# The CBMS Conference: Mathematical Methods for Novel Metamaterials

#### May 20 - 24, 2024

**Principal Lecture:**

The principal lecturer **Professor Habib Ammari** (ETH, Zurich) will deliver ten lectures and present a coherent mathematical theory for subwavelength physics to explain spectacular properties of metamaterials consisting of subwavelength resonators in various settings. The speaker will examine how quantum phenomena observed in condensed-matter physics can be transformed to classical waves at subwavelength scales and elucidate the similarities as well as the fundamental differences between them (such as long-range interactions between the quantum and classical worlds). Using subwavelength resonators as the building blocks for metamaterials, the lecturer will provide the mathematical foundation for a variety of unusual properties of metamaterials. Especially, the lecturer will focus on the design of complex structures whose localization, confinement and guiding properties of waves are robust with respect to fabrication imperfections and discuss phase transitions in metamaterials. He will also investigate Hermitian, non-Hermitian and time-modulated models for systems of subwavelength resonators and identify the degeneracies in the spectrum of the operators arising from those models, which are responsible for the exotic phenomena that have been observed experimentally. The main mathematical results for various models will be illustrated numerically. Finally, he will introduce open problems and discuss future directions in this important research area.

**Lecture 1: Introduction**

This lecture will cover (i) the applications of subwavelength metamaterials, espeically in biomedical super-resolution imaging, telecommunications, and quantum computing; (ii) examples of subwavelength resonators: plasmonic particles, high-index dielectric particles, Helmholtz resonators, high-contrast acoustic resonators; (iii) research goals: establish the mathematical theory for subwavelength physics and transfer demonstrated quantum phenomena to classical waves at subwavelength scales. The lecturer will introduce several key concepts in condensed-matter physics, such as topological defects, phase transitions, Hall e ect, localized states, and mathematical models of tight-binding and nearest-neighborhood approximations in condensed-matter physics. He would then discuss non-validity of the tight-binding approximation in subwavelength physics, especially in the strong and long-range interactions in three dimensions.

References

- H. Ammari, B. Li, and J. Zou, Mathematical analysis of electromagnetic scattering by dielectric nanoparticles with high refractive indices, Trans. Amer. Math. Soc., 376 (2023), 39-90.
- H. Ammari, P. Millien, M. Ruiz, and H. Zhang, Mathematical analysis of plasmonic nanoparticles: the scalar case, Arch. Ration. Mech. Anal., 224 (2017), 597-658.
- S. Yu and H. Ammari, Plasmonic interaction between nanospheres, SIAM Rev. 60 (2018), 356-385.
- H Ammari and H. Zhang, A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators, Comm. Math. Phys., 337 (2015), 379-428.

**Lecture 2: Single Subwavelength Resonator and Effective Medium Theory in the Dilute Regime**

The lecture discusses subwavelength resonance of a single resonator and the mathematical tools to analyze the resonances. The lecturer will present boundary integral formulation, pole pencil decomposition, and Muller's method to derive the asymptotic formula for the subwavelength resonance. He will also discuss e ective medium theory for a dilute system of resonators, single negative materials and double negative materials.

References

- H. Ammari, B. Fitzpatrick, Bria, D. Gontier, H. Lee, and H. Zhang, Minnaert resonances for acoustic waves in bubbly media, Ann. Inst. H. Poincar e C Anal. Non Lin eaire, 35 (2018), 1975-1998.
- H. Ammari, B. Fitzpatrick, H. Kang, M. Ruiz, S. Yu, and H. Zhang, Mathematical and Computational Methods in Photonics and Phononics, Mathematical Surveys and Monographs, Vol. 235, American Mathematical Society, Providence, RI, 2018.
- H. Ammari, B. Fitzpatrick, H. Lee, S. Yu, and H. Zhang, Double-negative acoustic metamaterials, Quart. Appl. Math., 77 (2019), 767-791.
- H. Ammari and H. Zhang, E ective medium theory for acoustic waves in bubbly fuids near Minnaert resonant frequency, SIAM J. Math. Anal., 49 (2017), 3252-3276.

**Lecture 3: Systems of Strongly Interacting Subwavelength Resonators**

The lecture discusses resonances for systems of strongly interacting subwavelength resonators. The lecturer will derive the capacitance matrix approximation for the resonant frequencies and the resonant modes, introduce modal decomposition for wave scattering by subwavelength resonators, and then study the properties of the capacitance matrix, quasiperiodic capacitance matrix for periodic systems of subwavelength resonators, and finally derive resonances in the first radiation continuum.

References

- H. Ammari, B. Davies, E. Hiltunen, Functional analytic methods for discrete approximations of subwavelength resonator systems, arXiv: 2106.12301.
- H. Ammari, B. Fitzpatrick, H. Lee, S. Yu, and H. Zhang, Subwavelength phononic bandgap opening in bubbly media, J. Di erential Equations, 263 (2017), 5610-5629.
- F. Feppon and H. Ammari, Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies, Stud. Appl. Math., 149 (2022), 164-229.

**Lecture 4: Hermitian Systems I: Subwavelength Trapping and Guiding of Waves**

The lecture discusses the band structure for a square lattice of subwavelength resonators. These include subwavelength bandgap opening and the two-scale behavior of Bloch modes, Bound states in the continuum and Fano-resonance, Dirac degeneracies for Honeycomb structures, and spectral convergence of defect modes for a finite resonator array.

References

- H. Ammari, B. Davies, and E. Hiltunen, Spectral convergence of defect modes infinite resonator arrays,
*arXiv:2301.03402*(2023) - H. Ammari, B. Davies, E. Hiltunen, H. Lee, and S. Yu, Bound states in the continuum and Fano resonances in subwavelength resonator arrays, J. Math. Phys., 62 (2021), 101506.
- H. Ammari, B. Fitzpatrick, E. Hiltunen, H. Lee, and S. Yu, Honeycomb-lattice Minnaert bubbles, SIAM J. Math. Anal., 52 (2020), 5441-5466.
- H. Ammari, E. Hiltunen, and S. Yu, A high-frequency homogenization approach near the Dirac points in bubbly honeycomb crystals, Arch. Ration. Mech. Anal., 238 (2020), 1559-1583.
- H. Ammari, E. Hiltunen, and S. Yu, Subwavelength guided modes for acoustic waves in bubbly crystals with a line defect, J. Eur. Math. Soc, 24 (2022), 2279-2313.
- H. Ammari, H. Lee, and H. Zhang, Bloch waves in bubbly crystal near the rst band gap: a high-frequency homogenization approach, SIAM J. Math. Anal., 51 (2019), 45-59.

**Lecture 5: Hermitian Systems II: Topological Defects and Anderson Localization**

The lecture discusses topologically protected edge modes for the Su-Schrieffer-Heeger model of a chain of resonator dimers, quantized topological invariants and topological phase transitions, and edge modes in a dislocated chain of resonator dimers. In addition, the lecture will discuss Anderson localization with long-range interactions, Laurent operator formulation, Toeplitz matrix formulation for compact defects, hybridization and level repulsion, phase transition and eigenmode symmetry swapping.

References

- H. Ammari, B. Davies, E. Hiltunen, and S. Yu, Topologically protected edge modes in onedimensional chains of subwavelength resonators, J. Math. Pures Appl., 144 (2020), 17-49.
- H. Ammari, B. Davies, and E. Hiltunen, Robust edge modes in dislocated systems of subwavelength resonators, J. Lond. Math. Soc., 106 (2022), 2075-2135.
- H. Ammari, B. Davies, and E. Hiltunen, Anderson localization in the subwavelength regime, arXiv:2205.13337.

**Lecture 6: Non-Hermitian Systems I**

This lecture will discuss non-Hermitian systems with subwavelength metamaterials. The lecturer will introduce exceptional point degeneracies for parity-time symmetric systems of subwavelength resonators, high-order exceptional point degeneracies, unidirectional reflection and extraordinary transmission in non-Hermitian systems.

References

- H. Ammari, B. Davies, E. Hiltunen, H. Lee, and S. Yu, Exceptional points in parity-timesymmetric subwavelength metamaterials, SIAM J. Math. Anal., 54 (2022), 6223-6253.
- H. Ammari, B. Davies, E. Hiltunen, H. Lee, and S. Yu, High-order exceptional points and enhanced sensing in subwavelength resonator arrays, Stud. Appl. Math., 146 (2021), 440-462.

**Lecture 7: Non-Hermitian Systems II**

This lecture will discuss band inversion and edge modes in non-Hermitian subwavelength resonators, non-quantized topological invariants, partial topological phase transitions, and condensation of bulk modes at one edge of the structure.

References

- H. Ammari and E. Hiltunen, Time-dependent high-contrast subwavelength resonators, J. Comput. Phys., 445 (2021), 110594.
- H. Ammari, E. Hiltunen, and T. Kosche, The asymptotic Floquet theory for rst order ODEs with finite Fourier series perturbation and its applications to Floquet metamaterials, J. Differential Equations, 319 (2022), 227-287.

**Lecture 8: Time-modulated Systems I**

The lecture focuses on time-modulated subwavelength metamaterials. The lecturer will introduce the capacitance matrix formulation for the subwavelength quasi-frequencies, space-time modulated systems of resonators and folding degeneracies, pseudo-spin e ect in trimer honeycomb lattices with phase-shifted time modulations and double-zero refraction.

References

- H. Ammari and E. Hiltunen, Edge modes in active systems of subwavelength resonators, arXiv: 2006.05719.

**Lecture 9: Time-modulated Systems II**

The lecture discusses non-reciprocal bandgaps in time-modulated subwavelength metamaterials, unidirectional guiding and broken time-reversal symmetry k-gaps and amplified emission.

References

- H. Ammari and J. Cao, Unidirectional edge modes in time-modulated metamaterials, Proc. A., 478 (2022), 395.
- H. Ammari, J. Cao, and E. Hiltunen, Nonreciprocal wave propagation in space-time modulated media, Multiscale Model. Simul., 20 (2022), 1228-1250.
- H. Ammari, J. Cao, X. Zeng, Transmission properties of space-time modulated metamaterials, Stud. Appl. Math. (2022).

**Lecture 10: Summary and Open Problems**

This lecture discusses similarities and fundamental di erences between the theory of condensedmatter physics and the theory of subwavelength physics, the one-dimensional model versus the three-dimensional model in subwavelength physics. The lecturer will also introduce open problems in this area, especially on the twisted honeycomb lattices, time-space localization, and high-order degeneracies in the band structure, etc.

References

- H. Ammari, F. Fiorani, E. Hiltunen, On the validity of the tight-binding method for describing systems of subwavelength resonators, SIAM J. Appl. Math., 82 (2022), 1611-1634.

**Other Lectures:**

**Professor Andrea Alu** (City University of New York) will deliver a lecture on “Exotic Linear and Nonlinear Light-Matter Interactions in Metamaterials”. He will discuss recent research in electromagnetics and nano-optics in his lab, showing how suitably tailored meta-atoms and their arrangements open exciting venues for enhanced wave-matter interactions. He will discuss unusual scattering, absorption and waveguiding responses, from cloaking and scattering suppression, to nonreciprocity and topological phenomena, enhanced nonlinear effects at subwavelength scales, and bound states in the continuum. Physical insights into the underlying phenomena and new devices based on these concepts will be presented.

**Professor Oscar Bruno**(Caltech) will deliver a lecture on “Advanced Computational Methods for Modeling, Simulation and Design”. He will present a novel ”Interpolated Factored Green Function” method (IFGF), including a massively parallel implementation, for the accelerated evaluation of the integral operators in scattering theory and other areas, with application to atmospheric propagation and metamaterial design. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of O(N log N) operations for an N-point surface mesh. Importantly, the proposed method does not utilize previously-employed acceleration elements such as the Fast Fourier transform (FFT), special-function expansions, high dimensional linear- algebra factorizations, translation operators, equivalent sources, or parabolic scaling. Instead, the IFGF strategy, which leads to an extremely simple algorithm, capitalizes on slow variations inherent in a certain Green-function ”analytic factor”, which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. In particular, the IFGF method runs on a small memory footprint, and, as it does not utilize the Fast Fourier Transforms (FFT), it is better suited than other methods for efficient parallelization in distributed memory computer systems. Application of related integral equation techniques and associated device-optimization problems will be described, including a novel time-domain scattering solver that effectively solves time-domain problems of arbitrary duration via Fourier transformation in time, with application to design of metamaterials with time-dependent material properties.

**Professor Graeme Milton** (University of Utah) will deliver a lecture on “Guiding Stress : From Pentamodes to Cable Webs to Masonry Structures”. He will discuss pentamode materials which have been proposed for acoustic cloaking by guiding stress around objects. A key feature of pentamode materials is that each vertex in the material is the junction of 4 double cone elements. Thus the tension in one element determines the tension in the other elements, and by extension uniquely determines the stress in the entire metamaterial. He will show how this key feature can be extended to discrete wire networks, supporting forces at the terminal nodes and which may have internal nodes where no forces are applied. The discrete networks provide an alternative way of distributing the forces through the geometry of the network. In particular the network can be chosen so it is uniloadable, i.e. supports only one set of forces at the terminal nodes. Such uniloadable networks provide the natural generalization of pentamode materials to discrete networks. They extend such a problem to compression-only ’strut nets’ subjected to fixed and variable nodal loads. These systems provide discrete element models of masonry bodies, which lie inside the polygon/polyhedron with vertices at the points of application of the given forces (’underlying masonry structures’). In particular, they solve the two-dimensional problem where one wants the strut net to avoid a given set of obstacles, and also allow some of the forces to be reactive ones.

**Professor John Schotland** (Yale University) will deliver a lecture on “Quantum Optics in Random Media”. He will discuss quantum optics, which is the quantum theory of the interaction of light and matter. He will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.

**Professor Hai Zhang** (Hong Kong University of Science and Technology) will deliver a lecture on “Mathematics of Edge Modes in Topological Photonic/Phononic Materials”. He will discuss the edge modes in topological photonic/phononic structures, which are essential in guiding the propagation of waves along the desired direction. The developments of topological insulators have provided a new avenue for creating edge modes in photonic/phononic structures, which are topologically protected and therefore stable against fabrication defects. He will present recent results on the existence of in-gap edge modes that are bifurcated from Dirac points in photonic/phononic structures.