Coupled-mode Formalism Research Articles

Low-power, ultrafast, and dynamic all-optical tunable plasmonic analog to electromagnetically induced transparency in two resonators side-coupled with a waveguide system

We theoretically and numerically investigate a low-power, ultrafast, and dynamic all-optical tunable plasmonic analog to electromagnetically induced transparency (EIT) in two nanodisk resonators side-coupled to a metal-insulator-metal plasmonic waveguide system. The optical Kerr effect is enhanced by the slow light effect of the plasmonic EIT-like effect and the plasmonic waveguide based on graphene-Ag composite material structures with giant effective Kerr nonlinear coefficient. The optical Kerr effect modulation method is applied to improve tuning rate with response time of subpicoseconds or even femtoseconds. With dynamically tuning the propagation phase of the plasmonic waveguide, π-phase shift of the transmission spectrum in the plasmonic EIT-like system is achieved under excitation of a pump light with an intensity as low as 5.85 MW/cm2. The group delay is controlled between 0.09 and 0.4 ps. All observed schemes are analyzed rigorously through finite-difference time-domain simulations and coupled-mode formalism. Results show a new direction toward the low power consumption and ultrafast responses of integration plasmonic photonic devices and all-optical dynamical storage of light devices in optical communication and quantum information processing.

Noise and dynamics of stimulated-Brillouin-scattering microresonator lasers

We use theoretical analysis and numerical simulation to investigate the operation of a laser oscillating from gain supplied by stimulated Brillouin scattering (SBS) in a microresonator. The interaction of the forward, backward, and density waves within the microresonator results in a set of coupled-mode equations describing both the laser's phase and amplitude evolution over time. Using this coupled-mode formalism, we investigate the performance of the SBS laser under noise perturbation and identify the fundamental parameters and their optimization to enable low-noise SBS operation. The intrinsic laser linewidth, which is primarily limited by incoherent thermal occupation of the density wave, can be of order hertz or below. Our analysis also determines the SBS laser's relaxation oscillation, which results from the coupling between the optical and density waves, and appears as a resonance in both the phase and amplitude quadratures. We further explore contributions of the pump noise to the SBS laser's performance, which we find under most circumstances to increase the SBS laser noise beyond its fundamental limits. By tightly stabilizing the pump laser onto the microcavity resonance, the transfer of pump noise is significantly reduced. Our analysis is both supported and extended through numerical simulations of the SBS laser.

Condition for Perfect Resonant Antireflection

ABSTRACTReflection occurs at an air-material interface. The development of antireflection schemes, which aims to cancel such reflection, is important for a wide variety of applications including solar cells and photodetectors. Recently, it has been demonstrated that a periodic array of resonant subwavelength objects placed at an air-material interface can significantly reduce reflection that otherwise would have occurred at such an interface. Here, we introduce the theoretical condition for complete reflection cancellation in this resonant antireflection scheme. Using both general theoretical arguments and analytical temporal coupled-mode theory formalisms, we show that in order to achieve perfect resonant antireflection, the periodicity of the array needs to be smaller than the free-space wavelength of the incident light for normal incidence, and also the resonances in the subwavelength objects need to radiate into air and the dielectric material in a balanced fashion. Our theory is validated using first-principles full-field electromagnetic simulations of structures operating in the infrared wavelength ranges. For solar cell or photodetector applications, resonant antireflection has the potential of providing a low-cost technique for antireflection that does not require nanofabrication into the absorber materials, which may introduce detrimental effects such as additional surface recombination. Our work here provides theoretical guidance for the practical design of such resonant antireflection schemes.

Condition for perfect antireflection by optical resonance at material interface

Reflection occurs at an air–material interface. The development of antireflection schemes, which aims to cancel such reflection, is important for a wide variety of applications including solar cells and photodetectors. Recently, it has been demonstrated that a periodic array of resonant subwavelength objects placed at an air–material interface can significantly reduce reflection that otherwise would have occurred at such an interface. Here, we introduce the theoretical condition for complete reflection cancellation in this resonant antireflection scheme. Using both general theoretical arguments and analytical temporal coupled-mode theory formalisms, we show that in order to achieve perfect resonant antireflection, the periodicity of the array needs to be smaller than the free-space wavelength of the incident light for normal incidence, and also the resonances in the subwavelength objects need to radiate into air and the dielectric material in a balanced fashion. Our theory is validated using first-principles full-field electromagnetic simulations of structures operating in the infrared wavelength ranges. For solar cell or photodetector applications, resonant antireflection has the potential for providing a low-cost technique for antireflection that does not require nanofabrication into the absorber materials, which may introduce detrimental effects such as additional surface recombination. Our work here provides theoretical guidance for the practical design of such resonant antireflection schemes.

A dynamic and ultrafast group delay tuning mechanism in two microcavities side-coupled with a waveguide system

We theoretically propose a dynamic and ultrafast group delay tuning mechanism in two microcavities side-coupled to a waveguide system through external optical pump beams. The optical Kerr effect modulation method is applied to improve tuning rate with response time of subpicoseconds or even femtoseconds. The group delay of an all-optical analog to electromagnetically induced transparency effect can be controlled by tuning either the frequency of photonic crystal microcavities or the propagation phase of line waveguide. Group delay is controlled between 5.88 and 70.98 ps by dynamically tuning resonant frequencies of the microcavities. Alternatively, the group delay is controlled between 1.86 and 12.08 ps by dynamically tuning the propagation phase of line waveguide. All observed schemes are analyzed rigorously through finite-difference time-domain simulations and coupled-mode formalism. Results show a new direction toward microstructure integration optical pulse trapping and all-optical dynamical storage of light devices in optical communication and quantum information processing.

Energy conservation via coupled modes in waveguides with an impedance boundary condition

A statement of energy conservation for a coupled mode formulation with real mode functions and eigenvalues has been demonstrated to be consistent with the statement of conservation derivable from the Helmholtz equation. The restriction to real mode functions and eigenvalues precludes coupled mode descriptions with waveguide absorption or untrapped modes. The demonstration, along with the derivation of the coupled mode range equation, relies on orthonormality in terms of a product of two modal depth functions integrated to infinite depth. This paper shows that energy conservation and the derivation of the coupled mode range equation can be extended to complex mode functions and eigenvalues, and that energy is conserved for ocean waveguides with a penetrable bottom boundary at a finite depth beneath any range dependence. For this, the penetrable bottom boundary is specified by an impedance condition for the mode functions. The new derivations rely on completeness and a modified orthonormality statement. Mode coupling is driven solely by waveguide range dependence. Thus, the form of the range equation and the values of the coupling coefficients are unaffected by a finite depth waveguide. Applications of energy conservation to examine the accuracy of a numerical coupled mode calculation are presented.

Phase shift multiplication effect of all-optical analog to electromagnetically induced transparency in two micro-cavities side coupled to a waveguide system

We propose phase shift multiplication effect of all-optical analog to electromagnetically induced transparency in two photonic crystal micro-cavities side coupled to a waveguide system through external optical pump beams. With dynamically tuning the propagation phase of the line waveguide, the phase shift of the transmission spectrum in two micro-cavities side coupled to a waveguide system is doubled along with the phase shift of the line waveguide. π-phase shift and 2π-phase shift of the transmission spectrum are obtained when the propagation phase of the line waveguide is tuned to 0.5π-phase shift and π-phase shift, respectively. All observed schemes are analyzed rigorously through finite-difference time-domain simulations and the coupled-mode formalism. These results show a new direction to the miniaturization and the low power consumption of microstructure integration photonic devices in optical communication and quantum information processing.

Can carbon nanotube fibers achieve the ultimate conductivity?—Coupled-mode analysis for electron transport through the carbon nanotube contact

Recent measurements of carbon nanotube (CNT) fibers electrical conductivity still show the values lower than that of individual CNTs, by about one magnitude order. The imperfections of manufacturing process and constituent components are described as culprits. What if every segment is made perfect? In this work, we study the quantum conductance through the parallel junction of flawless armchair CNTs using tight-binding method in conjunction with non-equilibrium Green's function approach. Short-range oscillations within the long-range oscillations as well as decaying envelopes are all observed in the computed Fermi-level (low bias) conductance as a function of contact length, L. The propagation of CNTs' Bloch waves is cast in the coupled-mode formalism and helps to reveal the quantum interference nature of various behaviors of conductance. Our analysis shows that the Bloch waves at the Fermi-level propagate through a parallel junction without reflection only at an optimal value of contact length. For quite a long junction, however, the conductance at the Fermi level diminishes due to the perturbation of periodic potential field of close-packed CNTs. Thus, a macroscopic fiber, containing an infinite number of junctions, forms a filter that permits passage of electrons with specific wave vectors, and these wave vectors are determined by the collection of all the junction lengths. We also argue that the energy gap introduced by long junctions can be overcome by small voltage (∼0.04 V) across the whole fiber. Overall, developing long individual all-armchair metallic CNTs serves as a promising way to the manufacture of high-conductivity fibers.

Derivation of energy loss via coupled modes

A coupled mode formalism can describe energy loss within an ocean waveguide due to all possible mechanisms, including bulk attenuation, scattering from rough surfaces, and volume inhomogeneities. First order corrections produced from mode-couplings can be incorporated into the modal loss. Accounting for these losses within an adiabatic mode approach would provide improvements over current standard modal propagation models. Highlights of the formalism for scattering from a rough surface will be provided.

Energy exchange and scattering loss within a two-way coupled-mode formulation

The loss of energy due to propagation in an ocean environment arises from several possible processes. In a two-way coupled-mode description scattering effects from a rough bottom Pekeris waveguide can be isolated by excluding bulk attenuation, and intermodal energy exchange can be examined. The range- and frequency-dependent interaction of a single trapped mode exchanging energy with multiple continuum modes is identified within this framework. The goal of the analysis is two-fold. First, it provides insight into additional loss mechanisms that can be incorporated into current local mode models (c.f. Koch and Stotts, A Derivation of Energy Loss via Coupled Modes, 165th Meeting of the Acoustical Society of America, June 2–7, 2013). Second, a comparison can be made to previous descriptions of additional attenuation, such as Kirchhoff scattering loss. Varying the bottom roughness permits tests of the applicability of the Born approximation.

Fundamental bounds on decay rates in asymmetric single-mode optical resonators

We derive tight upper and lower bounds of the ratio between decay rates to two ports from a single resonance exhibiting Fano interference, based on a general temporal coupled-mode theory formalism. The photon transport between these two ports involves both direct and resonance-assisted contributions, and the bounds depend only on the direct process. The bounds imply that, in a lossless system, full reflection is always achievable at Fano resonance, even for structures lacking mirror symmetries, while full transmission can only be seen in a symmetric configuration where the two decay rates are equal. The analytic predictions are verified against full-field electromagnetic simulations.

A coupled-mode method for sound propagation in a range-dependent penetrable waveguide

The coupled-mode method based on the direct global matrix (DGMCM) approach for sound propagation in range-dependent waveguides [Luo et al., "A numerically stable coupled-mode formulation for acoustic propagation in range-dependent waveguides," Sci. China-Phys. Mech. Astron. 55, 572 (2012)] is further developed. The normal mode model KRAKEN is adopted to provide local modal solutions and their associated coupling matrices. As a result, the model DGMCM is capable of providing full two-way solutions for the two-dimensional realistic problems characterized by a penetrable bottom and a depth-varying sound speed profile. In addition, the closed-form expressions of coupling matrices for sound propagation in a range-dependent, two-layer waveguide are proposed. The numerical solutions of the coupling matrices by DGMCM agree well with the analytical solutions. Sound propagation in a penetrable wedge is solved by the updated DGMCM model. The numerical results indicate that the updated DGMCM model is numerically stable and accurate, and can provide benchmark solutions for realistic range-dependent problems.

Coupled-mode formulation of two-parallel photonic-crystal waveguides

A self-contained coupled-mode formulation for coupled two-dimensional photonic-crystal waveguides (PCWs) is discussed. Using a perturbation theory, the first-order coupled-mode equations are systematically derived, which govern the evolution of the modal amplitude of individual PCWs in isolation. The coupled-mode equations are used to analyze the coupled symmetric PCWs consisting of a square lattice of circular dielectric rods or air holes. It is shown that the results are in good agreement with those obtained by the rigorous direct analysis of the coupled waveguide system.

A numerically stable coupled-mode formulation for acoustic propagation in range-dependent waveguides

In this paper, a stepwise coupled-mode model with the use of the direct global matrix approach is proposed. This method is capable of handling two-dimensional problems with either a point source in cylindrical geometry or a line source in plane geometry. With the use of the direct global matrix approach, this method is numerically stable. In addition, by introducing appropriately normalized range solutions, thismodel is free from the numerical overflow problem. Furthermore, we put forward source conditions appropriate for the line-source problem in plane geometry. As a result, this method is capable of addressing the scenario with a line source on top of a sloping bottom. Closed-form expressions for coupling matrices are derived and applied in this paper for handling problems with pressure-release boundaries and a homogeneous water column. The numerical simulations indicate that the proposed model is accurate, efficient, and numerically stable. Consequently, this model can serve as a benchmark model in range-dependent propagation modeling. Although this method is verified by an ideal wedge problem in this paper, the formulation applies to realistic problems as well.

Scattering in a Pekeris waveguide from a rough bottom using a two-way coupled mode approach

Scattering from a rough ocean bottom is described numerically with a two-way coupled-mode formalism that contains scattering effects to all orders and provides an exact solution to the wave equation. Both scattered field and direct blast components are computed within the formalism framework. A comparison of the scattered component solution from the coupled mode with the Born approximation (BA) solution for scattering from a rough bottom Pekeris waveguide shows that the BA predicts correctly the scattered field levels but not detailed structure. The transition from direct blast to scattered field dominance is identified in the total field time series.

Complex coupled-mode theory for tapered optical waveguides

A coupled-mode formulation based on complex local modes is developed for tapered and longitudinally varying optical waveguides. Different from the conventional coupled-mode theory that requires integration over the entire spectrum of radiation modes, the new formulation treats the radiation fields via discrete complex modes similarly to the guided modes. Accuracy, convergence, and scope of validity for the solutions of the complex coupled-mode equations are investigated in detail for a typical single-mode waveguide taper. It is demonstrated that the complex coupled-mode theory has overcome the difficulties of the conventional theory in simulation of radiation field effects while preserving the simplicity and intuitiveness of this popular method.

Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances

We present a theoretical analysis of lasing action in photonic crystal surface-emitting lasers (PCSELs). The semiclassical laser equations for such structures are simulated with three different theoretical techniques: exact finite-difference time-domain calculations, an steady-state ab-initio laser theory and a semi-analytical coupled-mode formalism. Our simulations show that, for an exemplary four-level gain model, the excitation of dark Fano resonances featuring arbitrarily large quality factors can lead to a significant reduction of the lasing threshold of PCSELs with respect to conventional vertical-cavity surface-emitting lasers. Our calculations also suggest that at the onset of lasing action, most of the laser power generated by finite-size PCSELs is emitted in the photonic crystal plane rather than the vertical direction. In addition to their fundamental interest, these findings may affect further engineering of active devices based on photonic crystal slabs.

Cylindrical Coordinates Coupled Mode Theory

We present a cylindrical coordinates coupled mode formulation for coupling between two degenerate modes of a traveling wave resonator. The resonator is assumed to have rotational symmetry and therefore two degenerate modes propagating in opposite directions. We analyze coupling of the resonance modes in space due to a perturbation in the resonator and derive first-order coupling equations. We also present an application of the method in modeling the mode coupling in a ring resonator with an integrated distributed Bragg reflector. For the presented structure, the results from coupled mode analysis agree with the finite elements method simulation results, and the method dramatically reduces the simulation time from 2 h to less than 5 min.

Light wheel buildup using a backward surface mode

When a guided mode is excited in a dielectric slab coupled to a backward surface wave at the interface between a dielectric and a left-handed medium, light is confined in the structure: this is a light wheel. Complex plane analysis of the dispersion relation and coupled-mode formalism give deep insight into the physics of this phenomenon (lateral confinement and the presence of a dark zone).

A Spectral Coupled-Mode Formulation for Sound Propagation around Axisymmetric Seamounts

A spectral coupled-mode solution of the three-dimensional (3D) acoustic field generated by a point source in the presence of an axisymmetric seamount is developed. Based on the same theoretical foundation as the formulation presented by Taroudakis [J. Comput. Acoust. 4 (1996) 101], the present approach combines a spectral decomposition in azimuth with a coupled-mode theory for two-way range-dependent propagation. However, the earlier formulations are severely limited in terms of frequency, size and geometry of the seamount, the seabed composition, and the distance between the source and the seamount, and are therefore severely limited in regard to realistic seamount problems. Without changing the fundamental theoretical foundation, the present approach applies a number of modifications to the formulation, leading to orders of magnitude improvement in numerical efficiency for realistic problems. Therefore, realistic propagation and scattering scenarios can be modeled, including effects of seamount roughness and realistic sedimentary structure.