We investigated the Talbot effect in an anti-parity-time(PT)symmetric synthetic photonic lattice composed of two coupled fiber loops.We calculated the band structures and found that with an increase in the gain-loss p...We investigated the Talbot effect in an anti-parity-time(PT)symmetric synthetic photonic lattice composed of two coupled fiber loops.We calculated the band structures and found that with an increase in the gain-loss parameter,the band transitions from a real spectrum to a complex spectrum.We study the influence of phase in the Hermitian operator on the Talbot effect,and the Talbot effect disappears when the period of the input field is N>8.Further study shows that the variation of Talbot distance can also be modulated by non-Hermitian coefficients of gain and loss.This work may find significant applications in pulse repetition-rate multiplication,temporal invisibility,and tunable intensity amplifiers.展开更多
We introduce an idea of producing an optical lattice relied on the Talbot effect. Our alternative scheme is based on the interference of light behind a diffraction grating in the near-field regime. We demonstrate 1D a...We introduce an idea of producing an optical lattice relied on the Talbot effect. Our alternative scheme is based on the interference of light behind a diffraction grating in the near-field regime. We demonstrate 1D and 2D optical lattices with the simulations and experiments. This Talbot optical lattice can be broadly used from quantum simulations to quantum information. The Talbot effect is usually used in lensless optical systems, therefore it provides small aberrations.展开更多
The plasmonics Talbot effect in metallic layer with infinite periodic grooves is presented in this study. Numerical approach based on the finite element method is employed to verify the derived Talbot carpet on the no...The plasmonics Talbot effect in metallic layer with infinite periodic grooves is presented in this study. Numerical approach based on the finite element method is employed to verify the derived Talbot carpet on the non-illumination side. The groove depth is less than the metallic layer thickness; however, for specific conditions, surface plasmons polaritons(SPPs)can penetrate through grooves, propagate under the metallic layer, and form Talbot revivals. The geometrical parameters are specified via groove width, gap size, period, and wavelength, and their proper values are determined by introducing two opening ratio parameters. To quantitatively compare different Talbot carpets, we introduce new parameters such as R-square that characterizes the periodicity of Talbot images. The higher the R-square of a carpet, the more coincident with non-paraxial approximation the Talbot distance becomes. We believe that our results can help to understand the nature of SPPs and also contribute to exploring this phenomenon in Talbot-image-based applications, including imaging, optical systems, and measurements.展开更多
A fast and simple method to extract phase-contrast images from interferograms is proposed, and its effectiveness is demonstrated through simulation and experiment. For x-ray differential phase contrast imaging, a stro...A fast and simple method to extract phase-contrast images from interferograms is proposed, and its effectiveness is demonstrated through simulation and experiment. For x-ray differential phase contrast imaging, a strong attenuation signal acts as an overwhelming background intensity that obscures the weak phase signal so that no obvious phase-gradient information is detectable in the raw image. By subtracting one interferogram from another, chosen at particular intervals,the phase signal can be isolated and magnified.展开更多
The use of a phase mask with 536 nm uniform pitch allowed the fabrication of a fiber Bragg grating for use at a Bragg wavelength of 785 nm. Reflection and transmission features at 1552 nm, twice the Bragg wavelength, ...The use of a phase mask with 536 nm uniform pitch allowed the fabrication of a fiber Bragg grating for use at a Bragg wavelength of 785 nm. Reflection and transmission features at 1552 nm, twice the Bragg wavelength, associated with the phase mask periodicity were observed. However, when phase mask orders other than +1 were absent during fabrication the features at 1552 nm were not evident.展开更多
Dual-energy X-ray absorptiometry(DEXA) has been widely applied to measure the bone mineral density(BMD) and soft-tissue composition of the human body. However, the use of DEXA is greatly limited for lowZ materials...Dual-energy X-ray absorptiometry(DEXA) has been widely applied to measure the bone mineral density(BMD) and soft-tissue composition of the human body. However, the use of DEXA is greatly limited for lowZ materials such as soft tissues due to their weak absorption, while X-ray phase-contrast imaging(XPCI) shows significantly improved contrast in comparison with the conventional standard absorption-based X-ray imaging for soft tissues. In this paper, we propose a novel X-ray phase-contrast method to measure the area density of low-Z materials, including a single-energy method and a dual-energy method. The single-energy method is for the area density calculation of one low-Z material, while the dual-energy method aims to calculate the area densities of two low-Z materials simultaneously. Comparing the experimental and simulation results with the theoretical ones, the new method proves to have the potential to replace DEXA in area density measurement. The new method sets the prerequisites for a future precise and low-dose area density calculation method for low-Z materials.展开更多
This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main clas...This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main classes of numerical methods used to invert the Laplace transform. Because the numerical inversion of the Laplace transform is a perturbed problem, rounding errors which are generated in numerical approximations can adversely affect the accurate reconstruction of the inverse transform. This paper demonstrates that working in a multi-precision environment can substantially reduce these errors and the resulting perturbations exist in transforming the data from the s-space into the time domain and in so doing overcome the main drawback of numerically inverting the Laplace transform. Our main finding is that both the Talbot and the accelerated Gaver functionals perform considerably better in a multi-precision environment increasing the advantages of using Laplace transform methods over time-stepping procedures in solving diffusion and more generally parabolic partial differential equations.展开更多
Achromatic Talbot lithography(ATL)with high resolution has been demonstrated to be an excellent technique for large area periodic nano-fabrication.In this work,the uniformity of pattern distribution in ATL was studied...Achromatic Talbot lithography(ATL)with high resolution has been demonstrated to be an excellent technique for large area periodic nano-fabrication.In this work,the uniformity of pattern distribution in ATL was studied in detail.Two ATL transmission masks with^50%duty cycle in a square lattice were illuminated by a spatial coherent broadband extreme ultraviolet beam with a relative bandwidth of 2.38%.Nonuniform dot size distribution was observed by experiments and finite-difference time-domain simulations.The sum of the two kinds of diffraction patterns,with different lattice directions(45°rotated)and different intensity distributions,results in the final nonuniform pattern distribution.展开更多
The nonlinear Talbot effect is a near-field nonlinear diffraction phenomenon in which the self-imaging of periodic objects is formed by the second harmonics of the incident laser beam. We demonstrate the first, to the...The nonlinear Talbot effect is a near-field nonlinear diffraction phenomenon in which the self-imaging of periodic objects is formed by the second harmonics of the incident laser beam. We demonstrate the first, to the best of our knowledge, example of nonlinear Talbot self-healing, i.e., the capability of creating defect-free images from faulty nonlinear optical structures. In particular, we employ the tightly focused femtosecond infrared optical pulses to fabricate LiNbO_(3) nonlinear photonic crystals and show that the defects in the form of the missing points of two-dimensional square and hexagonal periodic structures are restored in the second harmonic images at the first nonlinear Talbot plane. The observed nonlinear Talbot self-healing opens up new possibilities for defect-tolerant optical lithography and printing.展开更多
A type of scalable self-imaging capable of variable magnification or minification of periodic objects is demonstrated in the focal plane of a lens illuminated by a point source.The theory and the experimental results ...A type of scalable self-imaging capable of variable magnification or minification of periodic objects is demonstrated in the focal plane of a lens illuminated by a point source.The theory and the experimental results show that the self-imaging phenomenon can also be realized in the focal plane of a lens regardless of whether the distances satisfy the lens formula or not.The particular property of this scalable self-imaging effect is that the images in the focal plane can be controlled with different scaling factors only when the distances between the point source and the periodic object satisfy a certain condition.This discovery should open a new field of diffraction imaging and new application opportunities in precision measurement.展开更多
We investigate the Airy–Talbot effect of an Airy pulse train in time-dependent linear potentials.The parabolic trajectory of self-imaging depends on both the dispersion sign and the linear potential gradient.By impos...We investigate the Airy–Talbot effect of an Airy pulse train in time-dependent linear potentials.The parabolic trajectory of self-imaging depends on both the dispersion sign and the linear potential gradient.By imposing linear phase modulations on the pulse train,the Airy–Talbot effects accompanied with positive and negative refractions are realized.For an input composed of stationary Airy pulses,the self-imaging follows straight lines,and the Airy–Talbot distance can be engineered by varying the linear potential gradient.The effect is also achieved in symmetric linear potentials.The study provides opportunities to control the self-imaging of aperiodic optical fields in time dimension.展开更多
Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to sol...Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.展开更多
In this study,we report on the fractional Talbot ffect of nonpar axial self-accelerating beams in a multilevel electromagnetically induced transparency(EIT)atomic configuration,which,to the best of our knowledge,is th...In this study,we report on the fractional Talbot ffect of nonpar axial self-accelerating beams in a multilevel electromagnetically induced transparency(EIT)atomic configuration,which,to the best of our knowledge,is the first study on this subject.The Tallbot ffect originates from superposed eigenmodes of the Helmholtz equation and forms in the EIT window in the presence of both linear and cubic susceptibilities.The Talbot ffect can be realized by appropriately selecting the cofficients of the beam components.Our results indicate that the larger the radial difference between beam components,the stronger the interference between them,the smaller the Tallbot angle is.The results of this study can be useful when studying optical imaging.optical measurements,and optical computing.展开更多
We report the experimental observation of two-dimensional Talbot effect when a resonance plane wave interacts with a two-dimensional atomic density grating generated by standing wave manipulation of ultracold Bose gas...We report the experimental observation of two-dimensional Talbot effect when a resonance plane wave interacts with a two-dimensional atomic density grating generated by standing wave manipulation of ultracold Bose gases. Clear self-images of the grating and sub-images with reversed phase or fractal patterns are observed. By calculating the autocorrelation functions of the images, the behavior of periodic Talbot images is studied. The Talbot effect with two-dimensional atomic density grating expands the applications of the Talbot effect in a wide variety of research fields.展开更多
We demonstrate a conformally mapped multifunctional acoustic metamaterial Mikaelian lens.Mikaelian lens is a gradient medium with a hyperbolic secant refractive index profile that can realize functions like beam self-...We demonstrate a conformally mapped multifunctional acoustic metamaterial Mikaelian lens.Mikaelian lens is a gradient medium with a hyperbolic secant refractive index profile that can realize functions like beam self-focusing.Unlike the conventional design approaches,with a conformal transformation method,only isotropic material parameters with gradient refractive index profiles are required for the construction of such lens.To realize desired gradient index distribution,we carefully design a new type of crosschannel-shaped acoustic metamaterial,whose refractive index can be effectively modulated by simply changing the slit opening size.The distinct capabilities of the metamaterial Mikaelian lens in manipulating acoustic waves are experimentally verified with the fabricated sample.Simultaneous sound guiding and Talbot effects,which normally require respective geometrical and wave acoustic approximations,are observed in simulations and experiments.Furthermore,those effects of shaping acoustic wave propagations were validated within a relatively wide frequency range.Our study reveals how the conformal transformation method can help to bridge the ray acoustics with wave acoustics.It offers opportunities to the development of novel multifunctional acoustic devices for various applications,such as sound and particle manipulations.展开更多
基金supported by the National Key Research and Development Program of China(Nos.2022YFA1404800 and 2019YFA0705000)the National Natural Science Foundation of China(Nos.12104272,12274270,91950104,12192254,92250304,and 11974218)the Local Science and Technology Development Project of the Central Government(No.YDZX20203700001766)。
文摘We investigated the Talbot effect in an anti-parity-time(PT)symmetric synthetic photonic lattice composed of two coupled fiber loops.We calculated the band structures and found that with an increase in the gain-loss parameter,the band transitions from a real spectrum to a complex spectrum.We study the influence of phase in the Hermitian operator on the Talbot effect,and the Talbot effect disappears when the period of the input field is N>8.Further study shows that the variation of Talbot distance can also be modulated by non-Hermitian coefficients of gain and loss.This work may find significant applications in pulse repetition-rate multiplication,temporal invisibility,and tunable intensity amplifiers.
基金the office of the higher education commissionthe Thailand research fund(TRF)+3 种基金Faculty of Science,Burapha university under contract number MRG5380264the National Electronics and Computer Technology CenterNational Science and Technology Development Agency and Industry/University Cooperative Research Center(I/UCRC)in HDD Componentthe Faculty of Engineering,Khon Kaen University
文摘We introduce an idea of producing an optical lattice relied on the Talbot effect. Our alternative scheme is based on the interference of light behind a diffraction grating in the near-field regime. We demonstrate 1D and 2D optical lattices with the simulations and experiments. This Talbot optical lattice can be broadly used from quantum simulations to quantum information. The Talbot effect is usually used in lensless optical systems, therefore it provides small aberrations.
基金Project supported by the 111 Project,China(Grant No.D17021)the Changjiang Scholars and Innovative Research Team in University,China(Grant No.PCSIRT,IRT 16R07)
文摘The plasmonics Talbot effect in metallic layer with infinite periodic grooves is presented in this study. Numerical approach based on the finite element method is employed to verify the derived Talbot carpet on the non-illumination side. The groove depth is less than the metallic layer thickness; however, for specific conditions, surface plasmons polaritons(SPPs)can penetrate through grooves, propagate under the metallic layer, and form Talbot revivals. The geometrical parameters are specified via groove width, gap size, period, and wavelength, and their proper values are determined by introducing two opening ratio parameters. To quantitatively compare different Talbot carpets, we introduce new parameters such as R-square that characterizes the periodicity of Talbot images. The higher the R-square of a carpet, the more coincident with non-paraxial approximation the Talbot distance becomes. We believe that our results can help to understand the nature of SPPs and also contribute to exploring this phenomenon in Talbot-image-based applications, including imaging, optical systems, and measurements.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61101175,61571305,and 61227802)
文摘A fast and simple method to extract phase-contrast images from interferograms is proposed, and its effectiveness is demonstrated through simulation and experiment. For x-ray differential phase contrast imaging, a strong attenuation signal acts as an overwhelming background intensity that obscures the weak phase signal so that no obvious phase-gradient information is detectable in the raw image. By subtracting one interferogram from another, chosen at particular intervals,the phase signal can be isolated and magnified.
基金supported by the Australian Research Council under Grant No. DP0773999.
文摘The use of a phase mask with 536 nm uniform pitch allowed the fabrication of a fiber Bragg grating for use at a Bragg wavelength of 785 nm. Reflection and transmission features at 1552 nm, twice the Bragg wavelength, associated with the phase mask periodicity were observed. However, when phase mask orders other than +1 were absent during fabrication the features at 1552 nm were not evident.
基金Supported by Major State Basic Research Development Program(2012CB825800)Science Fund for Creative Research Groups(11321503)National Natural Science Foundation of China(11179004,10979055,11205189,11205157)
文摘Dual-energy X-ray absorptiometry(DEXA) has been widely applied to measure the bone mineral density(BMD) and soft-tissue composition of the human body. However, the use of DEXA is greatly limited for lowZ materials such as soft tissues due to their weak absorption, while X-ray phase-contrast imaging(XPCI) shows significantly improved contrast in comparison with the conventional standard absorption-based X-ray imaging for soft tissues. In this paper, we propose a novel X-ray phase-contrast method to measure the area density of low-Z materials, including a single-energy method and a dual-energy method. The single-energy method is for the area density calculation of one low-Z material, while the dual-energy method aims to calculate the area densities of two low-Z materials simultaneously. Comparing the experimental and simulation results with the theoretical ones, the new method proves to have the potential to replace DEXA in area density measurement. The new method sets the prerequisites for a future precise and low-dose area density calculation method for low-Z materials.
文摘This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main classes of numerical methods used to invert the Laplace transform. Because the numerical inversion of the Laplace transform is a perturbed problem, rounding errors which are generated in numerical approximations can adversely affect the accurate reconstruction of the inverse transform. This paper demonstrates that working in a multi-precision environment can substantially reduce these errors and the resulting perturbations exist in transforming the data from the s-space into the time domain and in so doing overcome the main drawback of numerically inverting the Laplace transform. Our main finding is that both the Talbot and the accelerated Gaver functionals perform considerably better in a multi-precision environment increasing the advantages of using Laplace transform methods over time-stepping procedures in solving diffusion and more generally parabolic partial differential equations.
基金supported by the National Key R&D Program of China(No.2017YFA0206001)the National Natural Science Foundation of China(Nos.11505275,51705369,11275255,11475251,and 11875314)the Shanghai Science and Technology Commission Research Projects(No.17JC1400802)
文摘Achromatic Talbot lithography(ATL)with high resolution has been demonstrated to be an excellent technique for large area periodic nano-fabrication.In this work,the uniformity of pattern distribution in ATL was studied in detail.Two ATL transmission masks with^50%duty cycle in a square lattice were illuminated by a spatial coherent broadband extreme ultraviolet beam with a relative bandwidth of 2.38%.Nonuniform dot size distribution was observed by experiments and finite-difference time-domain simulations.The sum of the two kinds of diffraction patterns,with different lattice directions(45°rotated)and different intensity distributions,results in the final nonuniform pattern distribution.
基金supported by the National Natural Science Foundation of China(Nos.61905124,11974196,and 61905125)the Australian Research Council(No.DP19010774)+2 种基金the Qatar National Research Fund(No.NPRP 12S-0205190047)the Yongjiang Scholar Foundation of Ningbothe K.C.Wong Magna Fund of Ningbo University。
文摘The nonlinear Talbot effect is a near-field nonlinear diffraction phenomenon in which the self-imaging of periodic objects is formed by the second harmonics of the incident laser beam. We demonstrate the first, to the best of our knowledge, example of nonlinear Talbot self-healing, i.e., the capability of creating defect-free images from faulty nonlinear optical structures. In particular, we employ the tightly focused femtosecond infrared optical pulses to fabricate LiNbO_(3) nonlinear photonic crystals and show that the defects in the form of the missing points of two-dimensional square and hexagonal periodic structures are restored in the second harmonic images at the first nonlinear Talbot plane. The observed nonlinear Talbot self-healing opens up new possibilities for defect-tolerant optical lithography and printing.
基金supported by the National Natural Science Foundation of China(NSFC)(Nos.61675093 and61705096)the Natural Science Foundation of Shandong Province(No.ZR2017MA035)+1 种基金the Shandong Province Higher Educational Science and Technology Program(No.J17KA176)the Taishan Scholars Project of Shandong Province(No.ts2015110055)。
文摘A type of scalable self-imaging capable of variable magnification or minification of periodic objects is demonstrated in the focal plane of a lens illuminated by a point source.The theory and the experimental results show that the self-imaging phenomenon can also be realized in the focal plane of a lens regardless of whether the distances satisfy the lens formula or not.The particular property of this scalable self-imaging effect is that the images in the focal plane can be controlled with different scaling factors only when the distances between the point source and the periodic object satisfy a certain condition.This discovery should open a new field of diffraction imaging and new application opportunities in precision measurement.
基金supported by the National Natural Science Foundation of China(Nos.11674117 and 11974124)。
文摘We investigate the Airy–Talbot effect of an Airy pulse train in time-dependent linear potentials.The parabolic trajectory of self-imaging depends on both the dispersion sign and the linear potential gradient.By imposing linear phase modulations on the pulse train,the Airy–Talbot effects accompanied with positive and negative refractions are realized.For an input composed of stationary Airy pulses,the self-imaging follows straight lines,and the Airy–Talbot distance can be engineered by varying the linear potential gradient.The effect is also achieved in symmetric linear potentials.The study provides opportunities to control the self-imaging of aperiodic optical fields in time dimension.
文摘Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.
基金the National Natural Science Foundation of China(Grant Nos.61805068,11747046,and 61875053)China Postdoctoral Science Foundation(CPSF)(Grant No.2017M620300)+1 种基金the Natural Science Fund of Shaanxi Province(Grant No.2018JQ6002)the Science and Technology Department of Henan Province(Grant No.202102210111).
文摘In this study,we report on the fractional Talbot ffect of nonpar axial self-accelerating beams in a multilevel electromagnetically induced transparency(EIT)atomic configuration,which,to the best of our knowledge,is the first study on this subject.The Tallbot ffect originates from superposed eigenmodes of the Helmholtz equation and forms in the EIT window in the presence of both linear and cubic susceptibilities.The Talbot ffect can be realized by appropriately selecting the cofficients of the beam components.Our results indicate that the larger the radial difference between beam components,the stronger the interference between them,the smaller the Tallbot angle is.The results of this study can be useful when studying optical imaging.optical measurements,and optical computing.
基金Supported by the State Key Development Program for Basic Research of China under Grant No 2016YFA0301501the National Natural Science Foundation of China under Grant Nos 11504328,61475007,11334001 and 91336103
文摘We report the experimental observation of two-dimensional Talbot effect when a resonance plane wave interacts with a two-dimensional atomic density grating generated by standing wave manipulation of ultracold Bose gases. Clear self-images of the grating and sub-images with reversed phase or fractal patterns are observed. By calculating the autocorrelation functions of the images, the behavior of periodic Talbot images is studied. The Talbot effect with two-dimensional atomic density grating expands the applications of the Talbot effect in a wide variety of research fields.
基金The work was supported by the National Natural Science Foundation of China(Grant No.11774297)the university research grant G-YBXM from the Hong Kong Polytechnic UniversityY.L.acknowledges the support from the National Natural Science Foundation of China(Grant No.11704284).
文摘We demonstrate a conformally mapped multifunctional acoustic metamaterial Mikaelian lens.Mikaelian lens is a gradient medium with a hyperbolic secant refractive index profile that can realize functions like beam self-focusing.Unlike the conventional design approaches,with a conformal transformation method,only isotropic material parameters with gradient refractive index profiles are required for the construction of such lens.To realize desired gradient index distribution,we carefully design a new type of crosschannel-shaped acoustic metamaterial,whose refractive index can be effectively modulated by simply changing the slit opening size.The distinct capabilities of the metamaterial Mikaelian lens in manipulating acoustic waves are experimentally verified with the fabricated sample.Simultaneous sound guiding and Talbot effects,which normally require respective geometrical and wave acoustic approximations,are observed in simulations and experiments.Furthermore,those effects of shaping acoustic wave propagations were validated within a relatively wide frequency range.Our study reveals how the conformal transformation method can help to bridge the ray acoustics with wave acoustics.It offers opportunities to the development of novel multifunctional acoustic devices for various applications,such as sound and particle manipulations.