Argument Principle Method Research Articles

Tunable Quasistationary States in a One-dimensional Quantum Heterostructure

In this work, we focus on the quasistationary states, lifetime, and transmittance in opened quantum wells with biased and unbiased. In order to solve the quasibound states, the complex eigenenergies are solved in our calculation model by adaptive finite element method. We have demonstrated the accuracy to exam the numerical convergence. In this case, the 1D quantum heterostructure is commonly composed of GaAs and AlxGa1-xAs. With the different applied bias, the resonant tunneling and transmittance profiles could be changed, respectively. Increasing the thickness of the outermost barrier can be prevented an electron penetrated through the barrier from the quasistationary state. This is a useful way to design easily the high-speed switch for semiconductor devices. Our results of numerical calculations are good agreement with the argument principle method approach. These results are useful and helped us to design quantum devices and quantum computations.

Robust location of optical fiber modes via the argument principle method

We implement a robust, globally convergent root search method for transcendental equations guaranteed to locate all complex roots within a specified search domain, based on Cauchy’s residue theorem. Although several implementations of the argument principle already exist, ours has several advantages: it allows singularities within the search domain and branch points are not fatal to the method. Furthermore, our implementation is simple and is written in MATLAB, fulfilling the need for an easily integrated implementation which can be readily modified to accommodate the many variations of the argument principle method, each of which is suited to a different application. We apply the method to the step index fiber dispersion relation, which has become topical due to the recent proliferation of high index contrast fibers. We also find modes with permittivity as the eigenvalue, catering to recent numerical methods that expand the radiation of sources using eigenmodes.Program summaryProgram Title: disproots (Dispersion Roots)Program Files doi:http://dx.doi.org/10.17632/32297nv5kk.1Licensing provisions: CC BY NC 3.0Programming language: MATLABNature of problem: Locating the complex roots of a general transcendental equation is often a non-trivial task. Iterative methods such as Newton’s method face numerous difficulties because the existence of roots with narrow and otherwise difficult attraction basins requires very accurate initial guesses to locate. Robust location of a complete set of roots thus becomes problematic. In optics, modes of a circular fiber are obtained from a transcendental equation, the dispersion relation. Several recent advancements in optics have necessitated its robust solution, including the fabrication of high index contrast fibers and analytical methods that expand radiating sources using eigenmodes.Solution method: We employ the argument principle method, a robust globally convergent method guaranteed to locate all roots in the specified search domain. It is based on the Cauchy residue theorem, and projects the locations of the roots on to a polynomial basis. Unlike previous implementations of the argument principle method [1,2] and related methods [3], our implementation has two features vital for solving the fiber dispersion relation. It allows isolated singularities within the search domain, and allows the search domain to approach arbitrarily close to branch points without experiencing failure. Furthermore, our simple MATLAB implementation is designed to be easily modified and integrated for a variety of applications.Additional comments including Restrictions and Unusual features: The specified search domain must be meromorphic, in other words be complex analytic containing at most isolated singularities. The locations and orders of these singularities must be known analytically, so we describe how they are determined for our fiber dispersion relation. Branch points and branch cuts must be avoided, though the search domain may be arbitrarily close. We exploit knowledge of our dispersion relation, specifically that roots are located close to singularities, to simplify the method. [1]L. C. Botten, M. S. Craig, R. C. McPhedran, Complex zeros of analytic functions, Comput. Phys. Commun. 29 (3) (1983) 245–259.[2]P. Kravanja, M. Van Barel, O. Ragos, M. N. Vrahatis, F. A. Zafiropoulos, ZEAL: A mathematical software package for computing zeros of analytic functions, Comput. Phys. Commun. 124 (2–3) (2000) 212–232.[3]C. J. Gillan, A. Schuchinsky, I. Spence, Computing zeros of analytic functions in the complex plane without using derivatives, Comput. Phys. Commun. 175 (4) (2006) 304–313.

Cable vibration control with both lateral and rotational dampers attached at an intermediate location

Lateral dampers have been extensively studied and implemented for supplementing modal damping in cable vibration mitigation. When considering the cable flexural stiffness that is actually present, albeit small, there is another degree of freedom of the cable at the lateral damper, namely the rotation, that can be constrained by a rotational damper to achieve larger additional damping. This is of particular significance for long cables where the near-anchorage lateral damper alone is usually insufficient. The problem of a cable with bending stiffness, attached with both lateral and rotational dampers at an intermediate point, is therefore considered in this study. The characteristic equation of the resulting system is formulated by assembling the dynamic stiffness from the two segments divided by the damper, which is subsequently solved using argument principle method. Dynamics of the controlled system is thus discussed in general through parametric analysis. For the case where the damper location is close to the anchorage, asymptotic solutions for complex frequency and damping ratio are provided; explicit formulas for determining the optimal damper coefficients are also derived. It is found that when the lateral and rotational damper coefficients are properly balanced, the proposed strategy can achieve up to 30 percent damping enhancement compared to the case with only the lateral damper, in practical cable bending stiffness range.

Extraction of all propagation constants in a specified region from the transcendental equation of a dispersion relation using the Sakurai-Sugiura projection method.

A transcendental equation occurs when we compute the dispersion relations of an electromagnetic waveguide, such as a planar multilayer waveguide. Without an initial guess, the Sakurai-Sugiura projection method (SSM) can obtain solutions to the transcendental equation in a region bounded by a contour integral path in the complex plane. In this paper, a criterion employing the condition number of eigenvalues as a simple index to distinguish physical solutions from spurious ones in the SSM is presented, and a transcendental equation of a multilayer waveguide obtained by the transfer matrix method is solved by the SSM. Numerical results show the usefulness of the index and good agreement with the results of the argument principle method and Newton's method.

Finding resonant frequencies for high loss dielectrics in cylindrical cavities

This article proposes the use of argument principle method (APM) to find all complex resonant frequencies in a three layer cylindrical cavity. APM guarantees that no root is lost and frequencies can be associated with the resonant mode. The roots can be used to find permittivity of a material inside a cavity. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:530–535, 2015.

Exact solution of free vibration of a uniform tensioned beam combined with both lateral and rotational linear sub-systems

A general exact solution of free vibration of a uniform tensioned beam with any number of both lateral and rotational sub-systems is addressed. This study is completed in complex domain, because of the presence of the non-proportional damping caused by the dampers utilized in sub-systems. The sub-systems constructed herein could degenerate to the conventional supports and sub-systems, and be arbitrarily placed. Accordingly, the solution is also suitable for multi-span beams. The argument principle method (APM), which does not need initial values and iteration, is employed to solve the characteristic equation in complex domain. Nevertheless, the presence of multi-valued square root functions due to the axial loading renders the function non-analytic in complex domain. A variable substitution method, in which the main branch of logarithmic function is chosen, is proposed to solve this problem. The results obtained are compared with those in literature. Furthermore, several typical case studies were completed, and the root loci of the frequency parameters attained are compared with those obtained by the finite element method (FEM). A good agreement is observed, which confirms the validity of the presented methodologies in this paper.

Superconductive Traveling-Wave Photodetectors: Fundamentals and Optical Propagation

This paper studies the theory of superconductive traveling-wave photodector devices (STWPDs), as a general platform for ultrafast, ultrasensitive, and ultralow-noise optoelectronic functions such as detection, modulation, photomixing, high-frequency electrical signal generation and amplification. Principles of operation of STWPDs are discussed based on the kinetic-inductance theory of superconductive thin films and a thorough investigation of the guiding mechanism of light within this class of devices is presented. The transfer matrix method is introduced to model the superconductive optical waveguide in TWDs and an efficient numerical method based on the Cauchy integral method and the argument principle method are developed for the analysis and design of the waveguides. Moreover, several regimes of device operation will be distinguished in terms of the modal characteristics of the optical waveguide and their effects on the overall performance of the device will be highlighted.

The cyclicity of period annuli of some classes of reversible quadratic systems

The cyclicity of period annuli of some classes of reversible andnon-Hamiltonian quadratic systems under quadratic perturbationsare studied. The argument principle method and the centroid curvemethod are combined to prove that the related Abelian integral hasat most two zeros.

Polynomial expansion for extraction of electromagnetic eigenmodes in layered structures

A polynomial expansion approach to the extraction of guided and leaky modes in layered structures including dielectric waveguides and periodic stratified media is proposed. To verify the method we compared the results of analysis of a typical test case with those reported in the literature and found good agreement. Polynomial expansion is a nonharmonic expansion and does not involve harmonic functions or intrinsic modes of homogenous layers. This approach has the benefit of leading to algebraic dispersion equations rather than to a transcendental dispersion equation; therefore, it will be easier to use than other methods such as the argument principle method, the reflection pole method, and the wave-vector density method, which solve the transcendental dispersion equation by means of integrals. Besides, an algebraic dispersion equation can be obtained without any problem of numerical instability, whereas an ordinary transcendental dispersion equation, which is usually derived by the transfer matrix method, is difficult to obtain because of instability in multiplying transfer matrices. A demonstration of the utility of the proposed method when the other methods mentioned are inferior or fail are also given.

Quantum-well infrared photodetector structure synthesis: methodology and experimental verification

A numerical method for global optimization of quantum-well infrared photodetector (QWIP) performance parameters is presented and experimentally verified. The single-band effective-mass Schroedinger equation is solved by employing the argument principle method (APM) to extract both the bound and quasibound eigen-energies of the quantum heterostructure. APM is combined with a simulated annealing algorithm to determine a set of device design parameters such as potential barrier height V/sub i/, layer thickness d/sub i/, number of material layers N, total device length, applied bias V/sub Bias/ etc., for which the QWIP performance is within a predetermined convergence criterion. The method presented incorporates the effect of energy-dependent effective mass of electrons in nonparabolic conduction bands. The present model can handle many optimization parameters and can incorporate fabrication constraints to achieve physically realizable devices. In addition, the method is not limited to the optimization of absorption structures, and can be used for other intersubband devices such as electron-wave Fabry-Perot filters and quantum-cascade lasers. The strength and versatility of the present method are demonstrated by the design of a bicolor equal-absorption-peak QWIP structure, and experimental verification of the zero-bias absorption spectrum is presented.

Spatially varying effective mass effects on energy level populations in semiconductor quantum devices

For the analysis and design of semiconductor intersubband devices, accurate values for the Fermi energy and the subband electron population are needed. The effect of the position-dependent electron effective massm* (z) is commonly neglected in the determination of these two intersubband device parameters. This approach is nearly valid for single-well devices in the GaAs/Al xGa1−x As material system. However, in multiple-coupled-well devices, the variable nature of the effective mass must be taken into account. In material systems other than the ones based on GaAs, failure to include the position-dependence of the electron mass may give rise to significant errors in the values of the Fermi energy and other device parameters such as the intersubband absorption coefficient. In this paper, the effects ofm* (z) on the Fermi level and the intersubband charge distribution are explored and quantified.Theoretical formulation for the intersubband Fermi energy and the subband electron distribution, with the inclusion of position-dependent electron mass, is presented. The eigenenergies of the intersubband structures are obtained by solving the single-band effective-mass Schroedinger equation using the argument principle method. The electron distribution and the Fermi energy are calculated using both the approximate method ( m0*) and the rigorous formulation [ m*(z)], and the relative differences in the corresponding values are presented. It is demonstrated that these differences are small in the GaAs/Al xGa1−x As material system, but can become very significant in other materials. In addition, the variable nature of carrier effective mass plays an important role in other types of devices such as interband quantum well photodetectors and lasers that employ optical transitions between the valence and the conduction bands. Electronic devices such as the resonant tunneling diode are also affected by the position-dependence of carrier mass and thus the results are applicable to both optoelectronic and electronic quantum devices.

Semiconductor intersubband laser/detector performance optimization using a simulated annealing algorithm

A numerical method for global optimization of semiconductor intersubband laser/detector performance parameters is presented. The single-band effective-mass Schroedinger equation is solved by employing the argument principle method (APM) to extract both the bound (B) and quasibound (QB) eigen-energies of the quantum heterostructure. APM is combined with a simulated annealing (SA) algorithm to determine a set of device design parameters such as potential barrier height Vi, layer thickness di, applied biasVBias , for which the intersubband device performance is within a predetermined convergence criterion. The method presented incorporates the energy-dependent effective mass of electrons in nonparabolic conduction bands. The performance of the method is evaluated for the design of an asymmetric Fabry–Perot electron-wave interference filter (laser structure) and a dual-band quantum well infrared photodetector (QWIP). Results with and without nonparabolic effects are presented. In addition, results from the present method are compared to results obtained via the optimization technique based on super-symmetric quantum mechanics (SUSYQM) for the case of an optically-pumped quantum cascade (QC) laser. The present method is shown to improve the device performance beyond that obtained via SUSYQM optimization. Further, the present model can handle many optimization parameters and can incorporate fabrication constraints to achieve physically realizable devices.

The quasibound state model for self-consistent characteristics of semiconductor intersubband devices

The quasibound state model (QBSM) for determining the self-consistent conduction band profile and space charge density of semiconductor intersubband devices is presented. This new method is based on the quasibound (QB) state resonances of quantum structures. For heterostructures, the traditional self-consistent energy continuum model (ECM) calculates space charge by integration over the entire energy continuum, weighted by Fermi–Dirac statistics. In the present approach, the continuum of energy states of the heterostructure is accurately represented by a small number of QB states, and the space charge calculations are performed only at these eigen-energies. This approach significantly reduces the computational burden associated with all self-consistent algorithms. Theoretical formulation of QBSM is compared with the traditional ECM approach. The bound (B) and QB eigenenergies of the structure are obtained by solving the single-band effective-mass Schrödinger equation using the argument principle method. The performance and the accuracy of the QBSM are evaluated for a double-barrier resonant structure and an asymmetric Fabry–Perot electron-wave interference filter. The self-consistent electron density and potential profiles calculated by the present method are shown to be in excellent agreement with the results obtained from the traditional ECM model. In addition to requiring less computational time, the present method is easily implemented and may be applied equally well to biased/unbiased, symmetric/asymmetric heterostructures.

Application of the argument principle method to the calculation of DFB laser diode modes

AbstractA convenient technique for the exact determination of the longitudinal cavity modes of a distributed feed‐back (DFB) laser is presented. The method employs the argument principle method (APM) to find the singular points of the characteristic equation derived from consideration of the counter propagating coupled modes within the cavity. The method is applied to both linear and circular grating laser diodes. Copyright © 2001 John Wiley & Sons, Ltd.

Dipole matrix elements of semiconductor intersubband quantum structures

A method for determining the dipole matrix element for an intersubband optical transition in multi-layered semiconductor quantum heterostructures is presented. The single-band effective-mass Schrödinger equation is solved by employing the argument principle method (APM) to extract the bound (B) and quasibound (QB) eigenenergies of the quantum heterostructure. The major types of optical transitions involving bound and QB states are defined and the corresponding dipole matrix elements are calculated for each type. The method presented incorporates the energy-dependent effective mass of electrons arising from conduction-band nonparabolicity. The performance and the accuracy of the method are evaluated for an asymmetric Fabry–Perot electron wave interference filter. The physical dimensions of the filter are varied to show their effect on the dipole matrix elements. Results with and without nonparabolic effects are presented and compared. Dipole matrix elements are also calculated for the filter with an applied electric field bias. In this case the eigenstate wavefunctions can be expanded as linear combinations of Airy and complementary Airy functions. In addition, results from the present method are compared to a Kronig–Penney and a multi-band model. The dipole matrix element values calculated by the present method are shown to be in excellent agreement with the values obtained from these models. Further, the present model is numerically efficient and easily implemented.

Lossy multilayer channel optical waveguides analyzed by the transmission line matrix method

We demonstrate that the three-dimensional vectorial transmission line matrix (TLM) method is applicable to the analysis of lossy multilayer optical waveguiding structures. Any lossy multilayer waveguide geometry, including sharp discontinuities in the transverse plane, can be treated taking into account the coupling between all optical field components. The complex propagation constants (propagation constants and the attenuation coefficients) for the fundamental TE-like and TM-like modes can be determined. These parameters of the fundamental TM-like mode of a typical lossy multilayer rib dielectric waveguide are obtained as functions of free-space wavelength. Calculation of the electric-field pattern is also included. Numerical comparisons with the argument principle method (for the case of lossy multilayer slab waveguides) and the spectral-index technique (for the case of lossy multilayer rib waveguides) are also included, and it is shown that the application of the TLM method to lossy multilayer optical waveguides is accurate.

Optimization of multilayer integrated optics waveguides

A numerical method is described that provides the means for optimizing any objective function representing a general multilayer integrated optics waveguide. In this way, the physical parameters of the multilayer structure can be set so that the performance of the device is optimized. The method uses any standard numerical minimization algorithm in conjunction with the argument principle method. The method has been successfully applied for the optimization of a multilayer immunosensor and a TE-mode polarizer. The advantages of the method are its generality, its efficiency, its accuracy, and its applicability to a wide range of planar integrated optics devices. >