Coupled-mode Equations Research Articles

Modulation instabilities in nonlinear two-core optical fibers with fourth order dispersion

We study in detail the modulation instabilities (MIs) in nonlinear two-core optical fibers (TCFs) in the presence of fourth-order dispersion (FOD) by considering the symmetric and antisymmetric CW states. The analytical instability gain spectra with the FOD are shown explicitly. For the symmetric CW state, in the normal (anomalous) dispersion regime, the positive (negative) FOD reduces the original MI band, and the negative (positive) FOD can generate a completely new dominate MI band in the very high frequency range even with quite small amplitude of FOD for typical TCF parameters. For the antisymmetric CW state, in the normal dispersion regime, there is no MI without FOD. With the negative FOD, MI is generated at any given total powers. In the anomalous dispersion regime, being similar with the behaviors for symmetric CW state, the positive (negative) FOD effect is to reduce the original MI band, and the negative (positive) FOD can generate a completely new dominate MI band in the very high frequency range. We also verify our MI analysis by solving the coupled-mode equations numerically.

Theoretical investigations of power fluctuations statistics in Brillouin erbium-doped fiber lasers.

We present theoretical investigations including simulations and statistical analyses on the fluctuations of the temporal output power in a Brillouin erbium-doped fiber laser. The generation of even Stokes waves up to the 5th order is considered by solving coupled-mode equations including SBS and Kerr nonlinearities. It is demonstrated that by increasing the EDFA pump power and generating a few orders of Stokes waves in such a laser, there are strong power fluctuations and rogue events are expected. Transition from Gaussian-like to levy-like regime is described as the power is increased from threshold resulting in the initial Stokes wave generation to well beyond threshold generating 5 effective even Stokes waves. Accordingly, phase portraits confirm increasing fluctuations as a function of the power. It is also shown that at the SBS lasing threshold, the output signals have the maximum correlation over replicas in round trips, nevertheless by enhancing the power, the correlation diminishes, which results in a full symmetry breaking and the system radiates in a chaotic manner and exhibits random laser behavior.

Worst crosstalk analysis in heterogeneous and trench assisted heterogeneous multicore fiber for different core counts and layouts

In order to obtain appreciably low worst crosstalk level in high core count multicore fiber, heterogeneous MCF with different core configurations and layouts have been investigated analytically using coupled mode equation. The variations in worst crosstalk behavior with respect to the bending radius, core configurations, core layouts, and outer cladding thickness have been analyzed. Further, the core count dependent variations in worst crosstalk level, core pitch, and peak bending radius in different core layouts have been presented. It has been observed that the selection of core configuration, core layout, and outer cladding thickness have significant impact on the reduction in worst crosstalk level. For example, in 12-core normal-index heterogeneous MCF with SLS structure, the core Configuration 1 can provide the worst crosstalk level reduced by 7.09 dB and 16.92 dB in comparison to the other two core configurations, i.e., Configuration 2, and Configuration 3, respectively. Further, for the same core count with TA Configuration 1, SLS core layout can achieve the reduced worst crosstalk level in comparison with circular ORS and DRS, hexagonal ORS and DRS respectively by 24.62 dB, 25.86 dB, 28.87 dB and 30.64 dB.

Analysis of a Fractional-Order Wireless Power Transfer System

This brief presents a fractional coupled model of fractional-order wireless power transfer (FOWPT) system based on the fractional calculus. By applying coupled-mode theory (CMT), dynamic characteristics of fractional systems can be described clearly while avoiding complex circuit models. The theoretical simulation results indicate that the FOWPT system has better characteristics and greater design freedom. Furthermore, the fractional coupled model is in good agreement with the circuit model and experiment, which proves the correctness of the generalized fractional coupled mode equations. Therefore, the fractional coupled model provides a valuable tool for the further analysis of FOWPT systems and may be used in other fractional-order resonant circuits.

Explicit solutions of coupled-mode equations for multiple waveguide devices: The optical analogy of a three-state geometric phase

Coupled waveguides mediated by the engineered refractive indexes are the central devices in integrated optical system. Generically, the coupled-mode equations describing the optical transports in the multiple waveguide-coupled structure are relatively complexity and hard to be analytically solved. In this paper we first show that the waveguides with the nearest-neighbor interactions possess simply a SU(2) dynamical symmetry, and then the analytical solutions of the relevant coupled-mode equations can be easily obtained. As a consequence, by setting properly the coupling coefficients between the waveguides, the optical transports in these waveguides can be controlled and the relevant optics devices can be designed robustly. Specifically, we show how the famous geometric phase acquired by the cyclic evolution of a parameter-driven three-level quantum system can be analogously demonstrated with the classical optical pulses transporting in a three-waveguide structure with the transverse interactions. Hopefully, the optical analogue of geometric phase might be applied to design and fabricate various novel phase-modulated integrated devices.

Study on mode coupling characteristics of multimode magneto-optical fibers

The magneto-optic (MO) coupled mode equations (CMEs) of linearly polarized (LP) guided light in MO fibers are deduced according to the coupled mode perturbation theory. The MO-CMEs are very useful for designing magnetically tunable multimode fiber devices. The modal dependency of MO coupling coefficient is derived, and the theoretical results are consistent with the simulated data. Our research shows that, the influence of the inter-mode-group (MG) coupling on the intra-MG polarization conversion efficiency (PCE) can be greatly reduced by appropriately adjusting the applied magnetic field and optimizing the MO-fiber length; For the application of spatial mode converters, we can further improve the inter-MG mode conversion efficiency (MCE) by using the MO fiber with a larger specific Faraday rotation.

Distributed feedback lasing based on a negative-index metamaterial waveguide.

This Letter lays the foundation of a new type of distributed feedback (DFB) laser whose optical feedback is due to the evanescent coupling between an active positive-index material (PIM) waveguide and a lossy negative-index metamaterial (NIM) waveguide. Active PIM-NIM coupled-mode equations are presented and solved to characterize the dispersion relation, resonant optical gain, and lasing. The photonic bandgap of this grating-less DFB laser does not depend on a Bragg wavenumber, but depends on the difference between the wavenumbers of the PIM and NIM waveguides; controlling this wavenumber difference allows for single-mode lasing and, ultimately, single-mode broadband lasing.

Enhanced coupling coefficients of a thin-film waveguide grating with a double negative cover

We study the propagation of waves in waveguide gratings with a double negative cover. The waveguide grating is considered as a small perturbation to a perfect waveguide and the effects of index variations on the propagating modes in the grating region are studied. To describe the energy transfer between the modes of the original structure, two coupled-mode equations are derived using the perturbation method. General formulas for both longitudinal and transverse coupling coefficients of thin-film waveguide gratings are obtained, which can be used for both conventional and left-handed thin-film waveguide gratings. To emphasize the unusual properties of the left-handed waveguide gratings, we numerically obtain the coupling coefficients of waveguide gratings with a double negative cover and compare them with those of conventional waveguides. We show that the coupling coefficients of the left-handed waveguide gratings enhance more than 3 orders of magnitude. While the longitudinal coupling coefficient of a conventional waveguide is zero, we report that it is nonzero for a waveguide grating with a double negative cover. In particular, we show that for especial ratios of the substrate to cover permittivity, the coupling coefficient of the fundamental mode (m=0) only exists within a small frequency range.

Research on stability and uniformity of multi-wavelength Brillouin-erbium fiber laser based on FWM

In this paper, we experimentally and theoretically investigate the stability and uniformity of multi-wavelength Brillouin-erbium fiber laser based on four-wave mixing (FWM). Multi-wavelength generation is limited by cross-gain saturation between different wavelengths from homogeneous broadening of erbium-doped fiber. Modeling coupled-mode equation for FWM and analyzing the influence of FWM on multi-wavelength generation. 16 orders Brillouin Stokes (BS) waves with a wavelength spacing of 0.084 nm are obtained experimentally within a 3 dB bandwidth by the FWM effect in a 100 m highly nonlinear fiber (HNLF), power distribution of each laser is uniform and peak power fluctuation is less than 1.184 dB within a 10 dB bandwidth, the outputs exhibit good time stability and uniformity. Compared with the same configuration, the laser without HNLF has only 9 orders stable BS waves within 3 dB bandwidth, and high-order Stokes wave shows significant waveform jitter and power fluctuation.

Super Gaussian pulse generation in a passively mode-locked fiber laser

The generation of self-similar super Gaussian-like pulses from a passively mode-locked fiber laser is numerically demonstrated. The laser cavity utilizes a long period fiber grating as a pulse-shaping element. The model is based on the normalized complex Ginzburg-Landau equation and the nonlinear coupled mode equations of the grating. The numerical results show stable super Gaussian pulses with a linear frequency chirp, a 1.5 ps pulse duration and 2 nJ energy. The generated pulses are scalable and exhibits no bifurcation or instability as the gain is increased.

Mesoscopic Hamiltonian for Josephson traveling-wave parametric amplifiers

We present a theory describing parametric amplification in a Josephson junction embedded transmission line. We will focus on the process of four-wave mixing under the assumption of an undepleted pump. However, the approach taken is quite general, such that a different parametric process or the process under different assumptions is easily derived. First the classical theory of the coupled-mode equations as presented by O'Brien et al. [Phys. Rev. Lett. $\mathbf{113}:157001$] is shortly reviewed. Then a derivation of the full quantum theory is given using mesoscopic quantisation techniques in terms of discrete mode operators. This results in a Hamiltonian that describes the process of parametric amplification. We show that the coupled-mode equations can be derived from this Hamiltonian in the classical limit and elaborate on the validity of the theory.

Intensity dependent spectral properties of long-period fiber grating for applications in lightwave communication

Using nonlinear coupled mode equations, the spectral properties of long-period fiber grating are studied analytically taking into account the Kerr type optical nonlinearity in the medium. The nonlinear coupled-mode equations were solved for co-propagating core and cladding mode amplitudes and expressions for transmittivity and phase factor at high excitation intensity are obtained. It is observed that the resonance wavelength of grating shifts towards higher wavelength side with increasing excitation intensity. Also, the minimum transmittivity of the grating shows oscillating behavior with increasing grating length, whereas the phase of the transmitted wave increases monotonically with length at high input intensities.

Generalized coupled-mode formalism in reciprocal waveguides with gain, loss, anisotropy, or bianisotropy

In anisotropic or bianisotropic waveguides, the standard coupled mode theory fails due to the broken link between the forward and backward propagating modes, which together form the dual mode sets that are crucial in constructing couple mode equations. We generalize the coupled mode theory by treating the forward and backward propagating modes on the same footing via a generalized eigenvalue problem that is exactly equivalent to the waveguide Hamiltonian. The generalized eigenvalue problem is fully characterized by two operators, i.e., $( \bar{\bm{L}},\bar{\bm{B}})$, wherein $\bar{\bm{L}}$ is a self-adjoint differential operator, while $\bar{\bm{B}}$ is a constant antisymmetric operator. From the properties of $\bar{\bm{L}}$ and $\bar{\bm{B}}$, we establish the relation between the dual mode sets that are essential in constructing coupled mode equations in terms of forward and backward propagating modes. By perturbation, the generalized coupled mode equation can be derived in a natural way. Our generalized coupled mode formalism can be used to study the mode coupling in waveguides that may contain gain/loss, anisotropy or bianisotropy. We further illustrate how the generalized coupled theory can be used to study the modal coupling in anisotropy and bianisotropy waveguides through a few concrete examples.

Electrically tunable temporal imaging in a graphene-based waveguide

We propose an electrically tunable temporal imaging system (TIS) based on four-wave mixing in a dispersion engineered graphene-based waveguide, which could realize a magnification factor of 1000× for a signal consisting of two 100-fs-wide pulses separated by 500 fs and a large working bandwidth of about 700 nm. The TIS was analyzed by solving the couple-mode equations in detail. It was demonstrated that the working wavelength range could be tuned via a small disturbed bias voltage applied to the graphene layer without changing the geometric structure of the waveguide. These results provide attractive insights for potential applications in integrated optics and optical communications.

Description of glass transition kinetics in 3D [formula omitted] model in terms of gauge field theory

A gauge theory of the glass transition in the frustrated XY model being simplest model containing topologically nontrivial excitations is considered. Investigation of the transition kinetics of this system finds that the three-dimensional system exhibits the Vogel–Fulcher–Tammann criticality going with its freezing into a spin glass. It is analytically shown that the system demonstrates all glass transition properties, like the logarithmic relaxation, and corresponding behaviour of linear and non-linear susceptibility. The mode-coupling theory equation in the Zwanziger–Mori representation also is derived in framework of our approach. The findings provide insights into the topological origin of glass formation, that allows to make progress in understanding glass-transition processes in more intricate systems.

Static and dynamic mode instabilities in dual-core fiber amplifiers

This paper provides a detailed derivation of coupled-mode equations for thermo-optic nonlinear effects in dual-core fiber amplifiers. The equations predict both static and dynamic modal deformations depending on amplifier design. The prediction of static deformations is confirmed by nonlinear beam-propagation simulations. The dependencies of instabilities and their thresholds on launch conditions are analyzed by numerical simulations and analytical arguments. It is shown that the output stability properties are strongly dependent on the relative phase of the input in the two cores. The instability power threshold for dual-core amplifiers with strongly coupled cores are found to be lower than for a comparable single-core amplifier. However, as the core separation is increased, the dual-core amplifier threshold rapidly increases when light is amplified in the odd supermode.

Analytical method for four wave mixing in space-frequency multiplexing optical fibers

In recent years, the transmission capacity of wavelength division multiplexing (WDM) communication systems has gradually approached to the nonlinear Shannon limit. To meet the increasing demand for communication capacity, space division multiplexing (SDM) has become one of the most concerned technologies. In this paper, the four-wave mixing process (FWM) in fibers is considered from the frequency domain to the mode division multiplexing (MDM) spatial domain under pump depletion and the exact analytical solution to the FWM coupled-mode equations in the space-frequency domain is in detail deduced. The analytical method is verified by numerically calculating the amplitude and phase evolution of the idler wave in non-degenerate four-wave mixing. We discuss three new applications of the analytical solution as follows. 1) Using the phase matching condition we select the terms in the multi-wave coupling equation, and only retain the coupling term that plays a major role. According to the analytical solution in this paper, the phase matching percentage parameter is introduced to determine the FWM coupling terms necessary for multi-wave coupling equations, thus simplifying the multi-wave coupling problem in the study. 2) Combining the analytical solution with the numerical calculation results, we find the initial phase relationship between the output idler and the input guided wave for phase-insensitive FWM, and we provide the analytical expression for a theoretical basis to efficiently design the FWM-based phase arithmetic devices in parallel operating at WDM and MDM systems. 3) We propose a nonlinear compensation algorithm based on analytical solution, which can be used in the few-mode transmission system. The algorithm can fast evaluate or compensate for the fiber nonlinearity by taking into account the pump depletion of the FWM effect. Compared with the traditional digital back propagation (DBP) algorithm, our algorithm has the advantage of lower complexity.

Relative intensity noise in a multi-Stokes Brillouin laser.

We investigate the relative intensity noise (RIN) properties of a multi-Stokes Brillouin fiber ring laser. We experimentally analyse the intensity noise of each Stokes wave and study the noise dynamics of the cascaded Brillouin scattering process. We observe up to 20 dB/Hz intensity noise reduction compared to that of the RIN input pump laser. We examine the impact of the fiber ring quality factor on the laser RIN features such as amplitude reduction and relaxation frequency. We also investigate the influence of the Brillouin gain detuning on the RIN reduction. A numerical model based on a set of coupled-mode equations replicates the experimental observations. Our study enables us to determine the optimal parameter values to operate the multi-Stokes laserin the low noise regime.

Resistive drift instabilities for thermal and non-thermal electron distributions in electron-ion plasma

Local dispersion relations for resistive drift mode in a nonuniform magnetize plasma are derived for thermal and non-thermal distribution of electrons. The coupled mode equations are obtained by using Braginskii's transport equations for ions and electrons with thermal as well as non-thermal (Cairns and kappa) distribution for electrons. The dispersion relations are then analyzed both analytically as well as numerically for all distributions. It is found that growth rate is highest for Maxwellian, Intermediate for kappa and lowest for Cairns distribution. It has been found that increasing values of Γ (which estimate population of non-thermal electrons) for Cairn distributed electrons are able to stabilize the mode. Furthermore, increasing the values of κ (which is spectral index) for the kappa distributed electrons have destabilizing effects on the mode. The result might be useful in the interpretation of electromagnetic fluctuations in nonuniform magneto-plasma in which resistivity is a key element in calculation of drift instabilities in the presence of thermal or nonthermal electron distributions, such systems are extensively observed in laboratory as well as space plasma.

Backward Third-Harmonic Pulse Generation in a One-Dimensional PIM/NIM Structure

In this work, a set of couple-mode equations for describing a backward third harmonic generation (BTHG) in a one-dimensional periodic structure of positive-index material (PIM) layers and third-order nonlinear negative-index material (NIM) layers was analysed using multiple-scale method. Due to the negative-index phase matching and band-edge field enhancement the intensities of backward third-harmonic pulsed generated from the PIM/NIM periodic structure is increased for 100 time of those generated from a single NIM medium.