Comparison transfer matrix methods and scattering matrix method for investigation the optical properties of multilayer structures

PyLlama: A stable and versatile Python toolkit for the electromagnetic modelling of multilayered anisotropic media

The possibility of using the ellipsometry method for investigation of the optical properties of multilayer films and structures is shown. The optical properties of structures HfO2/SiO2/Si, HfO2/Si, ZrO2/Si, Ta2O5/Si, and Al2O3/Si are studied. It is found that a layer of hafnium silicate is formed at the interface between the HfO2 film and Si. Annealing of the structures in oxygen shows that oxides studied are oxygen-permeable and that the thickness of SiO2 at the film-substrate interface increases. The growth rate of SiO2 layers depends on the chemical nature of an oxide. Al2O3 films are impermeable for oxygen diffusion. The production of layers of alloys (Al2O3) x (HfO2)1 − x is optimized, which allows one to obtain layers with a homogeneous distribution of elements over the thickness.

A new improved transfer matrix method (TMM) is presented. It is shown that the method not only overcomes the numerical instability found in the original TMM, but also greatly improves the scalability of computation. The new improved TMM has no extra cost of computing time as the length of the homogeneous scattering region becomes large. The comparison between the scattering matrix method (SMM) and our new TMM is given. It clearly shows that our new method is much faster than the SMM.

In this study, the value of transmission coefficient on InN/GaN semiconductor from a single barrier to five barriers was determined by using the propagation matrix method and the transfer matrix method. This study aims to see the effect of adding a barrier to the number of resonance tunneling that occurs, to see the difference in transmission coefficient values which was obtained with the two methods, and to determine the effectiveness of the program execution process time from the propagation matrix and transfer matrix methods using Matlab programming. The results obtained indicated that the value of the transmission coefficient obtained from the two methods was the same. As the number of barriers increases, the number of resonance tunneling that occurs will increase. These two matrix methods had differences in terms of the effectiveness of the program execution process time and calculation process. The propagation matrix method was considered more effective than the transfer matrix method.

So far the transfer matrix method (TMM) has been extensively used for the calculation of the propagation of an electromagnetic wave through planar stratified media and in particular for the calculation of the reflection coefficient and generation function in III-V solar cells. TMM however can present numerical instability when applied to MJ solar cells whose structures allow improving the cell photovoltage. It is shown that for such structures, in order to overcome the numerical instability of TMM, a simplified scattering matrix method can be successfully applied.

In this work, reflection and transmission of electromagnetic wave through a multilayered structure containing diamond-like carbon, porous silicon, and left-handed material (LHM) are investigated theoretically and numerically. The mentioned materials are described, and their main parameters are given in detail. After the construction of the problem, the reflection and transmission coefficients are derived in a closed form by a transfer matrix method. The reflected and transmitted powers of the structure are calculated using these coefficients. In the numerical results, the mentioned powers are computed and illustrated as a function of frequency, angle of incidence, and slabs thickness, when the damping coefficient of the LHM changes. The results obtained may be useful to the researchers and designer working in the area solar cells.

Legendre orthogonal polynomial method in calculating reflection and transmission coefficients of fluid-loaded functionally gradient plates

The elastic wave propagation through the finite and infinite periodic laminated structure of micropolar elasticity

A simple analytical calculation scheme to determine near field radiation through decomposing an emission domain into lots of thin thermal current sheets is presented. Through finding the orthogonal modes of thermal current of each thin layer, the thin current sheets can be treated as radiation sources of electromagnetic waves with determined analytical solutions. The outgoing electromagnetic waves from each thin current sheets can be either in transverse electric (TE) or transverse magnetic (TM) modes depending on the orientations of the current in the thin current sheets with respect to the directions of amplitude modulations of the orthogonal modes. Electromagnetic waves arriving to a collection domain are related to the electromagnetic waves leaving from each thin current thermal sheet with transfer coefficients. Transfer coefficient for TE and TM waves can be determined analytically with transfer matrix method or scattering matrix methods. Compared with existing dyadic Green's function method, the new calculation scheme allows material and temperature variations along one direction of the emission domain based on determined analytical expressions of TE and TM waves leaving from each thin current sheets. The simple calculation scheme is especially useful in near field radiation of layered structures with different material such as hyperbolic material with negative refractive indices. With this new approach, we recovered analytical solutions of near field radiation between two semi-infinite domains with uniform temperature and derived closed form solution of near field radiation between two semi-infinite domains with temperature profiles with/without laminated structures.

In this article, we rapidly and accurately solve out reflection and transmission coefficients in the multilayered fully anisotropic media (MFAM) based on the transfer-matrix method (TMM) so that we could find out the energy transmission course (ETC) of plane waves that transmit through MFAM in the air background. Beginning with Maxwell's equations, MFAM's Helmholtz equation could be obtained by the known transverse- k vectors of plane waves. Through a series of complicated manipulations applied to the derivation of Helmholtz equations, we can attain ordinary differential equation of electric fields that merely exist in a certain direction and then compute four different eigenvalues in different MFAM regions. Under 3-D cases, meanwhile, concrete expressions of other fields can be obtained by given electric fields in the fixed direction. The ETC of electromagnetic fields in the tangential continuity is mainly considered in the TMM, and hence, the transferring matrix of electromagnetic field between MFAM regions can be constructed to achieve reflection and transmission coefficients. With various plane-wave modes such as transverse electromagnetic (TEM), transverse electric (TE), and transverse magnetic (TM) field modes, respectively, computational results of MFAM models are solved and adopted to compare with those from the commercial software of COMSOL. Finally, to discuss deeply and efficiently achieve color images, we acquire those reflection and transmission coefficients of more complicated fully anisotropic materials under the different transverse- k vectors. Our work can be an efficient and reliable solver for excavating prospective applications to predict ETC of models with single-layered/multilayered materials possessing fully anisotropy.

We use the two-dimensional (2-D) scattering matrix method (SMM) to analyze the slot characteristics in slotted single-mode semiconductor lasers and compare the results with those calculated by the one-dimensional transfer matrix method (TMM). The analysis shows that the 2-D SMM is required to accurately account for the measured results. Using the 2-D SMM simulation, we find that there is almost no reflection at the interface from slot to waveguide while a large reflection exists at the interface from waveguide to slot, and the power loss is much larger than the power reflected. For a single slot, the slot width has little influence on the slot reflectivity, which coincides with the measured results. The reflection and transmission of the slot are found to be exponentially dependent on the slot depth

This paper focuses on the reflection and transmission (R/T) problem of elastic waves in multilayered anisotropic structures. Stability analysis of the mixed variable method (MVM) for computing the R/T coefficients of elastic waves in multilayered anisotropic structures is presented. For this purpose, a detailed comparison of the MVM with the other two widely used methods, namely the transfer matrix method (TMM) and the stiffness matrix method (SMM), is made. Although the TMM, the SMM and the MVM are mathematically equivalent, they are quite different in numerical stability. The theoretical analysis shows that the MVM is unconditionally stable for arbitrary wavenumber–thickness products, whereas the TMM and the SMM may become unstable for large or small wavenumber–thickness products, respectively. This conclusion is numerically verified by various examples. Finally, the R/T coefficients of elastic waves in generally anisotropic multilayered structures bounded by two semi-infinite spaces are calculated using the MVM for a quasi-longitudinal or quasi-transverse wave incidence, and the effects of incident angles and wavenumber–thickness products on the R/T coefficients are discussed in detail through an example.

The Fresnel reflection and transmission coefficients for a homogeneous isotropic layer of finite thickness can be derived by using the Airy summation method or by applying the boundary conditions for the electric and magnetic fields at the boundaries of the slab. The two methods result in different expressions for the transmission and reflection coefficients. The reflection coefficients can be reduced to the simplest form. However, the transmission Fresnel coefficients cannot be transformed to the simplest form. Instead, they can be reduced to two different forms which have different phase constants. The transmission coefficient derived based on the Airy summation method does not depend on the index of refraction of the medium behind the slab while the transmission coefficient derived based on the electrodynamics boundary condition method does depend on the index of refraction of the medium behind the slab. The difference between the two expressions for the transmission coefficients stems from the choice of the phase of the transmitted wave as assumed in the two methods. The two different forms for the transmission coefficient of a slab lead to quantitatively and qualitatively different dependences of the real and imaginary parts of the transmission coefficient as a function of the index of refraction of the medium behind the slab. The striking difference in the numerical results is the oscillating form of the real and imaginary parts of the transmission coefficients and the phase constant as a function of the index of refraction of the medium behind the slab. We propose a generalized form of the Fresnel transmission coefficient of a slab.

Reflection and transmission of plane elastic waves in porous piezo-thermoelastic laminated plate immersed in a fluid has been studied. A theoretical model is derived which is based on transfer matrix method. The layered structure is considered to be consisting of ‘n’ number of layers of porous piezo-thermoelastic materials immersed in a fluid. All the layers are considered to be 2 mm PPTE materials. The formal solution for the mechanical displacements, electric potential, mechanical stresses, electric displacements, temperature changes and temperature gradient are derived. The transfer matrix technique is used to study the layered materials. The closed form expressions of the elements of transfer matrix are derived. The analytical expressions for the reflection coefficient and transmission coefficient are derived. The effects of frequency, angle of incidence, number of layers, layer’s thickness and porosity on the reflection coefficient and transmission coefficient, reflection loss and transmission loss are investigated numerically for different configurations. Particular cases of the present study have also been obtained for the verification of the results.

The input impedance of a brass instrument provides information about quality and playing characteristics, such as intonation and response. Modeling the input impedance of an instrument with the transfer matrix and finite element methods based on its geometry is useful, as the calculations can be performed without a physical prototype. Previous work has determined that both the transfer matrix and finite element methods can determine the properties of the impedances of mouthpieces and flaring bells. These results will be more closely examined and compared with regard to relative peak locations. At present, the transfer matrix method can only be used for conical or cylindrical segments along a straight axis, so it is useful to add curvature as a parameter of the calculation. An adaptation to the transfer matrix calculation for curved pipes will be presented. Transfer matrix and finite element calculations will be compared with each other and with impedance measurements. The ultimate goal of this work is to accurately calculate the input impedance of a realistic brass instrument, with curved pipes, valves, water keys, and slides. This will allow for improvement of existing instruments and development of new instruments and instrument parts exhibiting particular acoustical characteristics.