Probe-field reflection on a plasma surface driven by a strong electromagnetic field

Transcription

1 J. Phys. B: At. Mol. Opt. Phys. 33 (2000) Printed in the UK PII: S (00)50056-X Probe-field refletion on a plasma surfae driven by a strong eletromagneti field Kazimierz Rz ażewski, Luis Plaja, Luis Roso and D von der Linde Center for Theoretial Physis and College of Siene, Polish Aademy of Sienes, Aleja Lotników 32/46, Warsaw, Poland Fakultät für Physik, Universität Bielefeld, D Bielefeld, Germany Departamento de Físia Apliada, Universidad de Salamana, E Salamana, Spain Institute für Laser- und Plasmaphysik, Universität Essen, D Essen, Germany Reeived 4 January 2000 Abstrat. We develop a theoretial approah to haraterize the dynamis of a plasma surfae irradiated by a high-intensity eletromagneti wave. The method is based on the analysis of the harmoni ontent in the refletion of a probe field, whih is aimed at the plasma simultaneously with the strong field. In partiular, a expliit formula for the angles of refletion of these harmonis is derived, showing a partiular distribution diretly related to the frequeny of osillation of the vauum plasma interfae. Also, a system of equations for the refleted field amplitudes is formulated, and solved numerially in some partiular ases. The experimental implementation of this sheme would provide a diret test of the so-alled moving mirror models, whih presently provide a basis for the understanding of the intense field plasma interation. 1. Introdution Over the past two deades, the onstant development of the hirped pulse amplifiation tehniques has led to the availability of oherent eletromagneti soures of unpreedented intensity. In lose relation to this, the branh of nonlinear optis beyond the perturbative regime has lost its exoti harater and beome a fully established field. Due to the strong oupling with the eletromagneti field, the dynamis of the matter targets under these irumstanes is extraordinarily diffiult to eluidate. In most ases, even a numerial solution of the equations of motion beomes umbersome, as the theoretial developments are strongly dependent not only on experimental progress in the field, but also on the progressive development of adequate omputational tools. The first steps in the exat solution of the problem were made towards the desription of one-dimensional single-eletron model atoms [1], whih were soon followed by the omplete three-dimensional (3D) omputation of the hydrogen atom [2]. These efforts resulted not only in the onfirmation of earlier analytial approahes, suh as the Keldysh Faisal Reiss (KFR) theory, but also in the lose reprodution of experimental phenomena, suh as above-threshold ionization, and the proposal of new effets (stabilization of ionization) whih might be diffiult to observe experimentally. A review on these phenomena may be found in [3] /00/ $ IOP Publishing Ltd 2549

2 2550 KRz ażewski et al In the quest to find situations whih ould help in the development of bright soures of high-frequeny radiation, the field of intense laser matter interations has merged with plasma physis to study the dynamis of a solid target, whih is almost instantaneously ionized by the eletromagneti field. The many-partile problem is inherent to suh a marosopi system and, therefore, the omplexity of its treatment in the general ase falls well beyond today s omputing apabilities. However, a plasma loud irradiated by suh high-energy eletromagneti radiation permits us to adopt a lassial mehanis approah, in whih eletrons are defined as partiles with well defined momentum and position and, therefore, are governed by the Newton Lorentz equation. Consequently, relativisti dynamis an be introdued to some extent in a quite straightforward manner, while in the quantum mehanial ase this is omplex even for a single-eletron atom. Even though the lassial approah simplifies extraordinarily the theory, the aurate following of the whole set of partiles ontained in a solid slab ( eletrons/m 3 ) is not possible. This traditional problem in plasma physis is solved basially by adopting one of two different point of views. The first approah is to fous on the statistial properties of the ensemble of partiles, by means of a harged-fluid desription. The seond approah, whih is generally preferred in the problem to whih we refer, is to retain the partile aspet of the problem by the definition of a pseudopartile, i.e. a model partile whih is onsidered to inlude a whole set of point harges. In the simplest approah, the partiles belonging to a pseudopartile move together as a whole, the latter beoming a finite-size harge that evolves aordingly to the equations of motion. Over reent years, so-alled partile-in-ell (PIC) odes based in this approah [4] have dominated the sene in intense laser plasma theory. Contrary to expetations, the interation of a plasma slab driven relativistially by a laser field admits a reasonable desription in terms of a very simple approah. The so-alled moving mirror models desribe the plasma interfae as a perfetly refleting surfae whih moves under the influene of the inident eletromagneti wave in the same way as would a single negative harge bounded to an equilibrium point by a harmoni restoring fore [5 7]. These models seem to be espeially suited to desribing the refletion from a plasma slab whih is thinner than the absorption depth of the eletromagneti field. In this ase, the omplete refletion at the plasma surfae may be substituted by the proper refletion oeffiient, the model giving aurate quantitative results in omparison with the PIC alulations and solving the paradox of assuming a surfae driven by a field whih must vanish should the perfet refletion ondition be attained [8]. These models are, nowadays, used in the predition of new phenomena in the strongly relativisti regime, suh as the generation of trains of attoseond pulses [9]. In this paper we would like to address the problem of the detetion and monitoring of the plasma surfae osillations in the laboratory, and therefore the potential reinforement of the moving mirror approah by experimental validation. To this end, we have developed some preditions on the modulation of the refletion of a probe field, whih is onsidered to be weaker than the pump field, responsible for the osillation of the plasma surfae. The following setion will be devoted to the statement of the problem and the strategy for its solution, whih will require a Lorentz transformation to a boosted frame. Setion 3 will state and disuss the equation desribing the distortion of the angles of refletion of the probe field, due to the spatial modulation of the osillating surfae. The next setion, the last before the onlusions, will ontain the derivation of a system of equations whih determines the amplitude of the refleted harmonis and a worked example for some partiular situation.

3 Probe-field refletion on a plasma surfae Definition of the problem Let us onsider the situation depited in figure 1: an s-polarized high-intensity eletromagneti wave (pump) is aimed at the surfae of a plasma at a given angle of inidene ϕ. Assuming the field to be a plane wave, it may be written as E P ( r,t) = E P exp { i ω m 2 { B P ( r,t) = E P exp i ω m 2 [t + 1 (x sin ϕ + z os ϕ) ]} e y (1) [t + 1 (x sin ϕ + z os ϕ) ]} (os ϕ e x sin ϕ e z ) (2) where ω m will refer later on to the frequeny of osillation of the moving surfae (mirror), whih doubles the pump field frequeny. Simultaneously with the pump field, a weaker s- polarized probe is also shone on the plasma surfae at an angle θ. Due to the nonlinear osillation indued by the pump field, the probe refletion (and also the pump refletion) will ontain harmonis of the inidene frequeny. We will desribe the probe field as follows: E i ( r,t) = E i exp { iω 0 [t + 1 ]} (x sin θ + z os θ) e y (3) B i ( r,t) = E i exp { iω 0 [t + 1 ]} (x sin θ + z os θ) (os θ e x sin θ e z ). (4) Figure 1. Geometry of the problem onsidered: (a) an intense pump laser is aimed at the surfae of a plasma, induing an inhomogeneous osillatory pattern in the surfae harge; (b) simultaneously, a weaker probe field is sattered by the surfae in the form of a fan of harmonis. The elongation of the plasma interfae follows the instantaneous phase of the pump field (equation (1)) at every point of the surfae. Consequently, the surfae irradiated is inhomogeneously driven when the pump field is not direted perpendiularly to it. Sine the philosophy of the moving mirror approah is to onsider the mirror as being omposed of individual eletrons bound harmonially to the ion bakground, it is apparent that for the spae-dependent fore due to the pump field on the surfae, the z omponent will affet mirror elongation, and the x omponent will indue inhomogeneous surfae displaements of the harges, giving rise to a surfae harge density modulation. Both effets have already been onsidered before separately [7, 10], and will be inluded together in the present study.

4 2552 KRz ażewski et al Figure 2. A boosted (primed) referene frame, moving along the x-axis, is hosen to math the ondition of perpendiular inidene of the pump wave. (a) The moving frame from the point of view of the laboratory, (b) the interation geometry from the point of view of the boosted frame. In any ase, both displaements give rise to a rippled surfae, whih behaves as an ative diffration grating. Due to this, the probe field refletion will not only deviate from the flatsurfae refletion angle, but also harmonis will be indued and sattered in different diretions (angles α n of figure 1(b)). In order to take into aount both spatial modulations, it is onvenient to jump onto a boosted referene frame in whih the pump field is seen to shine perpendiularly to the surfae. In this new referene frame the phase of the pump field is onstant along the entire surfae, and no ripples will be observed. Figure 2 shows the hoie of this primed referene frame. The veloity of the boosting is found by imposing the normal inidene ondition in the primed system, B P z = γ ( ) B P z βe P y = 0 (5) where B P z is the omponent of the magneti field of the pump wave in the boosted frame, perpendiular to the plasma surfae, and γ is the relativisti Lorentz fator. From this ondition and equation (1) one an immediately find the required boost veloity and Lorentz fator, β = B P z /E P y = sin ϕ γ = 1 os ϕ. (6) The pump field in this new referene frame reads, therefore, as E P = γ ( { ) E P y βb P z = E P exp i ω )} m (t + z e y (7) 2 B P = E P { i exp ω )} m (t + z e x (8) 2 with E P = E P os ϕ and ω m = ω m os ϕ. On the other hand, the Lorentz-transformed probe

5 E i Probe-field refletion on a plasma surfae 2553 field may be simply written as ( r,t ) { = E i exp iω 0 B i [ t + 1 ( x sin θ + z os θ )]} e y (9) ( r,t ) { [ = E i exp iω 0 t + 1 ( x sin θ + z os θ )]} ( os θ e x sin θ ) e z with E i = E i γ(1+β sin θ) = E i (1 sin ϕ sin θ)/os ϕ, ω 0 = ω 0(1 sin ϕ sin θ)/os ϕ, sin θ = (sin θ+β)/(1+β sin ϕ), os θ = os θ/γ (1+β sin θ) = os θ os ϕ/(1 sin θ sin ϕ). Although the transformation to a boosted frame is a general proedure in the mirror model alulations, and in the numerial simulation of oblique inidene of light on a plasma surfae [6], the validity of the mirror model in the boosted frame remains an open question. The motion of the mirror breaks the ovariane in the same sense that the ovariane is broken, for instane, when solving the Dira equation for an eletron moving in the Coulomb field produed by an infinitely heavy nuleus. However, one has to take into aount that the mirror plays a passive role here (just a boundary ondition). 3. The rainbow equation As disussed before, the nonlinear and inhomogeneous surfae osillation will indue a refletion of harmonis of the probe field, sattered at different angles α n (α n in the boosted frame). The total refleted probe will, therefore, be of the form E R = [ f n { iω exp n t + 1 ( x sin α n n z os α n ) ]} e y (11) [ B R = f n { iω exp n t + 1 ( x sin α n n z os α n ) ]} ( os α n e x sin α n e ) z. (12) As formulated in [7], the motion of the surfae along z is given by s (t ) = s 0 sin ( ω m t ) + ϕ 0. On the other hand, the perfet mirror ondition demands that the eletri field vanishes at the surfae (in its instantaneous rest frame). Combining this ondition with equations (9) (12), we have [ 1+ṡ (t ] [ ) os θ E i { iω exp 0 t + 1 ( x sin θ s (t ) os θ )]} = [ 1+ṡ (t ] { [ ) os α n f n n exp iω n t + 1 ( x sin α n s (t ) os α n ) ]}. (13) Sine s (t ) evolves sinusoidally, the frequenies present in this equation must be of the form ω n = ω 0 + nω m. Integrating in time both sides of equation (13) we may find the orresponding equality for the potential vetor E i { [ ω 0 exp iω 0 τ t ( x sin θ s 0 sin(ω m τ ) os θ )]} = f n n ω 0 + nω m { [ exp iω 0 (1+nξ ) τ t ( x sin α n s 0 sin(ω m τ ) os α n ) ]} (14) where t 0 = ϕ 0/ω 0, τ = t + t 0 and ξ = ω m /ω 0 = ω m os 2 ϕ/(1 sin ϕ sin θ). Equation (14) implies idential spatial variation for eah element, therefore, (10) sin α n = 1 1+nξ sin θ (15)

6 2554 KRz ażewski et al Figure 3. (a) Shows shematially the harmoni angular distribution of the probe refleted radiation. Negative orders are sattered at angles smaller than the pump angle of inidene, ϕ, and positive orders are spread at angles greater than ϕ. The law of refletion is still valid for the refletion at the fundamental frequeny, α 0 = θ. (b) Depits the result for the sattering angles in a realisti ase (see text). The presene of a shadowed order (n = 1) is highlighted. whih speifies the angular spread (rainbow) of the refleted probe field in the boosted frame. From the point of view of the laboratory variables, the rainbow equation (15) may be written as sin α n = 1 1+nξ sin θ + nξ sin ϕ (16) 1+nξ with ξ = ω m /ω 0. As speified by this equation, the harmonis of the probe laser are generated and sattered at different angles. One of the general lines of the sattering distribution is the angle ϕ, orresponding to the refletion of the pump field on a flat surfae, whih now is the limiting value for the diretion of propagation of the higher harmonis (n± ). On the other hand, the refletion at the fundamental frequeny of the probe field follows the standard

7 Probe-field refletion on a plasma surfae 2555 Figure 4. Universal plot showing the number of shadowed orders as a funtion of the pump and probe inidene angles. The ontours orrespond to the quantity Nξ, where N is the number of shadowed orders. lawofrefletion, being sattered at an angle equal to the inidene. As plotted shematially in figure 3(a), the negative orders are refleted at angles smaller than ϕ, or even negative, while the positive orders are sattered at greater angles, where θ is the maximum. Figure 3(b) shows the partiular angle distribution of the refletion for a pump laser of wavelength λ 800 nm (similar to Ti:sapphire) and intensity 100 au ( Wm 2 ), and a probe field of λ 260 nm (similar to KrF) and intensity 0.01 au ( Wm 2 ). The pump field is aimed at the plasma surfae at an inidene angle of ϕ = π/8 and the probe field at θ = π/4. In addition to the already mentioned struture, a lose inspetion of this latter figure reveals the absene of the lowest negative harmoni order n = 1. This surprising fat may be well understood in terms of equation (16) by noting the possibility of evanesent solutions of the kind sin α n > 1. For the partiular ase of figure 3(b), only one order belongs to these shadowed solutions. More generally, the interval of shadowed orders is bounded as 1 + sin θ 1 sin θ <nξ< 1 + sin ϕ 1 sin ϕ 1 sin θ 1 + sin θ <nξ< 1 sin ϕ 1 + sin ϕ if ϕ<θ (17) if ϕ>θ (18) yet, in all ases, will omprise only negative orders. The length of the shadowed region depends on the frequeny ratio between the pump and probe field, as well as on the respetive angles of inidene. It is, however, possible to plot a universal urve to estimate it, as is depited in figure 4. For every ordered pair (ϕ, θ) a orresponding value of L = Nξ is plotted, so that the number of shadowed orders may be easily found as N = L/ξ. From this piture, it is lear that the shadowing an affet several orders in the limit ϕ π/2 together with the ondition ϕ>θ.

8 2556 KRz ażewski et al 4. Intensities of the refleted field Substituting equation (15) into (14), the following equality holds: E i = n g n e iω o nξ τ 1+nξ { exp i ω 0 s 0 sin ( ω m τ )[ os θ + (1+nξ ) os α n] } (19) Figure 5. Intensities of the refleted orders for the partiular ase shown in figure 3, and different probe field inidene angles: (a) positive harmoni orders (b) negative harmoni orders.

9 Probe-field refletion on a plasma surfae 2557 with g n = f ( n e inω 0 t 0. Using the expansion exp iz sin ω m τ ) = k exp ( ikω m τ ) J k (z), a system of equations with g n as variables an be stated E i δ l,0 = g n e iω 0 nξ τ { ω J 0 s [ 0 n 1+nξ n+l os θ + (1+nξ ) os α ] } n. (20) One solved, the field amplitudes in the laboratory frame an be omputed from g n = g n os ϕ/(1 sin ϕ sin α n), with α n given by equation (16). The amplitude of the mirror osillation s 0 remains invariant in hanging between the boosted frame and the laboratory frame, and an be omputed using [7] s 0 = s 0 = λ ωm 2 a 2 8π ωp 2 P os ϕ sin δ s (21) with ω p ω m the plasma frequeny, a P = 2eE P /mω m the adimensional amplitude of the pump field potential vetor and δ s the Fresnel refletion phase. This latter parameter may be omputed from the Fresnel refletion oeffiient in the boosted frame (normal inidene) as sin δ s ω m /ω p and, therefore, s 0 = λ ωm 3 a 2 8π ωp 3 P os2 ϕ. (22) Figure 5 show the intensities of the refleted harmonis in the partiular ase studied in figure 3(b), but sanning the angle of inidene of the probe field. As is shown, the intensities tend to diminish quite rapidly as the absolute value of the harmoni order is inreased. This is basially a onsequene of the small amplitude s 0 of the osillation of the surfae due to the huge plasma restoring fore. The presene of shadowed refletions for negative orders is also apparent in figure 5(b) for the lowest orders, espeially when the angle of inidene of the probe field is smaller than the angle of inidene of the pump wave. 5. Conlusions We have developed a theoretial study of the refletion of a probe field by the surfae of a plasma driven by a strong pump eletromagneti wave. When this latter field is direted at the plasma surfae obliquely, the inhomogeneities indued in the surfae harge dramatially distort its refletion properties. By means of a moving mirror approah, we demonstrate that, under these irumstanes, a probe field will be refleted in the form of a fan of harmonis. An expliit formula for the angles of refletion of these harmonis has been derived, that predits the absene of ertain harmoni orders. In addition, the system of equations governing the amplitudes of the refleted probe harmonis is obtained, and solved numerially for some partiular ases. The distortion indued in the refleted probe, therefore, provides useful information about the surfae dynamis that ould be used to give experimental onfirmation of the moving mirror model. Aknowledgments We thank C Maros-Atxutegi for fruitful disussions. LP and LR aknowledge support form the Spanish DGICYT (grant PB ) and the Junta de Castilla y León (grant SA16/98). KR thanks the Humboldt Foundation (Bonn) and the Foundation for Polish Siene for their generous support. LP wishes to thank support from the Spanish Ministerio de Eduaión y Cultura (under grant EX ).

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