Analytical Transfer Matrix Method Research Articles

Lithography-free near-infrared broadband absorber based on a multilayer nanosystem

Broadband absorber has been intensively researched for decades because of its high absorption property which can be applied to various applications. Most designed absorbers are based on metasurfaces with complex surface-patterns and complicate manufacturing processes. In this paper, we propose a lithography-free, simple four-layer dielectric and metal nanosystem as a broadband absorber. The performance of the absorber is demonstrated by both numerical finite difference time domain (FDTD) method and analytical transfer matrix method (TMM). In the concerned wavelength range from 800 nm to 2300 nm, the average absorption is 95.81%. And the two maxima of absorptions are 99.99% at 970 nm and 99.31% at 1708 nm. The broadband absorption is due to the ohmic loss of the material Cr, the reflection of the Al layer, the destructive interference between the Cr and the Al layers, and the top Al2O3 helping much light to enter the absorber. And the simple multilayer absorber does not require precious metals, lithography processes, making a large-scale, low-cost manufacturing process possible.

Experimental, numerical and analytical studies on the attenuation of maglev train-induced vibrations with depth in layered soils

This paper investigates the attenuation characteristics of far-field environmental vibrations induced by the high-speed maglev train, especially the attenuation with depth. Field tests are carried out to measure the train-induced ground vibrations at different distances and soil depths. The vibration amplitude generally decreases as the distance increases according to Bornitz's formula. However, for the vibration along depth, there is an amplification zone where the vibration amplitude is much larger than the analytical prediction in a homogeneous soil. Numerical simulation using finite element method is also adopted to analyze the attenuation of vibrations and the results are consistent with the field measurements. Moreover, analytical transfer-matrix method is used to calculate the dispersion curve of Rayleigh wave in layered soils, and an analytical model is proposed to predict the vibration attenuation with depth. The vibration attenuation curve along depth obtained by the proposed model shows good agreements with the field measurement and numerical simulation. It reveals that the amplification of vibration at certain depths below the ground is attributed to the resonance of Rayleigh wave in layered soils.

Discrete Singular Convolution Method for Acoustic Transmission Lines

Discrete singular convolution (DSC) algorithm is an accurate dynamic analysis method for single structures. However, vibration analysis of connected structures via the DSC method is limited to step beams and plates. This paper extends the applicability of the DSC method in handling acoustic transmission lines composed of several duct elements with different diameter ratios. Connections of elements are handled by setting up continuity equations at geometric discontinuities. Conformity of DSC equations at these connections is carried out using Taylor series expansion. In this study, natural frequencies, mode shapes and power transmission coefficients of acoustic transmission lines are performed to test the proposed DSC implementation. Power transmission coefficients are obtained through the two-load method using the data provided by the DSC approach. The results are compared with finite element method and analytical solutions (if applicable), and the analytical transfer matrix method is also used for validating the power transmission coefficients. This paper shows that the DSC method is accurate and reliable, and so applicable for acoustic transmission line analysis.

Circular dichroism mode splitting and bounds to its enhancement with cavity-plasmon-polaritons

Abstract Geometrical chirality is a widespread phenomenon that has fundamental implications for discriminating enantiomers of biomolecules. In order to enhance the chiral response of the medium, it has been suggested to couple chiral molecules to resonant optical cavities in order to enhance the circular dichroism (CD) signal at the resonant frequency of the cavity. Here, we studied a distinctly different regime of chiral light-matter interaction, wherein the CD signal of a chiral medium splits into polaritonic modes by reaching the strong coupling regime with an optical microcavity. Specifically, we show that by strongly coupling chiral plasmonic nanoparticles to a non-chiral Fabry-Pérot microcavity one can imprint the mode splitting on the CD spectrum of the coupled system and thereby effectively shift the initial chiral resonance to a different energy. We first examined the effect with the use of analytical transfer-matrix method as well as numerical finite-difference time-domain (FDTD) simulations. Furthermore, we confirmed the validity of theoretical predictions in a proof-of-principle experiment involving chiral plasmonic nanoparticles coupled to a Fabry-Pérot microcavity.

Analysis of the ground-state energy eigenvalues of fractal quantum potentials

Traditional models in solid-state physics study the motion of electrons in a periodic lattice. Recent developments in the physics of graphene allow scientists to construct two-dimensional structures with fractal geometry and conduct experiments on them. Recently, some theoretical approaches have been developed to study the optical and electrical properties of semiconductor layers with self-similar characteristics. This newly emerged direction in solid-state physics inspired us to focus on studying the quantum mechanical properties of fractal potentials. We first introduce sequences of potential wells converging towards different fractal structures. Then, we calculate the ground-state energy eigenvalues of the time-independent Schrödinger equation for these potential functions using two numerical methods, the Numerov and analytical transfer matrix method, to demonstrate the effect of the potential structure morphology and properties on the behavior of the energy eigenvalues. Ground-state energies for the generalized Cantor set, the Smith–Volterra–Cantor set, a multi-level Cantor set and the Weierstrass function will be calculated and compared. We will deal with the question of how the ground state of these potential functions changes as the fractal generator is applied, and we show that properties such as the Lebesgue measure of the Cantor potentials and the Hausdorff dimension of the Weierstrass function strongly control the convergence of energy eigenvalues.

The analytical transfer matrix method for PT-symmetric complex potential

We have extended the analytical transfer matrix (ATM) method to solve quantum mechanical bound state problems with complex PT-symmetric potentials. Our work focuses on a class of models studied by Bender and Jones, we calculate the energy eigenvalues, discuss the critical values of g and compare the results with those obtained from other methods such as exact numerical computation and WKB approximation method.

Wind load response of offshore wind turbine towers with fixed monopile platform

Wind turbine towers are continuously subjected to wind force as their major dynamic load. Dynamic response of the wind turbine tower subjected to wind forces is usually estimated by numerical methods such as finite element method (FEM). In this paper, the analytical transfer matrix method (TMM) is used to determine wind load response of wind turbine. Therefore, various and general models including coupled springs (CS), distributed springs (DS) and apparent fixity length (AF) models are employed for modeling of wind turbine foundation and axial force is modeled as a variable force along the tower, as well. Wind force is divided into two parts containing thrust and distributed force. The transfer matrix is derived based on Euler-Bernoulli's beam differential equation. The global transfer matrix is obtained by applying boundary conditions of the tower and constructing the point matrix. Then, the factors affecting the tower dynamic response such as foundation models, axial induction factor, terrain factor, wind shear and gust condition are examined in several case studies. Finally, the TMM results are compared with the available results in the literature such as of the FEM which show good agreement. The comparisons can be concluded that the TMM can be used instead of the conventional numerical methods maintaining high accuracy for wind induced vibration analysis of wind turbine towers.

Heat diffusion in inhomogeneous graded media: Chains of exact solutions by joint Property & Field Darboux Transformations

This paper presents a formal method for constructing solvable effusivity profiles, i.e. those leading to closed-form analytic one-dimensional solutions in the Laplace or Fourier domain for transient or steady-periodic temperature. A Liouville transformation applied to the heat equation expressed either in temperature or in heat flux yields two stationary Schrödinger-type equations. They reveal that temperature and heat flux depend solely on the thermal effusivity profile expressed in terms of the heat diffusion-time variable. Each Schrödinger-type equation can be cast in a system of two homologous Schrödinger-type equations, it means equations with the same potential function. The Darboux transformation is known to change a solvable linear second order differential equation into another solvable, but more complex, equation of the same type. We show that when applied to the heat transfer equation for continuously heterogeneous media and starting from an elementary solution, it allows finding simultaneously a set of more complex solvable effusivity profiles and the associated temperature solutions. The whole procedure constitutes a joint Property & Field Darboux Transformation (PROFIDT). By iterating these transformations we may construct the temperature solution related to an arbitrarily complex effusivity profile. Another way consists in applying an n-order Darboux-Crum transformation simultaneously on a seed field-function and on its associated square root effusivity profile (or the reciprocal). We illustrate these procedures, starting from the elementary solutions associated to a linear, a hyperbolic or a trigonometric effusivity profile (or their reciprocal). The solvable profiles and the temperature obtained therefrom present the distinct advantage of involving only elementary functions. As an alternative to the n-order solutions, we introduce the concept of solvable splines. They are obtained by stitching profiles from a set of parsimonious basis functions obtained after the application of a single PROFIDT. The resulting interpolation spline may be C2 continuous at the nodes. The associated exact temperature solution is then obtained by the thermal quadrupole method (analytical transfer matrix method). The proposed methodology can be applied to a variety of physical problems where the dynamic field inside the graded material is described by a diffusion-like or a wave-like equation.

Prediction of the nonlinear pull-out response of FRP ground anchors using an analytical transfer matrix method

Fiber-reinforced polymer (FRP) rods have been increasingly used in grouted ground anchors due to their high strength-to-weight ratio, excellent corrosion resistance, and convenience in incorporating the fiber sensing technology. To establish their pull-out capacity, FRP rods are usually embedded within a grouted steel tube and then subjected to pull-out in the laboratory. The aim of this paper is to develop a numerical method for predicting the nonlinear pull-out response of FRP rods embedded in steel tubes filled with cement grout. In the method, the cement grout is assumed to be subject to simple shear, the local interfacial bond stress–slip model of the bar-to-grout interface is represented by a piece-wise curve comprising elastic, softening, and frictional stages, and the unloading effect is also taken into account. A set of two second-order ordinary differential equations are derived in terms of the displacements of the FRP rod and steel tube and solved analytically to formulate the element transfer matrix. When the thickness of the steel tube approaches infinity, this method can be applied to the problem of FRP rods embedded in rock. Based on the developed numerical method, the interfacial bond properties and snapback phenomenon are analyzed. After the method is validated by comparisons with four sets of experimental data, the effects of the radius and length of FRP rods, the local peak bond stress and the residual frictional strength on the maximum pull-out load are evaluated in a quantitative manner.

Dynamic response of a multi-span Timoshenko beam with internal and external flexible constraints subject to a moving mass

An analytical transfer matrix method is presented to determine the effect of intermediate flexible constraints on the dynamic behavior of a multi-span beam subject to a moving mass. By using the Timoshenko beam theory on separate beams and applying the compatibility requirements on each constraint point, the relationships between every two adjacent spans can be obtained. By using a transfer matrix method, eigensolutions of the entire system can be determined. The forced responses can then be calculated by the modal expansion theory using the determined eigenfunctions. Some numerical results are presented to show the effects of intermediate constraints and locations on the dynamic response of the multi-span beams. It will be seen that the general formulation developed here can cover a large array of problems such as cracked or intermediately constrained beams.

The rotating Morse potential energy eigenvalues solved by using the analytical transfer matrix method

We study the eigenvalues of the rotating Morse potential by using the quantization condition from the analytical transfer matrix (ATM) method. A hierarchy of supersymmetric partner potentials is obtained with Pekeris approximation, which can be used to calculate the energies of higher rotational states from the energies of lower states. The energies of rotational states of the hydrogen molecule are calculated by the ATM condition, and comparison of the results with those from the hypervirial perturbation method reveals that the accuracy of the approximate expression of Pekeris for the eigenvalues of the rotating Morse potential can be improved substantially in the framework of supersymmetric quantum mechanics.

Enhanced detection of virus particles by nanoisland-based localized surface plasmon resonance

In this paper, we investigate localized surface plasmon resonance (SPR) detection based on nanoislands. Theoretical calculation performed with rigorous coupled-wave analysis and analytical transfer matrix method using effective medium theory suggests improvement on nanoislands in the limit of detection and sensitivity over conventional thin film-based SPR detection. Experimental results obtained with non-specific detection of ambient adenovirus confirm the improvement by more than one order of magnitude increase in the limit of detection. The enhancement achieved with nanoislands was explored in connection with efficient overlap between target and near-field distribution produced by nanoislands.

Modified and Simplified Sectional Flexibility of a Cracked Beam

This paper presents a new sectional flexibility factor to simulate the reduction of the stiffness of a single-edge open cracked beam. The structural model for crack of the beam is considered as a rotational spring which is related to the ratio of crack depth to the beam height,a/h. The mathematical model of this single-edge open crack beam is considered as an Euler-Bernoulli beam. The modified factor,f(a/h), derived in this paper is in good agreement with previous researchers' results for crack depth ratioa/hless than 0.5. The natural frequencies and corresponding mode shapes for lateral vibration with different types of single-edge open crack beams can then be evaluated by applying this modified factorf(a/h). Using the compatibility conditions on the crack and the analytical transfer matrix method, the numerical solutions for natural frequencies of the cracked beam are obtained. The natural frequencies and the mode shapes with crack at different locations are obtained and compared with the latest research literature. The numerical results of the proposed cracked beam model obtained by this method can be extended to construct frequency contour. The natural frequencies measured from field can be used in solving the inverse problem to identify cracks in structures.

Generation and classification of the translational shape-invariant potentials based on the analytical transfer matrix method

For the conventional translational shape-invariant potentials (TSIPs), it has demonstrated that the phase contribution devoted by the scattered subwaves in the analytical transfer matrix quantization condition is integrable and independent of n. Based on this fact we propose a novel strategy to generate the whole set of conventional TSIPs and classify them into three types. The generating functions are given explicitly and the Morse potential is taken as an example to illustrate this strategy.

Dual-wavelength micro-resonator combining photonic crystal membrane and Fabry-Perot cavity

We propose a novel system of dual-wavelength micro-cavity based on the coupling between a photonic crystal membrane (PCM); operating at the Γ- point of the Brillouin zone, with a Fabry-Perot vertical cavity (FP). The optical coupling, which can be adjusted by the overlap between both optical modes, leads to the generation of two hybrid modes separated by a frequency difference which can be tuned using micro-opto-electromechanical structures. The proposed dual-wavelength micro-cavity is attractive for application where dual-mode behaviour is desirable as dual-lasing, frequency conversion. An analytical model, numerical (FDTD) and transfer matrix method investigations are presented.

Theoretical study of W-shaped optical fiber with a depression in core center by applying analytical transfer matrix method

Analytical transfer matrix method (ATMM) is a method for calculating the propagation constant in weakly guiding optical fiber. By using ATMM, the optical fiber with a depression in the index profile center and a valley in the cladding layer is analyzed. Compared with Wentzel–Kramers–Brillouin (WKB) method, the simulation result of differential mode delay (DMD) by using ATMM fits well with the experimental results obtained by Takahashi. Based on ATMM, by increasing the depth of central depression in a W-shaped index fiber, the improvement of DMD is also discussed.

Structural Damage Detection with Piezoelectric Wafer Active Sensors

Piezoelectric wafer active sensors (PWAS) are lightweight and inexpensive enablers for a large class of damage detection and structural health monitoring (SHM) applications. This paper starts with a brief review of PWAS physical principles and basic modelling and continues by considering the various ways in which PWAS can be used for damage detection: (a) embedded guided-wave ultrasonics, i.e., pitch-catch, pulse-echo, phased arrays, thickness mode; (b) high-frequency modal sensing, i.e., the electro-mechanical (E/M) impedance method; (c) passive detection, i.e., acoustic emission and impact detection. An example of crack-like damage detection and localization with PWAS phased arrays on a small metallic plate is given. The modelling of PWAS detection of disbond damage in adhesive joints is achieved with the analytical transfer matrix method (TMM). The analytical methods offer the advantage of fast computation which enables parameter studies and carpet plots. A parametric study of the effect of crack size and PWAS location on disbond detection is presented. The power and energy transduction between PWAS and structure is studied analytically with a wave propagation method. Special attention is given to the mechatronics modeling of the complete transduction cycle from electrical excitation into ultrasonic acoustic waves by the piezoelectric effect, the transfer through the structure, and finally reverse piezoelectric transduction to generate the received electric signal. It is found that the combination of PWAS size and wave frequency/wavelength play an important role in identifying transduction maxima and minima that could be exploited to achieve an optimum power-efficient design. The multi-physics finite element method (MP-FEM), which permits fine discretization of damaged regions and complicated structural geometries, is used to study the generation of guided waves in a plate from an electrically excited transmitter PWAS and the capture of these waves as electric signals at a receiver PWAS. Wave diffraction from a hole damage is illustrated through time-frame snapshots. The paper ends with conclusions and suggestions for further work.

Reflectionless Potentials via Complex Potentials

We point out that the complex potentials can be used to construct the reflectionless potentials and two new classes of reflectionless potentials are proposed. In particular, the robustness of these reflectionless potentials is thoroughly studied by the analytical transfer matrix method.

Photon tunneling and transmittance resonance through a multi-layer structure with a left-handed material

This paper investigates the photon tunneling and transmittance resonance through a multi-layer structure including a left-handed material(LHM). An analytical expression for the transmittance in a five-layer structure is given by the analytical transfer matrix method. The transmittance is studied as a function of the refractive index and the width of the LHM layer. The perfect photon tunneling results from the multi-layer structure, especially from the relation between the magnitude of the refractive index and the width of the LHM layer and those of the adjoining layers. Photons may tunnel through a much greater distance in this structure. Transmittance resonance happens, the peaks and valleys appear periodically at the resonance thickness. For an LHM with inherent losses, the perfect transmittance is suppressed.

Leak detection in complex series pipelines by using the system frequency response method

This research investigates the applicability of the transient-based frequency-response function (FRF) method for detecting leaks in complex series pipelines. The behaviour of transient waves in these pipelines with internal series junctions indicates that junction reflections modify the system resonant frequencies but have a small effect on the leak-induced information contained within the system frequency responses. The analogous method previously developed for single pipelines is extended to complex series pipe systems by using the analytical transfer matrix method herein and the extended method is validated by two simple numerical experimental cases consisting of 3-series pipes and 10-series pipes, respectively. The applied results indicate that the extended FRF method can be applied to detect single and multiple leaks in complex series pipelines as long as the location and size of the resonant peaks of system frequency responses are accurately determined.