Non-paraxial analytical solution for the generation of focal curves

Abstract

A solution for the transmittance of elements forming planar and nonplanar focal curves in the non-paraxial case is proposed in this paper. The treatment is based on the scalar diffraction theory and makes use of the stationary phase method in evaluation of the diffraction integral. The equation of a stationary point enables one to establish mapping relations between the geometrical regions of the element and the corresponding points of the focal curve. These regions turn out to be described by conical curves. The mapping relations can be manipulated by a phase function imposed onto the focal curve. The amplitude factors of the derived transmittances can be omitted and the desired intensity distribution along the focal curve can then be obtained by an appropriate change in the aperture of pure phase elements. The approach is illustrated by examples of elliptical, hyperbolic, conical and circular zone plates. The focus generations for both non-paraxial and paraxial cases of these elements are compared using numerical simulations.     

Nonredundant calculations for creating digital Fresnel holograms

In this paper we propose an alternative technique for producing digital Fresnel holograms. The evaluation of a diffraction pattern in a wide region is implemented in such a way as to avoid redundant calculations and preserve the precision. Because of the symmetry of the kernel, the complex amplitude is calculated at four points in the registration plane simultaneously. This algorithm decreases the required CPU time 4 times with respect to direct calculation. The digital Fresnel hologram is numerically and optically reconstructed, and some qualitative comparisons are made.     

Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones

The single-sideband method of holography, as is well known, cuts off beams that come from conjugate images for holograms produced in the Fraunhofer region and from objects with no phase components. The single-sideband method with half-zone-plate processing is also effective in the Fresnel region for beams from an object that has phase components. However, this method restricts the viewing zone to a narrow range. We propose a method to improve this restriction by time-alternating switching of hologram patterns and a spatial filter set on the focal plane of a reconstruction lens.     

四维画法几何中的曲线投射法

本文是研究以曲线作为辅助投射线,对四维空间中的点、直线、平面等几何元素进行投影变换的问题。文中采用了以平面为投射轴,以平面曲线为投射线的方法建立曲线投射的变换模型。并给出了两个实例,说明用此方法可解决些用一般四维画法几何的方法难以解决的问题。     

Synthetic display holography: Line segments through the hologram

Abstract

In synthetic display holography the reconstruction of 3D objects that are intersected by the hologram plane are of interest. Some characteristics and limitations of such a situation are investigated. We concentrate on images assembled of lines. Certain constraints exist for the calculation of distributions to be stored in holograms. Some depend on the properties of the lines and others are of a more general character, e.g. the hologram sampling. The influence of major parameters related to a reconstructed line are analysed.     

Complex Fresnel hologram display using a single SLM

We propose a novel optical method to display a complex Fresnel hologram using a single spatial light modulator (SLM). The method consists of a standard coherent image processing system with a sinusoidal grating at the Fourier plane. Two or three position-shifted amplitude holograms displayed at the input plane of the processing system can be coupled via the grating and will be precisely overlapped at the system's output plane. As a result, we can synthesize a complex hologram that is free of the twin image and the zero-order light using a single SLM. Because the twin image is not removed via filtering, the full bandwidth of the SLM can be utilized for displaying on-axis holograms. In addition, the degree of freedom of the synthesized complex hologram display can be extended by involving more than three amplitude holograms.     

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.     

Near-field light distributions propagated from human corneas: Determination of relevant patterns

Abstract

A procedure to determine the best image range of distances that can be obtained from an aphakic eye is presented. Corneal surfaces are calculated from videokeratographic data. Light distributions at different distances inside the eye are calculated with a near-field Fresnel diffraction algorithm. These patterns are evaluated using three different quality functions which quantify the appearance of the energy distributions in each plane. Local extremes in those functions will serve to locate relevant distributions as Sturm focal planes and even the best image plane which is affordable by the cornea. The study is applied to strongly aberrated corneas.     

Computational reconstruction of images from holograms

The equations to reconstruct an image plane from a hologram are developed. This development is carried out for planes parallel to the hologram, which allows fast computation through the use of fast Fourier transforms. Algorithms for a digital computer are developed so images can be reconstructed, both with and without the Fresnel approximation, from a digitized hologram without the need for three-dimensional optical reconstruction equipment. Examples of holographically recorded images of marine micro-organisms are shown. A computational method for counting the number of micro-organisms in the holographically recorded volume is developed, and an example is provided.     

Parallel computing of a digital hologram and particle searching for microdigital-holographic particle-tracking velocimetry

We have developed a parallel algorithm for microdigital-holographic particle-tracking velocimetry. The algorithm is used in (1) numerical reconstruction of a particle image computer using a digital hologram, and (2) searching for particles. The numerical reconstruction from the digital hologram makes use of the Fresnel diffraction equation and the FFT (fast Fourier transform), whereas the particle search algorithm looks for local maximum graduation in a reconstruction field represented by a 3D matrix. To achieve high performance computing for both calculations (reconstruction and particle search), two memory partitions are allocated to the 3D matrix. In this matrix, the reconstruction part consists of horizontally placed 2D memory partitions on the x-y plane for the FFT, whereas, the particle search part consists of vertically placed 2D memory partitions set along the z axes. Consequently, the scalability can be obtained for the proportion of processor elements, where the benchmarks are carried out for parallel computation by a SGI Altix machine.     

Modeling microlenses by use of vectorial field rays and diffraction integrals

A nonparaxial vector-field method is used to describe the behavior of low-f-number microlenses by use of ray propagation, Fresnel coefficients and the solution of Maxwell equations to determine the field propagating through the lens boundaries, followed by use of the Rayleigh-Sommerfeld method to find the diffracted field behind the lenses. This approach enables the phase, the amplitude, and the polarization of the diffracted fields to be determined. Numerical simulations for a convex-plano lens illustrate the effects of the radii of curvature, the lens apertures, the index of refraction, and the wavelength on the variations of the focal length, the focal plane field distribution, and the cross polarization of the field in the focal plane.     

微透镜列阵衍射效应所致的串扰

本文分析了微透镜列阵衍射效应的影响因素,推导出了微透镜焦平面上光强分布的解析表达式,对菲涅尔数评价衍射效应的物理含义给予了合理的解释.并利用ZEMAX软件对微透镜列阵进行仿真,基于惠更斯子波直接积分的算法计算得到了微透镜列阵焦平面上的光场强度分布.通过比较不同条件下所得到的计算结果,验证了以菲涅尔数作为微透镜列阵衍射效应评价依据的的合理性,同时验证了以菲涅尔数判断焦斑间串扰的可行性.     

Multiple-viewpoint projection holograms synthesized by spatially incoherent correlation with broadband functions

We present a theoretical framework for recording and reconstructing incoherent correlation holograms of real-existing three-dimensional scenes observed from multiple viewpoints. This framework is demonstrated by generating and reconstructing a modified Fresnel hologram as well as a new correlation hologram called a protected correlation hologram. The reconstructed scene obtained from the protected correlation hologram has a significantly improved transverse resolution for the far objects in the scene compared to the modified Fresnel hologram. Additionally, the three-dimensional information encoded into the protected correlation hologram is scrambled by a secretive point spread function and thus the hologram can be used for encrypting the observed scene. The proposed holography methods are demonstrated by both simulations and experiments.     

Spatial filtering for zero-order and twin-image elimination in digital off-axis holography

Off-axis holograms recorded with a CCD camera are numerically reconstructed with a calculation of scalar diffraction in the Fresnel approximation. We show that the zero order of diffraction and the twin image can be digitally eliminated by means of filtering their associated spatial frequencies in the computed Fourier transform of the hologram. We show that this operation enhances the contrast of the reconstructed images and reduces the noise produced by parasitic reflections reaching the hologram plane with an incidence angle other than that of the object wave.     

Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position

Introducing a microscope objective in an interferometric setup induces a phase curvature on the resulting wavefront. In digital holography, the compensation of this curvature is often done by introducing an identical curvature in the reference arm and the hologram is then processed using a plane wave in the reconstruction. This physical compensation can be avoided, and several numerical methods exist to retrieve phase contrast images in which the microscope curvature is compensated. Usually, a digital array of complex numbers is introduced in the reconstruction process to perform this curvature correction. Different corrections are discussed in terms of their influence on the reconstructed image size and location in space. The results are presented according to two different expressions of the Fresnel transform, the single Fourier transform and convolution approaches, used to propagate the reconstructed wavefront from the hologram plane to the final image plane.     

Off-axis Analysis of the Strehl Ratio Using the Wigner Distribution Function

Abstract

An approach based on the Wigner distribution function (WDF) is used for analysing the Strehl ratio behaviour for varying focus errors and object source locations. A relationship is found between the WDF of the pupil function and the WDF of the light field in the Fresnel region about the image plane. Tolerance criteria to defocus can be specified in terms of the ‘amplitude’ of the parageometrical rays that intersect a tilted optical axis defined by the locations of the object and image points.     

Focal swift for a Gaussian beam: an experimental study

The beam radius along a focused, unapertured Gaussian beam was measured and used to calculate the dependence of the geometrical Fresnel number on the effective Fresnel number of the beam as it emerged from the focusing lens. The resulting data clearly demonstrate a focal shift away from the focal plane given by geometrical optics. The data agree very well with a theory due to Carter. This theory indicates that a significant focal shift occurs in these beams if the focusing lens is placed within the near field of the focal plane.     

Computer holographic lens

Calculation of the optical path difference between the input and the focal planes of a lens is used to compute the phase introduced by the lens. This represents a new way to create computer-generated optical elements. The phase change that takes place during the light propagation between the computer-generated hologram and the image plane is taken into consideration. Two examples that show different ways to plot the synthetic lens are presented.     

GETFOCOS for imaging atmospheric Cherenkov telescopes—A GEant4 tool for optimization and characterization of an optical system

The aim of this article is to introduce a dedicated simulation package, named GETFOCOS, that combines Geant4 and ray-tracing algorithms that can be used on the characterization of the optics of a generic Imaging Atmospheric Cherenkov telescope as well as on the optimization of its focal plane geometry. The image spot size is evaluated for both cases in a large field-of-view observation scenario. This tool allows to perform fast but precise tests for different optical options. The specific case of a Fresnel lens, inspired by the concept developed for the GAW experiment, is analysed. However, this intends to be an universal and relevant tool for any kind of optical system. A complete characterization of the optics is presented together with a study for the optimization of the focal plane shape.