Intensity and Phase in Fresnel Diffraction by a Plane Screen Consisting of Parallel Strips

Using the Huygens-Fresnel principle the expressions for the intensity and the phase in Fresnel diffraction phenomena have been derived. The cases of spherical, cylindrical and plane scalar waves incident upon the plane diffraction screen have been investigated. The diffraction screen may consist of any number of parallel strips of different transmissivities and different phase shifts, the width of the individual strips being large compared with the wave-length. As special cases of the general formulae, the intensity and phase distributions in the diffraction patterns for the following forms of the diffraction screen have been calculated: opaque and partially transparent half-plane, slit, strip and double slit. By means of the derived expressions, the validity of Babinet's principle for Fresnel diffraction phenomena of this type has been verified.     

Rear-Surface Laser Damage on 355-nm Silica Optics Owing to Fresnel Diffraction on Front-Surface Contamination Particles

Light intensity modulations caused by opaque obstacles (e.g., dust) on silica lenses in high-power lasers often enhance the potential for laser-induced damage. To study this effect, particles (10-250 mum) with various shapes were sputter deposited on the input surface and irradiated with a 3-ns laser beam at 355 nm. Although a clean silica surface damages at fluences above 15 J/cm(2), a surface contaminated with particles can damage below 11.5 J/cm(2). A pattern that conforms to the shape of the input surface particle is printed on the output surface. Repetitive illumination resulted in catastrophic drilling of the optic. The damage pattern correlated with an interference image of the particle before irradiation. The image shows that the incident beam undergoes phase (and amplitude) modulations after it passes around the particle. We modeled the experiments by calculating the light intensity distribution behind an obscuration by use of Fresnel diffraction theory. The comparison between calculated light intensity distribution and the output surface damage pattern showed good agreement. The model was then used to predict the increased damage vulnerability that results from intensity modulations as a function of particle size, shape, and lens thickness. The predictions provide the basis for optics cleanliness specifications on the National Ignition Facility to reduce the likelihood of optical damage.     

Fresnel transform-based correlator

We present a new technique for information processing using Fresnel transform-based correlation. The main emphasis is on the design of a correlator involving the reference object and its near-field diffraction pattern at an optimized distance. The input-scene image and its diffraction pattern constitute the input pattern of the new correlator. The new technique shows a significant increase in discrimination ability and optical efficiency. Moreover, different encoding methods, such as the phase-only filter or the matched filter, can be used in conjunction with this method. A theoretical analysis as well as examples are given.     

Expansions for irradiance distribution near the focus in systems of different Fresnel numbers

It is shown in an earlier paper dealing with flat-topped light beams [Opt. Lett. 27, 1007 (2002)] that the profile of flat-topped beams can be expressed in the form 1-[1-exp(-xi2)]M, where xi is a dimensionless parameter and M is a nonnegative number. The expansion of the proposed expression is a finite series containing only the lowest-order Gaussian modes. This situation provides the possibility of reformulating the scalar theory of diffraction at an aperture in an opaque screen if the Gaussian mode expansion is employed to describe the boundary values of the light incident on the screen. As an example of this effort, an asymptotic model is established for three-dimensional irradiance distributions near the focus in systems of different Fresnel numbers. The proposed expansions contain only elementary functions and permit all elementary operations; therefore no special functions or special algorithms are needed in the evaluation of either irradiance distributions or the integrated energy in a focused field.     

Diffraction of a one-dimensional phase grating in the deep Fresnel field

We analyze theoretically the diffraction of phase gratings in the deep Fresnel field on the basis of the theory of scalar diffraction and Green's theorem and present the general formula for the diffraction intensity of a one-dimensional sinusoidal phase grating. The numerical calculations show that in the deep Fresnel region the diffraction distribution can be described by designating three characteristic regions that are influenced by the parameters of the grating. The microlensing effect of the interface of the phase grating provides the corresponding explanation. Moreover, according to the viewpoint that the diffraction intensity distribution is the result of the interference of the diffraction orders of the grating, we find that the diffraction patterns, depending on the carved depth of the phase grating, are determined by the contributing diffraction orders, their relative power, and the quasi-Talbot effect of the phase grating, which results from the second meeting of the diffraction orders carrying most of the power of the total field, as in the case of the amplitude grating.     

Diffraction of a gaussian beam around a strip mask

A theoretical treatment is given for the diffraction of a Gaussianbeam around an opaque strip mask. Such situations arise frequentlyin the diffraction of laser beams around wires and fibers. Scalarderivations are given for the Fraunhofer and Fresnel regions with bothdevelopments, leading to similar forms of rapidly convergent series forthe field at an observation plane. Predictions show good agreementwith measurements on the diffraction patterns from wires.     

Comprehensive focusing analysis of various Fresnel zone plates

A series-form expression for the individual diffracted field of a general annular ring is derived from the Rayleigh-Sommerfeld diffraction integral. It can be used for the accurate and fast simulation of any diffractive focusing element composed of concentric transparent rings. We present a comprehensive analysis, based on the leading term and the linear superposition principle, of the focusing performances of various Fresnel zone plates. Many problems, such as the equivalent aperture function, the diffraction efficiency, the focal spot pattern, the suppression of higher orders and the appearance of "fractional orders," and the explanation for the appearance of Fraunhofer diffraction patterns, are analytically investigated in detail. Because of the great similarity between Fresnel zone plates and multilevel diffractive lenses, most of the obtained results are also applicable to multilevel diffractive lenses.     

On Fresnel Diffraction by One-dimensional Periodic Objects,with Application to Structure Determination of Phase Objects

In the case of a periodic object, the wave amplitude in every point of a Fresnel diffraction pattern may be obtained from the values of the wave amplitude at a finite number of points in the object plane. This results is applied to the case of pure phase objects. A simple condition on the object structure is derived, which would allow the object phase distribution to be determined directly from a measured intensity profile for a certain defocusing distance.     

Fresnel number of a regular polygon and slit

The concept of the Fresnel number is generalized to make it possible to describe the regular polygon and slit illuminated by a homogeneous plane wave. The generalization is based on nonlinear regression of the axial intensity distribution curve. A useful analytical expression for the Fresnel number is presented. A simple experiment to show the different Fresnel numbers in one observation plane is illustrated.     

Theoretical description of fiber Bragg reflectors prepared by Fresnel diffraction images

The theory of Fresnel diffraction images is applied to Bragg-grating formation in a germanium-doped silica fiber. Fresnel diffraction images arise from the near-field diffraction at a periodic mask. The diffraction images are calculated as a function of the propagation distance for several mask configurations. The average of the diffraction-image intensities is calculated for a single longitudinal repetitive interval, and it is shown that the period of the resulting average intensity field is twice that of the original mask period. In some cases the periodic mask can be predicted for a desired average intensity field by calculation of the magnitude of its Fourier coefficients.     

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.     

Quasi-Talbot effect of a grating in the deep Fresnel diffraction region

Based on the theory of scalar diffraction, the diffraction of gratings in the deep Fresnel diffraction region is developed, and the general formula of the diffraction intensity of the one-dimensional grating is presented by using the Hankel function. Through numerical calculations, some interesting diffraction phenomena are found. In the deep Fresnel diffraction region, the dominant effects, with increasing propagation distance from the grating, are, in order, the geometrical effect, the quasi-geometrical effect, and the interference and diffraction effects. Furthermore, the diffraction intensities vary periodically in the diffraction effect region with increasing propagation distance. Quasi-Talbot imaging of the grating exists in the interference and diffraction regions, and the intensity distributions most similar to the structure of the grating are not at the exact Talbot distances. These phenomena in the deep Fresnel diffraction region are distinct from those in the Fresnel diffraction region. The formation origin of quasi-Talbot imaging of the grating is also discussed, and the numerical calculations powerfully verify the theoretical results.     

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.     

Electooptic Fresnel lens-scanner with an array of channel waveguides

A new type of beam scanner is discussed based on a 1-D Fresnel zone plate consisting of titanium-diffused channel waveguides on LiNbO3. By electrooptically controlling the guided-wave phase, both beam scanning and 1-D focusing are achieved without a condensing lens. It was experimentally confirmed using the scanner with twenty-one Fresnel zones that the beam spot with a diameter of approximately 50 microm at half-power level of diffraction pattern is scanned over a distance of +/-70 microm in the focal plane with an applied voltage of +/-40 V at 633 nm.     

Diffraction theory of optimized low-resolution Fresnel encoded lenses

A mathematical model describing the behavior of low-resolution Fresnel encoded lenses (LRFEL's) encoded in any low-resolution device (e.g., a spatial light modulator) has recently been developed. From this model, an LRFEL with a short focal length was optimized by our imposing the maximum intensity of light onto the optical axis. With this model, analytical expressions for the light-amplitude distribution, the diffraction efficiency, and the frequency response of the optimized LRFEL's are derived.     

Fresnel and Far-field Diffraction of a Truncated Elliptical Gaussian Beam Due to an Elliptical Aperture

Abstract

Using the Hopkins algorithm, expressions are derived for the intensity patterns, in both the Fresnel and far-field regions, associated with the diffraction of a plane-wave elliptical Gaussian beam truncated by an elliptical aperture. This is accomplished by evaluating the diffraction integral subject to the Fresnel approximation. Numerical results are presented that indicate how the truncation parameter affects the side-lobe level in the Fresnel and far-field regions.     

Fresnel diffraction as an aperture edge integral

Abstract

The standard Fresnel diffraction wave field from an aperture in a plane screen involves an integral over the aperture area. Here it is reduced to an aperture edge integral instead. The reduction really amounts to performing one integral but can be considered as the reduction of the ‘Maggi-Rubinovics’ edge formula from Kirchhoff optics to paraxial (i.e. Fresnel) optics.     

Observing the fractional Fourier transform by free-space Fresnel diffraction

The fractional Fourier transform of an object can be observed in the free-space Fresnel diffraction pattern of the object.     

Fresnel diffraction from curved fiber snippets with application to fiber diameter measurement

Fiber curvature is examined for its effect on apparent measured fiber diameter in a double-diffraction-based instrument that is in widespread use in the wool industry. The development uses a two-dimensional Fresnel diffraction model. The magnitude of the effect is studied for 2-mm-long snippets of various diameters from 8 to 50 μm and with radii of curvature of 1 m (straight), 600 μm, 280 μm, 200 μm, and 160 μm. The two-dimensional Fresnel model gives rise to deeply nested vector integrations that make computations with a straightforward approach exceedingly time consuming and impractical. A number of simplifying techniques are used to facilitate and speed up the numerical computations, thereby permitting investigations to be carried out on a personal computer.     

Elliptic Laguerre-Gaussian beams

An analytical expression for the diffraction of an elliptic Laguerre-Gaussian (LG) beam is derived and analyzed. We show that a beam with even singularity order has nonzero axial intensity for any degree of ellipticity and at any finite distance z from the initial plane, whereas at z = 0 and z = infinity the axial intensity is zero. We show that for a beam with a small degree of ellipticity and even order of singularity, two isolated intensity zeroes appear in the Fresnel zone on a straight line at an angle of 45 deg or -45 deg, depending whether the beam's spin is right or left. The theoretical conclusions are confirmed by numerical simulation and physical experiments.