1244 (2026). DOI 10.1038/s41565-026-02243-9, https://doi.org/10.1038/s41565-026-02243-9 (arXiv:2606.10534, https://arxiv.org/abs/2606.10534)