Hybrid mode phase matching and effective nonlinear coefficient forsecond harmonic generation in biaxial waveguides

Phase matching between fundamental and second harmonic hybrid guided modes in nonlinear biaxial planar waveguides is determined for the first time based on principal axes orientations of the film, substrate, and cover. We have derived an expression for the effective nonlinear coefficient D_eff of hybrid modes in a biaxial waveguide and have calculated this coefficient numerically along the effective index phase match loci for these modes supported by a POM waveguide surrounded by isotropic media. The optimum phase matching orientations are determined for a fixed waveguide thickness. Dispersion of the principal indexes is considered     

A Novel Configuration for Phase-Matched Second-Harmonic Generation in LiNbO3 Waveguides

We present a feasibility study of a novel technique for continuously phase-matched second-harmonic generation in proton-exchanged lithium niobate waveguides. We demonstrate that a periodic variation of the nonlinear coefficient along the transverse direction with respect to the propagation one permits for efficient conversions. Comparison with the conventional first-order quasi-phase-matching interactions are then presented in front of unavoidable imperfections of real devices whenever Czochralski off-center technique is exploited in the fabrication process     

Optically nonlinear s-polarized waves in dielectric superlattices

We study theoretically the s-polarized electromagnetic waves in two-component superlattices, in which one component has a Kerr-type nonlinear dielectric constant. Numerical results are obtained for the effective refractive index (and hence the dispersion relation) of these modes as a function of an effective nonlinear coefficient and the geometric parameters of the structure. Attention is focused on determining both edges of the nonlinear wave subbands, which are shown to form a rich spectrum in these periodic structures. A more than double-valued relation between the effective refractive index and the electric field intensity is obtained in some cases, contrasting with the situation in structures having a single nonlinear layer. This property offers a potential application to non resonant multi stable devices     

Coupling of periodic modes

The author examines the effect of a periodic perturbation on a periodic structure, taking linear twin-core couplers with an axially periodic refractive index as examples. The focus is on the coupling effect of the normal periodic modes of a periodic structure under a sinusoidal perturbation, since any periodic function can be represented by a Fourier series. A perturbation to an ideal waveguide structure, is shown to lead to coupling of the normal modes of the structure. For an ideal periodic structure (such as a twin-core nonlinear coupler at a fixed excitation and power), a periodic perturbation to the structure gives rise to coupling of the normal periodic modes of the structure     

Quasi-phase-matched second harmonic generation: tuning andtolerances

The theory of quasi-phase-matched second-harmonic generation is presented in both the space domain and the wave vector mismatch domain. Departures from ideal quasi-phase matching in periodicity, wavelength, angle of propagation, and temperature are examined to determine the tuning properties and acceptance bandwidths for second-harmonic generation in periodic structures. Numerical examples are tabulated for periodically poled lithium niobate. Various types of errors in the periodicity of these structures are then analyzed to find their effects on the conversion efficiency and on the shape of the tuning curve. This analysis is useful for establishing fabrication tolerances for practical quasi-phase-matched devices. A method of designing structures having desired phase-matching tuning curve shapes is also described. The method makes use of varying domain lengths to establish a varying effective nonlinear coefficient along the interaction length     

Enhanced transmission through two-period arrays of subwavelength holes

A two-period array of subwavelength holes in a perfect electrically conducting plate permits enhanced transmission of incident plane waves with wavenumbers near those of leaky waves supported by the array. A simple periodic impedance sheet model explains this phenomenon and provides an approximate expression for the transmission peak location. Transmission through thin perforated plates is studied using a full-wave solver to verify model-theoretic observations. Transmission coefficient peak splitting and narrowing phenomena, associated with thick perforated plates, are explored.     

Shaping of flat-topped element patterns in an array of slow-wavestrip structures excited by parallel-plate waveguides

A two-dimensional (2-D) model of a corrugated-rod antenna array in the form of an infinite periodic array of finite slow-wave strip structures arranged above the apertures of two-or three-mode parallel-plate waveguides excited by TEM modes is analyzed. The formulated problem is solved by the method of matching the waveguide and Floquet modes on the waveguide apertures in combination with the Galerkin method for calculation of the currents on the strips. The geometrical parameters of the strip structures are optimized for shaping the flat-topped element patterns. It is shown that high quality of the flat-topped patterns in the element spacing range from 0.65 to 1.3 of wavelength is achieved if the excitation of the optimized structures is performed using the second and third reflected waveguide modes returned back to the apertures with the corresponding optimum phases. The presented examples also show that the shape of the obtained patterns keeps well at least in a 10% frequency band     

Intermodal coupling by periodic microbending in dual-corefibers-comparison of experiment and theory

Grating coupling (caused by periodic microbending) between the quasi-even and quasi-odd normal modes of dual-core fibers is treated, and an accurate design formula is derived for the coupling coefficient. An analytical form for the intrinsic coupling coefficient of two dissimilar cores is also given. Good agreement is found between theory and experimental data on fibers perturbed either by acoustic flexural waves or fixed mechanical gratings. The paper provides useful design rules for dual-core fiber devices     

六角形结构光子晶体光纤的非线性效应研究

本文根据全矢量模型对六角形光子晶体光纤(PCF)的各种结构参数对基模非线性系数的影响进行数值研究,数值研究发现,增大空气孔的直径d或者减小空气孔的间距Λ都会使非线性系数增大,这将对设计不同要求的非线性微结构光纤提供有效的理论依据.     

Determination of bend mode characteristics in dielectric waveguides

A new and effective method for the electromagnetic analysis of generic bent dielectric waveguides is presented. The method is based on the expansion of the bend mode in modes of the straight waveguide and permits determination of the shape and the phase constant of the fundamental mode of the bend waveguide with great accuracy at a very low computational cost. Simple analytical expressions of the phase constant, coupling losses, and bending-induced birefringence are derived under very general conditions. The proposed method is useful for the design and optimization of each bent section of integrated optics components     

Generalized cable equation model for myelinated nerve fiber

Herein, the well-known cable equation for nonmyelinated axon model is extended analytically for myelinated axon formulation. The myelinated membrane conductivity is represented via the Fourier series expansion. The classical cable equation is thereby modified into a linear second order ordinary differential equation with periodic coefficients, known as Hill's equation. The general internal source response, expressed via repeated convolutions, uniformly converges provided that the entire periodic membrane is passive. The solution can be interpreted as an extended source response in an equivalent nonmyelinated axon (i.e., the response is governed by the classical cable equation). The extended source consists of the original source and a novel activation function, replacing the periodic membrane in the myelinated axon model. Hill's equation is explicitly integrated for the specific choice of piecewise constant membrane conductivity profile, thereby resulting in an explicit closed form expression for the transmembrane potential in terms of trigonometric functions. The Floquet's modes are recognized as the nerve fiber activation modes, which are conventionally associated with the nonlinear Hodgkin-Huxley formulation. They can also be incorporated in our linear model, provided that the periodic membrane point-wise passivity constraint is properly modified. Indeed, the modified condition, enforcing the periodic membrane passivity constraint on the average conductivity only leads, for the first time, to the inclusion of the nerve fiber activation modes in our novel model. The validity of the generalized transmission-line and cable equation models for a myelinated nerve fiber, is verified herein through a rigorous Green's function formulation and numerical simulations for transmembrane potential induced in three-dimensional myelinated cylindrical cell. It is shown that the dominant pole contribution of the exact modal expansion is the transmembrane potential solution of our generalized model.     

A New Formulation of Coupled Propagation Equations in Periodic Nanophotonic Waveguides for the Treatment of Kerr-Induced Nonlinearities

Nonlinear phenomena could be used to implement important signal processing functionalities in future nanophotonic integrated optical devices. In this paper, a semi-analytical model incorporating the influence of Kerr-induced nonlinearity in the propagation of an optical signal inside a periodic nanophotonic waveguide is derived. The approach consists of a system of nonlinear coupled mode propagation equations and is applicable to both single and multimode waveguides. The influence of the mode group velocity on the value of the self-phase modulation coefficient gamma is analyzed and the impact of higher order nonlinear terms is also investigated both at the middle and edge of the guided band. The model is also applied to estimate the nonlinear coupling coefficients of a photonic crystal waveguide coupler and provides an efficient method to analyze the influence of nonlinear phenomena in periodic nanophotonic waveguide devices.     

Modal analysis of nonlinear propagation in dielectric slabwaveguide

In this paper we present a study of nonlinear guided propagation based on the development of the actual electromagnetic field in terms of the modes of a linear dielectric guide. This approach permits in principle to consider propagation over finite lengths of waveguide of general nonuniform nonlinearity, with arbitrary excitation at a finite location and to retain the second derivative term that is neglected in the usual Schrodinger equation approximation to the wave equation     

Correlation between mode fluctuations in a multilongitudinal mode laser

A theoretical analysis of the correlation between mode fluctuations in a multilongitudinal mode laser is presented. The rate equations driven by Langevin noise sources are used to derive the correlation coefficients. By taking nonlinear terms into account, this analysis also applies to lasers with few modes and permits the derivation of the correlation coefficients from the the amplitude of these modes. They are found to be different from one pair of modes to another and to have not only negative, but also positive values. The fluctuations of a given mode are found to be concentrated in the low-frequency range, and the associated time constants are of great importance in the estimation of the partition noise for a signal propagating through a dispersive fiber.     

Static and dynamic potential integrals for linearly varying source distributions in two- and three-dimensional problems

A technique for the evaluation of static and dynamic potentials due to source distributions defined on domains with simple shape is presented. The domains considered are polyhedral regions and, in two-dimensional problems, plane polygons, on which uniform or linearly varying source distributions are defined. It is shown how three-dimensional (two-dimensional) potential integrals are always reducible to surface (line) integrals with nonsingular kernel, by use of a nonlinear transformation for the integration variables that permits analytic integration. In the static case the integration on the boundary is performed analytically and closed form results are given. In the dynamic case the expressions of the boundary integrals are given in a form suitable for numerical integration. The use of matrix notation allows for very compact expressions readily translatable into computer programs.     

Analysis and design of periodic leaky-wave antennas for the millimeter waveband in hybrid waveguide-planar technology

This work presents a full-wave integral equation approach specifically conceived for the analysis and design of laterally-shielded rectangular dielectric waveguides, periodically loaded with planar perturbations of rectangular shape. This type of open periodic waveguide supports the propagation of leaky-wave modes, which can be used to build leaky-wave antennas which exhibit many desirable features for millimeter waveband applications. The particularities of the leaky-mode analysis theory are described in this paper, and comparisons with other methods are presented for validation purposes. Using this leaky-mode analysis method, a novel periodic leaky-wave antenna is presented and designed. This novel antenna shows some important improvements with respect to the features of previously proposed antennas. The results of the designed radiation patterns are validated with three-dimensional electromagnetic analysis using commercial software.     

Transmission Distortion in Multimode Random Waveguides

We consider the coupled line equations for two-mode random media in which both modes travel in the same (forward) direction as a model for multimode millimeter waveguides and optical fibers, in which mode conversion at imperfections occurs primarily in the forward direction. Some exact general properties satisfied by the transfer function and the impulse response of such a system are given for an arbitrary coupling coefficient. A random stationary coupling coefficient with statistically independent successive values, and consequently a white spectrum (e.g., a white Gaussian or a Poisson noise), permits exact determination of transmission statistics; we obtain first- and second-order statistics in the time and frequency domains. No perturbation or other approximations are made in any of the above results, which are obtained directly from the coupled line equations. These results are used to study signal distortion in long guides. By straightforward extension of this work more complicated calculations can treat more forward modes, but not backward modes or nonwhite coupling coefficient spectra. In this paper the coupling coefficient is assumed frequency independent, and under certain conditions the signal distortion decreases as the mode conversion increases. In practical cases the coupling coefficients are frequency dependent and the above behavior is modified; the present work is extended to this important case in a companion paper.     

Multiple-wavelength quasi-phase-matched nonlinear interactions

Quasi-phase-matching allows one to arbitrarily phase match a single interaction by periodic modulation of the material nonlinear coefficient. A partial extension is obtained by Fibonacci-based quasi-periodic modulation of the nonlinear coefficient. These Fibonacci-based structures allow for simultaneously phase matching two interactions, provided that their wavevector mismatch ratio obeys selection rules, which are governed by the golden ratio τ=(1+√5)/2. In this paper, we present a novel method for simultaneously phase matching any two nonlinear interactions by general quasi-periodic modulation of the nonlinear coefficient. These quasi-periodic structures, which also include the Fibonacci-based structures as a subgroup, provide greater design flexibility. Our method can be useful for various nonlinear devices, such as multiple-peak frequency doublers, frequency triplers, and frequency quadruplers. We show for two specific devices that similar efficiency, compared to a cascaded device, can be obtained. Furthermore, in contrast to some cascaded devices, these structures can be used in double-pass and standing-wave configurations, since they operate with the same efficiency in both directions of propagation     

光在一维光子晶体中的传播特性分析

利用转移矩阵法分析具有线性和非线性缺陷层的一维光子晶体的传递函数并进行了数值模拟。由于缺陷层对周期性介质层的周期性破坏,带隙中产生了缺陷模。通过模拟仿真的结果,发现多缺陷层和非线性层都能够使带隙展宽,变得更加平坦。相比较之下,非线性层缺陷对带隙的作用更加强烈。     

功率平衡基础上的基波分析法

在非线性网络的适当端口施加正弦电压源U_s.用基波平衡原理求得注入网络的基波电流i_s1.当i_s1=0时,正弦电压源的频率和幅值被称为基波解.如果能求出n个基波解,则原网络必然也对应的存在有n个周期解.如果基波解不存在,则原网络的周期解也不存在.以考比兹电路为例,说明虚功平衡和周期振荡的密切关系,以蔡氏电路为例,用基波平衡原理说明混沌振荡的性状.结论的普遍性可以推广到三阶非线性微分方程.其正确性可以用SIMULINK仿真验证.