Copyright. Francesco Monticone Topological Embedded Eigenstates

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2 Copyright The use of this work is restricted solely for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author.

3 Abstract Light confinement is one of the most common and important operations in optics, photonics, and electromagnetics. Recently, a fundamentally new approach to light confinement has been explored by different groups, based on the concept of bound states in the radiation continuum (BICs), or embedded eigenstates. While conventional bound states in, e.g., fibers, waveguides, and resonators are created by preventing their coupling to radiation modes through symmetry incompatibility, momentum mismatch, or by directly suppressing outgoing waves, BICs are compatible with free-space radiation, but remain confined due to the destructive interference between different radiation channels. Two recent works experimentally demonstrated, for the first time, a defining feature of BICs in planar structures, namely, the fact that the light radiated by these modes, around the BIC, exhibits a vortex in its polarization far-field profile. This radiation feature is associated to an integer winding number, or topological charge, which is unaffected by continuous deformations of the structure. Hence, the presence of the polarization vortex bestows topological robustness to the BIC. For our experimental demonstration, we fabricated a dielectric metasurface supporting a BIC at optical frequencies. Using momentumspace polarimetry, we measured the full polarization state of the reflected light, highlighting the presence of the topological vortex. The essential physics can be captured in a general, new, theoretical framework that identifies BICs as radiationless distributions of induced currents of electric and magnetic types. This model fully accounts for the topological nature of the BIC, showing how the vortex must appear right at the point of destructive interference between two radiation channels. Keywords: Bound states in the continuum, embedded eigenstates, topological electromagnetics, optical vortices, scattering.

4 Biography Francesco Monticone is an Assistant Professor of Electrical and Computer Engineering at Cornell University. He received the B.Sc. and M.Sc. (summa cum laude) degrees from Politecnico di Torino, Italy, in 29 and 211, respectively, and the Ph.D. degree in Electrical and Computer Engineering from The University of Texas at Austin, Austin, TX, in 216, where he was advised by Prof. Andrea Alu. Dr. Monticone joined the faculty of Cornell University in January 217. Dr. Monticone s current research interests are in the areas of applied electromagnetics, metamaterials, plasmonics, and nanophotonics, with applications ranging from microwaves to optical frequencies. His research group at Cornell University investigates theoretically, numerically and experimentally innovative and extreme aspects of wave interaction with engineered metamaterials and nano-structures, which may have strong scientific and practical impact in the coming years. His recent research work spans a broad range of topics including extreme scattering engineering, cloaking and invisibility, nanoparticles, nanocircuits, nanoantennas, parity-time symmetry, non-hermitian electromagnetic systems, advanced metasurfaces, and topological electromagnetics. Dr. Monticone has authored and co-authored more than 1 scientific contributions in peer-reviewed journal papers, book chapters, and peer-reviewed conference proceedings, receiving more than 25 citations, and has given over 2 invited talks and seminars. His papers have appeared in several high-impact journals, including Physical Review Letters (three times selected as Editor s Suggestion ), Nature Nanotechnology, Proceedings of the IEEE, Optica, and Science. Some of his research work has been picked up by national and international media outlets, such as BBC, NBC News, The Huffington Post, and Time Magazine. Dr. Monticone received the U.S. Air Force Office of Scientific Research (AFOSR) Young Investigator Program Award (YIP) in 218. He was also the recipient of the Inaugural Margarida Jacome Dissertation Award, awarded by The University of Texas at Austin, the 217 Raj Mittra Travel Grant Award, a WNCG Student Leadership Award and the H.L. Bruce Graduate Fellowship, both from UT Austin, an IEEE Antennas and Propagation Society Doctoral Research Award, an IEEE Photonics Society Graduate Student Fellowship, and the Best Student Paper Award at the conference Metamaterials 213. Dr. Monticone is serving as a reviewer for several journals and international conferences, and has been a member of the organizing committee of the Metamaterials congress series since 215. Dr. Monticone is a member of the IEEE, the American Physical Society (APS), the Optical Society of America (OSA), and is a full member of the International Union of Radio Science (URSI).

5 Light confinement How do we confine light? How do we create an optical bound state? Conventional bound states in, e.g., optical fibers, waveguides, and resonators are created by preventing their coupling to radiation modes through symmetry incompatibility, momentum mismatch, or by directly suppressing outgoing waves mmwaveapplications1.htm

6 Embedded Eigenstates Different strategy: bound states within the radiation continuum, or embedded eigenstates (EE), which remain ideally confined, even though they are compatible with free-space radiation. EE manifest themselves as scattering Fano resonances with diverging Q factor Planar geometries Three-dimensional geometries SCS/λ 2 (db) 4 λ/λ c η = 2 a a c2 a c2 a c1 a ε 1 ε 2 ε 3 C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Observation of trapped light within the radiation continuum, Nature 499, 7457, (213). η 2 F. Monticone and A. Alù, Phys. Rev. Lett. 112, (214). M. G. Silveirinha, Phys. Rev. A 89, (214).

7 Nonradiating modes Embedded eigenstates correspond to nonradiating modes of an open cavity/waveguide Examples of extended distributions of accelerating charges that do not radiate (nonradiating sources): A. J. Devaney and E. Wolf, Phys. Rev. D 8, (1973). Pulsating sphere of charges (radial distribution of electric dipoles) H. A. Haus, Am. J. Phys. 54, 1126 (1986). G. Gbur, Prog. Opt. 45, (23). Electric Field image credit: Prof. G. Gbur, UNCC F. Monticone and A. Alù, Phys. Rev. Lett. 112, (214). Transversely-oscillating homogenous plane of charges (transverse distribution of electric dipoles) image credit: Prof. G. Gbur, UNCC Any other types of nonradiating modes?

8 z Radiationless traveling-wave current distribution Homogenous planar sheet of traveling-wave electric and magnetic currents y x J = J M = M ikx ( x + ky y ) e δ ikx ( x + ky y ) e δ ( z) ( z) J ˆ = J z knon-radiating x < k = ω c Fast conditions: wave M = Plane-wave M 1 M ˆ y kxradiation, k =± with angle η J 1 Radiated fields destructively interfere θ = cos ( k k ) x (for k y =) Polarization angle of the radiated field, around the nonradiating condition in k-space α ( k ) ( Jk x ± M k η) α ( k) = arg Ex ( k) + iey ( k) = atan kj y Multivalued function with a branch-point singularity at the nonradiating condition ky kx Vortex in the polarization profile of the radiated field, with topological charge: 1 q = d α π k k 2 C ( k) Integer! quantized topological invariant If an embedded eigenstate corresponds to such a radiationless magneto-electric current Francesco distribution, Monticone Topological it is Embedded topologically Eigenstates protected!

9 Embedded eigenstate in a dielectric metasurface Can we confirm the topological nature of an EE of this type experimentally? Experimental demonstration of an optical EE in a dielectric metasurface (in collaboration with Femius Koenderink s group, AMOLF Institute, Amsterdam) Reflectivity (RCWA) Reflectivity (experiment)

10 Embedded eigenstate in a dielectric metasurface λ λ EE H E z y Dipole moments per unit length m p m p Only one space harmonic radiates m p Magneto-electric character of the TM leaky mode x Average current sheets z z y x M = M J = J ikx ( x + ky y ) e δ ikx ( x + ky y ) e δ ( z) ( z) k x k 1 ± η J M J ˆ = J z M = M ˆ y Dipoles oscillate in phase Nonradiating distribution of induced traveling-wave currents of electric and magnetic type

11 Experimental demonstration of a polarization vortex Predicted polarization angle Measured polarization angle Embedded eigenstate = Eye of a polarization vortex in momentum space (with quantized vortex charge, i.e., winding number) Measurements by Hugo M. Doeleman, AMOLF Institute, Amsterdam, The Netherlands

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13 References 1. C. W. Hsu et al., Observation of trapped light within the radiation continuum., Nature, vol. 499, no. 7457, pp , Jul F. Monticone and A. Alù, Embedded photonic eigenvalues in 3D nanostructures, Phys. Rev. Lett., vol. 112, no. 21, M. G. Silveirinha, Trapping light in open plasmonic nanostructures, Phys. Rev. A, vol. 89, no. 2, p , Feb B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, Topological Nature of Optical Bound States in the Continuum, Phys. Rev. Lett., vol. 113, no. 25, p , Dec C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Bound states in the continuum, Nat. Rev. Mater., vol. 1, p. 1648, Jul F. Monticone and A. Alù, Bound states within the radiation continuum in diffraction gratings and the role of leaky modes, New J. Phys., vol. 19, no. 9, H. M. Doeleman, F. Monticone, W. Den Hollander, A. Alù, and A. F. Koenderink, Experimental observation of a polarization vortex at an optical bound state in the continuum, Nat. Photonics, vol. 12, pp , Y. Zhang et al., Observation of Polarization Vortices in Momentum Space, Phys. Rev. Lett., vol. 12, no. 18, p , F. Monticone, H. M. Doeleman, W. Den Hollander, A. F. Koenderink, and A. Alù, Trapping Light in Plain Sight: Embedded Photonic Eigenstates in Zero-Index Metamaterials, Laser Photon. Rev., vol. 12, no. 5, p. 1722, 218.