Paraxial Gaussian Beam Research Articles

Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics.

Paraxial Gaussian beams (GB's) are collimated wave objects that have found wide application in optical system analysis and design. A GB propagates in physical space according to well-established quasi-geometric-optical rules that can accommodate weakly inhomogeneous media as well as reflection from and transmission through curved interfaces and thin-lens configurations. We examine the GB concept from a broad perspective in the frequency domain (FD) and the short-pulse time domain (TD) and within as well as arbitrarily beyond the paraxial constraint. For the formal analysis, which is followed by physics-matched high-frequency asymptotics, we use a (space-time)-(wavenumber-frequency) phase-space format to discuss the exact complex-source-point method and the associated asymptotic beam tracking by means of complex rays, the TD pulsed-beam (PB) ultrawideband wave-packet counterpart of the FD GB, GB's and PB's as basis functions for representing arbitrary fields, GB and PB diffraction, and FD-TD radiation from extended continuous aperture distributions in which the GB and the PB bases, installed through windowed transforms, yield numerically compact physics-matched a priori localization in the plane-wave-based nonwindowed spectral representations.

Scalar field of nonparaxial Gaussian beams

A family of closed-form expressions for the scalar field of strongly focused Gaussian beams in oblate spheroidal coordinates is given. The solutions satisfy the wave equation and are free from singularities. The lowest-order solution in the far field closely matches the energy density produced by a sine-condition, high-aperture lens illuminated by a paraxial Gaussian beam. At the large waist limit the solution reduces to the paraxial Gaussian beam form. The solution is equivalent to the spherical wave of a combined complex point source and sink but has the advantage of being more directly interpretatable.

Nonparaxial gaussian beams: 1. Vector fields

Rigorous analytical solutions were obtained for the vector field of a propagating nonparaxial Gaussian beam in the vicinity of the beam focus. It is demonstrated that a nonparaxial beam of the lowest order has six modes. Upon the limiting transition kz0 ≫ 1, four of these modes exhibit degeneracy and convert into a paraxial Gaussian beam, while the other two modes form the azimuth-symmetric TE and TM fields. A characteristic feature of the nonparaxial modal beam is the absence of symmetry between (still mutually orthogonal) electric and magnetic fields. The deviation from symmetry results in the appearance of negative energy fluxes in the vicinity of the phase singularity manifested by Airy’s fringes, “light drops” (special states of the light field), and “singularity islands”. The obtained results are in good agreement with other relevant data published.

Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma

The beam tracing technique describes electromagnetic wave beam propagation in the short wavelength limit, including diffraction effects, not described by the usual geometric optics. It involves a set of ordinary differential equations: a central or reference ray is determined and around this the beam profile and the shape of the phase front are calculated. Here, these equations are obtained for a Gaussian wave beam. Analytic solutions in an inhomogeneous magnetized plasma are given for perpendicular propagation in a slab geometry. The beam is found to show a strong deformation of its cross section due to the anisotropy of the medium. The rotation of the cross section and that of the curvature of the phase front are discussed.

Geometry of the fringe field formed in the intersection of two Gaussian beams

A general expression is obtained for the exact computation of the fringe field in the intersection volume of two paraxial Gaussian beams for arbitrary beam waist positions and sizes. The expression is then simplified to allow easy fringe field computation while retaining good accuracy. By relating the simplified expression to the system parameters relevant to dual-beam laser Doppler velocimeters, simple design equations are obtained. These equations permit rapid evaluation of the fringe field variation along the major and minor axes of the intersection volume and clearly identify the system parameters controlling this variation.

Topological phase in Gaussian beam optics

The evolution of a complex beam width parameter g describing focusing or defocusing of a paraxial Gaussian beam is considered as the cause of an additional topological (Berry) phase of the electromagnetic field associated with the beam. It is pointed out that the well-known Gouy phase is a special case of such a phase that should arise in general in all symplectic systems.

Log-amplitude variance and wave structure function: a new perspective for Gaussian beams

Two naturally linked pairs of nondimensional parameters are identified such that either pair, together with wavelength and path length, completely specifies the diffractive propagation environment for a lowest-order paraxial Gaussian beam. Both parameter pairs are intuitive, and within the context of locally homogeneous and isotropic turbulence they reflect the long-recognized importance of the Fresnel zone size in the behavior of Rytov propagation statistics. These parameter pairs, called, respectively, the transmitter and receiver parameters, also provide a change in perspective in the analysis of optical turbulence effects on Gaussian beams by unifying a number of behavioral traits previously observed or predicted, and they create an environment in which the determination of limiting interrelationships between beam forms is especially simple. The fundamental nature of the parameter pairs becomes apparent in the derived analytical expressions for the log-amplitude variance and the wave structure function. These expressions verify general optical turbulence-related characteristics predicted for Gaussian beams, provide additional insights into beam-wave behavior, and are convenient tools for beam-wave analysis.

Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam

Fifth-order corrected expressions for the electromagnetic field components of a monochromatic fundamental Gaussian beam (i.e., a focused TEM00 mode laser beam) propagating within a homogeneous dielectric media are derived and presented. Calculations of relative error indicate that the fifth-order Gaussian beam description provides a significantly improved solution to Maxwell’s equations in comparison with commonly used paraxial (zeroth-order) and first-order Gaussian beam descriptions.

Modal, ray, and beam techniques for analyzing the EM scattering by open-ended waveguide cavities

The problem of high-frequency electromagnetic scattering by open-ended waveguide cavities with an interior termination is analyzed via three different approaches. When cavities can be adequately modeled by joining together piecewise separable waveguide sections, a hybrid combination of asymptotic high-frequency and modal techniques is employed. In the case of more arbitrarily shaped waveguide cavities for which modes cannot even be defined in the conventional sense, the geometrical optics ray approach proves to be highly useful. However, at sufficiently high frequencies, both of these approaches tend to become inefficient; hence, a paraxial Gaussian beam technique, which retains much of the simplicity of the ray approximation but is potentially more efficient, is investigated. Typical numerical results based on the different approaches are discussed. >

Amplitude and phase data inversions for phase velocity anomalies in the Pacific Ocean basin

Summary. Rayleigh wave phase velocities at periods 30-80 s in the Pacific Ocean are calculated by inverting phase and amplitude anomaly data using the paraxial ray approximation and the Gaussian beam method. The region is divided into 5x 5 blocks, and approximately 200 source-receiver pairs from 18 well-studied events around the Pacific Ocean are used. First, we assume phase anomalies for the lithospheric age-dependent model. Next, conventional phase data inversions are conducted assuming great circle paths so that the phase discrepancies are reduced to less than T. This procedure is essential for later inversions using amplitude data. We then determine the residuals of both amplitude and phase terms by calculating ray-synthetic seismograms. Using the Born approximation for a 2-D wave equation, a nonlinear iterative inversion for phase velocities is performed with both residuals. FrBchet derivatives for the inversion consist primarily of two wavefields: (1) the wavefield at the model point from the source, and (2) the Green's function from the model point to the receiver. These wavefields are also calculated by the paraxial ray approximation and Gaussian beam methods. In the inverse formulations, the simple use of the conventional Backus-Gilbert approach yields undesirable results in the non-linear iterative case and an extra term is necessary to control the model perturbations in order to minimize departures from the a priori model. The use of this additional term guarantees that we are able to obtain a fairly reliable phase velocity model even in the present non-linear problem. In most cases residual variances are significantly reduced after two or three iterations as far as the starting model is fairly correct. Compared with the phase data inversions, this inverse scheme gives more reliable resolution and most of the inverted features in phase velocities are significantly larger than the uncertainty level while some features obtained by the great circle phase data

Propagating pulsed beam solutions by complex source parameter substitution

The complex source point method is extended to the time domain by considering the field due to an impulsive source point with complex space coordinates and a complex initiation time. The resulting solution, which is a particular analytic continuation of the free space time-dependent Green's function, describes a propagating pulsed beam that has a moving peak along the beam axis. Near the pulse maximum in the paraxial region, this new field type is the Fourier transform into the time domain of the time-harmonic paraxial Gaussian beam field but the method also accommodates, in closed form, observation points far from the beam axis and observation times long before and after the peak has passed. By corresponding analytic continuation of available space-time Green's functions for various propagation and diffraction environments, one may generate directly the response due to the pulsed beam incident on the environment.

Complex argument Hermite–Gaussian and Laguerre–Gaussian beams

Hermite–Gaussian and Laguerre–Gaussian beams with complex arguments of the type introduced by Siegman [ J. Opt. Soc. Am.63, 1093 ( 1973)] are shown to arise naturally in correction terms of a perturbation expansion whose leading term is the fundamental paraxial Gaussian beam. Additionally, they can all be expressed as derivatives of the fundamental Gaussian beam and as paraxial limits of multipole complex-source point solutions of the reduced-wave equation.

Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams

Summary. The Gaussian beam method has recently been introduced into synthetic seismology to overcome shortcomings of the ray method, especially in transition regions due to focusing or diffraction where ray theory fails. One proceeds by discretizing the initial data as a superposition of paraxial Gaussian beams, each of which is then traced through the seismic environment. Since Gaussian beam fields do not diverge in ray transition regions, they are ‘uniformly regular’ although the quality of this regularity depends on the beam parameters and on the ‘numerical distance’ which defines the extent of the transitional domain. However, when Gaussian beam patches are used to simulate non-Gaussian initial data, there arise ambiguities due to choice of patch size and location, beam width, etc., which are at the user's disposal. The effects of this arbitrariness have customarily been explored by trial and error numerical experiment but no quantitative recommendations have emerged as yet. As a step toward a priori predictive capability, it is proposed here to perform a systematic study on analytically tractable prototype models of how the parameters and location of a single beam affect the quality of the observed seismic field, especially in ray transition regions. The conversion of ordinary ray fields into beam fields in canonical configurations can be accomplished conveniently by displacing a real source point into a complex coordinate space. Thus, the desired beam solutions can be obtained directly from available ray, and even paraxial ray, fields. Complex ray theory and its implications are reviewed here, with an emphasis on improvements of beam tracking schemes employed at present.

From Gaussian beam to complex-source-point spherical wave

It is shown that the paraxial Gaussian beam becomes the complex-source-point spherical wave when all-order corrections are made according to the method of Lax, Louisell, and McKnight. Apparent contradictions between previously published first-order corrections are also discussed.

Propagation of Gaussian beams in piano-curved optical waveguides with a parabolic refractive index profile

An analysis is made of the propagation of paraxial Gaussian beams in optical waveguides arbitrarily curved in one plane and having a parabolic refractive index profile. The propagation function is derived for this case and by analyzing this function, it is shown that the center of the beam moves along a specific trajectory. Moreover, the Gaussian intensity distribution with a periodically varying parameter characterizing the beam width is conserved along the trajectory in the transverse cross section of the beam.

Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking

Inhomogeneous waves behave locally like A(r) exp[ikS(r)], where A and S are spatially dependent complex amplitude and phase functions, and k is the (large) free-space wavenumber. A previously developed asymptotic theory for high-frequency propagation and scattering of such waves is here applied to the propagation and scattering of paraxial Gaussian beams. Attention is given to Gaussian beams in free space, to beams in a lens-like medium with parabolic variation of the refractive index, and to beam reflection by a cylindrical obstacle. In the latter instance, the obstacle size may be comparable to the incident beamwidth, thereby introducing substantial distortion into the reflected beam. The results obtained from the asymptotic theory are verified by comparison with rigorously derived solutions, thereby confirming the validity of the theory, which can also be applied to more general medium and obstacle configurations.