Kirchhoff Diffraction Integral Research Articles

Thermal-induced nonlinearities in rose, linseed, and chamomile oils using continuous wave visible laser beam

Self-diffraction rings or spatial self-phase modulation were observed in rose, linseed, and chamomile oils under 473 nm continuous wave laser irradiation. The measurements were performed by propagating the laser beam through a cell containing each sample. The number of rings as well as diameter of the outer-most ring in each pattern obtained increases monotonically with increasing input power. The diffraction ring patterns are theoretically simulated using Fresnel–Kirchhoff diffraction integral in the case of an optically thin medium. The experimental and simulation results show that when a laser beam with Gaussian profile is transmitted through an oil medium, a series of circular diffraction rings forms in the intensity distribution pattern in the far-field. The nonlinear refractive index, n2, was determined from the number of observed rings and by the Z-scan technique. The results obtained from self-diffraction rings experiment and Z-scan are compared and analyzed for the three different oils. A large value was obtained of the order of n2 = 1.32 × l0−6 cm2/W for chamomile oil using the diffraction ring pattern technique. This large nonlinearity is attributed to a thermal effect resulting from linear absorption. Moreover, the optical limiting characteristics of rose, linseed, and chamomile oils were investigated.

Effects of mask-alignment error on point spread function for multi-level Fresnel diffractive lenses

The full aperture complex amplitude transmittance function of a multi-level diffraction lens with mask-alignment errors was derived based on scalar diffraction theory. The point spread function (PSF) was calculated by the Kirchhoff diffraction integral. It is found that the radius of the Airy disk increases with the increase of the error in the direction of misalignment, and the image center shifts along the direction of misalignment. A four-level diffractive lens with a diameter of 80 mm was fabricated, and its PSF and diffraction efficiency of +1st order were calculated and measured. The distribution of PSF is consistent with the calculated results, and the tested diffraction efficiency is slightly smaller than the calculated value; the relative error is 5.71%.

From Airy to Abbe: quantifying the effects of wide-angle focusing for scalar spherical waves

Recent advances in optical microscopy have enabled imaging with spatial resolution beyond the diffraction limit. This limit is sometimes taken as one of several different criteria according to different conventions, including Rayleigh’s Abbe’s and Sparrow’s In this paper, we perform a parametric study, numerically integrating the scalar Kirchhoff diffraction integrals, and we propose new functional forms for the resolution limits derived from scalar focusing. The new expressions remain accurate under wide angle focusing, up to Our results could materially impact the design of high intensity focused ultrasound systems, and can be used as a qualitative guideline for the design of a particular type of planar optical element: the flat lens metasurface.

Role of the aperture in Z-scan experiments: A parametric study

In close-aperture Z-scan experiments, a small aperture is conventionally located in the far-field thereby enabling the detection of slight changes in the laser beam profile due to the Kerr-lensing effect. In this work, by numerically solving the Fresnel–Kirchhoff diffraction integrals, the amount of transmitted power through apertures has been evaluated and a parametric study on the role of the various parameters that can influence this transmitted power has been done. In order to perform a comprehensive analysis, we have used a nonlinear phase shift optimized for nonlocal nonlinear media in our calculations. Our results show that apertures will result in the formation of symmetrical fluctuations on the wings of Z-scan transmittance curves. It is further shown that the appearance of these fluctuations can be ascribed to the natural diffraction of the Gaussian beam as it propagates up to the aperture plane. Our calculations reveal that the nonlocal parameter variations can shift the position of fluctuations along the optical axis, whereas their magnitude depends on the largeness of the induced nonlinear phase shift. It is concluded that since the mentioned fluctuations are produced by the natural diffraction of the Gaussian beam itself, one must take care not to mistakenly interpret them as noise and should not expect to eliminate them from experimental Z-scan transmittance curves by using apertures with different sizes.

High-efficiency gold-coated cross-grating for heterodyne grating interferometer with improved signal contrast and optical subdivision

The gold-coated cross-grating is investigated to simultaneously improve the diffraction efficiency, signal contrast and optical subdivision in heterodyne grating interferometer (HGI). Based on the theories of Fresnel–Kirchhoff diffraction integral and rigorous coupled wave analysis (RCWA), the diffractive characteristics related to diffraction efficiency, wavelength selectivity and polarization dependency are analyzed. The analysis reveals that grating pitch less than twice of the incident wavelength ought to be adopted here to avoid the second-orders, and longer wavelength is preferred to achieve high diffraction efficiency in the visible region. Meanwhile, the equivalent efficiency for TE and TM polarization, probably up to 18.32% at the wavelength of 632.8nm, can be obtained with normal incidence which results in a signal contrast of 100% theoretically. Additionally, an eightfold optical subdivision method with cross-gratings is proposed for HGIs, and it is applicable to nearly all the existing two-dimensional (2D) optical configurations. The experimental tests on diffraction field, polarization dependency and signal contrast are performed. The results show reasonable consistency with the theoretical analysis, which validates the capability of cross-gratings for HGIs.

The multidimensional stationary-phase method for an asymptotic estimate of edge effects in the multiple Kirchhoff diffraction by curved surfaces

An analytical approach is proposed to study the contribution of edge effects in the multiple high-frequency diffraction, according to guidelines of classical Kirchhoff theory in (scalar) wave propagation. We start from a suitable asymptotic analysis of the Kirchhoff diffraction integral, here set up in a generalized (iterated) form to describe the multiple reflections from an arbitrary sequence of curved reflecting (smooth) surfaces. The explicit formula obtained for a concrete example of double reflection is compared with the results from direct numerical simulation.

Effect of tilt and astigmatism aberration of turbulent atmosphere on the intensity distributions of a LG11 beam

The effects of turbulent aberration such as z-tilt aberration, and astigmatism on the intensity distribution of double ring shaped LG11 beam have been investigated using the Fresnel–Kirchhoff diffraction integral and the quadratic approximation. The effect of turbulent aberration on the intensity distributions at the focal plane with different turbulent strength, and propagation distance is analyzed numerically. It is observed that the effect of tilt aberration gives major effect on the intensity distribution of the LG(1,1) beam in the turbulent atmosphere.

Subwavelength visualization of light in thin film waveguides with photoelectrons

We report the visualization and quantitative analysis of electromagnetic surface fields at solid surfaces with the potential for \ensuremath{\lambda}/50 resolution. To illustrate this capability, we investigate patterns in two-photon photoemission images of light-diffracting structures in waveguiding, transparent thin films. The obtained micrographs show interference patterns between incident and guided light with a remarkable sensitivity to subwavelength features. We demonstrate that photoemission rates are directly related to the surface field strengths and develop a subwavelength method to calculate the surface fields from optical properties and surface topology based on the two-dimensional Kirchhoff diffraction integral. Calculated images based on this theoretical approach compare favorably to experimental electron micrographs.

Analysis and Design of a Quasi-Optical Mode Converter for a 1-kW, 550-GHz, $ {\rm TE}_{15,2}$-Mode Gyrotron

A simple method is presented for the analysis and design of gyrotron output couplers based on a Vlasov-type launcher and a single focusing mirror. It is shown that a quasi-analytical solution to the Kirchhoff diffraction integral can be implemented to accurately model the diffraction spreading of the launched radiation with good accuracy relative to an Electric Field Integral Equation simulation. Spectral information is exploited in the design to appropriately size the mirror and to predict the conversion efficiency and Gaussian beam parameters of the converted beam. Additionally, a set of design equations, which is more general than any previously published, is derived for the reflector. A comparison of the conversion properties of a converter designed using the new equations and previously published equations is provided. For the case considered, the converter designed using the new equations generate a field distribution with almost complete isolation of the main radiation lobe at the window plane. For the reflector designed using the previous equations, the contamination of the main lobe by the side lobes is significant beginning at the 9-dB contour. The spot size of the gyrotron output beam has also been reduced by about 30% by using the new design equations.

An asymmetric optical vortex generated by a spiral refractive plate

Using a revised Kirchhoff diffraction integral and the finite-difference time-domain method we show that an optical vortex generated by a refractive spiral plate with a relief step has an asymmetric profile. The annular diffraction pattern in the optical vortex beam cross-section is found to be disturbed not only for near-field diffraction but also for the Fresnel zone. For a spiral phase plate with topological charge 3, fabricated on a resist, optical vortex asymmetry has been shown experimentally by near-field scanning optical microscopy.

A new representation of the Kirchhoff's diffraction integral

The diffraction integral of Kirchhoff is rearranged according to its integral boundaries. The new approach is based on the theory of the edge dislocation waves and provides a more correct field representation for the semi-infinite and infinite integrals in the direct numerical computation. The integral is studied on the diffraction problem of plane waves by a perfectly conducting half-plane. The correctness of the scattering diagrams is compared with the classical approach and the Fresnel integral representation of the scattered waves numerically.

Spatial self-phase modulation of a laser beam propagating through liquids with self-induced natural convection flow

In this work, the results of experimental and theoretical investigations of diffraction ring patterns are reported. Diffraction rings are formed due to the self-phase modulation of a continuous wave laser beam propagating through a solution of dye in ethanol. The self-phase modulation of the laser beam is the result of the changes in the refractive index due to the heating of the sample by a small absorbed fraction of the laser power. To find the thermal nonlinearity inside the liquid irradiated by a Gaussian laser beam, we have solved the heat transfer equation including conduction and convection effects in a liquid medium. The temporal dynamics and structural characteristics of the diffraction ring patterns are studied theoretically on the basis of a Fresnel–Kirchhoff diffraction integral in the approximation of an optically thin absorbing medium. The simulation results show that when a Gaussian laser beam is transmitted through a liquid medium the intensity distribution pattern in the far-field first forms a series of circular diffraction rings, but after a period of time the rings change to half circular symmetry due to convection.

Generation of negative pressure of underwater intensive acoustic pulse and cavitation bubble dynamics

In this paper, the high speed photography and the pressure measurement are used to study the propagation and focusing process of the underwater intensive acoustic pulse which is reflected from an ellipsoidal reflector. The experimental results indicate that the reflector contribute to both the sound energy focusing and the generation of a negative pressure portion of the intensive acoustic pulse, and the cavitation bubble cluster will be induced to grow accordingly. Based on the experimental results, a sound propagation model derived from Kirchhoff diffraction integral and the equation of bubble dynamics proposed by Qian and Xiao [Qian Z W and Xiao L 2003 Chin. Phys. Lett. 20 80; Qian Z W and Xiao L 2008 Chinese Physics B 17 3785] are used to numerically study the propagation of the intensive acoustic pulse and the bubble motion. The numerical results indicate that the "wake wave" and "edge wave" will evolve into the negative pressure portion of the intensive acoustic pulse in front of the far focus of the reflector and the "center wave" will evolve into the negative pressure part beyond the far focus of the reflector. When the bubble is subjected to the reflected wave, it will be compressed and oscillate periodically during the positive pressure phase, and it will expand during the negative pressure phase of the incident pulse. And after the reflected wave passes, the typical processes of expanding, collapsing and rebounding can be seen clearly from the photographs. The research results have practical significance for understanding the propagation characteristic of the underwater intensive acoustic pulse with an ellipsoidal reflector.

Fourth order correction to a paraxial Gaussian beam in a rotationally symmetric system of non-spherical surfaces

AbstractWe formulate the fourth order correction to a paraxial Gaussian beam propagating through a non-spherical surface system that is rotationally symmetric. First we represent the Gaussian beam by a complex-source-point spherical wave (CSPSW). Next we examine the evolution of the CSPSW through the non-spherical surface system by applying a Fresnel–Kirchhoff diffraction integral. We find a ray-optical solution to the diffraction integral in terms of both the conic constant and the aspheric coefficient of fourth order. Then we numerically evaluate the quality factor of the fourth order-corrected beam propagating through either an aspheric mirror or a thin lens made up of two non-spherical surfaces. The fourth order formulas derived here can be useful in determining the quality factor of the Gaussian beam degraded by a rotationally symmetric system of non-spherical surfaces.

Far-field temporal dynamics of a laser beam passed through a thin nanocolloidal layer

We report on the experimental results of the temporal dynamics and formation of far-field diffraction patterns obtained when a low-intensity laser beam passes through a thin cell with metallic nanocolloids. The steady state structure of the intensity distribution on a distant screen is shown to be formed in a few seconds starting from the moment of laser beam incidence on the colloid and consists of several coaxial bright and dark rings. The ring’s diameter and number are varied depending on the solvent type and optical thickness of the colloid. The diffraction patterns depend also on the particle concentration and laser power. The theoretical interpretation of the observable peculiarities of diffraction pattern dynamics is carried out based on the free-space Kirchhoff diffraction integral and the analytical solution of the heat transfer equation for a light absorbing medium.

The simulation of the origin and propagation of speckle field generated through a plane wave and Gaussian beam and its verification by speckle correlation method

This paper presents a simulation model of the origin and propagation of speckle field within the Fresnel approximation and verification of the model through a speckle correlation method for determination of in-plane object's translation. There is used a convolution form of the Fresnel–Kirchhoff diffraction integral in the simulation model instead of often used Fast Fourier Transform approach. A possible way of simulation of the light scattering by object's surface and in-plane translation of the illuminated surface is mentioned, whereas two types of illumination (plane wave and Gaussian beam) are used. Finally, results of numerical correlation of generated displaced speckle fields are shown and compared with the results obtained by theoretical relations.

Spatial phase modulation from permanent memory in doped glass

Diffraction rings are observed from photoinduced permanent memory of doped glass. The permanent memory is created by the high-intensity picosecond laser beam. A 1 mm spot size of laser beam creates spatially variable refractive index memory, which appears as a void located inside the glass. When a probe laser beam passes through the memory region, the diffraction rings arisen from spatial phase modulation of the transverse phase of the input beam are created. Agreement between the observed and calculated beam pattern using Kirchhoff's diffraction integral is satisfactory.

The effect of tilt aberration and astigmatism of turbulent atmosphere on the intensity distribution of a vortex carrying Gaussian beam

The effect of tilt and astigmatism aberration of the turbulent atmosphere on the intensity distribution of a focused vortex carrying Gaussian beam was investigated based on the extended Fresnel–Kirchhoff diffraction integral and the quadratic approximation of phase structure function. Our results have shown that the intensity distribution on the focal plane in the effect of tilt aberration changes with the turbulent strength, the propagation distance and the topological charge of the initial beam. The propagation distance is larger, the focal spot size will be larger, and the central dip will be less deep. It is also noticed that the intensity distribution of a beam with single topological charge is affected more by tilt aberration in comparison to the beam with double topological charge. The effect of astigmatism on the intensity distribution is quite less than that of tilt aberration. For focused beam propagation in atmosphere, the effect of the tilt aberration on the intensity distribution of optical vortex beam is the main effect of total turbulent aberration.

Apertured paraxial Bessel beams

The paraxial Bessel beam is obtained by applying an approximation in the wavenumbers. The scattering of the beams by a circular aperture in an absorbing screen is investigated. The scattered fields are expressed in terms of the Fresnel integrals by evaluating the Kirchhoff diffraction integral in the paraxial approximation. The results are examined numerically.

Electromagnetic Radiation in Accelerator Physics

This article reviews some fundamental concepts and presents several recent techniques used for calculation of radiation in various environments. They include properties of longitudinal and transverse formation lengths of radiation, usage of the parabolic equation and the Kirchhoff diffraction integral in radiation, coherent radiation and fluctuations in the beam, and the radiative reaction force resulting from coherent radiation.