Temporal Coupled-mode Theory Research Articles

Eigenmodes of finite length silicon-on-insulator microring resonator arrays

In this paper the eigenmodes of finite length microring resonator arrays have been systematically studied, both analytically using temporal coupled mode theory (CMT), and numerically using two-dimensional finite element method (FEM). The method for obtaining the values of parameters appearing in simplified CMT model using results of FEM calculations is presented. Calculations were carried out by COMSOL FEM packages for a wide range of distances between the rings. The obtained results reveal that the rotational degeneracy is preserved for a wide range of interrings distances. It is shown how the eigenvalue spectrum depends on the number of cavities in the system. The differences for the cases of odd and even numbers of rings, and its implications on actual applications, are discussed in details. The central branch appearing in odd-number arrays plays significant role for the delay-lines and optical buffering applications. Based on the first order perturbation theory, an analytical expressions for the eigenfrequencies of arbitrary (finite) length linear array of microring resonators are derived. The analytical expressions describing eigenfrequencies are useful for determining positions of the maxima in transfer characteristics in microring arrays with external buses.

Parity-time symmetry in coherently coupled vertical cavity laser arrays

We report parity-time (PT) symmetry breaking in electrically injected, coherently coupled, vertical cavity surface emitting laser arrays. We predict beam steering, mode evolution and mode hopping as a consequence of the non-Hermiticity of the array analyzed by temporal coupled mode theory with both asymmetric gain distribution and local frequency detuning. We present experimental confirmation of the predicted mode evolution, mode hopping and PT symmetry breaking with quantitative agreement with the theory.

Investigation of Fano resonance in compound resonant waveguide gratings for optical sensing

An optical sensor is designed to support the Fano effect based on a compound resonant waveguide grating (CRWG). The transmission spectra of the CRWG are investigated by utilizing a theoretical method that combines the temporal coupled mode theory with the eigenmode information of the grating structure. The theoretical results, which are observed to agree closely with those acquired by rigorous coupled-wave analysis, show that the linewidth of the transmission spectrum decreases upon increasing the distance between the grating strips, and the central resonance frequency decreases as the refractive index of the analyte increases. Here, the proposed CRWG structures will find potential uses in optical sensing.

Non-Hermitian coupled-mode theory for incoherently pumped exciton-polariton condensates

The generalized Gross-Pitaevskii equation (gGPE) is an effective phenomenological description for the dynamics of incoherently pumped exciton-polariton condensates. However, a brute force numerical simulation of the gGPE provides little physical insight into condensate formation under arbitrary pumping configurations, and is demanding in terms of computational resources. We introduce in this paper a modal description of polariton condensation under incoherent pumping of arbitrary spatial profile, based on eigenmodes of the non-Hermitian generator of the linearized dynamics. A pump-dependent basis is then introduced to formulate a temporal coupled-mode theory that captures condensate dynamics in the presence of all nonlinear interactions. Simulations using a single set of modes for a given pumping and trapping configuration agree very well with a full integration of the gGPE in diverse dynamical regimes, supporting the validity of this modal description, while also providing a speedup in simulation times.

Tailoring the Multiple Fano Resonances in Nanobelt Plasmonic Cluster

We theoretically explore the appearances and characteristics of Fano resonances in novel-designed nanobelt cluster, which shows strong modulation depths, and the Fano dips can be tailored independently. The underlying physical mechanisms contributing to Fano resonances are discussed with the electric field enhancement and charge distributions. The results demonstrate that dark-mode excitation is not a necessary condition for the existence of Fano resonances in the nanobelt cluster. The lineshape of the scattering spectra can be regulated effectively by adjusting the geometrical parameters. The effect of the incident illumination polarization orientation is also numerically investigated, indicating that Fano resonances can be completely switched off when the nanobelt cluster is induced by 90°. Moreover, multiple Fano resonances would occur as the number of the nanobelt is increased, which is explained by a physical mechanism based on temporal coupled-mode theory.

Ring modulators with enhanced efficiency based on standing-wave operation on a field-matched, interdigitated p-n junction

We propose ring modulators based on interdigitated p-n junctions that exploit standing rather than traveling-wave resonant modes to improve modulation efficiency, insertion loss and speed. Matching the longitudinal nodes and antinodes of a standing-wave mode with high (contacts) and low (depletion regions) carrier density regions, respectively, simultaneously lowers loss and increases sensitivity significantly. This approach permits further to relax optical constraints on contacts placement and can lead to lower device capacitance. Such structures are well-matched to fabrication in advanced microelectronics CMOS processes. Device architectures that exploit this concept are presented along with their benefits and drawbacks. A temporal coupled mode theory model is used to investigate the static and dynamic response. We show that modulation efficiencies or loss Q factors up to 2 times higher than in previous traveling-wave geometries can be achieved leading to much larger extinction ratios. Finally, we discuss more complex doping geometries that can improve carrier dynamics for higher modulation speeds in this context.

Temporal coupled mode theory linking to surface-wave dispersion relations in near-field electromagnetic heat transfer

We provide a detailed discussion of the use of coupled mode theory to describe near-field heat transfer. We consider a simple physical model system of coupled harmonic oscillators with each oscillator maintaining at a different temperature, where heat transfer between the oscillators can be analytically treated from first-principles using the Newton's equation and the fluctuation dissipation theorem. Applying a slowly varying envelope approximation to the Newton's equation, we derive a coupled mode theory formalism. We then apply this coupled mode theory formalism in the study of the near-field heat transfer between either silicon carbide plates or between two graphene sheets. The coupled mode theory provides a quantitative link between the dispersion relation of the coupled system and the heat transfer, and agrees with exact numerical results over all range of wavevectors. To obtain such complete agreement, the key observation here is that one should include the frequency shift, that is, the frequency of the individual mode used in the coupled mode theory should be different from the frequency of the mode of an isolated structure. Finally, we show that the coupled mode theory can be applied even when more than two modes are involved in the heat transfer. As an example, we extend our formalism to the near-field heat transfer in a four-layer graphene structure.

Sub-nanometer linewidth perfect absorption in visible band induced by Bloch surface wave

We demonstrate the unity absorption of visible light with an ultra-narrow 0.1 nm linewidth. It arises from the Bloch surface wave resonance in alternating TiO2/SiO2 multilayers. The total absorption and narrow linewidth are explained from the radiative and absorptive damping, which are quantitatively determined by the temporal coupled mode theory. When a silver film with proper thickness is added to the absorber, the perfect absorption is achieved with only 3 structural bilayers, in contrast with 8 bilayers required without Ag. Furthermore, significant field enhancement and an ultrahigh 2600/RIU sensing figure-of-merit are simultaneously obtained at resonance, which might facilitate applications in nonlinear optical devices and high resolution refractive index sensing.

Theoretical Investigation of a Plasmonic Demultiplexer in MIM Waveguide Crossing with Multiple Side-Coupled Hexagonal Resonators

A novel plasmonic demultiplexer in metal–insulator–metal (MIM) waveguide crossing with multiple side-coupled hexagonal resonators is proposed and numerically investigated. The operating principle of the structure is analyzed by using the temporal coupled-mode theory. It is found that wavelength demultiplexing can be realized by modulating locations of resonators, which is validated by finite-difference time-domain (FDTD) simulations. In addition, the influences of structural parameters on transmission characteristics are studied by simulations. Simulation results reveal that the demultiplexed wavelength, transmission efficiency, and bandwidth of each channel can be manipulated by adjusting structural parameters of the demultiplexer. The proposed demultiplexer will provide an alternative for the design of highly integrated optical circuits and complex waveguide networks.

Plasmonically Induced Absorption and Transparency Based on Stub Waveguide with Nanodisk and Fabry-Perot Resonator

A novel metal-insulator-metal (MIM) plasmonic waveguides structure, which is composed by stub waveguide with nanodisk and Fabry-Perot (F-P) resonator, has been proposed and numerically simulated with the finite-difference time-domain (FDTD). Based on the three-level system, the extreme destructive interference between bright and dark resonators gives rise to the distinct plasmonically induced absorption (PIA) response with the abnormal dispersion and novel fast-light feature. Simultaneously, the dramatic double plasmonically induced transparency (PIT) effect with slow-light characteristic can also be achieved in the system. The relationship between the transmission characteristics and the geometric parameters is studied in detail. By optimum design, the modulation depth of the PIA transmission spectrum of 90 % with 0.145 and 0.14 ps fast-light effect can be gained simultaneously, and the peak transmissivity of the double PIT system of 75.2 and 72.8 % with −0.38 ps slow light-effect can be achieved. The simulated transmission features are in agreement with the temporal-coupled mode theory (CMT). The characteristics of the system indicate an important potential application in integrated optical circuits such as slow-light and fast-light devices, high-performance filter, and optical storage.

Perfect single-sided radiation and absorption without mirrors

Highly directional radiation from photonic structures is important for many applications, including high power photonic crystal surface emitting lasers, grating couplers, and light detection and ranging devices. However, previous dielectric, few-layer designs only achieved moderate asymmetry ratios, and a fundamental understanding of bounds on asymmetric radiation from arbitrary structures is still lacking. Here, we show that breaking the 180$^\circ$ rotational symmetry of the structure is crucial for achieving highly asymmetric radiation. We develop a general temporal coupled-mode theory formalism to derive bounds on the asymmetric decay rates to the top and bottom of a photonic crystal slab for a resonance with arbitrary in-plane wavevector. Guided by this formalism, we show that infinite asymmetry is still achievable even without the need of back-reflection mirrors, and we provide numerical examples of designs that achieve asymmetry ratios exceeding $10^4$. The emission direction can also be rapidly switched from top to bottom by tuning the wavevector or frequency. Furthermore, we show that with the addition of weak material absorption loss, such structures can be used to achieve perfect absorption with single-sided illumination, even for single-pass absorption rates less than $0.5\%$ and without back-reflection mirrors. Our work provides new design principles for achieving highly directional radiation and perfect absorption in photonics.

Modeling quasi-dark states with temporal coupled-mode theory.

Coupled resonators are commonly used to achieve tailored spectral responses and allow novel functionalities in a broad range of applications. The Temporal Coupled-Mode Theory (TCMT) provides a simple and general tool that is widely used to model these devices. Relying on TCMT to model coupled resonators might however be misleading in some circumstances due to the lumped-element nature of the model. In this article, we report an important limitation of TCMT related to the prediction of dark states. Studying a coupled system composed of three microring resonators, we demonstrate that TCMT predicts the existence of a dark state that is in disagreement with experimental observations and with the more general results obtained with the Transfer Matrix Method (TMM) and the Finite-Difference Time-Domain (FDTD) simulations. We identify the limitation in the TCMT model to be related to the mechanism of excitation/decay of the supermodes and we propose a correction that effectively reconciles the model with expected results. Our discussion based on coupled microring resonators can be useful for other electromagnetic resonant systems due to the generality and far-reach of the TCMT formalism.

Modulating Spatial Light by Grating Slot Waveguides With Transparent Conducting Oxides

Transparent conducting oxides (TCOs) are emerging active materials for light modulating due to their special epsilon-near-zero effect. A novel spatial light modulating technique based on the guided mode resonance effect in a slot waveguide with TCOs is proposed, where both transmission and reflection can be manipulated by injecting or depleting carriers into the TCO layer. Temporal coupled mode theory and finite element method are used to analyze the light coupling and filtering. A modulation depth at 1550 nm as high as 0.85 is obtained due to the dramatic variations of modal properties in a TCO layer embedded slot waveguide. The wavelength bandwidth with a modulation depth above 0.5 is 8.6 nm. In addition, the resonant wavelength is sensitive to the bias on the TCO layer, which makes the device a tunable transmissive/reflective filter with a tuning range of 40 nm.

A wavelength demultiplexing structure based on graphene nanoribbon resonators

A wavelength demultiplexing (WDM) structure based on graphene nanoribbon resonators is proposed and numerically investigated by the finite-difference time-domain (FDTD) method. The demultiplexing wavelength can be easily derived by adjusting the length of the resonator, which is accurately explained by the Fabry–Perot (F–P) resonant theory. Meanwhile, the transmission characteristics of the WDM structure are influenced by the coupling distance between the resonator and drop/bus waveguides, and the performance of the WDM device is analyzed at different nanoribbon width and chemical potential. In addition, in order to improve the transmission efficiency from the drop waveguide, a reflection structure is introduced at the end of the bus waveguide. The exact mechanism for the WDM structure is analyzed in detail using the temporal coupled-mode theory. The proposed structure will have potential applications in the field of ultra-compact WDM systems in highly integrated optical circuits.

The design and test of dual/triple-mode bandpass filter based on temporal coupled-mode theory

In this article, the design and test of both a K-band dual-mode bandpass filter BPF pair and a K-band triple-mode BPF are presented based on N-mode temporal coupled-mode theory CMT. The expressions of transmission and reflection responses are analytically derived. All the parameters in the expressions have clear physical meanings and are easily optimized to reach the required filter performance. Aided by eigenmode simulations, concrete structures of the three integrated BPFs are designed and optimized to approach the calculated physical parameters. After the fabrications and measurements of the three BPFs, extended upper/lower stopband with high stopband rejections are achieved, and by increasing the number of resonant modes, improved frequency selectivity and better passband flatness are obtained. The analytical analysis well predicts the simulation and measurement results, which provides an efficient way for BPF designs. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:609-622, 2016.

Backscattering in silicon microring resonators: a quantitative analysis

AbstractSilicon microring resonators very often exhibit resonance splitting due to backscattering. This effect is hard to quantitatively and predicatively model. This paper presents a behavioral circuit model for microrings that quantitatively explains the wide variations in resonance splitting observed in experiments. The model is based on an in‐depth analysis of the contributions to backscattering by both the waveguides and couplers. Backscattering transforms unidirectional microrings into bidirectional circuits by coupling the clockwise and counterclockwise circulating modes. In high‐Q microrings, visible resonance splitting will be induced, but, due to the stochastic nature of backscattering, this splitting is different for each resonance. Our model, based on temporal coupled mode theory, and the associated fitting method, are both accurate and robust, and can also explain asymmetrically split resonances. The cause of asymmetric resonance splitting is identified as the backcoupling in the couplers. This is experimentally confirmed and its dependency on gap and coupling length is further analyzed. Moreover, the wide variation in resonance splitting of one spectrum is analyzed and successfully explained by our circuit model that incorporates most linear parasitic effects in the microring. This analysis uncovers multi‐cavity interference within the microring as an important source of this variation. image

Multispectral Detection with Metal-Dielectric Filters: An Investigation in Several Wavelength Bands with Temporal Coupled-Mode Theory

Multispectral infrared (IR) detection is of great interest to enhance our ability to gather information from a scene. Filtering is a low-cost alternative to the complex multispectral device architectures to which the IR community has devoted much attention. Multilayer dielectric filters are standard in industry, but they require changing the thickness of at least one layer to tune the wavelength. Here, we pursue an approach based on apertures in a metallic layer of fixed thickness, in which the filtered wavelengths are selected by varying the aperture geometry. In particular, we study filters made of at least one sheet of resonating apertures in metal embedded in dielectrics. We will discuss two interesting problems that arise when one attempts to design such filters. First, metallic absorption must be taken into account. Second, the form and size of the pattern is limited by lithography. We will present some design examples and an attempt at explaining the filtering behavior based on the temporal coupled mode theory. That theory models the filter as a resonator interacting with the environment via loss channels. The transmission is solely determined by the loss rates associated with those channels. This model allows us to give a general picture of the filtering performance and compare their characteristics at different wavelength bands.

Measurement of bound states in the continuum by a detector embedded in a photonic crystal.

We directly measure optical bound states in the continuum (BICs) by embedding a photodetector into a photonic crystal slab. The BICs observed in our experiment are the result of accidental phase matching between incident, reflected and in-plane waves at seemingly random wave vectors in the photonic band structure. Our measurements were confirmed through a rigorously coupled-wave analysis simulation in conjunction with temporal coupled mode theory. Polarization mixing between photonic crystal slab modes was observed and described using a plane wave expansion simulation. The ability to probe the field intensity inside the photonic crystal and thereby to directly measure BICs represents a milestone in the development of integrated opto-electronic devices based on BICs.

Design and simulation of linear logic gates in the two-dimensional square-lattice photonic crystals

We have proposed all-optical three-port AND, OR, XOR and NOT photonic crystal logic gates, based on modified symmetric T-branch combiners. Using the temporal coupled-mode theory the operation of the symmetric T-branch combiner has been investigated and the conditions which lead to a maximum power at the output port have been obtained. At the output port of the proposed devices, the logics 1 and 0 are defined as a more than 42% and a less than 14% of the transmission, respectively. The minimum intensity contrast ratio of the suggested logic gates is about 4.77dB. The operations of the proposed devices are examined using time-domain transition-response simulations which have been performed using two dimensional finite-difference time-domain method.

Double Directions Nanoscale Range Finding Using Fano Resonance in Coupled Gratings

In this paper, a theoretical demonstration is given of nanoscale range finding by exciting Fano resonance in coupled gratings. Metallic ridges induce oscillation mode, whose interference with surface plasmon polartions generate narrow Fano resonance. The concept of hybridization is employed to understand the coupling effect of surface plasmon polartions and the oscillation due to metallic ridges. Fano behavior in this structure is captured by using the temporal coupled-mode theory. The gained fundamental understanding opens up new ways to control nanoscale spacing distances and tailor Fano resonance, thus facilitating rational design of nanosensors to improve the performance of nanomotion control systems.