Far-field diffraction patterns of circular sectors and related apertures

In studies of scalar diffraction theory and experimental practice, the basic geometric shape of a circle is widely used as an aperture. Its Fraunhofer diffraction pattern has a simple mathematical expression easily determined by use of the Fourier-Bessel transform. However, it may require considerable mathematical effort to determine the far-field diffraction patterns of aperture shapes related to the circular geometry. From a computational point of view, the mathematical difficulties posed by other aperture geometries as well as more-general apertures with irregular shapes can be surpassed by means of optical setups or fast numerical algorithms. Unfortunately, no additional insight or information can be obtained from their exclusive application, as would be the case if mathematical formulas were available. The research presented here describes the far-field diffraction patterns of single-sector apertures as well as their extension to double symmetrical sectors and to sector wheels formed by interleaved transparent sectors of equal angular size; in each case, full or annular sectors are considered. The analytic solutions of their far-field amplitude distribution are given here in terms of a series of Bessel functions, some interesting properties are deduced from these solutions, and several examples are provided to illustrate graphically the results obtained from approximate numerical computations. Our results have been verified numerically with the fast-Fourier-transform algorithm and experimentally by means of a spherical wavefront-single-lens Fourier-transform architecture.     

Description of phase singularities and their application to focusing design

We describe the focusing region associated with transmittances, analyzing its associated phase function. We show that generic features can be studied from the differential equation for focusing geometry, which is obtained through angular representation for diffraction fields. With the treatment, we recover the results for circular zone plates, and by introducing a linear transformation into the transmittance function we generate structures that keep the ability to generate focusing. According to the choice of the parameters involved, the diffraction field presents new focusing regions, whose three-dimensional geometry and spatial evolution can be described in a selective fashion with analysis of only the phase singularities associated with the diffraction field and avoidance of the integral representation. The treatment is also applied to a simple lens. We recover the theoretical predictions obtained by Berry and Upstill [M. V. Berry and C. Upstill, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259], and these predictions are corroborated experimentally. The results obtained are shown.     

Generalized optical ABCD theorem and its application to the diffraction integral calculation

We generalize the transfer matrix ABCD theorem for paraxial rays of the optical system to skew rays propagated off axis, whether or not the system possesses rotational symmetry. Furthermore, we apply the generalized ABCD theorem to evaluate the diffraction integral matrix elements A-D expressed in terms of the angle eikonal T, with the primary aberrations included. Finally, analysis and numerical calculation are given for propagation of a light beam through the optical system in the case in which spherical aberration and coma are present.     

Direct focusing modifications of elliptical gaussian beam by non-circular apertures

Abstract

For focusing the elliptical Gaussian beam directly, the effects of a non-circular aperture on the focusing properties are studied. The focusing properties for different shapes of apertures, which include a circle, an ellipse and a rectangle, are calculated and compared. Moreover, for different elliptical Gaussian beams, an empirical aperture selection rule that can be used to circularize the focusing spot is proposed. The energy transmission ratios are also considered in this paper.     

Numerical calculations of the Bessel functions for complex argument and application to X-ray diffraction

A numerical procedure for calculating the Bessel functions for complex argument is proposed. An asymptotic expansion is used to calculate the Bessel function for large modulus of the argument. Application to X-ray diffraction is presented.     

Matrix functions and their application to problems of heat and mass transfer

The boundary-value problem is solved for a system of differential equations of heat and mass transfer with unsteady sources of heat and mass to which the theory of matrix functions is applied.     

A new synthetic aperture focusing method to suppress the diffraction of ultrasound

Spatial resolution of an ultrasound image is limited by diffraction of ultrasound as it propagates along the axial direction. This paper proposes a method for reducing the diffraction spreading effect of ultrasound by using a synthetic aperture focusing (SAF) method that uses plane waves instead of spherical waves. The new method performs data acquisition and beamforming in the same manner as conventional SAF methods. The main difference is that all array elements are used on each firing to generate a plane wave, the traveling angle of which varies with the position of a receive subaperture. On reception, each scan line is formed by synthesizing RF samples acquired by relevant receive subapertures with delays to force the plane waves to meet at each imaging point. Theoretical analysis and computer simulation with infinite transmit aperture show that the proposed method is capable of suppressing the diffraction of ultrasound and especially causing the lateral beam width to remain unchanged beyond a certain depth determined by the size of a receive subaperture and the maximum traveling angle of plane waves. It is demonstrated that the proposed method is realizable using a linear array transducer. It is also shown that the lateral radiation pattern produced by the proposed method has smaller beam width than that using conventional SAF methods in the region of interest because it suppresses the diffraction of ultrasound.     

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.     

Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures

The second-order intensity moments and beam-propagation factor (M2 factor) of partially coherent beams have been generalized to include the case of hard-edged diffraction. A laser beam with amplitude modulation and phase fluctuation and a Gaussian Schell-model beam are taken as two typical examples of partially coherent beams. Analytical expressions for the generalized M2 factor are derived.     

Expanded structural functions and their application to investigate random processes with power-law spectra

Translated from Izmeritel'naya Tekhnika, No. 4, pp. 31–32, April, 1989.     

Scalar and electromagnetic diffraction point-spread functions

We describe an accurate technique for computing the diffraction point-spread function for optical systems. The approach is based on the combined method of ray tracing and diffraction, which implies that the computation is accomplished in a two-step procedure. First, ray tracing is employed to compute the wave-front error in a reference plane on the image side of the system and to determine the shape of the vignetted pupil. Next the Rayleigh-Sommerfeld diffraction theory, combined with the Kirchhoff approximation and the Stamnes-Spjelkavik-Pedersen method for numerical integration, is applied to compute the field in the region of the image. The method does not rely on small-angle approximations and works well for a pupil of general shape. Both scalar and electromagnetic computations are discussed and numerical results are presented.     

Extraordinary terahertz transmission through a copper film perforated with circular and rectangular apertures

The extraordinary transmission (ET) due to localized surface plasmon (LSP) and propagating surface plasmon (PSP) resonances of terahertz wave through a copper film perforated with circular and rectangular apertures is investigated theoretically and experimentally. Considering that the field distributions of LSP and PSP resonances are determined by the shape and periods of the apertures on the film, the relations between extraordinary and the geometrical parameters of the apertures are investigated. The intensities of the ETs induced by the PSP resonances mode [1, 1] are much stronger than the fundamental ones [1, 0] and [0, 1]. Our finds provide another effective method to tailor the extraordinary THz transmission in sub-wavelength metallic aperture structures.     

Radial distribution functions of non-crystalline polymers and their application to the structural analysis of PMMA

The use of Radial Distribution Functions (RDFs) in the determination of the structure of non-crystalline polymers is briefly reviewed. Particular aspects of the procedure for preparing RDFs from X-ray scattering are discussed in detail; namely the employment of an energy dispersive detector to remove the Compton component of the scattered X-rays and the application of the method of sampled transforms. A RDF is presented for atactic polymethylmethacrylate (PMMA) and its precision and reliability are discussed. It is analysed by comparison with RDFs calculated from computer-generated atom co-ordinates for isolated lengths of PMMA chains in different conformations. Methods are introduced by which the calculated RDFs are smeared to account for random disorder in the real chain and normalized so that, despite the finite range of the model, they can be immediately compared with the difference RDF which is directly obtained by transforming the data. Comparison between experimental and calculated RDFs shows that reasonable agreement is only obtained for a very limited range of conformations corresponding to sequences of backbone bond rotation angles of (10, 10, –10, –10) and the bond angles alternately 110 and 128. The form of the RDF appears very sensitive to important aspects of the molecular structure. The results both confirm and refine an earlier proposal from this laboratory which was based on comparisons between experimental and calculated functions in reciprocal space.     

Imaging performance of annular apertures. 4: Apodization and point spread functions

Apodizations with decreasing transmissions and with central obstructions transmute the point spread functions in image size and in the distribution of image irradiance. The distributions are characterized by the envelope functions which are exponential functions descending with variable exponents. At high apodization levels and lower central obstruction (<0.1) new extended radial zones in the outer part of the central ring groups are formed. Such transmutations of the image functions are of more than theoretical interest, particularly if the irradiance levels in the outer ring zones are to be compared to background irradiance levels.     

Analysis of powder X-ray diffraction resolution using collimating and focusing polycapillary optics

Focusing X-ray optics can be used to increase the intensity onto small samples, greatly reducing data collection time. Typically, the beam convergence is restricted to avoid loss of resolution, since the focused beams broaden the resulting powder diffraction rings. However, with smooth Gaussian peaks, the resolution defined by the uncertainty in peak location can be much less than the peak width. Polycapillary X-ray optics were used to collimate and focus X-rays onto standard inorganic powder diffraction samples. Comparisons were made of system resolution and diffracted beam intensity with and without focusing and collimating optics using a standard small spot rotating anode system in point source geometry. The area detector and optics also allowed for the use of a low power 60 W source, without increasing either the collection time or the peak center error compared to the rotating anode no optic case. Resolution and intensity were in good agreement with those obtained from a simple geometrical model developed for the optics, which allows for system design and optimization for the desired sample characteristics. Foils and powders were used to model thin film samples while allowing both reflection and transmission measurements to more effectively verify theoretical modeling of beam parameters.     

Evaluation of transmission functions of slit apertures in testing by shadow and split images

Translated from Izmeritel'naya Tekhnika, No. 9, pp. 19–21, September, 1992.     

Imaging performance of annular apertures. 5: Total and partial energy integral functions

The total energy integral functions for apodization T1 = (1 - r2)n and T2 = (2r - r2)n are compared for unobstructed apertures. The energy values increase linearity for T2 and with the (2n + 3) power for T1. For obstructed apertures, these values for T1 increase with higher powers at small image radii and assume linear characteristics only at larger radii. This transition occurs as the obstruction approaches zero. Partial energy functions for a region centered some distance from the central region show the same general trend as the image irradiance functions.