Coupled-mode Equations Research Articles

Acousto-optic modulation and opto-acoustic gating in piezo-optomechanical circuits.

Acoustic wave devices provide a promising chip-scale platform for efficiently coupling radio frequency (RF) and optical fields. Here, we use an integrated piezo-optomechanical circuit platform that exploits both the piezoelectric and photoelastic coupling mechanisms to link 2.4 GHz RF waves to 194 THz (1550 nm) optical waves, through coupling to propagating and localized 2.4 GHz acoustic waves. We demonstrate acousto-optic modulation, resonant in both the optical and mechanical domains, in which waveforms encoded on the RF carrier are mapped to the optical field. We also show opto-acoustic gating, in which the application of modulated optical pulses interferometrically gates the transmission of propagating acoustic pulses. The time-domain characteristics of this system under both pulsed RF and pulsed optical excitation are considered in the context of the different physical pathways involved in driving the acoustic waves, and modelled through the coupled mode equations of cavity optomechanics.

Performance analysis for all-fiber polarization transformer with specific spun rate profile

All-fiber polarization transformer (AFPT) is a length of variably spun birefringence fiber which can transform the state of polarization as a wave plate. A set of complete analytic solutions for the coupled-mode equation of AFPT with a specific function of spun rate is deduced by a modified method through deliberate matrix transformation. And the analytical expressions of the transmitting eigen-modes from the zero spun-rate end to the maximum spun-rate end are derived. Moreover, when the linear polarized light is input as single eigen-mode, the ellipticity of quasi-circular polarization light at the output end is calculated as function of the fiber structure parameters. The equivalent retardation of AFPT is evaluated and the critical constraints between the fiber length and the maximum spun rate are illustrated according to practical requirements.

Dipolar effects on the critical fluctuations in Fe: Investigation by the neutron spin-echo technique MIEZE

Iron is one of the archetypical ferromagnets to study the critical fluctuations at a continuous phase transition thus serving as a model system for the application of scaling theory. We report a comprehensive study of the critical dynamics at the transition from the ferro- to the paramagnetic phase in Fe, employing the high-resolution neutron spin echo technique MIEZE. The results show that the dipolar interactions lead to an additional damping of the critical spin fluctuations at small momentum transfers $\bf q$. The results agree essentially with scaling theory if the dipolar interactions are taken into account by means of the mode-coupling equations. However, in contrast to expectations, the dipolar wavenumber $q_D$ that plays a central role in the scaling function $f(\kappa/q,q_D/\kappa)$ becomes temperature dependent. In the limit of small $\bf q$ the critical exponent $z$ crosses over from 2.5 to 2.0.

Dynamic Channel Modeling for Mode-Division Multiplexing

Environmental perturbations, such as wind, mechanical stress, temperature, and lightning, impose microsecond-time-scale changes to the transfer matrix of a multimode fiber (MMF), necessitating adaptive multiple-input multiple-output (MIMO) equalization to track the time-varying channel. It is of significant interest to accurately model channel dynamics so that adaptive MIMO equalization algorithms can be optimized and their impact on digital signal processing complexity and performance can be assessed. We propose a dynamic channel model using coupled-mode theory to describe time-varying polarization- and spatial-mode coupling in MMF caused by fast environmental perturbations. Our method assumes that the MMF has built-in refractive index and geometric defects that are responsible for random birefringence and mode coupling. Various time-varying perturbations modify the coupling coefficients in the coupled-mode equations to drive channel dynamics. We use our dynamic channel model to simulate an example aerial fiber link subject to a sudden gust of wind and show that increasing the strength of the perturbation, the number of modes, and the length of exposed fiber all lead to higher error at the output of an adaptive MIMO equalizer. We employ the model to study the convergence and tracking performance of the least-mean-squares adaptive equalization algorithm.

Justification of the coupled mode asymptotics for localized wavepackets in the periodic nonlinear Schrödinger equation

We consider wavepackets composed of two modulated carrier Bloch waves with opposite group velocities in the one dimensional periodic Nonlinear Schrödinger/Gross–Pitaevskii equation. These can be approximated by first order coupled mode equations (CMEs) for the two slowly varying envelopes. Under a suitably selected periodic perturbation of the periodic structure the CMEs possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity. This leads to a family of approximate solitary waves in the periodic nonlinear Schrödinger equation. Besides a formal derivation of the CMEs a rigorous justification of the approximation and an error estimate in the supremum norm are provided. Several numerical tests corroborate the analysis.

Discrete Changes Model of Inter-core Crosstalk of Real Homogeneous Multi-core Fibers

An analytical discrete changes model of the inter-core crosstalk (ICXT) in real homogeneous weakly-coupled multi-core fibers (MCF) (cores with similar but different parameters) with bending and twisting perturbations is proposed for operation in the phase-matching region, where the bending radii are smaller than the critical bending radius. We show that this model is a generalization of the ICXT model proposed by other authors for perfectly homogeneous MCF. We compare simulation results of the coupled mode equation for a weakly-coupled step-index MCF with seven cores with theoretical predictions from the proposed model, and much faster and accurate mean ICXT power estimates are shown in the whole range of bending radii and difference of intrinsic effective refractive indexes corresponding to operation in the phase-matching region. We show that differences of mean ICXT power exceeding 4 dB can occur between the estimates provided by the model considering perfectly homogeneous MCF and the proposed model.

Singly resonant second-harmonic-generation frequency combs

We consider frequency comb generation in dispersive singly resonant second-harmonic-generation cavity systems. Using a single temporal mean-field equation for the fundamental field that features a noninstantaneous nonlinear response function, we model the temporal and spectral dynamics and analyze comb generation, continuous wave bistability, and modulational instability. It is found that, owing to the significant temporal walk-off between the fundamental and second-harmonic fields, modulational instability can occur even in the complete absence of group-velocity dispersion. We further consider the relation of our model to a previously proposed modal expansion approach, and present a derivation of a general system of coupled mode equations. We show that the two models provide very similar predictions and become exactly equivalent in the limit that absorption losses and group-velocity dispersion at the fundamental frequency are neglected. Finally, we perform numerical simulations that show examples of the variety of comb states that are possible in phase-matched quadratic resonators, and discuss the dynamics of the comb formation process.

Nonlinear coupling in graphene-coated nanowires.

We propose and analyze nonlinear coupler based on a pair of single mode graphene-coated nanowires. Nonlinear wave interactions in such structure are analyzed by the coupled mode equations derived from the unconjugated Lorentz reciprocity theorem. We show that the routing of plasmons in the proposed structure can be controlled by the input power due to the third order nonlinear response of graphene layer. Our findings show that graphene nonlinearity can be exploited in tunable nanoplasmonic circuits based on low-loss, edgeless cylindrical graphene waveguides.

Cavity ring-up spectroscopy for dissipative and dispersive sensing in a whispering gallery mode resonator

In whispering gallery mode resonator sensing applications, the conventional way to detect a change in the parameter to be measured is by observing the steady state transmission spectrum through the coupling waveguide. Alternatively, cavity ring-up spectroscopy (CRUS) sensing can be achieved transiently. In this work, we investigate CRUS using coupled mode equations and find analytical solutions with a large spectral broadening approximation of the input pulse. The relationships between the frequency detuning, coupling gap and ring-up peak height are determined and experimentally verified using an ultrahigh \textit{Q}-factor silica microsphere. This work shows that distinctive dispersive and dissipative transient sensing can be realised by simply measuring the peak height of the CRUS signal, which might improve the data collection rate.

All Optical Scalable Logic Gates Using Si3N4 Microring Resonators

In this paper, microring resonator-based logic gates are discussed that operate on same input and output wavelength. Same input and output wavelength allows the cascaded operation of the developed logic gates and thus can be utilized for developing photonic integrated circuits. Logic operations such as and , or , nor , xor , and xnor are successfully demonstrated. Optical Kerr effect is utilized for operation and structures are numerically simulated using coupled mode equations. Output power levels of around 0.18 and 0.05 W are achieved for high- and low-power logic, respectively. The proposed structures perform well for data rate under 1 Gbps and are analyzed to determine the acceptable input range for high- and low-power logic, respectively. The cascaded operation of proposed logic gates is also demonstrated.

Parametric Characterization of Self-Heating in Depletion-Type Si Micro-Ring Modulators

The influence of self-heating on the static transmission characteristics of depletion-type Si micro-ring modulators (MRMs) is investigated. Self-heating, caused by free-carrier absorption of the input light inside the doped ring waveguide, increases the effective refractive index of the ring waveguide and results in the red-shifted resonance wavelength. This phenomenon is modeled based on the coupled-mode equation with a newly-introduced self-heating coefficient R . The accuracy of our model is confirmed by measurement. In addition, dependence of R on device size and doping concentration is experimentally investigated and the resulting dependence is explained.

Broadband higher order mode conversion using chirped microbend long period gratings.

We suggest a new scheme to create chirped microbend long period gratings. Employing this scheme, the bandwidth of mode conversion between LP01 to LP11 is increased 4.8-fold with a conversion efficiency of 20 dB. This scheme includes a first time demonstration of a non-linearly chirped long period grating. The scheme is investigated both numerically using coupled mode equations as well as experimentally.

Coupled mode analysis of a periodic one-dimensional multimodal fiber bundle

This contribution is a mathematical analysis of the coupled mode equations of a one dimensional infinite periodic lattice of multimodal adjacent fibers that are fused together (a “fiber bundle”). As such, it provides a systematic and detailed derivation of the coupled mode equations and their eigen (modal) solutions within a matrix-based framework and using Z-transform spectral-based formulation. The resulting solution is general in the sense that it is not restricted to a particular dielectric profile of the fibers. Moreover, under a weak coupling assumption, the modal solution clearly identifies the physical building blocks of the solution.

Effects of mode coupling between low-mode radiation flux asymmetry and intermediate-mode ablator roughness on ignition capsule implosions

The low-mode shell asymmetry and high-mode hot spot mixing appear to be the main reasons for the performance degradation of the National Ignition Facility (NIF) implosion experiments. The effects of the mode coupling between low-mode P2 radiation flux asymmetry and intermediate-mode L = 24 capsule roughness on the implosion performance of ignition capsule are investigated by two-dimensional radiation hydrodynamic simulations. It is shown that the amplitudes of new modes generated by the mode coupling are in good agreement with the second-order mode coupling equation during the acceleration phase. The later flow field not only shows large areal density P2 asymmetry in the main fuel, but also generates large-amplitude spikes and bubbles. In the deceleration phase, the increasing mode coupling generates more new modes, and the perturbation spectrum on the hot spot boundary is mainly from the strong mode interactions rather than the initial perturbation conditions. The combination of the low-mode and high-mode perturbations breaks up the capsule shell, resulting in a significant reduction of the hot spot temperature and implosion performance.

Exact scaling solution of the mode coupling equations for non-linear fluctuating hydrodynamics in one dimension

We obtain the exact solution of the one-loop mode-coupling equations for the dynamical structure function in the framework of non-linear fluctuating hydrodynamics in one space dimension for the strictly hyperbolic case where all characteristic velocities are different. All solutions are characterized by dynamical exponents which are Kepler ratios of consecutive Fibonacci numbers, which includes the golden mean as a limiting case. The scaling form of all higher Fibonacci modes are asymmetric Lévy-distributions. Thus a hierarchy of new dynamical universality classes is established. We also compute the precise numerical value of the Prähofer–Spohn scaling constant to which scaling functions obtained from mode coupling theory are sensitive.

Tonks-Dattner resonances in a quantum plasma bubble.

We establish the spectrum of Tonks-Dattner mode resonances in a quantum plasma bubble and consider the spectral changes associated with plasma quenching and plasma expansion. The quantum corrections associated with the mode spectrum are specified, which can be used as a diagnostic tool to identify the quantum regime. The frequency shifts associated with time-varying plasma bubbles correspond to time refraction and can also be used as a plasma diagnostic. We also study the energy mode coupling, in the presence of a low-frequency perturbation. It will be shown that the mode coupling equations take the form of generalized Bloch equations, where a nonlinear Rabi frequency can also be identified.

Electromagnetic field distribution in a quasi-ID rhombic waveguide array

The exact solution of the coupled mode equations is presented in the case of the continuous electromagnetic wave propagation in the quasi-one-dimensional rhombic array of the waveguides. In general case of the boundary conditions the discrete diffraction is described by these solutions. The flat band mode propagation is found as the special case.

Second harmonic generation in graphene-coated nanowires.

We study second harmonic generation in a pair of graphene-coated nanowires. We show that the phase matching condition for harmonic generation can be engineered in a wide range of frequencies by tuning the spacing between graphene nanowires. We derive coupled mode equations describing the process of second harmonic generation using an unconjugated Lorentz reciprocity theorem. We show that the highest harmonic generation efficiency can be achieved by phase matching the fundamental mode with the two lowest order symmetric modes at the second harmonic frequency. Despite losses in graphene, we predict that the efficiency can be further enhanced by reducing the radius of nanowires due to larger mode overlap and lower propagation loss.

Solitary wave solutions of the cubic-quintic-septic nonlinear Schrödinger equation in fiber Bragg gratings

We consider the nonlinear coupled mode equations with competing cubic-quintic-septic nonlinearities, describing propagation of femtosecond optical pulses through fiber Bragg grating structure. By means of the multiple scale analysis, the model is reduced into the perturbed nonlinear Schrödinger equation with non-Kerr terms and higher-order dispersions. The complex envelope function ansatz of Li is adopted to investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the resulting nonlinear Schrödinger type equation. The physical parameters of the shape preserving solutions such as power and pulse width are calculated in terms of the dependent model coefficients. The constraint conditions for the existence of exact solutions are presented.

Impact of Kerr nonlinearity and stimulated Raman scattering on the whispering gallery modes of an optical microsphere

We investigated the impact of Kerr nonlinearity and stimulated Raman scattering on the whispering gallery modes of an optical microsphere. When two resonant modes are nonlinearly interacting due to the Kerr effect, a resonant mode undergoes a frequency shift that depends not only on its energy, but also on the energy of the other mode. A mathematical theory is developed taking into account the tensorial nature of the third-order susceptibility and vector nature of whispering gallery modes. The technique is extended to include stimulated Raman scattering, and coupled mode equations are developed to describe the energy transfer between the pump and Stokes waves without ignoring self-phase modulation and cross-phase modulation. The analytical model is found to be in good agreement with the numerical results in situations where the pump depletion is negligible.