Coupled-mode Equations Research Articles

A novel surrounding refractive index sensor based on the polarization characteristics of thin-cladding long-period fiber grating

In this paper, a new method to measure the surrounding refractive index (SRI) is proposed. It is based on the core-only photoinduced birefringence of long-period fiber grating (LPFG). The relationship between Stokes parameters of output light polarization states and SRI is analyzed using full-vector coupled mode equations. Compared to the conventional LPFG SRI sensors, this sensing method features a fairly good linearity performance and reasonably high resolution over a much broader SRI measurement range from 1 to 1.46 which may be applied to the humidity and vapor sensing.

Directional Phase-Matched Interaction in the Saturated Parametric Mixer

We study the nonreciprocal nature of the saturated parametric amplifier response under different pump wavelengths and power levels. The coupled-mode equations are first employed to gain more insight into the origin of directional phase-matched response. We next characterize pump depletion bandwidth and signal induced pump depletion level to identify the optimum pump wavelength and propagation direction. The nonreciprocal behavior was clearly observed at all of the examined pump wavelengths and power levels.

Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires.

Semiconductor nanowires supporting leaky mode resonances have been used to increase light absorption in optoelectronic applications from solar cell to photodetector and sensor. The light conventionally illuminates these devices with a wide range of different incident angles from half space. Currently, most of the investigated nanowires have centrosymmetric geometry cross section, such as circle, hexagon, and rectangle. Here we show that the absorption capability of these symmetrical nanowires has an upper limit under the half-space illumination. Based on the temporal coupled-mode equation, we develop a reciprocity theory for leaky mode resonances in order to connect the angle-dependent absorption cross section and the radiation pattern. We show that in order to exceed such a half-space limit the radiation pattern should be noncentrosymmetric and dominate in the direction reciprocal to the illumination. As an example, we design a metal trough structure to achieve the desired radiation pattern for an embedded nanowire. In comparison to a single nanowire case the trough structure indeed overcomes the half-space limit and leads to 39% and 64% absorption enhancement in TM and TE polarizations, respectively. Also the trough structure enables the enhancement over a broad wavelength range.

Experimental evaluation of nonlinear crosstalk in multi-core fiber.

In this paper we evaluate experimentally and model theoretically the nonlinear crosstalk random process in multi-core fiber. The experimental results indicate that mode coupling in multi-core fibers is reduced in presence of fiber Kerr nonlinearities. An analytical study of the inter-core crosstalk probability density function in nonlinear regime is performed, where the theoretical distribution, derived from the nonlinear coupled-mode equation, is experimentally validated in homogeneous four-core fiber. The herein presented analysis includes the evaluation of the inter-core crosstalk probability density function, mean and variance evolution considering the optical power launched into the fiber.

Coupled Mode Theory Applied to Resonators in the Presence of Conductors

Using the method of images, Energy Coupled Mode Theory (ECMT), a coupled mode equation in the frequency domain, is extended to deal with important cases where resonators are in close proximity to conducting surfaces. Depending on the type of conductors and the orientation of the resonators, the method of images determines the relative phases of the images. Using the formed images, the coupled frequencies and fields can be determined by applying ECMT. Two cases are studied. In the first case, it is shown that a dielectric resonator inserted in a cavity couples with both the mirror image and the cavity. The frequency behavior is described by the interaction with the image which counteracts that with the cavity. The second case is that of resonators sandwiched between conducting plates. It is shown that an infinite array of stacked images is formed. The coupling of the resonator with its images determines the coupled frequencies and fields. In this context, the main advantage of ECMT is its ability to separate the effects of the walls from the uncoupled system. This means that the system parameters are independent of the separation distances and/or the type of conductors, which renders the post processing analyses easier and predictable. Provided that the behaviors of the uncoupled resonators are known, ECMT is general and can be applied to more complex systems.

A Geometrical Perspective on the Coherent Multimode Optical Field and Mode Coupling Equations

The generalization of the Poincare sphere to $N \ge 2$ modes is the ( $N-1$ )-dimensional complex projective space CP( $N-1$ ). There is a minimal set of $2N-2$ Stokes vector components that determine the coherent multimode optical field. These are obtained from the inverse stereographic projection of coordinate hyperplanes in CP( $N-1$ ) into a $2N-2$ sphere, just as in the $N=2$ case. We derive $N$ -mode analogs of Poole’s optical fiber polarization-mode dispersion (PMD) equations that involve only $2N-2$ independent variables. This is achieved by means of an explicit generalized coherent state representation of the optical field, which enables the components of the PMD vector to be expressed in terms of the optical state and its frequency derivatives. Poole’s equations describe mode coupling as a flow on CP( $N-1$ ). We give general constraints on the mode-coupling matrix and Stokes vector components. The group delay operator is shown to be a rank-2 perturbation of a diagonal matrix.

Optimization of one-third harmonic generation in the presence of nonlinear phase modulations and power attenuation

We investigate the one-third harmonic generation (OTHG) in optical microfibers with power attenuation considered by analytically analyzing and numerically solving the coupled mode equations (CMEs). Both the strength and effective length of signal power growing in nonlinear media, which are extremely sensitive to the relative phase between the interaction waves, contribute to the final conversion efficiency. The relative phase and its evolution along the propagating direction play crucial roles in highly efficient OTHG. In order to obtain high conversion efficiency, the general expressions of optical threshold conditions are derived and discussed for choosing proper initial parameters. Numerically simulations are performed with both partial and absolute phase matching, which are corresponding to the microfibers with uniform and non-uniform diameters, respectively. Optimizations of relative phase and phase compensation are suggested by the simulations and provide significant enhancement of conversion efficiencies.

All-optical ternary signal processing using uniform nonlinear chalcogenide fiber Bragg gratings

The fiber Bragg grating (FBG) has a huge capacity in optical communication, especially for showing optical multistability, which can be used in optical switching and optical memories. In this paper, we numerically solve the coupled mode equations governing the uniform FBG and obtain the output shapes of the bright and dark solitons as input pulses with different amplitudes. This paper shows that we can reach different output shapes using identical input shapes with different amplitudes. Hence we can control the output light waveform using light as the controlling input. For signal processing purposes, it is made clear that the input pulse characteristics, such as amplitude and shape, are distinguishable applying only the output shape. Also, we can predict the length of the FBG. The deliverables of this article will facilitate generating light pulses with application to optical fiber digital communication systems.

Ab initiomultimode linewidth theory for arbitrary inhomogeneous laser cavities

We present a multimode laser-linewidth theory for arbitrary cavity structures and geometries that contains nearly all previously known effects and also finds new nonlinear and multimode corrections, e.g. a bad-cavity correction to the Henry $\alpha$ factor and a multimode Schawlow--Townes relation (each linewidth is proportional to a sum of inverse powers of all lasing modes). Our theory produces a quantitatively accurate formula for the linewidth, with no free parameters, including the full spatial degrees of freedom of the system. Starting with the Maxwell--Bloch equations, we handle quantum and thermal noise by introducing random currents whose correlations are given by the fluctuation--dissipation theorem. We derive coupled-mode equations for the lasing-mode amplitudes and obtain a formula for the linewidths in terms of simple integrals over the steady-state lasing modes.

Nonlinear light propagation in cholesteric liquid crystals with a helical Bragg microstructure

Nonlinear optical propagation in cholesteric liquid crystals (CLC) with a spatially periodic helical molecular structure is studied experimentally and modeled numerically. This periodic structure can be seen as a Bragg grating with a propagation stopband for circularly polarized light. The CLC nonlinearity can be strengthened by adding absorption dye, thus reducing the nonlinear intensity threshold and the necessary propagation length. As the input power increases, a blue shift of the stopband is induced by the self-defocusing nonlinearity, leading to a substantial enhancement of the transmission and spreading of the beam. With further increase of the input power, the self-defocusing nonlinearity saturates, and the beam propagates as in the linear-diffraction regime. A system of nonlinear couple-mode equations is used to describe the propagation of the beam. Numerical results agree well with the experiment findings, suggesting that modulation of intensity and spatial profile of the beam can be achieved simultaneously under low input intensities in a compact CLC-based micro-device.

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.

Magnetically-controllable optical multi-stability in magneto-optic fiber Bragg gratings with potential applications to multi-level all-optical regeneration

Starting with the nonlinear coupled-mode equations of guided optical waves in the magneto-optic fiber Bragg grating (MFBG), the amplitude transfer curve of the transmitted light is numerically calculated for the incident right-circularly polarized wave, and the multi-stability is analyzed by introducing the parameter of jitter suppression. It is shown that, (i) the performance of amplitude jitter suppression in the stable states of high level is better than that of low level; (ii) the jitter suppression in the multi-stable regions can be enhanced when the magnetic field is applied to the MFBG in the opposite direction of the incident wave; and (iii) by adjusting the applied magnetic field, the multi-stable levels can be tuned flexibly, which is helpful for developing the intelligent all-optical devices for multilevel regeneration.

Reconfigurable optical-force-drive chirp and delay line in micro- or nanofiber Bragg grating

The emergence of optical micro/nano-fiber (MNF) with a subwavelength diameter, which has ultra-light mass and an intense light field, brings an opportunity for develop fiber based optomechanical systems. In this study, we theoretically show an optomechanical effect in silica MNF Bragg gratings (MNFBGs). The light-induced mechanical effect results in continuously distributed strain along the grating. It is shown that the power-related strain introduces an optically reconfigurable chirp in the grating period. We develop new optomechanical coupled-mode equations and theoretically analyze the influence of the optical-force-induced nonlinearity and chirp on the grating performance. Compared with weak Kerr effect, the optomechanics effect dominated in the properties evolution of MNFBGs and significant group velocity reduction and switching effect have been theoretically demonstrated at medium power level. This kind of optomechanical MNFBG with optically reconfigurable chirp may offer a path toward all-optical tunable bandwidth of Bragg resonance and may lead to useful applications such as all-optical switching and optically controlled dispersion and slow/fast light.

Coupled-mode Analysis of Plasmonic MIM Waveguide Coupled with a Resonant Cavity

The coupling coefficients of two parallel metal-insulator-metal (MIM) plasmonic waveguides are investigated using the analysis of the coupled-mode equations. The frequency characteristics of the power transmittance of a MIM waveguide coupled with a resonant cavity are studied and compared with the simulation results obtained from the FDTD method into which motion equations of free electrons are installed.

Loss-compensated nonlinear modes and symmetry breaking in amplifying metal-dielectric-metal plasmonic couplers

We theoretically investigate the propagation of surface plasmon polaritons in an amplifying plasmonic coupler (metal--amplifying dielectric--metal). We study the loss-compensated nonlinear stationary modes of the system by deriving coupled-mode equations for the optical amplitudes, predicting the existence of a mode with broken symmetry for gain values higher than a characteristic threshold. We analyze the stability of symmetric, antisymmetric, and nonsymmetric modes by solving the linearized system for small perturbations and by numerically integrating coupled-mode equations in propagation. We find that, while the antisymmetric mode stays always stable or marginally stable, the stability of symmetric and nonsymmetric modes is more involved.

Crosstalk analysis in homogeneous multi-core two-mode fiber under bent condition.

We analyze the inter-core crosstalk in homogeneous multi-core two-mode fibers (MC-TMFs) under bent condition by using the coupled-mode equations. In particular, we investigate the effects of the intra-core mode coupling on the inter-core crosstalk for two different types of MC-TMFs at various bending radii. The results show that the inter-core homo-mode crosstalk of LP(11) mode is dominant under the gentle fiber bending condition due to its large effective area. However, as the fiber bending becomes tight, the intra-core mode coupling is significantly enhanced and consequently makes all the inter-core crosstalk levels comparable to each other regardless of the mode. A similar tendency is observed at a reduced bending radius when the difference in the propagation constants between modes is large and core pitch is small.

Ray-based description of mode coupling by sound speed fluctuations in the ocean.

A traditional approach to the analysis of mode coupling in a fluctuating underwater waveguide is based on solving the system of coupled equations for the second statistical moments of mode amplitudes derived in the Markov approximation [D. B. Creamer, J. Acoust. Soc. Am. 99, 2825-2838 (1996)]. In the present work, an alternative approach is considered. It is based on an analytic solution of the mode coupling equation derived in the high frequency approximation [A. L. Virovlyanskii and A. G. Kosterin, Sov. Phys. Acoust. 35, 138-142 (1987)]. This solution, representing the mode amplitude as a sum of contributions from two geometrical rays, is convenient for statistical averaging. It allows one to easily derive analytical expressions for any statistical moments of mode amplitudes. The applicability of this approach is demonstrated by comparing its predictions for a deep water acoustic waveguide with results of a full wave numerical simulation carried out using the method of wide angle parabolic equation.

Supersymmetric Bragg gratings

The supersymmetric (SUSY) structure of coupled-mode equations that describe scattering of optical waves in one-dimensional Bragg gratings is highlighted. This property can find applications to the synthesis of special Bragg filters and distributed-feedback (DFB) optical cavities. In particular, multiple SUSY (Darboux–Crum) transformations can be used to synthesize DFB filters with any desired number of resonances at target frequencies. As an example, we describe the design of a DFB structure with a set of equally-spaced resonances, i.e. a frequency comb transmission filter.

Three-mode coupling of symmetric and antisymmetric Lamb waves in plates with finite corrugations.

Coupled-mode equations governing the amplitudes of the higher-order symmetric Lamb modes S1 and S2 with the antisymmetric mode A2 in an infinite elastic plate with sinusoidal surface corrugation over a finite length are obtained via multiple-scales analysis. This phenomenon of threemode coupling is observed when the wavenumbers k(s1) and k(s2) of the symmetric modes and k(A2) of the antisymmetric mode satisfy the simultaneous resonance conditions k(s1) - k(A2) = k(w) and k(A2) - k(s2) = k(w), where k(w) is the wavenumber of the sinusoidal corrugation. Near resonance, the coupled amplitude equations are solved exactly as an initial-value problem and it is seen that the modes are transmitted through the grating without reflection. Complete conversion from the symmetric modes into the antisymmetric mode is observed at periodic intervals along the grating when the resonance conditions are exactly satisfied. The effect of detuning away from resonance also shows propagation without reflection with periodic energy exchange. In the latter case, the modes couple without complete conversion. This phenomenon of mode conversion is confirmed by the results of an experiment on an aluminum plate with a triangular grating excited with the S2 symmetric mode at 2.7 MHz.

Dirac solitons in square binary waveguide lattices

We study optical analogs of two-dimensional (2D) Dirac solitons in square binary waveguide lattices with two different topologies in the presence of Kerr nonlinearity. These 2D solitons turn out to be quite robust. We demonstrate that with the found 2D solitons, the coupled mode equations governing light dynamics in square binary waveguide lattices can be converted into the nonlinear relativistic 2D Dirac equation with the four-component bispinor. This paves the way for using binary waveguide lattices as a classical simulator of quantum nonlinear effects arising from the 2D Dirac equation, something that is thought to be impossible to achieve in conventional (i.e., linear) quantum field theory.