**4. Wavelength scaling of laser‐matter interaction**

The visible spectrum of the hybrid femtosecond systems based on the XeF(C‐A) amplifiers allows for the λ‐scaling laser‐matter interaction to be studied. Just to illustrate the role of laser‐ driver wavelength in laser‐matter interaction we shall mention briefly some of applications in which the shorter laser wavelength provides advantages due to favorable wavelength scaling with special emphasis placed on the possible realization of a recombination soft X‐ray laser operating in the "water window". The examples considered below do not cover all the applications in which the shorter laser wavelength may be favorable as compared with near IR solid‐state femtosecond systems. Nevertheless, they show that the extension of laser wavelength to the visible region allows for wider experimental conditions to be realized providing different laser‐matter interaction parameters and better understanding of the underlying physics.

#### **4.1. Laser wake‐field acceleration**

300 J at U0 = 95 kV. As a result, the total small‐signal gain at 33 passes was enhanced up to (5–

**Figure 11.** (a) Dependence of the small‐signal gain vs the nitrogen pressure, p(XeF2) = 0.25 Torr, 1—U0 = 90 kV, 2—U0 = 

Amplification of a chirped pulse in the XeF(C‐A) amplifier was carried out with the use of the laser mixture containing 0.2 Torr XeF2 and 0.5 bar nitrogen at U0 = 95 kV. An output energy of 2.5 J was reached when a 2.4 ps seed pulse with an energy of about 1 mJ in a super‐Gaussian beam was injected into the XeF(C‐A) amplifier [51]. An autograph of the output laser beam on a photographic paper sheet is shown in **Figure 12**. The energy obtained in this experiment promises a peak power as high as ∼50 TW to be attained in the visible after pulse recompres‐

95 kV. (b) Dependence of the small‐signal gain vs the XeF2 pressure, p(N2) = 0.5 bar, U0 = 90 kV.

sion to the initial duration of 50 fs.

**Figure 12.** Imprint of the output laser beam with 2.5 J energy.

signal gain in arbitrary units vs the partial pressures of nitrogen and XeF2 vapor.

. **Figure 10b** shows the temporal behavior of the diode current (1) and total small‐signal gain (2) at 33 passes through the active medium. **Figure 11** shows the dependence of the small‐

6) × 10<sup>4</sup>

14616 High Energy and Short Pulse Lasers

Laser‐driven plasma wake‐field acceleration (LWFA) capable of producing high‐energy electron beams is widely studied both theoretically and experimentally. In plasma‐based acceleration, an intense laser beam drives large amplitude plasma waves via the ponderomo‐ tive force. The plasma wave can support very high longitudinal electric fields trapping and accelerating electrons. LWFA experiments have demonstrated acceleration gradients >100 GV  m-1 enabling electrons to be accelerated well beyond GeV energy on a distance of about 1 cm using a 100 TW‐class laser [55].

The electron energy gain ΔE is proportional to the acceleration length, Lacc, and longitudinal electric field, Ez, averaged over the acceleration length: ΔE = eEzLacc. Among the factors limiting the effective acceleration length, laser diffraction, electron dephasing, and pump depletion are the most important. In experiments, the limiting role of the first factor is usually mitigated due to relativistic self‐guiding or by using a preformed plasma channel. Electron dephasing originates from the difference of electron and plasma‐wave propagation velocities. As a result, highly relativistic electrons, accelerated up to a velocity approaching the speed of light, outrun the accelerating phase region of the plasma wave propagating with a phase velocity, vp, that is close to the laser pulse group velocity vg and less than the electron velocity. Electrons are accelerated until their phase slips by one‐half the plasma‐wave period. In the most promising and efficient high‐intensity limit corresponding to a nonlinear wake‐field acceleration, referred to the blow‐out, bubble, or cavitation regime, the radial ponderomotive force expels all the plasma electrons outward to create a electron density structure resembling a spherical ion cavity behind the laser pulse. Coulomb forces pull the electrons back to the axis in about a plasma period at the rear of the cavity to be trapped and accelerated by the wake‐field until they reach the cavity center where they diphase. The acceleration length strongly depends on the plasma‐wave phase velocity, which is close to the laser pulse group velocity, vg, obeying the plasma dispersion low: vg = dω/dk ≈ c(1‐ω<sup>p</sup> 2 /ω0 2 ) 1/2 with ω0 and ωp being the laser and plasma frequencies, respectively. Due to the plasma dispersion, shorter wavelength laser pulses propagating with higher group velocity provide longer dephasing length, Ld. Accord‐ ing to the estimations made in [51] with pump depletion taken into account, the dephasing length, given by Ld ≈ 4ca<sup>0</sup> 1/2 (ω<sup>0</sup> 2 /3ω<sup>p</sup> 3 ) with a0 = eA/mec2 being the relativistically normalized laser amplitude, scales as ω<sup>0</sup> 2 showing that shorter laser wavelengths are highly beneficial from the viewpoint of an increase in the acceleration length.

The detailed consideration [56] based on the phenomenological 3D theory for LWFA in the blowout regime, valid at laser power, P, exceeding the critical power Pc = 17(ω<sup>0</sup> 2 /ω<sup>p</sup> 2 ) [GW] for relativistic self‐focusing, predicts the electron energy gain

$$\Delta\text{E}\left[\text{GeV}\right] = 1.7 \left(\text{P}\left[\text{TW}\right]/100\right)^{1/3} \left(10^{18}/\text{n}\_p\left[\text{cm}^{-3}\right]\right)^{2/3} \langle 0.8/\lambda\_o[\text{\mu m}]\rangle^{4/3}$$

where np is the plasma density, and λ<sup>0</sup> stands for the laser wavelength. This indicates that the λ‐scaling of LWFA could be of great practical interest. Practically, the same λ‐scaling has been obtained in Ref. [57]. However, it should be noted that the gain in energy is achieved at the expense of reducing the number of accelerated electrons, which is proportional to the laser wavelength [56, 57].

#### **4.2. High‐order harmonic generation**

High‐order harmonic generation (HHG) is nowadays widely used to generate spatially and temporally coherent short‐wavelength radiation when an intense optical field interacts with a gas or solid target. HHG can provide a single burst or train of attosecond pulses, which allow for ultrafast dynamics of electrons in atoms, molecules, or even solids to be explored [58]. (For a detailed review of experimental and theoretical developments in HHG, see, e.g., [58, 59].)

According to the generally accepted semiclassical three‐step model of HHG in gases, the highest possible photon energy (cutoff energy, Ecutoff) in the high harmonic spectrum that can be generated from a single atom or ion is predicted by the universal law Ecutoff = Ip + 3.17U<sup>p</sup> [60]. Here, Ip is the ionization potential and Up = 9.33 × 10-14 I0λ<sup>0</sup> 2 is the ponderomotive energy, which is the cycle‐averaged kinetic energy of an electron in the laser electric field of intensity I0 and wavelength λ0. The λ<sup>0</sup> 2 dependence of U<sup>p</sup> implies that the use of long excitation wavelengths should result in extending the harmonic cutoff energy further into the X‐ray region. On the other hand, conversion efficiency of HHG in gases strongly depends on laser wavelength. The λ‐scaling at constant laser intensity has revealed the dependences of HHG efficiency to be between λ<sup>0</sup> -5 and λ<sup>0</sup> -6, which have been obtained experimentally and from numerical simula‐ tions [61–63]. General scaling analysis of HHG efficiency as a function of drive laser parameters and material properties is given in [64], which predicts the scaling of the HHG efficiency with the driving wavelength to be λ<sup>0</sup> -5 at the cutoff and λ<sup>0</sup> -6 at the plateau region for fixed harmonic wavelength. The severe wavelength dependence of the HHG efficiency is associated with the single‐atom dipole response and phase matching. Shorter driver wavelengths are advanta‐ geous for both of these factors, if the final objective is not to produce as high‐energetic photons as possible. The experimental results obtained in Ref. [65] for different noble gases confirm this wavelength scaling and show two orders of magnitude higher HHG intensity in the energy range of 20–70 eV with 400 nm pulses as compared with 800 nm laser driver.

However, HHG in gases has fundamental physical restrictions arising from the limitation on the laser intensity since plasma generation, caused by strong ionization of the gaseous medium at intensities above 1016 W/cm<sup>2</sup> , results in phase mismatch, thereby suppressing harmonic generation [66]. This limitation is not present in the case of HHG from solid. According to the oscillating plasma mirror model [67], the laser field produces a relativistic oscillation of the overcritical plasma surface with the laser frequency, inducing incident pulse modulation that gives rise to the high‐order harmonics in the spectrum of the reflected emission due to a transient Doppler frequency upshift. (For more detailed analysis of the basic generation mechanisms lying behind HHG from solids, see, e.g., [59, 68, 69].)

Using particle‐in‐cell (PIC) simulation to accurately model the HHG with a plasma target, Teubner and Gibbon [59] have obtained the laser‐to‐harmonic conversion efficiency, ηH, in the laser intensity range I0 = 1017–1019 W/cm<sup>2</sup> . Summarized by an empirical relation for high harmonic orders (N = λ0/λH >> 1, where λ<sup>H</sup> stands for harmonic wavelength), the results of the numerical simulation take the form [54]:

$$\eta\_{\rm H} \approx \, \text{9\,} \, 10^{-5} \, \text{(I}\_{0} \text{\AA}\_{0} \text{}^{2} \,/\, 10^{18} \,\text{W} \text{ cm}^{-2} \mu \text{m}^{2}\text{)}^{2} \left(\text{N} / 10\right)^{-\alpha}$$

with α depending on the laser intensity and ranging from α = 6 at I0 = 1017 W/cm<sup>2</sup> to α = 3.5 at I0 = 1019 W/cm<sup>2</sup> at λ0 = 1 μm. On the basis of this empirical scaling, the authors came to the conclusion that shorter wavelength lasers are highly beneficial from the viewpoints both of extending to shorter‐wavelength harmonics and harmonic efficiency enhancement. The wavelength scaling of the same form but with α = 5 independent of laser intensity was argued with the use of 1D‐PIC code in the earlier paper of Gibbon [70]. The analysis made in [59] of a variety of experimental results obtained with different laser wavelengths confirms the prediction of the strong harmonic yield increase with laser frequency. This makes powerful blue‐green hybrid systems to be very promising as the drivers for generation of soft X‐ray harmonics well within the water‐window spectral range.

#### **4.3. Soft X‐ray lasers**

ing to the estimations made in [51] with pump depletion taken into account, the dephasing

) with a0 = eA/mec2

The detailed consideration [56] based on the phenomenological 3D theory for LWFA in the

( ) ( ) - é ù éù é ù ëû ë <sup>=</sup> <sup>û</sup> <sup>D</sup> <sup>û</sup> <sup>l</sup> <sup>ë</sup><sup>m</sup>

p 0 E GeV 1.7 P TW / 100 10 / n cm 0.8 m[ ])/(

where np is the plasma density, and λ<sup>0</sup> stands for the laser wavelength. This indicates that the λ‐scaling of LWFA could be of great practical interest. Practically, the same λ‐scaling has been obtained in Ref. [57]. However, it should be noted that the gain in energy is achieved at the expense of reducing the number of accelerated electrons, which is proportional to the laser

High‐order harmonic generation (HHG) is nowadays widely used to generate spatially and temporally coherent short‐wavelength radiation when an intense optical field interacts with a gas or solid target. HHG can provide a single burst or train of attosecond pulses, which allow for ultrafast dynamics of electrons in atoms, molecules, or even solids to be explored [58]. (For a detailed review of experimental and theoretical developments in HHG, see, e.g., [58, 59].)

According to the generally accepted semiclassical three‐step model of HHG in gases, the highest possible photon energy (cutoff energy, Ecutoff) in the high harmonic spectrum that can be generated from a single atom or ion is predicted by the universal law Ecutoff = Ip + 3.17U<sup>p</sup> [60].

is the cycle‐averaged kinetic energy of an electron in the laser electric field of intensity I0 and

should result in extending the harmonic cutoff energy further into the X‐ray region. On the other hand, conversion efficiency of HHG in gases strongly depends on laser wavelength. The λ‐scaling at constant laser intensity has revealed the dependences of HHG efficiency to be

tions [61–63]. General scaling analysis of HHG efficiency as a function of drive laser parameters and material properties is given in [64], which predicts the scaling of the HHG efficiency with

wavelength. The severe wavelength dependence of the HHG efficiency is associated with the single‐atom dipole response and phase matching. Shorter driver wavelengths are advanta‐ geous for both of these factors, if the final objective is not to produce as high‐energetic photons as possible. The experimental results obtained in Ref. [65] for different noble gases confirm


2

dependence of U<sup>p</sup> implies that the use of long excitation wavelengths


blowout regime, valid at laser power, P, exceeding the critical power Pc = 17(ω<sup>0</sup>

being the relativistically normalized

is the ponderomotive energy, which


2 /ω<sup>p</sup> 2

) [GW] for

showing that shorter laser wavelengths are highly beneficial from

1/3 2/3 <sup>18</sup> <sup>3</sup> 4/3

length, given by Ld ≈ 4ca<sup>0</sup>

14818 High Energy and Short Pulse Lasers

wavelength [56, 57].

wavelength λ0. The λ<sup>0</sup>


the driving wavelength to be λ<sup>0</sup>

between λ<sup>0</sup>

**4.2. High‐order harmonic generation**

laser amplitude, scales as ω<sup>0</sup>

1/2 (ω<sup>0</sup> 2 /3ω<sup>p</sup> 3

2

the viewpoint of an increase in the acceleration length.

relativistic self‐focusing, predicts the electron energy gain

Here, Ip is the ionization potential and Up = 9.33 × 10-14 I0λ<sup>0</sup>

2

Development of coherent X‐ray sources is motivated by a variety of their applications in science and technology. X‐ray lasers offer new capabilities in understanding the nanoscale structure of complex materials, including biological systems, and X‐ray matter under extreme condi‐ tions. One of the greatest challenges is the high‐resolution 3D holographic microscopy of a wide range of biological objects in the living state. For this purpose, coherent ultrafast X‐ray sources of high power are required. One of the most important milestones for the high contrast X‐ray imaging of living biological structures in a natural aqueous environment is the "water window" lying between the K absorption edges of carbon (4.37 nm) and oxygen (2.33 nm), where carbon is highly opaque, while water is largely transparent. A primary challenge of X‐ ray exposure is the realization of "diffraction‐before‐destruction" approach allowing for diffraction patterns to be obtained on time scales shorter than the onset of radiation damage of samples. This approach has been successfully demonstrated in studies of biological samples with the use of X‐ray free electron lasers (XFEL) producing femtosecond pulses of high intensity (For example, see [71]).

There has been remarkable progress in the development of XFELs that hold the great promise for user experiments ranging from atomic physics to biological structure determination. Despite the fact that these lasers are powerful tools in studies of matter structure and physics of light‐matter interaction, they have limited accessibility because of their high cost and large‐ scale. This makes the current search for alternative X‐ray sources of laboratory scale to be of great importance.

Presently, there are two main approaches to the development of compact ultrashort‐pulsed sources of coherent soft X‐ray: above considered HHG by gas and solid targets, as well as the generation of coherent X‐ray radiation in the laser plasma. The first one is characterized by low‐intensity soft X‐ray radiation, insufficient for the realization of holographic imaging methods. The laser plasma enables generation of lasing in the soft X‐ray region with beam performances close to those of XFELs [72].

Actually, only collisional and recombination schemes of active media excitation to produce soft X‐ray in a laser plasma are of practical interest. The first of them was realized in a laser plasma with high electron temperature, providing a population inversion on transitions between excited states of ions, which typically lie in the range 10–50 nm [72]. The most promising way to extend the spectral range of the X‐ray lasers deeper into the X‐ray region, including the "water window," lies in the further developing recombination scheme of excitation of transitions to the ground state of recombining fully stripped ions.

The first observation of the amplification on the transition to the ground state dates back to 1983 [73] when hydrogen‐like lithium ions were excited in the laser plasma produced due to optical field ionization (OFI) by the UV radiation from a subpicosecond KrF laser. Later that year, this observation was confirmed in different experimental conditions [74–76]. The OFI approach to excitation of recombination soft X‐ray lasers is particularly attractive since it produces fully stripped ions on a time scale of one period of the incident laser electric field and enables formation of cold electrons with low residual energy (for a linearly polarized laser pulse) providing favorable conditions for high‐rate three‐body recombination. Residual energy is proportional to the square of a pump laser wavelength and can be reduced by using a short‐wavelength driver pulse.

An electron removed from an atom due to OFI interacts with the plane polarized laser field and acquires quiver energy of the coherent electron oscillation in the field and energy of electron drift along the laser field direction [77, 78]. For ultrashort pulses, the quiver energy is returned to the wave, and it does not contribute to residual energy. Most of the electrons are ionized within a narrow interval near the crest of the oscillating electric field because of the exponential dependence of the ionization rate on the electric field amplitude. Classically, the average drift energy, ε, of an electron depends on the phase mismatch, Δϕ, between the phase at which the electron is freed and the crest of the electromagnetic wave:

f <sup>2</sup> <sup>q</sup> ε = 2ε sin Δ

where εq is the quiver energy (εq = e<sup>2</sup> E0 2 /4meω<sup>2</sup> with E0 and ω being the peak amplitude and angular frequency of the laser electric field E = E0sinωt, respectively). Thus, the residual energy of electrons produced by OFI can be much lower than the electron quiver energy, and, secondly, shorter wavelength ionizing lasers are beneficial to achieve gain in the recombination scheme.

To demonstrate advantages of short‐wavelength pumping, λ‐scaling of the recombination excitation efficiency for the transitions to ground state was experimentally and numerically studied by different groups [76, 78, 79]. Numerical simulations of the small‐signal gain on the 4s1/2–3p3/2 transition at 23.2 nm in Ar7+ show that 400 nm pump laser radiation allows an increase in the small‐signal gain on the 4s1/2–3p3/2 transition at 23.2 nm in Ar7+ by more than an order of magnitude as compared with 800 nm pumping [80].

Simulation of the recombination gain formation on the 2 → 1 transition at 3.4 nm in H‐like CVI pumped with a 400 nm pump laser has been performed in [81]. It was shown that the recom‐ bination gain as high as 180 cm-1 can be achieved on this transition using the driving pulse duration of 20 fs with peak intensity of 8 × 1018 W/cm<sup>2</sup> and 10 μm diameter focal spot. The key factor playing important role in the recombination mechanism of pumping is the non‐ Maxwellian nature of the distribution function after OFI [79, 81], which is strongly peaked near the zero electron energy. Ultrashort pumping time (<100 fs) is required to minimize heating and Maxwellization of electron energy distribution at the time scale of three‐body recombi‐ nation.

Less encouraging results have been obtained in Ref. [82], indicating that there are a number of issues, which have to be investigated experimentally for better understanding of the physical processes lying behind the optical production of recombination plasmas. This requires the development of ultrashort (20–50 fs) multiterawatt lasers in the UV or visible range as pumping sources.
