Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space

Clarification for the fields of different radially polarized Laguerre-Gaussian light beams

Radially polarized light beams have found many applications in particle manipulation, laser processing, and microscopy. Just as with linear polarization, radially polarized light beams can have higher-order transverse modes that involve Laguerre polynomials. Fields of a radially polarized Laguerre-Gaussian light beam have been calculated before, even beyond the paraxial approximation. However, there are in fact multiple solutions to the paraxial wave equation that involve Laguerre polynomials with different properties and propagation characteristics. We therefore clarify the discrepancies among three valid radially polarized solutions to the paraxial wave equation that involve Laguerre polynomials.

Exact vectorial model for nonparaxial focusing of freeform wavefronts

We present a new formalism, based on Richards-Wolf theory, to rigorously model nonparaxial focusing of radially polarized electromagnetic beams with freeform wavefront. The beams can be expressed in terms of Zernike polynomials. Our approach is validated by comparing known results obtained by Richards-Wolf theory. Our integral representation is compliant with diffraction theory, is thoroughly discussed and solved for various freeform wavefront that, so far, have not been treated analytically. The extension of the method to other polarization states is straightforward.

Acceleration of electrons by tightly focused azimuthally polarized ultrashort pulses in a vacuum

Using the complex sink-source model (CSSM) and the Hertz potential method (HPM), the electromagnetic field expressions of tightly focused ultrashort azimuthally polarized pulses can be obtained. By numerically solving the relativistic Newton-Lorentz equation, the acceleration and confinement of electrons by the sub-cycle and few-cycle azimuthally polarized ultrashort pulses in vacuum are studied. Considering the radiation reaction force, it is found that electrons with an initial kinetic energy of less than 1MeV can be accelerated to hundreds of MeV and can be confined in the range of less than 1 micron for hundreds of femtoseconds in the direction perpendicular to the pulse propagation (transverse direction) by the pulses. With the increase of the beam waist and the intensity of the pulse, the electrons can obtain the exit kinetic energy exceeding 1GeV. When electrons are accelerated by the few-cycle pulses, the confined time of the electrons in the transverse direction is three times longer than that of the sub-cycle pulse. When the initial velocity of the electron points to a point in front of the focus, the electron can obtain the maximum exit kinetic energy. The change of the angular frequency corresponding to the spectral peak of the electromagnetic radiation from the electron acceleration with the electric field amplitude parameter E₀ of the pulse is studied. The phenomena of redshift and blueshift of the spectrum peak frequency of the electron radiation with the E₀ are found. These studies provide the methods to confine the movement of electrons in certain directions and accelerate electrons in the same time.

On the importance of frequency-dependent beam parameters for vacuum acceleration with few-cycle radially polarized laser beams

Tightly focused, ultrashort radially polarized laser beams have a large longitudinal field, which provides a strong motivation for direct particle acceleration and manipulation in a vacuum. The broadband nature of these beams means that chromatic properties of propagation and focusing are important to consider. We show via single-particle simulations that using the correct frequency-dependent beam parameters is imperative, especially as the pulse duration decreases to the few-cycle regime. The results with different spatio-spectral amplitude profiles show either a drastic increase or decrease of the final accelerated electron energy depending on the shape, motivating both proper characterization and potentially a route to optimization.

Analytical inversion of the focusing of high-numerical-aperture aplanatic systems

We propose a method for an analytical inversion of the electric and magnetic fields at the focus of a high-NA aplanatic system to obtain incident light beam distribution. Our approach is based on an inverse Fourier transform of the Richards-Wolf formalism for targeted longitudinal fields along the radial or axial directions at the non-paraxial focus. Analytical solutions are discussed for both axial and radial focal fields for a radially polarized incident light beam, and a criterion is defined to access a physically valid solution. We also validate the method according to results found in the literature. Finally, we show how the method can be generalized to other incident field distributions.

Influence of longitudinal chromatism on vacuum acceleration by intense radially polarized laser beams

We report with single-particle simulations that longitudinal chromatism, a commonly occurring spatio-temporal coupling in ultrashort laser pulses, can have a significant influence in the longitudinal acceleration of electrons via high-power, tightly-focused, and radially polarized laser beams. This effect can be advantageous, and even more so when combined with small values of temporal chirp. However, the effect can also be highly destructive when the magnitude and sign of the longitudinal chromatism is not ideal, even at very small magnitudes. This motivates the characterization and understanding of the driving laser pulses and further study of the influence of similar low-order spatial-temporal couplings on such acceleration.

Electron acceleration driven by sub-cycle and single-cycle focused optical pulse with radially polarized electromagnetic field

The space-time properties of the expressions of sub-cycle and single-cycle focused optical pulses with radially polarized electromagnetic field based on the Sink-Source model are studied. The self-induced blue shift of the center frequency of spectrum in the center of the pulse field is found to have an important impact on the electrons acceleration. When the electrons approach to the center of pulse, the electrons will obtain a large kinetic energy gain in a short time. The effect of radiation-reaction force can't be ignored if the net kinetic energy gain of electrons is more than GeVs. The electrons will deviate from the original acceleration channel and the gain of kinetic energy that electrons may gain will be greatly reduced if the radiation-reaction effect is considered. In contrast to the few-cycle laser pulse accelerating electrons, the gain of kinetic energy obtained by electrons is a few times higher and the corresponding peak optical power is one order of magnitude lower in the case of the sub-cycle laser pulses accelerating electrons. The maximal kinetic energy gain of electrons is robust against the variation of the incident angles.

Short pulse laser beam beyond paraxial approximation

Nonparaxial perturbative equations are derived from the scalar wave equation by taking into account spatiotemporal couplings. General solutions are obtained in Fourier space and further transformed back in direct space. They depend on parameters that can be used to match various boundary conditions and the perturbative expansion of any nonparaxial exact solutions. This parametrization is used to study the sensitivity of direct electron acceleration off an ultrashort tightly focused laser pulse to nonparaxial corrections of radially polarized electromagnetic fields.

Exact vectorial model for nonparaxial focusing by arbitrary axisymmetric surfaces

We present a new approach, based on Richards-Wolf formalism, to rigorously model nonparaxial focusing of radially and azimuthally polarized electromagnetic beams by axisymmetric systems without a single-point focus. Our approach is based on a combined method that uses ray tracing and diffraction integrals. Our method is validated by comparing known results obtained with a parabolic mirror. Our integral representation of the focused beams, compliant with diffraction theory, is thoroughly discussed and solved for various conics that, so far, have not been treated analytically. The extension of the method to other polarization states is straightforward.

Focused electromagnetic doughnut pulses and their interaction with interfaces and nanostructures

"The "focused doughnut", a single-cycle electromagnetic perturbation of toroidal topology with inseparable time and spatial dependencies propagates at the speed of light in vacuum, as was shown by Hellwarth and Nouchi in 1996. While normal incidence reflection and refraction of conventional electromagnetic pulses in isotropic media do not lead to polarization changes, "focused doughnut" pulses undergo complex field transformations owing to the toroidal field structure and the presence of longitudinal components. We also demonstrate that "focused doughnuts" can interact strongly with structured media exciting dominant dynamic toroidal dipoles in spherical dielectric particles."

Spatio-temporal modification of femtosecond focal spot under tight focusing condition

The focusing property of a focal spot of a femtosecond laser pulse is presented under tight focusing conditions (below f-number of 1). The spatial and temporal intensity distributions of a focused electric field are calculated by vector diffraction integrals and coherent superposition method. The validity of the calculation method is examined by comparing the intensity distribution obtained under a high f-number condition to that obtained with the fast Fourier transform method that assumes the scalar paraxial approximation. The spatial and temporal modifications under tight focusing conditions are described for a focused femtosecond laser pulse. The calculation results show that a peak intensity of about 2.5x10(24) W/cm2 can be achievable by tightly focusing a 12-fs, 10 PW laser pulse with a f/0.5 parabolic optic. The precise information on intensity distributions of a femtosecond focal spot obtained under a tight focusing condition will be crucial in assessing a focused intensity and in describing the motion of charged particles under an extremely strong electric field in ultra-relativistic and/or relativistic laser matter-interaction studies.

Femtosecond 240-keV electron pulses from direct laser acceleration in a low-density gas

We propose a simple laser-driven electron acceleration scheme based on tightly focused radially polarized laser pulses for the production of femtosecond electron bunches with energies in the few-hundreds-of-keV range. In this method, the electrons are accelerated forward in the focal volume by the longitudinal electric field component of the laser pulse. Three-dimensional test-particle and particle-in-cell simulations reveal the feasibility of generating well-collimated electron bunches with an energy spread of 5% and a temporal duration of the order of 1 fs. These results offer a route towards unprecedented time resolution in ultrafast electron diffraction experiments.

Validity of the paraxial approximation for electron acceleration with radially polarized laser beams

In the study of laser-driven electron acceleration, it has become customary to work within the framework of paraxial wave optics. Using an exact solution to the Helmholtz equation as well as its paraxial counterpart, we perform numerical simulations of electron acceleration with a high-power TM(01) beam. For beam waist sizes at which the paraxial approximation was previously recognized valid, we highlight significant differences in the angular divergence and energy distribution of the electron bunches produced by the exact and the paraxial solutions. Our results demonstrate that extra care has to be taken when working under the paraxial approximation in the context of electron acceleration with radially polarized laser beams.

Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent

Optical beams exhibiting a long depth of focus and a minimum spot size can be obtained with the tight focusing of a narrow annulus of radially polarized light, leading to a needle of longitudinally polarized light. Such beams are of increasing interest for their applications, for example in optical data storage, particle acceleration, and biomedical imaging. Hence one needs to characterize the needles of longitudinally polarized light obtained with different focusing optics and incident beams. In this paper, we present analytical expressions for the electric field of such a nearly nondiffracting, subwavelength beam obtained with a parabolic mirror or an aplanatic lens. Based on these results, we give expressions of the transverse and longitudinal full widths at half maximum of the focal lines as a function of the width of the incident annular beam and we compare the performances of the two focusing systems. Then, we propose a practical solution to produce a needle of longitudinally polarized light with a tunable axial extent and a transverse width reaching the theoretical limit of 0.36λ.

Generation and characterization of electron bunches with ramped current profiles in a dual-frequency superconducting linear accelerator

We report on the successful experimental generation of electron bunches with ramped current profiles. The technique relies on impressing nonlinear correlations in the longitudinal phase space using a superconducing radio frequency linear accelerator operating at two frequencies and a current-enhancing dispersive section. The produced ~700-MeV bunches have peak currents of the order of a kilo-Ampère. Data taken for various accelerator settings demonstrate the versatility of the method and, in particular, its ability to produce current profiles that have a quasilinear dependency on the longitudinal (temporal) coordinate. The measured bunch parameters are shown, via numerical simulations, to produce gigavolt-per-meter peak accelerating electric fields with transformer ratios larger than 2 in dielectric-lined waveguides.

Focusing a TM(01) beam with a slightly tilted parabolic mirror

A parabolic mirror illuminated with an incident collimated beam whose axis of propagation does not exactly coincide with the axis of revolution of the mirror shows distortion and strong coma. To understand the behavior of such a focused beam, a detailed description of the electric field in the focal region of a parabolic mirror illuminated with a beam having a nonzero angle of incidence is required. We use the Richards-Wolf vector field equation to investigate the electric energy density distribution of a beam focused with a parabolic mirror. The explicit aberration function of this focused field is provided along with numerically calculated electric energy densities in the focal region for different angles of incidence. The location of the peak intensity, the Strehl ratio and the full-width at half-maximum as a function of the angle of incidence are given and discussed. The results confirm that the focal spot of a strongly focused beam is affected by severe coma, even for very small tilting of the mirror. This analysis provides a clearer understanding of the effect of the angle of incidence on the focusing properties of a parabolic mirror as such a focusing device is of growing interest in microscopy.

Two-color-laser-driven direct electron acceleration in infinite vacuum

We propose a direct electron acceleration scheme that uses a two-color pulsed radially polarized laser beam. The two-color scheme achieves electron acceleration exceeding 90% of the theoretical energy gain limit, over twice of what is possible with a one-color pulsed beam of equal total energy and pulse duration. The scheme succeeds by exploiting the Gouy phase shift to cause an acceleration-favoring interference of fields only as the electron enters its effectively final accelerating cycle. Optimization conditions and power scaling characteristics are discussed.

Direct acceleration of an electron in infinite vacuum by a pulsed radially-polarized laser beam

We study the direct acceleration of a free electron in infinite vacuum along the axis of a pulsed radially-polarized laser beam. We find that net energy transfer from laser pulse to electron is maximized with the tightest focusing. We show that the net energy gain of an electron initially moving at a relativistic velocity may exceed more than half the theoretical limit of energy transfer, which is not possible with an initially stationary electron in the parameter space studied. We determine and analyze the power scaling of maximum energy gain, extending our study to include a relatively unexplored regime of low powers and revealing that substantial acceleration is already possible without the use of petawatt peak-power laser technology.

4π Focusing of TM(01) beams under nonparaxial conditions

The Richards-Wolf theory and the complex point-source method are both used to express the phasor of the electric field of tightly focused beams, but the connection between these two approaches is not straightforward. In this paper, the Richards-Wolf vector field equations are used to find the electromagnetic field of a TM(01) beam in the neighborhood of the focus of a 4π focusing system, such as a parabolic mirror with infinite transverse dimensions. Closed-form solutions are found for the distribution of the fields at any point in the vicinity of the focus; these solutions are identical to the electromagnetic field obtained with the complex source-point method in which sources are accompanied by sinks. This work thus establishes a connection between the Richards-Wolf theory and the complex sink/source model. The vector magnetic potential is introduced to simplify the computation of the six electromagnetic field components. The method is then used to find analytical expressions for the electromagnetic field of strongly focused TM(01) beams affected by primary aberrations such as curvature of field, coma, astigmatism and spherical aberration.

Power carried by a nonparaxial TM beam

In paraxial optics, the power carried by an optical beam can be accurately calculated by means of the integral of the squared modulus of its electric field over a plane transverse to the propagation axis. However, for nonparaxial electromagnetic beams, it is more appropriate to define the power carried by the beam by the integral of the longitudinal component of its time-averaged Poynting vector over a plane transverse to the propagation axis. In this paper, the expression of the power carried by a high-aperture transverse magnetic (TM) beam of any order is determined. The general expression of the power carried by a TM beam, which also applies for a transverse electric (TE) beam, is given in terms of a modified Struve function of order equal to an integer plus one-half.

Ultrashort focused electromagnetic pulses

In this paper we present a closed analytical description for few-cycle, focused electromagnetic pulses of arbitrary duration and carrier-envelope phase. Because of the vectorial character of light, not all thinkable one-dimensional shapes for the transverse electric field or vector potential can be realized as finite energy three-dimensional (3D) structures. We cope with this problem by using a second potential, which is defined as a primitive to the vector potential. This allows one to construct fully consistent 3D wave-packet solutions for the Maxwell equations, given a solution of the scalar wave equation. The wave equation is solved for ultrashort, Gaussian, and related pulses in paraxial approximation. The solution is given in a closed and numerically convenient form, based on the complex error function. All results undergo thorough numerical testing, validating their correctness and accuracy. A reliable and accurate representation of few-cycle pulses is, e.g., crucial for analytical and numerical theory of vacuum particle acceleration.

Relativistic attosecond electron pulses from a free-space laser-acceleration scheme

In this paper we describe how relativistic attosecond electron pulses could be produced in free space by ultrafast and ultraintense transverse magnetic (TM) laser beams. Numerical solutions of the time-dependent three-dimensional Maxwell-Lorentz equations reveal that electrons initially at rest at the waist of a multi-TW pulsed TM01 laser beam can be accelerated to multi-MeV energies. The use of a few-cycle laser beam and a compact initial electron cloud forces the particles to effectively interact with a single half-cycle of the laser field and form a pulse of attosecond duration.

Analytical calculation of the longitudinal electric field resulting from the tight focusing of an ultrafast transverse-magnetic laser beam

A purely time-domain approach is proposed for the propagation of vectorial ultrafast beams in free space beyond the paraxial and the slowly varying envelope approximations. As an example of application of this method, we describe in detail the vectorial properties of an ultrafast tightly focused transverse-magnetic (TM(01)) beam, where special attention is given to the longitudinal electric field component. We show that for spot sizes at the waist comparable to the wavelength, the beam diverges more rapidly than expected from paraxial theory. A consequence of this phenomenon is a faster decrease of the amplitude of the longitudinal field away from the waist and a faster evolution of the axial Gouy phase shift in the vicinity of the focus. It has been observed that the phase of the beam has an overall variation of 2pi from z=-infinity to infinity, independent of the beam spot size at the waist and pulse duration.