Optics in the relativistic regime

The advent of ultraintense laser pulses generated by the technique of chirped pulse amplification (CPA) along with the development of high-fluence laser materials has opened up an entirely new field of optics. The electromagnetic field intensities produced by these techniques, in excess of ${10}^{18}\mathrm{W}∕{\mathrm{cm}}^2$, lead to relativistic electron motion in the laser field. The CPA method is reviewed and the future growth of laser technique is discussed, including the prospect of generating the ultimate power of a zettawatt. A number of consequences of relativistic-strength optical fields are surveyed. In contrast to the nonrelativistic regime, these laser fields are capable of moving matter more effectively, including motion in the direction of laser propagation. One of the consequences of this is wakefield generation, a relativistic version of optical rectification, in which longitudinal field effects could be as large as the transverse ones. In addition to this, other effects may occur, including relativistic focusing, relativistic transparency, nonlinear modulation and multiple harmonic generation, and strong coupling to matter and other fields (such as high-frequency radiation). A proper utilization of these phenomena and effects leads to the new technology of relativistic engineering, in which light-matter interactions in the relativistic regime drives the development of laser-driven accelerator science. A number of significant applications are reviewed, including the fast ignition of an inertially confined fusion target by short-pulsed laser energy and potential sources of energetic particles (electrons, protons, other ions, positrons, pions, etc.). The coupling of an intense laser field to matter also has implications for the study of the highest energies in astrophysics, such as ultrahigh-energy cosmic rays, with energies in excess of ${10}^{20}\mathrm{eV}$. The laser fields can be so intense as to make the accelerating field large enough for general relativistic effects (via the equivalence principle) to be examined in the laboratory. It will also enable one to access the nonlinear regime of quantum electrodynamics, where the effects of radiative damping are no longer negligible. Furthermore, when the fields are close to the Schwinger value, the vacuum can behave like a nonlinear medium in much the same way as ordinary dielectric matter expanded to laser radiation in the early days of laser research.

Figure 1

Laser intensity vs years. This curve is obtained for amplifying beam of around ${\mathrm{cm}}^2$ beam size. Note the steep slope in intensities that occurred during the 1960s. This period corresponded to the discovery of most nonlinear optical effects due to the bound electron. We are today experiencing a similar rapid increase in intensity opening up a new regime in optics dominated by the relativistic character of the electron. Note that a few years ago we called it high intensity when the electron in a quiver energy was around $1\mathrm{eV}$. Today high intensity corresponds to electron quiver energies of the order of $m{c}^2\sim 0.5\mathrm{MeV}$. The dashed line corresponds to what could be obtained with significant increases in beam size (see Sec. II.I).

Figure 2

Pulse duration vs years. The laser-pulse duration has also rapidly changed from microsecond (free running), nanosecond ($Q$ switched), and picosecond mode locking. Here we show the pulse duration evolution since the 1990s after the invention of Ti:sapphire Kerr lens mode locking (512). Courtesy of F. Krausz, TU Vienna.

Figure 3

Amplifier efficiency. This illustrates the importance for the input pulse fluence $F_{\text{pulse}}$ to be few times the saturation fluence $F_{\mathrm{sat}}$ to obtain a good extraction efficiency.

Figure 4

Chirped pulse amplification concept. To minimize nonlinear effects the pulse is first stretched several thousand times lowering the intensity accordingly without changing the input fluence $(\mathrm{J}∕{\mathrm{cm}}^2)$. The pulse is next amplified by a factor of ${10}^6–{10}^{12}$ and is then recompressed by a factor of several thousand times close to its initial value.

Figure 5

Treacy and Martinez grating arrangements. The Martinez grating pair used as a stretcher and Treacy grating pair used as a compressor. It was discovered and demonstrated (426) that these two grating arrangements are in fact matched over all orders. The pulse can be stretched and recompressed arbitrarily keeping the initial pulse unchanged. This grating arrangement is used in most CPA systems.

Figure 6

Matching between the Martinez and Treacy grating pair arrangements. The input grating is imaged by a telescope of magnification 1, to form a “virtual” grating parallel to the second grating. The distance $b$ between the two gratings, real and virtual, can be continuously adjusted from positive to negative.

Figure 7

OPCPA concept. In the OPCPA the pulse is amplified by optical parametric amplification instead of regular optical amplification. Note that for efficiency the pump pulse and the stretched pulse must have approximately the same duration and the same spatial extend.

Figure 8

Pulse contrast. Prior to the main pulse, the base of the pulse needs to stay below a certain intensity level (broad shady line decreasing at 45°) to avoid the creation of a preformed plasma. Courtesy G. Cheriaux, LOA.

Figure 9

Third-order autocorrelation of a $27\mathrm{fs}$, full width at half maximum from the laser HERCULES at the University of Michigan. Note the very large dynamic range. $1\mathrm{ns}$ before the main pulse we can see the contribution of the amplified stimulated emission. The two prepulses at $-100\mathrm{ps}$ are due to measurement artifacts in the autocorrelator. The slow pedestal seen in (b) is due to incomplete compression, i.e., higher-order terms.

Figure 10

Polarization rotation used in a single-mode optical fiber to clean the prepulse energy (541). Efficient temporal cleaning can be obtained without sacrificing beam quality. (a) and (b) The second-order autocorrelation of the input under some slightly different conditions. (c) The beam cleaner output. It shows how effective polarization rotation can be. (d) After polarization rotation the pulse has increased its bandwidth. The autocorrelation trace shows the output pulse after recompression.

Figure 11

The use of a deformable mirror (DM) in conjunction with a low $f/0.6$ ellipsoid mirror can eliminate unwanted aberrations and produce a single-wavelength-focused spot size with a good Strehl ratio. A record intensity of ${10}^{22}\mathrm{W}∕{\mathrm{cm}}^2$ was obtained. The two figures, DM corrected and uncorrected, show the dramatic effect of a well-corrected laser beam. The additional benefit provided with a deformable mirror is the laser field can be determined anywhere in the beam by using the Fresnel-Kirckhoff integral.

Figure 12

Comparison between ruled and holographic gratings illustrating the difference in spot quality. In the case of ruled gratings the structure comes from the nonsinusoidal profile and ghosts produced by the imperfect and broken periodicity as they are ruling the grating.

Figure 13

Theoretical peak power per ${\mathrm{cm}}^2$ of beams for various amplifying media.

Figure 14

Plasma compression by Raman backscattering. This scheme is not sensitive to damage as it works with a plasma, a medium that is already broken down.

Figure 15

Average power versus repetition rate. This graph illustrates the fact that the average power is relatively independent of repetition rates over a wide range from Hz for small laser systems to mHz for the very large ones. At present the average power of any CPA systems regardless of their repetition rate is of the order of $1\mathrm{W}$. Systems at the $10\text{−}\mathrm{W}$ level have been demonstrated. Before CPA it was typically around $10\mathrm{mW}$. Many applications will require average powers greater than $1\mathrm{kW}$.

Figure 16

Quasimonoenergetic energy spectrum of electrons with the energy close to its maximal value.

Figure 17

Relativistic self-focusing. Isosurface of the electromagnetic energy density of a linearly polarized semi-infinite beam in projection to p and s plane, and its 3D view; cross section of the magnetic-field component.

Figure 18

Vortex row behind the laser pulse seen in the isocontours of the magnetic field.

Figure 19

Laser-pulse shaping.

Figure 20

Soliton formation and its development into the postsoliton in the interaction of an $s$-polarized laser pulse with the plasma: the $z$ component of the electric field (first column), the electron density (second column), and the ion density (third column) in the $x,y$ plane at (a) $t=30$, (b) $t=70$, and (c) $t=120$.

Figure 21

3D plot of the $z$ component (a) of the electric field and (b) of the ion density inside the postsoliton at $t=120$.

Figure 22

Isolated soliton and a soliton train behind the laser pulse.

Figure 23

Structure of (a),(b) electric and (c),(d) magnetic fields inside the soliton at (a),(c) $t=39.3$ and (b),(d) $t=40.2$. (e) The magnetic- and electric-field topology in the TE (with poloidal magnetic field and toroidal electric field) and in the TM (with poloidal electric field and toroidal magnetic field) solitons.

Figure 24

The postsoliton phase plane at different times.

Figure 25

Electrostatic $\phi$ and vector potential $a$ wave forms for bright solitons with $p=0,1,2$ and velocities close to breaking.

Figure 26

Shocklike front formation during laser-pulse propagation in underdense plasmas.

Figure 27

The reflected-wave electric field vs time.

Figure 28

Progress in laser development. The target chamber of the Helios system, the first laser that has shown relativistic effects like harmonic generation. This $\mathrm{C}{\mathrm{O}}_2$ laser had $a_0^2\sim 1$ with a repetition rate of $1\mathrm{mHz}$. Courtesy LANL.

Figure 29

Progress in laser development. The target chamber of the University of Michigan ${\lambda }^3$ laser. This compact laser has $a_0^2\sim 1$ with a repetition rate of $1\mathrm{kHz}$. This university-size system is the smallest relativistic intensity laser.

Figure 30

Laser-pulse compactification scheme. (a) The laboratory frame (L) before reflection of the laser pulse from the “flying mirror.” The laser pulse propagates from right to left. (b) The comoving reference frame (K). Laser-pulse reflection and focusing occurs in the focus spot with ${\lambda }^{\prime }\approx {\lambda }_0∕2{\gamma }_{\mathrm{ph}}$. (c) The laboratory frame (L). The reflected electromagnetic radiation has ${\lambda }_f\approx {\lambda }_0∕4{\gamma }_{\mathrm{ph}}^2$, and it propagates in a narrow angle $\theta \approx 1∕{\gamma }_{\mathrm{ph}}$.

Figure 31

(a) Paraboloidal modulations of the electron density in the wake behind the driver laser pulse. Projections of the electric-field components in the $x,y$ plane (the $x$ component of the wake wave) and in the $x,z$ plane of the $y$ component of the reflected pulse at $t=20$. (b) The laser-pulse driver is shown by contours on the right-hand side of the computation box.

Figure 32

The electromagnetic energy density of the reflected radiation $(E^2+B^2)$ at $t=11$. Numbers (1), (2), and (3) indicate the most intense pulses in the reflected radiation. Parameters of the simulation: $a=3$, $\tau =5\mathrm{fs}$, $n_0=1.5n_{\mathrm{cr}}$.

Figure 33

The incident electric field $E_{\xi }$ (dash-dotted line), electron velocity $v_x\left(\xi \right)$ (dotted line), and the reflected electric field $E_r\left({\xi }_r\right)$ (solid line) for the parameters $a_0=2.5$, $\theta =\pi ∕3$, and ${\epsilon }_0=0.5$ in the analytical model for pulse (a) compression and (b) decompression. The arrow shows the phase change for the reflected pulse.

Figure 34

High-quality proton beam generation. (a) Sketch of the double-layer target. Distribution of the electric charge inside the computation region at (b) $t=40$ and at (c) $t=80$.

Figure 35

The proton and the heavy-ion energy spectrum at $t=80$.

Figure 36

High efficiency ion acceleration in the radiation pressure dominant (RPD) regime. (a) The ion density isosurface (a quarter removed to reveal the interior) and the $x$ component of the normalized Poynting vector at $t=40$. (b) The isosurface of the ion density at $t=100$; the black curve shows the ion density along the laser pulse axis. (c) The maximum ion kinetic energy vs time.

Figure 37

Finite horizon and leakage of the wave function.

Figure 38

Use of a high-energy electron ring and a high-intensity laser to provide the conditions appropriate for nonlinear QED experiments.