First-order Born Research Articles

Finite-frequency Sensitivity Kernels in Spherical Geometry for Time–Distance Helioseismology

Abstract The inference of internal properties of the Sun from surface measurements of wave travel times is the goal of time–distance helioseismology. A critical step toward the accurate interpretation of shifts in travel time is the computation of sensitivity functions linking seismic measurements to internal structure. Here we calculate finite-frequency sensitivity kernels in spherical geometry for two-point measurements of travel time. We numerically build Green’s function by solving for it at each frequency and each degree of spherical harmonic and summing over all these pieces. These computations are performed in parallel (“embarrassingly”), thereby achieving significant speedup in wall-clock time. Kernels are calculated by invoking the first-order Born approximation connecting deviations in the wave field to perturbations in the operator. Validated flow kernels are shown to produce travel times within 0.47% of the true value for uniform flows up to . We find that travel time can be obtained with errors of 1 ms or less for flows having magnitudes similar to meridional circulation. Alongside flows, we also compute and validate a sensitivity kernel for perturbations in sound speed. These accurate sensitivity kernels might improve the current inferences of subsurface flows significantly.

Deterministic compressive sampling for high-quality image reconstruction of ultrasound tomography

BackgroundA well-known diagnostic imaging modality, termed ultrasound tomography, was quickly developed for the detection of very small tumors whose sizes are smaller than the wavelength of the incident pressure wave without ionizing radiation, compared to the current gold-standard X-ray mammography. Based on inverse scattering technique, ultrasound tomography uses some material properties such as sound contrast or attenuation to detect small targets. The Distorted Born Iterative Method (DBIM) based on first-order Born approximation is an efficient diffraction tomography approach. One of the challenges for a high quality reconstruction is to obtain many measurements from the number of transmitters and receivers. Given the fact that biomedical images are often sparse, the compressed sensing (CS) technique could be therefore effectively applied to ultrasound tomography by reducing the number of transmitters and receivers, while maintaining a high quality of image reconstruction.MethodsThere are currently several work on CS that dispose randomly distributed locations for the measurement system. However, this random configuration is relatively difficult to implement in practice. Instead of it, we should adopt a methodology that helps determine the locations of measurement devices in a deterministic way. For this, we develop the novel DCS-DBIM algorithm that is highly applicable in practice. Inspired of the exploitation of the deterministic compressed sensing technique (DCS) introduced by the authors few years ago with the image reconstruction process implemented using l1 regularization.ResultsSimulation results of the proposed approach have demonstrated its high performance, with the normalized error approximately 90% reduced, compared to the conventional approach, this new approach can save half of number of measurements and only uses two iterations. Universal image quality index is also evaluated in order to prove the efficiency of the proposed approach.ConclusionsNumerical simulation results indicate that CS and DCS techniques offer equivalent image reconstruction quality with simpler practical implementation. It would be a very promising approach in practical applications of modern biomedical imaging technology.

The correlation of intensity fluctuation of light waves on scattering from a particulate medium

The higher-order coherent effect of light waves on scattering from a collection of particles has been discussed within the accuracy of the first-order Born approximation. It is shown that the correlation of intensity fluctuation (CIF) of the scattered field is closely related to the characteristic of the collection of particles. An example of light waves on scattering from a collection of particles with Gaussian-centered distribution has been discussed, and the influence of the distribution characteristic on the normalized correlation coefficient of CIF of the scattered field has been studied. This result may provide a potential way to determine the distribution information of a collection of particles by measuring the intensity data of the scattered field.

Electromagnetic scattering of a polarized plane wave from an ellipsoidal particle in the near field

Within the validity of the first-order Born approximation, we study the near-zone evanescent wave properties for a polarized plane wave scattering upon an ellipsoidal particle. Integral expressions are obtained for the three-dimensional electromagnetic field of the near-zone scattered evanescent wave, and the dependences of the scattered intensity distributions on the degree of polarization of the incident wave and the scattering potential profile of the particle are presented. The scattered intensity from the particle can exhibit a focused pattern concentrated around the central scattering region, but the scattered intensity generated from a circularly polarized wave shows a smooth distribution for different scattering angles. Moreover, the scattered intensity also enhances when either the summation index or the effective radius of the particle increases. Our results can be utilized to generate near-field focused scattered patterns that can be tuned flexibly by controlling the degree of the polarization of the plane wave and the scattering potential parameters of the ellipsoidal particle.

Spectral shifts and spectral switches produced by scattering from a random hollow scatterer with adjustable shell thickness

Within the accuracy of the first-order Born approximation, the spectral shifts and spectral switches produced by the scattering of an electromagnetic light wave from a random hollow scatterer with adjustable shell thickness have been investigated. The effects of the properties of the scatterer and the incident light wave on the far-zone scattered spectrum have been discussed in detail by using numerical examples. It is shown that, as the scattering angle increases, the scattered spectrum will split into two peaks, and subsequently the two peaks will make a rapid transition as a spectral switch. The position at which the spectral switch occurs is affected by the shell thickness, the outer and inner correlation lengths of the scatterer as well as the polarization of the incident light. Besides, the polarization of the incident light also has an impact on the spectral width of the scattered field.

Reciprocal relations for light from Young’s pinholes scattering upon a quasi-homogeneous medium

Within the validity of the first-order Born approximation, a group of reciprocal relations are derived for light from Young’s pinholes scattering upon a spatially quasi-homogeneous (QH) medium. It is shown that the spectral degree of coherence (SDOC) of the scattered field is proportional to the Fourier transform of the strength of the scattering potential, and the spectral density of the scattered field is proportional to the Fourier transform of the normalized correlation coefficient of the scattering potential. By considering a particular case where the correlation function of the QH medium obeys the Gaussian distribution, we obtain the far-zone spectral density and SDOC of the scattered field which satisfy the derived reciprocal relations. Compared with previous reciprocal relations for scattering, our relations additionally introduce Young’s pinhole parameter that characterizes the geometry of Young’s pinholes.

Reciprocal Relations for Third-Order Correlation Between Intensity Fluctuations of Light Scattered From a Quasi-Homogeneous Medium

Within the validity of the first-order Born approximation, we introduce the third-order correlation between intensity fluctuation (CIF) of light scattered from a quasi-homogeneous medium. The third-order CIF is defined as the degree of intensity correlations specified at three position vectors in scattered field. Expressions are derived for the normalized CIF and variance in intensity fluctuations (VIF) of the scattered field. Two reciprocal relations are generalized for the third-order CIF of the scattered field: 1) The normalized CIF is proportional to the real portion of the Fourier transform products of the strength of the scattering potential; 2) the third-order VIF of the scattered field is proportional to the cube of the Fourier transform of the normalized correlation coefficient of the scattering potential.

Spectral shift of a light wave on scattering from an ellipsoidal particle with arbitrary orientation.

Within the accuracy of the first-order Born approximation, a general expression for the far-zone spectrum of a light wave on scattering from an arbitrarily orientated ellipsoidal particle is derived. We show that the spectrum of the scattered field, in general, changes with the scattering azimuthal angle, displaying rotational nonsymmetry. The influence of the orientation of the particle on the spectrum of the scattered field is discussed, and the relationship between the orientation of scattering particle and the distribution of the relative spectral shift of the scattered field is investigated.

Theoretical model for optical oximetry at the capillary level: exploring hemoglobin oxygen saturation through backscattering of single red blood cells.

Oxygen saturation ( sO 2 ) of red blood cells (RBCs) in capillaries can indirectly assess local tissue oxygenation and metabolic function. For example, the altered retinal oxygenation in diabetic retinopathy and local hypoxia during tumor development in cancer are reflected by abnormal sO 2 of local capillary networks. However, it is far from clear whether accurate label-free optical oximetry (i.e., measuring hemoglobin sO 2 ) is feasible from dispersed RBCs at the single capillary level. The sO 2 -dependent hemoglobin absorption contrast present in optical scattering signal is complicated by geometry-dependent scattering from RBCs. We present a numerical study of backscattering spectra from single RBCs based on the first-order Born approximation, considering practical factors: RBC orientations, size variation, and deformations. We show that the oscillatory spectral behavior of RBC geometries is smoothed by variations in cell size and orientation, resulting in clear sO 2 -dependent spectral contrast. In addition, this spectral contrast persists with different mean cellular hemoglobin content and different deformations of RBCs. This study shows for the first time the feasibility of, and provides a theoretical model for, label-free optical oximetry at the single capillary level using backscattering-based imaging modalities, challenging the popular view that such measurements are impossible at the single capillary level.

Scattering Properties of Near-Field Evanescent Wave From Medium With Ellipsoidal and Cylindrical Distributions

The evolution properties of the near-field evanescent wave scattered from two particles with the ellipsoidal and cylindrical scattering potentials are studied, respectively. Within the accuracy of the first-order Born approximation, the integrations are derived for the scattered fields from these two kinds of particles. It is found that the resultant scattered intensities are closely related to the scattering potential types. Compared with the ellipsoidal particle, the cylindrical one gives rise to less fluctuations of the scattered intensities. It also is shown that the scattered intensity strongly depends on the z-component of the effective radii and summation indices of both particles. Results indicate that the scattered intensity distribution can be tuned by controlling the Effective Radius of Scattering potentials (ERSPs) and summation indices of the medium. The obtained results are useful for determining the structural parameters of tiny particles and molecules by converting the scattering formulas.

Self-energy of cold atoms in a long-range disordered optical potential

We study the effect of correlation on the expansion of a Bose---Einstein condensate (BEC) released from a harmonic trap. We go beyond the first-order Born approximation (FBA) to use the self-consistent Born approximation (SCBA) to calculate the self-energy $$\Sigma (\varepsilon , k)$$Σ(?,k) of an atom in a speckle potential with constant amplitude of disorder. For very cold atoms ($$k= 0$$k=0), low chemical potential $$\mu \ll \varepsilon _{\xi }$$μ???, and $$U/ \varepsilon ^{2}_{\xi } = 1$$U/??2=1, the self-energy spectrum is wide when calculated in the SCBA compared with the FBA. The SCBA locates the band edge at somewhat higher energy, giving rise to many localized atoms. We focus mainly on the energy distribution at $$k = 0$$k=0 as expressed by the spectral function. Our numerical results show that the spectral function of the expanding atoms as calculated in the FBA at $$\varepsilon = 0$$?=0 has low weight and extends up to $$\varepsilon = 4 \varepsilon _{\xi }$$?=4??. When using the SCBA, the behavior of the energy distribution for low chemical potential is different, showing a continuum, while a large weight corresponding to $$\varepsilon $$? shifts the spectrum of negative energy values to $$\frac{\varepsilon }{\varepsilon _\xi } = -0.78$$???=-0.78 and the energy distribution is near unity. We show that the form of the density of states locates the mobility near $$\varepsilon = - \varepsilon _\xi $$?=-??. At $$\varepsilon = 0$$?=0, we obtain 35 % of delocalized atoms. We conclude that correlation disorder seems to help localize atoms with energy below zero. In particular, we show that the negative energy values observed in the energy spectrum imply that the probability of finding an atom with energy $$\varepsilon _\xi $$?? around $$\varepsilon $$? is conserved.

Weak Scattering of Young's Diffractive Light Wave From a Spatially Deterministic Medium

We study the modeling of the scattering of Young's diffractive light wave from a 3-D medium of which the refractive index is a deterministic function of position. Within the validity of the first-order Born approximation, the spectral density of the scattered field is derived as the analytic form. It is shown that the scattering angle and effective radius of the scattering potential account for the substantial effects on the spectral density profiles of the scattered field. It is also revealed that Young's pinhole parameter can affect the spectral density distribution of the scattered field. Our findings may be significant for the determination of structural parameters of an unknown scatterer.

Spectral shifts generated by scattering of Gaussian Schell-model arrays beam from a deterministic medium

According to the theory of first-order Born approximation, analytical expressions for Gaussian Schell-model arrays (GSMA) beam scattered on a deterministic medium in the far-zone are derived. In terms of the analytical formula obtained, shifts of GSMA beam's scattered spectrum are numerically investigated. Results show that the scattering directions sx and sy, effective radius σ of the scattering medium, the initial beam transverse width σ0, correlation widths δx and δy of the source, and line width Γ0 of the incident spectrum closely influence the distributions of normalized scattered spectrum in the far-zone. These features of GSMA beam scattered spectrum could be used to obtain information about the structure of a deterministic medium.

Correlation between intensity fluctuations of light generated by scattering of Young’s diffractive electromagnetic waves by a quasi-homogeneous, anisotropic medium

Based on the first-order Born approximation, formulas are derived for the correlation between intensity fluctuations (CIF) of light generated by a Young’s diffractive electromagnetic wave scattered by a spatially quasi-homogeneous (QH), anisotropic medium. It is shown that the CIF of the scattered field can be written as the summation of the Fourier transforms of the strengths and normalized correlation coefficients (NCCs) of the scattering potentials. The differences between our results and those obtained in the previous literature are discussed. Our results might be important in investigating the high-order intensity correlation of an electromagnetic wave scattered from a 3D anisotropic object.

Correlation between intensity fluctuations of electromagnetic waves scattered from a spatially quasi-homogeneous, anisotropic medium.

Within the validity of the first-order Born approximation, expressions are derived for the correlation between intensity fluctuations (CIF) of an electromagnetic plane wave scattered from a spatially quasi-homogeneous (QH), anisotropic medium. Upon establishing the correlation matrix of the scattering potential of the medium, we show that the CIF is the summation of Fourier transforms of the strengths and normalized correlation coefficients (NCCs) of the scattering potential matrix. Numerical results reveal that the CIF is susceptible to the effective width and correlation length of the medium, and degree of polarization of the incident electromagnetic wave. Our study not only extends the current knowledge of the CIF of a scattered field but also provides an important reference to the study of high-order intensity correlations of light scattered from a spatially anisotropic medium.

Creating von Laue patterns in crystal scattering with partially coherent sources

When spatially coherent radiation is diffracted by a crystalline object, the field is scattered in specific directions, giving rise to so-called von Laue patterns. We examine the role of spatial coherence in this process. Using the first-order Born approximation, a general analytic expression for the far-zone spectral density of the scattered field is obtained. This equation relates the coherence properties of the source to the angular distribution of the scattered intensity. We apply this result to two types of sources. Quasihomogeneous Gaussian Schell-model sources are found to produce von Laue spots whose size is governed by the effective source width. Delta-correlated ring sources produce von Laue rings and ellipses instead of point-like spots. In forward scattering, polychromatic ellipses are created, whereas in backscattering striking, overlapping ring patterns are formed. We show that both the directionality and the wavelength-selectivity of the scattering process can be controlled by the state of coherence of the illuminating source.

Spectral changes of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry scattered on a deterministic medium.

Based on the theory of first-order Born approximation, analytical expressions for rectangular cosine-Gaussian Schell-model (RCGSM) beams scattered on a deterministic medium in the far zone are derived. In terms of the analytical formula obtained, the changes of a RCGSM beam's scattered spectrum are numerically investigated. Results show that several parameters (including the scattering directions sx and sy, effective radius σ of the scattering medium, the initial beam's correlation widths δx and δy, and line width Γ0 of the incident spectrum) closely influence the distributions of the normalized scattered spectrum in the far zone. These features of a RCGSM beam scattered spectrum can be used to obtain information about the structure of a deterministic medium.

Correlation between intensity fluctuations induced by scattering of a partially coherent, electromagnetic wave from a quasi-homogeneous medium

Based on the first-order Born approximation, the correlation between intensity fluctuations is derived for a partially coherent, electromagnetic plane wave scattering from a spatially quasi-homogeneous medium. Young׳s pinholes are utilized to control the degree of coherence of the incident field. For the electromagnetic scattering case, it is shown that the CIF of the scattered field strongly depends on the degree of polarization of the incident wave, Young׳s pinhole parameter, effective radius and correlation length of the medium. The influences of these parameters on the CIF distributions are revealed by numerical calculations.

Scattering of an electromagnetic light wave from a quasi-homogeneous medium with semisoft boundary

Based on the first-order Born approximation, the scattering of an electromagnetic plane wave from a relatively more realistic random medium, a quasi-homogeneous medium with semisoft boundary, has been investigated. The analytic expressions for the spectral density, the spectral degree of coherence and the spectral degree of polarization have been derived, and the effects of the characteristics of the medium and the polarization of the incident light wave on the far-zone scattered field are determined. The numerical simulations indicate that, with the increasing of the edge softness M of the medium, the spectral density presents a pattern with interference fringes, and the number, position and width of interference fringes can be modified by the parameter. It is also found that there is an obvious value saltation in the coherence profile. Besides, unlike the intensity and the coherence are significantly affected by the properties of the medium, the polarization of the scattered field is irrelevant to them due to the quasi-homogeneity and isotropy of the medium, and it is only connected with the polarization of the incident light.

The intensity properties of a Multi-Gaussian Schell-model pulse scattering from a sphere with semisoft boundaries

The scattering of a Multi-Gaussian Schell-model pulse from a sphere with semisoft boundaries within the accuracy of the first-order Born approximation is investigated. The analytic expression for the average intensity of the scattered pulse in the far zone is derived. The numerical calculations are performed in order to show the dependence of the average intensity of the scattered field on the incident pulse duration, the temporal coherence length of the incident pulse, the upper summation index and the effective radius of the medium. The effect of the boundary’s softness on the scattered intensity distribution is emphasized. It is shown that, comparing with that of scattering from a Gaussian sphere, the average intensity distribution of the pulse scattering from a sphere with semisoft boundaries has two or three peaks under certain circumstances.