Dynamic Ray Tracing Method Research Articles

A Novel Universal Applicability Design Method for Low Cross-Polarization Terahertz Tri-Reflector CATR

In this paper, a novel design approach with universal applicability is proposed for realizing a low cross-polarized terahertz tri-reflector compact antenna test range (CATR). This approach not only enables the radiation direction of the feed relative to the main reflector to be designed arbitrarily but also enables the designed CATR systems corresponding to different feed rotation angles while still maintaining a low cross-polarization. By using the beam mode analysis method and cross-polarization elimination conditions, the geometric configurations of the tri-reflector CATR can be designed for feeds in any different rotation angles, and then according to the kinematic and dynamic ray-tracing method with frequency independence in geometric optics, shaped subreflectors can also be synthesized. Through the above design procedure, four tri-reflector CATRs corresponding to four different feed rotation angles have been achieved, respectively. Numerical simulation results show that the cross-polarizations of four tri-reflector CATRs are all less than –38 dB and the peak-to-peak amplitude (phase) ripples of the quiet zone (QZ) are all within 1 dB (10°) in the frequency range of 100–500 GHz. This demonstrates the effectiveness and universal applicability of the proposed design method in realizing the design of low cross-polarization and good QZ performance for tri-reflector CATRs with different feed rotation angles.

Implementation of a dynamic ray tracing method in CIVA HealthCare HIFU simulation platform—Application to the propagation in inhomogeneous and heterogeneous tissues

For several years, CEA-LIST has been developing in partnership with INSERM, a HIFU simulation platform, CIVA Healthcare. It provides specific tools for the development and optimization of probes and therapeutic protocols in order to target clinical problematic of specific organs. This platform allows to easily simulate 3D pressure field induced by HIFU, in linear and non-linear regimes, and the corresponding thermal dose in tissues and phantoms. In this communication, we propose a particular focus on the dynamic ray method for linear acoustic propagation. This model enables fast simulations of HIFU propagation in heterogeneous configurations, including soft and hard tissues, made of isotropic and anisotropic medium, with homogeneous or inhomogeneous elastic properties. The contribution of a source point located on the emitting surface is obtained considering the evolution of a pencil of rays centered on the geometrical path linking this source and the observation point. This models takes into account the propagations of bulk waves in each medium and the various reflections and refractions, with and without mode conversion, at each interface. Examples of computation of pressure fields through hard tissues (bones, skull) and in heated soft tissues will be presented. [Work supported by French Nation Research Agency (ANR SATURN -15-CE19-0016).]

Numerical modeling of Gaussian beam propagation and diffraction in inhomogeneous media based on the complex eikonal equation

Gaussian beam is an important complex geometrical optical technology for modeling seismic wave propagation and diffraction in the subsurface with complex geological structure. Current methods for Gaussian beam modeling rely on the dynamic ray tracing and the evanescent wave tracking. However, the dynamic ray tracing method is based on the paraxial ray approximation and the evanescent wave tracking method cannot describe strongly evanescent fields. This leads to inaccuracy of the computed wave fields in the region with a strong inhomogeneous medium. To address this problem, we compute Gaussian beam wave fields using the complex phase by directly solving the complex eikonal equation. In this method, the fast marching method, which is widely used for phase calculation, is combined with Gauss–Newton optimization algorithm to obtain the complex phase at the regular grid points. The main theoretical challenge in combination of this method with Gaussian beam modeling is to address the irregular boundary near the curved central ray. To cope with this challenge, we present the non-uniform finite difference operator and a modified fast marching method. The numerical results confirm the proposed approach.

Seismic triplication used to reveal slab subduction that had disappeared in the late Mesozoic beneath the northeastern South China Sea

We provided a new study of the seismic velocity structure of the mantle transition zone (MTZ) beneath the northeastern South China Sea using P-wave triplications from two earthquakes at the central Philippines recorded by the Chinese Digital Seismic Network. Through fitting the observed and theoretical triplications modeled by the dynamic ray tracing method for traveltimes, and the reflectivity method for synthetic waveforms using grid-searching method, best-fit velocity models based on IASP91 were obtained to constrain the P-wave velocity structure of the MTZ. The models show that a high-velocity anomaly (HVA) resides at the bottom of MTZ. The HVA is 215km to 225km thick, with a P-wave velocity increment of 1.0% between 450km and 665km or 675km transition and increase by 2.5–3.5% at 665km or 675km depth. The P-wave velocity increment ranges from approximately 0.3% to 0.8% below the 665km or 675km. We proposed that the HVA in the MTZ was caused by the broken fragments of a diving oceanic plate falling into the MTZ at a high angle, and/or by unstable thick continental lithosphere dropping into the MTZ sequentially or almost simultaneously.

Numerical synthesis of tri‐reflector CATR with high cross‐polarisation isolation

In this Letter, a tri-reflector compact antenna test range (CATR) with a paraboloidal main reflector and two shaped sub-reflectors is synthesised by the kinematic and dynamic ray tracing method, while the high cross-polarisation isolation is achieved through the proposed system geometry. Numerical simulations by general responsibility assignment software patterns demonstrate that a cross-polarisation isolation >40 dB and a peak-to-peak amplitude (phase) ripple <1 dB (10°) can be achieved on the principal cuts of the designed quiet zone at 90–325 GHz. This clearly shows that the high cross-polarisation isolation in such a tri-reflector CATR can be achieved by the proposed antenna geometry without depolarisation control in the shaping procedure.

Local algorithm for computing complex travel time based on the complex eikonal equation.

The traditional algorithm for computing the complex travel time, e.g., dynamic ray tracing method, is based on the paraxial ray approximation, which exploits the second-order Taylor expansion. Consequently, the computed results are strongly dependent on the width of the ray tube and, in regions with dramatic velocity variations, it is difficult for the method to account for the velocity variations. When solving the complex eikonal equation, the paraxial ray approximation can be avoided and no second-order Taylor expansion is required. However, this process is time consuming. In this case, we may replace the global computation of the whole model with local computation by taking both sides of the ray as curved boundaries of the evanescent wave. For a given ray, the imaginary part of the complex travel time should be zero on the central ray. To satisfy this condition, the central ray should be taken as a curved boundary. We propose a nonuniform grid-based finite difference scheme to solve the curved boundary problem. In addition, we apply the limited-memory Broyden-Fletcher-Goldfarb-Shanno technology for obtaining the imaginary slowness used to compute the complex travel time. The numerical experiments show that the proposed method is accurate. We examine the effectiveness of the algorithm for the complex travel time by comparing the results with those from the dynamic ray tracing method and the Gauss-Newton Conjugate Gradient fast marching method.

Numerical model for the deformation of nucleated cells by optical stretchers

In this paper, we seek to numerically study the deformation of nucleated cells by single diode-laser bar optical stretchers. We employ a recently developed computational model, the dynamic ray-tracing method, to determine the force distribution induced by optical stretchers on a cell encapsulating a nucleus of different optical properties. These optical forces are shape dependent and can deform real non-rigid objects; thus resulting in dynamically changing distributions with cell and nucleus deformation. A Chinese hamster ovary (CHO) cell is a common biological cell that is of interest to the biomedical community because of its use in recombinant protein therapeutics and is an example of a nucleated cell. To this end, we model CHO cells as two concentric three-dimensional elastic capsules immersed in a fluid where the hydrodynamic forces are calculated using the immersed boundary method. We vary the inner capsule size to simulate different nucleus sizes. Our results show that the presence of a nucleus has a major effect on the force distribution on the cell surface and consequently on its net deformation. Scattering and gradient forces are reported for different nucleus sizes and the effect of nucleus size on the cell deformation is discussed quantitatively.

Lateral migration of a microdroplet under optical forces in a uniform flow

The behavior of a microdroplet in a uniform flow and subjected to a vertical optical force applied by a loosely focused Gaussian laser beam was studied numerically. The lattice Boltzmann method was applied to obtain the two-phase flow field, and the dynamic ray tracing method was adopted to calculate the optical force. The optical forces acting on the spherical droplets agreed well with the analytical values. The numerically predicted droplet migration distances agreed well with the experimentally obtained values. Simulations of the various flow and optical parameters showed that the droplet migration distance nondimensionalized by the droplet radius is proportional to the S number (zd/rp = 0.377S), which is the ratio of the optical force to the viscous drag. The effect of the surface tension was also examined. These results indicated that the surface tension influenced the droplet migration distance to a lesser degree than the flow and optical parameters. The results of the present work hold for the refractive indices of the mean fluid and the droplet being 1.33 and 1.59, respectively.

Optical separation of ellipsoidal particles in a uniform flow

The behavior of an ellipsoidal particle subjected to a vertical optical force by a loosely focused laser beam in a uniform flow was studied numerically. The fluid flow and the particle motion were separately solved and coupled using the penalty immersed boundary method, and the optical force was calculated using the dynamic ray tracing method. The optical force and optically induced torque on the ellipsoidal particle varied according to the aspect ratio and initial inclination angle. The ellipsoidal particle, whose major axis was initially aligned with the laser beam axis, was more migrated as the aspect ratio increased. The migration distance also depended on the initial inclination angle, even for a given ellipsoidal particle shape. As the laser beam power increased and the flow velocity decreased, the effect of the initial inclination angle increased. The ellipsoidal particles with different aspect ratios could be effectively separated if the rotation along the spanwise direction was suppressed. Moreover, the migration distance could be predicted analytically by introducing a new dimensionless number Sc to represent the ratio of the optical force to the viscous force for the ellipsoidal particles.

Lateral migration of an elastic capsule by optical force in a uniform flow

The lateral migration of an elastic capsule under an optical force in a uniform flow was studied to show the separation characteristics according to the elastic properties in the cross-type optical separator. The initially spherical capsule was moved through the fluid flow using a laser beam with a Gaussian distribution focused along the direction normal to the flow device surface. To simulate such a system, a penalty immersed boundary method was adopted to enable fluid-membrane coupling, and a dynamic ray tracing method was applied to the optical force calculation. The effects of the elastic properties of the capsule membrane (the surface Young's modulus and the bending modulus) on the lateral migration were studied. By increasing the surface Young's modulus, the capsule deformed less and the migration distance increased; however, buckling occurred in the capsule with a high surface Young's modulus. Buckling could be suppressed by increasing the bending rigidity. The effects of the flow velocity and the laser beam power were also examined. In the simulation, the S number, i.e., the ratio of the optical force to the viscous force, was adjusted by decreasing the flow velocity or increasing the laser beam power. The migration distance increased as the S number increased, and a constant lateral migration distance was obtained for a rigid particle for a given S number. An elastic capsule under conditions intermediate between a fixed flow velocity and a fixed laser beam power, however, did not yield a constant lateral migration distance due to the extent of the deformation in the different situations. To predict the lateral migration distance of an elastic capsule, a nondimensional parameter, S_{e}, was defined to include the effects of the optical force, the elastic force, and the fluid viscous force. A unified tendency of the lateral migration distance with S_{e} was obtained for capsules with intermediate elasticity, by varying either the flow velocity or the laser beam power.

Optical levitation of a non-spherical particle in a loosely focused Gaussian beam

The optical force on a non-spherical particle subjected to a loosely focused laser beam was calculated using the dynamic ray tracing method. Ellipsoidal particles with different aspect ratios, inclination angles, and positions were modeled, and the effects of these parameters on the optical force were examined. The vertical component of the optical force parallel to the laser beam axis decreased as the aspect ratio decreased, whereas the ellipsoid with a small aspect ratio and a large inclination angle experienced a large vertical optical force. The ellipsoids were pulled toward or repelled away from the laser beam axis, depending on the inclination angle, and they experienced a torque near the focal point. The behavior of the ellipsoids in a viscous fluid was examined by analyzing a dynamic simulation based on the penalty immersed boundary method. As the ellipsoids levitated along the direction of the laser beam propagation, they moved horizontally with rotation. Except for the ellipsoid with a small aspect ratio and a zero inclination angle near the focal point, the ellipsoids rotated until the major axis aligned with the laser beam axis.

Gaussian beams in inhomogeneous media: A review

The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich’s method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich’s method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich’s method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names. The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.

Seismic velocities and Q-factors in the uppermost crust beneath the SVEKA profile in Finland

The SVEKA deep seismic sounding profile in Central Finland was shot in 1981 to study the crustal structure of the Fennoscandian shield in the contact zone between the Archaean and Svecokarelian provinces. Apart from P-and S-waves, well developed Rayleigh surface waves (R-waves) with periods of 0.5–2.0 s were recorded from all the shot points. In the frequency band 0.5–2.5 Hz the R-waves were clearly observed on the entire 325 km long profile. Amplitudes of P-, S- and R-waves corrected for charge size were used to study the attenuation of waves and Q-factors. From the records of body and R-waves the distributions of seismic velocities and quality factors were determined using the dynamic ray-tracing method (2-D modelling) and the reflectivity method (1-D modelling). In the 2-D model of the uppermost crust, the P-wave velocities vary from 5.8–5.9 km/s at the top to 6.2–6.3 km/s at a depth of 6 km. The velocity ratio solV p V s varies with depth and distance from 1.67 to 1.74, reaching its maximum value of 1.8–2.0 in the uppermost 200–500 m and values of 1.76–1.80 in the uppermost 1 km. The quality factors Q p and Q s vary in the uppermost 1 km from 20 to 120, reaching a higher value of 140 in the marginal zone of the Archaean basement. In the depth range of 1–2 km Q-values of 80–400 were found. In the depth range 2–6 km Q-values of 300–800 were found for the Svecokarelian province and 200–400 for the Archaean. The low values of V s and Q in the uppermost 1 km are thought to be the result of rock fracturing. The crack density, ϵ, was estimated for the uppermost crust along the profile. For the SVEKA profile we found ϵ-values of 0.35–0.54 in the uppermost 200 m, 0.10–0.32 at a depth of 0.2-1.0 km and ϵ < 0.1 for depths of 1–4 km. The comparison of our seismic data with results of theoretical and laboratory investigations implies that the uppermost crustal rocks are saturated. The zones of strong attenuation of R-waves coincide with faults and schist zones which are well known from surface geology.

Range of validity of seismic ray and beam methods in general inhomogeneous media - II. A canonical problem

Summary. The Green’s function, in a constant gradient medium, is derived for an explosive point source, in the frequency and the time domains. The a.nalytica1 dynamic ray tracing (DRT) solution is rederived with conditions stated in Part 1. The Gaussian beam (GB) solution is investigated. New beam parameters and conditions are defined. Comparisons between exact and approximate solutions are undertaken. For both methods, DRT and GB, conditions of validity are .explicit and quantitative. An accuracy criterion is defined in the time domain, and measures a global relative error. The range of validity is expressed in the form of two inequalities for the dynamic ray tracing method and of five inequalities for the Gaussian beam method. Results remain accurate at ray turning points. For the types of medium considered, the breakdown of the dynamic ray tracing method is smoother and better behaved than that of Gaussian beams. As examples, a vertical seismic profiling configuration, and a shallow earthquake are modelled, using Gaussian beams.