Reciprocity Principles for Focused Wavefields and the Modified Debye Integral

It is shown that focused wavefields satisfy a reciprocity principle concerning the roles of the focal point and the observation point. According to this principle, the diffracted field at a point P₂ of a converging spherical wave centred on the focal point P₁ is the complex conjugate of the diffracted field observed at P₁ if the focal point is at P₂. The commonly used Debye integral representation of focused wavefields violates the reciprocity principle. Therefore we derive a new diffraction formula for focused wavefields which we call the modified Debye integral (MDI). It is constructed so as to satisfy the reciprocity principle, and may be considered as a generalization of the Debye integral and also as a refinement on a diffraction integral derived by Hopkins. For focusing of a two-dimensional perfect wave we show that the MDI gives the same f-number dependent parabolic phase factor on the geometrical focal line as is obtained numerically in the Kirchhoff approximation. For focusing of three-dimensional waves the MDI and the Kirchhoff theory are shown to predict the same focal shift in the paraxial approximation, as long as the shift is small compared with the focal distance.     

Spectral anomalies in focused waves of different Fresnel numbers

Light propagation induces remarkable changes in the spectrum of focused diffracted beams. We show that spectral changes take place in the vicinity of phase singularities in the focal region of spatially coherent, polychromatic spherical waves of different Fresnel numbers. Instead of the Debye formulation, we use the Kirchhoff integral to evaluate the focal field accurately. We find that as a result of a decrease in the Fresnel number, some cylindrical spectral switches are geometrically transformed into conical spectral switches.     

Singular behavior of the spectrum in the neighborhood of focus

In a recent paper [Phys. Rev. Lett. 88, 013901 (2002)] it was shown that when a convergent spatially coherent polychromatic wave is diffracted at an aperture, remarkable spectral changes take place on axis in the neighborhood of certain points near the geometrical focus. In particular, it was shown that the spectrum is red-shifted at some points, blueshifted at others, and split into two lines elsewhere. In the present paper we extend the analysis and show that similar changes take place in the focal plane, in the neighborhood of the dark rings of the Airy pattern.     

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.     

Study of the focusing of a field refracted by an inhomogeneous slab into a uniaxial crystal by using Maslov's method

High-frequency fields refracted by a geometry that contains an inhomogeneous slab placed at a certain distance from a plane uniaxial interface are derived by using Maslov's method. The geometrical optics approximation that is generally valid for high-frequency fields fails in the vicinity of a caustic. Maslov's method is a systematic procedure for predicting the field in the caustic region combining the simplicity of rays and the generality of the transform method. Numerical computations are carried out for the field pattern around the caustic or focal point by using Maslov's method. The results are found to be in good agreement with those obtained using Kirchhoff's approximation.     

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.     

Image Formation and Processing with Toric Surfaces: I. Geometrical optic properties

This paper studies the toric surface as a representation of an astigmatic system in two directions. From a geometrical point of view the system splits the focus and its corresponding focal plane in two, and two straight lines are obtained, ?_S and ?_T, which characterize the direction on the respective focal planes. The image pattern has a circular configuration inscribed within an astroyd, itself characteristic of the third-order aperture. When a circular pupil is used, this pattern is found in the harmonic-mean position of the foci. With a Eulerian pupil this pattern is shifted to the arithmetic mean position.     

Single-scatter vector-wave scattering from surfaces with infinite slopes using the Kirchhoff approximation

This paper presents a new formulation of the 3D Kirchhoff approximation that allows calculation of the scattering of vector waves from 2D rough surfaces containing structures with infinite slopes. This type of surface has applications, for example, in remote sensing and in testing or imaging of printed circuits. Some preliminary calculations for rectangular-shaped grooves in a plane are presented for the 2D surface method and are compared with the equivalent 1D surface calculations for the Kirchhoff and integral equation methods. Good agreement is found between the methods.     

Debye representation of dispersive focused waves

We report a matrix-based diffraction integral that evaluates the focal field of any diffraction-limited axisymmetric complex system in the paraxial regime. This diffraction formula is a generalization of the Debye integral, here accommodated to broadband problems. The Fresnel number is reformulated to guarantee that the focal region is entirely within the region of validity of the Debye approximation when this parameter largely exceeds unity. Several examples are examined in detail, one of them exhibiting in-focal-plane compensation of the spatial dispersion. This simple formalism opens the door for the analysis and design of focused beams with arbitrary angular dispersion.     

Elastic large deflection analysis of plates subjected to uniaxial thrust using meshfree Mindlin-Reissner formulation

A meshfree approach for plate buckling/post-buckling problems in the case of uniaxial thrust is presented. A geometrical nonlinear formulation is employed using reproducing kernel approximation and stabilized conforming nodal integration. The bending components are represented by Mindlin–Reissner plate theory. The formulation has a locking-free property in imposing the Kirchhoff mode reproducing condition. In addition, in-plane deformation components are approximated by reproducing kernels. The deformation components are coupled to solve the general plate bending problem with geometrical non-linearity. In buckling/post-buckling analysis of plates, the in-plane displacement of the edges in their perpendicular directions is assumed to be uniform by considering the continuity of plating, and periodic boundary conditions are considered in assuming the periodicity of structures. In such boundary condition enforcements, some node displacements/rotations should be synchronized with others. However, the enforcements introduce difficulties in the meshfree approach because the reproducing kernel function does not have the so-called Kronecker delta property. In this paper, the multiple point constraint technique is introduced to treat such boundary conditions as well as the essential boundary conditions. Numerical studies are performed to examine the accuracy of the multiple point constraint enforcements. As numerical examples, buckling/post-buckling analyses of a rectangular plate and stiffened plate structure are presented to validate the proposed approach.     

On the Fraunhofer (Far Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background

The diffracting objects considered are opaque and semi-transparent objects of known cross section which are situated in an otherwise transparent area and illuminated with a collimated quasi-monochromatic beam of light. The plane in which the resultant diffraction pattern is considered is at a sufficient distance from the object for the approximation of Fraunhofer (far field) diffraction to be made whilst still remaining in the near field of the surrounding transparent area. The resultant intensity distribution is investigated both theoretically and experimentally for objects which present both circular and rectangular cross sections to the beam.     

Shaping the focal intensity distribution using spatial coherence

The intensity and the state of coherence are examined in the focal region of a converging, partially coherent wave field. In particular, Bessel-correlated fields are studied in detail. It is found that it is possible to change the intensity distribution and even to produce a local minimum of intensity at the geometrical focus by altering the coherence length. It is also shown that, even though the original field is partially coherent, in the focal region there are pairs of points at which the field is fully correlated and pairs of points at which the field is completely incoherent. The relevance of this work to applications such as optical trapping and beam shaping is discussed.     

Ab initio prediction of elastic and thermal properties of cubic TiO2

In this paper we focus on the novel solar material, namely cubic TiO₂. The full potential linearized augmented plane wave method in combination with the local density approximation (LDA) and the generalized gradient approximation (GGA) have been used. We calculated structural parameters, elastic constants, wave velocities and thermal properties of the material assuming the fluorite structure. The obtained values were in good agreement with the available theoretical and experimental data. Moreover, the pressure and temperature dependences of the bulk modulus, Debye temperature, Heat capacity and linear expansion coefficient have been addressed for the first time.     

Interaction of a monochromatic ultrasonic beam with a finite length defect at the interface between two anisotropic layers: kirchhoff approximation and fourier representation

This paper presents a fast computation method to simulate the interaction between a bounded acoustic beam and a 2-layered anisotropic structure with a finite defect on the internal interface. The method uses the classical Fourier decomposition of the fields into plane waves, and the Kirchhoff approximation is introduced to calculate the diffusion by the defect. The validity of the approximation is estimated by comparison with the Keller Geometrical Theory of Diffraction and with results obtained by boundary element methods. The quickness of the method allows testing several geometrical configurations (varying incident angle, thickness of the layers or the physical nature of the defect). These studies may be used to foresee what experimental configurations would be adequate to have a chance to detect the defect.     

Scattering by simple and nonsimple shapes by the combined method of ray tracing and diffraction: application to circular cylinders

The combined method of ray tracing and diffraction (CMRD) is an efficient and accurate technique for computing the scattered field in focal regions of optical systems. Here we extend the CMRD concept so it can be used to compute fields scattered by objects of simple as well as nonsimple shapes. To that end we replace the scattering object by an equivalent, planar phase object; use ray tracing to determine its location, aperture area, amplitude distribution, and phase distribution; and use standard Kirchhoff diffraction theory to compute the field scattered by the equivalent phase object. To illustrate the practical use of the CMRD we apply it to a two-dimensional problem in which a plane or cylindrical wave is normally incident upon a circular cylinder. For this application we determine the range of validity of the CMRD by comparing its results for the scattered field with those obtained by use of an exact eigenfunction expansion.     

Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures

We show that the contribution of the electric field components into the focal region can be controlled using binary phase structures. We discuss differently polarized incident waves, for each case suggesting easily implemented binary phase distributions that ensure a maximum contribution of a definite electric field component on the optical axis. A decrease in the size of the central focal spot produced by a high numerical aperture (NA) focusing system comes as the result of the spatial redistribution of the contribution of different electric field components into the focal region. Using a polarization conversion matrix of a high NA lens and the numerical simulation of the focusing system in Debye's approximation, we demonstrate benefits of using asymmetric to polar angle ? binary phase distributions (such as arg[cos ?] or arg[sin 2?]) for generating a subwavelength focal spot in separate electric field components. Additional binary structure variations with respect to the azimuthal angle also make possible controlling the longitudinal distribution of light. In particular, the contribution of the transverse components in the focal plane can be reduced by the use of a simple axicon-like structure that serves to enhance the NA of the lens central part, redirecting the energy from focal plane. As compared with the superimposition of a narrow annular aperture, this approach is more energy efficient, and as compared with the Toraldo filters, it is easier to control when applied to three-dimensional focal shaping.     

Establishment of the Maximum Encircled Energy in the Geometrical Focal Plane

The classical theory of the focusing of light predicts that light energy is highly concentrated in the geometrical focal plane; in other words, the geometrical focal plane contains more energy per unit area than any other plane parallel to it. However, it has recently been shown that the classical theory is valid only for focusing systems of large Fresnel number. This paper examines how the maximum encircled energy is accumulated in the geometrical focal plane as the Fresnel number of the diffracting aperture increases.     

Potential and limits of holographic reconstruction algorithms

For the purpose of ultrasonic nondestructive testing of materials, holography in connection with digital reconstruction algorithms has been proposed as a modern tool to extract crack sizes from ultrasonic scattering data. Defining the typical holographic reconstruction algorithm as the application of the scalar Kirchhoff diffraction theory to backward wave propagation, we demonstrate its general incapability of reconstructing equivalent sources, and hence, geometries of scattering bodies. Only the special case of a planar measurement recording surface, that is to say, a hologram plane, and a planar crack with perfectly rigid boundary conditions parallel to the hologram plane and perpendicular to the incident field yields a nearly perfect correlation between crack size and reconstructed image; the reconstruction algorithm is then referred to as the Rayleigh-Sommerfeld formula; it therefore represents the optimal case matched to that special geometrical situation and, hence, may be interpreted as a quasi-matched spatial filter. Using integral equation theory and physical optics, we compute synthetic holographic data for a linear cracklike scatterer for both plane and spherical wave incidence, the latter case simulating a synthetic aperture impulse echo situation, thus illustrating how the Rayleigh-Sommerfeld algorithm or its Fresnel approximation increasingly fail for cracks inclined to the hologram plane and excited nonperpendicularly. Furthermore, we point out how the physical data recording process may additionally influence the reconstruction accuracy, and, finally, guidelines for a careful and serious application of these holographic reconstruction algorithms are given. The theoretical results are supported by measurements.     

Scattering from anisotropic plasma-coated PEMC cylinder buried beneath a slightly rough surface

A general theoretical formulation is done to calculate the field scattered by perfect electromagnetic conductor (PEMC) cylinder coated with anisotropic plasma material. It is buried below a slightly rough surface. Spectral plane wave representation of fields and small perturbation method are used to calculate multiple reflections between coated cylinder and rough surface. To validate the present formulation, scattered field from a PEMC cylinder coated with double negative (DNG) material is obtained from the present formulation. Scattering pattern of non coated PEC/PEMC cylinder or coated with isotropic material can also be obtained by the proper selection of physical parameters such as anisotropy, admittance of PEMC cylinder and permittivity of hosting medium. Analytical expressions of scattered field for a sinusoidal rough surface are given along with their physical interpretation to get a good insight. Effect of geometrical and physical parameters on scattering pattern is observed.     

Shape identification by physical optics: the two-dimensional TE case

A method is provided for reconstruction of the shape of perfectly conducting objects in a homogeneous space starting from knowledge of the scattered far field under the incidence of TE-polarized plane waves. The Kirchhoff model of scattering permits linearization of the inverse problem, which is further simplified by adopting an asymptotic approximation. Thus the problem is tackled with an approach based on singular-value decomposition already developed for the TM case.