Rayleigh-Sommerfeld Integral Research Articles

Improvement of optical wave propagation simulations: the scaled angular spectrum method for far-field and focal analysis

In the field of optics, accurately simulating wave propagation is essential for both theoretical insights and practical applications. This study introduces the scaled angular spectrum method (Sc-ASM) for simulating far-field and focal patterns, overcoming the limitations inherent in the standard angular spectrum method (ASM) by allowing variable sampling intervals between source and observation planes. Comparative analysis with the Rayleigh–Sommerfeld integral demonstrates Sc-ASM's superior accuracy in estimating the far-field patterns of beams with complex phase structures. Our results underscore Sc-ASM’s potential to set a new benchmark in optical simulations, significantly advancing optical system design and the study of wave propagation.

Calculation of the intensity at the sharp focus of a cylindrical vector beam by three methods

We conduct a comparative analysis of diffraction fields upon sharply focusing vortex and non-vortex incident beams, calculated using three non-paraxial methods. The methods employed are a finite difference time domain (FDTD) method, a Rayleigh–Sommerfeld integral, and a Richards–Wolf transformation. The Richards–Wolf transformation is used with two apodization functions of the exit pupil, for a spherical lens and a thin diffractive lens. The numerical simulation shows that the Rayleigh–Sommerfeld integral and the Richards–Wolf transformation can produce almost the same result and save time significantly. Meanwhile, the root-mean-square deviation of the results of both methods from the FDTD method can be as low as 2%. If an ultra-short focal length is used, the Richards–Wolf transformation is found to be more accurate, whereas with increasing distance from the initial plane and outside the focal plane, the Rayleigh-Sommerfeld integral is more accurate.

Influence of optical "dipoles" on the topological charge of a field with a fractional initial charge.

In this work, using the Rayleigh-Sommerfeld integral and Berry formula, a topological charge (TC) of a Gaussian optical vortex with an initial fractional TC in the far field was calculated. It was found that, for diverse fractional parts of the TC, the beam contained different numbers of screw dislocations, which determined the TC of the entire beam. If a fractional part of the TC was small, the beam consisted of the main optical vortex centered on the optical axis, with the TC equal to the nearest integer (say n>0) and two edge dislocations located on the vertical axis (one above and the other below the center). When the fractional part of the initial TC increased, a "dipole" was formed from the upper edge dislocation, consisting of two vortices with TCs equal to +1 and -1. With a further increase in the fractional part, the additional vortex with TC=+1 moved to the center of the beam, and the vortex with TC=-1 moved to the periphery. When the fractional part of the TC increased further, another "dipole" was formed from the lower edge dislocation, in which, on the contrary, the vortex with TC=-1 was displaced to the optical axis (to the center of the beam) and the vortex with TC=+1 moved to the beam periphery. When the fractional part of the TC became equal to 1/2, the lower vortex with a TC=-1, which was earlier displaced to the center of the beam, began to shift to the periphery, and the upper vortex with a TC=+1 moved closer and closer to the center of the beam, eventually merging with the main vortex when the fractional part approached 1. Such dynamics of additional vortices with TCs above +1 and below -1 determined which whole TC the beam would have (n or n+1) for different values of the fractional part from the segment [n,n+1]. Our analysis has shown that, for any value of the fractional part of the initial topological charge, the TC of the beam in the far field will not be determined. Since, with an increase in the radius of the circle in the beam section on which the TC is calculated, more optical "dipoles" will appear, and the TC will be either n or n+1.

Topological charge of optical vortices in the far field with an initial fractional charge: optical "dipoles"

In this work, using the Rayleigh-Sommerfeld integral and the Berry formula, the topological charge (TC) of a Gaussian optical vortex with an initial fractional TC is calculated. It is shown that for different fractional parts of the TС, the beam contains a different number of screw dislocations, which determine the TС of the entire beam. With a small fractional part of the TС, the beam consists of the main optical vortex centered on the optical axis with the TС equal to the nearest integer (let be n), and two edge dislocations located on the vertical axis (above and below the center). With an increase in the fractional part of the initial TC, a "dipole" is formed from the upper edge dislocation, consisting of two vortices with TC+1 and –1. With a further increase in the fractional part, the additional vortex with TC+1 is displaced to the center of the beam, and the vortex with TC–1 is displaced to the periphery. With a further increase in the fractional part of the TC, another "dipole" is formed from the lower edge dislocation, in which, on the contrary, the vortex with TC–1 is displaced to the optical axis (to the center of the beam), and the vortex with TC+1 is displaced to the beam periphery. When the fractional part of the TC becomes equal to 1/2, the "lower" vortex with TC–1, which was displaced to the center of the beam, begins to shift to the periphery, and the "upper" vortex with TC+1 moves closer and closer to the center of the beam and merges with the main vortex when the fractional part approaches 1. Such dynamics of additional vortices with upper TC+1 and lower TC–1 determine the whole TC the beam have (n or n+1) for different values of the fractional part from the segment [n, n+1].

FOCUSING OF MODES WITH AN INHOMOGENEOUS SPATIAL POLARIZATION OF THE DIELECTRIC RESONATOR OF A TERAHERTZ LASER

Wave beams with non-uniform spatial polarization of radiation are necessary for solving important fundamental and applied problems related to the interaction of electromagnetic waves of the terahertz range with matter - surface diagnostics of materials, thin films, biological objects, information transmission, and processing systems, achieving subwave resolution in tomography, communication systems, etc. The results are presented in the literature on focusing on pulsed radiation beams in the terahertz range. Data on the focusing of continuous radiation beams are practically absent. The spatial-energy characteristics of laser beams having an inhomogeneous spatial polarization at their moderate and sharp focusing are theoretically investigated. As the radiation understudy, in the numerical simulation of the focusing of wave beams in the terahertz range, we used the modes of a laser waveguide dielectric resonator that coincide with the eigenmodes of a hollow dielectric waveguide. Symmetric and asymmetric modes with both spatially inhomogeneous azimuthal, radial, and homogeneous linear field polarizations are studied. The components of the electric field of laser beams during their propagation in free space were studied using the Rayleigh-Sommerfeld integrals in the nonparaxial approximation. The effect of the focusing lens on laser radiation was taken into account using the amplitude-phase correction function. The distributions of the total intensity of the resonator modes and their individual field components in the focal region of the lens are studied. The obtained results expand the knowledge about the features of focusing terahertz laser beams.

Non-paraxial propagation of a circular Lorentz-Gauss vortex beam.

The optical field of a circular Lorentz-Gauss vortex beam propagating in free space is derived by using the vectorial Rayleigh-Sommerfeld integrals. The expression of the longitudinal component of the angular momentum density of a circular Lorentz-Gauss vortex beam is also presented. The normalized intensity, the three components of intensity, and the longitudinal angular momentum density of a circular Lorentz-Gauss vortex beam are demonstrated in the reference plane, respectively. The effects of linearly polarized angle, topological charge, and Lorentzian width parameter on the intensity, the three components of intensity, and the longitudinal angular momentum density are analyzed, respectively. The normalized intensity and the longitudinal angular momentum density of a paraxial circular Lorentz-Gauss vortex beam are also shown in the same reference plane. The missing longitudinal component of the optical field will result in deficient patterns of the intensity and the longitudinal angular momentum density. In terms of the longitudinal angular momentum density, the longitudinal component of a circular Lorentz-Gauss vortex beam cannot be neglected under the condition that the Lorentzian width parameter and the Gaussian waist are of the order of the optical wavelength.

Non-paraxial multi-Gaussian beam model of Leaky Rayleigh waves generated by a focused immersion transducer

A non-paraxial multi-Gaussian beam (NMGB) model is proposed for Leaky Rayleigh Waves (LRWs) generated by a focused immersion transducer at oblique incidence. Using the NMGB model, the velocity fields are calculated and compared with the corresponding results obtained by the paraxial multi-Gaussian beam (MGB) model and the more exact Rayleigh–Sommerfeld integral (RSI) model. Numerical results show that the LRW beam behavior obtained using the NMGB model agrees well with that using the RSI model, but the NMGB model is much more efficient. Moreover the NMGB model overcomes the accuracy limitation of the MGB model. Good agreement between the NMGB model and experimental measurements for both on-axis and off-axis fields is obtained when an attenuation coefficient is introduced. In addition, this model can be used to measure the attenuation coefficient with consideration of the diffraction attenuation. It is observed that the attenuation coefficient of the LRW will increase when the acoustic impendence differences between the solid and fluid decrease. The NMGB model described in this article provides an efficient tool for calculating the velocity fields of the LRW and is therefore significant for practical applications of ultrasonic measurements.

Planar plasmonic lens based on circular nanohole in a gold film in visible light range

Plasmonic planar lens consisting of circular nanohole etched on the gold film have been investigated, the proposed plamonic lens was immersed in a high index medium to improve the focusing efficiency. The focal length and the full-width half-maximum beams width of the focal point were calculated in detail. The focal length calculated by FDTD simulation agrees well with Rayleigh–Sommerfeld integral. Moreover, the proposed lens structure is polarization-independent since circularly symmetric. The easy fabricate structure and excellent focusing performance of this plasmonic lens will open new avenues for optical storage and optical integration.

Acoustic Field Characterization of Medical Array Transducers Based on Unfocused Transmits and Single-Plane Hydrophone Measurements.

Medical ultrasonic arrays are typically characterized in controlled water baths using measurements by a hydrophone, which can be translated with a positioning stage. Characterization of 3D acoustic fields conventionally requires measurements at each spatial location, which is tedious and time-consuming, and may be prohibitive given limitations of experimental setup (e.g., the bath and stage) and measurement equipment (i.e., the hydrophone). Moreover, with the development of new ultrasound sequences and modalities, multiple measurements are often required to characterize each imaging mode to ensure performance and clinical safety. Acoustic holography allows efficient characterization of source transducer fields based on single plane measurements. In this work, we explore the applicability of a re-radiation method based on the Rayleigh–Sommerfeld integral to medical imaging array characterization. We show that source fields can be reconstructed at single crystal level at wavelength resolution, based on far-field measurements. This is herein presented for three practical application scenarios: for identifying faulty transducer elements; for characterizing acoustic safety parameters in focused ultrasound sequences from 2D planar measurements; and for estimating arbitrary focused fields based on calibration from an unfocused sound field and software beamforming. The results experimentally show that the acquired pressure fields closely match those estimated using our technique.

Estimation of methods for calculating the phase shift of wave fronts of own modes of a confocal resonator

Background: The redefinition of a unit of length - a meter - through a unit of time and a fundamental constant - the speed of light in vacuum - has opened up the fundamental possibility of a significant reduction in the uncertainty of its reproduction. Now progress in areas such as absolute ballistic gravimetry, control of large-sized aspherical optics, laser interferometry, and the production of electronic components in the semiconductor industry have made this feature extremely relevant. It is known that the measuring scale of laser interferometers used for precision distance measurement is non-linear, since the common-mode surfaces of any real radiation beam are located irregularly in space. To compensate for the effect of this irregularity on the measurement result, it is necessary to know the precision phase structure of real laser beams. Objectives of the work is comparing existing methods for studying the phase structure of optical radiation beams and estimating the distribution of the topological phase shift of a relatively uniform plane wave. Materials and methods:. The well-known theoretical methods for calculating the topological phase shift of in-phase surfaces of an optical beam are considered and compared - the Lommel-Debye method based on the Fresnel-Kirchhoff integral, the modified method based on the Rayleigh-Sommerfeld integral and the Gaussian beam method based on a parabolic equation. Results: Each method performed calculations of the accumulated phase lag of the focused radiation beam when moving the observation plane relative to the focal point. The distribution of the relative change in the distance between the in-phase surfaces in the range of displacements from λ to 106•λ was also calculated. The most adequate physical picture of the phenomenon was obtained by the Gaussian beam method based on a parabolic equation. Conclusion: The results will be used to reduce the systematic error of laser interferometers.

Target response in a focused, modulated sound field: Modeling and experiment

In a previous work, we had shown that low frequency flexural modes of circular plates and cylinders could be excited using modulated radiation pressure generated by focused ultrasound [ J. Acoust. Soc. Am. 139, 2053 (2016)]. We recently conducted experiments probing how the response of these targets varied as a function of position in the focused ultrasound field. Surprisingly, the response of the circular plate was found to change sign as it was moved away from the source. Two different models were developed to analyze the target response and understand this sign change. A purely geometric model based on ray optics and a semi-physical optics model that uses the incident intensity calculated for the focused source, using a Rayleigh-Sommerfeld integral. Both geometric and semi-physical optics models predict a sign change for the plate response at approximately the correct distance from the source. The sign change is due to a factor in a mode projection that more strongly weights points farther from the center of the plate. The physical optics model was also applied to cylindrical targets. [Work supported by ONR.]In a previous work, we had shown that low frequency flexural modes of circular plates and cylinders could be excited using modulated radiation pressure generated by focused ultrasound [ J. Acoust. Soc. Am. 139, 2053 (2016)]. We recently conducted experiments probing how the response of these targets varied as a function of position in the focused ultrasound field. Surprisingly, the response of the circular plate was found to change sign as it was moved away from the source. Two different models were developed to analyze the target response and understand this sign change. A purely geometric model based on ray optics and a semi-physical optics model that uses the incident intensity calculated for the focused source, using a Rayleigh-Sommerfeld integral. Both geometric and semi-physical optics models predict a sign change for the plate response at approximately the correct distance from the source. The sign change is due to a factor in a mode projection that more strongly weights points farther from the cen...

Measurement of surface position by focusing an interference pattern of multiple apertures with a confocal system.

We propose a method to measure the position of a flat mirror based on the identification of the best focused interferogram on the mirror. The interferogram is generated using a wavefront-splitting interferometer in a confocal configuration. The wavefront is sampled with an array of identical circular apertures. We analyze experimentally and theoretically the diffraction pattern for one, two, and three apertures when they are focused on the mirror. For the theoretical analysis we solve numerically the Rayleigh-Sommerfeld integral, and for the experimental analysis we measure the axial irradiance and the square power of the interferogram as a function of the axial displacement of the mirror. The maximum of these measurements corresponds to the best focused interferogram. The experimental and theoretical results are in good agreement. We found that the uncertainty associated with the axial position of the mirror is reduced significantly (about 161 times) when we use the interferogram generated by three apertures in comparison with the diffraction pattern for one aperture.

Calculation of Volumetric Sound Field of Pulsed Air-Coupled Ultrasound Transducers Based on Single-Plane Measurements.

Quantitative and reproducible air-coupled ultrasound (ACU) testing requires characterization of the volumetric pressure fields radiated by ACU probes. In this paper, a closed-form reradiation method combining the Rayleigh-Sommerfeld integral and time-reversal acoustics is proposed, which allows calculation of both near- field and far-field based on a single-plane measurement. The method was validated for both 3-D (circular, square) and 2-D (rectangular) planar transducers in the 50-230 kHz range. The pressure fields were scanned with a calibrated microphone. The measurement window was at least four times the size of the transducer area and the grid step size was one third of the wavelength. Best results were observed by acquiring the measurement plane at near-field distance. The method accurately reproduces pulsed ultrasound waveforms and pressure distributions (RMSE <2.5% in far field and <5.5% in near field), even at the transducer radiation surface. The effects of speed of sound drifts during the scan in the pressure were negligible (RMSE <0.3%). The reradiation method clearly outperforms conventional baffled piston models. Possible applications are transducer manufacture control (imperfections at radiation surface) and calibration (on-axis pressure, side lobes, and beamwidth) together with generation of accurate source functions for quantitative nondestructive evaluation inverse problems.

Simulation of hard x-ray focusing using an array of cylindrical micro-holes in a diamond film

The focusing of hard x-rays was numerically simulated using a beam propagation method implemented in the BeamPROP software package and a Rayleigh-Sommerfeld integral. It was shown that when focusing hard X-rays of wavelength 1.34 A with sequentially arranged circular cylindrical microlenses (shaped as 10-µm micro-holes) found in a polycrystalline diamond film, a nearly 10 times shorter focal length of the microlens array can be achieved, reducing from 14.4 cm to 15 mm, via increasing the number of microlenses from 1 to 20. Meanwhile, the x-ray focal spot size was found to decrease 7 times along the minor axis from 1.5 μm to 200 nm, while the maximum intensity in the focus experienced a 40-times increase.

Singular laser beams nanofocusing with dielectric nanostructures: theoretical investigation

We show that near-field nanoscale focusing is feasible using not only metallic but also dielectric sharp-edged structures. Based on vector Rayleigh–Sommerfeld integrals, we prove that the effect of an enhancement of the longitudinal electromagnetic field component occurs in the vicinity of the incident field phase discontinuities due to abrupt jumps in the optical element’s microrelief. Using a finite-element method, we simulate diffraction of the electromagnetic field on sharp edges of metallic and high-contrast dielectric structures. The resulting focal spot size is shown to be near directly proportional to the structure tip’s curvature radius. We propose a focusing arrangement that contains a radiation collector in the form of a conic axicon and a nanofocuser in the form of a nanosphere on the axicon’s apex. Using the finite-element method, we demonstrate that the focal spot size is approximately linearly proportional to the nanosphere radius. Thus, the proposed setup enables light to be confined to a thin focal spot with the size λ/373 when the silicon nanosphere radius is 2 nm.

Controllable design of super-oscillatory planar lenses for sub-diffraction-limit optical needles.

Sub-diffraction-limit optical needle can be created by a binary amplitude mask through tailoring the interference of diffraction beams. In this paper, a controllable design of super-oscillatory planar lenses to create sub-diffraction-limit optical needles with the tunable focal length and depth of focus (DOF) is presented. As a high-quality optical needle is influenced by various factors, we first propose a multi-objective and multi-constraint optimization model compromising all the main factors to achieve a needle with the prescribed characteristics. The optimizing procedure is self-designed using the Matlab programming language based on the genetic algorithm (GA) and fast Hankel transform algorithm. Numerical simulations show that the optical needles' properties can be controlled accurately. The optimized results are further validated by the theoretical calculation with the Rayleigh-Sommerfeld integral. The sub-diffraction-limit optical needles can be used in wide fields such as optical nanofabrication, super-resolution imaging, particle acceleration and high-density optical data storage.

Frequency considerations for deep ablation with high-intensity focused ultrasound: A simulation study.

The objective of this study was to explore frequency considerations for large-volume, deep thermal ablations with focused ultrasound. Though focal patterns, focal steering rate, and the size of focal clusters have all been explored in this context, frequency studies have generally explored shallower depths and hyperthermia applications. This study examines both treatment efficiency and near-field heating rate as functions of frequency and depth. Flat, 150 mm transducer arrays were simulated to operate at frequencies of 250, 500, 750, 1000, 1250, and 1500 kHz. Each array had λ2 interelement spacing yielding arrays of 2000-70 000 piston-shaped elements arranged in concentric rings. Depths of 50, 100, and 150 mm were explored, with attenuation (α) values of 2.5-10 (Np/m)/MHz. Ultrasound propagation was simulated with the Rayleigh-Sommerfeld integral over a volume of homogeneous simulated tissue. Absorbed power density was determined from the acoustic pressure which, in turn, was modeled with the Pennes bioheat transfer equation. Using this knowledge of temperature over time, thermal dose function of Sapareto and Dewey was used to model the resulting bioeffect of each simulated sonication. Initially, single foci at each depth, frequency, and α were examined with either fixed peak temperatures or fixed powers. Based on the size of the resulting, single foci lesions, larger compound sonications were designed with foci packed together in multiple layers and rings. For each depth, focal patterns were chosen to produce a similar total ablated volume for each frequency. These compound sonications were performed with a fixed peak temperature at each focus. The resulting energy efficiency (volume ablated per acoustic energy applied), near-field heating rate (temperature increase in the anterior third of the simulation space per unit volume ablated), and near- and far-field margins were assessed. Lesions of comparable volume were created with different frequencies at different depths. The results reflect the interconnected nature of frequency as it effects focal size (decreasing with frequency), peak pressure (generally increasing with frequency), and attenuation (also increasing with frequency). The ablation efficiency was the highest for α = 5 (Np/m)/MHz at a frequency of 750 kHz at each depth. For α = 10 (Np/m)/MHz, efficiency was the highest at 750 kHz for a depth of 50 mm, and 500 kHz at depths of 100 and 150 mm. At all sonication depths, near-field heating was minimized with lower frequencies of 250 and 500 kHz. Large-volume ablations are most efficient at frequencies of 500-750 kHz at depths of 100-150 mm. When one considers that near-field heat accumulation tends to be the rate limiting factor in large-volume ablations like uterine fibroid surgery, the results show that frequencies as low as 500 kHz are favored for their ability to reduce heating in the near-field.

A novel method for the design of diffractive optical elements based on the Rayleigh–Sommerfeld integral

The original design method for diffractive optical elements (DOEs) is limited to cases of small-angle diffraction due to the Fresnel or Fraunhofer diffraction integral. In this paper, we propose a new method based on the Rayleigh–Sommerfeld diffraction integral, which does not have this limit. In this method, the target intensity distribution is first modified via coordinate transformation and intensity adjustment. Then, the modified Gerchberg–Saxton algorithm is used to achieve the phase distribution of the DOE. To verify this method, simulations and experiments are performed. The results show that the original method is effective only when the full diffraction angle of the DOE is below 25°. Conversely, this method can achieve the target with both small and large diffraction angles.

Use of combined zone plates as imaging optics for hard x-rays

We have carried out modeling of an image formation in hard x-ray coherent radiation using a Rayleigh-Sommerfeld integral. Combined zone plates have been used to form the image. The resolution of the combined zone plate was numerically shown to increase up to 10 times (equaling 20 nm) when compared with a traditional Fresnel zone plate without any changes of the outermost zone width (205 nm). The zone plate consisted of 6 areas operating in 1, 3, 5, 7, 9 and 11 diffraction orders. An efficiency improvement of 42% has been numerically achieved in contrast to a traditional Fresnel zone plate used in the third diffraction order. For the combined zone plate, the efficiency of 3.7 % has been achieved.

Diffraction theory of high numerical aperture subwavelength circular binary phase Fresnel zone plate.

An analytical model of vector formalism is proposed to investigate the diffraction of high numerical aperture subwavelength circular binary phase Fresnel zone plate (FZP). In the proposed model, the scattering on the FZP's surface, reflection and refraction within groove zones are considered and diffraction fields are calculated using the vector Rayleigh-Sommerfeld integral. The numerical results obtained by the proposed phase thick FZP (TFZP) model show a good agreement with those obtained by the finite-difference time-domain (FDTD) method within the effective extent of etch depth. The optimal etch depths predicted by both methods are approximately equal. The analytical TFZP model is very useful for designing a phase and hybrid amplitude-phase FZP with high-NA and short focal length.