Average lattices and aperiodic structures

Statistically averaged lattices provide a common basis to understand the diffraction properties of structures displaying deviations from regular crystal structures. An average lattice is defined and examples are given in one and two dimensions along with their diffraction patterns. The absence of periodicity in reciprocal space corresponding to aperiodic structures is shown to arise out of different projected spacings that are irrationally related, when the grid points are projected along the chosen coordinate axes. It is shown that the projected length scales are important factors which determine the existence or absence of observable periodicity in the diffraction pattern more than the sequence of arrangement.     

Magneto-optical resonances in periodic dielectric structures magnetized in plane

We studied the magneto-optical properties of a magnetized diffractive structure composed of a binary diffraction grating and a homogeneous layer. The structure is magnetized in-plane and the magnetization vector is perpendicular to the grating slits. By the electromagnetic modeling, the structure is shown to produce magneto-optical effects in the form of the resonant change of transmittance and reflectance in response to changing magnetization. The said resonant effects are found to occur both for TE- and TM-polarization of the incident wave. The dispersion curves for the structure eigenmodes have been derived by calculating the complex poles of the scattering matrix. Analysis of the dispersion curves shows that frequencies of the magneto-optical resonances coincide with those of the structure eigenmodes. We studied in which way the dispersion curves vary when the structure material is being magnetized. We propose a classification of magneto-optical resonances based on their relation with the symmetry of eigenmodes excited in the structure.     

Design of aperiodic multilayer structures for attosecond pulses in the extreme ultraviolet

The design of aperiodic reflecting multilayer (ML) structures for attosecond physics in the extreme ultraviolet spectral region is presented. An optimization procedure based on "evolutive strategy" has been developed in order to get coating structures reflecting high photon fluxes in ultrashort duration pulses. The MLs are designed for a specific (75-105 eV) spectral interval with suitable reflectance and phase characteristics, in particular high total spectral reflectivity coupled with very wide bandwidth, spectral phase compensation, and amplitude reshaping. Furthermore, to take into account manufacturing tolerances, solutions stable with respect to random layer thickness variations are selected. To test the reliability of the proposed design procedure, examples of Mo/Si ML structures designed to reflect ultrashort pulses with different amplitude profiles and phase behavior are considered. The performances of the various structures are analyzed.     

Light scattering resonances in small particles with electric and magnetic properties

Lorenz-Mie resonances produced by small spheres are analyzed as a function of their size and optical properties (epsi > or < 0, mu > or < 0). New generalized (mu not equal to 1) approximate and compact expressions of the first four Lorenz-Mie coefficients (a1, b1, a2, and b2) are calculated. With these expressions and for small particles with various values of epsi and mu, the extinction cross section (Q(ext)) is calculated and analyzed, in particular for resonant conditions. The dependence on particle size of the extinction resonance, together with the resonance shape (FWHM), is also analyzed. In addition to the former analysis, a study of the scattering diagrams for some interesting values of epsi and mu is also presented.     

Narrow resonances and ripple fluctuations in light scattering by a spheroid

We develop a semiclassical theory to explain the rapid ripple fluctuations in the extinction efficiency of light scattering by a transparent prolate spheroid. The theory is based on uniform asymptotic expansion of spheroidal radial functions. We have calculated the extinction efficiency for normal and oblique incidence. Our results suggest that the excitation of resonant electromagnetic modes inside a spheroidal particle is an important factor in the ripple structure. To verify this assumption and based on a Breit-Wigner formula, we develop a method to fit the peaks that appear in the spheroid's extinction cross section when some scattering parameters vary. In other words, our calculations suggest that narrow resonances are related to ripple fluctuations, whereas broad resonances contribute to extinction cross-sectional background.     

Dispersion relation of guided-mode resonances in multimode grating waveguide structures

It is well known that the location of guided-mode resonance (GMR) in grating waveguide structures closely tracks the leaky mode dispersion curves. In this paper, taking Bragg reflection due to periodicity and interaction between different modes into account, we first present a schematic diagram of the dispersion relations of leaky modes in multimode grating waveguide structures, both for s-polarized (TE mode) and p-polarized (TM mode) incident waves. Due to the perturbation of the grating layer, the interaction between different resonance modes (transverse standing waves) is inevitable. This transverse interference will result in the non-Bragg nature resonance band gaps in the dispersion curves. Exploiting the characteristics of leaky mode dispersions over the full range of the first Brillouin zone, we hoped we could gain some insight into the relationship among the mode interactions, band gaps, and their benefits to optical elements utilizing the GMR effect in grating waveguide structures. Finally, a specific structure is analyzed.     

Mie resonances and Bragg-like multiple scattering in opacity of two-dimensional photonic crystals

The lowest (main) and high-order Mie resonances and the Bragg-like multiple scattering of electromagnetic (EM) waves are determined as three mechanisms of formation and frequency position of two opaque bands, with narrow peaks in one of the bands in the transmission spectra of 2D photonic crystals composed of dielectric cylinders arranged parallel to the EM wave's electric vector in the square lattice. The main Mie resonance in a single cylinder defines the frequency position of the main gap whose formation results from the Bragg-like scattering. An additional gap with narrow transmission peaks opens in the spectrum of a cylinder layer and becomes pronounced with the number of layers. It is argued that higher-order Mie resonances are responsible for the transmission peaks within the additional band of a perfect crystal. It is shown that 2D photonic crystals with a filling factor ranging from 3% to 20% at a fixed crystal period may be a good zero approximation to study wave transmission through a localizing 2D dense random medium slab.     

Random excitation of nonlinear elastic structures with internal resonances

The influence of random vibration on the design of mechanical components has been restricted to the linear theory of small oscillations. However, this theory is inadequate and fails to predict the complex response characteristics which may be observed experimentally and can only be predicted by employing the nonlinear theory. This paper presents a brief overview of the basic nonlinear phenomena associated with nonlinear random vibration. An example of a clamped-clamped beam under filtered white noise excitation in the neighbourhood of 1:1 internal resonance condition is considered. Three approaches are employed to examine the response and stochastic bifurcation of the beam coupled modes. These are the Fokker-Planck equation together with closure schemes, Monte Carlo simulation, and experimental testing. The analytical results are compared with those determined by Monte Carlo simulation. It is found that above a critical static buckling load the analytical results fail to predict the snap-through phenomenon, while both Monte Carlo simulation and experimental results reveal the occurrence of snap-through. The bifurcation of second mode is studied in terms of excitation level, internal detuning and damping ratios. It is found that below the critical load parameter, the response statistics do not significantly deviate from normality. Above the critical value, where snap-through takes place, the response is strongly non-Gaussian. This research is supported by a grant from the National Science Foundation under grant number MSS-9203733 and by additional funds from the Institute for Manufacturing Research at Wayne State University.     

Improved dynamic analysis of structures with mechanical uncertainties under deterministic input

This paper addresses the dynamic analysis of linear systems with uncertain parameters subjected to deterministic excitation. The conventional methods dealing with stochastic structures are based on series expansion of stochastic quantities with respect to uncertain parameters, by means of either Taylor expansion, perturbation technique or Neumann expansion and evaluate the first- and second-order moments of the response by solving deterministic equations. Unfortunately, these methods lead to significant error when the coefficients of variation of uncertainties are relatively large. Herein, an improved first-order perturbation approach is proposed, which considers the stochastic quantities as the sum of their mean and deviation. Comparisons with conventional second-order perturbation approach and Monte Carlo simulations illustrate the efficiency of the proposed method. Applications are discussed in order to investigate the influence of mass, damping and stiffness uncertainty on the dynamic response of the system.     

Studies on ultrasonic scattering from quasi-periodic structures

This study extends the work done on nonuniform phase statistics by including additional results based on quasi-periodic scattering. Three parameters are used to predict the presence of regular structure within the region of interest. The signal-to-noise ratio of phase and the chi (2) statistic resulting from conducting a goodness of fit test are two parameters used to verify whether the phase signal followed a uniform distribution. A third parameter, the power spectral density (PSD), was studied and its ability to provide information on the level of periodicity present was analyzed. Computer simulations and experiments on tissue mimicking phantoms were carried out, the results of which indicate that the parameters introduced in this paper have good potential in providing a better understanding of scattering from a collection of quasi-periodic scatterers.     

Diffraction and scattering at antiglare structures for display devices

Antiglare layers for display devices feature a rough surface structure that scatters the incident light and thereby reduces the specular reflectivity of the screen. The same structure will, however, also diffuse part of the light from the phosphor layer, resulting in a degradation of image quality. The microstructure, light-scattering properties, and effects on image contrast and resolution of some relevant antiglare structures are examined. A model based on scalar diffraction theory that relates the various optical properties to the surface microstructure is presented.     

Analysis of the influence of absorption on spatial field structures of morphology-dependent resonances in cylindrical microparticles

Morphology-dependent resonances in cylindrical microparticles are investigated and the properties of particle matter considered. The spatial structures of resonance modes inside microparticles are studied. It is shown that the resonant mode with a lesser quality factor can have a higher value of internal field intensity inside microparticles. Considering a small amount of absorption of particle matter permits more-or-less exact prediction of the value of the internal field intensity, which may increase or decrease, depending on the properties of the particle matter.     

A general n-fractal aperiodic zone plate

We propose a general n-fractal aperiodic zone plate generating a beam with the first-order foci with more chosen positions. The construction method of the general n-fractal aperiodic zone plate can apply to many aperiodic zone plates, e.g. fractal, Thue-Morse and Fibonacci zone plates, and the general n-fractal aperiodic zone plates can generate more adjustable first-order foci. Parameters of the n-fractal aperiodic zone plate such as the substitution rule and the fractal order are examined and found not to affect the fractal distribution of the first-order foci. The more adjustable first-order foci generated by the general n-fractal aperiodic zone plate have been verified with both simulations and experiments.     

Ultrasonic inverse scattering of multidimensional objects buried in multilayered elastic background structures

The authors focus on the multidimensional inverse scattering of objects buried in an inhomogeneous elastic background structure. The medium is probed by an ultrasonic force and the scattered field is observed along a receiver array. The goal is to retrieve both the geometry (imaging problem) and the constitutive parameters (inverse problem) of the object through an appropriate multiparameter direct linear inversion. The problem is cast in terms of a vector integral equation elastic scattering framework. The multidimensional inverse scattering problem, being nonlinear and ill-posed, is linearized within the Born approximation for inhomogeneous background, and a minimum-norm least-square solution to the discretized version of the vector integral formulation is sought. The solution is based on a singular value decomposition of the forward operator matrix. The method is illustrated on a 2-D problem where constrained least-square inversion of the object is performed from synthetic data. A Tikhonov regularization scheme is examined and compared to the minimum-norm least-square estimate.     

Plane-wave-time-domain-enhanced marching-on-in-time scheme for analyzing scattering from homogeneous dielectric structures

A novel and fast integral-equation-based scheme is presented for analyzing transient electromagnetic scattering from homogeneous, isotropic, and nondispersive bodies. The computational complexity of classical marching-on-in-time (MOT) methods for solving time-domain integral equations governing electromagnetic scattering phenomena involving homogeneous penetrable bodies scales as O(NtNs2). Here, Nt represents the number of time steps in the analysis, and Ns denotes the number of spatial degrees of freedom of the discretized electric and magnetic currents on the body's surface. In contrast, the computational complexity of the proposed plane-wave-time-domain-enhanced MOT solver scales as O(NtNs log2Ns). Numerical results that demonstrate the accuracy and the efficacy of the scheme are presented.     

Using modified nodal analysis in cavity-mode resonances printed circuit board power bus structures with the segmentation method

In this paper, we used the modified nodal analysis (MNA) method to obtain the cavity-mode equation for the segmentation method. In the experiment, the calculation result of MNA is more accurate than traditional double summation with discrete capacitors. The proposed approaches decrease computation and easily add discrete capacitors. The MNA and segmentation methods can be combined to conveniently calculate an irregularly shaped cavity by using computers. This approach can reduce the time required for deriving the impedance equation. The result can be used to build a mathematical model of the MNA method, which can suppress cavity-mode resonances within the power bus by using discrete capacitors. We used electromagnetic simulation software to verify these models.     

Simulation of scattering processes of optical radiation by nanodimensional structures

Mathematical simulation of the process of scattering of an optical wave by micro- or nano-scale structure on the basis of the vector equations of the electromagnetic field is performed. A rigorous analytic solution is obtained and the scattering problem solved by means of the method of finite elements.     

Reconstruction of periodic structures from optical scattering measurements using adjoint equations

A new numerical approach to efficiently reconstruct the profile of a grating from measurements of reflection coefficients is demonstrated. The problem is posed in the mathematical framework of an inverse scattering problem and solved using gradient-based algorithms. The gradient is computed efficiently using adjoint equations, which amounts to an extra scattering computation per iteration. For symmetric profiles it is shown that only knowledge of the scattered field is sufficient to compute the gradient. As a result, complex profiles can be reconstructed rapidly, and the method can be potentially used in metrology applications in semiconductors. The technique is demonstrated for the case of TE polarization.