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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. [ABSTRACT FROM AUTHOR]

We derive a coupled-mode theory (CMT) formulation for the fast analysis of periodic multi-element metasurfaces in the presence of radiation losses. Full-wave simulations of periodic multi-element metasurfaces are very time- and memory-consuming, especially as the size and complexity of the metasurface increase. The CMT formulation provides a considerably faster and efficient alternative. It results in a small system of equations with size equal to the number of supported resonator modes in the frequency range of interest, allowing to calculate the resonator mode amplitudes and, consequently, the metasurface response. Subsequently, we systematically derive analytical closed-form expressions for the coupling coefficients between two weakly coupled resonators in the presence of radiation losses and incorporate them into the CMT model, which is found important for the accurate description of the metasurface, while also providing insight into the underlying physics of complex metasurfaces. We validate the proposed formulation on benchmark examples of both metal- and dielectric-based metasurface absorbers (MSAs) by comparing the CMT results to spectral FEM simulations of the composing supercell. To further demonstrate the potential of the proposed formulation, as a proof of concept, we use the CMT to synthesize a larger optimized periodic multi-element MSA. A comprehensive comparison to full-wave FEM simulations of the composing supercell is included in terms of time and computational requirements, which shows that our method provides a valuable and efficient alternative solver for synthesizing complex metasurfaces. [ABSTRACT FROM AUTHOR]

A nanophotonic passively Q‐switched lasing element in the near infrared is proposed and theoretically investigated. It consists of a silicon‐rich nitride disk resonator enhanced with the contemporary MoS2/WSe2$\left(\text{MoS}\right)_{2} / \left(\text{WSe}\right)_{2}$ hetero‐bilayer and a graphene monolayer to provide gain and saturable absorption, respectively. The two‐dimensional materials are placed on top of the disk resonator and are separated by a spacer of hexagonal boron nitride. MoS2/WSe2$\left(\text{MoS}\right)_{2} / \left(\text{WSe}\right)_{2}$ emits at 1128 nm due to the radiative recombination of interlayer excitons after being optically pumped at 740 nm. Optical pumping is conducted in a guided‐wave manner aiming at achieving a high overall efficiency by critically coupling to a cavity mode near the pump transition. The response of the proposed pulsed laser is assessed by utilizing a coupled‐mode theory framework fed with linear finite‐element method simulations, rigorously derived from the Maxwell–Bloch equations. Following a meticulous design process and exploiting the guided pumping scheme, an ultralow lasing threshold of just 24.2 $24. 2 \text{ } \mu \text{W} $μW is obtained. Overall, the Q‐switched laser delivers pulsed light inside an integrated bus waveguide with mW peak power, ps duration, and GHz repetition rates requiring sub‐mW continuous wave pumping. These properties are highly promising for communication applications and highlight the potential of two‐dimensional materials for nanophotonic light sources. [ABSTRACT FROM AUTHOR]

A whispering-gallery mode nanophotonic laser cavity having as active medium a transition-metaldichalcogenide (TMD) bilayer is examined. The proposed system is analysed and designed utilizing a strict and rigorous computational framework based on the coupled-mode theory. Our framework is capable of accurately and efficiently handling the gain properties of two-dimensional materials, such as contemporary TMD monolayers, multilayers, and heterostructures. The presented lasing cavity exhibits an adequately low pump threshold and light emission in the order of milliwatts is predicted. Exploiting the capabilities of the developed framework, we were in position to efficiently design the cavity as well as to estimate quantitative lasing parameters such as the pumping threshold and the lasing frequency.

We assess the continuous wave and dynamic routing performance of a compact silicon-on-insulator disk resonator overlaid with a graphene monolayer at telecommunication wavelengths. Switching action is enabled by saturable absorption in graphene, controlled by a pump wave of only a few milliwatts. Graphene saturable absorption is modeled through a carrier rate equation that incorporates both the finite relaxation time and diffusion of photo-generated carriers, providing a realistic account of carrier dynamics. The overall nonlinear response of the resonator is evaluated with a rigorous mathematical framework based on perturbation theory and temporal coupled-mode theory. We thoroughly investigate the effects of carrier diffusion and finite relaxation time, both separately and together. We also take into account nonlinear refraction via a Kerr effect term and quantify its impact on the overall response. In order to suppress the Kerr effect, we replace silicon with silicon-rich nitride, allowing for the individual contributions of the resonator core and graphene (of opposite sign) to exactly compensate each other. Our results contribute to the understanding of carrier dynamics and their impact on the performance of practical graphene-based switching components. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A silicon disk resonator overlaid with a uniform graphene layer in an add-drop configuration is proposed as an all-optical routing element. Operation is based on the saturable absorption effect provided by the graphene layer. The element is thoroughly analyzed as a two-channel device in the context of an appropriate nonlinear framework combining perturbation theory and temporal coupled-mode theory. Taking into consideration the primary nonlinear effect, which is graphene saturable absorption, a design path is carefully developed that eventually leads to a traveling-wave resonant element with low-power requirements, low insertion loss, high extinction ratio, and sufficient bandwidth. In a subsequent step, other important nonlinear effects originating from graphene and the silicon disk, including the Kerr effect and free-carrier effects, are considered and means for counterbalancing their action are demonstrated. A low control power of 9 mW together with a bandwidth of 20 GHz is shown possible, with the insertion loss of almost 3 dB and an extinction ratio over 10 dB in both ports (add and drop). [ABSTRACT FROM AUTHOR]

Practical graphene-comprising resonant structures are proposed for high-quality Kerr-induced bistability with a low input power. Two structures are designed for operation in the far-infrared (FIR) and near-infrared (NIR) frequency regimes, respectively. The nonlinear response is studied by utilizing a theoretical framework combining perturbation theory and coupled-mode theory, capable of accurately and efficiently modeling resonant structures with dispersive bulk and sheet materials. The FIR system is based on a side-coupled graphene-nanoribbon ring resonator, formed by applying a bias voltage between a uniform graphene sheet and an uneven silicon substrate. By optimizing the system geometry, we demonstrate bistable response with a theoretically infinite extinction ratio between states and an operating power of only 400 μW at 10 THz. In the NIR circuit, a silicon photonic ring resonator is overlaid with a graphene sheet. The silicon-slot geometry is judiciously chosen to maximize the field overlap with graphene, resulting in low power requirements of 90 mW at 1:55 μm. In both cases, nonlinearity stems from the instantaneous Kerr effect in graphene, allowing for ultra-fast response. Combining low input power and fast response times, the proposed components highlight the potential of graphene for nonlinear applications over a broad spectral range. [ABSTRACT FROM AUTHOR]

In this paper, a technique introducing the generalized-pencil-of-function (GPOF) algorithm as a post processing procedure in the finite-difference time-domain (FDTD) method is presented to facilitate the study of electromagnetic bandgap (EBG) devices. Correction criteria are also developed to deal with erroneous eigenfrequencies associated with computational noise. The numerical results obtained are considered sufficiently accurate, while the potentials of this approach are thoroughly discussed. Index Terms--Eigenfrequencies, electromagnetic bandgap (EBG) structures, finite-difference time-domain (FDTD) method, generalized-pencil-of-function (GPOF) method.

The spectral properties of three devices, which are based on dielectric and metallic photonic crystals and contain defects filled with a nematic liquid crystal material is studied. The placement of a defect mode relative to the bandgap edges greatly influences both its linewidth as well as its tuning range.

The first-order diffraction efficiency [[eta].sub.1] of surface-stabilized ferroelectric liquid-crystal (SSFLC) phase gratings is calculated for device thicknesses in the range d = 1 to 5 [micro]m and for pitches p of 5 to 20 [micro]m assuming incident light at 633 nm. The peak value of [eta.sub.1] as a function of d has negligible dependence on the incoming polarization when p = 20 [micro]m. For smaller pitch values the peak value of [[eta].sub.1] decreases and becomes increasingly dependent on the orientation of the incoming polarization owing to the influence of the domain walls that occur between the SSFLC pixels. OCIS codes: 050.1950, 160.3710, 200.4650, 260.1180.

A rigorous analysis is presented of the diffraction efficiency of a polarization-insensitive surface-stabilized ferroelectric liquid-crystal (SSFLC) phase grating, taking full account of the internal structure of the ferroelectric liquid-crystal layer. When no field is applied, the twisted director profile in the relaxed state gives an optimum diffraction efficiency for a device thickness between the half-wave-plate and the full-wave-plate conditions. The influence of the magnitude of the spontaneous polarization and applied ac fields are investigated, and it is shown that the diffraction efficiency of a binary SSFLC phase grating can be strongly enhanced with the technique of ac stabilization. OCIS codes: 050.1950, 160.3710, 200.4650, 260.1180.

A composite scheme based on the finite-difference time-domain method and a plane-wave expansion method is developed and applied to the optics of periodic liquid-crystal microstructures. This is used to investigate three-dimensional light-wave propagation in grating-induced bistable nematic devices with double periodicity. Detailed models of realistic devices are analyzed with emphasis on two different underlying surface-relief grating structures: a smooth bisinusoidal grating and a square-post array. The influence of the grating feature size is quantified. Device performance is examined in conjunction with an appropriate compensation layer, and the optimum layer thickness is determined for the different grating geometries.

The characteristic spectral response of an inclined strip grating, either free-standing or embedded in a dielectric slab, is quite sensitive to variations of the cant angle. It is an attractive frequency-selective surface which allows considerable design flexibility through the cant angle and/or the refractive index of the host dielectric. The inclined strip grating is analyzed through an indirect mode-matching technique based on Green's second theorem. The analytical approach is simple and numerically efficient. Calculations focus on the effect of cant angle on the magnitude of all propagating Floquet harmonics on both sides of the structure. Interesting features of the scattering behavior are discussed.

We describe a rigorous formalism for the calculation of the nonlinear parameter of arbitrary three-dimensional nanophotonic graphene-comprising waveguides. Graphene is naturally implemented as a zero-thickness conductive sheet, modeled solely by complex linear and nonlinear surface conductivity tensors, whose values are extracted from theoretical models. This representation is compared to the more commonly employed equivalent bulk-medium representation and is found superior. We numerically calculate the nonlinear parameters of several optical waveguide archetypes overlaid with infinite graphene monolayers, including silicon-wire and plasmonic metal-slot and metal-stripe configurations. The metal-slot configuration offers the most promising performance for Kerr-type nonlinear applications. Finally, we apply the same formalism to probe the potential of graphene nanoribbon waveguide nonlinearity in the terahertz band. [ABSTRACT FROM AUTHOR]