2. FOCUSED OPTICAL FIELD MODEL WITH LONGITUDINAL-FIELD AND SHORT-PULSE CORRECTIONS

Consider the Maxwell equations (Coulomb gauge) for the laser pulse vector potential in vacuum

(2) $$\Delta {\bf A} - \partial _t^2 {\bf A} = 0,\quad (\nabla, {\bf A}) = 0,$$

where $\nabla = (\partial _x, \partial _y, \partial _z )$ . The coordinates and time in Eq. (2) are normalized by w ₀ and w ₀/c, c standing for the speed of light and w ₀ being a parameter to be ultimately equated to the focused laser pulse waist. The vector potential in Eq. (2) is normalized by mc ²/e. Assuming that the electromagnetic field is a linearly polarized envelope such that A _y  ≡ 0, we seek solutions to Eq. (2) of the form

(3) $$A_x = \exp \left( {i\displaystyle{{t - z} \over {\rm \epsilon}}} \right)\left( {a({\rm \tau}, x,y,s) + \sum\limits_{m = 1}^\infty {{\rm \epsilon} ^m} a_{x,m} ({\rm \tau}, x,y,s)} \right) + {\rm c}{\rm. c}.,$$

(4) $$A_z = \exp \left( {i\displaystyle{{t - z} \over {\rm \epsilon}}} \right)\sum\limits_{m = 1}^\infty {{\rm \epsilon} ^m} a_{z,m} ({\rm \tau}, x,y,s) + {\rm c}{\rm. c}{\rm.}, $$

with τ = 2εz, s = t − z, and ε = (λ/2πw ₀), where λ is the laser pulse wavelength. In line with the envelope approximation, ε is assumed to be a small parameter. The lowest and first-order results derived by substituting Eqs. (3) and (4) into the first of Eq. (2) are Schröedinger equations for the corresponding envelopes

(5) $$4i\partial _{\rm \tau} a + \triangle _ \bot a = 0,$$

(6) $$- 4i\partial _{\rm \tau} a_{x,1} + \triangle _ \bot a_{x,1} = 4\partial _{{\rm \tau} s}^2 a$$

with $\Delta_ \bot = \partial _x^2 + \partial _y^2 $ . Solutions to Eqs. (5) and (6), along with the expression for the first-order envelope defined by Eq. (4), are easily found to be of the form

(7) $$a({\rm \tau}, x,y,s) = a_0 (s)u({\rm \tau}, x,y),$$

(8) $$a_{x,1} ({\rm \tau}, x,y,s) = ia^{\prime}_0 (s)\partial _{\rm \tau} \left( {{\rm \tau} u({\rm \tau}, x,y)} \right),$$

(9) $$a_{z,1} ({\rm \tau}, x,y,s) = - ia_0 (s)\partial _x u({\rm \tau}, x,y),$$

where u(τ, x, y) is a solution to Eq. (5). Importantly, the term given by Eq. (8) provides a first-order correction within the overall expression (3), which governs the transverse field. This correction is duration-related, is of the same magnitude as the longitudinal field, and should be taken into account for short laser pulses. Obviously, the duration-related correction vanishes in the long-pulse limit, in which the above model reverts to the one used previously by Galkin et al. (Reference Galkin, Korobkin, Romanovsky and Shiryaev2008).

The solution which resulted from earlier derivations (Quesnel and Mora, Reference Quesnel and Mora1998; Salamin et al., Reference Salamin, Mocken and Keitel2002) aimed at describing the transverse component of the vacuum optical field was developed under the a priori assumption that it can be represented asymptotically as a sum of terms proportional to even powers of the small parameter and, hence, ignored the short-pulse effects.

In the case of axial symmetry, the simplest case is embodied in the solution of the form

$$\eqalign{u({\rm \tau}, x,y) = & \displaystyle{{\Lambda ({\rm \tau}, r)} \over {\sqrt {{\rm \tau} ^2 + 1}}} \exp \left( {i{\rm \psi} ({\rm \tau}, r)} \right), \cr \Lambda ({\rm \tau}, r) = & \exp \left( { - \displaystyle{{r^2} \over {{\rm \tau} ^2 + 1}}} \right),\quad {\rm \psi} ({\rm \tau}, r) = - \displaystyle{{{\rm \tau} r^2} \over {{\rm \tau} ^2 + 1}} + \arctan \left( {\rm \tau} \right).} $$

Laguerre beams are a more general example of solutions to Eq. (5)

(10) $$\eqalign{& u({\rm \tau}, x,y) \cr &= \displaystyle{\matrix{2^{l/2} \big(\big[r/(\sqrt {{\rm \tau} ^2 + 1} )\big]\big)^l \sin \left( {l{\rm \varphi} + {\rm \varphi} _0} \right) L_{\rm \delta} ^l \big(\big[2r^2 /({\rm \tau} ^2 + 1)\big]\big)\cr \qquad\times\exp \big(i(2{\rm \delta} + l + 1) \hfill \tan ^{ - 1} ({\rm \tau} ) - \big(\big[r^2 (1 + i{\rm \tau} )\big]/[{\rm \tau} ^2 + 1]\big)\big) \hfill} \over {\sqrt {{\rm \tau} ^2 + 1}}}.}$$

Here ${\rm \varphi} = \arctan (y/x)$ and φ₀ is an integration constant. A superposition of Laguerre modes constitutes the general solution describing a focused optical field in vacuum. The above Laguerre mode is reduced to a Gaussian one when l = 0 and δ = 0.

Altogether, Eqs. (3), (4) and (7)–(10) define the lowest and first-order approximations to a solution to the Maxwell equations governing the propagation of a short, focused electromagnetic envelope in vacuum. The approach can be extended to obtain higher approximations, but the corrections are unlikely to contribute substantially to the electron dynamics for a realistic combination of the values of laser wavelength and beam waist.

It should be noted that efforts have been made to study charged particle acceleration by optical fields with structures more complex than Gaussian pulses. Vacuum acceleration by Airy beams (Li et al., Reference Li, Zang and Tian2010) and by Gaussian modes (Wang et al., Reference Wang, Ho, Tang and Wang2007) was examined. The issue of high-order corrections to the expressions for the field of an ultrashort laser pulse in vacuum was also addressed (Hua et al., Reference Hua, Ho, Lin, Chen, Xie, Zhang, Yan and Xu2004).

The electromagnetic field being defined by Eqs. (3), (4) and (7)–(9), the relativistic Newton equation for an electron driven by the corresponding Lorentz force is

(11) $${\bf p}_t = - \left( {{\bf E} + {\rm \gamma} ^{ - 1} {\bf p} \times {\bf H}} \right),$$

the electron momentum being normalized by mc. Here ${\bf E} =$ $- \partial _t {\bf A}$ and ${\bf H} = \nabla \times {\bf A}$ are the electric and magnetic fields normalized by mc ²/(ew ₀), and ${\rm \gamma} = \sqrt {1 + {\bf p}^2} $ is the relativistic mass factor. Eq. (11) is solved numerically below for an array of initial conditions to calculate the energy spectra of a sparse electron ensemble scattered by a relativistically intense laser field. The simulations detailed below fully take into account the three components of the electron coordinate and momentum, of the electric and magnetic field, and of the corresponding Lorentz force.

3. ELECTRON ENERGY SPECTRA FOR SPARSE TARGETS SCATTERED BY RELATIVISTICALLY INTENSE LASER PULSES

The laser radiation temporal profile adopted in the study detailed in this paper is

$$a_0 (s) = ge^{ - (1/2){\kern 1pt} ([(s - z_{\rm d} )/{\rm \tau} _0 ])^2}, $$

where τ₀ and z _d denote the laser pulse duration and the initial distance to the target, normalized respectively by w ₀/c and by w ₀. In the simulations described below, z _d is chosen to be large. No data accumulation over multiple laser shots is being modeled. The parameter g and the peak laser intensity I are related as $g = \sqrt {I/I_{\rm r}} /{\rm \epsilon} $ . The simulations described below apply to targets consisting of electrons injected into the focal area or born within it due to the ionization of low-density gas by the pulse front.

Provided that the target is sufficiently sparse, the resulting electron dynamics can be modeled by solving Newton's equations for each particle independently as the Coulomb forces within it are negligible compared with the optical field Lorentz force (Galkin, Reference Galkin, Korobkin, Romanovsky and Shiryaev2008). Assuming that the above condition is met, which is the case for the gas density in the experimental settings described by Kalashnikov et al. (Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015), the electron scattering is simulated in the present study by solving Eq. (11) for an array of initial conditions. The electrons are initially distributed uniformly over large target space and launched with noise-level start momenta, with no attempt being made to assign statistical weights to individual particles.

The electrons were born via the optical field ionization in the case of the study by Kalashnikov et al. (Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015). The above approach based on treating electrons as a pre-existing environment is warranted for the simulation of the experimental conditions of the study as, for the intensity achieved in it, ionization occurs far at the front of the pulse, with the high-intensity temporally central part of the optical field running into the thus preformed electron target.

The electron energy spectra were measured under same conditions for Ar and Kr to diagnose the potential influence of the optical field ionization and proved to be practically identical (Kalashnikov et al., Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015), demonstrating that the experiment for the given intensity did not extend beyond full ionization of Ar. An estimate using the Coulomb barrier suppression model (Augst et al., Reference August, Strickland, Meyerhofer, Chin and Eberly1989, Reference Augst, Meyerhofer, Strickland and Chin1991) shows that, in the case of Ar, achieving the two maximally ionized states should take the intensity of around (4–5) × 10²¹ W/cm², which is slightly out of reach under the experimental conditions discussed by Kalashnikov et al. (Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015), while the third-highest ionized state only requires an intensity close to 10¹⁹ W/cm² and should materialize ahead of the advent of the high-intensity part of the laser pulse. Consequently, the major strong-field scattering action of the laser radiation occurred, under the given conditions, in an electron environment already created at the pulse rise. It should be noted that, therefore, the effect of exceptionally efficient acceleration of electrons due to their being discharged via ionization right into the focal spot, as described elsewhere (Hu & Starace, Reference Hu and Starace2002, Reference Hu and Starace2006), was absent. Not being injected right into the focal spot due to the above effect or to pre-acceleration, the electrons tend to be scattered by the laser pulse before the focal spot covers them, which explains the fact that the electron energies recorded in the recent experimental study (Kalashnikov et al., Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015) and reproduced in computations below appear modest compared to the energy of the oscillations an electron would exhibit in a field with the amplitude corresponding to the peak laser intensity considered.

As shown in previous studies (Galkin et al., Reference Galkin, Korobkin, Romanovsky and Shiryaev2008), an electron interacting with a focused intense laser pulse is captured by the optical field, oscillates in it, and, finally, drops out of the focal spot due to dephasing as the pulse propagating with the speed of light overtakes the accelerated particle. The focused optical field being transversely inhomogeneous, the electron gets displaced radially relative to its original position over the interaction time and, therefore, is eventually released at an angle to the propagation axis with a kinetic energy, which can make a considerable fraction of the energy acquired in the process of laser-driven oscillations. For an electron ensemble, the outputs combine into electron energy spectra, which are recorded within various angular ranges relative to the laser pulse propagation axis.

Representative simulation results for the electron energy spectra produced by the interaction between a sparse target and relativistically intense laser radiation are shown in Figures 1 and 2. The laser pulse was assumed to have the peak intensity of 50I _r, the duration of 11.24 optical cycles, and the focal spot radius of 2.75 laser wavelengths [roughly, these are the laser system parameters reported by Kalashnikov et al. (Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015)]. The amplitude transverse distributions in the optical fields correspond to a Gaussian pulse in Figure 1 and to the first Laguerre mode [l = 0 and δ = 1 in Eq. (10)] comprising a Gaussian core and a ring – in Figure 2, the purpose being to assess how the electron energy spectra are affected by the optical field transverse structure. The angular ranges for which the spectra are depicted are chosen in each case to highlight the patterns specific to the scattering induced by Gaussian and Laguerre laser pulses. The data in Figure 1 correspond to (a) 40°–50°; (b) 50°–60°; (c) 60°–70°; (d) 70°–80°, and in Figure 2 – to (a) 10°–20°; (b) 30°–40°; (c) 40°–50°. Electrons with noise-level post-interaction energies (≤ 0.001mc ²) were filtered out of the spectra, and the energy mesh was slightly adapted to the character of the distributions by using finer divisions to visualize the cold fringes where the numbers of particles tend to be much higher. The resultant discrete spectra were interpolated.

The key finding illustrated by Figures 1 and 2 is that significantly different patterns arise in the energy spectra of the electron ensembles scattered by relativistically intense laser pulses with Gaussian and Laguerre transverse distributions of the laser field amplitude. For a Gaussian pulse, the angular range where the maximal electron energy is attained exhibits a distribution localized around the peak value, as can be seen in Figure 2a. The localization effect holds within a neighboring angular range with a lower peak energy (Fig. 2b), but the type of behavior changes as greater deviations from the laser propagation axis are examined. For those, the energy spectra, exemplified by Figures 2c and 2d, are dominated by cold electrons and have the form of evanescent distributions. Their near-exponential character over large parts of the energy diapason is illustrated by the insets in Figures 2c and 2d showing the same data in semi-logarithmic scale.

In a crucial contrast, the energy spectra of the electrons scattered by a Laguerre-mode laser pulse combine cold and high-energy components within many of the angular ranges. For some of them, the relativistic parts of the spectra have complex shapes, such as the twin-maxima structure visible in Figure 3b. The insets in Figures 3a–3c show uncurtailed zooms of the cold components, while their semi-logarithmic plots, which are also presented in Figure 3, attest to their exponential character.

Fig. 3. The energy spectra of electrons scattered into various angular ranges relative to the propagation axis by a laser pulse with a Laguerre transverse distribution of amplitude and the peak intensity, duration, and focal spot radius respectively equal to 50I _r [where I _r stands for the relativistic intensity given by Eq. (1)], 11.24 optical cycles, and 2.75 laser wavelengths. The results are shown for the angular ranges making (a) 10°–20°; (b) 30°–40°; (c) 40°–50°. For every angular range, the figures include the overall spectrum with the cold part capped to accentuate the features of the relativistic component, an inset showing the cold part in full range, and a semi-logarithmic plot of the cold part illustrating its near-exponential character.

With these patterns prominent in the scattering picture for the Laguerre-mode transverse structure of the optical field, the corresponding simulated energy spectra bear strong semblance of the experimental findings (Kalashnikov et al., Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015) which are reproduced in Figures 1b and 1c. Note that the focal spot structure seen in the photo in Figure 1a, taken from the study by Kalashnikov et al. (Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015), supports the assumption that the optical field in the case had a first-order Laguerre transverse distribution of amplitude.

Fully quantitative comparison between the experimental results (Kalashnikov et al., Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015) and the above computations is problematic as the observations data underwent statistical processing in the former case and is fairly sensitive to computational parameters in the latter. Moreover, a plethora of contributing factors including the laser pulse contrast, temporal profile, and so on may not be fully taken into account. Nevertheless, there appears to be reasonable agreement between the observations and the simulation outcomes shown respectively in Figures 1b and 3b as well as between Figures 1c and 3c. In Figure 1b, the sub-relativistic part is an exponentially decreasing distribution followed by a plateau, which altogether fit within the energy range of under 0.5mc ², a structure similar to the one depicted in Figure 3b. The characteristic twin-peak formation in the relativistic diapason is present in both figures, with the maxima in the computational picture positioned approximately at 2.2mc ² and 4.6mc ². The values differ from the corresponding ones in Figure 1b by around 0.5mc ², which is considerably less than the widths of the peaks [in computations, the widths come out to be close to those observed experimentally by Kalashnikov et al. (Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015)]. An exponential cold part somewhat wider than in the previous pair of figures is found in both Figures 1c and 3c, while the scarcely distinguishable relativistic-range peaks in Figure 1c are matched by a single pileup of relativistic electrons in the computational data given in Figure 3c. The onset of the relativistic part of the distribution in Figure 3c is also shifted in comparison with that of Figure 1c by around 0.5mc ², a value much less than the combined width of the peaks.

It should be noted that, overall, the shapes of the simulated energy spectra appear to be sensitive to the full number of energy scale divisions and to how the total angle is partitioned into ranges, so that the more meaningful aspects of the description of the scattering of electrons by laser pulses are reflected by the data on the widths of the energy distributions.

4. SUMMARY

The post-interaction energy spectra of electrons scattered by relativistically intense laser pulses into various ranges of angles relative to the optical field propagation axis are examined. The concentration in the electron ensemble is assumed to be low enough to neglect the Coulomb interactions between electrons as well as between electrons and the ion carcass if target ionization by the laser pulse front is implied.

The electron dynamics is modeled by treating the relativistic Newton equation for an array of initial conditions, the charged particles being driven by the Lorentz force exerted by a focused electromagnetic envelope. An approximate solution to the Maxwell equations in vacuum is spelled out to describe the envelope, with the longitudinal field and short-pulse corrections (which are of the same order) taken into account. In the general case, the solution can be represented as a series of Laguerre modes, the lowest one corresponding to the Gaussian transverse intensity distribution.

Simulation results are presented for the Gaussian mode and for the first Laguerre mode, which can be interpreted as a superposition of a Gaussian distribution and an optical field ring. The latter case appears to reproduce the features seen in the laser focus photo from a recent experimental study (Kalashnikov et al., Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015).

Computations demonstrate that the electron energy spectra within particular angular ranges tend to be considerably non-monoenergetic for both Gaussian and first-order Laguerre laser pulses, but distinct patterns emerge in each case. Depending on the angular range, the spectra of the electrons ejected from the target by a Gaussian pulse either are peaked around a maximal value or have the shape of evanescent distributions stretching from the cold to the chiefly sub-relativistic zone of the spectrum. In contrast, pulses with the first-order Laguerre transverse amplitude profile produce, within certain angular ranges, grossly non-monoenergetic electron energy spectra combining cold near-exponential components and markedly relativistic parts. Furthermore, the relativistic-energy parts of the electron spectra may exhibit characteristic twin-maxima structure. Both of the above patterns emerging in the simulations for the first-order Laguerre transverse intensity distribution are recognizable in the experimentally measured electron energy spectra, which are reported by Kalashnikov et al. (Reference Kalashnikov, Andreev, Ivanov, Galkin, Korobkin, Romanovsky, Shiryaev, Schnuerer, Braenzel and Trofimov2015).