Showing papers on "Paraxial approximation published in 1992"

Ray perturbation theory for traveltimes and ray paths in 3-D heterogeneous media

Abstract: SUMMARY Both paraxial ray tracing and two-point ray tracing are powerful tools for solving wave propagation problems. When a slowness model is mildly perturbed from a reference model, one can use perturbation theory for the determination of the ray positions and the traveltimes. An extension of Fermat’s theorem is presented, which states that the traveltime is stationary with respect to the perturbations in the ray position provided that the endpoints of the ray are perturbed along the wavefront of the unperturbed ray. It is shown that when the ray perturbation satisfies this condition the second-order traveltime perturbation can be computed from the first-order ray perturbation. A perturbation analysis of the equation of kinematic ray tracing leads to a simple second-order differential equation for the ray deflection expressed in ray coordinates. This constitutes a perturbation method based on a Lagrangian formulation, and leads to a first-order expression for the ray deflection and a second-order expression for the traveltime perturbation. This is of relevance to non-linear traveltime tomography because it leads to an efficient method for evaluating the lowest order ray deflection and the non-linear effect this has on the traveltimes. The theory is applicable both to two-point ray tracing and to the determination of paraxial rays. The derivations in this paper are completely self-contained. All expressions, including the transformation to ray coordinates, are derived from first principles. In this way one obtains insight in the approximations that are actually made. A scale analysis leads to dimensionless numbers that give an indication whether the theory is applicable to a specific problem. For the special case of a layered reference medium the final equations are particularly simple. Plane discontinuities in the reference model and the slowness perturbation are incorporated in the theory. The final expressions for the ray deflection and the traveltime perturbation can be implemented numerically in a simple way. It is indicated how applications to very large-scale problems can be achieved. Several examples, including the propagation of waves through a quasi-random model of the earth’s mantle illustrate the theory.

Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams

Abstract: Two related basic transition operators, T1 and T2, are found that transform arbitrary solutions of the parabolic equation of the paraxial approximation into exact monochromatic solutions of the scalar wave equation or of the corresponding Helmholtz equation. The operators realize different boundary conditions. The operator T1 preserves the transverse field distribution of the paraxial approximation in the plane z = 0 for the obtained exact solution. The method is applied to calculate the complete corrections to the paraxial approximation of the fundamental Gaussian beam solutions of the n-dimensional wave equation. The lowest-order correction to the paraxial approximation in the three-dimensional case is found to be in agreement with the result of Agrawal and Pattanayak [ J. Opt. Soc. Am.69, 575 ( 1979)]. The complete series of corrections on the axis is summed up to a transcendental function and discussed for the three-dimensional case.

Fresnel zone inversion for lateral heterogeneities in the earth

Abstract: We propose a different kind of seismic inversion from travel-time or waveform inversion for lateral heterogeneities in the earth: Fresnel zone inversion. Amplitude and phase delay of data in several frequency ranges are inverted for model space around ray paths with a width corresponding to the considered frequency so that primary effect of finiteness of wavelength be included. For vertically heterogeneous media, Frechet derivatives for inversion are obtained very efficiently using the paraxial ray approximation, with nearly similar amounts of computation compared to travel-time inversion. As an example, Frechet derivatives are computed for a teleseismic observation system for a three-dimensional structure in the lithosphere beneath an array of seismic stations. Even if the used frequency is around 2 Hz, the width of Frechet derivatives cannot be neglected, particularly near the bottom of the lithosphere. Sensitivity of model parameters to observations is, moreover, different in our approach from conventional travel-time inversion: it is zero along ray paths but large slightly away from them. Some model calculations show that travel-time inversion, particularly with models divided into very fine meshes or blocks, might give misleading results. An example of inversion for a simple Camembert model, in the event that travel-time inversion gives no reliable results, shows how this technique works with much smaller data sets and computation than waveform inversions.

Wave field determination using three-dimensional intensity information.

Abstract: The Wigner distribution of the electric fields in a quasimonochromatic light wave is equivalent, in the paraxial approximation, to the cross-spectral density function of the wave. The intensity distribution in a plane may be described as a projection across this Wigner distribution. Three-dimensional intensity information can be used to generate a set of projections from which it is possible to reconstruct the second-order statistics of the partially coherent wave field.

A wide-angle vector beam propagation method

Abstract: A wide-angle vector beam propagation method is developed and presented. Considerable improvement in accuracy over the paraxial vector propagating algorithms is achieved with virtually no additional computation. >

Gaussian Schell-model beams generated with synthetic acousto-optic holograms

Abstract: The recently introduced concept of a synthetic acousto-optic hologram [ J. Appl. Phys.67, 49 ( 1990)] is applied to convert a Gaussian laser beam into a partially coherent anisotropic Gaussian Schell-model (AGSM) beam. Real-time reconfigurability of the coherence properties is achieved by this technique, which features scattering of the laser beam by an electronically synthesized, digitally phase-modulated volume grating that propagates in an acousto-optic Bragg cell. The coherence and intensity distributions of the fields obtained by different types of phase modulation are investigated theoretically. We demonstrate some particularly interesting AGSM sources and fields: a secondary elliptical AGSM source with a circularly symmetric far-field intensity distribution and an AGSM field that retains the eccentricity of its intensity profile in the propagation through any centrosymmetric (paraxial) optical system.

Some historical and technical aspects of beam quality

Abstract: A survey is given on the various algorithms to describe the propagation of radiation fields in the paraxial approximation, starting with the Kirchhoff-Fresnel-Collins-integral and ending with the intensity moments and the generalizedABCD-law. The importance of characterizing beam quality for technical applications by suitable parameters is pointed out. The two competing definitions of beam parameters, intensity moments and power content values are discussed.

Derivation and simulation of higher numerical aperture scalar aerial images

Abstract: The application of scalar diffraction theory to a projection optics system is examined here for somewhat higher numerical aperture conditions upon removing the paraxial, or small angle, approximation that is typically made. A detailed derivation is given that notes the key physical assumptions originally contained in work by others on vector imaging theory. The asymptotic limit of large lenses and focal length sizes to object and image sizes is explicitly carried out, while keeping numerical apertures and magnification fixed. Numerical results of the resulting equations are presented for a variety of imaging conditions, including phase shift masks and modified illumination.

Accuracy of a Gaussian beam

Abstract: The accuracy of a multimode Gaussian beam is discussed. Novel modifications are given to the conventional Gaussian beam formulation derived using the paraxial approximation. With the modifications, the field can be calculated accurately with the Gaussian beam. Modified multimode Gaussian beam techniques are used for calculation of the beam of a corrugated horn antenna. Phase and amplitude patterns given by this method are compared with measurements and other theoretical methods, namely Huygen's principle and the modal matching technique. The meaning of the phase and phase center is also discussed. >

Comparison of the paraxial-ray approximation and the variational method solutions to the numerical results for a beam propagating in a self-focusing Kerr medium.

Abstract: The analytic solutions for the amplitude and phase of a cw cylindrically symmetric beam propagating in a self-focusing Kerr medium obtained from the aberrationless paraxial approximation and the variational method are compared with the numerical results of the theory. For beam energies close to the critical focusing value, the variational method expression for the longitudinal phase is shown, except for an extremely small longitudinal distance of propagation, to give better qualitative and quantitative agreement with the numerical solutions. In particular, we confirm the variational method prediction that the regularized phase does not change sign.

Extending scalar aerial image calculations to higher numerical apertures

Abstract: The usual formula for the scalar aerial image of an isolated object due to a projection lens system has been generalized beyond the paraxial approximation in an attempt to extend scalar diffraction theory to include numerical aperture (NA) values up to about 0.6. Beyond this regime, or certainly beyond NA=0.7, polarization effects need to be included, thereby demanding a full vector treatment and invalidating the present scalar formulation. A key point to the present scalar result without the paraxial approximation is the predicted functional dependence of the aerial image on magnification as NA increases. A second key point is that the usual scaling of λ/NA for the object dimensions and λ/NA2 for defocus become invalid for high NA systems. Numerical results of illustrative test cases are shown.

Relativistic effects for Doppler measurements near solar conjunction

Abstract: The authors consider the relativistic corrections to the Doppler effect in a metric theory of gravity, in the weak field and slow motion approximation, for a source and a receiver at great distances from, but near alignment with, the perturbing body (i.e. near conjunction). They also work in the approximation of a thin screen and paraxial rays, and apply this formalism to an ideal experiment, related to that of the deflection of light rays by the solar mass. They then introduce a differential Doppler technique, which allows some new and interesting developments in the field of experimental gravitation. With a spacecraft equipped with a hydrogen maser on board, one can measure the difference between the fractional frequency shifts in both directions. Near conjunction it is possible to have information, for example, on the angular momentum of the Sun or on a possible effect of a privileged cosmological frame of reference. This technique also allows a drastic reduction of the effect due to the plasma of the solar corona. The maximum fractional frequency change induced by the angular momentum of the Sun is about 7*10-16, which is barely consistent with the stability of hydrogen masers currently available.

MoL-BPM Algorithms for Waveguide Bends and Vectorial Fields

Abstract: : The hitherto used BPM algorithm based on the MoL was first demonstrated at the Hilton Head Conference and than described in. The MoL-BPM overcomes the disadvantages of the BPM based on the paraxial approximation, e.g. small refractive index steps, artificial absorbers, numerical loss. Special characteristics of the MoL-BPM are the use of Helmholtz (or Sturm-Liouville) Equation, numerical absorbing or radiation conditions, use of eigensolution, implementation of the interface conditions for the fields. In this contribution two new algorithms will be presented. First a special algorithm for waveguide bends is presented. Then an algorithm for the vectorial field is described.

Approximation of the scalar focus wave modes

Abstract: The physical-realizability issues of scalar focus wave modes that are solutions of the homogeneous wave equation with the boundary data exp(−r2w0−2) given on a characteristic are addressed. The Gaussian beams are solutions of the paraxial equation with the same boundary data in a fixed plane. A comparison of these two situations suggests that the focus wave modes can be approximated by giving the exponential exp(−r2w0−2) on some moving plane. This suggestion is implemented.

Electromagnetic Gaussian beam beyond the paraxial regime

Abstract: A modified electromagnetic beam technique is derived analytically from the asymptotic field expressions for the quasi-optical, millimeter-wave, plane polarized Laguerre-Gaussian modes of arbitrary order. The method makes use of the paraxial-wave formulas only, yet it accurately gives the full electric field vector at any point far beyond the conventional Gaussian beam regime. The technique is assessed numerically against the exact Green's function results. >

Physical interpretation and stability of paraxial boundary conditions

Abstract: We derive in this paper the dynamic impedance matrices associated with some paraxial boundary conditions for wave motions in unbounded homogenous elastic media and use them to establish the existence of directions of propagation of waves for which the boundaries supply rather than dissipate energy. Also, we explore the existence of solutions of exponential growth and discuss conditions under which they can arise. These considerations may be used to provide a physical interpretation to the paraxial boundaries and to understand possible sources of instability that can develop in time-domain implementations of these schemes with finite differences.

Laser beam probing in capillary tubes

Abstract: In this technical note, we present the general formalism for paraxial capillary optics using transformation matrices. Furthermore, experimental results show that the required geometrical conditions are satisfied for some fused-silica capillaries employed in μ-HPLC and in CE

High speed method for predicting radio-wave propagation

Abstract: Radio-wave propagation is rapidly predicted by a hybrid computational method that uses Ray Optics techniques to calculate radio field strength above a limiting radio-wave ray, Parabolic Equation techniques to calculate radio field strength below the limiting radio-wave ray and below a predetermined altitude, and a newly created Extended Optics method to compute radio field strength in an area of the atmosphere below the limiting radio-wave ray and above the predetermined altitude. Rays in the extended optics area are initialized from the elevation angle that rays traced through the parabolic equation area make with the predetermined altitude. Where reflected, direct or origin created rays do not exist within the parabolic equation area, the elevation angle for the ray that does exist at the furthest range (optical limit) is used to initiate ray tracing in the extended optics area for ranges beyond the optical limit. Where the refractive index varies along the predetermined altitude, adjustments to the elevation angle used to initialize rays within the extended optics area are made. Propagation factors, determined through the Parabolic Equation method, are assigned to the rays traced through the extended optics area based upon the propagation factors in existence at the predetermined altitude. Field strength or propagation loss within the extended optics area is calculated from an interpolation between propagation factor values assigned to the extended optics rays.

Scalar optical beams with helical symmetry.

Abstract: Using a nonorthogonal helical coordinate system, we obtain exact free-space solutions for both the Helmholtz and paraxial Helmholtz equations. At optical frequencies the helical Helmholtz solutions can be interpreted as helical beams characterized by a constant pitch angle and beam radius. These solutions are shown to be generalizations of the family of scalar nondiffracting beams known as Bessel beams. They are similar to Bessel beams in some ways, such as an invariant intensity distribution profile in any plane normal to their axis of propagation, but have a nonconstant order

Image aberrations of poloidal toroid electrostatic analyzers

Abstract: The transfer matrix of poloidal electrostatic sector fields is determined by tracing ion trajectories through the calculated potential distribution To first order this is achieved by numerical ray tracing calculations and to higher order by tracing the expansion coefficients of a paraxial trajectory

Three-dimensional Coherent Transfer Function in Reflection-mode Confocal Microscopy Using Annular Lenses

Abstract: The three-dimensional (3D) coherent transfer function (CTF) in a reflection-mode confocal scanning microscope using two equal annular lenses has been derived analytically under the paraxial approximation. This analytic 3D CTF is then generalized to a confocal system using optical fibres as an illumination source and a signal collector. Various 3D- and two-dimensional (2D) numerical plots are presented to reveal the dependence of the 3D CTF on the size of the central obstruction and on the fibre spot size. As expected, using annular lenses may result in improved transverse resolution but poorer axial resolution compared with those for circular lenses, and using optical fibres of finite cross-section may result in degraded resolution in both axial and transverse directions. The relationship of the 3D CTF to the 2D in-focus CTF is discussed.

Seismic prospecting model determn. - gives optimum underground seismic wave propagation velocity distribution model

Abstract: Determn. of an opt. distribution model for underground seismic wave propagation velocities is carried out by (a) emitting seismic waves from a source placed successively at two or more shot points (14,14') on the ground surface; (b) recording, for each source position, the trace of the seismic waves reflected by the interface between subjacent formations by means of receivers (16) located at several sites on the ground surface; (c) devising an initial velocity distribution model from the recorded traces; (d) displaying the shot points (14,14') by a paraxial equation using the model to create a seismic image of the earth body interior; (e) modelling each shot point with the paraxial equation by downwardly propagating a wave resulting from a surface source; (f) re-modelling the seismic data of each slot point with the paraxial equation by extrapolation of a wave towards the surface, initiated by interaction of the source wave with the seismic image; (g) comparing the recorded seismic data (56) of each shot point with the re-modelled data (58) of each shot point and, if the data do not coincide, calculating the gradient of the `cost' function (mismatch) w.r.t. the initial velocity model; (h) modifying the velocity model with the gradient to reduce the difference between the observed and re-modelled seismic data; and (i) repeating these operations until the gradient is zero or sufficiently small to satisfy the requirements of inversion, which defines the optimal velocity model. ADVANTAGE - The `cost' function is not dependent on imaged reflections, so that the method is more sensitive to low frequency components of the velocity model, and corresponds better to seismic inversion.

A review of Fresnel zone plate moire patterns obtained by translations

Abstract: We present equations describing the basic grids that when superposed with themselves and then mutually translated create moire patterns in the form of spherical, equilateral hyperbolic, and linear zone plates. The family of basic grids described forms a complete set of solutions. Moire zone plates can serve as imaging elements with very long, variable focal lengths and have an application in alignment in the three-point technique. The derived solutions are then checked for cases of undesired residual translations and rotations. Evaluations of the focus based on paraxial ray tracing equations are also performed.

Three‐dimensional time‐domain paraxial approximations for ocean acoustic wave propagation

Abstract: One‐way narrow‐ and wide‐angle three‐dimensional time‐domain paraxial approximations to the wave equation are developed to model acoustic propagation. The approximate equations are designed to be appropriate for ocean applications including pulse propagation with volume attenuation and variable density. First, differential equations for acoustic pressure are obtained from a thermodynamic model, which incorporates attenuation due to the presence of acoustically absorbing chemical species. The method of multiple scales is then used to generate partial differential equations with paraxial characteristics. Comparisons are made to corresponding previous results. Results from an operator formalism using Pade approximants are subsequently compared with the corresponding multiscaling results. Appropriate boundary, initial, and interface conditions are described for the model equations. Stability of the three‐dimensional narrow‐angle approximation is demonstrated by means of an energy integral. Analytical expressions for pulse‐type solutions to a special case of the narrow‐angle equation are obtained. Comparisons with exact solutions to the full‐wave equation solution demonstrate the validity of the model when certain asymptotic constraints are observed.

Graphical depiction of the electromagnetic fields of Hermite-Gaussian modes

Abstract: The theory of Hermite-Gaussian beam waveguide modes is briefly reviewed, leading to expressions for the paraxial vector electric and magnetic fields for wave modes of arbitrary order. The variation of field amplitude in the transverse xy-plane is depicted by three-dimensional graphs, and constant-amplitude contour plots are derived. Also, the electric and magnetic field lines in longitudinal xz- and yz-planes are plotted for transverse modal orders ranging from (0,0) to (2,2).

A note on the derivation of paraxial equation in nonhomogeneous media

Abstract: The paraxial approximation of the two-dimensional and three-dimensional wave equations in nonhomogeneous media is discussed. It is shown that the preferred direction of propagation is along the bicharacteristics of the wave equation. The paraxial equation is obtained by assuming that in this direction, the variation of the propagating wave is slow enough to make negligible its second derivative. The best way to get bicharacteristics is to consider them as the null geodesics of some Riemannian space.

Path-integral analysis of an arbitrarily tapered, multimode, graded-index waveguide; the inverse-square-law and parabolic tapers

Abstract: The method of path integration is used to study tapered, graded-index waveguides in the context of paraxial, scalar wave optics. Closed form analytic results are obtained for the propagator, or Green's function, and coupling efficiencies of such structures. The results of this general theory are applied to tapers of parabolic and inverse-square-law shapes to derive closed form expressions for the lowest-order mode-coupling efficiency of these tapers. The results are compared with those obtained for the linear taper (discussed in a previous paper) in order to establish the suitability of each taper geometry for use in practical optical components. The inverse-square-law taper is found to be the least suitable one for single-mode devices.

Cone emission from laser-pumped two-level atoms. I. Quantum theory of resonant light propagation

Abstract: This is the first of a series of papers on the theory of cone emission from a laser-pumped two-level atomic medium. Starting from microscopic quantum theory for both the active medium and the field, equations for the slowly varying macroscopic quantities are derived. In the steady-state limit, which experimentally corresponds to cw excitation, we find that the sources for the paraxial fields of interest have a long coherence length despite the random nature of spontaneous emission

Analytical expressions for the paraxial parameters of a single lens with a spherical distribution of refractive index

Abstract: The concept of isoindicial surfaces and position variables is used to develop analytical expressions for the effective focal length and back focal length of a single gradient lens with a spherical distribution of refractive index, where the square of the refractive index is a quadratic function of the distance from the center of symmetry.