Fresnel-Kirchhoff Diffraction Formula Research Articles

Effect of sample position on formation of spatial-self phase modulation ring patterns in poly(azaneylylidene-acylene)

Poly(azaneylylidene-acylene) "DAZA" polymer has been synthesized and characterized using different spectroscopic techniques. Self-diffraction ring patterns at far field were observed, when the DAZA polymer was irradiated with a CW laser beam. The influence of sample position, geometrical pumping and laser power has been investigated on the formation of the far field self-diffraction ring patterns of the Gaussian laser beam passing through the DAZA polymer as nonlinear medium. Then, the Fresnel-Kirchhoff diffraction formula was applied to perform simulation of the self-diffraction ring patterns with different laser power at two sample positions before and after the focal point (z = 0) with respect to the geometrical pumping. Our results have shown the effect of type of geometrically pumping on the sample cell (horizontally /vertically) with respect to the distortion of the SSPM patterns. It has been found that, good consistent agreements between experimental and theoretical patterns.

Spatiotemporal characteristics of dynamic speckle from a 3D target in atmospheric turbulence

Spatiotemporal correlation function is the basic characteristic of dynamic laser speckle and the basis of various applications. The correlation function of the speckle intensity from a 3D target in turbulent atmosphere is derived based on the model of random phase screen and Fresnel Kirchhoff Diffraction Formula, and a fast algorithm based on FFT is developed. The particularity of dynamic speckle in turbulence is numerically analyzed and discussed. The results show that the speckle intensity fluctuates at two independent scales both in space and time domain, which are affected by target size and atmospheric turbulence respectively. In particular, the time scale caused by turbulence is also affected by the target translating velocity and the wind direction. The theory and algorithm developed in this paper can play important roles in applications of laser speckle such as remote detection in atmospheric environment.

Doppler Analysis of Forward Scattering Radar With Opportunistic Signals From LEO Satellites

Recently microwave communication signals from low Earth orbit (LEO) satellites have been proposed to be used to detect flying objects in a ground-based forward scattering passive radar setup. In this work the Doppler shift and the Doppler spread due to the flying object in such a system are derived directly from the Fresnel-Kirchhoff diffraction formula using a rotational Cartesian coordinate system to accommodate the continuously moving signal source from the LEO satellites. Coordinate transformations for flying objects including those in the atmosphere with rectilinear motion relative to the ground and those in space with circular orbits are investigated. Both analytical and simulation results demonstrate that the Doppler shift and Doppler spread are related to the velocity and volumetric profile of the flying object, thereby in future they may be used to help identify its size and velocity.

Light propagation in anisotropic and metamaterial media by a Finslerian vector eikonal method

A systematic study of light propagation in anisotropic and metamaterial media is presented by using a Finslerian vector eikonal method, that is, by considering the approximation of very short wavelengths of light in Maxwell’s equations and using particular metric tensors of the general Finsler metric tensor induced by an anisotropic medium. Green’s vector functions, and therefore general solutions, are obtained as a product of a complex scalar function with a local polarization unitary vector. The phase (wave surface or eikonal function) of such a complex scalar function is calculated using a Fermat integral derived from the Maxwell equations. This integral can be considered as the integral form of the eikonal equation and provides a Finslerian metric tensor which allows, together with energy conservation in anisotropic media, the derivation of a general expression depending on the eikonal function for obtaining the scalar amplitude of such a complex function. Likewise, the local polarization unitary vector u ( r ) is calculated by solving an approximated homogeneous linear equations system also depending on the eikonal function. Exact solutions are obtained for uniaxial materials and hyperbolic metamaterials, and approximate solutions are proposed for biaxial materials. Finally, an approximated vector Fresnel–Kirchhoff diffraction formula is presented by using vector Green’s functions in order to study optical propagation. For the sake of completeness, interference patterns produced by anisotropic materials are presented using the results obtained. This systematic study of anisotropic and metamaterial media by a Finslerian vector eikonal method is intended in particular for university physics teachers and as an advanced topic for undergraduate and first year postgraduate students.

Atmospheric coherent X-ray diffraction imaging for in situ structural analysis at SPring-8 Hyogo beamline BL24XU.

Coherent X-ray diffraction imaging (CXDI) is a promising technique for non-destructive structural analysis of micrometre-sized non-crystalline samples at nanometre resolutions. This article describes an atmospheric CXDI system developed at SPring-8 Hyogo beamline BL24XU for in situ structural analysis and designed for experiments at a photon energy of 8 keV. This relatively high X-ray energy enables experiments to be conducted under ambient atmospheric conditions, which is advantageous for the visualization of samples in native states. The illumination condition with pinhole-slit optics is optimized according to wave propagation calculations based on the Fresnel-Kirchhoff diffraction formula so that the sample is irradiated by X-rays with a plane wavefront and high photon flux of ∼1 × 1010 photons/16 µmø(FWHM)/s. This work demonstrates the imaging performance of the atmospheric CXDI system by visualizing internal voids of sub-micrometre-sized colloidal gold particles at a resolution of 29.1 nm. A CXDI experiment with a single macroporous silica particle under controlled humidity was also performed by installing a home-made humidity control device in the system. The in situ observation of changes in diffraction patterns according to humidity variation and reconstruction of projected electron-density maps at 5.2% RH (relative humidity) and 82.6% RH at resolutions of 133 and 217 nm, respectively, were accomplished.

Wavefronts, caustic, and intensity of a plane wave refracted by an arbitrary surface: the axicon and the plano spherical lenses.

The aim of the present work is to obtain an integral representation of the field associated with the refraction of a plane wave by an arbitrary surface. To this end, in the first part we consider two optical media with refraction indexes n1 and n2 separated by an arbitrary interface, and we show that the optical path length, ϕ, associated with the evolution of the plane wave is a complete integral of the eikonal equation in the optical medium with refraction index n2. Then by using the k function procedure introduced by Stavroudis, we define a new complete integral, S, of the eikonal equation. We remark that both complete integrals in general do not provide the same information; however, they give the geometrical wavefronts, light rays, and the caustic associated with the refraction of the plane wave. In the second part, using the Fresnel-Kirchhoff diffraction formula and the complete integral, S, we obtain an integral representation for the field associated only with the refraction phenomena, the geometric field approximation, in terms of secondary plane waves and the k-function introduced by Stavroudis in solving the problem from the geometrical optics point of view. We use the general results to compute: the wavefronts, light rays, caustic, and the intensity associated with the refraction of a plane wave by an axicon and plano-spherical lenses.

Supplementary zones-surrounded Fresnel zone plate with enhanced optical resolution

The supplementary zones surrounding an ordinary Fresnel zone plate (FZP) are utilized to improve the optical resolution at focus. Based on the Fresnel–Kirchhoff diffraction formula, an optimization model is used to design the supplementary zones-surrounded FZP (SZFZP). We justified this optimization model by comparing the numerical simulations and experimental measurements in the visible light region. As expected, the comparison shows a good fit between the simulation and measurement. Mostly, both results showed that the optical resolution and the focal intensity in the visible light region are improved by applying the optimized SZFZP. Afterwards, this model is used to design the SZFZP for applications in extreme ultraviolet (EUV) light. At this point, we demonstrated the ability to fabricate the optimized SZFZP in the EUV region. In addition, we simulated the focusing properties for the designed SZFZP. Consequently, the numerical results for EUV light focused by the designed SZFZP exhibited better performances in optical resolution and intensity.

Padrão de difração de um conjunto de <italic>n</italic> fendas não simétricas e de larguras arbitrárias

<p>Neste trabalho investigamos o padrão de difração no regime de Fresnel (campo próximo) e de Fraunhofer (campo distante) para um conjunto de <italic>n</italic>fendas não simétricas e de larguras diferentes. Partindo da fórmula de difração de Fresnel-Kirchhoff conseguimos obter uma expressão para a intensidade da onda difratada por um conjunto de <italic>n</italic> fendas de tamanhos arbitrários onde é possível observar a transição do regime de Fresnel para o regime de Fraunhofer.</p>

Spectra restoration of a transmissive periodic structure in near-field diffraction (Talbot spectra).

The free-space near-field diffraction of a transmissive periodic structure is studied with Fresnel-Kirchhoff diffraction formula. A new effect named Talbot spectra is proposed theoretically from the results, showing that under some conditions the spectra of an incident coherent polychromatic light can be completely restored after the diffraction. It can be contrasted with its monochromatic counterpart, Talbot images, which rebuild the images of a periodic structure at specific places in the near field. Taking a sinusoidal amplitude grating as an example for illustrating the idea, the general conditions and specific locations to find the restored spectra are obtained. The spectral switches involved are also presented.

Wave optics and image formation in gravitational lensing

We discuss image formation in gravitational lensing system using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from waves without using a lens equation for gravitational lensing.

Wave Optics and Image Formation in Gravitational Lensing

We discuss image formation in gravitational lensing systems using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from wave functions without using a lens equation for gravitational lensing.

An accurate SISAR imaging method of ground moving target in forward scatter radar

In order to accurately extract the profile image of the ground moving targets (GMT) in the forward scatter radar (FSR), a forward scatter signal model of GMT is built based on the Fresnel-Kirchhoff diffraction formula, the multipath propagation model and the high order phase approximation method. The classical shadow inverse synthetic aperture radar (SISAR) imaging algorithm is modified using the proposed signal model. In the modified imaging algorithm, the impacts of ground reflection, non-linear variation of target Doppler frequency and the target motion direction on imaging accuracy are discussed. Meanwhile, non-linear and time-dependent parameters are used to compensate for the motion phase of the GMT forward scatter signal, and a novel estimation method of the height difference and median line height of the target profile is put forward under the multipath interference. Finally, both the accuracy of the proposed signal model and the effectiveness of the modified imaging algorithm are validated by the simulation results.

Three-dimensional radiated sound field display system using directional loudspeakers and wave field synthesis

Three-dimensional (3D) radiated sound field display systems are important toward realizing ultrarealistic communications systems such as 3D television. In this paper, a 3D radiated sound field display system using directional loudspeakers and wave field synthesis is proposed. The proposed system is based on the Fresnel-Kirchhoff diffraction formula, which is the approximate form of the Kirchhoff-Helmholtz integral equation. The proposed system was developed by constructing the surrounding microphone array and radiated loudspeaker array. To evaluate the performance of the developed system, the positions of sound images were estimated. The results indicated that the 3D sound field is accurately controlled outside of the radiated loudspeaker array by sharpening the radiation directivity of the loudspeaker units and that the sound images are not correctly estimated in the developed system because the radiation directivity of loudspeaker units is not sufficiently sharp.

Investigation on accurate signal modelling and imaging of the moving target in ground-based forward scatter radar

The existing signal model and imaging algorithm of the moving target in the forward scatter radar (FSR) are built based on the assumptions that the baseline is long, diffraction angle is small and velocity direction of the target is approximately perpendicular to the baseline. However, the ground-based FSR system is characterised by short baseline and large diffraction angle, and the velocity direction of the target is not always perpendicular to the baseline. Therefore in many cases, the above assumptions introduce significant errors to the imaging results in the ground-based FSR. In the light of the imaging requirements of the moving target in the ground-based FSR, firstly, the current forward scatter (FS) signal model is modified using the high-order approximation method based on the Fresnel-Kirchhoff diffraction formula. The modified signal model considered the effect of high-order terms of Doppler phases, quadratic term of the target profile length, velocity direction of the target and diffraction angle changes, and therefore FS signal of the moving target in the ground-based FSR can be more precisely described. Secondly, the current imaging algorithm is modified based on the modified signal model, and the expressions applicable to precise reconstruction of the target profile in the ground-based FSR are analytically deduced. Finally, the effectiveness of the modified imaging algorithm is validated through the simulation results.

Various evaluations of a diffractive transmitted field of light through a one-dimensional metallic grating with subwavelength slits

Abstract We compare the Fresnel-Kirchhoff diffraction formula, the superposition of cylindrical waves, and the twodimensional (2D) Green’s function diffraction formula with a rigorous vector algorithm in calculating the near and intermediately transmitted field of light through a one-dimensional metallic grating with subwavelength slits. It is found that the results calculated by the 2D Green’s function diffraction formula coincide well with the precise result. The other evaluations deviate from the exact result by varying proportions. Our findings may provide a useful and precise way to analyze the transmitted field features of a metallic grating and subsequent possibility of achieving optimal designs for metallic optical elements with subwavelength scale.

Incoming inclination factor for scalar imaging theory

The conventional scalar imaging theories that are represented in the Fresnel-Kirchhoff diffraction formula or the Rayleigh-Sommerfeld diffraction formulae involve contradictions. By introducing incoming inclination factors not only for the diffraction surface but also for the image plane, we propose a new equation that fulfills both the self-consistency and the reciprocity theorem. We also confirm the validity of this new equation by both theoretical discussions and numerical calculations.

ホログラフィ入力に適したインテグラルフォトグラフィ撮影のための複数要素画像の重複記録低減に関する検討

We describe how to reduce interference between elemental images suitable for production of holograms during integral photography pickup that uses a lens array composed of typical convex lenses used as elemental lenses. We propose the use of an optical system that consists of both a field lens and an aperture that is located in the focal plane of the field lens. The system is arranged between the lens array and the camera used for taking elemental images. Diffraction patterns in the focal plane of the field lens of beams going through elemental lenses were calculated with the Fresnel-Kirchhoff diffraction formula. Consequently, we clarified that the system was able to limit the incident angle of the beam passing through the elemental lenses with the size of the aperture located in the focal plane. In addition, we carried out an optical experiment and proved that using the optical system reduced the interference between the elemental images.

Optimizing design of high-resolution optical encoder using a point-source light-emitting diode with slits

We discuss an interference fringe type encoder using a point source LED equipped with a slit; this encoder offers high resolution, a wide temperature range, and high reliability. To optimize the design of this encoder, we develop a simulator for an interference-fringe-type encoder using the Fresnel-Kirchhoff diffraction formula, and we analyze the following elements of the optical system of the encoder: (1) the effect of the LED wavelength band, (2) the effect of the size of the LED radiation area, and (3) the effect of the shape of the diffraction grating surface. The interference fringes of this encoder are generated with 60% greater efficiency than an encoder using an LD featuring a very narrow slit width.

Spectral and Temporal Distributions of Ultrashort Laser Pulses Diffracted by a Slit

We present numerical images for frequency-space and time-space distributions of cycle laser pulses diffracted by a slit with Fresnel-Kirchhoff diffraction formula. The wide frequency bandwidth with respect to the cycle pulse results in spectral redistribution of diffracted laser pulses and dispersion effect subsequently. Although spatial patterns still retain their asymmetry, spectral or temporal distributions lose their asymmetry gradually with increasing distance from the beam center for few-cycle pulses.

Three-dimensional behavior of apodized nontelecentric focusing systems.

The scalar field in the focal volume of nontelecentric apodized focusing systems cannot be accurately described by the Debye integral representation. By use of the Fresnel-Kirchhoff diffraction formula it is found that, if the aperture stop is axially displaced, the focal-volume structure is tuned. We analyze the influence of the apodizing function and find that, whereas axially superresolving pupil filters are highly sensitive to the focal-volume reshaping effect, axially apodizing filters are more inclined to the focal-shift effect.