Droplet profiles obtained from the intensity distribution of refraction patterns

A noninterferometric method for obtaining profiles of axially symmetric transparent liquid droplets is described. The drops are illuminated along the symmetry axis by a uniform parallel beam whose intensity distribution is recorded at the focal plane of a lens placed behind the drop. In some conditions and within the geometrical optics approach, it is possible to reconstruct the profile of the drop from this intensity distribution except for the length scale factor, which, if necessary, may be provided by an additional simple measurement. Because of CCD cameras and digital image processing, this method is an interesting alternative technique for measuring drop profile shapes with considerable accuracy when interferometry is unwieldy. We also analyze the diffraction features of the intensity distribution to clarify the extent that they affect the approach that we used and to establish additional information that they may provide.     

Light intensity distributions of the Gaussian laser beam through an aperture team

Based on the propagating theory of the laser beam, the propagating characteristics of the Gaussian beam through an aperture team that comprises two apertures and a convergent lens, are studied. The approximate expressions for the field distribution are derived by the diffracted integral equation in detail under the condition of approximations. In comparison with the approximate expression and the precise expression, we know that there are the approximate same results for the two expressions if the radius of the second aperture is not too large. The numerical examples are given to confirm the correctness of our calculated results.     

Axial irradiance distribution throughout the whole space behind an annular aperture: comments

We comment on the recent paper by Harvey and Krywonos [Appl. Opt. 41, 3790-3795 (2002)], in which approximate irradiance calculations along the axis of a circular aperture illuminated by a plane wave are performed. As the starting point of their calculations, an approximated version (valid for z > lambda) of the Rayleigh-Sommerfeld diffraction integral is used. They based their numerical conclusions on a misleading "near field criterion," which guides the readers to the wrong idea that their calculations are valid even for the very near field behind the aperture. Their ideas are not original; the exact calculations of diffracted fields behind a circular aperture have been known for 40 years.     

A nonlinear theory of wake development

Summary A simple new series, using an expansion of the velocity profile in parabolic cylinder functions, has been developed to describe the nonlinear evolution of a steady, laminar, incompressible wake from a given arbitrary initial profile. The first term in this series is itself found to provide a very satisfactory prediction of the decay of the maximum velocity defect in the wake behind a flat plate or aft of the recirculation zone behind a symmetric blunt body. A detailed analysis, including higher order terms, has been made of the flat plate wake with a Blasius profile at the trailing edge. The same method yields, as a special case, complete results for the development of linearized wakes with arbitrary initial profile under the influence of arbitrary pressure gradients. Finally, for purposes of comparison, a simple approximate solution is obtained using momentum integral methods, and found to predict satisfactorily the decay of the maximum velocity defect.     

The Intensity Distribution of a Gaussian Schell-model Beam Focused by an Aperture Lens

Abstract

Starting from the generalized Huygens-Fresnel diffraction integral, we study the intensity distribution of a Gaussian Schell-model (GSM) beam diffracted at an aperture lens. A great number of numerical calculations have been performed to illustrate the focused field characteristics. Isophote diagrams are given for systems of different Fresnel numbers, which focus GSM beams, and the related analysis is presented.     

Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation

Practical collimating diffractive cylindrical lenses of 2, 4, 8, and 16 discrete levels are analyzed with a sequential application of the two-region formulation of the rigorous electromagnetic boundary-element method (BEM). A Gaussian beam of TE or TM polarization is incident upon the finite-thickness lens. F/4, F/2, and F/1.4 lenses are analyzed and near-field electric-field patterns are presented. The near-field wave-front quality is quantified by its mean-square deviation from a planar wave front. This deviation is found to be less than 0.05 free-space wavelengths. The far-field intensity patterns are determined and compared with the ones predicted by the approximate Fraunhofer scalar diffraction analysis. The diffraction efficiencies determined with the rigorous BEM are found to be generally lower than those obtained with the scalar approximation. For comparison, the performance characteristics of the corresponding continuous Fresnel (continuous profile within a zone but discontinuous at zone boundaries) and continuous refractive lenses are determined by the use of both the BEM and the scalar approximation. The diffraction efficiency of the continuous Fresnel lens is found to be similar to that of the 16-level diffractive lens but less than that of the continuous refractive lens. It is shown that the validity of the scalar approximation deteriorates as the lens f-number decreases.     

Coherent and incoherent combination of Gaussian beams employing lens array distributed on the spherical chamber

Coherent and incoherent combination of Gaussian beams employing a lens array distributed on the spherical chamber is theoretically analyzed. The output field of each source in the array is coupled through an individual optical system whose local optical axis coincides with the radial direction of the chamber. The resulting intensity profile near the origin is derived. The intensity profile and power in the bucket on the target for rectangular and hexagonal arrangement are numerically calculated. The influences of the center-to-center separation and the ring number of the focusing lens array are given. The synthetic intensity profile of incoherent combination changes little for a lens array scale much smaller than the chamber size. In contrast, the synthetic intensity profile of coherent combination shows an interference pattern with a sharp central peak and sidelobes.     

Spectral shifts and spectral switches of a pulsed Gaussian beam from a rectangular aperture in the far field

The far-field anomalous spectral behaviours of a space–time-dependent Gaussian pulsed beam passing through a rectangular aperture are studied. By expanding a hard aperture function into a finite sum of complex Gaussian functions and starting from the Fresnel diffraction integral, the approximate analytical expression for the spectral intensity of a space–time-dependent Gaussian pulsed beam passing through a rectangular aperture is derived. Meanwhile, the corresponding closed-forms for the slit and the unapertured cases are also given as special cases of the general results. The red and blue shifts and the spectral intensity distribution are studied and illustrated with numerical calculations. Specifically, it is shown that the spectral switch takes place when the truncation parameter is equal to particular values or the observation position is at the critical diffraction angle. The possibility of tunable spectral switching in the far field with an apertured pulsed beam by varying the size of the rectangular aperture is highlighted.     

Fraunhofer diffraction of laser beams with amplitude modulations and phase fluctuations by a lens with a central obscure aperture

Abstract

By using the statistical-optics model and generalized Huygens-Fresnel diffraction integral, Fraunhofer diffraction of high-power laser beams with amplitude modulations (AMs) and phase fluctuations (PFs) focused by a lens with a central obscure aperture has been studied. Detailed numerical calculation results have been given, showing the dependence of the intensity distribution at the geometrical focal position and the power (energy) focus-ability of high-power laser beams not only on the obscure ratio, but also on the truncation parameter, AMs and PFs of beams.     

Numerical analysis of the efficiency of multilayer-coated gratings using integral method

An analysis is made of a rigorous and an approximate approach to the solution of the diffraction problem for a multilayer-coated X-ray grating by the integral equation formalism. Whereas a rigorous analysis involving the integral method requires a lot of computer resources, even for gratings with a small number of layers, the approximate approach based on a modification of the solution of the integral equation at the lower boundary with a finite conductivity is practically independent of the number of layers and is readily tractable with the use of a standard PC. The efficiencies of multilayer gratings measured at grazing angles with synchrotron soft X-ray radiation are compared with the values calculated using the integral approaches for ideal groove profiles.     

High-efficiency diffractive waveguide lenses by parametric optimization

The first-order diffraction efficiency of a waveguide diffraction grating is maximized, for a wide range of grating periods, by optimization of the effective-index modulation profile. Three different values of effective-index modulation in the range of 0.02-0.11 are considered. The analysis is performed with the thin-grating-decomposition method. The results are verified by electromagnetic grating theory and applied to the construction of diffractive waveguide lenses with an improved overall efficiency. In the neighborhood of the optical axis, the optimized lens structure is a close approximation of a gradient-thickness Fresnel lens. Significant deviations from this shape appear when the local grating period reduces below ~ 15-25λ, where ~ is the wavelength of the guided mode. Near the edges of a high-numerical-aperture lens, where the local period is ~ 3-6λ, an approximate Bragg grating structure is obtained.     

Focal shifts in focused beams with an elliptical diffracting screen

The approximate analytical formula of the focal shift has been derived for an elliptical diffracting screen that consists of a circular lens and an elliptical aperture. It is found that the focal shift of the beam focused by this elliptical diffraction screen depends not only on the product of the Fresnel number of the focusing geometry and the standard deviation of a mapped function, but also on the ellipticity (the ratio between the minor and the major axes) of the elliptical aperture. The focal shift increases as the decrease in the ellipticity of elliptical aperture. By using an approximate analytical formula and the diffraction integral formula, some numerical simulation comparisons are done. The presented analyses show that the actual analytical expression for the focal shift of a rotationally asymmetric diffraction screen cannot generally be obtained by using the azimuthal average of the screen amplitude transmittance.     

Scattering of light by a periodic structure in the presence of randomness. V. Detection of successive peaks in a periodic structure

We address the problem of detecting periodic structures hidden behind roughness. We have shown that if r(0) is the coherence length of the scattered radiation, due to the random part of the surface, and Lambda is the wavelength of the periodic part of the surface, then with a matched filtering method that we introduce, and by using simple computations with the intensity data, it is possible to detect the hidden first-order peak even when (r(0)/Lambda) approximately 0.11. Here we advance the method to bring out very weak second-order peaks, which we demonstrate for what we believe is the first time. The unmistakable presence of both the first- and second-order peaks, which have identical shapes as the zeroth-order peak, is strong evidence of the hidden periodicity and serves as a novel method for the detection of weak periodicities hidden behind strong randomness.     

Femtosecond laser-induced thermal lens effect in chromium film

The thermal lens (TL) effect induced by femtosecond laser pulses in chromium film is reported. A Fresnel diffraction theory is used to explain the TL effect. The intensity profile of the TL calculated by the theoretical model is in agreement with the experimental results. The contrast ratio of the TL is defined to describe the TL effect, and we find that the maximum contrast ratio of the TL effect is obtained when the probe beam is recorded at a characteristic distance. The dependence of the contrast ratio of the TL on different pump laser power levels and delay times is also investigated. Numerical simulations are also consistent with the experimental results.     

High-efficiency production of propagation-invariant spot arrays

Tailoring of the transverse intensity profiles of propagation-invariant optical fields is considered. The design of diffractive elements capable of realizing such fields by Fourier synthesis is discussed. High-efficiency realization of finite-aperture approximations of the constructed fields is demonstrated in a system consisting of two multilevel diffractive elements. The first element is a diffractive toroidal lens, which focuses the incident field into a ring pattern. The second diffractive element, located at the focal plane of the first element, introduces the phase modulation necessary to realize the desired transverse intensity profile behind a separate collimating lens. The influence of the fabrication errors of the diffractive elements on the fidelity of the propagation-invariant spot array is simulated, and system-integration aspects based on substrate-mode planar-integrated optics are considered.     

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

Summary The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.     

Midwave infrared optical zooming design and kinoform degrading evaluation methods

In this study, an optical zooming design method is constructed by ray tracing. The loci of each thin lens is determined utilizing algebraic relationships. A mechanical compensation structure is adapted to stabilize the position of the focal plane. The Gaussian design result is applied for the midwave infrared spectrum, and aberrations can be reduced by controlling the geometric parameters of the thick lens. One hybrid achromatic singlet is introduced utilizing a diffraction optical element. The kinoform surface relief is calculated being the same as its microfabrication process. The effects of the discontinuous zonal profile and the thermal degradation are evaluated.     

Calculation of diffraction patterns on a spatial surface

An approximated formulation of the Fresnel function is put forward and is used in the approximate evaluation of the Fresnel diffraction integral. By comparing the approximate formulation with the experimental measurements and calculations in the fast Fourier transform (FFT) method of the diffraction integral, we demonstrate that the proposed method is sufficiently accurate for calculating the Fresnel diffraction. For the diffraction field calculation on a spatial surface, the calculation speed of this method is usually higher than that of the FFT method.     

Evaluation of the errors in equations for stress intensity factors of elliptical type cracks

A method of evaluation of the integral error in the approximate equations for stress intensity factors based on the energy balance equation is described. A review of the equations for calculation of stress intensity factors for cracks of elliptical, semielliptical, and quarter-elliptical form is given. The boundaries of applicability of the equations are given, the errors in approximation of the corresponding numerical solutions are noted, and the integral errors in the equations for stress intensity factors are also calculated.Institute of Problems of Strength, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Problemy Prochnosti, No. 10, pp. 53–58, October, 1989.     

Design of diffractive axicons producing uniform line images in Gaussian Schell-model illumination

We present a design method for diffractive axicons in spatially partially coherent Gaussian Schell-model illumination. The method of stationary phase applied to the Fresnel diffraction integral for on-axis intensity leads, on requiring a uniform axial image profile, to a second-order differential equation for the optimal axicon phase function. The first integral can be formally performed, and the phase function is subsequently obtained numerically. The correctness of the synthesized phase profiles is confirmed by numerical simulations using partially coherent Fresnel diffraction theory. The effects of input-beam spot size and coherence width are assessed, and influences of different forms of apodization, including asymmetric functions for narrow incident beams, in annular-aperture diffractive axicons are examined.