Paraxial Theory Research Articles

Propagation of high power electromagnetic beam in relativistic magnetoplasma: Higher order paraxial ray theory

This paper presents an analysis of self-consistent, steady-state, theoretical model, which explains the ring formation in a Gaussian electromagnetic beam propagating in a magnetoplasma, characterized by relativistic nonlinearity. Higher order terms (up to r4) in the expansion of the dielectric function and the eikonal have been taken into account. The condition for the formation of a dark and bright ring derived earlier by Misra and Mishra [J. Plasma Phys. 75, 769 (2009)] has been used to study focusing/defocusing of the beam. It is seen that inclusion of higher order terms does significantly affect the dependence of the beam width on the distance of propagation. Further, the effect of the magnetic field and the nature of nonlinearity on the ring formation and self-focusing of the beam have been explored.

Paraxial theory of electromagnetic waves in plane inhomogeneous media

The problem of electromagnetic waves propagating in inhomogeneous media is formulated within the paraxial approximation. The analysis is restricted to a medium with a plane and smooth inhomogeneity. The general form of a Gaussian beam is obtained in terms of the permittivity and permeability of the medium. Particular attention is paid to the case of internal reflection, where a short-wavelength approximation breaks down; a possible treatment of the problem is proposed. A simple model is worked out to illustrate the formalism.

Propagation regimes of intense circularly polarized laser beam in magnetoactive plasma

This paper presents an analytical and numerical investigation of an intense circularly polarized wave propagating along the static magnetic field parallel to oscillating magnetic field in magnetoactive plasma. In the relativistic regime such a magnetic field is created by pulse itself. The authors have studied different regimes of propagation with relativistic electron mass effect for magnetized plasma. An appropriate expression for dielectric tensor in relativistic magnetoactive plasma has been evaluated under paraxial theory. Two modes of propagation as extraordinary and ordinary exist; because of the relativistic effect, ultra-strong magnetic fields are generated which significantly influence the propagation of laser beam in plasma. The nature of propagation is characterized through the critical-divider curves in the normalized beam width with power plane For given values of normalized density ( ω p / ω) and magnetic field ( ω c / ω) the regions are namely steady divergence (SD), oscillatory divergence (OD) and self-focusing (SF). Numerical computations are performed for typical parameters of relativistic laser–plasma interaction: magnetic field B = 10–100 MG; intensity I = 10 16 to 10 20 W/cm 2; laser frequency ω = 1.1 × 10 15 s −1; cyclotron frequency ω c = 1.7 × 10 13 s −1; electron density n e = 2.18 × 10 20 cm −3. From the calculations, we confirm that a circularly polarized wave can propagate in different regimes for both the modes, and explicitly indicating enhancement in wave propagation, beam focusing/self-guiding and penetration of E-mode in presence of magnetic field.

Bistable dark solitons of a cubic-quintic Helmholtz equation

We provide a report on exact analytical bistable dark spatial solitons of a nonlinear Helmholtz equation with a cubic-quintic refractive-index model. Our analysis begins with an investigation of the modulational instability characteristics of Helmholtz plane waves. We then derive a dark soliton by mapping the desired asymptotic form onto a uniform background field and obtain a more general solution by deploying rotational invariance laws in the laboratory frame. The geometry of the new soliton is explored in detail, and a range of new physical predictions is uncovered. Particular attention is paid to the unified phenomena of arbitrary-angle off-axis propagation and nondegenerate bistability. Crucially, the corresponding solution of paraxial theory emerges in a simultaneous multiple limit. We conclude with a set of computer simulations that examine the role of Helmholtz dark solitons as robust attractors.

Free-space evolution of focused Gaussian beams transformed by conical diffraction in a biaxial crystal

The intensity and polarization pattern of circularly and linearly polarized Gaussian beams transformed by the effect of conical diffraction in a biaxial crystal has been studied experimentally, and compared with the predictions of paraxial theory. Some general regularities in the relative intensity and geometry of beam structures were analyzed. The formation of Mathieu beam profiles from linearly polarized input beams was demonstrated.

Nonlinear propagation of intense electromagnetic beams with plasma density ramp functions

The relativistic oscillation of the mass of the electrons and also the relativistic electron ponderomotive force are shown to have a effect on the nonlinear propagation of intense electromagnetic waves. Based on WKB and paraxial ray theory, the steady state nonlinear refraction of relativistically intense, circularly polarized, Gaussian electromagnetic beams in an inhomogeneous plasma is studied. The nature of propagation of the beam depends on the power, width of the beam and 'Ω', the ratio of plasma and wave-frequency. It can be concluded from the calculations that electromagnetic waves can propagate in different regimes. The regions are steady divergence, oscillatory divergence and self-focusing. Numerical estimates are made for typical values of relativistic laser-plasma interaction process with electron density varying between 1017 -1020 particles per cm 3 and radiation intensities in the range 1016 -1020 W/cm 2 for different plasma density ramp functions.

A strip relativistic electron beam in a narrow cavity

In the scope of the paraxial theory, propagation of a beam in a narrow curved cavity is studied for the general case of nonmonoenergetic injection and a nonuniform current density distribution over the cross section. It is shown that, in a simpler problem on a beam in crossed fields, which was studied earlier by numerical methods, there are no reasons for asymmetric pulsations of the flow boundaries if the external forces on the rectilinear axis of the beam are compensated. Conditions preventing the formation of focuses and realization of aperiodic modes are formulated. It is noted that the description of the force action on the beam by external fields has different characters in the hydrodynamic and numerical models.

Nanostructured GRIN microlenses for Gaussian beam focusing

Gaussian beam propagation through a novel high refractive index contrast nanostructured Gradient-Index (GRIN) microlens with an arbitrary refractive index distribution is investigated. The concept of the nanostructured GRIN microlenses is based on the effective medium theory and we show numerically, using the FDTD method, that nanostructured GRIN microlenses can focus Gaussian beams in an identical manner to conventional GRIN microlenses. Nanostructured GRIN microlenses have a reduced thickness by almost two orders of magnitude compared to conventional GRIN lenses. We also investigate the accuracy of the standard scalar and paraxial GRIN theory in the case of a very high contrast nanostructured GRIN microlens. The paraxial GRIN lens theory is shown only to give an approximate description of Gaussian beam propagation in such a high contrast nanostructured GRIN microlens.

Propagation modes and regimes of intense laser beam in magnetized plasma

This paper presents an analytical and numerical investigation of a circularly polarized beam propagating along the static magnetic field parallel to oscillating magnetic field in plasma at relativistic intensities. In the high intensity regime, such a magnetic field is created by the pulse itself. The authors have identified three regimes of propagation taking into account the relativistic mass correction. Based on WKB and paraxial ray theory, an appropriate expression for a dielectric tensor has been evaluated in the presence of an externally applied magnetic field. The natural electromagnetic modes are circularly polarized. Consequently, extraordinary and ordinary modes propagate, which are significantly affected due to the relativistic mechanism. The regimes are characterized by dimensionless power and beamwidth, characterizing the nature of propagation as steady divergence, oscillatory divergence and self-focusing. Numerical computations are presented and discussed for typical parameters of laser plasma interaction; defined through critical parameters, namely cyclotron-to-beam frequency (Ω c ), plasma-to-beam frequency (Ω p ) and beam power for arbitrary large intensities.

Effect of vectorial nature on spectral anomalies of ultrashort pulsed beams from a hard-edged aperture

Based on the vectorial Rayleigh–Sommerfeld diffraction integrals, the analytical expression for the spectral intensity of a vectorial nonparaxial ultrashort pulsed Gaussian beam diffracted at a rectangular hard-aperture is derived. The results of the far-field and paraxial cases can be regarded as special cases of the general expression. Effect of vectorial nature on spectral anomalies of ultrashort pulsed beams passing through a hard-edged aperture is analyzed in detail. It is shown that the spectral switch near the phase singularity, which is predicated under the condition of the scalar paraxial theory, disappears with the increasing effect of the vectorial and nonparaxial nature.

Effect of self-generated axial magnetic field and on propagation of intense laser radiation in plasmas

The present paper deals with an analytical study of a self-generated axial magnetic field (SGAMF) through the inverse Faraday effect (IFE) and its influence on the propagation of circularly polarized light wave for relativistic intensities. As a first step, the non-linear dielectric constant incorporating a magnetic field in the relativistic factor within the framework of WKB (for Wentzel, Kramers, and Brillouin) and a paraxial ray theory is formulated. It is noticed that for intensities (>1018 W cm−2), circularly polarized radiation can propagate in electron plasma whose density is greater than the critical density as well as a strong flow of relativistic electrons, axially co-moving with the pulse rise. The above generates a magnetic field up to 100 MG and strongly influences the light propagation. Two modes of propagation exist, namely, extraordinary and ordinary, and critical power for focusing is different for the two modes. The non-linear dielectric tensor, propagation equation, and the self-trapped radius are evaluated incorporating an induced magnetic field. The focusing conditions strongly depend on the power of the beam, strength of the magnetic field as well as on the density of the medium. Numerical calculations are made for a typical set of relativistic laser plasma interaction processes.

Transformation properties of an indefinite media slab for paraxial beams

On the basis of paraxial beam propagation theory in indefinite media, the effective refraction index for both of the two types of polarized electromagnetic wave, TE and TM waves, are introduced. According to the definition of effective refraction index, we obtain the conditions that paraxial beams pass through the indefinite media. We propose to employ a slab of indefinite media to construct a polarizer or polarizing beam splitter by choosing appropriate anisotropic parameters. Under certain conditions, the indefinite media slab exhibits polarization-selective focusing effect.

Effect of laser self-defocusing on third harmonic generation in a tunnel ionizing gas

A paraxial theory of third harmonic generation by a finite spot size laser in a tunnel ionizing gas is developed. The laser, with intensity close to tunnel ionization threshold, creates plasma whose density rises in a stepwise manner in every half wave period, giving a strong second harmonic component in electron density. The density beats with the oscillatory velocity at the fundamental to produce a nonlinear current, driving the third harmonic. The region of harmonic generation is localized near the laser axis. Further, the strong radial nonuniformity, created in electron density by the ionization process, causes defocusing of the laser, limiting the efficiency of harmonic generation.

Time-to-depth conversion and seismic velocity estimation using time-migration velocity

The objective was to build an efficient algorithm (1) to estimate seismic velocity from time-migration velocity, and (2) to convert time-migrated images to depth. We established theoretical relations between the time-migration velocity and seismic velocity in two and three dimensions using paraxial ray-tracing theory. The relation in two dimensions implies that the conventional Dix velocity is the ratio of the interval seismic velocity and the geometric spreading of image rays. We formulated an inverse problem of finding seismic velocity from the Dix velocity and developed a numerical procedure for solving it. The procedure consists of two steps: (1) computation of the geometric spreading of image rays and the true seismic velocity in time-domain coordinates from the Dix velocity; (2) conversion of the true seismic velocity from the time domain to the depth domain and computation of the transition matrices from time-domain coordinates todepth. For step 1, we derived a partial differential equation (PDE) in two and three dimensions relating the Dix velocity and the geometric spreading of image rays to be found. This is a nonlinear elliptic PDE. The physical setting allows us to pose a Cauchy problem for it. This problem is ill posed, but we can solve it numerically in two ways on the required interval of time, if it is sufficiently short. One way is a finite-difference scheme inspired by the Lax-Friedrichs method. The second way is a spectral Chebyshev method. For step 2, we developed an efficient Dijkstra-like solver motivated by Sethian’s fast marching method. We tested numerical procedures on a synthetic data example and applied them to a field data example. We demonstrated that the algorithms produce a significantly more accurate estimate of seismic velocity than the conventional Dix inversion. This velocity estimate can be used as a reasonable first guess in building velocity models for depth imaging.

An automated submicron beam profiler for characterization of high numerical aperture optics

An automated, interferometrically referenced scanning knife-edge beam profiler with submicron resolution is demonstrated by directly measuring the focusing properties of three aspheric lenses with numerical aperture (NA) between 0.53 and 0.68, with spatial resolution of 0.02 microm. The results obtained for two of the three lenses tested were in agreement with paraxial Gaussian beam theory. It was also found that the highest NA aspheric lens, which was designed for 830 nm, was not diffraction limited at 633 nm. This process was automated using motorized translation stages and provides a direct method for testing the design specifications of high numerical aperture optics.

Lens theory of non-paraxial rays for electron gun characterization

It has been attempted to characterize the cathode rays inside electron guns by utilizing the optical concepts familiar in the paraxial lens theory. We have proposed the Canonical Mapping Transformation (CMT) theory to obtain the gun optical parameters. The method is based on the Lagrange differential invariant theorem known in analytical mechanics. It has been found that major electron source properties, such as the crossover position, crossover size, and the angular current intensity, are all deducible from the four CMT optical parameters ( zco , f , Csg , and Ccg ), which in turn can be estimated by calculating the normal electron rays, whose emanation vectors on the cathode surface are in the surface normal direction. Since the normal electron rays can in many cases be regarded as paraxial, a scheme has been proposed to calculate the relevant optical parameters by a modification of the conventional paraxial trajectory calculation. It is shown that the normal electron ray of the CMT theory corresponds to (1, 0) principal trajectory in the paraxial method, g(z) . The conventional perturbation characteristic function integral method can be employed for the evaluation of the CMT aberration coefficients. Two realistic electron gun models (a single-crystal LaB6 cathode gun and a Schottky emitter source) were analyzed by use of the CMT optical parameters. Both the ray tracing of the normal electrons and the modified paraxial calculation method were employed for the analyses. It has been found that the guns with quite different nature in the source properties can well be described by the CMT optical parameters proposed. The paraxial calculations have been shown to produce accurate enough results and the authors hope their use would help electron optical column designers both in reducing the work load and in having clear physical images of their gun characteristics. © 2008 Elsevier B.V. All rights

Finite‐frequency tomography in a crustal environment: Application to the western part of the Gulf of Corinth

ABSTRACTIn this paper we investigate finite‐frequency effects in crustal tomography. We developed an inversion procedure based on an exact numerical computation of the sensitivity kernels. In this approach we compute the 3D travel‐time sensitivity kernels by using (1) graph theory and an additional bending to estimate accurately both rays and travel‐times between source/receiver and diffraction points and (2) paraxial ray theory to estimate the amplitude along theses rays. We invert both the velocity and the hypocentre parameters, using these so‐called banana‐doughnut kernels and the LSQR iterative solver. We compare the ray‐theoretical and the finite‐frequency tomography to image the intermediate structures beneath the Gulf of Corinth (Greece), which has long been recognized as the most active continental rifting zone in the Mediterranean region. Our dataset consists of 451 local events with 9233 P‐ first‐arrival times recorded in the western part of the Gulf (Aigion area) in the framework of the 3F‐Corinth European project. Previous tomographic images showed a complex velocity crustal model and a low‐dip surface that may accommodate the deformation. Accurate velocity models will help to better constrain the rifting process, which is still a subject of debate. The main results of this study show that finite‐frequency tomography improves crustal tomographic images by providing better resolved images of the 3D complicated velocity structure. Because the kernels spread the information over a volume, finite‐frequency tomography results in a sharpening of layer boundaries as we observed for the shallower part of the crust (down to 5 km depth) beneath the Gulf of Corinth.

Propagation of high power electromagnetic beams in overdense plasmas: Higher order paraxial theory

This article presents the paraxial theory of the propagation of an initially Gaussian electromagnetic beam in an inhomogeneous plasma with an overdense region; in contrast to earlier work on penetration in overdense plasma, higher order terms (up to r4) in the expansion of the dielectric function and the eikonal have been taken into account. Three types of nonlinearities, viz., collisional, ponderomotive, and relativistic, have been considered. As expected the higher order terms do not affect the critical curves, corresponding to initial propagation without convergence or divergence. It is seen that the inclusion of higher order terms does significantly affect the dependence of the beam width on the distance of propagation. Corresponding to the case of ponderomotive nonlinearity numerical results for the dependence of beam width parameter and the axial dielectric function on the distance of propagation have been presented for specific values of the initial beam width and axial irradiance and specific spatial dependence of the electron density in the absence of the beam. Both the situations, viz., formation of bright or dark rings in the transverse irradiation pattern, have been considered. From a parametric analysis the dependence of the maximum penetration (when the axial dielectric function tends to zero) on the axial irradiance and an inhomogeneity parameter has been graphically illustrated.

Paraxial imaging and walk-off effects with birefringent media in refracting surfaces, lenses, and slabs

The paraxial theory of spherical refracting surfaces, spherical lenses, and slabs with one birefringent medium is investigated analytically: using walk-off effects in the paraxial domain, a number of relations between objects and images are deduced, along with cardinal elements, in the case where the optic axis is parallel to the optical axis. This method naturally shows that in some cases first-order astigmatism appears. An argument based on the wavefront (and phase) transformation shows that any spherical birefringent thin lens is stigmatic in the paraxial domain, because the first-order astigmatisms due to the two surfaces of such a lens compensate each other. This is a priori not the case with thick birefringent lenses -- but two such cases are detailed.

Inverse problem in seismic imaging

AbstractWe address the problem of estimating sound speeds (seismic velocities) inside the earth which is necessary for obtaining seismic images in regular Cartesian coordinates. The main goals are to develop algorithms to convert time migration velocities to true seismic velocities, and to convert time‐migrated images to depth images in regular Cartesian coordinates.Our main results are three‐fold. First, we establish a theoretical relation between the seismic velocities and the time migration velocities using the paraxial ray tracing theory. Second, we formulate an appropriate inverse problem describing the relation between time migration velocities and depth velocities and show that this problem is mathematically ill‐posed, i.e., unstable to small perturbations. Third, we develop numerical algorithms to solve regularized versions of these equations which can be used to recover smoothed velocity variations. Our algorithms consist of efficient time‐to‐depth conversion algorithms based on Dijkstra‐like Fast Marching Methods, as well as level set and ray tracing algorithms for transforming Dix velocities into seismic velocities. Our algorithms are applied to both two‐dimensional and three‐dimensional problems and we test them on a collection of both synthetic examples and field data. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)