Focusing of electromagnetic waves by paraboloid mirrors. I. Theory

We derive a solution to the problem of a plane electromagnetic wave focused by a parabolic mirror. The solution is obtained from the Stratton-Chu integral by solving a boundary-value problem. Our solution can be considered self-consistent. We also derive the far-field, i.e., Debye, approximation of our formulas. The solution shows that when the paraboloid is infinite, its focusing properties exhibit a dispersive behavior; that is, the structure of the field distribution in the vicinity of the focus strongly depends on the wavelength of the illumination. We show that for an infinite paraboloid the confinement of the focused energy worsens, with the energy distribution spreading in the focal plane. 2000 Optical Society of America [S0740-3232(00)01309-0] OCIS codes: 260.0260, 260.2110, 050.1960, 260.5430.     

Focal shifts in diffracted converging electromagnetic waves. II. Rayleigh theory

Part II of this study is an application of the Rayleigh vector diffraction integrals to an investigation of the effect of focal shifts in converging spherical waves diffracted in systems of arbitrary relative aperture. The results are compared numerically with those obtained in Part I [J. Opt. Soc. Am. A 22, 68 (2005)] from the Kirchhoff vector diffraction theory. The effect of the numerical aperture (NA) on focal shifts can be considered in two regions: When NA < or = 0.5 the system behaves like an paraxial system, and the Fresnel number is the dominant factor. When 0.5 < NA < or = 0.9 the absolute value of the relative focal shift decreases with increasing value of NA.     

Focusing diffractive cylindrical mirrors: rigorous evaluation of various design methods.

The performance characteristics of focusing diffractive mirrors designed with various methods are evaluated by using the rigorous boundary element method. Quantitative results are presented for (1) conventional-zero-thickness mirror designs, (2) alternative-zero-thickness designs that incorporate an off-axis correction factor and (3) finite-thickness designs. For TM polarization, the mirrors designed by using the alternative-zero-thickness method perform considerably worse than those designed by using the conventional-zero-thickness method, which contradicts predictions made in an earlier paper.     

Focusing of electromagnetic radiation by hyperboloidal and ellipsoidal lenses

A solution to the problem of plane electromagnetic waves focused by an ellipsoidal or a hyperboloidal lens is derived from the Stratton-Chu integral by solving a boundary-value problem. The current method is more rigorous than those hitherto published in the literature. Results show that for linearly polarized incident illumination and in the vicinity of the focus, the distribution of the time-averaged electric energy density is almost fully transverse electric.     

Scattering of electromagnetic waves by a perfect electromagnetic conducting sphero

An exact solution to the problem of scattering of electromagnetic waves from a perfect electromagnetic conducting spheroid is presented, using the method of separation of variables. The formulation of the problem is realised by expanding the incident as well as the scattered electromagnetic fields in terms of appropriate spheroidal vector wave functions and imposing the appropriate boundary conditions at the surface of the spheroid. This generates a set of simultaneous equations, the solution of which yields the unknown coefficients associated with the expansion of the scattered electromagnetic field. Results are presented in the form of normalised bistatic and backscattering cross-sections for spheroids of different axial ratios, sizes and admittances, for both transverse electric and transverse magnetic polarisations of the incident wave.     

Laser beam scanning by rotary mirrors. II. Conic-section scan patterns

Part II of this study is an application of the general theory of Part I to the following scanners: the galvanometer-based scanner, the paddle scanner, and the regular polygon. The scan field produced by these scanners is (or approximates) a circular cone. Therefore the scan pattern on the plane of observation can be one of the following curves, circle, ellipse, parabola, or hyperbola, depending on the position and orientation of the plane. Special topics to be addressed are (1) the effect of input offset, (2) the locus of the instantaneous scan center and the waist of the scan field, (3) the scanning on curved surfaces, and (4) the generalization of the scan-field expression. In Part III, X-Y scanning will be studied.     

Focusing properties of hemispherical mirrors for total integrating scattering instruments

Focusing properties are derived of a hemispherical mirror (also called a Coblentz sphere) used in total integrated scattering (TIS) instruments for the measurement of surface roughness. Analytical formulas for the scattered-light dependence on scattering angle and on the position of the photodetector and sample (with respect to the hemisphere) are given. From these formulas one can estimate useful parameters related to the construction and performance evaluation of TIS instruments, including spherical aberration of the hemispherical mirror, solid angle of the scattered light collected by a photodetector of a given size, and optimal size of the photodetector and its position with respect to the basal plane of the mirror.     

Modified fundamental solutions for the scattering of elastic waves by a cavity: Numerical results

The boundary integral equation method is very often used to solve exterior problems of scattering of waves (elastic waves, acoustic waves, water waves and electromagnetic waves). It is known, however, that this method fails to provide a unique solution at the so-called irregular frequencies. This difficulty is inherent to the method used rather than the nature of the problem. In the context of elastodynamics. we proposed, in a recent work¹, two methods for eliminating these irregular frequencies. Both are based on modifying the fundamental solution. Here we present numerical results pertaining to the solutions of the modified and unmodified integral equations.     

Scattering of plane electromagnetic waves by a chiral-metal cylinder

The problem of scattering of E-and H-polarized plane electromagnetic waves by a metal cylinder covered with a chiral layer is solved by the method of partial regions. The scattering field is studied in the near and far zones. The correlation between the type of polarization of the incident electromagnetic wave and the magnitude of the depolarized component of the scattered field is considered.     

Diffraction of electromagnetic waves by a multielement diffraction grating

A rigorously posed electrodynamical problem of diffraction of a plane wave by a multielement diffraction grating that consists of cylindrical screens with arbitrary curvature is solved by the method of integral equations. We construct an integral representation of the Green function of the problem under consideration which gives a significant increase in the accuracy of calculations and investigate the phenomenon of resonance propagation of an electromagnetic wave along the axis of a two-layer grating irradiated with aE-polarized wave which propagates along the normal to the grating.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 2, pp. 87–98, March – April, 1995.     

Diffraction of electromagnetic waves by periodic arrays of rectangular cylinders

Reflection, transmission, and absorption of electromagnetic waves by periodic arrays of conducting or dielectric rectangular cylinders are studied by a finite-difference time-domain technique. Truncated gratings made of lossless and lossy conducting and dielectric elements are considered. Results for surface current density, transmission, and reflection coefficients are calculated and compared with corresponding results in the literature, which are obtained by approximate or rigorous methods applicable only to idealized infinite models. An excellent agreement is observed in all cases, which demonstrates the accuracy and efficacy of our proposed analysis technique. Additionally, this numerical method easily analyzes practical gratings that contain a finite number of elements made of lossless, lossy, or even inhomogeneous materials. The results rapidly approach those for the idealized infinite arrays as the number of elements is increased. The method can also solve nested gratings, stacked gratings, and holographic gratings with little analytical or computational effort.     

Generation of electromagnetic waves by a rotating cylindrical electron layer

Generation of electromagnetic waves by electrons orbiting in crossed radial electrostatic and axial magnetic fields was studied. Frequencies and increments of the generated waves were calculated. The wave increments increase with the frequency for sufficiently large values of the radial electrostatic field strength. Strong radial electrostatic fields may considerably magnify the frequencies of waves generated in the systems.     

Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals

We present a numerical study of bidimensional photonic crystals with an emphasis on the behavior of the gaps versus the polarization and the conicity of the incident plane wave. We use a rigorous modal theory of diffraction at oblique incidence by a set of arbitrarily shaped parallel fibers. This theory allows the study of the refractive properties of bidimensional photonic crystals. We develop a heuristic method of homogenization that allows us to predict the position of the gaps and their behavior with respect to the polarization and the conicity angle. With this homogenization scheme, we also present some important elements for obtaining full gaps.     

Reflection and transmission properties of holographic mirrors and holographic Fabry-Perot filters. II. Holographic mirrors with partially coherent light

The reflection and transmission properties of holographic mirrors (HM's) under partially coherent illumination are investigated with emphasis on the properties of reflected light. The effects of a HM on the spectrum and on the coherence properties of partially coherent incident light are studied. We show that within angular and frequency-selectivity ranges of HM's (where their reflectivities are nearly uniform), the changes in the spectrum and in the coherence properties are negligible. Yet changes are expected when the spectrum of the incident light falls beyond frequency-selectivity ranges of the HM's or when the spectrum of the reflected light is observed beyond the angular-selectivity ranges of the mirrors. We illustrate the results by considering in detail the performance of HM's when the incident light is produced by planar secondary Gaussian Schell-model sources. Some computed curves are presented.     

Focusing of Rayleigh waves: simulation and experiments

A linear array of surface wave transducers has been developed to generate focused surface wave motion. A novel theoretical approach, whereby time-harmonic surface wave motion is represented by a carrier wave that satisfies a reduced wave equation on the surface of the body and supports the subsurface motion, is used to model the beam generated by a single element of the array. Comparison of theoretical and experimental results show that, for a single element, the opening angle of the beam is about 20 degrees and its cross-section can be represented by a Gaussian distribution of the normal displacements. For an eight-element array, the focused beam is subsequently obtained by superposition considerations. For the focused beam comparisons of theoretical and experimental results, in which the latter have been obtained by the use of a laser interferometer, show excellent agreement both for the normal displacements along a radial line and across the width of the beam. The array can be used for self-focusing of surface waves on a surface defect.     

A study of mathematical programmingmethods for structural optimization. Part II: Numerical results

Various mathematical programming methods for structural optimization are studied. In a companion paper, these methods have been studied based on certain theoretical considerations. In this paper, the methods are studied based on solving a set of test problems. The methods that are studied include recursive QP, feasible directions, gradient projection, SUMT and multiplier methods. Various computer codes have been developed, and are studied together with some existing programs such as CONMIN and OPTDYN. The test problems considered have 3–47 design variables and 3–252 constraints. The evaluation criteria consist of studying the accuracy, reliability and efficiency of a code. It turns out that globally convergent algorithms (multiplier methods, in particular) are very reliable but not efficient. Primal algorithms (like CONMIN), which are not proved to be globally convergent, are efficient but not reliable.     

On the theory of elastic scattering of electromagnetic waves by atoms

A Hamiltonian describing the elastic interaction of electromagnetic radiation (EMR) with an atom is obtained using the invariant theory of perturbations in the limit of EMR wavelengths λ significantly exceeding the atom size a₀. An exact expression for the interaction amplitude is obtained, and the probability of EMR scattering on the atom is calculated. It is established that the scattering probability at large λ is proportional to the squared frequency of monochromatic EMR. It is shown that, in the limit of large wavelengths, the formula h∼(ω/c)⁴ v ₀ for the extinction coefficient is inapplicable and the relation h=Aω² becomes valid, where A is a definite coefficient.