Longitudinal field components for laser beams in vacuum

Revisiting the Design Strategies for Metasurfaces: Fundamental Physics, Optimization, and Beyond

Over the last two decades, the capabilities of metasurfaces in light modulation with subwavelength thickness have been proven, and metasurfaces are expected to miniaturize conventional optical components and add various functionalities. Herein, various metasurface design strategies are reviewed thoroughly. First, the scalar diffraction theory is revisited to provide the basic principle of light propagation. Then, widely used design methods based on the unit-cell approach are discussed. The methods include a set of simplified steps, including the phase-map retrieval and meta-atom unit-cell design. Then, recently emerging metasurfaces that may not be accurately designed using unit-cell approach are introduced. Unconventional metasurfaces are examined where the conventional design methods fail and finally potential design methods for such metasurfaces are discussed.

Design of a Metasurface with Long Depth of Focus Using Superoscillation

Longitudinal optical field modulation is very important for applications such as optical imaging, spectroscopy, and optical manipulation. It can achieve high-resolution imaging or manipulation of the target object, but it is also limited by its depth of focus. The depth of focus determines whether the target object can be clearly imaged or manipulated at different distances, so extending the depth of focus can improve the adaptability and flexibility of the system. However, how to extend the depth of focus is still a significant challenge. In this paper, we use a super-oscillation phase modulation optimization method to design a polarization-independent metalens with extended focal depth, taking the axial focal depth length as the optimization objective. The optimized metalens has a focal depth of 13.07 μm (about 22.3 λ), and in the whole focal depth range, the transverse full width at half maximum values are close to the Rayleigh diffraction limit, and the focusing efficiency is above 10%. The results of this paper provide a new idea for the design of a metalens with a long focal depth and may have application value in imaging, lithography, and detection.

Inelastic electron scattering at a single-beam structured light wave

In free space, electrons undergo inelastic scattering in the presence of ponderomotive potentials generated by light pulses and standing light waves. The resulting modulated electron energy spectrum can exhibit the formation of discrete energy sidebands when multiple light beams are employed. Here, we demonstrate the inelastic scattering of slow-electron wavepackets at a propagating Hermite-Gaussian light beam. The pulsed Hermite-Gaussian beam thus forms a ponderomotive potential for the electron with sufficient momentum components, leading to the inelastic scattering and subsequent formation of discrete energy sidebands. We show that the resulting energy-gain spectra after the interaction are strongly influenced by the self-interference of the electrons in this ponderomotive potential. This effect is observable across various wavelengths, and the energy modulation can be controlled by varying the electron velocity and light intensity. By utilizing the vast landscape of structured electromagnetic fields, this effect introduces an additional platform for manipulating electron wavepackets.

Generation of soft annular beams with high uniformity, low ring width increment, and a smooth edge

In this paper, soft-edge toroidal amplitude filter (STAF) and soft-edge toroidal complex amplitude filter (STCAF) are designed according to the principle that soft-edge structures can eliminate diffraction. Based on the Mach-Zehnder interference principle, a double optical path compound interference modulation method that can generate soft annular beams is proposed by using STAF and STCAF. The 1/e² radius and peak-to-average ratio (PAR) were used to evaluate the ring width and uniformity of the annular beam. Compared with the annular beams generated by STAF and hard-edge toroidal amplitude filter (HTAF), it can be found that the soft annular beam generated by this proposed method has the advantages of high uniformity, small ring width increment, and smooth edges. By analyzing the influence of the number and height of the sawtooth structures on the annular beam propagation performance, the relationship between the PAR and the structure parameters of the STAF was established. Moreover, three kinds of toroidal filters were designed by lithography processing, and an experimental system was built to generate the soft annular beam. In the experiment, the average value of the ring width increment of the soft annular beam is 0.0125, the PAR is less than 1.5, and the root mean square error of the PAR curve is 0.0865, which indicates that the soft annular beam maintains high uniformity during propagation.

Optimization-free customization of optical tightly focused fields: uniform needles and hotspot chains

An optimization-free method based on an inverse problem of nonlinear equations is employed to design the binary phase diffraction optical element (BPDOE) that could modulate the incident light of a high-numerical-aperture (NA) objective lens so that the axisymmetric focal fields can be customized on demand. For example, a 43λ-long optical longitudinally polarized needle with its lateral size beyond diffraction limit is reported by using a 27-belt BPDOE, where the cost evaluated by the ratio of the belt number of BPDOE to the length of needle is record small compared with other optimization algorithms. Moreover, another longitudinal field with multiple hotspots along the propagation direction of light is also achieved with a 10-belt BPDOE. These achieved focal fields are verified doubly by using a finite-difference time-domain (FDTD) method, indicating the validity of Richards-Wolf vector diffraction theory. This optimization-free approach makes the design of BPDOEs with numerous belts viable to generate the expected focal fields, which might benefit various applications such as optical trapping, super-resolution imaging, and lithography.

On the Dressed Photon Constant and Its Implication for a Novel Perspective on Cosmology

As an important follow-up report on the latest study of the first author (H.S.) on an off-shell quantum field causing a dressed photon and dark energy, we further discuss a couple of intriguing subjects based on the new notion of simultaneous conformal symmetry breaking. One is the dressed photon constant. If we use it, in addition to h¯ and c, as the third component of natural units, it is defined as the geometric mean of the smallest and the largest lengths: Planck length and that relating to the cosmological constant. Interestingly, this length (≈50 nanometers) seems to give a rough measure of the Heisenberg cut for electromagnetic phenomena. The other is a new perspective on cosmology that combines two original notions, i.e., twin universes and conformal cyclic cosmology, proposed, respectively, by Petit and Penrose, into one novel picture where universes expand self-similarly. We show the possibility that twin universes having a dual structure of (matter with (dark energy and matter)) vs. corresponding anti-entities, separated by an event horizon embedded in the geometric structure of de Sitter space, undergo endless cyclic processes of birth and death, as in the case of the pair creation and annihilation of elementary particles through the intervention of a conformal light field.

Multipole decomposition for interactions between structured optical fields and meta-atoms

Interactions between structured optical fields (SOFs) and meta-atoms have been intensively studied, and stimulated by recent advancements on the generation of SOFs and on the synthesis of exotic meta-atoms. Multipole expansion is an efficient and accurate theoretical framework for studying such problems. In this work, explicit expressions of SOFs and their beam-shape coefficients are provided, and their properties are also briefly discussed; the considered SOFs include Laguerre-Gaussian (LG) beams, tightly-focused LG beams, Bessel beams, and cylindrical vector beams. Using the multipole expansion, selective excitations of multipolar resonances of a sphere is discussed. In addition, angular momentum dichroisms of a chiral sphere and an anisotropically chiral meta-atom are calculated to demonstrate selective excitation of multipoles with the desired order, parity, and orientation using engineered SOFs with angular momentum.

Aerosol scattering of vortex beams transmission in hazy atmosphere

Mie theory is widely used for the simulation and characterization of optical interaction with scattering media, such atmospheric pollutants. The complex refractive index of particle plays an important role in determining the scattering and absorption of light. Complex optical fields, such as vortex beams, will interact with scattering particulates differently to plane wave or Gaussian optical fields. By considering the three typical aerosol particles compositions that lead to haze in the atmosphere, distinctive scattering dynamic were identified for vortex beams as compared to Gaussian beams. Using parameters similar to real world atmospheric conditions, a new aerosol particle model is proposed to efficiently and concisely describe the aerosol scattering. Numerical simulations indicate unique signatures in the scattering dynamics of the vortex beams that can indicate particles composition and also suggest that potentially there is higher optical transmission of vortex beams propagating in certain hazy environments.

Off-Shell Quantum Fields to Connect Dressed Photons with Cosmology

The anomalous nanoscale electromagnetic field arising from light–matter interactions in a nanometric space is called a dressed photon. While the generic technology realized by utilizing dressed photons has demolished the conventional wisdom of optics, for example, the unexpectedly high-power light emission from indirect-transition type semiconductors, dressed photons are still considered to be too elusive to justify because conventional optical theory has never explained the mechanism causing them. The situation seems to be quite similar to that of the dark energy/matter issue in cosmology. Regarding these riddles in different disciplines, we find a common important clue for their resolution in the form of the relevance of space-like momentum support, without which quantum fields cannot interact with each other according to a mathematical result of axiomatic quantum field theory. Here, we show that a dressed photon, as well as dark energy, can be explained in terms of newly identified space-like momenta of the electromagnetic field and dark matter can be explained as the off-shell energy of the Weyl tensor field.

Vector fields in a tight laser focus: comparison of models

We assess several widely used vector models of a Gaussian laser beam in the context of more accurate vector diffraction integration. For the analysis, we present a streamlined derivation of the vector fields of a uniformly polarized beam reflected from an ideal parabolic mirror, both inside and outside of the resulting focus. This exact solution to Maxwell's equations, first developed in 1920 by V. S. Ignatovsky, is highly relevant to high-intensity laser experiments since the boundary conditions at a focusing optic dictate the form of the focus in a manner analogous to a physical experiment. In contrast, many models simply assume a field profile near the focus and develop the surrounding vector fields consistent with Maxwell's equations. In comparing the Ignatovsky result with popular closed-form analytic vector models of a Gaussian beam, we find that the relatively simple model developed by Erikson and Singh in 1994 provides good agreement in the paraxial limit. Models involving a Lax expansion introduce a divergences outside of the focus while providing little if any improvement in the focal region. Extremely tight focusing produces a somewhat complicated structure in the focus, and requires the Ignatovsky model for accurate representation.

Creation of a longitudinally polarized photonic nanojet via an engineered microsphere

Dielectric microspheres exhibit the ability to focus an incident beam to a subwavelength spot with strong localized field intensity. In this Letter, a high beam quality of a longitudinally polarized electromagnetic component is created by decorating the surface of the microsphere with engineered structures. By changing the design of the engineered microspheres, the relative contribution of the longitudinal and radial components of a radially polarized incident beam to the photonic nanojet can be modified efficiently, leading to a sharp spot size which exceeds the optical diffraction limit. More importantly, a high conversion efficiency of 0.89 is achieved. At a wavelength of 633 nm, a focal spot of 266 nm (0.42λ) is achieved numerically by illuminating the engineered microsphere with a focusing beam at a numerical aperture of 0.7.

Lorentz drift compensation in high harmonic generation in the soft and hard X-ray regions of the spectrum

We present a semi-classical study of the effects of the Lorentz force on electrons during high harmonic generation in the soft and hard X-ray regions driven by near- and mid-infrared lasers with wavelengths from 0.8 to 20 μm, and at intensities below 10¹⁵ W/cm². The transverse extent of the longitudinal Lorentz drift is compared for both Gaussian focus and waveguide geometries. Both geometries exhibit a longitudinal electric field component that cancels the magnetic Lorentz drift in some regions of the focus, once each full optical cycle. We show that the Lorentz force contributes a super-Gaussian scaling which acts in addition to the dominant high harmonic flux scaling of λ^-(5-6) due to quantum diffusion. We predict that the high harmonic yield will be reduced for driving wavelengths > 6 μm, and that the presence of dynamic spatial mode asymmetries results in the generation of both even and odd harmonic orders. Remarkably, we show that under realistic conditions, the recollision process can be controlled and does not shut off completely even for wavelengths >10 μm and recollision energies greater than 15 keV.

Trapping metallic particles under resonant wavelength with 4π tight focusing of radially polarized beam

Here we propose a new method for trapping the resonant metallic particles with the 4π tight focusing (high numerical-aperture (NA)) system, which is illuminated by radial polarization light. Numerical simulations have indicated the maximum total optical force is 16.1pN while with nearly zero scattering force under axis trapping, which keeps the gradient force predominant. Furthermore, the distribution of total force is centrosymmetric and odd. We also gain stable 3D trap with an equilibrium point along z axis and r axis as in normal optical tweezers. What's more, we obtain the nearly pure longitudinal field. The maximum transverse intensity is only 2.3 × 10^-3 and the transverse spot size reaches 0.36λ, which is below Abbe's diffraction limit.

Generation of vectorial optical fields with slot-antenna-based metasurface

A transmission-type metasurface composed of carefully designed rectangular slot antennas for the generation of vectorial optical fields is proposed and demonstrated. Acting as local linear polarizers, these slot antennas enable the spatial modulation of optical fields in amplitude, phase, and polarization for the cross-polarized component of the scattered field. As an illustration, a metasurface capable of forming a radially polarized scattered field with specific vectorial beam patterns with appropriate excitation at normal incidence is designed, fabricated, and tested. The radially polarized scattered field is designed to be further tightly focused by a high numerical aperture objective lens in order to obtain a uniform longitudinally polarized optical needle field along the propagation direction. Characterization experiments demonstrate that its overall extinction ratio satisfies the amplitude modulation requirement, and a corresponding π phase modulation is realized as proposed.

Vortex generation and inhomogeneous Faraday rotation of a nonparaxial Gaussian beam in isotropic magneto-optic crystals

We study the propagation of a nonparaxial Gaussian beam in the isotropic Bi₄Ge₃O₁₂ crystal under an external magnetic field. The analytical solutions in the paraxial limit are derived, which show that a vortex with topological charge m=2 (m=1) generates in the E_ (E(z)) component. Numerical calculations in the nonparaxial regime confirm that, with the assistance of an external magnetic field, the spin-to-orbital angular momentum coupling effect can occur even in the isotropic crystal. We also study the Faraday rotation of the nonparaxial Gaussian beam. At the center of the transverse plane, the rotation of the plane of polarization is linearly proportional to the magnetic field. However, the rotation is not uniform on the whole transverse plane, manifesting the nonparaxial effect.

Efficient generation of ultra-intense few-cycle radially polarized laser pulses

We report on efficient generation of millijoule-level, kilohertz-repetition-rate few-cycle laser pulses with radial polarization by combining a gas-filled hollow-waveguide compression technique with a suitable polarization mode converter. Peak power levels >85  GW are routinely achieved, capable of reaching relativistic intensities >10(19)  W/cm2 with carrier-envelope-phase control, by employing readily accessible ultrafast high-energy laser technology.

Creation of a 50,000λ long needle-like field with 0.36λ width

It is a great challenge to create a needle-like field with properties of long beam length, narrow lateral width, uniformity, and high optical efficiency. Here we show a method that can realize these properties all at once. The key element is a 90° apex-angle concave conical mirror. By using this condenser along with a radially polarized incident beam of a specific field distribution, we numerically created a super slim, uniform, pure needle-like axially polarized field. This axially polarized field has a length of 50,000λ along the optical axis, and its lateral width still maintains a minimum 0.36λ size.

Modeling the tight focusing of beams in absorbing media with Monte Carlo simulations

A severe drawback to the scalar Monte Carlo (MC) method is the difficulty of introducing diffraction when simulating light propagation. This hinders, for instance, the accurate modeling of beams focused through microscope objectives, where the diffraction patterns in the focal plane are of great importance in various applications. Here, we propose to overcome this issue by means of a direct extinction method. In the MC simulations, the photon paths' initial positions are sampled from probability distributions which are calculated with a modified angular spectrum of the plane waves technique. We restricted our study to the two-dimensional case, and investigated the feasibility of our approach for absorbing yet nonscattering materials. We simulated the focusing of collimated beams with uniform profiles through microscope objectives. Our results were compared with those yielded by independent simulations using the finite-difference time-domain method. Very good agreement was achieved between the results of both methods, not only for the power distributions around the focal region including diffraction patterns, but also for the distribution of the energy flow (Poynting vector).

Review on Recent Developments in Laser Driven Inertial Fusion

Discovery of the laser in 1960 hopes were based on using its very high energy concentration within very short pulses of time and very small volumes for energy generation from nuclear fusion as “Inertial Fusion Energy” (IFE), parallel to the efforts to produce energy from “Magnetic Confinement Fusion” (MCF), by burning deuterium-tritium (DT) in high temperature plasmas to helium. Over the years the fusion gain was increased by a number of magnitudes and has reached nearly break-even after numerous difficulties in physics and technology had been solved. After briefly summarizing laser driven IFE, we report how the recently developed lasers with pulses of petawatt power and picosecond duration may open new alternatives for IFE with the goal to possibly ignite solid or low compressed DT fuel thereby creating a simplified reactor scheme. Ultrahigh acceleration of plasma blocks after irradiation of picosecond (PS) laser pulses of around terawatt (TW) power in the range of 1020 cm/s2was discovered by Sauerbrey (1996) as measured by Doppler effect where the laser intensity was up to about 1018 W/cm2. This is several orders of magnitude higher than acceleration by irradiation based on thermal interaction of lasers has produced.

Generation of a strong uniform transversely polarized nondiffracting beam using a high-numerical-aperture lens axicon with a binary phase mask

We present a theoretical approach to generate a nondiffracting beam with extended depth of focus (DOF) and a smaller focal spot along the optical axis, by tight focusing of an azimuthally polarized beam with a circular symmetrical binary phase mask and an interference effect over a high-numerical-aperture (NA) lens axicon system. We find a general azimuthal diffraction integral for the circularly symmetric binary phase mask and examine it in two special cases: a high-NA lens and a high-NA lens axicon. The azimuthally polarized beam remains well behaved in both cases. We verify that the longitudinal component generated by azimuthally polarized illumination produces the narrowest spot size for a wide range of geometries. Finally, we discuss the effects of tight focusing on a dielectric interface and provide some ideas for circumventing the effects of the binary phase mask interface and even utilize them for spot size reduction.

Laser-driven plasma beat-wave propagation in a density-modulated plasma

A laser-driven plasma beat wave, propagating through a plasma with a periodic density modulation, can generate two sideband plasma waves. One sideband moves with a smaller phase velocity than the pump plasma wave and the other propagates with a larger phase velocity. The plasma beat wave with a smaller phase velocity can accelerate modest-energy electrons to gain substantial energy and the electrons are further accelerated by the main plasma wave. The large phase velocity plasma wave can accelerate these electrons to higher energies. As a result, the electrons can attain high energies during the acceleration by the plasma waves in the presence of a periodic density modulation. The analytical results are compared with particle-in-cell simulations and are found to be in reasonable agreement.

Orbital angular momentum of Laguerre-Gaussian beams beyond the paraxial approximation

We derive a full field solution for Laguerre-Gaussian beams consistent with the Helmholtz equation using the angular spectrum method. Field components are presented as an order expansion in the ratio of the wavelength to the beam waist, f=λ/(2πw₀), which is typically small. The result is then generalized to a beam of arbitrary polarization. This result is then used to reproduce the signature angular momentum properties of Laguerre-Gaussian beams in the paraxial limit. The subsequent higher-order term is similarly obtained, which does not display a clear separation of orbital and spin angular momentum components.

Analytic solution of flat-top Gaussian and Laguerre-Gaussian laser field components

We generalized the nonparaxial field components of Laguerre-Gaussian and flattened Gaussian beams obtained using the angular spectrum method to include symmetric radial and angular expansions and simplified them using an approximate evaluation of the integral equations for the field components. These field components possess series expressions in orders of a natural expansion parameter, which clarifies the physical interpretation of the series expansion. A connection between Laguerre-Gaussian and flat-top Gaussian profiles is obtained.

Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation

We report a new method to create high purity longitudinally polarized field with extremely long depth of focus in the focal volume of a high numerical aperture (NA) objective lens. Through reversing the radiated field from an electric dipole array situated near the focus of the high-NA lens, the required incident field distribution in the pupil plane for the creation of an ultra-long optical needle field can be found. Numerical examples demonstrate that an optical needle field with a depth of focus up to 8λ is obtainable. Throughout the depth of focus, this engineered focal field maintains a diffraction limited transverse spot size (<0.43λ) with high longitudinal polarization purity. From the calculated pupil plane distribution, a simplified discrete complex pupil filter can be designed and significant improvements over the previously reported complex filters are clearly demonstrated.

Design of DOE for generating a needle of a strong longitudinally polarized field

A needle of strong longitudinally polarized field with homogeneous intensity along the optical axis, long depth of focus, and subdiffraction beam size can be generated by focusing a radially polarized light with a high-NA lens and a diffractive optical element (DOE) with belts. A method combining the global-search-optimization algorithm and the tight focusing properties of the radially polarized light is proposed to design the DOE. Based on the tight focusing properties, the light incident on the lens is divided into two parts: areas A and B. We discover that the longitudinal field in the focal region is mainly dependent on the number of belts in area B but not the total number of belts in the DOE.

Analytical beam propagation model for clipped focused-Gaussian beams using vector diffraction theory

Vector diffraction theory is applied to the case of focused TEM(00) Gaussian beams passing through a spatially limiting aperture in order to investigate the propagation of these clipped focused-Gaussian beams. Beam distributions at different axial distances show that a traditional M(2) propagation model cannot be used for the propagation of clipped focus-Gaussian beams. Using Luneberg's vector diffraction theory and Fresnel approximations, an analytical model for the on-axis transverse and longitudinal electric fields and intensity distributions is presented including predictions of the maximum obtainable intensity. In addition, an analytical expression is provided for the longitudinal component of the electric field of a TEM(00) mode unperturbed Gaussian beam. Experimental results are also presented and compared to the model's predictions.

Nonparaxial polarizers

We develop a theoretical description for polarizers that goes beyond the paraxial approximation. By combining existing theories for fields with nonplanar wavefronts, we are able to derive a simple power series expansion expressing the electric field of a light beam after a polarizer as a linear function of the field and its spatial derivatives evaluated before the polarizer. The first few terms of such expansion are explicitly given, and their physical meaning is discussed.

Directed acceleration of electrons from a solid surface by Sub-10-fs laser pulses

Electrons have been accelerated from solid target surfaces by sub-10-fs laser pulses of 120 microJ energy which were focused to an intensity of 2x10;{16} W/cm;{2}. The electrons have a narrow angular distribution, and their observed energies exceed 150 keV. We show that these energies are not to be attributed to collective plasma effects but are mainly gained directly via repeated acceleration in the transient field pattern created by incident and reflected laser, alternating with phase-shift-generating scattering events in the solid.

Application of the multiscale singular perturbation method to nonparaxial beam propagations in free space

Starting from the vector Maxwell's equations and applying the multiscale singular perturbation method, the nonparaxial beam propagation is studied in free space. Two new equations have been derived for transverse and longitudinal electric fields of an arbitrary polarized electromagnetic wave even in the case of tightly focused nonparaxial laser beams. By using the analogy of the optical beam in the space domain and the optical pulse in the time domain, the higher-order diffraction term is introduced. For strongly nonparaxial beams that are characterized by large values of the perturbative parameter, our correction solutions yield an accurate description of the field in the near-field region and are consistent with all other correction results obtained by others in the far-field region. For weakly nonparaxial beams, our correction solutions can be expressed in a very simple form that is proved to be exactly consistent with the solutions obtained by Cao and Deng [J. Opt. Soc. Am. A15, 1144 (1998)]. In addition, the lowest-order correction to the paraxial approximation can be found to be in good agreement with the results of Lax et al. [Phys. Rev. A11, 1365 (1975)] and Seshadri [J. Opt. Soc. Am. A19, 2134 (2002)].

Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes

The exact vector integral solution for all the electromagnetic field components of a general flattened Gaussian laser mode is derived by using the angular spectrum method. This solution includes the pure and annular Gaussian modes as special cases. The integrals are of the form of Gegenbauer's finite integral and are computed analytically for each case, yielding fields satisfying the Maxwell equations exactly in the form of quickly converging Fourier-Gegenbauer series.

Annular pupils, radial polarization, and superresolution

An annular pupil, which can be used to produce a Bessel beam, when combined with radially polarized illumination promises improvements in microscope resolution, increased packing density for optical storage, and finer optical lithography. When combined with a circular detection pupil in confocal microscopy a point-spread function 112 nm wide results (lambda = 488 nm). Radially polarized annular illumination of a solid-immersion lens can yield a focal spot smaller than 100 nm for lambda = 488 nm. Use of radially polarized illumination with pupil masks is discussed.

Phase dependence of relativistic electron dynamics and emission spectra in the superposition of an ultraintense laser field and a strong uniform magnetic field

The phase dependence of the dynamics and emission spectra of a fully relativistic electron in the superposition of an ultraintense plane wave laser field and a strong uniform magnetic field has been investigated. It is found that the effect of changing the initial laser phase is quite different for circularly and linearly polarized laser fields. For circular polarization only the axis of the helical trajectory is changed with variation of the initial laser field phase. However, for linear polarization, the effect of changing the initial phase is opposite in the two parameter regions divided by the resonance condition r=1 (r stands for the ratio between the reduced cyclotron frequency and laser frequency). When r<1, with increase in the initial laser field phase eta(0) from 0 to pi/2, both the radius of the electron's helical trajectory and the height of the peak related to the uniform magnetic field are decreased, and these two physical values are increased with an increase in the laser initial phase when r>1. The phase dependence of the electron's energy and velocity components was also studied. Some beat structure is found when eta(0)=0 and this structure is absent when eta(0)=pi/2.

Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam

In this paper, we extend the work of Barton and Alexander [J. App. Phys. 66, 2800 (1989)] on the fifth-order corrected field expressions for a Hermite-Gaussian (0,0) mode laser beam to more general cases with adjustable parameters. The parametric dependence of the electron dynamics is investigated by numerical methods. Finally, the fifth-order corrected field equations for the Hermite-Gaussian (0,1) mode are also presented.

High-aperture beams

Beams of a high angle of convergence and divergence are called high-aperture beams. Different ways of defining high-aperture generalizations to paraxial beams are reviewed for both scalar beams and electromagnetic beams. The different approaches are divided into three types. The particular examples of Gaussian beams and Bessel beams are discussed. For Gaussian beams, beams that exhibit a Gaussian variation in the waist necessarily include evanescent components, which rules out their use in describing propagation over all space. Generalizations of the definitions of beam width and the beam-propagation factor M2 for high-aperture beams are described. The similarities among the three types of high-aperture beams and the different models of ultrashort-pulsed beams are discussed.