Rewriting the rules governing high intensity interactions of light with matter

The trajectory of discovery associated with the study of high-intensity nonlinear radiative interactions with matter and corresponding nonlinear modes of electromagnetic propagation through material that have been conducted over the last 50 years can be presented as a landscape in the intensity/quantum energy [I-ħω] plane. Based on an extensive series of experimental and theoretical findings, a universal zone of anomalous enhanced electromagnetic coupling, designated as the fundamental nonlinear domain, can be defined. Since the lower boundaries of this region for all atomic matter correspond to ħω ~ 103 eV and I  ≈  1016 W cm−2, it heralds a future dominated by x-ray and γ-ray studies of all phases of matter including nuclear states. The augmented strength of the interaction with materials can be generally expressed as an increase in the basic electromagnetic coupling constant in which the fine structure constant α  →  Z2α, where Z denotes the number of electrons participating in an ordered response to the driving field. Since radiative conditions strongly favoring the development of this enhanced electromagnetic coupling are readily produced in self-trapped plasma channels, the processes associated with the generation of nonlinear interactions with materials stand in natural alliance with the nonlinear mechanisms that induce confined propagation. An experimental example involving the Xe (4d105s25p6) supershell for which Z  ≅  18 that falls in the specified anomalous nonlinear domain is described. This yields an effective coupling constant of Z2α  ≅  2.4  >  1, a magnitude comparable to the strong interaction and a value rendering as useless conventional perturbative analyses founded on an expansion in powers of α. This enhancement can be quantitatively understood as a direct consequence of the dominant role played by coherently driven multiply-excited states in the dynamics of the coupling. It is also conclusively demonstrated by an abundance of data that the utterly peerless champion of the experimental campaign leading to the definition of the fundamental nonlinear domain was excimer laser technology. The basis of this unique role was the ability to satisfy simultaneously a triplet (ω, I, P) of conditions stating the minimal values of the frequency ω, intensity I, and the power P necessary to enable the key physical processes to be experimentally observed and controllably combined. The historical confluence of these developments creates a solid foundation for the prediction of future advances in the fundamental understanding of ultra-high power density states of matter. The atomic findings graciously generalize to the composition of a nuclear stanza expressing the accessibility of the nuclear domain. With this basis serving as the launch platform, a cadenza of three grand challenge problems representing both new materials and new interactions is presented for future solution; they are (1) the performance of an experimental probe of the properties of the vacuum state associated with the dark energy at an intensity approaching the Schwinger/Heisenberg limit, (2) the attainment of amplification in the γ-ray region (~1 MeV) and the discovery of a nuclear excimer, and (3) the determination of a path to the projected super-heavy nuclear island of stability.

The trajectory of discovery associated with the study of high-intensity nonlinear radiative interactions with matter and corresponding nonlinear modes of electromagnetic propagation through material that have been conducted over the last 50 years can be presented as a landscape in the intensity/quantum energy [I-ħω] plane. Based on an extensive series of experimental and theoretical findings, a universal zone of anomalous enhanced electromagnetic coupling, designated as the fundamental nonlinear domain, can be defined. Since the lower boundaries of this region for all atomic matter correspond to ħω ~ 10 3 eV and I ≈ 10 16 W cm −2 , it heralds a future dominated by x-ray and γ-ray studies of all phases of matter including nuclear states. The augmented strength of the interaction with materials can be generally expressed as an increase in the basic electromagnetic coupling constant in which the fine structure constant α → Z 2 α, where Z denotes the number of electrons participating in an ordered response to the driving field. Since radiative conditions strongly favoring the development of this enhanced electromagnetic coupling are readily produced in self-trapped plasma channels, the processes associated with the generation of nonlinear interactions with materials stand in natural alliance with the nonlinear mechanisms that induce confined propagation. An experimental example involving the Xe (4d 10 5s 2 5p 6 ) supershell for which Z ≅ 18 that falls in the specified anomalous nonlinear domain is described. This yields an effective coupling constant of Z 2 α ≅ 2.4 > 1, a magnitude comparable to the strong interaction and a value rendering as useless conventional perturbative analyses founded on an expansion in powers of α. This enhancement can be quantitatively understood as a direct consequence of the dominant role played by coherently driven multiply-excited states in the dynamics of the coupling. It is also conclusively demonstrated by an abundance of data that the utterly peerless champion of the experimental campaign leading to the definition of the fundamental nonlinear domain was excimer laser technology. The basis of this unique role was the ability to satisfy simultaneously a triplet (ω, I, P) of conditions stating the minimal values of the frequency ω, intensity I, and the power P necessary to enable the key physical processes to be experimentally observed and controllably combined. The historical confluence of these developments creates a solid foundation for the prediction of future advances in the fundamental understanding of ultra-high power density states of matter. The atomic findings graciously generalize to the

Introduction
The history of nonlinear high-intensity interactions, that commenced in 1961 with the observation of second harmonic radiation [1] at 347.2 nm in crystalline quartz, records a span of ~10 18 in experimental intensity that enabled the discovery of an expansive range of entirely new phenomena. As a consequence of a multitude of salient advances, this general field of inquiry remains a stable and highly robust province of fundamental laser-based research after a half century. Empowered by many developments in short wavelength generation [2][3][4][5][6][7][8][9][10][11], and assisted particularly by key results stemming from the dynamics of nonlinear modalities of confined propagation [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], the study of broad classes of nonlinear radiative phenomena now steps up into the x-ray spectral region, an advance that perforce lifts the interactions to a new and enormously increased level of power density. The present work discusses the nature of this transition, gives an explicit experimental example that represents a defining characteristic of future research activity, and concludes with the statement of three grand challenge problems for prospective solution.

Intensity-frequency interaction landscape: experimentally determined zone of anomalous coupling
Founded on the analysis of a long sequence of exper imental and theoretical studies , it is possible to identify a zone of anomalous enhanced electromagnetic coupling that stands in sharp contrast to a corresponding region of conventionally described weaker interaction. These two areas can be presented in the intensity (I ) -quantum energy (ħω) plane, as illustrated in figure 1, and the boundaries between these zones designate the transition to a fundamentally new regime of physical interaction that stands as the green shaded quadrant in the upper right. We will find below that the picture presented in figure 1 can be sharpened to be a universal statement about all atomic matter, a refinement that modestly reshapes the zone shown by raising the lower quantum energy bound to ħω ~ 1 keV.
Extant data are used below to establish the boundaries of the anomalous region that we designate as the fundamental nonlinear domain in figure 1. Stating the conclusion of the analysis at the outset, it is found that lower limit of the intensity for the enhanced region of coupling shown in figure 1 is given approximately by I ~ 3 × 10 15 -10 16 W cm −2 , a narrow range of values corresponding to an electric field slightly less than one atomic unit (e/a 0 2 ~ 5.14 × 10 9 V cm −1 ). The corresponding lower bound on the frequency ω, estimated on the basis of measurements with N 2 , Kr, Xe, and U, is represented by ħω ~ 5 eV-6 eV, a magnitude slightly less than one half of a Rydberg. Therefore, as a gross overall measure, the region of anomalous interaction is physically bounded by the characteristic values of electric field strength (intensity) and frequency (energy) expressed by the structure of the hydrogen atom, the paramount fundamental atomic entity. The main region of interest, as developed below, involves the x-ray range with quantum energies ħω ⩾ 10 3 eV and intensities I ⩾ 10 16 W cm −2 , since this terrain evidences properties that enable it to uniformly and universally reflect the anomalous interaction for all materials regardless of their atomic constituents or physical state. Radiative coupling in this zone will be completely dominated by anomalous enhanced coupling strengths that will be particularly prominent for high-Z materials [27,30,46,49]. In the description below, new data are presented that confirm the existence of this predicted zone through measurements made in the keV region with xenon (Z = 54).
We now describe the experimental history that enabled the construction of the physical map shown in figure 1 and the placement of the boundaries defining the fundamental nonlinear domain. The story commences in the infrared. The discovery of the CO 2 laser [68] in 1964 with a wavelength λ ~ 10 μm enabled the development of new high-resolution spectroscopic techniques that could be readily performed on infrared molecular vibrational-rotational transitions. These methods combined very high experimental precision with intensities sufficiently elevated to observe nonlinear two-photon (2γ) amplitudes quantitatively. An example is given by the detailed measurements [69] that were conducted in 1976 on specific vibrational-rotational transitions of the υ 2 -mode of 14 NH 3 . The quantitative analysis of both the 2γ coupling strength and the intensity dependent shift of the resonance observed in these measurements demonstrated solid agreement with a conventional perturbative analysis. The datum associated with this study is placed on figure 1 with the identification [NH 3 (2γ)].
The invention of the KrF * (248 nm) and ArF * (193 nm) excimer laser systems [70,71] in 1975 enabled the extension of the quantitative nonlinear spectroscopic methods successfully demonstrated in the infrared to the ultraviolet spectral range.
The first examples of high resolution spectroscopy utilizing nonlinear coupling at an ultraviolet wavelength (193 nm) were the studies in 1978 and 1980 of the 2γ excitation [72,73] of the double-minimum E, F Σ + 1 g state of H 2 , a level that possesses the same symmetry as the ground X Σ + 1 g term. Accordingly, the X → E, F excitation is strictly forbidden as a single-photon amplitude. The quantitative analysis of these measurements again clearly showed that the measured coupling strength conformed well to a conventional perturbative treatment for a second order process. The matrix elements and state energies of H 2 are well known, so that the effective cross section could be reliably computed and the presence of a significant anomalous component of the interaction could be ruled out. This study is designated on figure 1 as the datum [H 2 / E,F(2γ)] at an intensity of ~10 9 W cm −2 .
The subsequent development of short-pulse excimer laser technology [74] in the picosecond range (~10 ps) in 1982 immediately enabled the attainment of sharply elevated intensities at an ultraviolet wavelength. Specifically, electric field strengths at values corresponding approximately to one tenth of an atomic unit (~e/10a 0 2 ≈ 5 × 10 8 V cm −1 ) could be achieved. The initial experimental studies conducted with an ArF* (193 nm) source immediately gave an indication of anomalous coupling; this work produced the first hint of an unconventional and substantially enhanced strength of interaction. These measurements analyzed the atomic number (Z ) dependence of collision-free multiple ionization of atoms [27] with 193 nm radiation (ħω ≈ 6.3 eV) at an intensity of ~10 14 W cm −2 . The outcome was highly significant, since the customary theoretical approaches clearly could not describe the chief feature of the observations, specifically, the production of stages of ionization far above the conventional predictions. On the basis of these results, it was concluded (1) that conventional treatments of multiquantum ionization did not Global physical scaling in the intensity-quantum energy plane. Based on the series of studies at different intensities and wavelengths, the I-ħω plane is partitioned into two separate regions, one associated with conventional weak electromagnetic coupling (red) and a second (green) designating the fundamental nonlinear domain that is characterized by an anomalous enhanced coupling strength. Several specific experimental studies described in the text are indicated. A prospectively identified direct coupling nuclear zone is shown in the region bounded by I ⩾ 10 27 W cm −2 and ħω ⩾ 10 3 eV. The U(L 1 ) Auger width at ~124 eV is marked on the abscissa. The intensities corresponding to field strengths of an atomic unit E a = e/a 0 2 and its muonic counterpart E a (m μ /m e ) 2 are marked on the ordinate for reference. Significantly, the junction of the vertical and horizontal boundaries defining the transition between the conventional weak coupling zone and the anomalous region of enhanced coupling falls nearly coincident with the intensity corresponding to one atomic unit (~3.5 × 10 16 W cm −2 ) and a quantum energy of one Rydberg (ħω ≅ 13.6 eV), parameters that are defined by the hydrogen atom. Accordingly, the red star is placed on the map to designate these coordinates. The Schwinger/Heisenberg limit described in [101] determines the upper boundary at I ≅ 4.6 × 10 29 W cm −2 above which nonlinear effects of the vacuum become appreciable. The inset illustrates the ascending series of inner-shell thresholds discussed in the text that define the location of the vertical boundary at ħω ≅ 5 eV and the close correspondence to the position characterized by the hydrogen atom. correspond to the experimental findings for high-Z materials and (2) that the atomic shell structure was an important factor in determining the magnitude of the coupling. Furthermore, based on the Z-dependence of the experimental results, a collective motion of the d and f shells, essentially a plasma model, that can be imagined as the atomic counterpart of the nuclear giant dipole [75], was inferred [27]. In the cases of I and Xe, the experimental evidence pointed particularly to a role played by the 4d-shell. This study is denoted on figure 1 as [193-Z atom] at an intensity of ~10 14 W cm −2 , a value that is a factor of ~10 5 greater than that used in the studies of the E, F Σ + g state of the hydrogen molecule described above. In time additional results appeared [35,50,52,53,59,61] that supported this conclusion.
It should be noted that this conclusion did not receive uniform support at this time and was disputed by other findings [76,77] that involved studies of N 2 and HI, noting specifically that the latter system is isoelectronic (Z = 54) with Xe. However, as determined by direct comparative studies several years later [34,41,42], the wavelength of irradiation was found to be a key factor in the non-linear interaction, an outcome that was initially unexpected, and accordingly, not considered as relevant. This feature of the coupling is illustrated in figure 1. In the end, this controversial aspect of the research has been resolved. The earlier conflict can now be seen as two sets of different experiments that gave correspondingly different results.
Elaboration on the nature of these possible ordered electronic motions appeared [28][29][30]78] shortly thereafter, two involving analogies with fast atom-atom collisions [28,78]. The basic idea can be summarized as an ordered electronic motion that achieves a 'phase space control' of the energy. A simple picture aimed at the evaluation of an upper bound σ m of the effective nonlinear cross section in the high-Z and highintensity limit was developed [46] that predicted the existence of a universal limiting value of (1) that would be approached for intensities >10 20 W cm −2 . The universality of equation (1) is astounding; it depends only upon the electron mass through its Compton wavelength c and is completely independent of the conventional notion of the nonlinear order of the amplitude, the electronic charge, the frequency of the radiation, and any atomic properties. In addition, the magnitude of σ m was highly significant, since it indicated that specific power densities greater than ~1 W/atom could be possible as high-intensity laser technology continued to develop and enable the attainment of experimental intensities in the ~10 20 W cm −2 range. Experimental intensities above this level are now routinely produced in laboratories worldwide. The next major development was the introduction of a molecular concept and the study of a suitable model system at an intensity sufficiently high to reveal the anomalous coupling in a dominant manner. Studies of N 2 with subpicosecond 248 nm radiation in 1989 at intensities in the 10 16 -10 17 W cm −2 range, for which the optical electric field is approximately one atomic unit (e/a 0 2 ), quickly provided the necessary data [32].
The key observation was the selective production of a molecular inner-shell 2σ g vacancy that was detected by its characteristic radiative decay at λ ~ 55.8 nm. The detection of this line was revelatory on the dynamics of the interaction. Specifically, it led unexpectedly to the identification [32] of a new previously unknown state (A′ 3 Π u ) of + N 2 2 and the unambiguous assignment [32] of the 55 nm emission to the A′ 3 Π u → D 3 Π g transition of + N 2 2 . Hence, the strong generation of the 55 nm line clearly demonstrated that it was possible to produce an inner-shell vacancy without simultaneously removing a large fraction of the outer shell electrons. An electronically hollow system was the prompt result of the excitation. The radiative transition observed at λ ~ 55.8 nm involving the 2σ g vacancy likewise possessed several highly unusual properties. Although a fully electric dipole (E1) allowed amplitude and the most intense feature in the recorded spectrum [32], this line was unknown and had never been reported during approximately a century of spectroscopic study under conditions that had utilized a wide range of methods for excitation of this elementary homonuclear diatomic molecule [79,80]. This puzzling observation, in an extensively studied simple system, in alliance with the high intensity of the 55 nm emissions, was doubly significant. These results signified (1) that special conditions were required to produce the particular excited state that was the source of the 55.8 nm emission and (2), if those conditions were met, efficient production of the A′ 3 Π u state was the outcome, an indication that the channel was dynamically favored. The conclusion was clear; inner-shell excited states could be selectively and efficiently produced in molecules.
The analysis [32] of the observed spectrum demonstrated that these exceptional characteristics were matched by corresponding unexpected and unusual properties of the upper A′ 3 Π u level and the electronic character of the A′ → D transition. The excited A′ level was found to be a bound 'charge transfer' state whose asymptote correlates with the charge asymmetric N 2+ + N configuration; the polarization-attraction of the N 2+ ion to the neutral N atom accounts for the binding. In contrast, the lower D 3 Π g state is well known and correlates to the obviously repulsive unbound symmetric N + + N + asymptote. Therefore, the A′ → D transition at λ ~ 55.8 nm is 'excimerlike' and corresponds to an electronic motion that accomplishes the transfer of an electron from the neutral (N) end of the molecule to the doubly charged (N 2+ ) side. It follows that the production of the A ′ 3 Π u state involves the orderly motion of an electron induced by the external electric field along the axis of the molecule in the opposite direction. With an interatomic spacing d ≈ 1.1 Å and an electric field of ~5.1 × 10 9 V cm −1 , the work W associated with this electron transfer is a magnitude sufficient to liberate a 2σ g electron whose binding energy [79,81] is ~37.7 eV. The point on figure 1 denoted as [N 2 / A′ 3 Π u ] represents this experimental study of N 2 . The chief conclusion derived from the experiments on N 2 was that ordered motions of electrons could be induced in molecules that can efficiently and selectively remove innerorbital electrons while leaving outer-shell electrons largely undisturbed. Hence, the combination (1) of this induced ordered motion with (2) the collective plasma picture associated with sufficiently high-Z atoms, that was supported by the studies of multiphoton ionization [27][28][29][30] described above, presented itself as an obvious possibility for efficient short wavelength generation. Specifically, with these molecular dynamics, the use of a high-Z molecule could enable the scaling of the inner-shell excitation observed in N 2 with the 2σ g orbital at ~40 eV to far more deeply bound corresponding states in the multikilovolt range [33].
At this stage in the development of the study of highintensity interactions, a new centrally important nonlinear optical phenomenon was experimentally established that enabled the stable compression of power in macroscopic volumes to be achieved. The initial demonstration [12] of self-trapped channeling of 248 nm radiation in 1992 has been recently extended with a quantitative experimental comparison of channels produced in Kr and Xe cluster targets [26,82]. The availability of this physical mechanism for power compression had a profound influence on the study of nonlinear processes, since it enabled three conditions to be simultaneously and simply produced [14-16, 18, 20, 24]. They are (1) the production of intensities in the 10 20 -10 21 W cm −2 region, (2) a sharp rise time of the propagating pulse of ~3 fs that is perforce generated by the dynamics of channel formation, and (3) a greatly extended length of the high intensity region provided by the channel that readily reaches several millimeters.
The visualization of these channels, as shown in figure 2 for the case of Xe, illustrates many properties of the self-trapped propagation [26,82]. The comparison of the Thomson image of the electron density with the simultaneously recorded transversely observed morphology of the Xe(M) x-ray (ħω ~ 1 keV) emission zone, as presented in figure 2, is revealing. The correspondences between the Thomson image in panel (a) and the Xe(M) x-ray image in panel (b) manifestly show that the detailed channel dynamics at the position Z ≅ 0 mm, where the large abrupt transverse expansion occurs, and clearly signal in both images the termination of the confined propagation and the concomitant release of the trapped 248 nm pulse energy from the channel. The collapsed narrow zone of the channel observed before this terminus has a length ℓ Xe ≅ 0.8 mm. Furthermore, in this compressed region betwe en −0.8 mm ⩽ Z ⩽ 0.0 mm, the small structural features visible in the Thomson recording are directly mirrored by matching variations in the x-ray emission at the exact corresponding axial positions in the x-ray image. The intensity in this region is estimated to be ~10 20 W cm −2 with an atom-specific power density approaching ~1 W per atom.
The concept of the high-Z molecular system [33] was tested in 1994 with Kr n and Xe n clusters (n < 10 3 ) [34,83] and the profound effectiveness of the molecular dynamics was immediately apparent. The key findings were (1) the copious generation Simultaneously recorded single-pulse images of (a) the Thomson scattered signal from the electron density and (b) the transverse Xe(M) ~ 1 keV x-ray emission zone (log scale) of a stable 248 nm channel produced in a Xe cluster target. Experimental parameters: pulse energy 217 mJ, Xe plenum pressure 185 psi, and Xe plenum temperature 285 K. The x-ray camera utilized a pinhole with a diameter of 10 μm, a size that gives a limiting spatial resolution estimated to be 20-30 μm. The direction of propagation is left to right and the center of the nozzle corresponds to the coordinate Z ≅ 0 mm. The Xe cluster target was produced by a cooled high-pressure pulsed-valve fitted with a circular nozzle having a diameter of 2.5 mm. These data correspond to pulse #19 (24 January 2011). The locations of corresponding features in these images are indicated by the vertical connecting lines. The sharp expansion at Z ≅ 0 mm in panel (a) signals the termination of the self-trapped channel. A broad elliptical halo of ionization with a diameter of ~1 mm is seen in the −0. of multikilovolt x-rays on Kr(L), Xe(M), and Xe(L) transitions involving 2p and 3d vacancies spanning the ~1-4.5 keV range, (2) the discovery that the Xe(L) emission corresponded exclusively to hollow atom Xe q+ charge state arrays [34,38] for 30 ⩽ q ⩽ 36, and (3) the observation of an exceptional brightness of the recorded emission that clearly signaled the action of an anomalously enhanced coupling strength [35,52,53,84].
We note here that the dependence of the coupling on the cluster size was developed at the outset [33] and produced a highly significant and unexpected finding. Specifically, the analysis predicted the existence of a critical cluster size n c that stated for clusters of n atoms with n ⩾ n c that every atom in the cluster would emit an inner-shell x-ray. This critical size was the direct analog of the nuclear concept of a critical mass. Accordingly, in the formulation of the critical cluster size n c , the cross section for electronic collisional inner-shell ionization σ ei replaced the nuclear fission cross section σ nf in the critical mass. Since n c scales as the inverse cube of the relevant cross section σ ei , there is a strong atomic shell dependence that generally favors heavier systems. Based on the presently known parameters for Xe(M) radiation at ~1 keV, n c ≅ 30 atoms. Since this size falls well below that maximum cluster size estimate of ~50-100, it properly corresponds to both the high Xe(M) yields observed [26] and the relatively weak temper ature dependence [64] measured for the x-ray emission.
In 1995, the cluster mechanism and the self-trapped channels were successfully combined [37], an important result that demonstrated the mutual compatibility of these two highly nonlinear processes. The importance of the clusters was underscored, since an earlier study on Xe atoms [85] unambiguously demonstrated that Xe(M) and Xe(L) x-rays were not produced by ultraviolet radiation at the intensities used in the cluster studies [34,83]. The cluster formation was essential in order to involve the molecular dynamics, initially observed in N 2 [32], that produced the strong efficient excitation of the A ′3 Π u state of + N 2 2 , and enabled the generation of the deep 2p and 3d inner-shell excitations. In addition, the direct hollow atom production [34] was a huge bonus, since it eliminated the need for the relatively slow process of recombination to form the excited Xe*(L) states. Hence, excited levels exhibiting prompt x-ray emission were efficiently produced, an outcome that automatically produced inverted population densities and subsequently enabled amplification in the multikilovolt x-ray spectral region ∼4.5 keV on 3d → 2p transitions to be achieved [38,39,[61][62][63]86]. Recent work [85] has extended these findings with the demonstration of amplification on the Kr(L) 3s → 2p Kr 26+1 P 1 → 1 S 0 line at λ ≅ 7.5 Å.  [26] and the magnitude of the x-ray yield that it could produce. The measurements demonstrated >50 mJ/pulse on the Xe(M) band [50] and a corresponding efficiency of production >20% with 248 nm excitation. Since the 248 nm KrF* technology [88] is presently scalable to pulse energies of ~2 J and the Xe(M) emission spectrum at ħω ≅ 1 keV can be generated [26] with an efficiency of ~20-30% with a bandwidth [66] of ~100 eV, the door is now open to the production of attosecond pulses of spectrally bright coherent x-ray pulses at the ~500 mJ level [89], a point echoed in section 3.
In sharp contrast to the 2p and 2s2p inner-shell excitations readily produced with ultraviolet radiation at ħω = 5 eV with Xe clusters, the performance of directly comparable experiments with Xe at 800 nm (ħω ~ 1.5 eV) produced no significant Xe(L) production [41,51]. The low level of Xe(L) emission detected had a far different spectral envelope and was quantitatively ~3000-fold weaker. The point on figure 1 that designates the experiments performed with 800 nm excitation is denoted by [Xe(L)/800 nm]. The net conclusion from the set of studies described above is that two factors were simultaneously essential for efficient inner-shell excitation; they are (1) the molecular environment provided by the cluster and (2) irradiation with a sufficiently high frequency.
These comparative measurements [41,51] of the Xe(L) spectra and corresponding x-ray yields were a direct demonstration that the wavelength of excitation was an important factor in the interaction that produced the inner-shell 2p vacancies. Indeed, an earlier analysis [30], based on an analogy with atom-atom collisions, had estimated the optimum frequency ω for excitation as in which α denotes the fine structure constant and c is the reduced Compton wavelength, λ c /2π. This relationship yields a wavelength of λ ≈ 200 nm, a magnitude not far from the experimental values of 193 nm and 248 nm used in the studies  [56] of N 2 and the Kr and Xe clusters described above [32,34,38,83,87]. It is also considerably shorter than the 800 nm wavelength used in the comparative studies [41,51] that showed practically no x-ray generation. These findings led to important insight on the mechanism of inner-shell excitation. On the basis of several subsequent analyses [35,36,42,52], it was concluded that an ordered phase-dependent interaction was present. Furthermore, it was found that the experimental findings could be understood [42] if the L 1 -subshell Auger widths [90][91][92][93][94][95][96][97][98] were used as a measure of the dephasing time that destroyed the ordered motion. A new analysis, that explores the excitation of the Xe(L) hollow atom states in self-trapped plasma channels [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]40] and is presented in section 4.1.1 below, illustrates the dynamical details of the attosecond character of this interaction that are dependent on the confined propagation [26].
The observational pattern developed by this sequence of experimental studies was sufficient to establish prospectively two fundamental divisions in the I-ħω plane. The main lesson delivered by the experimental findings discussed above was that, at sufficiently high intensity at a sufficiently short wavelength, an anomalously enhanced coupling strength governs the interaction. The ascending series of six points [193- With the physical Ansatz that the minimum electric field required to develop the anomalous interaction is independent of the frequency, or equivalently, that a minimum level of force on the electrons must be exerted, the line marking this estimated transition was initially drawn horizontally with a vanishing slope. This assumption recalls the result given in equation (1) in which the value of the upper bound of the cross-section σ m in the high-intensity high-Z limit was found to be independent of the frequency ω. This physical choice was subsequently confirmed by the results of two independent studies. One involved the ionization of clusters [45] in the soft x-ray (ħω = 350 eV) regime and the other [44] concerned the excitation of the prominent Xe 4d-resonance at ħω ≈ 90.5 eV. The representative points on figure 1 associated with the findings of those analyses and experiments are respectively denoted by [CLU/350] and [Xe/4d*]. Indeed, we note that these data fall precisely on the originally estimated flat contour that connects with the point designating the Xe(M) threshold at ħω = 5 eV. Hence, the results are uniformly consistent with the Ansatz of frequency independence introduced above.
The I-ħω plane presented in figure 1 is consequently divided by these experimentally anchored contours into two separate interaction zones, the green quadrant standing at the upper right corresponding to the dynamically enhanced coupling strength, and its contrasting complement, the red zone associated with conventionally described and generally weaker interactions.
The location of the vertical boundary erected at ħω ~ 5 eV in figure 1, however, requires further consideration. The phase-dependent enhancement of the interaction can occur only if the period of the driven electron motions is sufficiently short [35,36,42] in comparison with the electronic dephasing time τ e associated with the electron-electron interactions in the material. In order to estimate this relaxation time, the L-shell Auger widths of atomic innershell vacancies, as noted above, have been successfully used as an indicator [42] of this condition. With this criterion, Xe is a particularly favorable case, since the L 1 -subshell Auger width of Xe sits at a strong local minimum of ~ 2-3 eV for atomic numbers in the 50 ⩽ Z ⩽ 60 range [42,[90][91][92][93][94][95][96][97][98]. In contrast, the greatest L 1 Auger width known is for uranium [98] at ~124 eV. This datum is indicated on figure 1 by [U(L 1 )] on the abscissa, the quantum energyaxis. It is accordingly expected that the lower quantum energy ħω boundary of the anomalous region, based on the known range of Auger widths, will depend on the material, particularly in the 5 eV ⩽ ħω ⩽ 124 eV interval. Hence, in order to define a zone of anomalous interaction that is fully universal, we require the condition ħω 124 eV and set ħω ≅ 10 3 eV as the lower bound, a value that ensures the observance of this requirement. This step defines the universal region for enhanced coupling illustrated in figure 3 that is designated as the fundamental nonlinear domain.
Key supplementary data are also placed on figure 3. They include (a) the line The upper intensity limit is taken as the Schwinger/Heisenberg limit [98,99] at I = 4.6 × 10 29 W cm −2 . The zone estimated for direct coupling to nuclei is shown extending upward from ∼10 27 W cm −2 . The contour associated with the onset of relativistic motion of an electron, given by eE/mωc = 1, is shown for reference. The datum associated with saturation of the Xe(L) hollow atom system [38,39,[61][62][63][64][65][66] is placed at ħω ≈ 4.5 keV is seen to fall well within the nonlinear domain. The intensity associated with the threshold of channeled propagation of Xe(L) radiation in U established by the requirement of a critical power in a channel whose characteristic diameter is ~100 Å at I ∼ 10 26 W cm −2 is shown [104]. The purple shaded zone, designed as Xe(M) → Xe(L) located in the lower left, is associated with experiment described in sections 2.2.1 and 2.2.2 that illustrates the new aspects of the enhanced coupling characteristic of the fundamental nonlinear domain in the x-ray region. In comparison with the conventional coupling, the strength of the interaction was augmented by a factor of ~3 × 10 12 .
that defines the peak electric field E for the onset of relativistic motions for elections with mass m and charge e driven at angular frequency ω, (b) the vacuum pair creation limit, the upper value bounding the achievable intensity that is associated with ultrarelativistic electron/positron cascades [97], (c) the intensity I ≅ 4.6 × 10 29 W cm −2 corresponding to the Schwinger/ Heisenberg limit [100,101], and (d) the point characterizing the intensity of saturated amplification with the Xe(L) hollow atom system [38,39,[61][62][63][64][65][66].
The creation of x-ray channels in solids offers the possibility to produce intensities >10 27 W cm −2 in high aspect ratio geometries [102,103] and penetrate into the zone shown in figure 3 estimated to produce direct coupling to nuclei. These interactions would represent the x-ray analogue of the indirectly induced nuclear reactions generated at optical wavelengths [104][105][106][107][108]. The intensity associated with the threshold of channeled propagation of Xe(L) radiation in solid U at ∼ 10 26 W cm −2 is indicated in figure 3 with the green region above it commencing at ~10 27 W cm −2 illustrating the entrance into the region associated with the direct induction of nuclear interactions and excitations. Also shown in figure 3, as a purple shaded zone in the lower left, is the region associated with the experimental study demonstrating the anomalously enhanced interaction that is discussed in sections 2.2.1 and 2.2.2 below. The analysis indicates that the overall enhancement in the coupling strength for the nonlinear amplitude considered corresponds to a factor of ~3 × 10 12 .

Experimental evidence.
Experimental studies at high intensities in the kiloelectronvolt x-ray range are beginning to demonstrate the new features of the enhanced coupling predicted to be characteristic of the fundamental nonlinear domain defined in section 2.1 above and illustrated in figure 3. In parallel with the earliest detection of nonlinear excitation at visible wavelengths [109], a two-photon amplitude with ħω ≅ 1.8 eV, the first process observed in the new zone of interaction is a corresponding multiphoton excitation that is boosted into the keV spectral region. Specifically, it is the observed production of Xe 2p vacancies in Xe atoms undergoing irradiation by intense (∼7 × 10 15 W cm −2 ) Xe(M) emission at ħω ∼ 1 keV, a process that minimally requires a 5-photon amplitude on energetic grounds. Accordingly, the net reaction is the absorption of Xe(M) radiation γ M at ħω ≅ 1 keV by Xe atoms with the liberation of a 2p electron that is signaled by the subsequent emission of a Xe(L) quantum on a 3d → 2p transition yielding the detection of photons L γ at ħω ≈ 4.5 keV. Hence, γ M is converted to γ L through the multiphoton amplitude in which the charge state q represented by the Xe q+ ion accounts for the level of ionization generated in the complex nonlinear mechanism. The overall reaction represented by equations (5a) and (5b), that yields the Xe q+ + qe − + γ L exit channel, can be properly considered as the x-ray analogue of the previously observed [50] nonlinear excitation of Xe(M) emission generated by intense 248 nm irradiation, the mechanism represented by reaction (6a) that produces the corresponding parallel exit channel In direct analogy with reactions (5a) and (5b), this process also involves complex multi-electron dynamics with the collateral production of a substantial level of ionization [50].
The initial evidence for the nonlinear amplitude given by equation (5) was obtained from simultaneously recorded single-pulse x-ray pinhole camera images that showed remarkably common spatial morphologies of the Xe(M) and Xe(L) emissions produced from Xe clusters in a self-trapped 248 nm channel [26]. These observational data were supplemented by simultaneously recorded Xe(L) spectra and corresponding spatially resolved Xe(L) longitudinal (Z-axis) emission profiles [38,39] that mapped the distribution of radiation from the zone of excitation. Overall, the Xe(L) emission γ L observed was attributed to the radiative cascade involving the pair of reactions in which the Xe(M) radiation γ M produced through reaction (6a) is absorbed by neutral Xe atoms in the immediately neighboring spatial region of the channel [26] that is unaffected directly by the incident 248 nm wave. The key process generating the γ L emission is reaction (6b) with the effective nonlinear cross section σ 5 . Since the x-ray camera is calibrated, the quantitative analysis of the Xe(M) and Xe(L) signal strengths experimentally measured showed that 5 σ ~ 2 × 10 −21 cm 2 at the ambient Xe(M) intensity [26] associated with the propagation. The specific experimental conditions and results of this analysis are given below in section 2.2.2.
The experimental configuration utilized for these observations is illustrated schematically in figure 4. The key instruments are the multiple pinhole x-ray camera, that simultaneously records images the Xe(M) and Xe(L) emissions, and the von Hámos x-ray spectrometer that transversely records the Xe(L) spectrum with spatial resolution along the axial (Z ) direction. Initially, we consider the x-ray pinhole camera images shown in figure 5 that present the spatial morphologies of the Xe(M) and Xe(L) emissions generated from selftrapped plasma channels [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] produced by a femtosecond 248 nm pulse in the Xe cluster target [26]. Prominently exhibited by the Xe(M) and Xe(L) images are corresponding spatially matching extended emission zones in the Z > −1 mm longitudinal region. Significantly, the transverse spatial width of the observed Xe(L) emission is δ L ≅ 250 μm. Since the threshold intensity for Xe(L) production with 248 nm excitation is known [63] to be I L ∼ 2 × 10 17 W cm −2 , as given in if direct excitation by the 248 nm wave were to generate the observed Xe(L) emission. However, since these experiments were conducted with a 248 nm source unquestionably incapable of delivering a power greater than 3-5 TW, as described below in section 3, the previously established mechanism producing Xe(L) hollow atom emission with 248 nm excitation [34,38] is completely ruled out as the source of the excitation leading to the broadly extended emission seen in the Z > −1 mm region. Conversely, since it is experimentally established [26] that Xe(M) radiation produced by the channels saturates the Xe absorption under the conditions of this experiment and thereby readily propagates for extended lengths, the nearly perfect spatial overlap of the Xe(M) and Xe(L) emissions evidences the strong direct coupling that occurs through the reactions represented by equation (5) that convert Xe(M) radiation to Xe(L).
The key data from three experimental files, documented in figures 5-7 following, illustrate the properties of the Xe(M) to Xe(L) conversion represented by the mutual radiative coupling specifically expressed by equations (5a) and (5b). The axial (Z) traces pictured in figure 6 of the isometric views of the Xe(M) and Xe(L) spectra shown in figure 5 illustrate a refined visualization of these longitudinal profiles. Three characteristics are prominent in figures 5 and 6. First, the main Xe(M) and Xe(L) signals respectively shown in figures 5(a) and (b) for Z < −1 mm exhibit nearly identical spatial contours with closely matching transverse widths of ∼250 μm. Second, the corresponding Xe(L) and the Xe(M) extensions in the Z > −1 mm region in these two panels both have lengths ∼500 μm. Hence, the correspondence of the Xe(M) and Xe(L) spatial morphologies in figure 5, both laterally and longitudinally, speak to a direct physical coupling as the origin of the Xe(L) emission. This is the expected spatial signature associated with reaction (6b).
The third important observation is shown in figure 6 in the Z > −1 mm zone, where the strong sharp modulations seen in the Xe(L) signal are very greatly reduced in the corresp onding Xe(M) recording. This observation is the canonical behavior of a high-order nonlinear amplitude of the kind given by equation (6b) in which a fifth-order mechanism produces the Xe(M) → Xe(L) conversion; specifically small variations in the ambient Xe(M) intensity, peak 'M' in figure 5(a), generate proportionally much larger fluctuations in the Xe(L) excitation corresponding to the peak 'L' in figure 5 The Xe(L) spectra illustrated in figure 7 reveal additional important characteristics of the emission. The single-pulse spectrum shown in panel (a) corresponds very well to the canonical Xe(L) 3d → 2p hollow atom spectrum produced  figure 5 and the transverse von Hámos spectrometer enabling the recording of a spatially resolved Xe(L) spectrum along the axis of the channel with a spectral resolution of ∼3 eV and a spatial resolution of ∼50 μm. The Thomson imaging system used for measurements of the electron density is shown along with axial von Hámos spectrometer. The incident 248 nm beam with a pulse rate of ~0.1 Hz and diameter of ~10 cm enters from the right and is focused by the off-axis parabolic mirror to the position of the Xe cluster target. The circular orifice of the 2.65 mm diameter sonic nozzle that furnishes the (Xe) n target with upward vertically directed flow is indicated. The single-pulse data from the two cameras are collected under computer control along with the 248 nm pulse energy, the Xe nozzle plenum pressure, and the Xe gas temperature. Additional experimental details are available in [26] and [87].
with 248 nm pulses from Xe clusters [38] presented by the recording from film #3 in panel (c); the two spectra are essentially identical. Likewise, the Xe(L) spectrum illustrated in panel (b), that is attributed to the nonlinear Xe(M) to Xe(L) conversion expressed by equation (6b), exhibits good correspondence with the two Xe(L) spectra shown in panels (a) and (c). Hence, all three spectral profiles stand as equivalent. The conclusion that all spectra represent Xe(L) 3d → 2p hollow atom emissions follows.

Formulation of estimated cross section.
We presently turn to an analysis of the nonlinear amplitude expressed by equation (6b). The process is essentially a 5-photon ioniz ation of the 2p-shell with Xe(M) radiation at ħω M ~ 1 keV that subsequently yields Xe(L) quanta at ħω L ≅ 4.5 keV. In order to evaluate approximately the scale of this 5γ cross section σ 5 , we can represent the magnitude initially as the conventional product of two terms given by We will subsequently introduce a radical modification into this conventional statement in order to account for the dynamically enhanced coupling that arises in the nonlinear amplitude expressed by equation (6b). In equation (8), σ γ denotes a characteristic cross section for the reaction which is taken for this estimate as the measured peak value for the linear photoionization [111] of Xe atoms; hence, we put σ γ ≅ 5 × 10 −18 cm 2 . The energy denominator Δħω in equation (8), based on the characteristic energies of the xenon atom [111], is assigned the value of ~10 3 eV, a magnitude that falls near the peak of the Xe(M) range. On the basis of the signal strengths observed with the calibrated x-ray cameras [26], the measured value of σ 5 was found to be Accordingly, we calculate below the estimated value σ 5 from equation (8) for n = 5 and compare this computed magnitude with the observed value σ 5m . The value of the electric field strength E in equation (8) representing the ambient Xe(M) radiation is experimentally established by studies [26] of 248 nm self-trapped channel production in a Xe cluster medium. The Xe(M) intensity found in that work was ~7.5 × 10 15 W cm −2 , a value that places the field at E ≅ 2.4 × 10 9 V cm −1 , a magnitude that is roughly equivalent to one half of an atomic unit (e/a 0 2 ).

Fundamental ansatz.
In order to perform the computation of the value of 5 σ given by equation (8), it remains to estimate the effective dipole moment μ of the electronic motion induced by the electric field E associated with the ambient Xe(M) ~ 1 keV radiation. At this point, we introduce a fundamental Ansatz that represents a gross departure from standard perturbative treatments that are associated with expansions involving terms of the kind given by equation (8). Accordingly, we write the dipole moment as with Z designating the number of coherently driven electrons, e the electronic charge, and x the radial scale size of active electrons. With α designating the fine structure constant, this modification of the dipole moment imports the replacement α → Z 2 α for the basic coupling constant of the system. This key step is physically equivalent to the selection in the state manifold of the Z-fold multiply excited states [27,28,30,47,112] as the dominant contribution to the amplitude. Since it is known from an abundance of studies examining the mechanisms of photoionization of Xe that the 18 electrons comprising the three relatively weakly bound outer atomic shells (5p 6 5s 2 4d 10 ) are strongly coupled [113,114] and act as a single 'super-shell', we put Z = 18.
The value of the dipole scale length x can be naturally associated with the orbital size of the 4d orbital, since the 4d electrons are the dominant members of the 'super-shell'. The spatial properties of the Xe 4d-orbital [115] are shown in figure 8, a result that was computed from consideration of electron-impact ionization. On the basis of these data, we assign the value It is now possible to evaluate the estimated magnitude of σ 5 from equation (8) and a summary of this computation is presented in table 2. The estimated value of σ 5 detailed in table 2 falls in very good correspondence with the measured magnitude stated in equation (9). Furthermore, without the incorporation of the Ansatz giving Z = 18, a gross disagreement of ~10 12 in the magnitudes would have been found. The values of both σ 5 and 5m σ can furthermore be compared to an earlier general study aimed at the determination of limiting cross sections for high-order multiphoton coupling [46]. The result found in those computations is illustrated in figure 9. In parallel with the Xe(M) profile, a strong peak is also associated with a structured extension into the Z > −1 mm zone. The signals shown in panels (a) and (b) exhibit remarkably similar spatial morphologies that indicate a direct physical coupling. In the extended region (Z > −1 mm), the Xe(L) signal exhibits strong modulation with features of ~100 μm in width, a characteristic that contrasts significantly with the Xe(M) signal that shows perceptible, but far weaker modulation, even though the spatial resolution is approximately two-fold greater; the conclusion that the Xe(L) modulations are considerably sharper and more pronounced than those illustrated by the Xe(M) counterpart follows.
We observe that the values for 5 σ correspond quite well with the magnitude of the effective nonlinear cross section estimated in that work whose goal was the determination of an upper-bound. As shown in figure 9, the limiting cross-section m σ = 8πλ c 2 , depends only upon the mass of the electron and is independent of the frequency ω and any atomic parameters. It is accordingly possible to interpret the abscissa in figure 9 as an intensity that is independent of wavelength. We also note that these results fall in good accord with recent findings of Richter [47] concerning experiments that used a quantum energy of ħω ~ 100 eV and an intensity of ~10 15 W cm −2 in which a similar physical picture is used. In all cases, the organized motion involving Z 1 is necessary to reconcile the experimental observations with a corresponding theoretical value for the amplitude.
The significance of the nonlinear coupling associated with equations (5) and (6b) is highlighted by the location of the purple Xe(M) rectangle shown in figure 3; it falls essentially at the lower limit of the zone given by ~10 15 W cm −2 in figure 1. Furthermore, on the basis of the measured threshold [63] of ~2 × 10 17 W cm −2 for the corresponding production of Xe(L) emission with 248 nm radiation ħω ≅ 5 eV, we see that the threshold value for 2p-shell excitation has decreased by a factor of ~10 2 at the higher quant um energy of ħω ~1 keV. It follows that the circumstances for the production of deep inner-shell vacancies could hardly be more favorable; the minimally required intensity that provides strong excitation of the key states of main interest for the production of highly energetic radiation has been strongly lowered. Furthermore, since it is experimentally known [26] that Xe(M) radiation can be efficiently produced (>20%), as noted in section 2.1, this greatly reduced threshold now enables the consideration of the excitation of correspondingly increased volumes of material with result that x-ray pulse energies in the 1-10 J range can now be practically contemplated, since the technological base of high pulse energy KrF* system is very solidly established. This magnitude for the x-ray energy in a single pulse of femtosecond duration is utterly out of reach by accelerator technology. Although a complete theoretical picture of the complex phenomenon producing the inner-shell vacancies is not in hand, the present experiments combine two conditions in an incomparably new situation; specifically, it is the alliance of (1) a strong radiation field on the order of e/a 0 2 and (2) a high frequency ω whose period is less than 5 as. Since a time of ~5 as is far less than an electron dephasing time in all materials, this situation is well suited for the production of unconventional highly organized electronic motions in all materials. Hence, in forthcoming studies we can anticipate the production of numerous new exceptional forms of energetically excited matter that are analogous to the examples [32,34] provided by N 2 and the Xe hollow atom states. And, as explained below in section 4, these favorable developments may possibly be extended to nuclear systems.

Introduction
The data presented in section 2, that enabled the construction of figure 1 with the corresponding definition of the boundaries of the fundamental nonlinear domain, were exclusively obtained with the use of excimer laser technology. The basis of this unique role was the ability to satisfy simultaneously a triplet (ω, I, P) of conditions stating the minimal values of the frequency ω, intensity I, and the power P necessary for the two key physical processes involved to be experimentally observed and controllably combined [37]. Specifically, these phenomena are (1) the excited hollow atom excitation from clusters [34,83] and (2) the relativistic charge-displacement self-channeling [12][13][14][15][16]. The values of the (ω, I, P) triplet required are given approximately by ħω ≅ 5 eV, I ≅ 10 16 W cm −2 , and P ≅ 1 TW. Three additional physical conditions were also simultaneously satisfied that contributed importantly to the efficacy of the excimer technology in the study of the nonlinear interactions. They are (a) the close match of the optimum wavelength of ~200 nm estimated in equation (3) with those characteristic of the 248 nm (KrF*) and 193 nm (ArF*) transitions, (b) the ease of mode-matching the ~2 μm focal diameter readily produced with high-brightness KrF* (248 nm) technology with the naturally formed plasma channel diameter (~1.5-2 μm) that results automatically from the self-focusing action, and (c) the strikingly similar electrostatic force structures [116] associated with the 3D Xe(L) hollow atom states and the 2D hollow plasma channel. The structural correspondence associated with latter point explains the remarkable compatibility of these two phenomena; specifically, the radiative conditions suitable for the excitation of the channeled propagation are automatically and perforce perfectly fit for the excitation of the Xe(L) hollow atom states.
The sextuple confluence formed by the satisfied requirements stemming from the (ω, I, P) triplet and the favorable set of the three supplemental physical conditions produced a situation comparable to that of the lucky surfer perfectly positioned on a huge wave; the only possibility is a rapid forward advance propelled by the accumulating momentum of the enormous sea of force behind him. Accordingly, we describe in this section the concepts, design considerations, and architecture of the femtosecond excimer laser technology and conclude with a presentation of predictions concerning its forthcoming scaling to greater peak powers and higher pulse energies.

Basic femtosecond excimer laser properties and design logic
Two distinct options exist for the construction of a terawatt (TW) femtosecond high-brightness source in the ultraviolet (ħω ⩾ 5 eV) spectral region. These alternative approaches are illustrated schematically in figure 10. One possible method is the conversion of radiation from a TW-class IR laser directly to the UV with harmonic production, as schematically represented in figure 10(a). However, it is extremely difficult technically to achieve this frequency conversion, if UV pulse energies up to ~100 mJ with a femtosecond (~200 fs) duration at high brightness are desired. Overall, the barriers to implementation are sufficiently convincing that this method has not been successfully developed.
An alternative hybrid strategy involves the conversion of the radiation of a commercially available IR femtosecond (~100 fs) laser system in order to produce a high-brightness UV seed beam that is subsequently amplified by excimer laser modules in the UV, as schematically shown in figure 10(b). The use of this method enables the wavelength λ, pulse duration τ, spatial beam divergence (ΔΩ), and phase control all to be established with small aperture optics at relatively low energy and power. This situation greatly facilitates the ability to control precisely the characteristics of the system output. Furthermore, the power amplification required to reach the TW regime can be accomplished in the UV with large aperture (10-30 cm diameter) excimer laser amplifiers that are available. However, this approach generally requires the implementation of a system of amplification stages that incorporates suitable processing of the propagating amplified beam to maintain the spatial quality of the radiation and obtain high-brightness output pulses. Hence, the architecture of the Figure 8. Plots of the 4f(CA) (solid curves) and 4f( 1 p) (dotted curves) effective potentials and radial wave functions for the 4d 9 5s 2 5p 5 4f configuration in Xe + . A logarithmic scale is used for the radius. The bar indicates the range of the radii over which the second antinode of the 4d radial wave function (in the ground-state configuration) occurs. The figure is adapted with permission from [115]. Copyright by the American Physical Society. system must be carefully optimized. This method has enabled the generation of pulse energies of ~400-500 mJ at a power of ~1-2 TW to be achieved at very high brightness, as further described below.
The media available for amplification in the UV spectral region are few. Indeed, the only presently known media for the efficient amplification of UV radiation are representatives of the class of rare gas-halide (excimer) gas systems. Excimer lasers derive their emission from molecules that exist only in an electronically excited bound state, as illustrated below in figure 32 for the KrF* excimer. These levels are radiatively unstable and decay rapidly by emission of radiation into Figure 9. Plot illustrating measured, estimated, and extrapolated cross sections for total energy deposition versus 248 nm intensity for subpicosecond irradiation under collision-free conditions as presented originally in [46]. Data marked with (T ) were determined from threshold measurements. The symbol ED refers to total energy deposition. Note the two-photon resonance applying to the Xe + datum. I c denotes the Compton intensity, E is the ultraviolet electric field strength, and for σ m refer to [46]. The dashed line ---represents an extrapolation of the data to the value σ m in the high field limit and the value corresponding to total photoabsorption cross section for the K-edge of Cf is indicated for reference in the high intensity region. The locations of σ 5 and σ 5m are shown at the experimental Xe(M) intensity of ~7.5 × 10 15 W cm −2 . A good overlap with the extrapolated curve originally estimated [46] is seen at the value of ~2 × 10 −21 cm 2 , as indicated by the vertical purple rectangle. Since the limiting cross section σ m ≅ 8πλ c 2 is independent of the frequency ω, the abscissa can be generally interpreted to represent validly the intensity of the Xe(M) x-rays. µ ω µ ω Note: The final estimated value is 5 σ ≅ 2.1 × 10 −21 cm 2 for Z = 18. This magnitude is in excellent correspondence with the measured value of 5m σ ≅ 2 × 10 −21 cm 2 . We note that without the factor of Z 10 ≅ 3.6 × 10 12 , the estimated value of 5 σ would be reduced to the nanobarn range, a level that would have been experimentally unobservable by a margin greater than 10 9 . a repulsive or loosely bound ground state from which the molecules rapidly dissociate. The transition dipole moment is maximally large, since an electron moves essentially the full interatomic separation of the diatomic molecular state, a property that maximizes the radiation rate. Since the gain of an amplifying transition is directly proportional to the rate of spontaneous emission, this is a very favorable characteristic. In addition, the spectral width of the molecular boundfree transition is large, a property that enables short duration (~100-200 fs) pulses to be amplified with minimal distortion. The most important molecules employed in commercially available excimer laser systems are ArF*, KrF*, XeCl*, and XeF*. Only the ArF* (193 nm) and KrF* (248 nm) systems were used in the experiments represented in figure 10(b).
The unbound dissociative electronic ground state naturally confers a four-level structure on these systems. Generally, four-level laser media can be operated in a continuous manner. However, this modality of operation is not practically possible with rare-gas halide excimer lasers because of the physical and technical restrictions fundamentally associated with both the ambient high power densities needed for excitation and the likewise mandated large heat transfer rates for cooling.
These technical restrictions relate to basic considerations governing excimer laser operation. Chief among them is the role of the rate of spontaneous emission with respect to the corresponding rate of stimulated emission in the ultraviolet spectral range; specifically, the ratio of the Einstein coefficients A and B that is proportional to the third power of the transition frequency ν through the relation [117] = A B hv c 2 .
Obviously, in the ultraviolet range, appreciable amplification through stimulated emission can only compete with losses from spontaneous emission, if the radiation intensity stimulating the transition is sufficiently high. Physically, this requires that the system must be excited at a correspondingly high value of local power density. Consequently, electrical power densities corresponding to greater than 1-2 GW per liter must be deposited in the gas volume. A high specific power density is a fundamental attribute of the excimer media.
In the initial developmental phase of excimer laser technology, the excitation required was furnished by electron beam pumping [118,119]. Presently, however, precisely controlled short-pulse transverse electrical discharges are generally being used, if the aperture of the system has a transverse width less than ~10 cm. An extension of the electrical pulse duration in these discharges beyond approximately 1 μs is limited by the development of instabilities that cause arcing, thereby destroying the homogeneity of the excitation and the optical quality of the medium. Furthermore, the repetition rate is likewise constrained, since detrimental reaction products and heat must be removed before the next cycle of excitation; accordingly, an exchange of the laser gas is required between pulses.
The electrical energy for the discharge is supplied by a storage capacitor or, more effectively, produced from a lowimpedance pulse-forming line. For homogeneous ignition and excitation of the laser volume, a uniform preionization is generally used. Glow discharges, especially in halogen-containing gas mixtures, tend to transform into an arc or unstable sparking discharge due to electron capture by the halogens, so that the laser emission and its spatial uniformity deteriorate rapidly. Hence, considerable effort has been devoted historically to the stabilization of glow discharges [120,121]. For example, in order to supply the electrical energy sufficiently fast to avoid the unwanted growth of discharge instabilities, the technique of magnetic pulse compression was introduced,  an approach that was implemented mainly with thyratrons serving as the high voltage switches. From the physical requirements described above, it follows that electrical energy at high specific density, on the order of 64 J/liter [122], must be delivered homogeneously into the gas medium on a time scale of a few nanoseconds, so that a highpressure (up to 0.5 MPa) glow discharge can be maintained for a sufficiently long duration. In practice, the discharge terminates after a few tens of nanoseconds. Consequently, the resulting pulse duration of the amplifying state of a standard excimer laser module is of the same order of magnitude, typically 20-30 ns.
This limit of the time duration to 20-30 ns has clear implications for the generation of femtosecond amplified pulses. Specifically, it prohibits the customary implementation of the mode-locking that would be necessary for the direct generation of the desired ultrashort pulses in the excimer amplifier. Conversely, the ArF* and KrF* excimers are especially well suited to the amplification of femtosecond UV seed pulses that are introduced from an independent external source, the configuration illustrated in figure 10(b). With appropriate excitation, the gain constant of the medium is appreciable, satur ation can be achieved yielding favorable energy extraction, and the broad bandwidths of excimer bound-free transitions can support pulse durations down to 100-200 fs, in which the lower limit of the latter parameter arises from gain narrowing. In sum, the natural properties of excimer systems facilitate the generation of (1) high output pulse energies, (2) pulse widths in the femtosecond domain, (3) high peak powers, and (4) very high focal intensities, since the wavelength is short and operation near the diffraction limit is achievable. Laboratory systems can routinely produce a focal intensity of ~10 20 W cm −2 at a pulse rate of ~0.1 Hz. A laser system is described below in section 3.4 that exhibits this performance.

Gain dynamics.
Measurements of the gain dynamics have been performed for XeF* [123], XeCl* [124,125], and KrF* [126,127] using the established pump-and-probe technique. These determinations indicated a duration of ~50-90 ps for recovery of the gain at 78%, 65%, and 25% relative amplitude, respectively, for these three systems. This recovery is attributed to a combined response involving the rotational relaxation within the B-state and the C → B-state relaxation, with a significant additional contribution arising from the relaxation of the ground state for XeF* and XeCl* [123,125]. It was found that the extractable fraction of the stored energy for pulses shorter than ~5 ps was 22% in XeF*, 38% in XeCl*, and 75% in KrF*. These results correspond to the measured values of the saturation energy density corresponding to ε sat = 0.2 mJ cm −2 [123], 0.85 mJ cm −2 [125,128], and 2 mJ cm −2 [125][126][127]129], respectively, for these media.
Based on these findings, the KrF*(248 nm) system is expected to show similar amplifications for both the short pulse (~100-200 fs) and the long pulse component comprised of the amplified spontaneous emission (ASE). This fact has great importance for minimizing the ASE background in highenergy amplifier chains. Furthermore, these results solidly confirm that KrF* is superior to both XeF* and XeCl* for short pulse amplification. Figure 11 illustrates in combined form several important operational characteristics of the KrF* system.
Since applications generally desire both a maximal pulse energy and a high extraction efficiency, the operation of KrF* amplifiers must be simultaneously optimized for these two properties. However, it is seen from figure 11, which shows the dependence of the extraction efficiency and the gain contrast coefficient on the energy density, that these twin requirements cannot be optimally fulfilled simultaneously. These variables trade-off when operation of the amplifier is close to the saturation energy density. Accordingly, optimized operation can be best attained by an amplifier for which the overall local energy density is limited to a given maximal value, a circumstance corresponding to a limited gain-length product gL, due to the steep slopes of the curves near the optimum range. These physical relationships establish the power limit of an amplifier module.

Amplification in media having non-saturable absorp-
tion. Since no effective saturable absorbers are available [131] at the excimer wavelengths, spatial filtering is the easiest way to avoid gain depletion. However, this approach preserves the ASE content in the mode of the spatial beam comprising the signal. This ASE content can only be reduced by proper choice of the operational conditions of the amplifier, generally at the expense of efficiency [132][133][134][135]. Unfortunately, discharge-pumped KrF* amplifiers are only efficient when the electrical excitation is fast, necessitating a small discharge loop [136,137]. This constraint limits the cross-section of the discharge, since the system prefers elongated geometries. However, this resulting geometry is not optimal for short-pulse amplifiers possessing significant non-saturable absorption [138][139][140]. The consideration of these constraints points to a modified architecture for the system.

Off-axis and multiple-pass off-axis amplification geom-
etries. An asymmetrical geometrical configuration can be introduced to optimize the properties for femtosecond pulse amplification. Figure 12 illustrates the implementation of this arrangement and shows how the effective cross-section of the elongated discharge pumped excimer amplifier is increased while the gain-length product (gL) is simultaneously decreased Figure 12. Illustration of the off-axis amplification geometry that shows the angular tilt β between the two axes. by tilting the input beam with respect to the geometrical axis of the amplifier [135,141].
An important property of the off-axis geometry for amplification is that it allows the optimization of the operational conditions, not only for a single amplification pass, but also for multiple pass configurations using the same amplifier module [135,141]. Using a geometry with the appropriate divergence, the beam can be optimally matched to the increased effective cross section of the amplifier by the proper choice of the offaxis angle β. This configuration enables optimal extraction of the stored energy of the amplifier.
The arrangement shown in figure 13 illustrates the practical implementation of a high-brightness UV laser system. In this case, a large-aperture dual-channel KrF* excimer amplifier is used to boost the final pulse energy to ~60 mJ. Relay imaging of the third-harmonic generation (THG) crystal to the output windows of the subsequent amplifier stages maintains high beam quality. A spatial filter positioned between the discharge tubes provides efficient ASE suppression. The system shown in figure 13 can deliver a (248 nm) pulse energy of ~30 mJ with a pulse width of ~100 fs having a final peak brightness exceeding 10 20 W cm −2 sr. The use of an f/1.4 off-axis parabolic mirror to focus the output beam yields the intensity distribution shown in figure 14; the peak intensity is I ⩾ 10 19 W cm −2 , a value corresponding to an electric field strength of ~17( ) e a / 0 2 ≅ 8.7 × 10 10 V cm −1 .
In the system described below in section 3.4, this source served as the preamplifier.

Interferometric multiplexing.
Multiplexing can be employed to optimize the efficiency of energy extraction in amplifier systems. Specifically, due to the short lifetime of the excitation in the excimer medium, successive replenishment of the momentarily stored energy is the only efficient way to extract the entire stored energy [140][141][142][143], whose value is given approximately by (13) in which T designates the gain window and τ represents the recovery time of the gain. Since T/τ ⩾ 10 for KrF*, a significant increase of the extractable energy is obtainable by successive depletion of the gain [140]. However, although multiple-pass amplification is technically simpler, in the conventional arrangement for amplification it does not allow the amplifier to operate under the optimum operational conditions for each pass. The off-axis configuration for amplification [141] significantly ameliorates this problem for discharge-pumped excimer amplifiers [135,140,141] up to a limited number (n ≈ 3) of amplification passes. Since the ideal number of the successive amplification steps in KrF* amplifiers is T/τ ⩾ 10 [140,144], the use of optical multiplexing is clearly necessary to enhance the optical efficiency of the system.  In optical multiplexing, the beam to be amplified is divided into partial beams, which are arranged in the multiplexer to form a pulse train with a separation comparable to the recovery time τ of the amplifier. The key issue in the multiplexing process is the level of spatial accuracy that can be achieved in the recombination of the partial beams after amplification has been performed. In the conventional multiplexing geometries, this recombination is far from interferometric accuracy, a result that imposes severe limitations on the focusability of the final beam. However, with the use of a properly revised multiplexing method, a rather common optical arrangement can be used both for multiplexing and for demultiplexing [144] that eliminates this problem. Hence, automatic (phase-locked) synchronization of the partial beams can be achieved for any alignment of the multiplexer (demultiplexer), and any form of distortion and/or misalignment is automatically compensated. With this restorative arrangement, optimum focusability can be obtained for the output beam. Indeed, the minimum focal spot size is determined by the diffraction limit of the entire recombined beam [140,144]. This significantly increases the source brightness and, accordingly, the maximum focused intensity. Figure 15 shows a possible realization for the case of two multiplexed beams, using a Sagnac-interferometer with polarization splitting. Furthermore, this arrangement can be adapted to the multiple-pass off-axis amplification geometry [144]. Detailed considerations governing the accuracy of the superposition of the beams and information concerning preliminary experiments for two and four multiplexed beams are presented in [144].

Pulse cleaning methods
The configuration shown in figure 10(b) indicates that shortpulse (femtosecond) excimer laser systems are generally dualwavelength lasers. Therefore, diffraction-limited beam quality and a well-determined intensity distribution of the frequencyconverted beam are essential. Since phase distortions and intensity modulations are readily magnified in conventional frequency-conversion approaches, the demands on the quality of the frequency converter and the beam spatial profile at the fundamental wavelength are accordingly key. Normally, spatial filtering is a widely used technique to improve the directional properties by allowing the transmission of only the first spatial-component at the position of the Fourier plane. A new special beam filtering method [145,146] that is based on replacing the pinhole by a nonlinear component, which, due either to the intensity or the energy-density, provides a nonlinear transmission that automatically selects only the intense central spatial-component can be employed. This dynamical method automatically generates an output beam with a highbrightness distribution. Moreover, the temporal dependence of the energy flow through the nonlinear element can also be used for spectral modification (spectral filtering) of the pulse [144]. Both spatial and spectral properties of the beam are thereby put under effective and precise control.
Recent development of Ti:sapphire lasers combined with optical parametric amplifiers (OPA) has allowed the technology enabling the generation powerful few-cycle pulses [147] to enter the attosecond regime [148][149][150][151][152]. There are several methods available to reduce the temporal background of these solid-state systems. One effective way is the construction of a laser system with two chirped pulse amplification stages, separated by an intermediate pulse recompression, in which a suitable nonlinear interaction improves the contrast (double CPA, DCPA) [153]. As the effective nonlinear component, a saturable absorber, a nonlinear Sagnac interferometer, a nonlinear ellipse rotator, and, the most promising, a crosspolarized wave generation (XPW) process [154][155][156][157][158] have been both suggested and experimentally verified.
However, it is important to note the inherent problems associated with CPA configurations. For example, the relatively high temporal background in the ~100 ps range of the main pulse is present [157], which has been proven to be critically dependent on the spectral distortions caused by the amplifier chain [159]. The spatial quality is likewise degraded by the limited optical quality of the solid-state material and by the associated dynamic distortions. For these reasons, the temporal profiles of these intense ultrashort laser pulses generally suffer from the presence of prepulses that may originate either from the ASE of the IR or UV amplifier chain, or from the pedestals that arise from imperfect pulse compression. Prepulses are exceptionally detrimental for high-intensity laser-material interactions, since a long prepulse generates a preplasma [160] that significantly alters the interaction and damages the target. In such cases, the main laser pulse interacts not with the intended solid target, but with a preformed plasma. It is shown [161] that even prepulses with an intensity in the range of 10 7 -10 8 W cm −2 can change laser-matter interactions considerably. The net outcome is that the requirements for the peak intensity to ASE ratio (temporal contrast) scale with the focused intensities. Hence, since the exper imentally achieved and the planned intensities are already in the 10 21 -10 25 W cm −2 range, the necessary temporal contrast is set beyond a factor of 10 13 -10 18 .  One of the most efficient and energy-scalable methods used to remove prepulses is based on the self-induced plasma shuttering or plasma mirror technique [162,163], an approach that has also been successfully demonstrated for short-pulse KrF* laser systems [164]. If the intensity of the laser pulse falling onto a suitable transparent solid material is chosen so that only the leading edge of the main pulse is above the plasma formation threshold, prepulses on the pedestal at lower intensity will be sharply suppressed, thereby improving the contrast up to several orders of magnitude. This method, however, normally allows only a limited contrast improvement and/or limited energy throughput. Moreover, the optical quality of the plasma front influences the phase front of the beam, and a fresh target area is perforce needed for each shot. For these reasons, a novel pulse-cleaning technique is presently introduced below. This approach is generally applicable both for single and dual wavelength lasers, and does not suffer from the shortcomings of the standard plasma mirror method described above.
In the new arrangement, a nonlinear component is situated at the center of a confocal telescope surrounded by a conjugated beam-block filter pair, as shown in figure 16. The basic concept behind this method is similar to that of active spatial filtering [145]; specifically, an intensity dependent modulation introduced at the Fourier-plane leads to directional modulation. For implementation, an annular input beam, introduced by the input filter with the intensity profile shown in figure 17(a), is ideal, since it has an 'Airy-like' intensity distribution in the Fourier-plane (see figure 17(b)) whose modulation is known to generate a 'Gaussian-like' output beam [145]. As long as no modulation occurs in the focal plane, the arrangement corresponding to figure 16 has a vanishing transmission, allowing full exclusion of any prepulses. However, for the intense main pulse, which introduces amplitude and/or phase modulation in the focal plane, finite transmission is expected to occur. As described fully below, the overall behavior produced by this arrangement is illustrated by comparison of the spatial profiles illustrated in figures 17 and 18.
The comparative spatial profiles shown in figures 17 and 18 can be readily understood. Indeed, the best results are obtained, together with simultaneous temporal and spatial filtering, when the phase modulation is introduced in the focal plane with no amplitude modulation. Figure 18 shows the results of the corresponding calculation when the phase of the 0th order is shifted by λ/2 by the nonlinear interaction. Practically the 'inverse' of the annular input beam shown in figure 17(a) emerges at the output; as high as 55% of the energy of the input beam is diffracted to the central hole of the output aperture. Another advantage of this method, beyond the high contrast improvement and high overall throughput, is that spatial filtering occurs in the central (transmitted) part of the beam. According to both numerical calculations and corresp onding experimental observations, any high order modulation (noise) of the input beam is only present in the low intensity ring-shaped part of the output, which is blocked by the output filter as shown in figure 19. The net outcome is that pure phase control produces essentially a perfect beam. Under proper experimental conditions, self-focusing in a self-generated laser plasma produced with a noble gas jet is capable of shifting the phase of the central lobe of the diffraction-pattern by λ/2 without significant absorption. Using a pulsed gas jet and a noble gas as the nonlinear phase shifter, constructive interference after passing through the nonlinear medium with a well-defined length allows simultaneously (a) the minimization of the losses and (b) the suppression of the unwanted spatial and temporal components of the laser beam. Importantly, this method is also self-aligning; no matching of the 0th order to a pinhole is needed.  In summary, the main features of the nonlinear plasma filter pictured schematically in figure 16 are the following: ▪ high improvement of the temporal contrast (>10 3 ), steepening of the leading edge (temporal filtering), ▪ beam smoothing (spatial filtering), ▪ self-adjustment (no need for precise alignment), ▪ very high overall transmission (>40% obtained experimentally), ▪ applicable to a broad wavelength range.
It is important to observe that the method based on the phase-shift introduced by the plasma of an ionized noble gas is applicable in a broad wavelength range. However, we note the practical limitation of the application to CPA systems that arises, since the use of the nonlinear temporal filter necessitates the presence of the compressed (minimum) pulse duration, which is normally only available at the end of the amplifier chain. On the other hand, with high-intensity excimer systems, due to the direct amplification of femtosecond pulses, the required pulse of short duration is available in all sections of the amplifier chain. Therefore, the use of the nonlinear filter with its high intrinsic throughput, together with saturated operation of the following amplifier(s), leads to a negligible reduction of the output energy of an excimer system.

Power and energy scalability of high-brightness discharge pumped excimer amplifier systems
We now turn to issues associated with the power and energy scalability of excimer systems, since this issue is directly relevant to the three grand challenge problems introduced in section 4. Since most applications of high-power, high-brightness laser systems require excellent spatial and temporal quality of the pulses, further amplification to boost the output energy from the 100 mJ level is a sound concept only if the contrast of the pulses can be maintained without significant optical losses or the introduction of excessive optical complexity. With the availability of the new pulse cleaning method described above, we now show that further amplification based on the discharge-pumped technology is feasible.
In order to obtain an increase of the short-pulse energy, a corresponding increase of the effective cross-section of the amplifier is necessary. This, however, requires a corresponding increase of the voltage and the transferred energy of the electric pumping circuit. Since, the complexity of the corresponding power system rises sharply above 30-40 kV, the discharge gap separation is limited to d ≈ 4 cm. This bounds the achievable beam cross-section to ~16 cm 2 and, therefore, sets a pulse energy limit of ~100 mJ.
We presently describe a solution that extends the energy scaling and thereby enables the short-pulse energy to be lifted to the Joule level. We limit these initial considerations to discharge-pumped KrF* amplifier technology with special emphasis on table-top design, ease of operation, and maximum repetition rate.
We commence this discussion with a specific model represented by a currently functioning system called 'PROMETHEUS' that operates in the x-ray Microimaging and Bioinformatics Laboratory of the University of Illinois at Chicago. The parameters of this system represent the present limit of femtosecond high-brightness discharge pumped KrF* technology. The power amplifier is a double-discharge gain module with an aperture of A = 10 2 cm 2 for the discharge cross-section. The performance yields a maximum gain-length product of g 0 L ≈ 16 over a duration of ~50 ns (FWHM). This existing system concept cannot be scaled to a larger effective cross-section. Analysis shows that its stored energy E m = E sat •g 0 •L•A = 32 J is significantly higher than the maximum extractable energy. The maximal amount allowed by the bounded effective beam cross-section (A) is E out = 2.2 E sat A ≈ 450 mJ [135]. The explanation of this limitation along with the general consideration of the optimum discharge geometry, with the use of the off-axis amplification modification, is given in [141].
In figures 20 and 21, the beam line overview and corresponding laboratory physical layout are illustrated respectively for the present Prometheus system. The nominal operating parameters are power P ≅ 1-2 TW, pulse energy E = 300-500 mJ, pulse length 200-300 fs, and repetition rate ~0.1 Hz. With an off-axis f/2 parabolic mirror on a 6-axis stage for precision alignment [201], as shown in figure 22, the system can produce the focal spot of ~2 μm in diameter that is presented in figure 23. This performance represents ~1.65 times the diffraction limit and enables focal intensities of ~ 6-8 × 10 19 W cm −2 to be produced, a value that corresponds to an electric field amplitude of ~40 ( ) e a / 0 2 . We now illustrate how the current extraction limitation can be reduced and permit the generation of a considerably elevated pulse energy. In order to achieve this improvement, a novel arrangement is introduced that is capable of increasing the optically achievable 'effective' cross-section of 'rectangular' discharges, essentially improving the aspect ratio associated with the geometry of the discharge. Discharges with an aspect ratio of 1:1 are complicated to create and normally require the use of x-ray preionization. Up to a distance d ~ 5 cm for the gap separation, the standard HV technology based on the use of thyratrons and magnetic pulse compression can be applied. If such a discharge technology is available, the  effective cross-section can be multiplied by a factor of 4 by geometrical combination of two discharge sections, as shown in figure 24.
In this configuration, the two discharges are 'matched' to each other through their flat electrodes. As a result, in 1D the combination of the two discharges enlarges the volume, while, in the other dimension, the off-axis geometry increases the effective beam size to be amplified. This corresponds to the dashed line shown in the figure 24. The arrangement uses a common window for the two discharges; hence, the effect of the masking from the flat electrode in the center is negligible.
The use of this geometry alters the scaling relationships and associated limitations; specifically, the optically usable cross-section is significantly increased, while the speed of the discharge loop remains unchanged. This modification would raise the achievable energy of a PROMETHEUS-class amplifier to the ~1.5 J range. A further technical advantage is that the two discharge sections can be preionized by a common preionizer unit. By the use of the experimentally proven, twobeam interferometric multiplexing of polarization splitting, an additional increase of the final energy, approximately by a factor of 1.7, is possible [165].
An alternative that would be operable up to an aperture scale of ~30 cm and enable an additional boost of output pulse energies to the multi-joule level involves the use of the e-beam KrF* technology that has been extensively developed for laser/  fusion applications [166]. With the efficiency of kilovolt x-ray production at the demonstrated [26] level of ~30%, x-ray pulse energies of ~1 J could be anticipated. Pulse energies of this magnitude would be sufficient to apply to the three grand challenge problems introduced in section 4.

Grand challenge problems
The trajectory of advance over the past 50 years serves as a convincing prognostication of future developments. In this section, we briefly introduce three foreseen areas of future study that are perceived as topics of major significance. The landscape associated with these prospective areas is shown in figure 25, which is a modestly modified extension of figure 1. The essential point shown is illustrated by the ascending blue contour that connects several key findings of the last 50 years and projects predictively upward to the region associated with intensities >10 20 W cm −2 , quantum energies >10 4 eV, and terminates in the zone defined by the Schwinger/Heisenberg limit.
The grand challenge problems introduced below are (1) a direct experimental probe of the vacuum state with intensities approaching the Schwinger/Heisenberg limit, (2) the achievement of amplification in the γ-ray region, and (3) the establishment of a navigable path to the prospective super-heavy nuclear island of stability. The latter two require the creation of conditions suitable for controlled coupling to nuclei while the former demands capability minimally at that level as indicated by the zone of direct nuclear coupling that lies adjacent to the Schwinger/Heisenberg limit in figure 25; the chief outcome is that the physical conditions are comparable for all these problems. Hence, with this consideration in mind, we accordingly provide a preamble that addresses the basis of the potential to achieve nuclear excitation as an introduction to the following discussions on the triplet of specific problems named above.

Atomic inner-shell excitation.
A striking result of the excitation of Xe clusters with intense 248 nm femtosecond pulses in self-trapped plasma channels is the copious production of Xe(L) emission from hollow atom states [34]. The key to the Xe(L) emission is the selective production of 2p vacancies in a manner that enables the retention of a subset of electrons in the 3d shell, thereby enabling the prompt generation of ~4.4 keV x-rays from Xe q+ ions on many (30 ⩽ q ⩽ 37) 3d → 2p transition arrays [38]. Experimental studies have demonstrated several important properties of the mechanism leading to the production of the Xe(L) hollow atom radiation. Salient among them are (a) the need for cluster formation [33,34], confirmed by the complete lack of x-ray emission from Xe monomer targets under similar conditions of excitation [85], and (b) the inability of comparably intense femtosecond 800 nm pulses to produce from Xe clusters the characteristic Xe(L) hollow atom emissions readily generated with 248 nm ultraviolet excitation [41,51], as shown in figure 1. It is also experimentally established [38] that bright Xe(L) emission is directly correlated with the production of self-trapped 248 nm plasma channels [12-16, 18, 20, 23, 24, 26], since the dynamics of the confined propagation perforce produces a very rapidly ascending leading edge on the pulse whose peak intensity reaches a value of ∼10 20 W cm −2 .
Previous studies have shown how the presence of an ordered phase-sensitive driven electron motion, such as that discussed in section 2.2, can provide an interaction capable of producing the observed behavior [36], a concept initially introduced in an intra-atomic context [28][29][30]. The present discussion (a) specifically illustrates the attosecond dynamics involved in the 2p vacancy excitation in Xe clusters that yields the Xe(L) emission and (b), based on this model, introduces the concept of scaling these processes in both frequency and medium density to the neutron-free induction of nuclear fission through the direct coupling to nuclear states with channeled x-rays in high-Z solids. The latter possibility could be considered as the zeptosecond 'nuclear echo' of the intracluster attosecond atomic inner-shell vacancy production. Hence, the atomic inner-shell excitation that is experimentally anchored is related to the corresponding nuclear context. This approach thereby furnishes an experimentally based estimate that indicates the feasibility of direct nuclear excitation.
The hollow atom Xe(L) spontaneous emission spectrum, shown in figure 26, conspicuously reveals the unusual nature of the excited electronic configurations that characterize these states. Normally, an atomic L-shell spectrum in a highly ionized plasma consists of a few narrow lines; in sharp contrast, the spectrum illustrated in figure 26 is a broad doublet that expresses perceptible structure. This atypical emission profile is the canonical presentation of a hollow atom system [110], namely, a high spectral density quasi-continuum of overlapping transitions whose average energy separation is less than the corresponding characteristic radiative width of the lines. For example, the prominently observed Xe 31+ array at λ ∼ 2.92 Å involves an electronic configuration with a half full 3d-shell whose corresponding spectrum consists of more  [39,[61][62][63]66].
A previous analysis of the intracluster dynamics [36] attributed the 2p-shell ionization to a driven 4p-2p coherent scattering. We give below dynamical estimates of this interaction as it occurs on the incident edge of a propagating self-trapped 248 nm pulse in a plasma channel [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] in order to illustrate the attosecond nature of this mechanism. Using the development of the first lobe of the 248 nm wave as a model for the profile of the rapidly rising field strength, the panels shown in figure 27 illustrate the sequence of interactions involved in this intracluster process.
Four time scales are identified in figure 27 that characterize the field ionization, interatomic transport, and exit of the electron from the cluster. The key time associated with the 2p vacancy production from the nearest neighbor electronic coupling is τ = + + ≅ t t t 95 as, the interval that governs the collisional interaction time of the free 4p electron liberated from one atom with the bound 2p electron in the adjacent system. Linear polarization of the 248 nm radiation is assumed so that the incident wave communicates no angular momentum to the electron in this interaction. We note that the time found for t 2 of 37 as is fully consistent with the moderately longer intervals that are naturally expected to be characteristic of the tunneling regime [169]. We note that the work of Pfeiffer et al [170], that involves a more complicated experimentally tested model of electron escape, yields comparable dynamical times. With the canonical atomic time τ a ≅ 24 as, we have hence, τ corresponds equivalently to an energy of ∼6.8 eV, a value that is only slightly above the quantum energy (ħω = 5 eV) associated with the incident wavelength of 248 nm. An earlier analysis [42] showed that the presence of an ordered phase-sensitive interaction could be correlated with the L 1 -subshell Auger width of atoms [94][95][96]. Specifically, if the inverse of this Auger width is used as a measure of the electron-electron interaction dephasing time, the experimentally established minimum in the L 1 level widths in the atomic number range 50 ⩽ Z ⩽ 70 favors Xe (Z = 54) atoms for a coherent interaction of the envisioned type. The corresponding L 1 -width for Xe is [42] Figure 25. Fifty year trajectory of advance in nonlinear studies serving as a prognostication of future developments. This graphic, adapted from figure 1 and commencing with the CO 2 laser (ħω ≅ 10 −1 eV), which subsequently extends through the excimer systems (e.g. KrF*) and the Xe(L) amplifier (ħω ≅ 4.5 keV), presents a physically anchored extrapolation that intersects the zone predicted for direct nuclear coupling and continues to the vicinity of the Schwinger limit (I ≅ 4.6 × 10 29 W cm −2 ) at ħω ≅ 1 MeV, the quantum energy associated with + − e e / pair production. The analysis supporting this projection was initially delivered at the 'Laser Optics Berlin' Conference in March 2010. The 'time-line' presented in the 'supplementary data' provides insight into the interpretation of the data.
a range corresponding to a characteristic time τ A ∼ 159-216 as that yields the result It follows from equation (17) that the estimated time scales of the interactions illustrated in figure 27 are consistent with the requirements for an ordered phase-sensitive interaction that can selectively generate 2p vacancies in Xe. We also note that, given the characteristic attosecond times represented in figure 27, sufficiently sensitive measurements of high harmonic generation in clusters [171] could potentially provide complementary information on these dynamics.

Extension to direct nuclear excitation.
The stable physical mechanism that compresses radiative energy into narrow channels [12-16, 18, 20, 23, 24, 26] with the propagation of 248 nm pulses in self-guided under-dense channels at a plasma electron density of ρ e ∼ 10 21 cm −3 , and thereby dynamically generates an extremely sharply rising incident wave profile, admits the possibility of scaling the interaction to greatly reduced time scales at shorter wavelengths, higher plasma densities, enormously augmented power densities, and nuclear interactions [104,200]. This scaling of the selfchanneling phenomenon from the ultraviolet region [12-16, 18, 20, 23, 24, 26] to the x-ray range is called 'Photon Staging' [102][103][104]. This step represents a transition in parameter space that raises the quantum energy ħω into the multikilovolt regime and the plasma density to ρ e ∼ 4.4 × 10 24 cm −3 in the case of solid uranium. The resulting correspondence is that both ω and ρ e are increased approximately by a factor of ∼10 3 and leads to the production of channels with characteristic diameters of ∼100 Å, a dimension less than the size of a typical virus. In general terms, the creation of such channels with multikilovolt x-rays in high-Z solids yields projected power densities [102,104] on the order of ∼10 28 -10 30 W cm −3 . In order to illustrate the characteristic intensities and time durations of this scaling, we will use the example of nuclear fission [105][106][107]172] generated in an x-ray channel that is produced in solid uranium. Previous calculations [104] of self-trapped channels of Xe(L) radiation at ħω ≅ 4.4 keV in solid uranium provide the information required to estimate the zeptosecond time scale to reach an intensity of ∼10 27 W cm −2 , the level associated with the projected onset of direct coupling [104] of the incident radiation to nuclear matter shown in figure 28. The peak intensity generated in these channels [104] is ∼10 29 W cm −2 . Using the same procedure that produced the intensity profile illustrated in figure 27(d), the resulting envelope of the leading edge of the channeled x-ray pulse, as shown in figure 28, gives a rise time of τ x ≅ 15 zs to the intensity I ≅ 10 27 W cm −2 . This interval is compared in figure 29 to the time scales associated with the process of nuclear fission. We observe that the scission dynamics can occur nominally in the τ f = 10-30 zs range [173,174], thereby giving the approximate equivalence a relationship that mirrors equation (17), given our present level of knowledge of the details of nuclear dynamics [175]. We also note that the corresponding nuclear dimensional size δ during this critical stage of nuclear evolution on the reaction coordinate is δ ∼ 7-25 × 10 −12 cm. Certain heavy strongly deformed triaxial nuclei that possess multiple and complex fission pathways [176] may be favorable candidate systems for the observation of this direct nonlinear nuclear process, since the gross nuclear spatial reorganization required to reach the fission exit configuration is expected to lengthen the dynamical time scale beyond that shown for τ f in figure 29, thereby strengthening the inequality given by equation (18). The fast pulse rise time that is perforce dynamically developed by the self-trapped propagation of 248 nm pulses in underdense plasma channels induces an attosecond scale inter-atomic electronic motion in Xe clusters that produces an ordered phase-dependent interaction capable of selectively generating hollow atom states with 2p-vacancies. The estimates given above indicate that this dynamical behavior may potentially be extended into the nuclear zeptosecond region with the concept of 'Photon Staging', a process that is the analogous production of self-trapped channels in high-Z solids with x-rays.  [61]. For further spectral details, see [38,39,[61][62][63]66].
probe of this entity would doubtlessly yield many informative findings giving insight on the nature of the vacuum. If appropriately developed laser technology could reach an intensity in the vicinity of the Heisenberg/Schwinger Limit [100,101,179] of ~4.6 × 10 29 W cm −2 , explosive + − e e / pair production is expected. However, studies [180][181][182][183][184][185] indicate that the achievement of intensities as low as ~10 25 W cm −2 would yield detectable signals stemming from the influence of vacuum polarization. Hence, experimental intensities in the 10 25 -10 29 W cm −2 range, a span that fully overlaps the zone designated as 'direct nuclear coupling' in figure 25, are indicated as sufficient to serve as a probe of the character of the vacuum state.
Two different technological proposals are presently under evaluation. One involves large infrared (λ ~ 1 μm) lasers, but faces a serious limit from electron cascades [99]. A second approach [104] utilizes x-rays, involves the concept known as 'photon staging' introduced above, and is immune to the electron cascade phenomenon. Studies have shown that the use of x-rays [104,186,187] has the advantage that the requisite intensities may be reached with a pulse energy as low as ~100 mJ. Accordingly, this approach has the advantage of small scale, relatively low cost, and a correspondingly more rapid rate of implementation. In connection with the approach involving x-rays, we outline below the concept of 'Photon Staging'. The detailed description of these proposed efforts concerning the study of the vacuum state are contained in [104,186,187].
The use of coherent x-rays for the attainment of intensities approaching the Schwinger limit [101] of ~4.6 × 10 29 W cm −2 , a condition initially discussed by Sauter [179] and Heisenberg and Euler [100], is supported by very powerful scaling relationships. The key concept for the production of ultrahigh intensities with x-rays is 'Photon Staging'. Simply stated, this is the channeling process outlined in figure 30 elevated in both frequency ω and electron density n e . The governing scalings are illustrated in figure 30; basically, it is the channeling of x-rays in solids, a phenomenon that raises the critical power P cr to values into the range of ~0.1-1 PW. Since the maximum plasma density is n e ~ 4-5 × 10 24 cm −3 in high-Z solids, the corresponding channel diameters are compressed to ~100 Ǻ in materials like Fe, Au, and U. This nonlinear focusing mechanism (1) obviates the need for x-ray optics, (2) Figure 27. The sequence of dynamical processes in the mechanism of 2p inner-shell ionization by a driven 4p-2p collisional interaction with a pulse of 248 nm radiation whose rapidly rising incident edge is dynamically generated by self-trapped propagation in a plasma channel is portrayed. The peak intensity achieved is estimated to be ∼10 20 W cm −2 . (a) Electronic potentials V(r) as a function of radial distance in Å units associated with two adjacent atoms in a Xe cluster are represented. The potentials shown correspond to atomic structural information derived from the analyses of Herman and Skillman [167]. Atomic subshell ionization data are given in [168]. (b) With the electric field amplitude shown in panel (d), that depicts the fast rise of the corresponding 248 nm intensity that develops on the leading incident edge of the propagating pulse, the estimated time for the Stark-induced continuum lowering to release a 4p electron with a binding energy of ∼572 eV is estimated to be t 1 ≅ 13 as. The corresponding time estimated for the escape of the unbound 4p state from the central region of the atom is t 2 ≅ 37 as at a field strength of 870 V Å −1 . (c) The transit of the 4p electron accelerated by the optical field through the Xe cluster is represented. The nearest-neighbor electronic interatomic transit time t 3 ≅ 45 as and a cluster transit time t 4 ≅ 264 as are estimated. The sums τ = t 1 + t 2 + t 3 = 95 as and T = τ + t 4 = 359 as represent the nearest-neighbor interatomic and cluster time scales, respectively, the latter for a cluster nominally containing ∼100 atoms. (d) The contour of the electric field amplitude of the pulse on the leading edge of the 248 nm wave propagating in a self-trapped plasma channel is depicted. The maximum intensity is ∼10 20 W cm −2 , the pulse peak power is ∼1 TW, and the plasma density of the channeling medium is ∼10 21 cm −3 , a level that is ∼5% of the critical plasma density at 248 nm. has demonstrated [26] an efficiency of energy transport in the ultraviolet of >90%, and (3) produces a focal diameter that is governed mainly by the plasma density. The key requirement [12][13][14][15][16] for the production of a channel in the underdense regime is the generation of a peak power P o exceeding the critical power P cr necessary for the development of the confined propagation with the relativistic/charge-displacement mechanism. The overall scaling of this mechanism is such that, for x-rays with ħω > 1 keV in any solid target, the onset of channeled propagation perforce generates an intensity of ~10 26 W cm −2 , or greater. Hence, these dynamics strongly favor the production of intensities in the desired range.
The stability of the propagation is assured by the existence of a robust eigenmode [13,15,16,23,25,26]. For the case of λ x ≅ 2.9 Å and fully ionized uranium, P cr ≅ 49 TW. With a pulse length τ x ~ 50 as, a value that is well within the projected performance [65,66] of the Xe(L) system, the critical power corresponding to uranium can be achieved with a pulse energy as small as E x ~ 3.0 mJ. The corresponding energy for Xe(M) at ħω ≅ 1 keV, from the scaling relations shown in figure 30 and the assumption of a comparable pulse width, is actually less than 1 mJ. The power and intensity scaling properties of a channel produced in uranium by Xe(L) radiation, at ħω ≅ 4.5 keV are represented in figure 31. The calculations of the propagation of Xe(L) radiation in uranium channels show that a power of ~0.8 PW is sufficient to reach the peak channel intensity of ~10 27 W cm −2 , a value enabling entrance to the direct nuclear coupling zone illustrated in figure 25. As Figure 28. Electric field profile of the incident leading edge of a self-trapped 2.8 Å x-ray pulse in solid uranium [104]. As illustrated, the peak intensity produced by the pulse is ∼10 29 W cm −2 and the intensity of ∼10 27 W cm −2 , the value estimated for direct radiative coupling to nuclear states, is reached in a time of τ x ≅ 15 zs. developed below, the photon staging concept, through its ability to produce intensities in the 10 27 W cm −2 -10 29 W cm −2 regime, plays a central role in both of the remaining grand challenge problems that (1) involve amplification in the γ-ray region and (2) the attainment of a route to the super-heavy nuclear island of stability. The 'Photon Staging' process is the key mechanism that can provide the high intensities generally required to address these advanced technical problems.

Amplification in the gamma-ray region.
Excimer laser technology was the key to providing all of the data included in figure 1 with the exception of two points; specifically, they are the NH 3 studies conducted with CO 2 laser radiation [69] and the experiment conducted at the FLASH facility [47]. Hence, the definition of the fundamental nonlinear domain historically rests entirely on experimental studies conducted with excimer lasers, specifically the KrF* and ArF* systems. Therefore, in light of the background discussed in section 3 and the conceptual extension developed explicitly below, we examine the possibility that the excimer concept could potentially be transferred into the nuclear region in a manner that would enable the amplification of γ-rays on nuclear transitions [188]. Hence, we explore below the basic question; can a 'nuclear excimer' be achieved?  'Photon staging', channel formation in solids with x-rays. Plasma channel formation becomes possible with pulse energies of ~10-100 mJ for corresponding x-ray pulse lengths of ~10-100 as. Enormously enhanced power compression is the leading outcome. The exact expression for the critical power for relativistic and ponderomotive self-channeling P cr has been derived in [13].
As a first step in considering this possibility, we observe that there exists a structural/dynamical analogy between excimers and a wide class of nuclear states. Consider the KrF* system and the energy level structure illustrated in figure 32. The excited KrF* state arises from the binding of the + Kr and − F ions. For both of these individual species, respectively, the removal (Kr + ) or addition ( ) − F of an electron affects quite weakly the orbits and energy states of the other bound electrons in the system. In the limiting case of the pure independent electron model, there is no influence; to wit, the orbital excitations of Kr and Kr + are highly comparable, both structurally and energetically. Conversely, the ground state and lowest excited states of the bound KrF* molecule differ greatly, as apparent from the potential energy curves shown in figure 32. The basis of this difference is that the ionic bond formation in the excited state manifold requires a self-consistent reconfiguration of the charge density that differs grossly from those characteristic of the participating atomic states. Hence, the redistribution of the atomic charge densities associated with the bonding mechanism deviates significantly from the independent single electron picture. This feature of the dynamics of the bond formation resembles the collective interactions that are important in nuclei, particularly systems removed from closed shell configurations [188,189]. In analogy with the self-consistent solution of the electron density associated with the KrF* bond formation, nuclei seek a similar corresponding self-consistent arrangement of the nucleons in order to form the bound state of the system. This structural/dynamic analogy can be explicitly illustrated by the comparison of two well documented reactions, specifically the neutron (n) induced fission of 235 U and the KrF* excimer formation and radiative decay. The fission channel of 235 U is represented by in which the unstable excited 236 U* configuration rapidly fissions with the production of energetic excited fission fragments f 1 * and f 2 * that can further deexcite yielding gamma quanta γ 1 and γ 2 with corresponding multiplicities n 1 and n 2 together with free neutrons n whose average multiplicity is ν ≅ 2.7. The addition of a single neutron stimulates a massive rearrangement of the nucleons in the unstable 236 U* system as it seeks a self-consistent solution to the nuclear force structure, a process that yields two fission fragments, that are generally internally excited, and a set of free neutrons. Hence, a relatively small initial perturbation arising from a single neutron is enormously amplified by the collective nuclear dynamics and produces an outcome that is a radical change in the state of the system, specifically, an energetic disintegration of the nucleus. The dynamics of KrF* excimer formation and radiative decay can be viewed analogously. Consider the reaction in which the − F and + Kr ions produce the bound radiatively unstable KrF* intermediate that rapidly decays yielding the production of the unbound neutral Kr and F atoms and the quantum γ, the mechanism shown in figure 32. In parallel with the fission reaction represented in equation (19), the large rearrangements of the charge distribution caused by the development of the self-consistent electron configuration associated with the KrF* bond formation and the subsequent radiative decay of the KrF* excimer lead to a radical change in the state of the system. Therefore, in both reactions, the strong reorganization arising from the self-consistent bond formation produces energetic decay of the entity involving the emission of particles and photons. The unstable 236 U* system and the KrF* excimer play analogous roles in equations (19) and (20).
From this structural/dynamical analogy, it follows that the molecular 'excimer' concept may be directly transferable to  ). The blue background shows the calculated shell-correction energies. The orangeshaded lines indicate known and predicted shell closures. The squares represent presently known or claimed nuclides. The nobelium and lawrencium isotopes whose masses are reported here [193] are indicated by red squares. The yellow and green squares represent nuclides whose masses are determined by use of these new mass values, respectively, as anchor points in combination with experimental α-decay energies. Hatched squares show nuclides with unknown or ambiguous excited states. Figure and caption adapted from [193] and used with permission. Copyright 2012 AAAS. nuclear systems. Accordingly, many nuclei could potentially appear as attractive 'excimer' candidates in the sense that their dynamic response to a relatively small initial perturbation could efficiently produce excited species capable of supporting amplification on nuclear transitions in the γ-ray region. It can be imagined that the initial interaction triggering the reaction could arise from direct electromagnetic coupling to the nucleus. As a model of this potential behavior, the selective production of excited fission fragments stimulated by an external electromagnetic interaction with a suitable nucleus, perhaps a presently unknown super-heavy system X A Z N of the kind discussed below in section 4.2.3. This general process, that appears particularly prominently in the heavy fission fragment distribution [190] of many nuclei, would appear to be a fertile area of study. Following the analogy given by equations (19) and (20), the illustration shown in figure 33 states this hypothesis as a mirror of the representation of the dynamics of the KrF* excimer in figure 32.

Path to nuclear super-heavy Island of stability.
The 'island of stability', first proposed [191] by Seaberg in 1969, has origins stemming from the properties of the nuclear 'shell model' initially discussed [192] by Göppert-Mayer in 1948. At present, the predicted island of stability is estimated [193] to have proton number Z ≅ 114 and neutron number N ≅ 184, giving A ≅ 298 and a corresponding neutron excess of N − Z ≅ 70. The exceptionally neutron-rich nuclei occupying this mass range are expected to be highly and variably distorted [194], express elevated fission energy yields (~1 GeV), and the corresponding neutral atoms are anticipated to exhibit unusual chemical properties. Essentially, everything is predicted to be anomalous.
A key step along the route to the super-heavy island of stability illustrated in figure 34 would be the ability to produce an abundance highly neutron-rich species from an available stable nuclear system. This desired feature naturally highlights many unusual properties of nuclear systems [195] and particularly the dynamics of the class of internal nuclear motions that produce a separation of protons and neutrons; the classic example is the giant dipole [196] in which the protons and neutrons oscillate against each other as two organized separate groups of particles. Ideally, a dynamic configuration leading selectively to enhanced proton emission from the system that would simultaneously retain the neutrons while leading to a sufficiently metastable or stable exit channel is sought. Some experimental evidence for this possibility exists in connection to the giant dipole resonance; proton emission has been observed [197] and multiple nucleon emission is experimentally known [198]. As a conceptual picture, imagine an initial maximally neutron-rich target nuclear system located just inside the boundary defined by the neutron drip-line, that is excited by direct interaction with an external field in such a way that a piece of the nuclear surface is dynamically reconfigured to pass through the proton drip-line, a step that would selectively open the door to rapid proton emission. In order to take advantage of the anisotropy of the nucleus for a deformed target system, nuclear alignment could be performed to orient the interaction. In addition, circular polarization could also be employed to communicate angular momentum in the coupling. In the 10 27 -10 29 W cm −2 range of intensity designated as the direct nuclear coupling zone in figure 25, simple estimates indicate a magnitude for the electromagnetic interaction energy for a system with A ≅ 240 in the range of 100-300 keV, a value comparable to level splittings in many heavy nuclei [194,199].

Conclusions
The research conducted over the last five decades covered by this article supports several important conclusions and points predictively toward expectations in the next half century.

New zone of interaction
A new zone of electromagnetic interaction has been identified. It is universal in its applicability to all materials and represents a dynamical region characterized by an enhanced coupling strength. This region is denoted as the fundamental nonlinear domain and has lower limit boundaries in the intensity (I)quantum energy (ħω) plane that correspond respectively to I ⩾ 10 16 W cm −2 and ħω ⩾ 10 3 eV. The augmented interaction can be considered as an increase in the basic electromagnetic constant α in which α → Z 2 α, where Z designates the number of electrons participating in a ordered response to the incident driving field. The magnitude Z 2 α > 1 has been observed and Z 2 α > 50 is projected to be attainable.

Identification of key technology
The data acquired over decades of research activity conclusively show that excimer laser technology was the clear champion in the experimental campaign leading to the establishment of the fundamental nonlinear domain. The basis of this role rests upon a triplet of conditions that could be simultaneously satisfied with this technology concerning the frequency ω, intensity I, and power P. The achievable values of the (ω, I, P) triplet were sufficiently high that the key processes associated with the organized electron motion leading to inner-shell excitation and the plasma channeling process could be satisfied, thereby permitting these phenomena to be controllably combined under conditions enabling rigorous and detailed experimental study.

Scaling relations and predictions
The derivation of important physically anchored scaling relations was achieved that enable the formulation of predictive assessments. A central relationship named 'photon staging', that is supported with a quantitative analysis, describes the extension of the self-channeling mechanism to x-rays in condensed matter. The predicted maximal intensities and power densities associated with these channels respectively are ~10 29 W cm −2 and ~10 30 W cm −3 , levels of increase that represent an advance by a factor of ~10 9 -10 10 . It is also estimated that direct coupling to nuclei will be achievable under these conditions, a step that will enable quantitative experimental exploration of many nuclear properties and processes. The key prediction that the nuclear arena is accessible is an outcome that connects directly to two of the three grand challenge problems introduced in this work.

Grand challenge problem formulation
On the basis of the present experimental and theoretical understanding, a triplet of grand challenge problems for future solution can be suggested. They are (1) the performance of an experimental probe of the vacuum state associated with the dark energy Ω Λ , (2) the attainment of the amplification in the γ-ray region (ħω ~ 1 MeV), accompanied by the discovery of a nuclear excimer, and (3) the determination of a navigable path to the projected super-heavy nuclear island of stability. Finally, we represent the view that all three goals will be achieved in the coming decades.

Projected technological expectations
The anticipated technical advances projected for future capability can be framed in an approximate pictorial representation. With the parameters P, I, τ, and ω denoting respectively the characteristic powers, intensities, time-scales, and frequencies associated with the expected developments, this picture, reading temporally from left to right, appears above in table 3.