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Stable relativistic/chargedisplacement channels in ultrahigh power density (≈10^{21 }W/cm^{3}) plasmas

Communicated by Charles H. Townes, University of California, Berkeley, CA (received for review February 27, 1998)
Abstract
Robust stability is a chief characteristic of relativistic/chargedisplacement selfchanneling. Theoretical analysis of the dynamics of this stability (i) reveals a leading role for the eigenmodes in the development of stable channels, (ii) suggests a technique using a simple longitudinal gradient in the electron density to extend the zone of stability into the high electron density/high power density regime, (iii) indicates that a situation approaching unconditional stability can be achieved, (iv) demonstrates the efficacy of the stable dynamics in trapping severely perturbed beams in single uniform channels, and (v) predicts that ≈10^{4} critical powers can be trapped in a single stable channel. The scaling of the maximum power density with the propagating wavelength λ is shown to be proportional to λ^{−4} for a given propagating power and a fixed ratio of the electron plasma density to the critical plasma density. An estimate of the maximum power density that can be achieved in these channels with a power of ≈2 TW at a UV (248 nm) wavelength gives a value of ≈10^{21} W/cm^{3} with a corresponding atomic specific magnitude of ≈60 W/atom. The characteristic intensity propagating in the channel under these conditions exceeds 10^{21} W/cm^{2}.
The development of methods for the compression of power in materials is one of the oldest endeavors of mankind with an origin that predates the Stone Age. From the use of a wooden club to the contemporary production of vigorous thermonuclear environments, the achievable power density (W/cm^{3}) has been advanced by approximately a factor of 20 orders of magnitude (≈10^{20}). New processes, involving the nonlinear interaction of intense (≈10^{18}–10^{21} W/cm^{2}) fs pulses of radiation with matter, currently are being explored to enhance further the controlled production of these environments to a new ultrahigh level (≈10^{19}–10^{21} W/cm^{3}), a range that can approach ≈100 W/atom. These conditions provide new possibilities for the production and regulation of many highly energetic physical processes, including hard xray generation, the initiation of nuclear reactions, particle acceleration, and the fast ignition of fusion targets. The key to the production of these exceptional conditions is the stable compression of the spatial distribution of powerful (P_{0} ≈ 1 TW–1 PW) pulses of radiation into very narrow plasma channels. Specifically, a complex mechanism, which is triggered by pulses whose power exceeds a critical value P_{cr} and involves both relativistic electron motions and the relative spatial separation of the electron and ion densities caused by the radiation pressure of the intense wave, produces the conditions necessary for channel formation. In brief (1), the ponderomotive radial displacement of the electrons and the contrasting inertial confinement of the ions cooperate to produce the two chief characteristics of the channels. They are (i) the refractive selffocusing action of the displaced electrons, which confines the propagating radiation, and (ii) the high spatial stability of the channels, the feature produced by the immobile electrostatic spine formed by the fixed ions. These narrow channels, which typically have a diameter of a few microns, represent an example of a new, largely unexplored class of strongly nonequilibrium excited matter that combines a very high energy density with a wellordered structure.
The existence of dynamic stability is essential for the control of high power density plasmas. Of particular importance are the physical limits of stable behavior and the corresponding implications on the maximum achievable power density. The overall result of this study is that exceptionally robust stability is a chief characteristic of the relativistic/chargedisplacement selfchanneling mechanism. Specifically, the six key findings are: (i) the discovery of the leading role played by the eigenmodes in the development of stable channels, (ii) the evaluation of a simple technique using a longitudinal gradient in the electron density to extend the zone of stability into the high electron density/high power density regime, (iii) the indication that a situation approaching unconditional stability can be achieved, an outcome reflecting the wellordered structure of the excited plasma, (iv) the demonstration of the efficacy of the dynamics in efficiently trapping severely spatially perturbed beams in single uniform channels, (v) an estimate showing that an extraordinary power density (≈10^{21} W/cm^{3}) can be produced in the channels with UV radiation, and (vi) the prediction that ≈10^{4} critical powers (P_{cr}) can be trapped in a single stable channel.
Theoretical work (1) has predicted that the channeled propagation can exhibit a large domain of stability. Initial experimental studies (2–5), conducted close to the threshold condition of the channeling phenomenon (6–15), have furnished evidence supporting this conclusion. Measurements of the spatial properties of the propagation, using both xray (4) and Thomson (16, 17) images, have clearly established the formation of long channels of the form shown in Fig. 1. Fig. 1 illustrates a singleexposure xray [Xe(M), ≈1 keV] image of the longest channel (>50 Rayleigh ranges) that was experimentally produced in a gaseous target containing (Xe)_{n} clusters (18) with a fs (≈250 fs) UV (248 nm) pulse (19) having a peak power of ≈1.4 TW. The salient characteristic of the image is a long, stable, and uniform channel of high power density, the magnitude (20) of which was estimated to be ≈2 × 10^{19} W/cm^{3}, or equivalently ≈1 W/atom. Other work (17), which examined the channeled region with images of the Thomson scattered 248 nm radiation, complemented the data shown in Fig. 1 and demonstrated that the channeling mechanism efficiently compresses the incident power into a single filament whose diameter does not exceed the resolution of the imaging system (5–6 μm), a result consistent with the corresponding theoretical (2, 4, 12) figure of ≈1–2 μm. The principal issues discussed herein are the determination of the conditions limiting the stability of the confined propagation illustrated in Fig. 1, and the evaluation of the corresponding upper bound on the power density.
Following conventional notation (1), we introduce the definitions of the coordinates of the ηρ_{0} plane given by 1 with P_{0} denoting the incident peak power and with the critical power (P_{cr}) given by (1, 12) 2 in which m_{e,0}, c, and e have their customary identifications, g_{0}(ρ) is the Townes mode (21), and ω, ω_{p,0}, and r_{0}, are the angular frequency corresponding to the propagating radiation, the angular frequency of the unperturbed plasma, and the radius of the incident intensity profile, respectively. In addition, lowest eigenmodes (1, 10, 12) exist with the dimensionless radius 3 in which U_{s,0}(ρ) represents the eigenmode (1, 10, 12) with index s. The present analysis was confined to electron densities N_{e} less than onequarter of the critical electron density (N_{cr}) to eliminate resonant plasma wave production, and forward Raman scattering (22, 23) was not included, because it is known experimentally that it can be suppressed (24).
Fig. 2 illustrates the geography in the ηρ_{0} plane of the stable and unstable regions characteristic of channel formation in initially homogeneous plasmas (1). The essential features are the locus of the eigenmode curve ρ_{e,0}(η) defined by Eq. 3, the existence of a region of stable propagation that includes the eigenmode curve, and a large zone of unstable propagation involving strong filamentation.
To increase the power density (W/cm^{3}) in the channel, both high electron density (N_{e}) and high radiative intensity (I) must simultaneously exist. This condition, which can be achieved by raising both the electron density (N_{e}) and the power (P_{0}) propagating in the channel, has the direct consequence of a correlated increase in both ρ_{0} and η, a trend that naturally displaces the operating point of the system directly toward and eventually into the unstable zone. Therefore, the attempt to increase the power density in this straightforward manner is immediately blocked by the onset of unstable plasma dynamics. However, this limitation is overcome if there exists a mechanism maintaining the location of the operating point in the stable zone regardless of the fact that sufficiently high values of η and N_{e} normally exhibit unstable behavior. Analysis described below indicates that the use of an appropriate longitudinal (z) electron density gradient can achieve this goal, thereby, simply and effectively providing a large extension of the zone of stability and a corresponding major increase in the power density. Moreover, it is found that the electron density profile can be arranged in such a way that the locus of the operating point of the system becomes dynamically trapped in the stable zone by the rapid adjustment of the system to a neighborhood close to the eigenmode curve. Therefore, if the initial condition corresponds to stability, the strong dynamical preference for the eigenmode isolates the system in the stable region and the trajectory of the operating point is prevented from entering the unstable zone.
The results shown in Figs. 2–4 both illustrate the details of the propagation and provide a basis for estimating the corresponding bound on the power density. Fig. 3A presents an initial (z = 0) transverse intensity distribution that corresponds to point A_{UNSTABLE} (η = 21.1, ρ_{0} = 20.6) in Fig. 2. Because this condition falls well within the unstable zone, the beam develops rapidly (z = 400 μm = 3.95 L_{R}, L_{R} ≡ Rayleigh range) into the fragmented multichannel form shown in Fig. 3B.
The use of an appropriate longitudinal electron density profile, such as that given in the inset of Fig. 2, can be used to achieve the stable propagation depicted in Fig. 3C at both the high electron density and incident power associated with point A_{UNSTABLE} given in Fig. 2. Basically, the use of the longitudinal electron density profile shown in the inset of Fig. 2 maps the point A_{UNSTABLE} to the point A_{STABLE} in the stable zone of Fig. 2, thereby enabling the system to evolve to B_{STABLE}, the point corresponding to the intensity distribution shown in Fig. 3C. The profile of the pulse, as it evolves from A_{STABLE} to B_{STABLE}, is illustrated in Fig. 3D.
The remapping of the initial condition fundamentally alters the dynamics of the propagation. In the comparison of Fig. 3 B and C, the strongly unstable propagation in Fig. 3B is converted into the formation of a single stable channel that contains more than 85% of the incident power. This dramatic shift in the behavior is driven by the dynamics of the stable region, namely, the strong proclivity of the system to seek the lowest eigenmode. This characteristic of the dynamics is clearly illustrated in the form of the trajectory of the operating point in Fig. 2, which connects the initial conditions corresponding to point A_{STABLE} with the final evolved channel denoted by datum B_{STABLE}. The path of this trajectory demonstrates emphatically that the system aggressively moves toward the eigenmode and remains virtually locked in a small neighborhood of the eigenmode curve as the electron density (N_{e}) and corresponding value of η both rise to the conditions of high power density. This dynamical behavior contrasts sharply with other mechanisms of selffocusing (e.g., Kerr and relativistic), which generally manifest very poor characteristics of stability (11, 25, 26).
The powerful tendency for attraction to the eigenmode, illustrated by the trajectory pictured in Fig. 2, suggests that the confined propagation may be highly robust against large spatial perturbations of the incident intensity profile. To evaluate this possibility, particularly for high power (≫TW) infrared (λ = 1 μm) pulses, the severely azimuthally aberrated intensity distribution of a 1 PW pulse shown in Fig. 4A was used to replace the weakly aberrated counterpart presented in Fig. 3A. The corresponding results are presented in Fig. 4 B–D. With the initial condition given by point A_{UNSTABLE} in Fig. 4B, the expected filamentation rapidly develops. However, the remapping of the launching point of the wave from A_{UNSTABLE} to A_{STABLE} with the longitudinal electron density profile depicted in the inset of Fig. 2 fully restores the stable pattern of propagation as illustrated in Fig. 4C. The corresponding trajectory of the operating point arising from the initial condition A_{STABLE} is illustrated in Fig. 4B. We note again the rapid convergence with the eigenmode curve and the efficient achievement of a single channel with the high power density corresponding to B_{STABLE}, a point representing a stable channel containing approximately 10^{4} critical powers. It is significant that the gross spatial restructuring of the pulse shown in Fig. 4D occurs with a modest loss of power, in this case, about 30%. The exceptional stability demonstrated by this result indicates that incident beam profiles deviating greatly from ideal spatial form can be efficiently converted into high brightness configurations.
The results discussed above provide the basis for an estimate of the upper bound of the controlled power density that can be achieved and the dependence of that limit on the wavelength (λ) of the propagating radiation. For λ = 248 nm, N_{e}/N_{cr} ≅ 1/6, and an incident P_{0} = 2 TW, it was found (cf. Fig. 3C) that the peak intensity in the channel is I_{ch} ≅ 1.62 × 10^{21} W/cm^{2}. To estimate the effective cross section σ_{nγ} for coupling of the radiation to the atomic or molecular material in the channel, we use a previous estimate (27) of the upper bound of σ_{nγ} valid in the limit of sufficiently high intensity (>10^{20} W/cm^{2}) and sufficiently high atomic number (Z). This analysis (27) led to a universal magnitude given by σ_{nγ} = 8π ƛ_{c}^{2} in which ƛ_{c} is the Compton wavelength of the electron. In arriving at this value for σ_{nγ}, appeal was made to a picture involving an extreme form of ordered driven electronic motion in atoms (28), a model that bears an analogy to certain atomatom and ionatom collisional processes (29). We note that this value of σ_{nγ} also has an experimental basis, because it gives good agreement (20) for power densities (≈1 W/atom) derived from images of Xe xray spectra produced in channels (30, 31). If we further assume that the channel contains uranium atoms at an average density N_{U} that experience ionization to the level Z = 70, the state of ionization predicted at an intensity ≈1.7 × 10^{21} W/cm^{2} by the Coulomb suppression model (32), we can write the corresponding power density (P/V) approximately as 4 or equivalently ≈60 W/atom.
Because the selfchanneling causes the transverse intensity profile of the laser beam to stabilize near a lowest eigenmode (1, 3, 12), the peak intensity I_{ch} in the channel can be expressed from Eqs. 1–3 as (1, 4) 5 where η_{ch} = P_{ch}/P_{cr}, and P_{ch} represents the power trapped in the channel. For the range of 1.4 ≲ η_{ch} ≲ 10, the normalized radius of the lowest eigenmodes is nearly constant (1, 3), ρ_{e,0} ≅ 1.7, and Eq. 5 reduces to the simple expression 6 with λ given in units of micrometers.
The selfchanneling of ultrapowerful PW laser pulses in high density plasmas generally involves a trapped power in the channel P_{ch} that is the order of 10^{3}–10^{4 }P_{cr}. For this range (10^{3} ≲ η_{ch} ≲ 4×10^{4}), the normalized radius of the lowest eigenmodes is ρ_{e,0} ≃ 3, and Eq. 5 for this range reads 7 with λ given in units of micrometers.
It follows from the definition of N_{cr} that I_{ch} can be written in the form 8 Because both ρ_{e,0} and P_{ch}/P_{0} vary slowly over the range 10 ≲ η_{ch} ≲ 10^{4} (ρ_{e,0} ≈ const, P_{ch}/P_{0} ≈ const), the peak intensity in the channel scales as 9 a result, which gives for a constant ratio of N_{e}/N_{cr} the simple scaling 10 From the obvious relation N_{e} ≈ λ^{−2} for N_{e}/N_{cr} ≡ const, together with Eqs. 4 and 10, we conclude that at a constant ratio of N_{e}/N_{cr} the power density P/V is expected to vary as 11 Within a margin of ≈10 percent, the results of our computations conform to this strong expected scaling favoring the UV.
In conclusion, detailed studies of the stability of relativistic/chargedisplacement selfchanneling have revealed two chief characteristics of this nonlinear mechanism of propagation: a dominant role for the lowest eigenmode for pulses launched in the stable zone and an exceptional robustness of the stability of single channels. As a consequence, strongly azimuthally perturbed incident intensity profiles can undergo efficient confinement to stable channeled distributions. The results demonstrate how a simple gradient in the electron density can be used to augment the effectiveness of the stable region and extend the channeling process into a high powerdensity regime that unites high propagating intensities with high plasma densities.
Acknowledgments
Support for this research was provided under contracts with the Strategic Defense Initiative/Naval Research Laboratory (N0001493K2004), the Army Research Office (DAAH0494G0089 and DAAG55971–0310), the Department of Energy at the Sandia National Laboratories (DEAC0494AL85000), the University of California/Lawrence Livermore National Laboratory (B328353), and the Japanese Ministry of Education, Science, Sport, and Culture (08405009 and 08750046).
 Received February 27, 1998.
 Accepted May 8, 1998.
 Copyright © 1998, The National Academy of Sciences
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