Electro-optic modulators are an indispensable part of photonic communication systems, largely dictating the achievable transmission rate. Recent advances in materials and fabrication/processing techniques have brought new elements and a renewed dynamic to research on optical modulation. Motivated by the new opportunities, this Perspective reviews the state of the art in integrated electro-optic modulators, covering a broad range of contemporary materials and integrated platforms. To provide a better overview of the status of current modulators, an assessment of the different material platforms is conducted on the basis of common performance metrics: extinction ratio, insertion loss, electro-optic bandwidth, driving voltage, and footprint. The main physical phenomena exploited for electro-optic modulation are first introduced, aiming to provide a self-contained reference to researchers in physics and engineering. Additionally, we take care to highlight topics that can be overlooked and require attention, such as the accurate calculation of carrier density distribution and energy consumption, the correct modeling of thin and two-dimensional materials, and the nature of contact electrodes. Finally, a future outlook for the different electro-optic materials is provided, anticipating the research and performance trends in the years to come.

The correct numerical calculation of the resonance characteristics and, principally, the quality factor Q of contemporary photonic and plasmonic resonant systems is of utmost importance, since Q defines the bandwidth and affects nonlinear and spontaneous emission processes. Here, we comparatively assess the commonly used methods for calculating Q using spectral simulations with commercially available, general-purpose software. We study the applicability range of these methods through judiciously selected examples covering different material systems and frequency regimes from the far-infrared to the visible. We take care in highlighting the underlying physical and numerical reasons limiting the applicability of each one. Our findings demonstrate that in contemporary systems (plasmonics, 2D materials) Q calculation is not trivial, mainly due to the physical complication of strong material dispersion and light leakage. Our work can act as a reference for the mindful and accurate calculation of the quality factor and can serve as a handbook for its evaluation in guided-wave and free-space photonic and plasmonic resonant systems.

Temporal coupled-mode theory (CMT) is an acclaimed and widely used theoretical framework for modeling the continuous-wave response and temporal dynamics of any integrated or free-space photonic resonant structure. It was initially employed to understand how energy is coupled into and out of a cavity and how it is exchanged between different resonant modes. In the 30 years that followed its establishment, CMT has been expanded to describe a broad range of nonlinear interactions as well (self- and cross-phase modulation, saturable absorption, frequency generation, gain, etc.). In this Tutorial, we thoroughly present the basic principles and the evolution of CMT throughout the years, showcasing its immense capabilities for the analysis and design of linear and nonlinear resonant photonic systems. Importantly, we focus on the examples of modern, open nanophotonic resonators incorporating contemporary bulk or sheet (2D) materials that may be lossy and dispersive. For each linear/nonlinear effect under study, we follow a meticulous, step-by-step approach, starting from an accurate model of the physical phenomenon and proceeding to its introduction in the CMT framework all the way to the efficient solution of the resulting system of equations. Our work highlights the merits of CMT as an efficient, accurate, and versatile theoretical tool. We envision that it can serve both as an introductory reference for any reader and as a comprehensive handbook on how to incorporate a broad range of linear and nonlinear effects in the CMT framework.

A general framework combining perturbation theory and coupled-mode theory is developed for analyzing nonlinear resonant structures comprising dispersive bulk and sheet materials. To allow for conductive sheet materials, a nonlinear current term is introduced in the formulation in addition to the more common nonlinear polarization. The framework is applied to model bistability in a graphene-based traveling-wave resonator system exhibiting third-order nonlinearity. We show that the complex conductivity of graphene disturbs the equality of electric and magnetic energies on resonance (a condition typically taken for granted), due to the reactive power associated with the imaginary part of graphene's surface conductivity. Furthermore, we demonstrate that the dispersive nature of conductive materials must always be taken into account, since it significantly impacts the nonlinear response. This is explained in terms of the energy stored in the surface current, which is zeroed-out when linear dispersion is neglected. The results obtained with the proposed framework are compared with full-wave nonlinear finite-element simulations with excellent agreement. Very low characteristic power for bistability is obtained, indicating the potential of graphene for nonlinear applications.

Two-dimensional (2D) or sheet materials have been recently recognized as fascinating materials for nonlinear photonics. Here, we develop a rigorous mathematical framework based on perturbation theory and temporal coupled-mode theory capable of analyzing third-order, ${\chi }^{\left(3\right)}$, multichannel nonlinear processes in resonant systems comprising 2D materials. The framework is applied to model degenerate four-wave mixing in a guided-wave graphene plasmon-polariton resonant structure, consisting of a standing-wave resonator directly coupled to access waveguides. The results obtained with the proposed framework are compared with full-wave finite-element simulations revealing excellent agreement. Aside from being accurate and efficient, our framework allows for selectively incorporating different nonlinear phenomena, identifying their unique impact on the nonlinear response and providing valuable physical insight. We are, thus, able to specify the optimal operating point leading to maximum conversion efficiency for the generated wave in a multiparameter space. In addition, we identify unstable operating regimes exhibiting optical bistability or limit cycles, thoroughly characterizing the component performance. Our framework enables the study of diverse multichannel phenomena (frequency generation, frequency mixing, and parametric amplification) in the thriving field of 2D material photonics, thus allowing for assessing the potential of these exciting materials for practical nonlinear applications.

A side-coupled disk-waveguide system based on a long-range hybrid plasmonic waveguiding configuration is theoretically investigated for Kerr bistability and self-pulsation. The nonlinear response is studied with a theoretical framework combining perturbation theory and temporal coupled-mode theory, where all relevant effects, including two-photon absorption (TPA) as well as free-carrier dispersion (FCD) and absorption (FCA), are taken into account. We show that bistable operation with input powers as low as 40 mW is possible; a consequence of the significant reduction in resistive losses made possible by the long-range waveguiding principle. The effect of TPA, FCD, and FCA on the nonlinear response is thoroughly assessed and it is shown that carrier lifetime must drop to 8 ps in order to suppress free-carrier effects and obtain a high-quality bistable response, which is subsequently exploited for demonstrating ultrafast memory operation with low power requirements and high extinction ratio between states. Finally, by appropriate tuning of the carrier lifetime, FCD can lead to self-pulsation. The operating regimes in the detuning-power space are identified and the characteristics of the spontaneous oscillation discussed.