Values Of Propagation Constant Research Articles

The formation and stability of single-peak and multi-peak gap nematicons in a periodic potential

ABSTRACT The formation and stability of single-peak and multi-peak gap nematicons in a periodic potential are investigated using numerical methods. Both the stationary solutions and the dynamic solutions have been studied in the first band gap. Single-peak and multi-peak gap nematicons can be found in the first band gap. The single-peak gap nematicons are obtained for a wide range of propagation constants. Multi-peak gap nematicons can exist only above a certain minimum value of propagation constant. The intensity distribution among various peaks in a multi-peak gap nematicon highly depends on input beam intensity. The stability of the stationary solution against small perturbations has been studied using Bogoliubov-de Gennes equations. Single-peak gap nematicons are stable. The multi-peak gap nematicons having the same amplitudes for various peaks are stable, whereas those with unequal amplitudes among peaks are unstable.

Dynamics of stable solitons in Complex Ginzburg–Landau equation with [formula omitted]-symmetric Gaussian potential

In this paper, we investigate the stable propagation of soliton in Complex Ginzburg–Landau (CGL) equation with self focusing nonlinear mode in the presence of PT-symmetric Gaussian potential. In our model, we find the required condition to obtain the stable solution is that the value of spectral filtering is negative and the positive values of diffraction, Kerr nonlinearity and nonlinear gain/loss. By satisfying this condition, one can manipulate the stable propagation, intensity and power conservation of the soliton by simply varying the strength of the imaginary part of the complex PT-symmetric potential. In other words, the dynamics of the soliton depend on the strength of the imaginary part of potential irrespective of the value of propagation constant.

Silicon Waveguide Analysis

The Maxwell’s equations are a basic formulation using for solve propagation equations within a waveguide. In this research, the wave propagation inside the waveguide was studied and some basic parameters of wave propagation were measured through MATLAB 2013. Silicon with a refractive index 3.46 was chosen to study this propagation and the use of core and cladding with a lower refractive index 2.46 and some parameters were extracted for the waveguide including propagation constant, cutoff wavenumber, cutoff frequency, cutoff thickness and normalized parameter. These parameters are studied at the wavelength 0.6 μm for transvers electric TE and transvers magnetic TM we changed the core thickness at wavelength constant and found that the number of modes increases as the increase of core thickness and the waveform is thinner in TM than it is in TE. We note from the measurements that the value of cutoff frequency and cutoff thickness are equal in the two types TE and TM. Moreover, when two modes (m=0, 1) and core thickness 0.24 μm the value of propagation constant in TE is equal (34.8901, 30.7737) while in TM (30.6440, 26.1967) respectively, we find that the values in TE are greater than TM. Also, the results obtained from Finite Difference Method were compared with the method used (Maxwell’s equation through wave equation solutions). This work represents a short pathway for the theoretical analysis concerning the electromagnetic waves propagating through Si planar waveguide with both modes (TE and TM) considered.

Wave Interaction with the Defect Characterized by Nonlinearity of General Form

Possible types of stationary states and waves in linear media separated by a nonlinear interface are analyzed. The mathematical formulation of the model is reduced to a one-dimensional boundary value problem for the nonlinear Schrodinger equation. The nonlinearity of the equation in the form of an arbitrary function of the desired field is taken into account only inside the waveguide. It is shown that there are stationary states of three types for different ranges of propagation constant values. The dispersion dependences of the propagation constant as functions of the parameters of the medium and the interface have explicitly been obtained for stationary states of all types, and conditions of their existence have been indicated. It is shown that total wave transition through the interface is possible. It has been established that the total transition of wave through the interface with nonzero parameters can occur only if the nonlinear response of the medium is taken into account.

Propagation of transverse electric mode in a non-integer dimensional dielectric slab waveguide

In this paper, solution for the transverse electric (TEz) mode propagating in a non-integer dimensional (NID) dielectric slab waveguide has been derived. For region occupied by the slab, the medium is assumed to be NID along the direction normal to interfaces of the slab. Transcendental equations yield solution in form of non-integer order Hankel functions. The order of the Hankel functions depends on parameter describing the dimension of the NID space. Graphical results has been used to depict that the behavior of modes changes substantially with respect to NID parameter. It also affects the value of propagation constant, attenuation constant and cutoff frequency. The classical result is reproduced taking NID parameter equal to two. It is observed that the value of propagation constant for TM modes is higher than the value of propagation constant for TE modes. Moreover, the value of attenuation constant for TE modes is higher than the value of attenuation constant for TM modes.

Propagation of transverse magnetic mode in a non-integer dimensional dielectric slab waveguide

In present paper, solutions for the transverse magnetic modes propagating in a non-integer dimensional (NID) dielectric slab waveguide have been derived. Medium within the slab is NID in direction normal to the dielectric interfaces of the slab. Transcendental equations in terms of non-integer order Hankel functions have been derived. The order of non-integer order Hankel functions depends on parameter describing dimension of the NID space. Using graphical treatment, it has been shown that behavior of propagating mode changes substantially when value of the parameter describing dimension of the NID space changes. It has also been noted that NID parameter significantly effects the value of propagation constant, attenuation constant and cutoff frequency. The classical results have also been recovered when dimension of the NID slab has been taken equal to the dimension of the medium surrounding the slab.

Optimized nonlinear SOI slot optical waveguides

Nonlinear-optic (NLO) effects provide many device functions such as wavelength conversion and signal processing. The use of waveguides allows the implementation of efficient and compact devices. The importance of exploring nonlinear effects in Silicon-on-Insulator (SOI) slot optical waveguides is discernible from the facts that, it is CMOS technology compatible, thus ensuring low-cost large-scale integration, moreover, this structure of optical waveguides mingle the advantages of both silicon and silica to explore the promising nonlinear interactions of light in the waveguides, such as rare-earth material doping to provide optical gain. In current research work, we have optimized nonlinear parameter (γ) of the basic SOI slot optical waveguide by reducing the effective area (Aeff). The effective area has been reduced by using Photonic Crystal Fiber (PCF) type cladding; where enough evidence is available that, SOI slot optical waveguide can be fabricated in a PCF type cladding. Wavelength dependent propagation constant values of slot optical waveguide are dominant over the band-gap effect (in case of PCF type cladding). Novelty lies in reducing the effective area (Aeff) by a considerable amount, with respect to already established results. The results, besides opening further venues for researchers to find out nonlinear optimization techniques of slot optical waveguide structures can also help in enhancing lab on chip circuits, where in-built optical source is mandatory.

P-polarized nonlinear surface polaritons near the surface of an epsilon-near-zero metamaterial with saturable permittivity

We study p-polarized nonlinear surface polaritons (NLSP) propagating along the interface of an epsilon-near-zero metamaterial with saturable nonlinearity, very small negative linear permittivity and an optically linear medium. On the basis of the exact solution of Maxwell's equations, we investigate the relationship between the NLSP propagation constant and the NLSP energy flux for self-focusing and self-defocusing cases. In both cases, for the chosen value of propagation constant, calculations due to Kerr model lead to the lowest NLSP energy flux compared with the energy flux for metamaterial with saturable permittivity.

Symmetric Plasmonic Slot Waveguides with a Nonlinear Dielectric Core: Bifurcations, Size Effects, and Higher Order Modes

We study the nonlinear stationary waves propagating in metal slot waveguides with a Kerr-type dielectric core. We develop two independent semianalytical models to describe these waves in such waveguides. Using these models, we compute the dispersion curves for about ten first modes of a nonlinear slot waveguide. For symmetric wave- guides, we find symmetric, antisymmetric, and asymmetric higher order modes which are grouped in two families. In addition, we study the influence of the slot width on the first symmetric and asymmetric modes. We also show that the dispersion curve of the first asymmetric mode is invariant with respect to the slot width for high propagation constant values and we provide analytical approximations of this curve. Using realistic values for the optogeometric parameters of the structure, we are able to obtain the bifurcation at a realistic power.

Guided Modes in Uniaxial Chiral Waveguide of Circular Cross-Section under PEC Boundary

Propagation of electromagnetic waves through a circular waveguide of uniaxial anisotropic chiral medium is studied with the emphasis on the energy flux patterns. The outer surface of the guide is assumed to be bounded by a perfect electric conductor (PEC) medium. The dispersion relation of guide is derived by applying suitable boundary conditions, and the allowed values of propagation constants are computed. Energy flux patterns corresponding to three different types of uniaxial anisotropic chiral metamaterials are taken into account. The propagation of negative energy flux is observed, and attributed to the presence of backward waves in the waveguide structure.

Solitons in a medium with linear dissipation and localized gain

We present a variety of dissipative solitons and breathing modes in a medium with localized gain and homogeneous linear dissipation. The system possesses a number of unusual properties, like exponentially localized modes in both focusing and defocusing media, existence of modes in focusing media at negative propagation constant values, simultaneous existence of stable symmetric and antisymmetric localized modes when the gain landscape possesses two local maxima, as well as the existence of stable breathing solutions.

Reactivities of carbocations and monomers in carbocationic polymerization and copolymerization

In carbocationic polymerization and copolymerization, a recent publication concluded that the substituent effect on carbocation reactivity is much larger than its effect on monomer reactivity, and this by a factor 106 in the case of the rate constant k12capp for p-methylstyrene addition (monomer M2) on, respectively, poly(p-methoxystyrene)± or poly(p-methylstyrene)± (M). This conclusion is disputed, as well as the assumption that the rate constants of capping (k12capp) obtained in deactivation reactions of poly(p-methoxystyrene)± are identical with cross propagation rate constants in copolymerization (k12copol). It is shown that the large calculated k12capp are based on propagation constant values for p-methylstyrene (k ≈ 109) obtained by the diffusion-clock method. They are 104 times smaller as found for all styrenes, that is, between 104 and 105 when they are based on the ionic species concentrations. In such a case, the available data are still in agreement with an approximate compensation between the reactivities of a monomer and of the corresponding carbocation. It is also shown that copolymerization data for styrenes are not compatible with k values near to diffusion control, and that variations of log k12capp and log k12copol with the nucleophilicity parameter N of the monomers indicate a much lower selectivity of the monomers in the case of copolymerization. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 2666–2680, 2010

Gap solitons and Bloch waves in nonlinear periodic systems

We comprehensively investigate gap solitons and Bloch waves in one-dimensional nonlinear periodic systems. Our results show that there exists a composition relation between them: Bloch waves at either the center or edge of the Brillouin zone are infinite chains composed of fundamental gap solitons(FGSs). We argue that such a relation is related to the exact relation between nonlinear Bloch waves and nonlinear Wannier functions. With this composition relation, many conclusions can be drawn for gap solitons without any computation. For example, for the defocusing nonlinearity, there are $n$ families of FGS in the $n$th linear Bloch band gap; for the focusing case, there are infinite number of families of FGSs in the semi-infinite gap and other gaps. In addition, the stability of gap solitons is analyzed. In literature there are numerical results showing that some FGSs have cutoffs on propagation constant (or chemical potential), i.e. these FGSs do not exist for all values of propagation constant (or chemical potential) in the linear band gap. We offer an explanation for this cutoff.

AN ANALYTICAL INVESTIGATION OF FOUR-LAYER DIELECTRIC OPTICAL FIBERS WITH AU NANO-COATING --- A COMPARISON WITH THREE-LAYER OPTICAL FIBERS

An analytical investigation has been presented of Au nano- coated dielectric optical flbers. The propagation constants of difierent transverse TE and hybrid EH modes are obtained corresponding to varying nano-coating thickness. It has been observed that the Au-layer has its profound efiect on the number of propagating modes in the flber, and the number of sustained modes is much reduced with the increase in Au-layer thickness. For the sake of comparative investigation, the modal behavior of three-layer dielectric flbers is also taken into account together with the Au nano-coated four-layer flber. It is reported that the Au-layer has the efiect of mode proliferation with simultaneous reduction in their propagation constant values.

Electrodynamical Analyses of Dielectric and Metamaterial Hollow-core CylindricalWaveguides

Development of computer algorithms for electrodynamical analyses of different waveguides is important nowadays due to the fact that these can be created from the newest materials with complicated dependences of the permittivity and the permeability on frequencies and waveguides can have nanometers sizes. We present the dispersion equation for analyzes of dielectric and metamaterial hollow-core cylindrical waveguides (HCWs). The main objective of our work is to analyze dispersion characteristics of the dielectric and metamaterial HCWs. We have analyzed dispersion characteristics of single−mode and many−mode dielectric and metamaterial HCWs. Dependencies of normalized propagation constant values on waveguide radiuses have been analyzed for dielectric and metamaterial HCWs. We have compared properties of the dielectric HCWs with metamaterial HCWs using obtained graphical results. Ill. 11, bibl. 10 (in English; summaries in English, Russian and Lithuanian).

Gaussian-Induced Rotation in Triangular Photonic Lattices

Time-dependent rotation of counterpropagating mutually incoherent self-trapped Gaussian beams in periodic optically induced fixed photonic lattices is numerically investigated. Rotation occurs for some values of control parameters. For parameters of such rotation, the solitonic solutions are found using modified Petviashvili’s method. It is shown that they correspond to the lowest values of propagation constant in the power diagrams and relation between observed rotation and less confined discrete solitonic solutions are demonstrated.

Universally‐Convergent Squared‐Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Three new iteration methods, namely the squared‐operator method, the modified squared‐operator method, and the power‐conserving squared‐operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast‐converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared‐operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher‐gap vortex solitons in the two‐dimensional nonlinear Schrödinger equations with periodic potentials, and isolated solitons in Ginzburg–Landau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared‐operator method delivers the best performance among the methods proposed in this article.

A Proposal of Simplified Eigenvalue Equation for an Analysis of Dielectric Slab Waveguide

In dielectric waveguide analysis and synthesis, we often encounter an awkward task of solving the eigenvalue equation to find the value of propagation constant. Since the dispersion equation is an irrational equation, we cannot solve it directly. Taking advantage of approximated calculation, we attempt here to solve this irrational dispersion equation. A new type of eigenvalue equation, in which guide index is expressed as a function of frequency, has been developed. In practical optical waveguide designing and in calculating the propagation mode, this equation will be used more conveniently than the previous one. To expedite the design of the waveguide, we then solve the eigenvalue equation of a slab waveguide, which is sufficiently accurate for practical purpose.

Mode Transfigurations in Chirowaveguides

it has been analyzed physical features of the modes behavior in a waveguide filled with the chiral medium. Both mathematical and physical models of their propagation have been defined and modes classification has been suggested. It has been shown that the same root feature in dispersion is typical both for chirowaveguide modes and for the unchiral waveguide: however the mode transfiguration is peculiar to all chirowaveguide modes and this determines the complex character of the final dispersion curves behavior. The following features are typical for chirowaveguide modes: connections between polarizations and between wave types, intersections of the dispersion curves, the spatial beatings and consecutive changes of the eigen function while moving the operating point along the dispersion curve (the mode transfiguration). The chirowaveguide performs polarization selection of propagating waves in such a way that only right-polarized waves can exist under big values of propagation constant in the chirowaveguide; the mode transfiguration is the reason of this.

MODE TRANSFIGURATIONS IN CHIROWAVEGUIDES - ABSTRACT

It has been analyzed physical features of the modes behavior in a waveguide filled with the chiral medium. Both mathematical and physical models of their propagation have been defined and modes classification has been suggested. It has been shown that the same root feature in dispersion is typical both for chirowaveguide modes and for the unchiral waveguide; however the mode transfiguration is peculiar to all chirowaveguide modes and this determines the complex character of the final dispersion curves behavior. The following features are typical for chirowaveguide modes: connections between polarizations and between wave types, intersections of the dispersion curves, the spatial beatings and consecutive changes of the eigen function while moving the operating point along the dispersion curve (the mode transfiguration). The chirowaveguide performs polarization selection of propagating waves in such a way that only right-polarized waves can exist under big values of propagation constant in the chirowaveguide; the mode transfiguration is the reason of this.