We demonstrate, both theoretically and experimentally, the concept of nonlinear second-order topological insulators, a class of bulk insulators with quantized Wannier centers and a bulk polarization directly controlled by the level of nonlinearity. We show that one-dimensional edge states and zero-dimensional corner states can be induced in a trivial crystal insulator made of evanescently coupled resonators with linear and nonlinear coupling coefficients, simply by tuning the intensity. This allows global external control over topological phase transitions and switching to a phase with nonzero bulk polarization, without requiring any structural or geometrical changes. We further show how these nonlinear effects enable dynamic tuning of the spectral properties and localization of the topological edge and corner states. Such self-induced second-order topological insulators, which can be found and implemented in a wide variety of physical platforms ranging from electronics to microwaves, acoustics, and optics, hold exciting promises for reconfigurable topological energy confinement, power harvesting, data storage, and spatial management of high-intensity fields.

The Fano resonance is a widespread wave scattering phenomenon associated with a peculiar asymmetric and ultrasharp line shape, which has found applications in a large variety of prominent optical devices. While its substantial sensitivity to geometrical and environmental changes makes it the cornerstone of efficient sensors, it also renders the practical realization of Fano-based systems extremely challenging. Here, we introduce the concept of topological Fano resonance, whose ultrasharp asymmetric line shape is guaranteed by design and protected against geometrical imperfections, yet remaining sensitive to external parameters. We report the experimental observation of such resonances in an acoustic system, and demonstrate their inherent robustness to geometrical disorder. Such topologically protected Fano resonances, which can also be found in microwave, optical, and plasmonic systems, open up exciting frontiers for the generation of various reliable wave-based devices including low-threshold lasers, perfect absorbers, ultrafast switches or modulators, and highly accurate interferometers, by circumventing the performance degradations caused by inadvertent fabrication flaws.

Optical computing has emerged as a promising candidate for real-time and parallel continuous data processing. Motivated by recent progresses in metamaterial-based analog computing [Science 343, 160 (2014) [CrossRef] ], we theoretically investigate the realization of two-dimensional complex mathematical operations using rotated configurations, recently reported in [Opt. Lett. 39, 1278 (2014) [CrossRef] ]. Breaking the reflection symmetry, such configurations could realize both even and odd Green’s functions associated with spatial operators. Based on such an appealing theory and by using the Brewster effect, we demonstrate realization of a first-order differentiator. Such an efficient wave-based computation method not only circumvents the major potential drawbacks of metamaterials, but also offers the most compact possible device compared to conventional bulky lens-based optical signal and data processors.

Analog signal processors have attracted a tremendous amount of attention recently, as they potentially offer much faster operation and lower power consumption than their digital versions. Yet, they are not preferable for large scale applications due to the considerable observational errors caused by their excessive sensitivity to environmental and structural variations. Here, we demonstrate both theoretically and experimentally the unique relevance of topological insulators for alleviating the unreliability of analog signal processors. In particular, we achieve an important signal processing task, namely resolution of linear differential equations, in an analog system that is protected by topology against large levels of disorder and geometrical perturbations. We believe that our strategy opens up large perspectives for a new generation of robust all-optical analog signal processors, which can now not only perform ultrafast, high-throughput, and power efficient signal processing tasks, but also compete with their digital counterparts in terms of reliability and flexibility.