Showing papers on "Paraxial approximation published in 1988"
TL;DR: In this paper, it was shown that the paraxial approximations to the wave equation used by Clayton and Engquist can be improved, allowing more accurate finite-difference formulations at internal boundaries and corners, and formulations that are stable over a wider range of the parameters.
Abstract: It is shown that the paraxial approximations to the wave equation used by Clayton and Engquist can be improved upon, allowing more accurate finite-difference formulations at internal boundaries and corners, and formulations that are stable over a wider range of the parameters.
173 citations
TL;DR: This article presents coefficients for L2, L infinity, and other alternative classes of approximants that have better wide-angle behavior.
Abstract: A one‐way wave equation, also known as a paraxial or parabolic wave equation, is a differential equation that permits wave propagation in certain directions only. Such equations are used regularly in underwater acoustics, in geophysics, and as energy‐absorbing numerical boundary conditions. The design of a one‐way wave equation is connected with the approximation of (1−s2)1/2 on [−1,1] by a rational function, which has usually been carried out by Pade approximation. This article presents coefficients for L2, L∞, and other alternative classes of approximants that have better wide‐angle behavior. For theoretical results establishing the well posedness of these wide‐angle equations, see the work of Trefethen and Halpern [‘‘Well‐posedness of one‐way wave equations and absorbing boundary conditions,’’ Math. Comput. 47, 421–435 (1986)].
154 citations
TL;DR: In this article, a B-spline representation is adopted for lateral velocity heterogeneity and interfaces, where the spline parameters have a geometrical interpretation and they may be explicitly used as unknowns in the inverse problem.
Abstract: Summary
Reflection tomography, the determination of velocity distribution and reflector position from reflection travel-time data, is a very non-linear inverse problem. Unlike in transmission tomography, ray paths have to be iteratively updated, because travel-time variations cannot be computed by integration of slowness along the original unperturbed ray paths. From a study of parameter sensitivity we conclude that in seismic reflection experiments the vertical variation of slowness inside layers is poorly resolved from travel-time data. For this reason, in each layer the slowness was modelled with functions varying only in the horizontal direction. A B-spline representation is adopted for lateral velocity heterogeneity and interfaces. These splines are local and well adapted for tomography because the spline parameters have a geometrical interpretation and they may be explicitly used as unknowns in the inverse problem. For each iteration, and for every source-receiver pair, two-point ray tracing was performed by paraxial ray tracing, and the inverse problem was solved by iterative least-squares. A priori data, necessary to stabilize the inverse problem, were introduced by a penalty function approach. This method is equivalent to using a priori convariance matrices, but it has a simpler physical interpretation and is faster to use. Damping was used to improve the convergence. the method was first tested in the inversion of synthetic data. These synthetic examples illustrate the limitations of reflection tomography: non-linearity effects, poor vertical resolution of the velocity, and decrease of the resolution with the ratio of maximum offset to interface depth. Finally, we inverted a data set from the Paris Basin. the inversion method reduces the residual norm to 6 ms, which is less than the expected error on the data.
136 citations
TL;DR: In this paper, the authors present a summary of new methods employing Lie algebraic tools for characterizing beam dynamics in charged-particle optical systems, which are applicable to accelerator design, charged particle beam transport, electron microscopes, and also light optics.
Abstract: The purpose of this paper is to present a summary of new methods employing Lie algebraic tools for characterizing beam dynamics in charged- particle optical systems. These methods are applicable to accelerator design, charged-particle beam transport, electron microscopes, and also light optics. The new methods represent the action of each separate element of a compound optical system, including all departures from paraxial optics by a certain operator. The operators can then be concatenated following well defined rules to obtain a resultant operator that characterizes the entire system. (AIP)
120 citations
ARCO1
TL;DR: In this paper, a rapid nonrecursive prestack Kirchhoff migration is implemented by computing the Green's functions (both traveltimes and amplitudes) in variable velocity media with the paraxial ray method.
Abstract: A rapid nonrecursive prestack Kirchhoff migration is implemented (for 2-D or 2.5-D media) by computing the Green’s functions (both traveltimes and amplitudes) in variable velocity media with the paraxial ray method. Since the paraxial ray method allows the Green’s functions to be determined at points which do not lie on the ray, two‐point ray tracing is not required. The Green’s functions between a source or receiver location and a dense grid of thousands of image points can be estimated to a desired accuracy by shooting a sufficiently dense fan of rays. For a given grid of image points, the paraxial ray method reduces computation time by one order of magnitude compared with interpolation schemes. The method is illustrated using synthetic data generated by acoustic ray tracing. Application to VSP data collected in a borehole adjacent to a reef in Michigan produces an image that clearly shows the location of the reef.
92 citations
TL;DR: In this paper, the authors proposed a solution for the fast solution of ray tracing in three-dimensional laterally heterogeneous media, which is based on the paraxial ray theory.
Abstract: We propose a technique for the fast solution of ray tracing in three-dimensional laterally heterogeneous media. Analytical solutions for both ray tracing and paraxial ray tracing for a medium with an arbitrary gradient of the square of slowness are used to develop a systematic procedure to construct exact solutions. Complex heterogeneous media are divided into tetrahedral finite elements inside which the square of the slowness has a simple linear distribution. Arbitrarily complex media may be studied by judicious choice of the elements. We develop appropriate boundary conditions for interfaces with zeroth or first-order velocity discontinuities. Two-point ray tracing is performed by a Newton method based on paraxial ray theory. With a few iterations the ray trajectories through the source and the observer are calculated with the same overall speed as with ray bending. The paraxial method has the additional advantage that it can be used to identify caustics and to separate travel time branches. We used the previously described method to replace the ray tracing algorithm of the HYPO71 program, probably the most commonly used earthquake location program. The ray tracing routine may be easily modified to adapt it to other earthquake location environments. With the modified HYPO71 program, we relocated the aftershock sequence of the November 23, 1980, Irpinia (Italy) earthquake. The introduction of a low-velocity zone proposed by different authors shifts aftershock locations to the NE. This horizontal displacement of hypocenters is not very sensitive to the thickness of the low-velocity zone.
74 citations
TL;DR: In this article, closed-form expressions for various statistical functions that arise in optical propagation through arbitrary optical systems that can be characterized by a complex ABCD matrix in the presence of distributed random inhomogeneities along the optical path are derived.
Abstract: Closed-form expressions are derived for various statistical functions that arise in optical propagation through arbitrary optical systems that can be characterized by a complex ABCD matrix in the presence of distributed random inhomogeneities along the optical path. Specifically, within the second-order Rytov approximation, explicit general expressions are presented for the mutual coherence function, the log-amplitude and phase correlation functions, and the mean-square irradiance that are obtained in propagation through an arbitrary paraxial ABCD optical system containing Gaussian-shaped limiting apertures. Additionally, we consider the performance of adaptive-optics systems through arbitrary real paraxial ABCD optical systems and derive an expression for the mean irradiance of an adaptive-optics laser transmitter through such systems.
61 citations
TL;DR: The method is well suited for testing even strong aspheric optics and other reflecting or transmitting objects and 3-D shape measurement of nonplanar glass plates, e.g., car windshields.
Abstract: We demonstrate a method for testing optics (spherical and aspheric) and other reflecting or transmitting objects. We call this experimental ray tracing. A laser beam is sent through the sample, and its propagation is determined with a lateral effect photodiode. A modified Hartmann test can be performed by measurement of beam location within two planes. Measurement in one plane close to the focus delivers a spot diagram. The method is well suited for testing even strong aspheric optics. As a further use we demonstrate 3-D shape measurement of nonplanar glass plates, e.g., car windshields.
57 citations
TL;DR: An approximation to the radiative transfer equation is derived for the propagation of narrow light beams and shows very good agreement over a wide range of extinction coefficients, scattering albedos, and particle size parameters.
Abstract: An approximation to the radiative transfer equation is derived for the propagation of narrow light beams. The main feature is the representation of the power flux in the direction normal to the propagation axis by a diffusion process. Coupled partial differential equations are obtained for the forward and backward flux densities. They are valid in the paraxial approximation. The solutions are written in general analytic formulas applicable to inhomogeneous aerosol media. Comparisons with Monte Carlo calculations and laboratory data reported in a companion paper [ Appl. Opt.27, 2485 ( 1988)] show very good agreement over a wide range of extinction coefficients, scattering albedos, and particle size parameters.
52 citations
TL;DR: The results indicate that accommodation has an effect on the level of these aberrations at the larger angles of eccentricity, with paraxial schematic eyes showing the same trend.
Abstract: Oblique astigmatism and field curvature of the human eye were measured at various levels of accommodation up to a maximum of SD and angles of eccentricity up to 60°. The results indicate that accommodation has an effect on the level of these aberrations at the larger angles of eccentricity. Comparisons with paraxial schematic eyes are favourable, with paraxial schematic eyes showing the same trend.
51 citations
TL;DR: Panchang et al. as mentioned in this paper proposed a method for solving the combined refraction-diffraction equation in large domains by modifying the reduced wave equation, and the elliptic, boundary-value problem is solved by the marching or "Error Vector Propagation" method.
TL;DR: In this paper, the extended focus and auto-focus techniques of confocal micro-scopy, which can be used to give greatly increased depth of field, are compared and an improved approximation is discussed and compared with exact high-angle calculations.
Abstract: The extended focus and auto-focus techniques of confocal micro- scopy, which can be used to give greatly increased depth of field, are compared . These methods rely on the use of high numerical apertures, so that the paraxial approximation is found to give inaccurate theoretical image predictions . An improved approximation is discussed and compared with exact high-angle calculations . The confocal surface-profiling technique, which allows non- invasive investigation surface topography is also considered .
TL;DR: In this article, the scattering of sound by sound from Gaussian beams that interact at small angles is investigated theoretically with a quasilinear solution of the Khokhlov-Zabolotskaya nonlinear parabolic wave equation.
Abstract: The scattering of sound by sound from Gaussian beams that interact at small angles is investigated theoretically with a quasilinear solution of the Khokhlov‐Zabolotskaya nonlinear parabolic wave equation. The analytical solution, which is valid throughout the entire paraxial field, is a generalization of the result obtained for parametric receiving arrays by Hamilton, Naze Tjotta, and Tjotta [J. Acoust. Soc. Am. 82, 311–318 (1987)]. Significant levels of scattered difference frequency sound are shown to exist outside the nonlinear interaction region. An asymptotic formula reveals that difference frequency sound is scattered in the approximate direction of k1‐k2, where ki is the wave vector associated with the direction and frequency of the ith primary beam. Computed propagation curves and beam patterns demonstrate the dependence of the scattered radiation on source separation, frequency ratio, interaction angle, and focusing. Results are also presented for the scattered sum frequency sound. Comparisons ar...
Abstract: Optical systems produce canonical transformations on phase space that are nonlinear. When a power expansion of the coordinates is performed around a chosen optical axis, the linear part is the paraxial approximation, and the nonlinear part is the ideal of aberrations. When the optical system has axial symmetry, its linear part is the symplectic group Sp(2, R) represented by 2 × 2 matrices. It is used to provide a classification of aberrations into multiplets of spin that are irreducible under the group, in complete analogy with the quantum harmonic-oscillator states. The “magnetic” axis of the latter may be chosen to adapt to magnifying systems or to optical fiberlike media. There seems to be a significant computational advantage in using the symplectic classification of aberrations.
01 Nov 1988-Nuclear Instruments & Methods in Physics Research Section A-accelerators Spectrometers Detectors and Associated Equipment
TL;DR: In this paper, the effect of finite dimensions of crystal and source and their interplay with a finite velocity vector of the source is discussed for Johann and Johansson spectrometers.
Abstract: A method of ray tracing is presented which allows a detailed analysis of the X-ray line profile observed with a curved-crystal spectrometer for both a stationary or a fast X-ray source. In particular, the effect of finite dimensions of crystal and source and their interplay with a finite velocity vector of the source is discussed for Johann and Johansson spectrometers. It is shown how some parameters cause distortions and shifts of the reflection and how far they can be resolved by using a two-dimensional position-sensitive X-ray detector.
TL;DR: It is pointed out that the nondiA'racting Jo beams of Durnin, Miceli, and Eberly are subgroup-reduced basis functions for the group of paraxial transformations of optical phase space that allows the action of the group on such and similar beams to be computed through 2 x 2 matrix algebra.
Abstract: We point out that the nondiffracting ${\mathrm{J}}_{0}$ beams of Durnin, Miceli, and Eberly are subgroup-reduced basis functions for the group of paraxial (symplectic, linear) transformations of optical phase space. This allows the action of the group on such and similar beams to be computed through 2\ifmmode\times\else\texttimes\fi{}2 matrix algebra.
TL;DR: In this paper, a geometrical optics approach is used to calculate the intensity distribution in the image of a small object centered on the axis, taking into account the size of the object as well as the spherical and chromatic aberrations of the lens.
Abstract: A geometrical optics approach is used to calculate the intensity distribution in the image of a small object centered on the axis. The calculations take into account the size of the object as well as the spherical and chromatic aberrations of the lens. The intensity distribution curves, calculated as a function of depth in the image, reveal the existence of a compact high intensity image peak in the caustic region between the paraxial image plane and the plane of least confusion. The intensity distribution curves show that the geometrical resolution is better in the plane with the high intensity peak than in the plane of least confusion. As the object radius approaches zero the plane in which the high intensity peaks occurs moves toward the paraxial plane, and the geometrical resolution approaches zero. For an object of finite brightness the intensity in the image peak also approaches zero. A geometrical error rg is introduced, which depends on the size of the smallest object providing the required intens...
TL;DR: In this paper, a theoretical analysis is carried out to synthesize acoustic material signatures (AMS) of solid plates immersed in water, which avoids three major simplifying assumptions of the presently available techniques, which are, paraxial approximation, assumption of perfect reflection and Gaussian summation of the incident field.
Abstract: A theoretical analysis is carried out to synthesize acoustic material signatures (AMS) of solid plates immersed in water. The distinctive feature of this analysis is that it avoids three major simplifying assumptions of the presently available techniques, which are, paraxial approximation, assumption of perfect reflection and Gaussian summation of the incident field. Presently available techniques can avoid some but not all of these simplifying assumptions for computing the AMS. In this paper the analysis is carried out for lowfrequency acoustic waves generated by a cylindrical transducer without a lens rod. Reasons for these changes in the conventional acoustic microscope geometry is given. The AMS is synthesized for an aluminium plate in presence as well as in absence of water on its one side. As expected a significant difference is observed between the signatures generated under these two situations.
TL;DR: An inhomogeneous rotation-translation matrix is developed to describe the path of a ray through the lens elements with quadratic radial index gradients, which is used to predict the paraxial properties of the array.
Abstract: Matrix methods have been used to examine the paraxial performance of homogeneous systems. This method has been extended to include the presence of gradient lens elements in the optical array. To accomplish this, an inhomogeneous rotation-translation matrix is developed to describe the path of a ray through the lens elements with quadratic radial index gradients. The system matrix elements are then used to predict the paraxial properties of the array. Three examples are examined: the Wood lens, two thin lenses coupled to a radial gradient index rod, and a model of the human eye which contains a radial gradient index crystalline lens.
TL;DR: In this article, the effects of anisotropy on the propagation of elastic waves in a structure which simulates a continental subcrustal lithosphere are studied and P- and S-wave synthetic record sections along several profiles and fans are presented and discussed.
TL;DR: It is shown that in the limit of a small step length a ray equation can be attributed to the beam-propagation method, which is valid in the nonparaxial case but does not agree with the standard geometrical optics ray equation even in the paraxial limit.
Abstract: It is shown that in the limit of a small step length a ray equation can be attributed to the beam-propagation method. The equation is valid in the nonparaxial case but does not agree with the standard geometrical optics ray equation even in the paraxial limit. The beam-propagation method will not therefore predict the correct focal length of a lens. This is illustrated by a numerical example.
TL;DR: In this article, the analysis and comparison between different planar Fresnel lenses have been carried out numerically using the beam-propagation method, which allows one to evaluate relevant parameters, such as focal point, spot size, efficiency, paraxial and wide-angle diffraction effects, taking into account the diffraction produced by the refractive-index discontinuities.
Abstract: The analysis and the comparison between different planar Fresnel lenses has been carried out numerically using the beam-propagation method. This technique allows one to evaluate relevant parameters, such as focal point, spot size, efficiency, paraxial and wide-angle diffraction effects, taking into account the diffraction produced by the refractive-index discontinuities. Comparisons with some of the published experimental results are presented.
01 Jan 1988
TL;DR: In this article, the optical propagator for the Helmholtz equation is given by the Fourier transform of a path integral; here, the authors try to express it directly by path integral, and the usual limiting cases (short wavelength, asymptotic approximation, paraxial approximation) are recovered.
Abstract: The optical propagator for the Helmholtz equation is given by the Fourier transform of a path integral; here we try to express it directly by a path integral. The usual limiting cases (short wavelength, asymptotic approximation, paraxial approximation) are recovered. © 1988 Societa Italiana di Fisica.
09 Jun 1988
TL;DR: In this paper, a simple model based on paraxial ray tracing was proposed to approximate the narcissus effect in a thermal imaging system, where each optical refracting surface can reflect internal radiation down onto the detector and the spurious signal generated by the composite effect of these reflections was derived.
Abstract: In a thermal imaging system each optical refracting surface can reflect internal radiation down onto the detector Narcissus is the spurious signal generated by the composite effect of these reflections A function is derived which represents the narcissus effect as an equivalent temperature differences in the scene The function can be approximated by a simple model based on paraxial ray-tracing The effectiveness of the model is demonstrated by four examples
TL;DR: In this paper, a second-quantized theory of propagation in laser-active media was formulated and applied to the description of amplified spontaneous emission for the case of homogeneously broadened three-level atoms in a rodlike geometry with arbitrary Fresnel number.
Abstract: We formulate a second-quantized theory of propagation in laser-active media and apply it to the description of amplified spontaneous emission for the case of homogeneously broadened three-level atoms in a rodlike geometry with arbitrary Fresnel number. The electromagnetic field is treated in paraxial approximation by an ad hoc quantization scheme, and spontaneous emission into off-axis modes is described by noise operator methods. We show by a scaling (dimensional) analysis how to derive the characteristic fields and lengths for superfluorescence and amplified spontaneous emission. The results show that dye media can be used as experimental analogs for x-ray lasers.
TL;DR: It is described how optical system defects propagate through an arbitrary paraxial optical system that can be described by an ABCD ray transfer matrix.
Abstract: We describe how optical system defects (tilt/jitter, decenter, and despace) propagate through an arbitrary paraxial optical system that can be described by an ABCD ray transfer matrix. A pedagogical example is given that demonstrates the effect of alignment errors on a typical optical system.
01 Oct 1988-Nuclear Instruments & Methods in Physics Research Section A-accelerators Spectrometers Detectors and Associated Equipment
TL;DR: In this paper, a method for analytic computation of the reflection matrix is developed by taking the convolution of the source function at the surface of the mirror with the paraxial propagator.
Abstract: The transformations of Gaussian radiation beams caused by reflection off mirrors is an important issue for FELs operating as oscillators. The reflected radiation from a single incident Gaussian mode will contain other modes due to the finite mirror size, the deflection of the beam and mismatches in the curvature. A method for analytic computation of the reflection matrix is developed by taking the convolution of the source function at the surface of the mirror with the paraxial propagator. The mirror surface that reflects spherical incoming wavefronts into spherical outgoing is found to be a paraboloid. Integral expressions for the reflection coefficients R pq mn for any incoming mode u mn into the outgoing u pq are obtained as functions of the deflection angle φ, the reflected beam spot size W o and the mirror size. The coefficient R 00 00 for the lowest-to-lowest mode reflection is determined analytically. The spot size W o can then be selected, depending on the mirror size, to maximize R 00 00 . The ratio of the mirror size to the spot size is the dominant factor determining the reflection coefficient. The effects of deflecting the light beam enter as small corrections, of first order in the diffraction angle θ d ⪡ 1.
TL;DR: In this article, the optical propagator for the Helmholtz equation is given by the Fourier transform of a path integral, and the usual limiting cases (short wavelength, asymptotic approximation, paraxial approximation) are recovered.
Abstract: The optical propagator for the Helmholtz equation is given by the Fourier transform of a path integral; here we try to express it directly by a path integral. The usual limiting cases (short wavelength, asymptotic approximation, paraxial approximation) are recovered.
TL;DR: A simple expression for the propagator of a dielectric waveguide structure with an arbitrarily graded refractive-index profile is presented, based on the use of a short-distance approximation to simplify the formal path integral solution to the paraxial optical wave equation.
Abstract: A simple expression for the propagator of a dielectric waveguide structure with an arbitrarily graded refractive-index profile is presented. This expression simplifies considerably the calculation of the propagation properties of dielectric waveguide structures described by z-independent paraxial optical Hamiltonians. The expression is based on the use of a short-distance approximation to simplify the formal path integral solution to the paraxial optical wave equation. The expression is incorporated into a simple and efficient numerical method for calculation of the propagation properties of dielectric waveguides with arbitrarily graded refractive-index profiles. The accuracy of this approach and its potential for use in optical design are illustrated by calculating the propagation constants and optical field distributions for a graded-index slab waveguide and a slab waveguide array.
06 Jun 1988
TL;DR: In this paper, a transition region theory (TRT) code is described for dynamic ray field amplitude and polarization tracing. But the code is not applicable to the case of the radio observatory in Arecibo.
Abstract: The antenna of the radio observatory in Arecibo is currently fed by line feeds that correct for the spherical aberrations of the reflector. To improve the bandwidth and the sensitivity of the radiotelescope the line feeds will be replaced with a dual-reflector feed system. A computer code is described that is based on an efficient geometrical optics (GO) ray tracing technique. The code, called the TRT code, is related to the procedure of dynamic ray tracing used in seismics, which means that the complete complex ray field amplitude (and polarization for EM waves) is traced. The code also makes use of the paraxial-ray approximation, applied by considering each individual ray as an axis. In addition to the GO analysis, the code gives asymptotic information about the diffraction losses. This is obtained by the diffraction tracing techniques of the transition region theory (TRT). The characters of the TRT code are noted. >