Zone plates performing generalized Hankel transforms and their metrological applications (original) (raw)

Projection method for the Hankel transform

Integral Transforms and Special Functions, 2011

Аннотация Numerical projection method for the zero-order Hankel transform inversion for the case of the data given on a finite interval has been developed. The justification of the convergence of the projection method has been done for the general case that also includes the sine-Fourier transform inversion. An inequality on the norms of the zeroth order Laguerre functions has been proved. Efficiency of the method was illustrated with the test data and for the problem of cylindrical distribution function calculation using melt surface layer diffraction data.

Focal Shift and Astigmatism for a Circular Zone-Plate

1992 International Conference on Lasers and Optoelectronics, 1993

Focal shift of zone plate is completely different from corresponding circular and annular apertures. Focal powers of zone plate are constant under some condition but for the latter they are always variables.

Imaging by zone plates: axial stigmatism at a particular order

Journal of the Optical Society of America A, 1994

A Fresnel zone plate (FZP) produces simultaneously multiple images, both real and virtual. In general, none of the images is stigmatic beyond the paraxial domain. Axially stigmatic imaging by FZP's calls for unique values of zone radii corresponding to each combination of the order number of the image and the object distance. In a recent paper [J. Opt. Soc. Am. A 10, 69 (1993)] we presented a set of nomographs for ready estimation of zone-plate parameters. The treatment dealt with the special case of a positive first-order image produced by a FZP. We now extend the study to incorporate imaging of any specific order. It is shown that, with a simple redefinition of the variables, the nomographs presented in the earlier paper retain their practical usefulness.

Zone plates with black focal spots

Applied Optics, 1992

Computer-designed linear and circular zone plates are considered that utilize a Tr-phase jump in order to create destructive interference in the focus. Intensity distributions in the focal plane as well as along the optical axis are calculated for a few examples. A significant decrease of the black spot diameter in comparison with the dimensions of the ordinary focal spot is obtained. Further reduction is achieved when the central region of the zone plate is obstructed. Some applications to alignment and the schlieren technique are suggested. Experimental results that confirm the calculated distributions are presented.

The Hankel Transform in n-dimensions and Its Applications in Optical Propagation and Imaging

Wave propagation is considered in multidimensional reciprocal space. For the first Rayleigh-Sommerfeld diffraction integral, the propagating field can be represented by homogeneous and inhomogeneous components. These add up to give a propagating component on a hemispherical surface in reciprocal space, and an evanescent component that lies totally outside the corresponding sphere. If evanescent waves can be neglected, the 3D angular spectrum method, entailing inverse Fourier transformation of the weighted hemisphere, can be used to calculate efficiently the propagated field. This basic concept is applied in spaces of different dimensionality. For functions displaying hyperspherical symmetry in nD space, the corresponding Hankel transformation leads to Hankel-transform pairs. Tables of functions relevant in wave propagation, diffraction, and information optics are presented. The two- dimensional (2D) case is particularly important as it can be applied to propagation in planar wave gu...

Second generation of grazing-incidence-phase Fresnel zone plates

Optics Communications, 1999

Second generation of grazing-incidence-phase Fresnel zone plates is presented in this paper. The term grazing-incidence Ž . discrete Fresnel lens GIDFL for these zone plates is proposed. This lens performs a multilevel phase modulation of the incident X-ray radiation. With the increase of the number of levels, the GIDFL virtually becomes an ideal kinoform zone plate. X-ray optical properties of a four-level GIDFL are studied here. It is found that this lens is capable of focusing X-ray radiation with an efficiency of 75%. A model is proposed which describes the diffraction properties of these lenses and allows one to calculate the focal spot intensity. q

Fractal generalized zone plates

Journal of The Optical Society of America A-optics Image Science and Vision, 2009

The construction of fractal generalized zone plates (FraGZPs) from a set of periodic diffractive optical elements with circular symmetry is proposed. This allows us to increase the number of foci of a conventional fractal zone plate (FraZP), keeping the self-similarity property within the axial irradiance. The focusing properties of these fractal diffractive optical elements for points not only along but also in the close vicinity of the optical axis are investigated. In both cases analytical expressions for the irradiance are derived. Numerical simulations of the energetic efficiency of FraGZPs under plane wave illumination are carried out. In addition, some effects on the axial irradiance caused by the variation in area of their transparent rings are shown.

Equilateral hyperbolic moir� zone plates with variable focus obtained by rotations

Optics Express, 2005

We present equilateral hyperbolic zone plates with variable focal length, which are formed as moiré patterns by a mutual rotation of two identical basic grids. Among others, all principal zone plates, except of the spherical one, can be used as these basic transmittances. Three most important advantages of the proposed moiré zone plates are: a constant aperture of the created element during the mutual movement of basic grids, lack of aberrations due to their undesired mutual lateral displacements and high diffraction efficiency of the binary phase version. To obtain clearer moiré fringe pattern, a radial carrier frequency can be added additionally to the transmittances of basic grids. The destructive interference between both arms of the focal cross of the equilateral hyperbolic moiré zone plate can be obtained by a constant phase shift introduced in the transmittances of the basic grids. Potential applications of discussed elements are indicated, including the most promising one in the three-point alignment technique.

Angular spectrum simulation of X-ray focusing by Fresnel zone plates

Journal of Synchrotron Radiation, 2013

A computing simulation routine to model any type of circularly symmetric diffractive X-ray element has been implemented. The wavefield transmitted beyond the diffractive structures is numerically computed by the angular spectrum propagation method to an arbitrary propagation distance. Cylindrical symmetry is exploited to reduce the computation and memory requirements while preserving the accuracy of the numerical calculation through a quasidiscrete Hankel transform algorithm, an approach described by Guizar-Sicairos & Gutierrez-Vega [J. Opt. Soc. Am. A, (2004), 21, 53-58]. In particular, the code has been used to investigate the requirements for the stacking of two highresolution Fresnel zone plates with an outermost zone width of 20 nm.