Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results

We present results of numerical computations obtained from a theory described in Part 1 of our current investigations [J. Opt. Soc. Am. A 17, 2081 (2000)]. We show that a segment of a paraboloid mirror produces an intensity distribution identical to that of a high-aperture lens. It is shown that when the convergence angle of the paraboloid is increased beyond the pi/2 limit, the lateral resolution in the direction orthogonal to the incident polarization improves, whereas in the other direction the resolution worsens. Numerical results show that paraboloid mirrors of high convergence angle exhibit dispersion; that is, when the focal length is altered by a quarter of the wavelength the intensity in the focus changes from its maximum to its minimum value. A focal shift is observed that, in the case of a paraboloid of low convergence angle is identical to the Fresnel shift. However, a focal shift is also observed at large convergence angles.     

Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory

Starting with the vector formulation of the Kirchhoff diffraction theory, expressions for the total energy density distribution along the axis are presented without using any of the usual assumptions except the assumption made by Kirchhoff for the boundary conditions of a black screen. To make the Kirchhoff integral compatible with Maxwell's equations, a line integral around the edge of the aperture is added in the analysis. The consequence of ignoring the contribution of this line integral to the axial field distribution is examined numerically. The focal shift effect is investigated for both aplanatic systems and parabolic mirrors having an arbitrary numerical aperture (NA) and finite value of the Fresnel number. The combined effects of the Fresnel number and NA on the focal shift are evaluated, and the validity of the results is carefully checked by comparing the wavelength with the system dimensions.     

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. I. Modeling mirror-scanning devices

Avector approach to tracing the path of a laser beam through an optical system containing movable plane mirrors is described, which permits a unified treatment of a number of basic mirror-scanning devices. We show that the scan field produced by the mirror-scanning system is a curved surface with a straight line as its generating element. The cross section of the scan field can be a circle, an ellipse, or a curve in the shape of an egg. Based on this understanding, some advanced topics are addressed, e.g., the relationship between the scan field and the scan pattern, the dependence of the scan pattern on the location and orientation of the observation surface, optical distortions in a scan pattern, spot-size enlargement caused by non-normal incidence of the scan beam on the observation plane, and so on. Design equations and curves are derived for the mirror-scanning devices that most frequently exist in linear and circular scan technology. Part II contains an analysis of the galvanometer-based optical scanner paddle scanner and the regular polygon. In Part III, X-Y scanning systems are studied.     

Efficient testing of segmented aspherical mirrors by use of reference plate and computer-generated holograms. I. Theory and system optimization

Telescopes with large aspherical primary mirrors collect more light and are therefore sought after by astronomers. Instead of large mirrors as a single piece, they can be made by use of numerous smaller segments. Because the segments must fit together to create the effect of a single mirror, segmented optics present unique challenges to fabrication and testing that are absent for monolithic optics. We have developed a new method for measuring large quantities of segments accurately, quickly, and economically using an interferometric test plate and computer-generated hologram (CGH). In this test, the aspheric mirror segments are interferometrically measured by use of a test plate with a best-fit spherical surface. The aspherical departure is accommodated with a small CGH that is imaged onto the test plates. The radius of curvature is tightly controlled by maintaining the gap between the test plate and the segment. We present a summary of the test and give the basic design tradeoffs for using a single system to measure all of the segments of a large aspheric mirror.     

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.     

Hydrodynamics of rotating liquid mirrors. I. Synchronous disturbances

The effects of axis alignment errors, planetary rotation, and tidal forces on rotating liquid mirrors are analyzed. These produce a surface distortion that decreases exponentially with distance inward from the rim with a characteristic length l= square root of 3hf/2, where h is the thickness of the fluid and f is the focal length. Even a small tilt of the rotation axis can produce a significant deformation of the optical surface. The maximum surface height error is 3epsilonl, where epsilon is the tilt angle and is typically of the order of 1.5 microm for a 1 arc sec tilt. The main optical effect of the wave is to produce a ring, with angular diameter 6epsilon, offset by half of the diameter in the direction opposite the tilt. This diamond ring aberration can be avoided by accurate alignment of the rotation axis or by masking the outer few centimeters of the mirror. Planetary rotation produces a small deformation of the order of 100 nm for a 10 m telescope at low latitude on Earth. This deformation can be canceled by a small tilt of the rotation axis. Tidal forces produced by the Moon, or by the Earth in the case of a lunar telescope, produce an inconsequential, subnanometer, surface deformation.     

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.     

Reflection and transmission properties of holographic mirrors and holographic Fabry-Perot filters. I. Holographic mirrors with monochromatic light

The reflection and transmission coefficients of holographic mirrors (HM's) and holographic Fabry-Perot filters (HFPF's) are studied. The diffraction efficiencies, the angular selectivities, and the wavelength sensitivities of HM's and HFPF's and their effects on the spectrum and on the coherence properties of the incident light are investigated. The results show that holographic techniques can indeed be used to construct high-quality reflecting mirrors and Fabry-Perot filters. HM's illuminated by a monochromatic plane wave are investigated in detail. The reflection and transmission coefficients are first studied as functions of frequency (wavelength) of the incident light and also as functions of the angle of incidence. It is shown that HM's can be used as highly reflecting mirrors around their Bragg angle of incidences. Their reflectivity curves have fairly wide angular- and frequency-selectivity ranges.     

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.     

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.     

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.