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Guiding-Center Equations for Electrons in Ultraintense Laser Fields Joel E

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Guiding-Center Equations for Electrons in Ultraintense Laser Fields Joel E ARTICLES Guiding-center equations for electrons in ultraintense laser fields Joel E. Moore and Nathaniel J. Fisch Princeton Plasma Physics Laboratory, Princeton University, P.O. Box 4.51, Princeton, New Jersey 08543 (Received 25 October 1993, accepted 31 January 1994) The guiding-center equations are derived for electrons in arbitrarily intense laser fields also subject to external fields and ponderomotive forces. Exhibiting the relativistic mass increaseof the oscillating electrons, a simple frame-invariant equation is shown to govern the behavior of the electrons for sufficiently weak background fields and ponderomotive forces. The parameter regime for which such a formulation is valid is made precise, and some predictions of the equation are checked by numerical simulation. I. INTRODUCTION but a different form of the equations derived here can be applied to this case as well. The increasing degree of interest in high-intensity la- Various accelerator schemes- attempt some sort of sers (a = eEdmcw - 1) motivates a theoretical examina- conversion of the intense transversefields of a laser into a tion of the behavior of electrons oscillating in the fields of more useful form, and constraints on when such conver- such lasers. The electron motion is well understood when sion can occur are given here. The general constraints the only forces presentare those from the wave,’ this paper given here confirm and generalizethe constraint found by examines the motion of electrons when other fields are Apollonov ef aL4 for a specific accelerator design, and also present in addition to the wave. explain the optimal parameters Kawata er al. found nu- The nonlinearity parameter a can be understood as the merically for two accelerator designs.5’6Essentially these ratio of the momentum imparted by the wave field in a accelerator designs convert some of the relativistic quiver single oscillation to mc. (For il= 1 pm, a=0.84 at an in- velocity into drift velocity, and this can only be done with tensity of I=cE387r= 10” W/cm*.) As a- 1, the quiver certain types of background fields. Powerful laser pulsesin velocity of an electron in the wave becomes relativistic, the presenceof background fields may also have applica- and, as a result, the magnetic component of the wave be- tions as diagnostic tools in plasmas.’ gins to affect the motion. Becausethe electron velocity is Electrons struck by a laser pulse experiencea ponder- relativistic over much of the orbit, the electrons can exhibit omotive acceleration at the beginning of the pulse that can result in a relativistic drift velocity (distinct from the an increased “effective mass” or increased inertia to ap- quiver velocity) during the body of the pulse.8It turns out plied forces. While effective mass equations do appear in that the ponderomotive acceleration appears naturally the literature, there remains a need for a general and sys- through an analysis of these drift velocities. We derive a tematic derivation of such equations, including precise simple equation, Eq. (41), which governs the time- statements concerning the limitations of such approxima- averagedbehavior of oscillating electronsin the presenceof tions. We show here that the nonoscillatory part of the weak background fields and/or weak ponderomotive gra- electron motion is given by a simple guiding-center equa- dients, for incident plane waves of arbitrary intensity. We tion, which predicts some interesting results, verified here show that, in some ways, the character of the ponderomo- by numerical simulation. tive force in the presenceof additional fields is changed. The formulation developedhere is useful in many sit- For example, in the absenceof background fields, the pon- uations of practical importance where electromagnetic deromotive force, taken over the whole wave, only dis- fields in addition to the wave field are present, and where placesthe particle and doesnot changeits energy, while, in these background fields are much weaker than the wave the presenceof a background electric or magnetic field, field and vary only slightly over oscillation time and space energy transfer from the ponderomotive force can take scales. For example, longitudinal electron oscillations in a place. plasma occur with characteristic time l/w,> l/o and space It will be assumedthroughout that the pulse has no scale roughly the Debye length ;1,, which may be larger or transverse variation and has phase velocity equal to the smaller than the beam wavelength. Plasma oscillations in speed of light in uacuo. The small spot sizes required to the presenceof laser pulses play key roles in the beat wave achieve very high intensity do create some transversevari- accelerator2 and wakefield accelerator.3 Close-encounter ation of the pulse, but in many casesthis variation is small collisions involving oscillating electrons can occur on time over a single electron’s orbit. For vacuum applications the and spacescales smaller than those of the oscillations, and assumption that the phasespeed is equal to c clearly poses in this case the enhanced-masspicture ceasesto be valid, no problems, although in plasma applications the phase Phys. Plasmas 1 (5), May 1994 1070-664X/94/1(5)/1105/12/$6.00 @ 1994 American Institute of Physics 1105 Downloaded 23 Nov 2005 to 198.35.4.75. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp speedmay vary. Often plasmas irradiated by intense lasers ence of external forces, as we now show. The goal is an are quite underdense, so that the deviation of the phase approximate solution of the equation of motion velocity from c is small. Longitudinal variation of the pulse dp leads to the well-known ponderomotive effect, which is ~=dEwave+BX~w,ve) considered here in some detail, Transverse variation of the pulse leads both to transverse ponderomotive effects and to waveguide-like longitudinal fields; while these effects can -te(Ebackground+BXBbackground). (1) be significant and useful, as in ponderomotive focusing’ First, note that there are exact analytical solutions for and longitudinal acceleration,” in many applications they the motion of individual electrons in a plane, monochro- are either small in magnitude or affect only a small fraction matic wave of any polarization (see, e.g., Landau and of the irradiated electrons. The analysis is specific to the Lifshitz13 for the case of linearly and circularly polarized case of nearly monochromatic laser light, because in the waves; the case of elliptical polarization can be handled nonlinear regime the superposition of frequency compo- similarly). Integrability is a consequenceof the existenceof nents becomes very complicated.” three integrals of the motion.14 The canonical momenta Some predictions of the analytical framework derived Py =p,, -I- eA,,/c, P, =pz + eA,/c here are checked with a short computer code that inte- (2) grates the Lorentz force equations numerically. The code is are conserved, becauseof the system’s symmetry with re- also used to examine regimes beyond the scope of the equa- gard to translations perpendicular to the direction of the tions here, in order to understand qualitatively the changes wave’s propagation. Here p denotes the kinematic momen- in the electron motion. tum and P the canonical momentum, with the direction 1; In Sec. II the guiding-center equations are found by of the wave taken along the x axis. In addition, there is a using special features of the drift solutions of electromag- third invariant of the motion mcy-p, (here y is the Lor- netic waves. Section III details the assumptions leading to entz factor of the electron), associated with the depen- the guiding-center equations and relates the conditions for dence of the problem on x and t only through the phase their validity to optimal designs for certain accelerators. In v=w(t--x/c). The exact solution for the motion can be Sec. IV, numerical examples are used to show, when con- derived from these three constants of the motion. In the ditions for their validity are not satisfied, the breakdown of presence of background fields, however weak, these three the guiding-center equations and the subsequent interest- quantities are not necessarily conserved, raising the ques- ing dynamics. Section V examines the classesof fields that tion of how the motion is modified. allow the guiding-center equations to be averaged to a sim- Second, note that a plane wave in one frame appears as pler form. Section VI derives guiding-center equations that a plane wave in all other frames, though possibly with a are frame invariant, including the ponderomotive force in different direction, frequency, and amplitude. Monochro- the presence of background fields. Our main results are matic waves remain monochromatic, however, and polar- summarized in Sec. VII. ization (i.e., eccentricity of polarization ellipse) is also in- variant. The norm of the vector potential v ( =cE/o II. DERIVATION OF GUIDING-CENTER EQUATIONS for a linearly polarized wave) is clearly invariant, so that a To derive the guiding-center equations, we treat a strength parameter a can be invariantly defined by background force as a series of closely spaced impulses. Between the impulses, the electron is subject only to the (3) intense electromagnetic wave, The size of each impulse is approximated as the instantaneous force multiplied by the with ( > denoting an average with regard to phase. (Note time interval between impulses. Then the effect of the back- that some other authors define a without the factor of 2.) ground force is updated at each impulse, and such an ap- Thus, given a drift velocity vd relative to the lab frame and proximation converges in the limit of infhritesimally spaced a plane monochromatic wave in the lab frame, there exists impulses. (This approach is similar to the Picard method a solution of the equations of motion for which the average used in the theory of differential equationst2) Certain fea- value of the electron velocity, as calculated in the lab tures of plane waves make it possible to find the response frame, is vd.
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