Three-dimensional electromagnetic diffraction of a Gaussian beam by a perfectly conducting half-plane

Formulas are derived for the diffraction of a three-dimensional electromagnetic Gaussian beam by a perfectly conducting half-plane. The beam can be incident from any direction, and the main component of the electric field can point in any direction on the plane of the beam waist. The center of the beam waist is on the edge of the half-plane. The incident beam is constructed as a superposition of plane waves, and the total diffracted field is obtained from a superposition of the diffracted fields that are due to each plane wave. Physical constraints that limit the size and direction of the beam relative to the half-plane are described and incorporated into the theory. The scattered field in the far zone is obtained by asymptotic evaluation of the general formulas. Graphical results for the near-field as well as far-field patterns are presented and discussed.     

High-frequency diffraction of a electromagnetic plane wave by an imperfectly conducting rectangular cylinder

We shall consider the problem of determining the scattered far-wave field produced when a plane E-polarized wave is incident on an imperfectly conducting rectangular cylinder. On the basis of the uniform asymptotic solution for the problem of the diffraction of a plane wave by a right-angled impedance wedge, in conjunction with Keller’s method and multiple diffraction, a high-frequency far-field solution to the problem is given for two edge diffractions.     

Vector diffraction analysis of optical disk readout

The optical disk readout signals from ROM disks are presented by use of a rigorous three-dimensional vector diffraction method. The optical disk is modeled as a crossed metal grating without restriction on the form of the information marks, and the permittivity of the metal is taken into account. The diffracted field from the disk is obtained by means of decomposing the focused incident beam into a spectrum of plane waves and then calculating the diffracted plane waves for each respective incident component. The readout signal is obtained by integration of the energy-flux density of the diffracted field according to the detection scheme of the optical disk system. A typical digital versatile disk (DVD) system is applied with this theory, and the result is far from that of scalar diffraction theory.     

Analysis of Far-field Diffraction Characteristics of Wavelength-sized Surface Relief Gratings

The rigorous boundary matching technique was used to analyse the far-field diffraction from wavelength-sized surface relief gratings for TE polarization. Diffraction characteristics of such surface relief gratings were studied as functions of complex refractive index, groove depth and the angle of incidence. The surface field at the bottom of the grooves is more severely influenced by the edges and walls for metallic material than dielectric ones. The reflectivities of gold gratings show significant fluctuation for different incident wavelengths. This can be clearly seen from the fact that the zeroth-order diffraction efficiency DE ₀ of gold is lower than that of aluminium at u = 0·4 wm but higher at u = 1·0 wm. It is also shown that the depth of the grooves has significant influences on the diffraction efficiencies, an optimum depth can be chosen to minimize the specular component. However, this optimum depth value is found to be different for the gratings of different materials, e.g. aluminium and gold, although the gratings have exactly the same size and shape. The effect of the incident angle, particularly at critical angles when certain diffracted waves are just about to be cut off was discussed. The zeroth-order diffraction efficiencies are found to change noticeably at the critical angles.     

Scattering of inhomogeneous plane waves by a slit formed with impedance and perfectly electric conducting half planes

Scattering of an inhomogeneous plane wave with an arbitrary angle of incidence travelling through a slit made of perfectly electric conducting and impedance half planes is investigated. For the investigation of the scattering phenomena evaluation of the modified theory of physical optics integrals are evaluated asymptotically. An inhomogeneous plane wave is taken into consideration by assuming the incident angle of a homogeneous plane wave as a complex parameter. Uniform asymptotic results will be employed for the correct solution of an incoming inhomogeneous plane wave scattering problem. Asymptotic evaluation is carried out for the reflected and diffracted fields. Diffracted fields are uniformly expressed in terms of the Fresnel functions. To obtain correct plots of the diffracted fields, complex detour parameter decomposition method is applied. Obtained resultant fields are plotted numerically.     

Optical Diffraction in Close Proximity to Plane Apertures. I. Boundary-Value Solutions for Circular Apertures and Slits

In this paper the classical Rayleigh-Sommerfeld and Kirchhoff boundary-value diffraction integrals are solved in closed form for circular apertures and slits illuminated by normally incident plane waves. The mathematical expressions obtained involve no simplifying approximations and are free of singularities, except in the aperture plane itself. Their use for numerical computations was straightforward and provided new insight into the nature of diffraction in the near zone where the Fresnel approximation does not apply. The Rayleigh-Sommerfeld integrals were found to be very similar to each other, so that polarization effects appear to be negligibly small. On the other hand, they differ substantially at sub-wavelength differences from the aperture plane and do not correctly describe the diffracted field as an analytical continuation of the incident geometrical field.     

Diffraction of homogeneous and inhomogeneous plane waves by a planar junction between perfectly conducting and impedance half-planes

The problem of diffraction of homogeneous and inhomogeneous plane waves at the discontinuity formed by perfectly conducting and impedance half-planes is examined by the method of modified theory of physical optics (MTPO). The MTPO integral of the reflected scattered waves by the perfectly conducting half-plane is reconstructed in order to include the effect of the diffracted wave coming from the edge of the impedance half-plane. The integrals are evaluated by a uniform asymptotic method. The results are plotted numerically and compared with the literature.     

Quasi-sudden brittle fracture at both edges of a finite crack

The diffraction of a plane horizontally polarized shear wave by a crack of finite length is analyzed and the extension of both crack edges prior to the arrival of the first diffracted waves, i.e. quasi-sudden fracture, is studied. In light of an energy rate balance criterion it is found that for an incident step-stress pulse, quasi-sudden fracture may occur but always at both crack edges, often initiating at the trailing edge first. For an incident wave whose stress vanishes at the wavefront, however, quasi-sudden fracture may occur only at the leading crack edge, or if at both edges, at the leading edge first. For both waveforms, the rate of crack extension is non-constant and increases rapidly so that crack branching may be expected. Finally instantaneous crack extension at a uniform rate is possible only if the incident wave stress possesses a square-root sinularity at the wavefront. This result agrees with earlier work by Achenbach.     

Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization

A rigorous modal theory for the diffraction of Gaussian beams from N equally spaced slits (finite grating) in a planar perfectly conducting thin screen is presented. The case of normal incidence and TE polarization state is considered; i.e., the electric field is parallel to the slits. The characteristics of the far-field diffraction patterns, the transmission coefficient, and the normally diffracted energy as a function of several optogeometrical parameters are analyzed within the so-called vectorial region, where the polarization effects are important. The diffraction pattern of an aperiodic grating is also considered. In addition, one diffraction property known to be valid in the scalar region is generalized to the vectorial region: the existence of constant-intensity angles in the far field when the incident beam wave is scanned along the N slits. The classical grating equation is tested for incident Gaussian beams under several conditions.     

Peculiarities of the angular selectivity of 3D holograms in the direction of nondispersive angular mismatch

Influence of the angular mismatch between the hologram and the reconstructing beam in the direction perpendicular to the plane of dispersion on the efficiency of diffraction was studied. It is demonstrated that angular detuning in this direction can lead to substantial increase of the intensity of the diffracted beam if the Bragg conditions are met. The results of calculation of the “anomalous” behavior of the diffraction efficiency are in good agreement with the experimental data for holograms recorded in the layers of a photopolymer.     

The theory of the boundary diffraction wave for wedge diffraction

A new potential function, line integration which gives the edge diffracted fields, is constructed for wedge diffraction by using the method of modified theory of physical optics. The surface integrals are transformed into line integrals by the technique of asymptotic reduction. As an application of the novel potential function, the diffracted field is obtained for the geometry of a wedge for arbitrary incidence of plane waves.     

Diffraction of an e.m. Wave by an Absorbing Circular Cylinder

Some expressions for the e.m. field diffracted by an absorbing circular cylinder of large radius in the Fresnel region are developed. Until recently results were available only for the far field of ‘good conductors’. The derived equations agree with Young's theory of diffraction since the diffracted field may be expressed as two cylindrical waves which seem to arise from the two ‘edges’ of the cylinder. Numerical calculations are also presented and compared with previous experimental results.     

Transmission of an inhomogeneous plane wave through an electrically small aperture in a perfectly conducting plane screen

Most solutions for electromagnetic diffraction by a circular aperture in a perfectly conducting plane screen are for an incident homogeneous (propagating) plane wave. When the aperture is electrically small (dimensions small compared to the wavelength), the well-known transmission coefficient behaves as the fourth power of the diameter/wavelength. We consider the case in which the incident field is an inhomogeneous (evanescent) plane wave. Numerical calculations for the electrically small circular aperture show that the transmission coefficient for an inhomogeneous plane wave can be substantially greater than for a homogeneous plane wave at the same frequency. This observation may be helpful in explaining the increased transmission recently reported for electrically small apertures in plane screens with modifications. The numerical calculations for the electrically small aperture are in agreement with results from approximate analytical expressions that are based on the equivalent electric and magnetic dipole moments for the electrically small complementary disk.     

Uniform theory for the diffraction of evanescent plane waves

A uniform diffracted field is obtained in terms of Fresnel functions with complex argument by subtracting the unit step function from the Fresnel integral. The method is applied to the problem of diffraction of inhomogeneous plane waves by a perfectly conducting half-plane and wedge. The results are plotted numerically and compared with results reported in the literature.     

Theory of electromagnetic monitoring of conducting conical bodies

We present the solution of an axisymmetric problem of diffraction of electromagnetic fields on perfectly conducting conical shells. By assuming that the required solution is representable as a Kontorovich-Lebedev integral, we reduce the problem under investigation to a Fredholm integral equation of the first kind, whose solution has the form of a series in Bessel functions. For a radial electric dipole placed on the symmetry axis of the cone, the problem is reduced to two coupled infinite linear systems in the unknown expansion coefficients. We consider the case of low frequencies. The expression for the scattering diagram is obtained for the case of diffraction of plane waves by passing to the limit. The characteristics of the diffracted field and the possibility of control over certain parameters of the conical shell are studied numerically.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 30, No. 5, pp. 49–58, September–October, 1995.     

Analysis of diffracted fields with the extended theory of the boundary diffraction wave for impedance surfaces

Uniform diffracted fields from impedance surfaces are investigated by the extended theory of boundary diffraction wave (ETBDW). The new vector potential of the ETBDW is constructed by considering the pseudoimpedance boundary condition. The method is applied to the diffraction problem from an impedance half-plane. It is shown that the total fields from an impedance half-plane reduce to the case of a perfectly electric or magnetic conducting and opaque half-plane for special values of surface impedance. The total and diffracted fields are compared numerically with the exact solution for the impedance half-plane and modified theory of physical optics (MTPO) solution for an impedance wedge. The numerical results show that the field expressions are in very good agreement with the exact and MTPO solutions.     

Determination of longitudinal and transverse shifts of three-dimensional Gaussian beams reflected at a grating near an anomaly

Spatial displacements of three-dimensional Gaussian beams diffracted at a reflection grating are studied theoretically and numerically. The complex diffraction coefficients (amplitudes and phases) of the grating diffracted plane waves calculated by a rigorous method for conical diffraction are the basis for this investigation. The classical analytical formula for the longitudinal shift depending on the gradient of the reflection phase is generalized to the simultaneous analytical treatment of the longitudinal Goos—Hänchen like shift as well as a transverse shift. A second method uses the full integration on the whole spectrum of plane waves of the diffracted beam.     

Polarizationally Non-sensitive Pseudo-holographic Computer Reconstruction of Random Rough Surface

Abstract

The function, describing a profile of a random rough surface (RRS) is expanded in a Fourier series, i.e. the surface is considered as a composition of sinusoidal gratings. The total diffracted optical field from this RRS is a sum of the fields due to all harmonic gratings, since Kirchhoff's condition for ‘locally flat surface’ is realized for each harmonic grating at a given light wavelength and at an appropriate choice of the basic grating period. The registered s and p components of the diffracted (+1 diffraction orders of each harmonic gratings), incident and mixed optic fields are separated with an optical analyser. These fields are experimentally measured and from these values the phase and the amplitude of each grating are determined. The profile of the surface is reconstructed for s and p polarization of the light scattered field, when the electric vector of the incident light concludes an arbitrary angle with the incidence plane. The mean roughness is determined in both cases. It is shown, that both reconstructions of the profile and the determination of the mean roughness are not dependent on the polarization of the incident light. The separation of the s and p components is of great importance at the two-dimensional reconstruction, when independent of incident light polarization (s or p), the scattered optical field is always depolarized. In this case the profile of the two-dimensional surface can be easily reconstructed with s or p component of the mixing and diffracted fields.     

Fringe pattern of the field diffracted by axicons

The far-field intensity pattern of laser beams diffracted by axicons is extensively characterized both theoretically and experimentally. The regular structure of the pattern, consisting of high-contrast fringes, is explained. The experimental results have been interpreted by representing the diffracted field as generated by an extended virtual source shaped as a circle centered on the optical axis of the incident laser beam. The simulations include modifications to the diffraction pattern arising from the laser radiation diffraction limit at the axicon tip, and they reproduce well the measured intensity profile at different distances from the axicon.     

Computer Reconstruction of a Random Rough Surface in the Presence of Higher Diffraction Orders

Abstract

The profile of a random rough surface (RRS), whose mean roughness R_a is greater than the light wavelength, is visualized by computer processing. The surface is presented as a sum of sinusoidal gratings. The light diffracted from this surface field is registered by a photodiode array. The second and third diffraction orders from each grating are taken into account in computer processing of the diffracted field and the mixing field–the field obtained at the mixing of the reference and the diffraction fields. The criterion for taking into account higher diffraction orders is the asymmetry of the diffraction pattern to the left and to the right relative to the central peak (the field of zero diffraction orders obtained from each grating) The number of the diffraction orders higher than the first is defined from the average intensity distribution between the central peak and the diffraction orders to the left and to the right at arbitrary light wavelength. The surface profile is reconstructed by a computer program and the mean roughness R_a is calculated. The obtained value of R_a is in satisfactory agreement with that measured by the contact pin method.