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Fig. 20. Measured helium angular-resolved photoelectron spectrum for four different emission angles at the saturation intensity. The polar plot: show the measured angular distributions (crosses) at the indicated energies and the solid lines are only to guide the reader [224]. Reprinted witl permission from [224]; Copyright 1996 by the American Physical Society.

Figure 20 Measured helium angular-resolved photoelectron spectrum for four different emission angles at the saturation intensity. The polar plot: show the measured angular distributions (crosses) at the indicated energies and the solid lines are only to guide the reader [224]. Reprinted witl permission from [224]; Copyright 1996 by the American Physical Society.

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E-mail address: keitel@mpi-hd.mpg.de (C.H. Keitel). 0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi: 10.1016/j.physrep.2006.01.002 Recent advances in laser technology have pushed the frontier of maximum intensity achieved to about 10?2 W / cm” and investi- gators currently believe even higher intensities may be reached in the near future. This, combined with other breakthroughs on the fronts of short pulse generation and high repetition rates, have stimulated considerable progress, theoretical as well as experimental, in the field of laser—matter interactions. It is now possible to laser-accelerate electrons to a few hundred MeV and laser-induced pair-production and nuclear physics experiments have made significant progress. This article is devoted to a review of the recent advances in the field and stresses quantum phenomena that require laser field intensities in excess of the relativistic threshold of ~ 1018 w / cm”. Interactions with free electrons, with highly-charged ions and with atoms and clusters, are reviewed. Electron laser acceleration, atomic quantum dynamics, high harmonic generation, quantum electrodynamical effects and nuclear interactions in plasmas and ions, are among the important topics covered in the article. © 2006 Elsevier B.V. All rights reserved.
E-mail address: keitel@mpi-hd.mpg.de (C.H. Keitel). 0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi: 10.1016/j.physrep.2006.01.002 Recent advances in laser technology have pushed the frontier of maximum intensity achieved to about 10?2 W / cm” and investi- gators currently believe even higher intensities may be reached in the near future. This, combined with other breakthroughs on the fronts of short pulse generation and high repetition rates, have stimulated considerable progress, theoretical as well as experimental, in the field of laser—matter interactions. It is now possible to laser-accelerate electrons to a few hundred MeV and laser-induced pair-production and nuclear physics experiments have made significant progress. This article is devoted to a review of the recent advances in the field and stresses quantum phenomena that require laser field intensities in excess of the relativistic threshold of ~ 1018 w / cm”. Interactions with free electrons, with highly-charged ions and with atoms and clusters, are reviewed. Electron laser acceleration, atomic quantum dynamics, high harmonic generation, quantum electrodynamical effects and nuclear interactions in plasmas and ions, are among the important topics covered in the article. © 2006 Elsevier B.V. All rights reserved.
Fig. 1. (a) Measured projection of, and (b) cross section through, the tomographically reconstructed intensity distribution for a radially polarized input field focused using an annular aperture (d; = 3.3 mm). The calculated power contained in the longitudinal field amounts to 72.8% of the total beam power. Open squares: experimental data; calculated curves: solid black line: total field; dashed line: transverse field; solid triangles: longitudinal field [31]. Reprinted with permission from [31]; Copyright 2003 by the American Physical Society.
Fig. 1. (a) Measured projection of, and (b) cross section through, the tomographically reconstructed intensity distribution for a radially polarized input field focused using an annular aperture (d; = 3.3 mm). The calculated power contained in the longitudinal field amounts to 72.8% of the total beam power. Open squares: experimental data; calculated curves: solid black line: total field; dashed line: transverse field; solid triangles: longitudinal field [31]. Reprinted with permission from [31]; Copyright 2003 by the American Physical Society.
Fig. 2. 3D PIC simulation of isolated attosecond pulse generation. Isosurface of the electromagnetic energy density demonstrates efficient generation of isolated attosecond pulse (left). Electron density slices show anisotropic electron activity at the time of attosecond pulse generation (right) [37]. Reprinted with permission from [37]; Copyright 2005 by the Taylor & Francis Ltd.; web site: http://www.tandf.co.uk/journals.
Fig. 2. 3D PIC simulation of isolated attosecond pulse generation. Isosurface of the electromagnetic energy density demonstrates efficient generation of isolated attosecond pulse (left). Electron density slices show anisotropic electron activity at the time of attosecond pulse generation (right) [37]. Reprinted with permission from [37]; Copyright 2005 by the Taylor & Francis Ltd.; web site: http://www.tandf.co.uk/journals.
The major drawback of the Dirac treatment is the temporal step size At<Sf/ E required, which has to be significantly smaller than for Schrédinger treatments, because of the large rest mass energy mc” that is contained in the particle’s total energy E. Again, just as in the Schrédinger case, fast Fourier transforms are an appropriate means to evaluate the action of the exponential momentum space operator that contains Ha. In addition, the matrix character has to be taken care of. This is most conveniently accomplished with the help of a series expansion. The result can be written as follows
The major drawback of the Dirac treatment is the temporal step size At<Sf/ E required, which has to be significantly smaller than for Schrédinger treatments, because of the large rest mass energy mc” that is contained in the particle’s total energy E. Again, just as in the Schrédinger case, fast Fourier transforms are an appropriate means to evaluate the action of the exponential momentum space operator that contains Ha. In addition, the matrix character has to be taken care of. This is most conveniently accomplished with the help of a series expansion. The result can be written as follows
ee Rr ens ee: Se ARNPRERTES OR EEM Neer ser ern nnn ae nn Unfortunately, the equations of motion (24), or their covariant counterpart, admit analytic solutions only for plane- wave fields and under very specialized conditions [141—150]. For example, let the laser field be modeled by a vector potential A(7), where 7 = wt —k-r, with t and r the time and space coordinates of the electron and k the wave vector of the laser field (@ = ck). Then let an electron be incident from the point with a position vector rp, and sideways at an arbitrary angle 09 to the beam direction of propagation, with an arbitrary initial velocity Bp = Vvo/c. Then, its subsequent energy and momentum and position vectors, respectively, may be found from [148] subject to initial conditions appropriate to the situation at hand. In Eqs. (24) the quantities have their usual definitions: the momentum p = ymcf, the energy & = ymc’, the Lorentz factor y = (1 — B’)- 1/2 and the velocity, normalized by the speed of light c in vacuum, is PB.
ee Rr ens ee: Se ARNPRERTES OR EEM Neer ser ern nnn ae nn Unfortunately, the equations of motion (24), or their covariant counterpart, admit analytic solutions only for plane- wave fields and under very specialized conditions [141—150]. For example, let the laser field be modeled by a vector potential A(7), where 7 = wt —k-r, with t and r the time and space coordinates of the electron and k the wave vector of the laser field (@ = ck). Then let an electron be incident from the point with a position vector rp, and sideways at an arbitrary angle 09 to the beam direction of propagation, with an arbitrary initial velocity Bp = Vvo/c. Then, its subsequent energy and momentum and position vectors, respectively, may be found from [148] subject to initial conditions appropriate to the situation at hand. In Eqs. (24) the quantities have their usual definitions: the momentum p = ymcf, the energy & = ymc’, the Lorentz factor y = (1 — B’)- 1/2 and the velocity, normalized by the speed of light c in vacuum, is PB.
Fig. 3. Electron wave-packet dynamics: (a) the trajectory of the maximum of the wave packet. The labels (i)—-(vi) mark the positions for which the wave packets are plotted in (b). The sides of each of the squares have a width of 70 a.u. In (iii) and (v), the principal axes of the wave packets are shown. The directions deviate from the diagonals by an angle of A(&jj/ty)/ (4c?), where t;ii/ty represent the times when the electron is at points (iii) and (v), respectively. The other wave packets in (b) are axially symmetric [151]. Reprinted with permission from [151]; Copyright 2005 by the Springer Science and Business Media.
Fig. 3. Electron wave-packet dynamics: (a) the trajectory of the maximum of the wave packet. The labels (i)—-(vi) mark the positions for which the wave packets are plotted in (b). The sides of each of the squares have a width of 70 a.u. In (iii) and (v), the principal axes of the wave packets are shown. The directions deviate from the diagonals by an angle of A(&jj/ty)/ (4c?), where t;ii/ty represent the times when the electron is at points (iii) and (v), respectively. The other wave packets in (b) are axially symmetric [151]. Reprinted with permission from [151]; Copyright 2005 by the Springer Science and Business Media.
Fig. 4. Evolution of a free Dirac electron driven by an intense ultraviolet laser field of frequency equal to w = 2 a.u. (the photon energy of 54.4 eV) at three different laser intensities. Each trajectory is viewed using nine different plots, at intervals of an eighth of a period. The Schrédinger-like case (trajectory on the left) corresponds to A = 4000a.u., where A is the amplitude of the laser field vector potential, (i.e. to an intensity of 1.2 x 1070 W/cm”). Contour plots of the total probability density are drawn, using a linear scale for the contours. The horizontal axis is the propagation direction and the vertical axis is the electric field direction, meaning that the magnetic field is perpendicular to the paper. A relativistic case (trajectory in the center) corresponds to A = 8000 a.u. (i.e. to an intensity of 4.8 x 1020 W/cm?). The trajectory for A = 20000 a.u. (i.e. an intensity of 3 x 102! W/ cm?) corresponds to a strongly relativistic case (trajectory on the right) [139]. Reprinted with permission from [139]; Copyright 2000 by the IOP Publishing Limited.
Fig. 4. Evolution of a free Dirac electron driven by an intense ultraviolet laser field of frequency equal to w = 2 a.u. (the photon energy of 54.4 eV) at three different laser intensities. Each trajectory is viewed using nine different plots, at intervals of an eighth of a period. The Schrédinger-like case (trajectory on the left) corresponds to A = 4000a.u., where A is the amplitude of the laser field vector potential, (i.e. to an intensity of 1.2 x 1070 W/cm”). Contour plots of the total probability density are drawn, using a linear scale for the contours. The horizontal axis is the propagation direction and the vertical axis is the electric field direction, meaning that the magnetic field is perpendicular to the paper. A relativistic case (trajectory in the center) corresponds to A = 8000 a.u. (i.e. to an intensity of 4.8 x 1020 W/cm?). The trajectory for A = 20000 a.u. (i.e. an intensity of 3 x 102! W/ cm?) corresponds to a strongly relativistic case (trajectory on the right) [139]. Reprinted with permission from [139]; Copyright 2000 by the IOP Publishing Limited.
Fig. 5. The Wigner function of a free electron initially (at t = 0) at rest at the origin of coordinates and subsequently subjected to a plane-wave las: field. The plots are for a wave packet that has interacted with a field of frequency m= | a.u. for a time equal to 0.5 T, where T is a laser—field cycl The snapshots correspond to x = 0 = px, and an initial wave packet modeled by a two-dimensional Gaussian of width equal to a = 100 a.u. in eith spatial dimension. The dipole approximation has not been made [152]. Reprinted with permission from [152]; Copyright 2005 by the Americz Physical Society.
Fig. 5. The Wigner function of a free electron initially (at t = 0) at rest at the origin of coordinates and subsequently subjected to a plane-wave las: field. The plots are for a wave packet that has interacted with a field of frequency m= | a.u. for a time equal to 0.5 T, where T is a laser—field cycl The snapshots correspond to x = 0 = px, and an initial wave packet modeled by a two-dimensional Gaussian of width equal to a = 100 a.u. in eith spatial dimension. The dipole approximation has not been made [152]. Reprinted with permission from [152]; Copyright 2005 by the Americz Physical Society.
Fig. 6. A schematic of the interaction of a free electron wave packet (solid lines) with a laser wave (dotted lines). (a) In the dipole approximation (DA) (A> a), where / is the laser wavelength and a is the size of the wave packet, (b) In the non-dipole approximation (NDA), but in the regime in which the magnetic force (v x B) is neglected, and (c) In the NDA case, where 2 < a and with the v x B force playing a significant role. The arrows indicate possible free electron quantum pathways [152]. Reprinted with permission from [152]; Copyright 2005 by the American Physical Society.
Fig. 6. A schematic of the interaction of a free electron wave packet (solid lines) with a laser wave (dotted lines). (a) In the dipole approximation (DA) (A> a), where / is the laser wavelength and a is the size of the wave packet, (b) In the non-dipole approximation (NDA), but in the regime in which the magnetic force (v x B) is neglected, and (c) In the NDA case, where 2 < a and with the v x B force playing a significant role. The arrows indicate possible free electron quantum pathways [152]. Reprinted with permission from [152]; Copyright 2005 by the American Physical Society.
Fig. 7. Harmonic intensities as functions of the harmonic order N for the H™ ion in a COp laser with intensity J; =5 x 10!0w / cm2. The magnetic field induction is B = 0 T (squares), 1000 T (triangles), and 2000 T (circles) [163]. Reprinted with permission from [163]; Copyright 1999 by the American Physical Society.
Fig. 7. Harmonic intensities as functions of the harmonic order N for the H™ ion in a COp laser with intensity J; =5 x 10!0w / cm2. The magnetic field induction is B = 0 T (squares), 1000 T (triangles), and 2000 T (circles) [163]. Reprinted with permission from [163]; Copyright 1999 by the American Physical Society.
Fig. 8. Harmonic intensities vs. B for fixed harmonic order: N = 11 (solid line), 13 (dot-dashed line), 15 (dashed line), 17 (dotted line), and 19 (long-dashed line). The laser field and the H~ ion parameters are the same as in Fig. 7. (a) 0 < B < 8000 T, and (b) 50 < B < 500 T [163]. Reprinted with permission from [153]; Copyright 1999 by the American Physical Society.
Fig. 8. Harmonic intensities vs. B for fixed harmonic order: N = 11 (solid line), 13 (dot-dashed line), 15 (dashed line), 17 (dotted line), and 19 (long-dashed line). The laser field and the H~ ion parameters are the same as in Fig. 7. (a) 0 < B < 8000 T, and (b) 50 < B < 500 T [163]. Reprinted with permission from [153]; Copyright 1999 by the American Physical Society.
Fig. 9. Comparison of the spatial probability distributions predicted by classical and quantum mechanics. The nine contour lines are chosen at heights 10-7/2, where n= —1,0,1,..., 7. The parameters are Ey = 800, wy = 80, 2= 96 and T = 0.66; all in a.u. [167]. Reprinted with permission from [167]; Copyright 2001 by the American Physical Society.
Fig. 9. Comparison of the spatial probability distributions predicted by classical and quantum mechanics. The nine contour lines are chosen at heights 10-7/2, where n= —1,0,1,..., 7. The parameters are Ey = 800, wy = 80, 2= 96 and T = 0.66; all in a.u. [167]. Reprinted with permission from [167]; Copyright 2001 by the American Physical Society.
Fig. 10. Steady-state charge distributions. Top: the “triangular” spatial electron charge distribution with parameters: Eg = 150, a7 =5, Q = 1.57, and 10 000 quasiparticles with initial uncertainties Ax = 0.5, Av = 1. Bottom: the “four-bladed propeller” spatial electron charge distribution with parameters: Ey = 150, wy = 5, 2 = 1.257, and using the same number of quasiparticles and uncertainties as for the triangular case. All axes and parameter values are given in a.u. [169]. Reprinted with permission from [169]; Copyright 2002 by the American Physical Society.
Fig. 10. Steady-state charge distributions. Top: the “triangular” spatial electron charge distribution with parameters: Eg = 150, a7 =5, Q = 1.57, and 10 000 quasiparticles with initial uncertainties Ax = 0.5, Av = 1. Bottom: the “four-bladed propeller” spatial electron charge distribution with parameters: Ey = 150, wy = 5, 2 = 1.257, and using the same number of quasiparticles and uncertainties as for the triangular case. All axes and parameter values are given in a.u. [169]. Reprinted with permission from [169]; Copyright 2002 by the American Physical Society.
Fig. 11. The tightly-focused Gaussian-beam geometry and single-beam acceleration configuration. See the text for definitions [182]. Reprinted with permission from [182]; Copyright 2003 by the Springer Science and Business Media. Similarly, the magnetic field components are given by
Fig. 11. The tightly-focused Gaussian-beam geometry and single-beam acceleration configuration. See the text for definitions [182]. Reprinted with permission from [182]; Copyright 2003 by the Springer Science and Business Media. Similarly, the magnetic field components are given by
Fig. 12. Energy gain vs. the injection angle 0; corresponding to a scaled injection energy 9 = 10, z9 = —5 mm, and an initial phase constant Wo = 3°. The remaining parameters are: power P = 10 PW, wavelength 2 = 1 pm, and waist radius wg = 10 pm. Injection is into the laser focus and the interaction time, starting from injection at t = 0, is equivalent to Ay = 10650z, with 4 = w(t — z/c). In (b) we only zoom on a small portion of (a) [182]. Reprinted with permission from [182]; Copyright 2003 by the Springer Science and Business Media. where y is the Lorentz factor at the end of integration. A scheme like this may obviously be used as a booster for particles already accelerated to MeV energies before they are submitted to the high-intensity focus of a laser beam. The initial injection energy and, hence, initial forward momentum, is required to allow the electron to penetrate the laser focus and reach regions of maximum intensity. It seems to acquire momentum from the field in the form of a small number of violent impulses, to get captured by the beam fields, and to travel slightly off of the beam axis. Otherwise, it would be reflected with little energy gain (or even loss).
Fig. 12. Energy gain vs. the injection angle 0; corresponding to a scaled injection energy 9 = 10, z9 = —5 mm, and an initial phase constant Wo = 3°. The remaining parameters are: power P = 10 PW, wavelength 2 = 1 pm, and waist radius wg = 10 pm. Injection is into the laser focus and the interaction time, starting from injection at t = 0, is equivalent to Ay = 10650z, with 4 = w(t — z/c). In (b) we only zoom on a small portion of (a) [182]. Reprinted with permission from [182]; Copyright 2003 by the Springer Science and Business Media. where y is the Lorentz factor at the end of integration. A scheme like this may obviously be used as a booster for particles already accelerated to MeV energies before they are submitted to the high-intensity focus of a laser beam. The initial injection energy and, hence, initial forward momentum, is required to allow the electron to penetrate the laser focus and reach regions of maximum intensity. It seems to acquire momentum from the field in the form of a small number of violent impulses, to get captured by the beam fields, and to travel slightly off of the beam axis. Otherwise, it would be reflected with little energy gain (or even loss).
Fig. 13. A schematic diagram of electron injection into the region of overlap of two identical laser beams. Note that the coordinate axes of the individual beams are not marked explicitly. The understanding here, and elsewhere, is that E;, points in the direction of +x, Ez, points in the direction of +z, and so on [182]. Reprinted with permission from [182]; Copyright 2003 by the Springer Science and Business Media. It has recently been shown that, in the configuration discussed here, regions develop around the beam focus, due to the mechanism of focusing, over which the velocity of the electron is sub-luminous [186,187]. This lends support to the interpretation that the underlying mechanism of acceleration is most dominantly one of capture and violent acceleration [188,189]. The capture is mainly due to the transverse electric and longitudinal magnetic field components and the acceleration relies on the axial (longitudinal) electric field component.
Fig. 13. A schematic diagram of electron injection into the region of overlap of two identical laser beams. Note that the coordinate axes of the individual beams are not marked explicitly. The understanding here, and elsewhere, is that E;, points in the direction of +x, Ez, points in the direction of +z, and so on [182]. Reprinted with permission from [182]; Copyright 2003 by the Springer Science and Business Media. It has recently been shown that, in the configuration discussed here, regions develop around the beam focus, due to the mechanism of focusing, over which the velocity of the electron is sub-luminous [186,187]. This lends support to the interpretation that the underlying mechanism of acceleration is most dominantly one of capture and violent acceleration [188,189]. The capture is mainly due to the transverse electric and longitudinal magnetic field components and the acceleration relies on the axial (longitudinal) electric field component.
Fig. 14. Electron energy gain vs. the off-axis injection distance xg, as a result of interaction with two identical beams. The injection energy is yg = 10, the crossing half-angle is 0; = 6.5°, zg = —5 um, and the initial phase constant Wy = 3°. The remaining parameters and initial conditions are the same as in Fig. 12. The integration has been carried out over a time equivalent to 10* laser periods [182]. Reprinted with permission from [182]; Copyright 2003 by the Springer Science and Business Media.
Fig. 14. Electron energy gain vs. the off-axis injection distance xg, as a result of interaction with two identical beams. The injection energy is yg = 10, the crossing half-angle is 0; = 6.5°, zg = —5 um, and the initial phase constant Wy = 3°. The remaining parameters and initial conditions are the same as in Fig. 12. The integration has been carried out over a time equivalent to 10* laser periods [182]. Reprinted with permission from [182]; Copyright 2003 by the Springer Science and Business Media.
Fig. 15. Electron energy gain in a vacuum beat wave laser accelerator as a function of the initial scaled injection energy. The two beams copropagate along +z, 127° out of phase. The other parameters are: 7; = 1 um, A = 4; /1.1, laser power = 1 PW, wo, = wo2 = 6 Lum, injection is sideways a an angle 6; = 10° with the propagation direction and from a point in the polarization plane with a z9 = —5 mm coordinate, and integration is taker over a time equal to 10° laser periods [183]. Reprinted with permission from [183]; Copyright 2005 by the Elsevier.
Fig. 15. Electron energy gain in a vacuum beat wave laser accelerator as a function of the initial scaled injection energy. The two beams copropagate along +z, 127° out of phase. The other parameters are: 7; = 1 um, A = 4; /1.1, laser power = 1 PW, wo, = wo2 = 6 Lum, injection is sideways a an angle 6; = 10° with the propagation direction and from a point in the polarization plane with a z9 = —5 mm coordinate, and integration is taker over a time equal to 10° laser periods [183]. Reprinted with permission from [183]; Copyright 2005 by the Elsevier.
Fig. 16. The solid curves display temporal evolution of the atom scaled velocity. The dashed curves show variation of the scaled ponderomotiv potential sensed by the atom along its trajectory: (a) the laser intensity is below the critical value: Vj = 0.8 Vc, and (b) above the critical value Vo = 1.3 Ver. The initial velocity of the atom is zero. d@ = @, — @ 7, @ 1 and Wy are the frequencies of the counter-propagating laser wave: (1 — @2)t = 10, with t the pulse duration [202]. Reprinted with permission from [202]; Copyright 2003 by the Springer Science and Busines Media.
Fig. 16. The solid curves display temporal evolution of the atom scaled velocity. The dashed curves show variation of the scaled ponderomotiv potential sensed by the atom along its trajectory: (a) the laser intensity is below the critical value: Vj = 0.8 Vc, and (b) above the critical value Vo = 1.3 Ver. The initial velocity of the atom is zero. d@ = @, — @ 7, @ 1 and Wy are the frequencies of the counter-propagating laser wave: (1 — @2)t = 10, with t the pulse duration [202]. Reprinted with permission from [202]; Copyright 2003 by the Springer Science and Busines Media.
Fig. 17. Electron energy in a plane-wave laser field and a uniform magnetic field in the autoresonance laser acceleration regime vs. the forward distance over which interaction takes place. The continuous lines are linear polarization cases, and the dotted lines represent circular polarization. The parameters are wavelength 2 = 800 nm, number of field cycles 1.5, and Bs = 30T: (a) intensity J ~ 10/8 W/cm? org=1;(b) l= 1020 W/cm? or g = 10; and (c) J ~ 1022 W/ cm? or q = 100. Effectively, these are plots of &(1) vs. z(y) parameterized by 7. In other words, data for these plots have been produced by evaluating the functions &(7) and z(7) over values ranging between 0 and 3z [150]. Reprinted with permission from [150]; Copyright 2000 by the American Physical Society. Some ideas have been also suggested to achieve acceleration by breaking the plane-wave symmetry, and hence violating the Lawson—Woodward theorem, via the addition of extra static electric and magnetic fields. One proposal suggested adding a static electric field perpendicular to the laser propagation direction [204] more than a decade ago. The case of an electric field oriented anti-parallel to the propagation direction of the plane-wave laser beam has been
Fig. 17. Electron energy in a plane-wave laser field and a uniform magnetic field in the autoresonance laser acceleration regime vs. the forward distance over which interaction takes place. The continuous lines are linear polarization cases, and the dotted lines represent circular polarization. The parameters are wavelength 2 = 800 nm, number of field cycles 1.5, and Bs = 30T: (a) intensity J ~ 10/8 W/cm? org=1;(b) l= 1020 W/cm? or g = 10; and (c) J ~ 1022 W/ cm? or q = 100. Effectively, these are plots of &(1) vs. z(y) parameterized by 7. In other words, data for these plots have been produced by evaluating the functions &(7) and z(7) over values ranging between 0 and 3z [150]. Reprinted with permission from [150]; Copyright 2000 by the American Physical Society. Some ideas have been also suggested to achieve acceleration by breaking the plane-wave symmetry, and hence violating the Lawson—Woodward theorem, via the addition of extra static electric and magnetic fields. One proposal suggested adding a static electric field perpendicular to the laser propagation direction [204] more than a decade ago. The case of an electric field oriented anti-parallel to the propagation direction of the plane-wave laser beam has been
Fig. 18. Three-dimensional Monte Carlo simulation results for v22+ located at the center of the Gaussian laser beam of wavelength 1054 nm anda peak intensity of 8 x 1027) w / cm2. The beam waist is assumed to be 10 lim, and its time envelope comprises a 5-cycle turn-on and turn-off, as well as a 5-cycle flat top with constant amplitude: (a) electron energies attained vs. the ejection angles with respect to the z axis; and (b) electron energy distribution normalized to the peak of the distribution [50]. Reprinted with permission from [50]; Copyright 2002 by the American Physical Society. ti ett VEEULS LUIS UUL LU UY idler. We stress here that the plane-wave-based discussions of the acceleration schemes using added static electric and magnetic fields have not led to the conclusion that such schemes might be plausible in the near future. However, the analytic work done in this area has been good in building an intuitive picture in each case. The equations may be quite useful in discussions involving configurations of the sort encountered in contexts different from acceleration per se, like atomic laser-induced ionization and control of the associated HHG. The equations have certainly some role to play in guiding the numerical analysis that must ultimately be resorted to in the study of laser acceleration employing tightly-focused beams.
Fig. 18. Three-dimensional Monte Carlo simulation results for v22+ located at the center of the Gaussian laser beam of wavelength 1054 nm anda peak intensity of 8 x 1027) w / cm2. The beam waist is assumed to be 10 lim, and its time envelope comprises a 5-cycle turn-on and turn-off, as well as a 5-cycle flat top with constant amplitude: (a) electron energies attained vs. the ejection angles with respect to the z axis; and (b) electron energy distribution normalized to the peak of the distribution [50]. Reprinted with permission from [50]; Copyright 2002 by the American Physical Society. ti ett VEEULS LUIS UUL LU UY idler. We stress here that the plane-wave-based discussions of the acceleration schemes using added static electric and magnetic fields have not led to the conclusion that such schemes might be plausible in the near future. However, the analytic work done in this area has been good in building an intuitive picture in each case. The equations may be quite useful in discussions involving configurations of the sort encountered in contexts different from acceleration per se, like atomic laser-induced ionization and control of the associated HHG. The equations have certainly some role to play in guiding the numerical analysis that must ultimately be resorted to in the study of laser acceleration employing tightly-focused beams.
Fig. 19. Simulation results showing electron ejection angle as a function of electron energy for Ar!7+ ionized with an 800 nm, 30 fs laser pulse at an intensity of 5 x 102! w / cm2. The dashed line shows the prediction of Eq. (1): (a) results when longitudinal fields are excluded from the calculation; and (b) results when longitudinal fields are included [51]. Reprinted with permission from [51]; Copyright 2003 by the American Physical Society.
Fig. 19. Simulation results showing electron ejection angle as a function of electron energy for Ar!7+ ionized with an 800 nm, 30 fs laser pulse at an intensity of 5 x 102! w / cm2. The dashed line shows the prediction of Eq. (1): (a) results when longitudinal fields are excluded from the calculation; and (b) results when longitudinal fields are included [51]. Reprinted with permission from [51]; Copyright 2003 by the American Physical Society.
Fig. 21. Calculated ATI spectrum of helium for a 0.1291 a.u. field (585 TW/cm2) at 800 nm, along the polarization vector. At this intensity there is a strong plateau between 4 and 8 hartree. The fat curve shows the spectrum at a slightly different field (0.1274 a.u.), which is nearly identical at low energy, but has a much weaker plateau. The inset is a blowup of the indicated region, revealing how the ATI peaks shift ponderomotively througt a nearly constant envelope. For clarity the main plot shows only the envelope for the fat spectrum [225]. Reprinted with permission from [225] Copyright 1999 by the American Physical Society.
Fig. 21. Calculated ATI spectrum of helium for a 0.1291 a.u. field (585 TW/cm2) at 800 nm, along the polarization vector. At this intensity there is a strong plateau between 4 and 8 hartree. The fat curve shows the spectrum at a slightly different field (0.1274 a.u.), which is nearly identical at low energy, but has a much weaker plateau. The inset is a blowup of the indicated region, revealing how the ATI peaks shift ponderomotively througt a nearly constant envelope. For clarity the main plot shows only the envelope for the fat spectrum [225]. Reprinted with permission from [225] Copyright 1999 by the American Physical Society.
Fig. 22. Angular distribution of photoelectrons as measured from the direction of propagation of the circularly polarized field. The circularly polarized laser has a wavelength of 10.6 jim and an intensity of 1.1 x 10!4w / cm2. The non-relativistic distribution is symmetric about 90°, and the relativistic distribution is displaced forward [92]. Reprinted with permission from [92]; Copyright 1990 by the American Physical Society. electrons. Electron energies of hundreds of MeVs or even a few GeVs can be reached [50,51]. Usually, when an ultra-intense laser pulse blasts a solid target or a gas jet, a plasma consisting of HCIs and electrons is formed. Once a plasma is produced, the subsequent dynamics is governed by the laser—plasma interactions. Laser—plasma physics is well reviewed as one of the active branches of modern physics [231]. Since neutral targets may be quickly blasted, atomic physicists are interested in bringing the tightly-bound systems to interact with relativistic laser fields [98]. Highly charged ions are such systems in which the strongly-bound electrons may sustain the peak intensity of a super-strong laser pulse. HCIs can be generated either via many atomic collision processes [49] or directly by the super-strong laser pulse itself [233]. With the currently used techniques of producing HCIs one can generate any charge state of any atomic species [234]. Measuring a harmonic (radiation) spectrum requires a high-enough target density (10!7-10!8 cm7?). However, since the density of currently available HCIs is not high enough to allow for measurement of HHG radiation spectra from intense laser-driven ions, experiments with ions are suitable for the study of the relativistic tunneling ionization spectra [98,99].
Fig. 22. Angular distribution of photoelectrons as measured from the direction of propagation of the circularly polarized field. The circularly polarized laser has a wavelength of 10.6 jim and an intensity of 1.1 x 10!4w / cm2. The non-relativistic distribution is symmetric about 90°, and the relativistic distribution is displaced forward [92]. Reprinted with permission from [92]; Copyright 1990 by the American Physical Society. electrons. Electron energies of hundreds of MeVs or even a few GeVs can be reached [50,51]. Usually, when an ultra-intense laser pulse blasts a solid target or a gas jet, a plasma consisting of HCIs and electrons is formed. Once a plasma is produced, the subsequent dynamics is governed by the laser—plasma interactions. Laser—plasma physics is well reviewed as one of the active branches of modern physics [231]. Since neutral targets may be quickly blasted, atomic physicists are interested in bringing the tightly-bound systems to interact with relativistic laser fields [98]. Highly charged ions are such systems in which the strongly-bound electrons may sustain the peak intensity of a super-strong laser pulse. HCIs can be generated either via many atomic collision processes [49] or directly by the super-strong laser pulse itself [233]. With the currently used techniques of producing HCIs one can generate any charge state of any atomic species [234]. Measuring a harmonic (radiation) spectrum requires a high-enough target density (10!7-10!8 cm7?). However, since the density of currently available HCIs is not high enough to allow for measurement of HHG radiation spectra from intense laser-driven ions, experiments with ions are suitable for the study of the relativistic tunneling ionization spectra [98,99].
Fig. 23. The center-of-mass evolution of the electronic wave packet for the cases with B-field (“full circles”) and without B-field (“open diamonds’) The laser parameters involve a wavelength of 248 nm, an intensity of 10!7 w / cm?,a three-cycle linear turn-on and a ten-cycle constant amplitude duration. A hole around the ionic core for Be>+ (Z =4, s = 1, k = 10.7) as indicated by the arrow is clearly observable in the time-evolution of the center-of-mass of the electronic wave packet, when the B-field is considered [235]. Reprinted with permission from [235]; Copyright 1999 by the EDP Sciences.
Fig. 23. The center-of-mass evolution of the electronic wave packet for the cases with B-field (“full circles”) and without B-field (“open diamonds’) The laser parameters involve a wavelength of 248 nm, an intensity of 10!7 w / cm?,a three-cycle linear turn-on and a ten-cycle constant amplitude duration. A hole around the ionic core for Be>+ (Z =4, s = 1, k = 10.7) as indicated by the arrow is clearly observable in the time-evolution of the center-of-mass of the electronic wave packet, when the B-field is considered [235]. Reprinted with permission from [235]; Copyright 1999 by the EDP Sciences.
Fig. 24. Lifetime (in femtoseconds) of atomic hydrogen as a function of intensity (in a.u., Jg = 3.51 x 10/6 Wiem?), for various c (in a.u.). The numbers adjacent to the points give the corresponding value of ao (in a.u.) [237]. Reprinted with permission from [237]; Copyright 2002 by the IOP Publishing Limited.
Fig. 24. Lifetime (in femtoseconds) of atomic hydrogen as a function of intensity (in a.u., Jg = 3.51 x 10/6 Wiem?), for various c (in a.u.). The numbers adjacent to the points give the corresponding value of ao (in a.u.) [237]. Reprinted with permission from [237]; Copyright 2002 by the IOP Publishing Limited.
Fig. 25. Ionization probability as a function of electric-field strength for Coulombic hydrogen (dashed line), helium (+) (thin solid line), boron (4+) (dotted line), and neon (9+) (thick solid line). The laser pulses used were sin? (3-4-3) type. In both graphs, the frequency has been scaled proportional to Z? in an attempt to reproduce the conditions necessary for stabilization to occur. In graph (a) @ = Z? and in (b) w = 5Z? [248]. Reprinted with permission from [248]; Copyright 2002 by the American Physical Society.
Fig. 25. Ionization probability as a function of electric-field strength for Coulombic hydrogen (dashed line), helium (+) (thin solid line), boron (4+) (dotted line), and neon (9+) (thick solid line). The laser pulses used were sin? (3-4-3) type. In both graphs, the frequency has been scaled proportional to Z? in an attempt to reproduce the conditions necessary for stabilization to occur. In graph (a) @ = Z? and in (b) w = 5Z? [248]. Reprinted with permission from [248]; Copyright 2002 by the American Physical Society.
Fig. 26. Schematic of the laser setup consisting of two equal handed, circularly polarized, counter-propagating laser pulses with equal intensities. Electric and magnetic fields are parallel, point along the arrows, rotate in space along the propagation axis, and vary sinusoidal in time and 90° out of phase [256]. Reprinted with permission from [256]; Copyright 2004 by the American Physical Society.
Fig. 26. Schematic of the laser setup consisting of two equal handed, circularly polarized, counter-propagating laser pulses with equal intensities. Electric and magnetic fields are parallel, point along the arrows, rotate in space along the propagation axis, and vary sinusoidal in time and 90° out of phase [256]. Reprinted with permission from [256]; Copyright 2004 by the American Physical Society.
Fig. 27. Weakly relativistic motion of the center-of-mass of the tunneled electron wavepacket. (x) and (z) denote the center-of-mass in the laset polarization direction and the laser propagation direction, respectively. The arrows indicate the direction of motion of the center-of-mass of the electron wavepacket [252]. Reprinted with permission from [252]; Copyright 2002 by the American Institute of Physics.
Fig. 27. Weakly relativistic motion of the center-of-mass of the tunneled electron wavepacket. (x) and (z) denote the center-of-mass in the laset polarization direction and the laser propagation direction, respectively. The arrows indicate the direction of motion of the center-of-mass of the electron wavepacket [252]. Reprinted with permission from [252]; Copyright 2002 by the American Institute of Physics.
ere Hg denotes the standard non-relativistic Hamiltonian in the Schrédinger form, where p is the momentum operato id A(z, f) is the time-dependent vector potential of the olarized along the x-axis and propagating in the +z-direct eld component and do not make the dipole approximatio ie xz plane. They considered atoms in the single active e ion. For the vector potential, the authors inc ) electrons and, thus, are easily available today via lasers [233] or to highest accuracy via shooting t ‘in foils [49]. A soft-core potential [262] was used to model the Coulomb field experienced by the active electron of : ultiply-charged ion, i.e., V(x, z) = —k/Vs + x24 22.1 [he parameters k and s are functions of the f positive charges Z sensed by the electron. On the other hand, s compensates for the effect of possib id reduced distances of the electronic wavepacket to the ionic core in two- rather than three-dimensional calculations Iso, k is adapted in such a way as to allow one to obtain the correct ionization energy of the system of interest witl 1 effective charge Z on the ionic core. The static field of the ionic core is expressed by the gradient of the potentia VV (x, z) and E’ (x, z, t) stands for this field plus the laser field E(z, t). The following term Hp in Eqs. (73) indicate: ie coupling of the laser magnetic field B to the electron spin as described by the Pauli matrix o. The sum Ho + Hy ives the Hamiltonian in the well known Pauli equation. Further, in Eq. (73) Hin denotes the leading term for the ‘lativistic mass increase, and Hp is the well-known Darwin term. Finally the term in the Hamiltonian, of most interes sre, iS Hg , which stands for the spin-orbit coupling. aser field, with an electric field component F(z, f) linearly ude the magnetic n, in performing a two-dimensional numerical integration i ectron approximation which are pre-ionized by the order o he atoms throug! effective numbe: e inner electron: Coanciderino the central notential Vy 7) the firect term in Hl. in Ba (72) dicanneare hecance Vv (—_VV(r 7))\)—(
ere Hg denotes the standard non-relativistic Hamiltonian in the Schrédinger form, where p is the momentum operato id A(z, f) is the time-dependent vector potential of the olarized along the x-axis and propagating in the +z-direct eld component and do not make the dipole approximatio ie xz plane. They considered atoms in the single active e ion. For the vector potential, the authors inc ) electrons and, thus, are easily available today via lasers [233] or to highest accuracy via shooting t ‘in foils [49]. A soft-core potential [262] was used to model the Coulomb field experienced by the active electron of : ultiply-charged ion, i.e., V(x, z) = —k/Vs + x24 22.1 [he parameters k and s are functions of the f positive charges Z sensed by the electron. On the other hand, s compensates for the effect of possib id reduced distances of the electronic wavepacket to the ionic core in two- rather than three-dimensional calculations Iso, k is adapted in such a way as to allow one to obtain the correct ionization energy of the system of interest witl 1 effective charge Z on the ionic core. The static field of the ionic core is expressed by the gradient of the potentia VV (x, z) and E’ (x, z, t) stands for this field plus the laser field E(z, t). The following term Hp in Eqs. (73) indicate: ie coupling of the laser magnetic field B to the electron spin as described by the Pauli matrix o. The sum Ho + Hy ives the Hamiltonian in the well known Pauli equation. Further, in Eq. (73) Hin denotes the leading term for the ‘lativistic mass increase, and Hp is the well-known Darwin term. Finally the term in the Hamiltonian, of most interes sre, iS Hg , which stands for the spin-orbit coupling. aser field, with an electric field component F(z, f) linearly ude the magnetic n, in performing a two-dimensional numerical integration i ectron approximation which are pre-ionized by the order o he atoms throug! effective numbe: e inner electron: Coanciderino the central notential Vy 7) the firect term in Hl. in Ba (72) dicanneare hecance Vv (—_VV(r 7))\)—(
Fig. 28. (i) Radiation spectrum of the laser-driven ion close to its lowest resonances. The first row corresponds to the Pauli modeled system, and the second row is for the case where spin-orbit coupling including the relativistic mass shift and Zitterbewegung is taken into account. Figs. (a), (b) and (c) are associated, respectively, with transitions from |le) to |g), |3e) to |g), and |3e) to |le) (see also (ii)). The spectral lines split into doublets ((a) and (b)) and four-line structure (c) configurations due to the spin-orbit interaction. A line shift arising from the mass shift Hxjn is also observable. The laser parameters involve a wavelength of 527 nm, an intensity of 7 x 10!6w / cm?,a 5-cycle linear turn-on and a 100-cycle duration with constant amplitude. The parameters for the ionic core are s = 1, k = 93.5. (ii) The schematic diagram of state-splitting induced by the intense laser enhanced spin-orbit interaction. Note that non-symmetric states split as opposed to symmetric states. Transitions (a), (b) and (c) are associated with the corresponding spectral lines in (i) [254]. Reprinted with permission from [254]; Copyright 2001 by the American Physical Society.
Fig. 28. (i) Radiation spectrum of the laser-driven ion close to its lowest resonances. The first row corresponds to the Pauli modeled system, and the second row is for the case where spin-orbit coupling including the relativistic mass shift and Zitterbewegung is taken into account. Figs. (a), (b) and (c) are associated, respectively, with transitions from |le) to |g), |3e) to |g), and |3e) to |le) (see also (ii)). The spectral lines split into doublets ((a) and (b)) and four-line structure (c) configurations due to the spin-orbit interaction. A line shift arising from the mass shift Hxjn is also observable. The laser parameters involve a wavelength of 527 nm, an intensity of 7 x 10!6w / cm?,a 5-cycle linear turn-on and a 100-cycle duration with constant amplitude. The parameters for the ionic core are s = 1, k = 93.5. (ii) The schematic diagram of state-splitting induced by the intense laser enhanced spin-orbit interaction. Note that non-symmetric states split as opposed to symmetric states. Transitions (a), (b) and (c) are associated with the corresponding spectral lines in (i) [254]. Reprinted with permission from [254]; Copyright 2001 by the American Physical Society.
Fig. 29. Multiply charged ion yield for Xe at a laser intensity of about 10!8 W/cm?. The “simultaneous” results include contribution fron non-sequential multiple-ionization, while the “sequential” do not [109]. Reprinted with permission from [109]; Copyright 2001 by the Ameri can Physical Society.
Fig. 29. Multiply charged ion yield for Xe at a laser intensity of about 10!8 W/cm?. The “simultaneous” results include contribution fron non-sequential multiple-ionization, while the “sequential” do not [109]. Reprinted with permission from [109]; Copyright 2001 by the Ameri can Physical Society.
Fig. 30. Classical trajectory calculation results for the ionization probability of Xe!9+ 5 Xe29+ (and return to the Xe29+ core) as a function of the initial phase of the field at the ionization time. The curves have been scaled to | at their peak values [109]. Reprinted with permission from [109]; Copyright 2001 by the American Physical Society. in which the Schrédinger dynamics of the electron is still valid while the magnetic field of the laser radiation i: fully taken into account without the dipole approximation. However, Ref. [110] deals with the tunneling regime whil Ref. [271] takes up the barrier suppression regime of ionization. The ionization dynamics of a model helium atom it both references is considered based on the extended version of a model suggested in Ref. [274]. Electrons are assumec distinguishable in this model, exchange interaction is neglected, and a two-electron wavefunction is approximated by the product ansatz: ¥(r1, r2, t) = ¢, (11, t)b> (12, t), where the ¢; (rj, f) with i = | or 2, are the respective one-electrot wavefunctions, and the antisymmetry requirement is neglected. In Ref. [110] the outer electron is modeled to move in time-independent potential of the ionic core screened by the inner electron in its initial state, neglecting the polarizatior effect of the ionic core on the dynamics of the first-ionized outer electron. The inner electron is assumed to be influencec by the electron-electron interaction in order to describe the re-collision process. Thus, the Schrédinger equation fo: the tunneling dynamics during the non-sequential ionization in Ref. [110] reads (in atomic units)
Fig. 30. Classical trajectory calculation results for the ionization probability of Xe!9+ 5 Xe29+ (and return to the Xe29+ core) as a function of the initial phase of the field at the ionization time. The curves have been scaled to | at their peak values [109]. Reprinted with permission from [109]; Copyright 2001 by the American Physical Society. in which the Schrédinger dynamics of the electron is still valid while the magnetic field of the laser radiation i: fully taken into account without the dipole approximation. However, Ref. [110] deals with the tunneling regime whil Ref. [271] takes up the barrier suppression regime of ionization. The ionization dynamics of a model helium atom it both references is considered based on the extended version of a model suggested in Ref. [274]. Electrons are assumec distinguishable in this model, exchange interaction is neglected, and a two-electron wavefunction is approximated by the product ansatz: ¥(r1, r2, t) = ¢, (11, t)b> (12, t), where the ¢; (rj, f) with i = | or 2, are the respective one-electrot wavefunctions, and the antisymmetry requirement is neglected. In Ref. [110] the outer electron is modeled to move in time-independent potential of the ionic core screened by the inner electron in its initial state, neglecting the polarizatior effect of the ionic core on the dynamics of the first-ionized outer electron. The inner electron is assumed to be influencec by the electron-electron interaction in order to describe the re-collision process. Thus, the Schrédinger equation fo: the tunneling dynamics during the non-sequential ionization in Ref. [110] reads (in atomic units)
Fig. 31. (a) and (b) Angular distribution of the ionized inner and outer electrons including the influence of the magnetic field. (c) and (d) Angular distribution of the inner electron with and without the influence of the magnetic field while excluding the force of the laser field on the inner electron. The laser intensities used are 2.19 x 10/5 W/cm? (a), (c) and 1.4 x 10/5 W/cm? in (b), (d), respectively, with wavelength of 1053 nm and a 3-cycle pulse shape with a 1|-cycle linear turn-on and turn-off in (a) and (c) and a 5-cycle sin? pulse-shape in (b) and (d), respectively. The positive polarization and propagation directions are 0° and 90°, respectively. The total area has been normalized to one and the inner electron yield in (a) has been multiplied by four [110]. Reprinted with permission from [110]; Copyright 2002 by the IOP Publishing Limited.
Fig. 31. (a) and (b) Angular distribution of the ionized inner and outer electrons including the influence of the magnetic field. (c) and (d) Angular distribution of the inner electron with and without the influence of the magnetic field while excluding the force of the laser field on the inner electron. The laser intensities used are 2.19 x 10/5 W/cm? (a), (c) and 1.4 x 10/5 W/cm? in (b), (d), respectively, with wavelength of 1053 nm and a 3-cycle pulse shape with a 1|-cycle linear turn-on and turn-off in (a) and (c) and a 5-cycle sin? pulse-shape in (b) and (d), respectively. The positive polarization and propagation directions are 0° and 90°, respectively. The total area has been normalized to one and the inner electron yield in (a) has been multiplied by four [110]. Reprinted with permission from [110]; Copyright 2002 by the IOP Publishing Limited.
Fig. 32. Probability density of the inner electron. The inset depicts the three different observation times for the employed pulse shape of three cycle: with one-cycle linear turn-on and turn-off each. The intensity used is 2.19 x 10)5 W/cm? and the wavelength is 1053 nm [110]. Reprinted witk permission from [110]; Copyright 2002 by the IOP Publishing Limited.
Fig. 32. Probability density of the inner electron. The inset depicts the three different observation times for the employed pulse shape of three cycle: with one-cycle linear turn-on and turn-off each. The intensity used is 2.19 x 10)5 W/cm? and the wavelength is 1053 nm [110]. Reprinted witk permission from [110]; Copyright 2002 by the IOP Publishing Limited.
Fig. 33. Center-of-mass motion of the inner and outer electrons. In (a) and (c) the repulsion between two electrons is taken into account, while it is neglected in (b) and (d). The laser frequency is | a.u., the peak electric field strength 5 a.u. for (a) and (b), and 12 a.u. for (c) and (d) [271]. Reprinted with permission from [271]; Copyright 2003 by the EDP Sciences.
Fig. 33. Center-of-mass motion of the inner and outer electrons. In (a) and (c) the repulsion between two electrons is taken into account, while it is neglected in (b) and (d). The laser frequency is | a.u., the peak electric field strength 5 a.u. for (a) and (b), and 12 a.u. for (c) and (d) [271]. Reprinted with permission from [271]; Copyright 2003 by the EDP Sciences.
Fig. 34. Angular distribution of the ionized inner and outer electrons. The positive laser polarization direction is along 0°, the propagation direction 90°. In (a) and (c) the repulsion between two electrons is taken into account, while it is neglected in (b) and (d). The laser frequency is | a.u. and the peak electric field strength is 5 a.u. for (a) and (b), and 12 a.u. for (c) and (d). The total area of the sum of both angular distributions is normalized to unity for each picture [271]. Reprinted with permission from [271]; Copyright 2003 by the EDP Sciences.
Fig. 34. Angular distribution of the ionized inner and outer electrons. The positive laser polarization direction is along 0°, the propagation direction 90°. In (a) and (c) the repulsion between two electrons is taken into account, while it is neglected in (b) and (d). The laser frequency is | a.u. and the peak electric field strength is 5 a.u. for (a) and (b), and 12 a.u. for (c) and (d). The total area of the sum of both angular distributions is normalized to unity for each picture [271]. Reprinted with permission from [271]; Copyright 2003 by the EDP Sciences.
Fig. 35. Harmonic yield obtained with the single active electron (SAE) approximation and with two active electrons (TAE). The intensity employed is2x 105 w / cm? ata laser wavelength of 798 nm [280]. Reprinted with permission from [280]; Copyright 2001 by the American Physical Society.
Fig. 35. Harmonic yield obtained with the single active electron (SAE) approximation and with two active electrons (TAE). The intensity employed is2x 105 w / cm? ata laser wavelength of 798 nm [280]. Reprinted with permission from [280]; Copyright 2001 by the American Physical Society.
Fig. 36. (a) Overview: contour plots at tf = 7.645, 8.195, 9.495, 12.095 a.u. corresponding to times before the first scattering, shortly after it, at the lower point of return, and after the second scattering, close to the upper point of return. The four dashed rectangles (some only partly visible) mark the dynamical grid boundaries for each of these situations. The thick dots indicate the positions of two Sn°+ ions, which are located right on the horizontal axis at positions 16.5 and 30.0a.u. The solid line depicts the trajectory of the expectation value of the particle’s spatial coordinate. We omit any enlarged view of the initially Gaussian packet at t = 7.645 a.u., but provide them for the other three cases as follows; (b) enlarged view at tf = 8.195 a.u. showing single scattering fringes shortly after the scattering event; (c) enlarged view at t = 9.495 a.u. that depicts the same single scattering fringes somewhat later in order to illustrate the growth of the distance between any two fringes (compare with (b)); and (d) enlarge view at tf = 12.095 a.u. which displays interference from two separate scattering events with crossed fringes on the left side and the unperturbed structure from single scattering at the first ion only on the right side. In all cases 20 contour lines are shown for ak with constant steps within —1.15 2 log IW? = — 4 [257]. Reprinted with permission from [257]; Copyright 2003 by the American Physical Society.
Fig. 36. (a) Overview: contour plots at tf = 7.645, 8.195, 9.495, 12.095 a.u. corresponding to times before the first scattering, shortly after it, at the lower point of return, and after the second scattering, close to the upper point of return. The four dashed rectangles (some only partly visible) mark the dynamical grid boundaries for each of these situations. The thick dots indicate the positions of two Sn°+ ions, which are located right on the horizontal axis at positions 16.5 and 30.0a.u. The solid line depicts the trajectory of the expectation value of the particle’s spatial coordinate. We omit any enlarged view of the initially Gaussian packet at t = 7.645 a.u., but provide them for the other three cases as follows; (b) enlarged view at tf = 8.195 a.u. showing single scattering fringes shortly after the scattering event; (c) enlarged view at t = 9.495 a.u. that depicts the same single scattering fringes somewhat later in order to illustrate the growth of the distance between any two fringes (compare with (b)); and (d) enlarge view at tf = 12.095 a.u. which displays interference from two separate scattering events with crossed fringes on the left side and the unperturbed structure from single scattering at the first ion only on the right side. In all cases 20 contour lines are shown for ak with constant steps within —1.15 2 log IW? = — 4 [257]. Reprinted with permission from [257]; Copyright 2003 by the American Physical Society.
Fig. 37. (a) Overview: contour plots at time t = 12.270 a.u. after two scattering at an ensemble of six Sn°9+ ions (thick dots) centered symmetrically around a further one at position (0,30) a.u. The dashed rectangle marks the grid boundary and the solid line depicts the trajectory of the expectation value of the electron’s spatial coordinate; and (b) this enlarged view of the wave packet in (a) illustrates how the two scattering events at seven ions modify the regularity in the interference pattern. In both graphs, 20 contour lines are shown for | |? with constant steps within —1.15 > log | 2 >-4 [257]. Reprinted with permission from [257]; Copyright 2003 by the American Physical Society.
Fig. 37. (a) Overview: contour plots at time t = 12.270 a.u. after two scattering at an ensemble of six Sn°9+ ions (thick dots) centered symmetrically around a further one at position (0,30) a.u. The dashed rectangle marks the grid boundary and the solid line depicts the trajectory of the expectation value of the electron’s spatial coordinate; and (b) this enlarged view of the wave packet in (a) illustrates how the two scattering events at seven ions modify the regularity in the interference pattern. In both graphs, 20 contour lines are shown for | |? with constant steps within —1.15 > log | 2 >-4 [257]. Reprinted with permission from [257]; Copyright 2003 by the American Physical Society.
ig. 38. Schematic diagram showing a thin layer in a strong laser field propagating parallel to its surface [297]. Reprinted with permission from 297]; Copyright 2002 by the IOP Publishing Limited. regime whenever R, <L_. Usually, strong collective effects during laser—solid interaction arise due to the strong ponderomotive pressure of the laser pulse on the solid surface that launches a collisionless shock wave into the bulk of the medium and causes rippling and chaotic oscillations of the solid surface after the wave-breaking. Due to the applied condition (81) a transverse Gaussian distribution of the laser intensity has only a very small gradient in the vicinity of the layer that significantly reduces the ponderomotive forces. In fact, the ponderomotive force due to the laser pulse’s transverse profile can be estimated to be in ~ ZUp/Iz, with Up = mc~q? /4 the ponderomotive energy, Z the effective ion charge, and /; a length determining the transverse gradient of the laser intensity, and q is the dimensionless intensity parameter. In the case of the bulk medium it is equal to the laser waist size. In the case of a layer obeying condition (81) this length is much larger, /, = wy /L, and the gradient is much smaller. It causes ion displacements that may be determined from
ig. 38. Schematic diagram showing a thin layer in a strong laser field propagating parallel to its surface [297]. Reprinted with permission from 297]; Copyright 2002 by the IOP Publishing Limited. regime whenever R, <L_. Usually, strong collective effects during laser—solid interaction arise due to the strong ponderomotive pressure of the laser pulse on the solid surface that launches a collisionless shock wave into the bulk of the medium and causes rippling and chaotic oscillations of the solid surface after the wave-breaking. Due to the applied condition (81) a transverse Gaussian distribution of the laser intensity has only a very small gradient in the vicinity of the layer that significantly reduces the ponderomotive forces. In fact, the ponderomotive force due to the laser pulse’s transverse profile can be estimated to be in ~ ZUp/Iz, with Up = mc~q? /4 the ponderomotive energy, Z the effective ion charge, and /; a length determining the transverse gradient of the laser intensity, and q is the dimensionless intensity parameter. In the case of the bulk medium it is equal to the laser waist size. In the case of a layer obeying condition (81) this length is much larger, /, = wy /L, and the gradient is much smaller. It causes ion displacements that may be determined from
Fig. 39. The dynamics of the ions (a) and electrons (b) in propagation (X) and polarization direction (Y). In the considered thin layer regime o! interaction, the transversal size of the electron oscillation amplitude exceeds the layer width. The electron and the ion trajectories are traced up tc 5.3 fs after switching on the interaction with the laser pulse. The layer consists of a Pb sample with density nj; = 3 x 1022 cm~3 and ion mas: M =3.8 x 10°m. The layer width is L | =5 x 10~6 cm, and the length L=4x 10~4 cm. The laser radiation parameters are: wavelength 27, = 248 nv (KrF), pulse duration t; = 10 fs, intensity J = 102! w / cm?. The electrons are extracted from the layer and oscillate with large amplitude while th motion of the ions is not noticeable [298]. Reprinted with permission from [298]; Copyright 2005 by the American Physical Society.
Fig. 39. The dynamics of the ions (a) and electrons (b) in propagation (X) and polarization direction (Y). In the considered thin layer regime o! interaction, the transversal size of the electron oscillation amplitude exceeds the layer width. The electron and the ion trajectories are traced up tc 5.3 fs after switching on the interaction with the laser pulse. The layer consists of a Pb sample with density nj; = 3 x 1022 cm~3 and ion mas: M =3.8 x 10°m. The layer width is L | =5 x 10~6 cm, and the length L=4x 10~4 cm. The laser radiation parameters are: wavelength 27, = 248 nv (KrF), pulse duration t; = 10 fs, intensity J = 102! w / cm?. The electrons are extracted from the layer and oscillate with large amplitude while th motion of the ions is not noticeable [298]. Reprinted with permission from [298]; Copyright 2005 by the American Physical Society.
Fig. 40. Dynamics of the ion (a), and electron (b) for the same situation as in Fig. 39 but with tracing the trajectories for 20 fs after switching on the 10 fs laser pulse. The laser pulse has traveled up to the end of the layer and the ionic motion is still not noticeable [298]. Reprinted with permission from [298]; Copyright 2005 by the American Physical Society.
Fig. 40. Dynamics of the ion (a), and electron (b) for the same situation as in Fig. 39 but with tracing the trajectories for 20 fs after switching on the 10 fs laser pulse. The laser pulse has traveled up to the end of the layer and the ionic motion is still not noticeable [298]. Reprinted with permission from [298]; Copyright 2005 by the American Physical Society.
Fig. 41. The ion (a) and electron (b) dynamics in the thick layer regime. Here the layer width exceeds the transversal size of the electron oscillation amplitude. The electron and the ion trajectories are traced up to 143 fs after switching on the interaction with the laser pulse. The laser radiation parameters are: wavelength 17, = 800 nm (Ti-Sapphire), pulse duration tT; = 30 fs, intensity J =5 x 102! W/cm?. The layer is assumed to consist of a potassium sample with density nj = 1.3 x 1022 cm~?. The layer width is L; = 1.4 x 10~3 cm, and the length L = 10-2 cm. The electrons oscillate mainly within the layer while ionic motion is still not noticeable [298]. Reprinted with permission from [298]; Copyright 2005 by the American Physical Society.
Fig. 41. The ion (a) and electron (b) dynamics in the thick layer regime. Here the layer width exceeds the transversal size of the electron oscillation amplitude. The electron and the ion trajectories are traced up to 143 fs after switching on the interaction with the laser pulse. The laser radiation parameters are: wavelength 17, = 800 nm (Ti-Sapphire), pulse duration tT; = 30 fs, intensity J =5 x 102! W/cm?. The layer is assumed to consist of a potassium sample with density nj = 1.3 x 1022 cm~?. The layer width is L; = 1.4 x 10~3 cm, and the length L = 10-2 cm. The electrons oscillate mainly within the layer while ionic motion is still not noticeable [298]. Reprinted with permission from [298]; Copyright 2005 by the American Physical Society.
Fig. 42. The schematic picture depicting the resonant multi-photon absorption of a multiply charged ion (ground state |0) and excited states |1) and |2)) in a super-strong laser field with angular frequency @ [253]. Reprinted with permission from [253]; Copyright 2002 by the EDP Sciences.
Fig. 42. The schematic picture depicting the resonant multi-photon absorption of a multiply charged ion (ground state |0) and excited states |1) and |2)) in a super-strong laser field with angular frequency @ [253]. Reprinted with permission from [253]; Copyright 2002 by the EDP Sciences.
Fig. 43. The radiation spectra for a model ion of o/t+ exposed to intense laser pulses. The laser wavelength is 248 nm and the intensities are: (a 10!7 W/cm?; (b) 5 x 10!7 W/cm?; and (c) 10!8 W/cm?. The laser pulse has a shape of a trapezoid with five-cycle turn-on and a flat section of ter cycles [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 43. The radiation spectra for a model ion of o/t+ exposed to intense laser pulses. The laser wavelength is 248 nm and the intensities are: (a 10!7 W/cm?; (b) 5 x 10!7 W/cm?; and (c) 10!8 W/cm?. The laser pulse has a shape of a trapezoid with five-cycle turn-on and a flat section of ter cycles [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 44. The segment of the multi-photon spectrum from Z = 12 model ions which are exposed to an intense laser pulse of intensity 7 x 10!6w / cm? and the laser wavelength 527 nm. The laser pulse has a 5.25-cycle turn-on, followed by a 100-cycle period with constant amplitude. The segment shows the transition between the bound states |4e) < |g). The solid and dotted lines represent the cases without and with the leading relativistic correction, respectively. A large redshift of 0.82@ for this resonant transition is visible when the leading relativistic correction is taken into account. It can be attributed to the relativistic Stark shift of the energy levels of the ion [254]. Reprinted with permission from [254]; Copyright 2001 by the American Physical Society.
Fig. 44. The segment of the multi-photon spectrum from Z = 12 model ions which are exposed to an intense laser pulse of intensity 7 x 10!6w / cm? and the laser wavelength 527 nm. The laser pulse has a 5.25-cycle turn-on, followed by a 100-cycle period with constant amplitude. The segment shows the transition between the bound states |4e) < |g). The solid and dotted lines represent the cases without and with the leading relativistic correction, respectively. A large redshift of 0.82@ for this resonant transition is visible when the leading relativistic correction is taken into account. It can be attributed to the relativistic Stark shift of the energy levels of the ion [254]. Reprinted with permission from [254]; Copyright 2001 by the American Physical Society.
Fig. 45. The region of parameters g and Z where substantial HHG is possible using a hydrogen-like ion. The region above the dashed curve correspond: to the condition of Eq. (90), below the thick curve to the condition of Eq. (91), and below the thin curve to the condition of Eq. (92). Therefore tunneling harmonics from a hydrogen-like ion will be generated in principle for those values of g and Z that belong to the shaded region. The HHC intensity is maximal near the thick curve. However, it is significantly damped away from this thick curve, i.e., with increasing Z. The maximal lase: intensity when optimal HHG (i.e. in the region close to the OBI regime) is still possible, corresponds to the point marked by O.
Fig. 45. The region of parameters g and Z where substantial HHG is possible using a hydrogen-like ion. The region above the dashed curve correspond: to the condition of Eq. (90), below the thick curve to the condition of Eq. (91), and below the thin curve to the condition of Eq. (92). Therefore tunneling harmonics from a hydrogen-like ion will be generated in principle for those values of g and Z that belong to the shaded region. The HHC intensity is maximal near the thick curve. However, it is significantly damped away from this thick curve, i.e., with increasing Z. The maximal lase: intensity when optimal HHG (i.e. in the region close to the OBI regime) is still possible, corresponds to the point marked by O.
Fig. 46. The harmonic spectrum emitted from Z = 3 ions, with parameters k = 6.48 and ge = 1 characterizing the potential (soft-core Coulomb) The laser intensity is 2.5 x 1016 w / cm”, and the KrF laser wavelength with 248 nm is applied. The laser pulse has a linear 10-cycle turn-on and i: followed by a 10-cycle duration with constant amplitude. The 131st harmonic is visible in the cutoff frequency. For these laser parameters considered the usual plateau range 3.17Up is equal to ~ 32.8keV [254]. Reprinted with permission from [254]; Copyright 2001 by the American Physica Society.
Fig. 46. The harmonic spectrum emitted from Z = 3 ions, with parameters k = 6.48 and ge = 1 characterizing the potential (soft-core Coulomb) The laser intensity is 2.5 x 1016 w / cm”, and the KrF laser wavelength with 248 nm is applied. The laser pulse has a linear 10-cycle turn-on and i: followed by a 10-cycle duration with constant amplitude. The 131st harmonic is visible in the cutoff frequency. For these laser parameters considered the usual plateau range 3.17Up is equal to ~ 32.8keV [254]. Reprinted with permission from [254]; Copyright 2001 by the American Physica Society.
Fig. 47. The harmonic spectrum from intense laser-driven Be++ (soft-core parameter k = 10.7 and ge = 1) ina laser pulse of intensity 10!7 w / cm? ions [254]. Reprinted with permission from [254]; Copyright 2001 by the American Physical Society. where A, is the vector potential of the laser field. As in the treatment of the non-relativistic case, the full Green’s function is approximated by the Klein—Gordon positive-energy Volkov propagator
Fig. 47. The harmonic spectrum from intense laser-driven Be++ (soft-core parameter k = 10.7 and ge = 1) ina laser pulse of intensity 10!7 w / cm? ions [254]. Reprinted with permission from [254]; Copyright 2001 by the American Physical Society. where A, is the vector potential of the laser field. As in the treatment of the non-relativistic case, the full Green’s function is approximated by the Klein—Gordon positive-energy Volkov propagator
Fig. 48. Harmonic emission rates as functions of the harmonic order for Li2+ (Ip = 122.45 eV) and Be3+ (Ip = 217.72 eV) ions, exposed to a Ti:sapphire laser (800 nm) with the intensities given in each panel. The blue curves indicate the non-relativistic calculation for comparison, while the black curves represent the applicable relativistic results [314]. Reprinted with permission from [314]; Copyright 2002 by the Springer Science and Business Media.
Fig. 48. Harmonic emission rates as functions of the harmonic order for Li2+ (Ip = 122.45 eV) and Be3+ (Ip = 217.72 eV) ions, exposed to a Ti:sapphire laser (800 nm) with the intensities given in each panel. The blue curves indicate the non-relativistic calculation for comparison, while the black curves represent the applicable relativistic results [314]. Reprinted with permission from [314]; Copyright 2002 by the Springer Science and Business Media.
Fig. 49. The spectrum of the “non-tunneling harmonics” for an electron in the soft-core model potential (V(x, z) = —k/Vqe + x2 +22), with k = 37.45 and qe = 1, exposed to a laser with an intensity of 1.9 x 108 W/cm? (q = 0.29) and a wavelength 2 = 248 nm. The arrow at the lower right marks the harmonic order having an energy equal to the ionization potential, [yp = 870.72 eV, corresponding to that of hydrogen-like ion of O7* [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 49. The spectrum of the “non-tunneling harmonics” for an electron in the soft-core model potential (V(x, z) = —k/Vqe + x2 +22), with k = 37.45 and qe = 1, exposed to a laser with an intensity of 1.9 x 108 W/cm? (q = 0.29) and a wavelength 2 = 248 nm. The arrow at the lower right marks the harmonic order having an energy equal to the ionization potential, [yp = 870.72 eV, corresponding to that of hydrogen-like ion of O7* [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 50. The average position (x) of the bound electronic wave packet along the laser-polarization direction (x axis) as a function of the interaction time. The driving electric field is sinusoidal in time. The parameters are the same as in Fig. 49, but the calculation is for a small box in order to eliminate the effect of the ionized part of the wave function [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited. by visual inspection of Fig. 50, determining the frequency by counting the number of small wiggles within the time interval 6.47 ~ 6.6T, where T is the laser period. This yields a harmonic order of about 60, which agrees very well with the cutoff position of Fig. 49.
Fig. 50. The average position (x) of the bound electronic wave packet along the laser-polarization direction (x axis) as a function of the interaction time. The driving electric field is sinusoidal in time. The parameters are the same as in Fig. 49, but the calculation is for a small box in order to eliminate the effect of the ionized part of the wave function [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited. by visual inspection of Fig. 50, determining the frequency by counting the number of small wiggles within the time interval 6.47 ~ 6.6T, where T is the laser period. This yields a harmonic order of about 60, which agrees very well with the cutoff position of Fig. 49.
Fig. 51. Plot of the absolute square of the electron acceleration ax (t, «) (cf. Eq. (13) in [255]) vs. t for 6 < t < 7 in units of the laser period T. The harmonic signal is proportional to the time-integration of |ax (f, c)|?. Results are shown for three of the harmonics: the 29th (upper panel), 53rd (middle panel) and 61st (lower panel) [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 51. Plot of the absolute square of the electron acceleration ax (t, «) (cf. Eq. (13) in [255]) vs. t for 6 < t < 7 in units of the laser period T. The harmonic signal is proportional to the time-integration of |ax (f, c)|?. Results are shown for three of the harmonics: the 29th (upper panel), 53rd (middle panel) and 61st (lower panel) [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 52. Position (a) and total energy gain (b) of a classical electron as a function of time for the conditions of Fig. 49 with initial conditions as described in the text: namely, the electron starts just outside of the saddle point of the effective potential (97) at a time when the field is maximal [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 52. Position (a) and total energy gain (b) of a classical electron as a function of time for the conditions of Fig. 49 with initial conditions as described in the text: namely, the electron starts just outside of the saddle point of the effective potential (97) at a time when the field is maximal [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 53. Illustration of the surfing mechanism described in the text. The effective potential (97) is plotted for the parameters of Fig. 49 at three different times. The time ¢ = 0.257 corresponds to the maximum of the field. The solid dots represent the positions as a function of time of an electron that is released with zero velocity just outside of the effective potential saddle at the time t = 0.25T, whose orbit is depicted in Fig. 52. Notice how from t =0.25T = to tot = 0.42T = 1, the electron moves from just outside to just inside the saddle point [255]. Reprinted with permission from [255] Copyright 2002 by the IOP Publishing Limited.
Fig. 53. Illustration of the surfing mechanism described in the text. The effective potential (97) is plotted for the parameters of Fig. 49 at three different times. The time ¢ = 0.257 corresponds to the maximum of the field. The solid dots represent the positions as a function of time of an electron that is released with zero velocity just outside of the effective potential saddle at the time t = 0.25T, whose orbit is depicted in Fig. 52. Notice how from t =0.25T = to tot = 0.42T = 1, the electron moves from just outside to just inside the saddle point [255]. Reprinted with permission from [255] Copyright 2002 by the IOP Publishing Limited.
Fig. 54. Harmonic spectrum for the soft-core potential with k = 15.32 and ge = 0.1. The lower panel shows a blow-up of the central part of the spectrum. It illuminates the transition from the purely odd harmonics of the low-energy part of the spectrum to the two series of harmonics characteristic of the upper part. The laser parameters are same as those of Fig. 49 [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 54. Harmonic spectrum for the soft-core potential with k = 15.32 and ge = 0.1. The lower panel shows a blow-up of the central part of the spectrum. It illuminates the transition from the purely odd harmonics of the low-energy part of the spectrum to the two series of harmonics characteristic of the upper part. The laser parameters are same as those of Fig. 49 [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 55. Comparison of the harmonic spectrum of Fig. 49 (filled circles) to a standard (tunneling-recombination) harmonic spectrum in neutral helium for the same wavelength 1 = 248 nm (open squares) as well as for the longer wavelength of 800 nm (filled diamonds), the latter two at intensities near saturation, 10!5 W/cm2 and 6 x 10!4 W/cm2, respectively. All spectra have been obtained by running the same code and are plotted on the same scale [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 55. Comparison of the harmonic spectrum of Fig. 49 (filled circles) to a standard (tunneling-recombination) harmonic spectrum in neutral helium for the same wavelength 1 = 248 nm (open squares) as well as for the longer wavelength of 800 nm (filled diamonds), the latter two at intensities near saturation, 10!5 W/cm2 and 6 x 10!4 W/cm2, respectively. All spectra have been obtained by running the same code and are plotted on the same scale [255]. Reprinted with permission from [255]; Copyright 2002 by the IOP Publishing Limited.
Fig. 56. The ionization wave function probability from Net? ion at the center of a # f/1.5 laser beam with intensity of 2 x 10!7 W/cm? at peak focus and wavelength 800 nm (¢ = 0.3) [323]. Reprinted with permission from [323]; Copyright 2004 by the Optical Society of America.
Fig. 56. The ionization wave function probability from Net? ion at the center of a # f/1.5 laser beam with intensity of 2 x 10!7 W/cm? at peak focus and wavelength 800 nm (¢ = 0.3) [323]. Reprinted with permission from [323]; Copyright 2004 by the Optical Society of America.
Fig. 57. The ionization wave function probability from Art! ion at the center of an laser beam # f/1.5, with intensity 1 x 1019 W/cm? at peak focus and wavelength 800 nm (q = 2.15). The x-axis extends to the e~? beam radius and the z-axis to one Rayleigh length [323]. Reprinted with permission from [323]; Copyright 2004 by the Optical Society of America.
Fig. 57. The ionization wave function probability from Art! ion at the center of an laser beam # f/1.5, with intensity 1 x 1019 W/cm? at peak focus and wavelength 800 nm (q = 2.15). The x-axis extends to the e~? beam radius and the z-axis to one Rayleigh length [323]. Reprinted with permission from [323]; Copyright 2004 by the Optical Society of America.
Fig. 58. The radiation field component along the laser polarization for three trajectories from ionization of Art! at laser intensity of 10!9w /om2 and wavelength 800 nm (q = 2.15) in (a). In (b) the frequency-resolved power spectrum per atom? is shown for | classical electron (gray, dashed) and the ionization from | (solid), 10 (gray, solid), and 1000 (dotted) atoms along the laser propagation axis [324]. Reprinted with permission from [324]; Copyright 2005 by the IOP Publishing Limited.
Fig. 58. The radiation field component along the laser polarization for three trajectories from ionization of Art! at laser intensity of 10!9w /om2 and wavelength 800 nm (q = 2.15) in (a). In (b) the frequency-resolved power spectrum per atom? is shown for | classical electron (gray, dashed) and the ionization from | (solid), 10 (gray, solid), and 1000 (dotted) atoms along the laser propagation axis [324]. Reprinted with permission from [324]; Copyright 2005 by the IOP Publishing Limited.
Fig. 59. Schematic diagram displaying positronium dynamics in an intense laser field. The bound system depicted by the density of its wave function may ionize in the laser field. Once free, both electron e~ and positron et could be described as classical particles. Their trajectories are shown by the solid lines. The electric field E accelerates the particles in opposite directions while the Lorentz force, due to the magnetic field H, leads to an identical drift for both in the propagation direction ky. Without an initial center-of-mass motion these trajectories are symmetric and thus both particles overlap periodically giving rise to possible recombination and thus high-frequency light emission [111]. Reprinted with permission from [111]; Copyright 2004 by the American Physical Society.
Fig. 59. Schematic diagram displaying positronium dynamics in an intense laser field. The bound system depicted by the density of its wave function may ionize in the laser field. Once free, both electron e~ and positron et could be described as classical particles. Their trajectories are shown by the solid lines. The electric field E accelerates the particles in opposite directions while the Lorentz force, due to the magnetic field H, leads to an identical drift for both in the propagation direction ky. Without an initial center-of-mass motion these trajectories are symmetric and thus both particles overlap periodically giving rise to possible recombination and thus high-frequency light emission [111]. Reprinted with permission from [111]; Copyright 2004 by the American Physical Society.
Fig. 60. Differential high-harmonic spectral intensities (dJ, /dQ) sin~2@ at odd harmonic numbers n as a function of the photon energy of the respective harmonic; 0 being the harmonic emission angle with respect to the polarization direction: (a) displays the spectrum of a single positronium with ca. 4.1 million harmonics via a coherent field of intensity J = 8.63 x 10/2 w if cm? ata wavelength of 2 = 100 um close to that provided by the CLIO free-electron laser [330], ¢g = 0.25 and positronium atom is initially at rest; and (b) displays the spectrum of an Ne’*+ ion in an Nd:YAG laser field with 2 = 1054nm at J = 1.5 x 10!” W/cm? (q = 0.35). At those parameters both systems yield approximately the same cutoff frequency, though with a yield in the spectral intensity of the cutoff harmonics differing by about 15 orders of magnitude [111]. Reprinted with permission from [111]; Copyright 2004 by the American Physical Society. where |0) is the ground state, and the coefficients are
Fig. 60. Differential high-harmonic spectral intensities (dJ, /dQ) sin~2@ at odd harmonic numbers n as a function of the photon energy of the respective harmonic; 0 being the harmonic emission angle with respect to the polarization direction: (a) displays the spectrum of a single positronium with ca. 4.1 million harmonics via a coherent field of intensity J = 8.63 x 10/2 w if cm? ata wavelength of 2 = 100 um close to that provided by the CLIO free-electron laser [330], ¢g = 0.25 and positronium atom is initially at rest; and (b) displays the spectrum of an Ne’*+ ion in an Nd:YAG laser field with 2 = 1054nm at J = 1.5 x 10!” W/cm? (q = 0.35). At those parameters both systems yield approximately the same cutoff frequency, though with a yield in the spectral intensity of the cutoff harmonics differing by about 15 orders of magnitude [111]. Reprinted with permission from [111]; Copyright 2004 by the American Physical Society. where |0) is the ground state, and the coefficients are
Fig. 61. Feynman diagrams displaying y-photon emission via electron—positron annihilation: (a) multiphoton annihilation in super strong laser fielc atq ~ (m2/oz Q)!/3 with emission of n ~ m2 /or Q laser photons, along with one y-quantum; (b) annihilation ina laser field at g < (m2/oz Qy' with emission of laser photons, along with two y-quanta; and (c) annihilation of ortho-positronium with emission of three photons without the las: field. The bold lines correspond to the electron (positron) Volkov states, i.e., involving a laser field, and the dashed lines correspond to the emitte photons [111]. Reprinted with permission from [111]; Copyright 2004 by the American Physical Society.
Fig. 61. Feynman diagrams displaying y-photon emission via electron—positron annihilation: (a) multiphoton annihilation in super strong laser fielc atq ~ (m2/oz Q)!/3 with emission of n ~ m2 /or Q laser photons, along with one y-quantum; (b) annihilation ina laser field at g < (m2/oz Qy' with emission of laser photons, along with two y-quanta; and (c) annihilation of ortho-positronium with emission of three photons without the las: field. The bold lines correspond to the electron (positron) Volkov states, i.e., involving a laser field, and the dashed lines correspond to the emitte photons [111]. Reprinted with permission from [111]; Copyright 2004 by the American Physical Society.
J,(¢) is an ordinary Bessel function, ro is the classical electron radius, and p = ag > with the Bohr radius ap. In a weak laser field with g <1 the formula for dwind given above matches the known result in Ref. [328]. The total probability of the laser induced annihilation for ortho-positronium is of the order of wind — > wird x = 16 pcq’, and that of para-positronium is Wind x x me pc. For the sake of comparison, we also invoke the corresponding spontaneous annihilation probabilities Wo? ~ are pc and Wp? x nro pc, with the fine structure constant « = e*/fic [326]. The ratio of the ortho-positronium lifetime in the presence of a laser field to that without it is «/q7. In laser fields with <1 of interest here, the ortho-positronium lifetime is of order 10~? s (10~!° s for para-positronium), which makes it well suited for HHG. Laser-induced annihilation according to Fig. 61(b) can be a source of high-frequency radiation. The spectrum is expected to be narrow and to be higher than the corresponding spontaneous radiation spectrum. In fact, the spectral bandwidth of the usual radiation from the spontaneous annihilation is w,.. Let us estimate the bandwidth of the radiation due to the laser-induced annihilation. With annihilation being accompanied by stimulated emission of laser photons, energy and momentum conservation allow emission of yy quanta only at certain frequencies in each direction. The spectral bandwidth Aw = max{Aq@p, va - co} of the induced y radiation due to annihilation is determined either by the laser Doppler line-width Awp ~ 10~* at room temperature, or by the range of the harmonic order As of the involved stimulated photons. This leads to a bandwidth much narrower than that of the spontaneous process, A@ < @<.. The ratio of stimulated to spontaneous y radiation probabilities
J,(¢) is an ordinary Bessel function, ro is the classical electron radius, and p = ag > with the Bohr radius ap. In a weak laser field with g <1 the formula for dwind given above matches the known result in Ref. [328]. The total probability of the laser induced annihilation for ortho-positronium is of the order of wind — > wird x = 16 pcq’, and that of para-positronium is Wind x x me pc. For the sake of comparison, we also invoke the corresponding spontaneous annihilation probabilities Wo? ~ are pc and Wp? x nro pc, with the fine structure constant « = e*/fic [326]. The ratio of the ortho-positronium lifetime in the presence of a laser field to that without it is «/q7. In laser fields with <1 of interest here, the ortho-positronium lifetime is of order 10~? s (10~!° s for para-positronium), which makes it well suited for HHG. Laser-induced annihilation according to Fig. 61(b) can be a source of high-frequency radiation. The spectrum is expected to be narrow and to be higher than the corresponding spontaneous radiation spectrum. In fact, the spectral bandwidth of the usual radiation from the spontaneous annihilation is w,.. Let us estimate the bandwidth of the radiation due to the laser-induced annihilation. With annihilation being accompanied by stimulated emission of laser photons, energy and momentum conservation allow emission of yy quanta only at certain frequencies in each direction. The spectral bandwidth Aw = max{Aq@p, va - co} of the induced y radiation due to annihilation is determined either by the laser Doppler line-width Awp ~ 10~* at room temperature, or by the range of the harmonic order As of the involved stimulated photons. This leads to a bandwidth much narrower than that of the spontaneous process, A@ < @<.. The ratio of stimulated to spontaneous y radiation probabilities
Fig. 62. (a) Gain in the EI regime according to Eq. (112) (thick line). The electron and ion densities are ne = 10nj; = 1073 cm73, the laser wavelength is Ay, = 3.5 x 10~6 cm, the laser intensity parameter is g = 6, the ion structure period a = 3.5 x 10~8 cm, the transversal size of the layer is L) =7 x 10-6 cm, u/a = 1;/a = 0.087 and the momentum spread of the electron beam is 4/mc = 10~*. The thin line describes the corresponding gain for the multi-photon Compton radiation amplification in a plasma [338] for the same parameters. The dotted line depicts the function J, (Q) +» Ky—n(© multiplied by 2.5 x 107 which peaks at the multi-photon parameter 7 in Eq. (111); and (b) a window of the fine structure of the gain corresponding to the thick line in (a) [296]. Reprinted with permission from [296]; Copyright 2001 by the American Physical Society.
Fig. 62. (a) Gain in the EI regime according to Eq. (112) (thick line). The electron and ion densities are ne = 10nj; = 1073 cm73, the laser wavelength is Ay, = 3.5 x 10~6 cm, the laser intensity parameter is g = 6, the ion structure period a = 3.5 x 10~8 cm, the transversal size of the layer is L) =7 x 10-6 cm, u/a = 1;/a = 0.087 and the momentum spread of the electron beam is 4/mc = 10~*. The thin line describes the corresponding gain for the multi-photon Compton radiation amplification in a plasma [338] for the same parameters. The dotted line depicts the function J, (Q) +» Ky—n(© multiplied by 2.5 x 107 which peaks at the multi-photon parameter 7 in Eq. (111); and (b) a window of the fine structure of the gain corresponding to the thick line in (a) [296]. Reprinted with permission from [296]; Copyright 2001 by the American Physical Society.
Fig. 63. (a) Sketch of a spectrum with periodic phase matching; and (b) HHG spectral intensity from a crystal layer W.. = dW/dadQ scaled by the respective peak value for one ion wo ) = (dw) /d@ dQ) max, Where x = w/o, — 10°. The narrow features (thin lines) correspond to the coherent HHG from the periodic structure, while the wide features (thick lines) correspond to the HHG from one ion multiplied by number of ions and magnified 3 x 10!4 times, corresponding by intensity to the incoherent HHG from disordered ions. The solid lines correspond to Nz = 2 x 105 and the dashed lines to Nz = 10°. The parameters employed are: Ly = Ly = 20 um, Ay = 800nm, tz, = 30fs, 0 = 0, ay = ay = az = 4.5 A, n — ny = 0.057 [297]. Reprinted with permission from [297]; Copyright 2002 by the IOP Publishing Limited.
Fig. 63. (a) Sketch of a spectrum with periodic phase matching; and (b) HHG spectral intensity from a crystal layer W.. = dW/dadQ scaled by the respective peak value for one ion wo ) = (dw) /d@ dQ) max, Where x = w/o, — 10°. The narrow features (thin lines) correspond to the coherent HHG from the periodic structure, while the wide features (thick lines) correspond to the HHG from one ion multiplied by number of ions and magnified 3 x 10!4 times, corresponding by intensity to the incoherent HHG from disordered ions. The solid lines correspond to Nz = 2 x 105 and the dashed lines to Nz = 10°. The parameters employed are: Ly = Ly = 20 um, Ay = 800nm, tz, = 30fs, 0 = 0, ay = ay = az = 4.5 A, n — ny = 0.057 [297]. Reprinted with permission from [297]; Copyright 2002 by the IOP Publishing Limited.
Fig. 64. Scattered electron energy spectra of the nonlinear Compton scattering experiment at the final focus test beam facility at SLAC. (a) Data and simulations for 42-mJ infrared laser pulses. (b) and (c) Data and simulations (solid curves) for infrared (b) and green (c) laser pulses, scaled to standard values of the interaction geometry. The dashed lines show the simulation for multiple linear Compton scattering only [352]. Reprinted with permission from [352]; Copyright 1996 by the American Physical Society.
Fig. 64. Scattered electron energy spectra of the nonlinear Compton scattering experiment at the final focus test beam facility at SLAC. (a) Data and simulations for 42-mJ infrared laser pulses. (b) and (c) Data and simulations (solid curves) for infrared (b) and green (c) laser pulses, scaled to standard values of the interaction geometry. The dashed lines show the simulation for multiple linear Compton scattering only [352]. Reprinted with permission from [352]; Copyright 1996 by the American Physical Society.
Fig. 65. Schematic diagram of the setup (top), Lorentz and Dirac—Lorentz electron dynamics (middle), and Lorentz and Dirac—Lorentz backscattered spectra (bottom). The parameters used in our notation reads A = g, € = 7r9/Ao, where Ao is the laser wavelength in the rest frame of electron, is the laser phase, and A¢@ is the total phase change in the laser pulse, the electron energy is 48 GeV, laser wavelength 1 ttm [145]. Reprinted with permission from [145]; Copyright 1996 by the American Physical Society.
Fig. 65. Schematic diagram of the setup (top), Lorentz and Dirac—Lorentz electron dynamics (middle), and Lorentz and Dirac—Lorentz backscattered spectra (bottom). The parameters used in our notation reads A = g, € = 7r9/Ao, where Ao is the laser wavelength in the rest frame of electron, is the laser phase, and A¢@ is the total phase change in the laser pulse, the electron energy is 48 GeV, laser wavelength 1 ttm [145]. Reprinted with permission from [145]; Copyright 1996 by the American Physical Society.
‘ig. 66. (a)-left column: (i) The contribution F(r) of the radiation-reaction force to the acceleration in the polarization direction of the laser field y-direction) (full curve) and the corresponding evolution of the y component (broken curve) of the electron as a function of time ¢ during interaction vith a seven-cycle constant-amplitude laser pulse with intensity 102° w / cm? and angular frequency @ = 0.086 a.u. for arbitrary scaling. Parts (ii) nd (ili) show the two contributions F\,q(¢) and F'ys(¢) to the acceleration in the y-direction from the radiation-reaction force in the same scaling as n (i) from the terms in the derivative of the laser field and the square of the laser field, respectively. (b)-right column: same as (a) with y replaced x (propagation direction of the laser pulse) [373]. Reprinted with permission from [373]; Copyright 1998 by the IOP Publishing Limited.
‘ig. 66. (a)-left column: (i) The contribution F(r) of the radiation-reaction force to the acceleration in the polarization direction of the laser field y-direction) (full curve) and the corresponding evolution of the y component (broken curve) of the electron as a function of time ¢ during interaction vith a seven-cycle constant-amplitude laser pulse with intensity 102° w / cm? and angular frequency @ = 0.086 a.u. for arbitrary scaling. Parts (ii) nd (ili) show the two contributions F\,q(¢) and F'ys(¢) to the acceleration in the y-direction from the radiation-reaction force in the same scaling as n (i) from the terms in the derivative of the laser field and the square of the laser field, respectively. (b)-right column: same as (a) with y replaced x (propagation direction of the laser pulse) [373]. Reprinted with permission from [373]; Copyright 1998 by the IOP Publishing Limited.
Fig. 67. The Monte Carlo radiation spectrum as a function of the multiple number n of the applied laser frequency wy ((i) Spy; S_x without radiation-reaction; (ii) S;-y; S;x with radiation-reaction; and (iii) Say; Sax the difference, the latter minus the former, for a hydrogen atom in a laser field with 10!9 Wcm~? intensity and j= 0.086 a.u. angular frequency. The initial state of hydrogen is simulated using a micro-canonical ensemble of 1000 members, while the time evolution is evaluated without the influence of the nucleus due to essentially instantaneous ionization. The spectrum is evaluated in the direction perpendicular to the polarization and propagation directions of the laser pulse, to show most clearly the effect on the red-shift. Part (a)-left column, shows the spectrum arising from the radiated electric field polarized along the polarization direction y of the laser field, while (b)-right column, considers the components polarized in the propagation direction x of the laser pulse [373]. Reprinted with permission from [373]; Copyright 1998 by the IOP Publishing Limited.
Fig. 67. The Monte Carlo radiation spectrum as a function of the multiple number n of the applied laser frequency wy ((i) Spy; S_x without radiation-reaction; (ii) S;-y; S;x with radiation-reaction; and (iii) Say; Sax the difference, the latter minus the former, for a hydrogen atom in a laser field with 10!9 Wcm~? intensity and j= 0.086 a.u. angular frequency. The initial state of hydrogen is simulated using a micro-canonical ensemble of 1000 members, while the time evolution is evaluated without the influence of the nucleus due to essentially instantaneous ionization. The spectrum is evaluated in the direction perpendicular to the polarization and propagation directions of the laser pulse, to show most clearly the effect on the red-shift. Part (a)-left column, shows the spectrum arising from the radiated electric field polarized along the polarization direction y of the laser field, while (b)-right column, considers the components polarized in the propagation direction x of the laser pulse [373]. Reprinted with permission from [373]; Copyright 1998 by the IOP Publishing Limited.
Fig. 68. (a) Number of positrons vs. momentum for laser-on and for laser-off pulses scaled to the laser-on pulses; (b) spectrum of signal positrons obtained by subtracting the laser-off from the laser-on distribution. The curve is the model calculation; and (c) and (d) are the same as (a) and (b) but choosing g > 0.216 data [353]. Reprinted with permission from [353]; Copyright 1997 by the American Physical Society.
Fig. 68. (a) Number of positrons vs. momentum for laser-on and for laser-off pulses scaled to the laser-on pulses; (b) spectrum of signal positrons obtained by subtracting the laser-off from the laser-on distribution. The curve is the model calculation; and (c) and (d) are the same as (a) and (b) but choosing g > 0.216 data [353]. Reprinted with permission from [353]; Copyright 1997 by the American Physical Society.
Fig. 69. Dependence of the positron rate on the field strength parameter g normalized to the number of Compton scatters. Lines are simulations fot two-step processes (solid line) and trident processes (dashed line) [353]. Reprinted with permission from [353]; Copyright 1997 by the American Physical Society.
Fig. 69. Dependence of the positron rate on the field strength parameter g normalized to the number of Compton scatters. Lines are simulations fot two-step processes (solid line) and trident processes (dashed line) [353]. Reprinted with permission from [353]; Copyright 1997 by the American Physical Society.
Fig. 70. Time evolution of the number density of pairs. In strong fields, particles accumulate, leading to the almost complete occupation of available momentum space. In weak fields, creation and annihilation occur in tune with the laser frequency [393]. Reprinted with permission from [393]; Copyright 2001 by the American Physical Society. the logarithm of the fermion determinant [386], were applied in investigating this problem. A review of most of this work may be found in the book [387]. The short-pulse and focused-laser-beam effects in this process are considered in [388]. Conditions for experimental realization of this process are very demanding, with the feasibility analysis in [389] indicating that the best hopes are connected with X-FEL.
Fig. 70. Time evolution of the number density of pairs. In strong fields, particles accumulate, leading to the almost complete occupation of available momentum space. In weak fields, creation and annihilation occur in tune with the laser frequency [393]. Reprinted with permission from [393]; Copyright 2001 by the American Physical Society. the logarithm of the fermion determinant [386], were applied in investigating this problem. A review of most of this work may be found in the book [387]. The short-pulse and focused-laser-beam effects in this process are considered in [388]. Conditions for experimental realization of this process are very demanding, with the feasibility analysis in [389] indicating that the best hopes are connected with X-FEL.
Fig. 72. Feynman diagrams for: (a) light-by-light scattering; and (b) photon splitting. Fig. 71. Peak particle number density versus laser field strength. There is a striking qualitative change at Ey ~ 0.25E¢, which marks the onset of particle accumulation [394]. Reprinted with permission from [394]; Copyright 2002 by the American Physical Society.
Fig. 72. Feynman diagrams for: (a) light-by-light scattering; and (b) photon splitting. Fig. 71. Peak particle number density versus laser field strength. There is a striking qualitative change at Ey ~ 0.25E¢, which marks the onset of particle accumulation [394]. Reprinted with permission from [394]; Copyright 2002 by the American Physical Society.
Fig. 73. Temporal distribution of the number of detected pulses at several distances of the detector from the target. The expected delays for neutrons with two different energies are indicated by the two straight lines. The second peak in each readout represents 2.45 MeV D-D fusion neutrons (adapted from [432] by its authors). Reprinted with permission from [432]; Copyright 1998 by the American Physical Society.
Fig. 73. Temporal distribution of the number of detected pulses at several distances of the detector from the target. The expected delays for neutrons with two different energies are indicated by the two straight lines. The second peak in each readout represents 2.45 MeV D-D fusion neutrons (adapted from [432] by its authors). Reprinted with permission from [432]; Copyright 1998 by the American Physical Society.
Fig. 74. (a) Typical arrangement for experiment on ion-induced reactions and photonuclear activation. Characteristic y peaks from a BT; secondary target when exposed to ions from a 27Al primary target, which was (b) unheated and (c) heated to 390 °C are shown [438]. Reprinted with permission from [438]; Copyright 2003 by the AAAS.
Fig. 74. (a) Typical arrangement for experiment on ion-induced reactions and photonuclear activation. Characteristic y peaks from a BT; secondary target when exposed to ions from a 27Al primary target, which was (b) unheated and (c) heated to 390 °C are shown [438]. Reprinted with permission from [438]; Copyright 2003 by the AAAS.
Fig. 75. Layout of the deuterium cluster fusion experiment [288]. Reprinted with permission from [288]; Copyright 1999 by the Macmillan Publishers Ltd.
Fig. 75. Layout of the deuterium cluster fusion experiment [288]. Reprinted with permission from [288]; Copyright 1999 by the Macmillan Publishers Ltd.
Fig. 76. The time-of-flight spectrum of particles detected on the neutron detectors for three plasma-detector separations (1, 2.5 and 3.2 m). Particle flight time is shown along the top axes. The presence of 2.45 MeV neutrons is clearly visible in all these spectra [288]. Reprinted with permission from [288]; Copyright 1999 by the Macmillan Publishers Ltd.
Fig. 76. The time-of-flight spectrum of particles detected on the neutron detectors for three plasma-detector separations (1, 2.5 and 3.2 m). Particle flight time is shown along the top axes. The presence of 2.45 MeV neutrons is clearly visible in all these spectra [288]. Reprinted with permission from [288]; Copyright 1999 by the Macmillan Publishers Ltd.
Fig. 77. Average deuterium ion energy (a) and total ion yield (b) as a function of laser energy for both D2 (open circles and squares) and CD, (filled circles and squares) cluster plasmas. The ion number and energy dependence on pulse energy were each fit to a power law dependence (shown by lines) [23]. Reprinted with permission from [23]; Copyright 2004 by the American Institute of Physics.
Fig. 77. Average deuterium ion energy (a) and total ion yield (b) as a function of laser energy for both D2 (open circles and squares) and CD, (filled circles and squares) cluster plasmas. The ion number and energy dependence on pulse energy were each fit to a power law dependence (shown by lines) [23]. Reprinted with permission from [23]; Copyright 2004 by the American Institute of Physics.
in Cluster targets. The work of Ref. [447] is devoted to the experimental an for peak laser intensities above 10'* W/cm?. It studies the pulse energy was 10J and the pulse duration was varied aser—cluster experiments in which the neutron yield follow is mainly due to the increase of ion yield with increasing shown, experimentally and numerically, that the ion energi pulse duration (see Fig. 80). This suggests that the cluster ex aser intensity, even in the d determined only by the ratio of the cluster ionization time to The ionization time is defined as the time during which the threshold up to the second threshold value for the complet exceeds the space charge field of the fully stripped cluster. W. d computational investigation of laser—cluster interactions resultant fusion yield as a function of the laser and target parameters to find optimal conditions for performance. The experiment used the JanUSP laser of LLNL. The maximum from 100fs to 1 ps. The work explains a feature of the s a quadratic power law dependence on pulse energy. This aser pulse energy. What is more important is that it was es vary only weakly by changing pulse energy at a given plosion mechanism in this regime is unaffected by the peak omain of relativistic intensities. Moreover, the ion energies, due to the cluster explosion, are its expansion time due to Coulomb explosion (see Fig. 81). aser intensity rises from the value of the atomic ionization e ion charge uncovering, i.e. when the laser electric field hen the ionization time is shorter than the cluster expansion time then in the rising edge of the laser pulse the cluster wil be completely stripped, after which a Coulomb explosion will occur. While in the opposite case, the cluster ionization process will continue during expansion, which will result in lower kinetic energy for the ions. The ionization time depends on the laser pulse duration (see Fig. 82). Consequently, the cluster explosion energy strongly depends on the pulse duration. Fig. 81 indicates also that the explosion energy is only weakly enhanced by the pulse duration reduction below 200 fs, when the cluster’s complete ionization occurs instantaneously. Meanwhile, the cluster explosion energy is rather insensitive to changes of the laser intensity above Fig. 78. Fusion neutron yield from exploding D> (open circles) and CD, cluster (filled circles) plasmas as a function of pulse energy. The solid lines are the least mean squares fit to a power law dependence of the yield on pulse energy [23]. Reprinted with permission from [23]; Copyright 2004 by the American Institute of Physics.
in Cluster targets. The work of Ref. [447] is devoted to the experimental an for peak laser intensities above 10'* W/cm?. It studies the pulse energy was 10J and the pulse duration was varied aser—cluster experiments in which the neutron yield follow is mainly due to the increase of ion yield with increasing shown, experimentally and numerically, that the ion energi pulse duration (see Fig. 80). This suggests that the cluster ex aser intensity, even in the d determined only by the ratio of the cluster ionization time to The ionization time is defined as the time during which the threshold up to the second threshold value for the complet exceeds the space charge field of the fully stripped cluster. W. d computational investigation of laser—cluster interactions resultant fusion yield as a function of the laser and target parameters to find optimal conditions for performance. The experiment used the JanUSP laser of LLNL. The maximum from 100fs to 1 ps. The work explains a feature of the s a quadratic power law dependence on pulse energy. This aser pulse energy. What is more important is that it was es vary only weakly by changing pulse energy at a given plosion mechanism in this regime is unaffected by the peak omain of relativistic intensities. Moreover, the ion energies, due to the cluster explosion, are its expansion time due to Coulomb explosion (see Fig. 81). aser intensity rises from the value of the atomic ionization e ion charge uncovering, i.e. when the laser electric field hen the ionization time is shorter than the cluster expansion time then in the rising edge of the laser pulse the cluster wil be completely stripped, after which a Coulomb explosion will occur. While in the opposite case, the cluster ionization process will continue during expansion, which will result in lower kinetic energy for the ions. The ionization time depends on the laser pulse duration (see Fig. 82). Consequently, the cluster explosion energy strongly depends on the pulse duration. Fig. 81 indicates also that the explosion energy is only weakly enhanced by the pulse duration reduction below 200 fs, when the cluster’s complete ionization occurs instantaneously. Meanwhile, the cluster explosion energy is rather insensitive to changes of the laser intensity above Fig. 78. Fusion neutron yield from exploding D> (open circles) and CD, cluster (filled circles) plasmas as a function of pulse energy. The solid lines are the least mean squares fit to a power law dependence of the yield on pulse energy [23]. Reprinted with permission from [23]; Copyright 2004 by the American Institute of Physics.
Fig. 79. (a) Single-shot ion spectrum registered on a microchannel-plate (Thomson parabola spectrometer) from a heavy-water DO spray of submicron droplets at 135° emission direction. Blob at the right: zero-point and spectrometer axis; blurred OT and 07+ traces due to propagation through ambient target material, and deuteron trace. (b) Determined deuteron spectrum. The low-energy cut for E <0.1 MeV is due to limited size of the microchannel-plate [25]. Reprinted with permission from [25]; Copyright 2005 by the American Institute of Physics.
Fig. 79. (a) Single-shot ion spectrum registered on a microchannel-plate (Thomson parabola spectrometer) from a heavy-water DO spray of submicron droplets at 135° emission direction. Blob at the right: zero-point and spectrometer axis; blurred OT and 07+ traces due to propagation through ambient target material, and deuteron trace. (b) Determined deuteron spectrum. The low-energy cut for E <0.1 MeV is due to limited size of the microchannel-plate [25]. Reprinted with permission from [25]; Copyright 2005 by the American Institute of Physics.
Fig. 80. Ion yield (a), average ion energy (b), and fusion neutron yield (c) as a function of laser energy for 100 fs pulse duration. The ion yield and energies, as well as the expected fusion neutron yield, were determined from two Faraday cup signals: cup A 0.52 m behind the laser focus and aligned 15° from the laser propagation axis (open circles) and cup B 0.36m from the laser focus and aligned 15° from the polarization axis and 90° from cup A (open squares). The measured fusion neutron yield (Y) at each energy (£) is also shown (filled circles) and fitted to a power law dependence (dotted line) with Y = QE” where Q =7.1 x 10* and o = 1.6 [447]. Reprinted with permission from [447]; Copyright 2004 by the American Physical Society.
Fig. 80. Ion yield (a), average ion energy (b), and fusion neutron yield (c) as a function of laser energy for 100 fs pulse duration. The ion yield and energies, as well as the expected fusion neutron yield, were determined from two Faraday cup signals: cup A 0.52 m behind the laser focus and aligned 15° from the laser propagation axis (open circles) and cup B 0.36m from the laser focus and aligned 15° from the polarization axis and 90° from cup A (open squares). The measured fusion neutron yield (Y) at each energy (£) is also shown (filled circles) and fitted to a power law dependence (dotted line) with Y = QE” where Q =7.1 x 10* and o = 1.6 [447]. Reprinted with permission from [447]; Copyright 2004 by the American Physical Society.
Fig. 81. Average ion energy from a cluster explosion plotted against the ratio g = fexp/tion. The cluster expansion time fexp ~ 10.7 fs, and the cluster ionization time was calculated at various pulse durations between 25 and 3200 fs (the results at 100 and 800 fs are marked) for cluster: of sizes N = 100 (empty circles and squares) and N = 1000 (filled circles and squares). The peak laser intensities were 2.8 x 10!7 (circles) anc 1.12 x 10:8 w / cm? (squares). The inset shows a calculation of the final explosion energy for a cluster of size N = 100, a pulse duration of 100 fs. and various peak laser intensities [447]. Reprinted with permission from [447]; Copyright 2004 by the American Physical Society.
Fig. 81. Average ion energy from a cluster explosion plotted against the ratio g = fexp/tion. The cluster expansion time fexp ~ 10.7 fs, and the cluster ionization time was calculated at various pulse durations between 25 and 3200 fs (the results at 100 and 800 fs are marked) for cluster: of sizes N = 100 (empty circles and squares) and N = 1000 (filled circles and squares). The peak laser intensities were 2.8 x 10!7 (circles) anc 1.12 x 10:8 w / cm? (squares). The inset shows a calculation of the final explosion energy for a cluster of size N = 100, a pulse duration of 100 fs. and various peak laser intensities [447]. Reprinted with permission from [447]; Copyright 2004 by the American Physical Society.
Fig. 82. Schematic of cluster ionization and expansion dynamics during the laser pulse. The ionization time fjon is determined by the time required for the laser intensity to rise from the first threshold, at which atomic ionization begins, to the second threshold, at which the electric field of the laser exceeds the threshold for complete ion charge uncovering (approximately when the laser electric field exceeds the space charge electric field of the fully stripped cluster). The second threshold, which is shown for a cluster of radius a = 5 nm, depends on the size and density of the clustet and will fall from its initial value as the cluster expands. Because of the finite duration of the cluster ionization time, the cluster will explode more slowly than its characteristic expansion time fexp resulting in a final kinetic energy below the characteristic energy of the expansion [447]. Reprinted with permission from [447]; Copyright 2004 by the American Physical Society.
Fig. 82. Schematic of cluster ionization and expansion dynamics during the laser pulse. The ionization time fjon is determined by the time required for the laser intensity to rise from the first threshold, at which atomic ionization begins, to the second threshold, at which the electric field of the laser exceeds the threshold for complete ion charge uncovering (approximately when the laser electric field exceeds the space charge electric field of the fully stripped cluster). The second threshold, which is shown for a cluster of radius a = 5 nm, depends on the size and density of the clustet and will fall from its initial value as the cluster expands. Because of the finite duration of the cluster ionization time, the cluster will explode more slowly than its characteristic expansion time fexp resulting in a final kinetic energy below the characteristic energy of the expansion [447]. Reprinted with permission from [447]; Copyright 2004 by the American Physical Society.
Neutron yield from recent table-top and medium-scaled fusion sources [447]
Neutron yield from recent table-top and medium-scaled fusion sources [447]
Fig. 83. Snapshots of the electron wavepacket taken at four different times t from 0.00055 to 0.04675 a.u. the contour lines indicate the logarithm of the probability density as shown by the labels in (a). In (d) the thick line which starts at the origin marks the trajectory of the position operator’s expectation value. The marker ’+’ indicates the position of the ionic core with the charge Z = 8. The Lorentz transformed amplitude and frequency of the laser field are Ey = 4.334 x 10° a.u. and @o = 1208.5 a.u., respectively [317]. Reprinted with permission from [317]; Copyright 2004 by the IOP Publishing Limited.
Fig. 83. Snapshots of the electron wavepacket taken at four different times t from 0.00055 to 0.04675 a.u. the contour lines indicate the logarithm of the probability density as shown by the labels in (a). In (d) the thick line which starts at the origin marks the trajectory of the position operator’s expectation value. The marker ’+’ indicates the position of the ionic core with the charge Z = 8. The Lorentz transformed amplitude and frequency of the laser field are Ey = 4.334 x 10° a.u. and @o = 1208.5 a.u., respectively [317]. Reprinted with permission from [317]; Copyright 2004 by the IOP Publishing Limited.
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