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Thomson Scattering and Ponderomotive Intermodulation Within Standing Laser Beat Waves in Plasma

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Thomson Scattering and Ponderomotive Intermodulation Within Standing Laser Beat Waves in Plasma PHYSICAL REVIEW E 72, 026501 ͑2005͒ Thomson scattering and ponderomotive intermodulation within standing laser beat waves in plasma Scott Sepke,* Y. Y. Lau, James Paul Holloway, and Donald Umstadter† Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, Michigan 48109-2099, USA ͑Received 12 April 2005; published 12 August 2005͒ Electrons in a standing electromagnetic wave—an optical lattice—tend to oscillate due to the quiver and ponderomotive potentials. For sufficiently intense laser fields ͑I␭2 Շ5ϫ1017 Wcm−2 ␮m2͒ and in plasmas with sufficiently low electron densities ͑nՇ1018 cm−3͒, these oscillations can occur faster than the plasma can respond. This paper shows that these oscillations result in Thomson scattering of light at both the laser and ponderomotive bounce frequencies and their harmonics as well as at mixtures of these frequencies. We term this mixing ponderomotive intermodulation. Here, the case of counterpropagating laser beams creating a one-dimensional ͑1D͒ optical lattice is analyzed. The near-equilibrium electron orbits and subsequent Thomson scattering patterns are computed in the single-particle limit. Scaling laws are derived to quantify the range of validity of this approach. Finally, collective plasma and laser focusing effects are included by using particle- in-cell ͑PIC͒ techniques. This effect resulting in light-frequency conversion has applications both as an infrared light source and as a means to diagnose high laser intensities inside dense plasmas. DOI: 10.1103/PhysRevE.72.026501 PACS number͑s͒: 41.60.Ϫm, 42.65.Ky, 42.65.Re, 52.20.Dq I. INTRODUCTION along the laser polarization direction and is, therefore, inher- ently separated from the bright laser source, eliminating the The interaction of beating electromagnetic waves within problem of filtering out the laser wavelength without simul- plasma has been of interest for some time, with Rosenbluth taneously removing the desired wavelength. and Liu having discussed copropagating lasers in plasma as ͓ ͔ In this paper, we take up the topic of equal frequency and early as 1972 1 . Recently, the idea of overlapped laser amplitude colliding laser beams and the light scattered by a pulses has received much attention and has been suggested test electron within these fields. This is the same laser geom- for several applications. Such waves are created in many of etry employed in the classic Kaptiza-Dirac problem, in the proposed laser accelerators such as the colliding beam ͓ ͔ ͓ ͔ which energetic electrons are directed across the standing accelerator 2–6 as well as in Raman amplification 7–10 , light wave and subsequently diffracted ͓23,24͔. In this study, optical traps ͓11͔, plasma gratings ͓12͔, and internal confine- ͓ ͔ however, we consider low-energy electrons trapped in the ment fusion Hohlraums 13 . Laser pulses also overlap in laser fields’ ponderomotive potential wells. In both cases, the high-density proton acceleration experiments when light is ͑ ͒ ͓ ͔ lasers set up a one-dimensional 1D standing optical lattice reflected from the critical surface 14 . with a high potential near the magnetic-field nodes and a low Owing to their wide applicability, infrared light sources potential near the electric-field nodes. Low-energy electrons have also garnered much attention recently. Free electron tend to bunch in the low-energy region, oscillating about the lasers, fast diodes, thermal sources, and even tabletop laser ␶ minimum with a period z. The individual electron dynamics sources have been both proposed and demonstrated as infra- in such a system are, of course, then dominated by both the red generators ͓15–21͔. Such sources have been proposed for ␶ ␶ laser period, 0, and the ponderomotive bounce period, z, use to image proteins, water, and even metals contained ␶ Շ␶ Ӷ␶ ␲␻−1 and as will be shown later, 0 z p =2 p for plasma within substrate materials through direct imaging as well as 19 −3 ͓ ͔ densities less than 10 cm . This implies that these elec- pump probe experiments 22 . trons respond on a time scale faster than the plasma and such Generation of an ultrafast infrared light burst is a chal- single-particle effects can exist independent of the plasma. lenging problem. In this article, we demonstrate theoretically This approach illustrates a phenomenon that we have termed that by employing an optical lattice generated by the beat ponderomotive intermodulation—that is, intermodulation of pattern of two interfering laser beams, electrons Thomson the laser and ponderomotive oscillation frequencies through scatter the laser light into longer wavelengths. The wave- the optical lattice, manifested in the Thomson-scattered ra- length generated is tunable with the laser intensity, I. For the diation patterns. case of counter propagating laser pulses, the emitted fre- 1/2 We begin in the first three sections by analytically calcu- quency is proportional to I . Moreover, this light is emitted lating the electron orbits near the magnetic- and electric-field nodes, as these are equilibria. Using these results, the subse- quently Thomson-scattered light is then computed as both a *Present address: Department of Physics and Astronomy, Univer- function of frequency and direction of observation analyti- sity of Nebraska, Lincoln, Nebraska 68588-0111, USA. Electronic cally and numerically by utilizing the Larmor formula and address: [email protected] applying Parseval’s theorem, following the model used pre- †Present address: Department of Physics and Astronomy, Univer- viously for both Thomson ͓25,26͔ and Compton ͓27–33͔ sity of Nebraska, Lincoln, Nebraska 68588-0111, USA. scattering. These results are then extended to an arbitrary 1539-3755/2005/72͑2͒/026501͑8͒/$23.00026501-1 ©2005 The American Physical Society SEPKE et al. PHYSICAL REVIEW E 72, 026501 ͑2005͒ initial electron phase, and simple scaling laws are derived to ␶ ͑␻͒ ␻␻ ͵ ␤ជ ͑ ͒ ͕ ␻͓ ͑ ͔͖͒ bound the range of validity of this single-particle approach. F = 1 t exp i t − nˆ · x t dt Plasma and more realistic laser fields are then added using 0 particle-in-cell ͑PIC͒ simulation techniques. Good agreement ϱ 19 −3 ϫ ␦͑␻ ␻ ͒ ͑ ͒ is, indeed, found for electron densities as high as 10 cm ͚ − m 1 , 5 and for 1-␮m-wavelength laser pulses as short as 300 fs fo- m=−ϱ cused to 12 ␮m as predicted by the derived scalings. This ␻ ␲ ͑␶ ͒ where 1 is the base frequency given by 2 / −nˆ ·x0 . For paper then concludes by illustrating the experimentally sig- ഛ ␻ ␻ ␲ ␶ a0 0.2, the regime studied in this paper, 1 = =2 / nificant consequences of this phenomenon, including the tun- ͓26,35͔. able infrared light, anomalous spectral broadening, a laser- To understand the conversion of laser energy to Thomson- intensity diagnostic, and tunable line emission. scattered light, the orbits must be computed for all initial conditions, and these velocities Fourier-transformed in the II. ELECTRON DYNAMICS detector-retarded time as shown earlier. Unfortunately, this is a complicated set of coupled, nonlinear equations, which are The simplest model of a plasma neglects all collective and difficult to solve, in general, even numerically ͓14,36͔. For edge effects. Thus, to begin this study, the laser fields are low- to moderate, intensity lasers—e.g., IՇ5 modeled as monochromatic plane waves propagating in the ϫ1017 Wcm−2 for ␭=800 nm—a test electron quivers ͑ ͒ ͑ ͒ ±zˆ direction such that E=xˆ2a0 cos z cos t and B within the lattice due to the sinusoidal ponderomotive poten- ͑ ͒ ͑ ͒ =yˆ2a0 sin z sin t . Throughout this paper, the axial displace- tial given by ␻ ment z is normalized by the laser wave number k= 0 /c and ␸ 2 2͑ ͒͑͒ ␻ p = a0 cos z 6 time t and frequency by the laser angular frequency 0. Also, the laser field strength E0 has been scaled to the normalized in units of electron rest-mass energy. This implies two equi- ␻ ͑ ϫ −9͒ͱ ␭ vector potential a0 =eE0 /m0 0c= 0.85 10 I , where e libria, one unstable at the nodes of the magnetic field ͑z Ͼ 0 is the unit charge, m0 is the electron rest mass, c is the =n␲͒ and the other stable at the electric-field nodes ͓z speed of light, I is the focused laser intensity in W cm−2, and =͑2n+1͒␲/2͔. ␭ is the laser wavelength in ␮m. The equation of motion for ជ a test electron within these fields F=q͑E+␤ϫB͒ can then be III. MAGNETIC-FIELD-NODE EQUILIBRIUM ␲ written and rearranged to yield The nodes of the magnetic-field wave occur at ¯zN =N for integer N. The electron motion here is purely along the laser ␥␤ ͑ ͒ ͑ ͒ ␥␤ ͑ ͒ ͑ ͒ x =−2a0 cos z sin t , y =0, 1 polarization xˆ and is known exactly for arbitrary a0 −2a cos͑¯z ͒sin͑t͒ ␤ ͑ ͒ 0 N ͑ ͒ x t = . 7 d dz 2 2 ͱ 2 2͑ ͒ ␥ ␥ =2a sin͑2z͒sin ͑t͒, ͑2͒ 1+4a0 sin t dt dt 0 This velocity corresponds to the special case of counter- where zero initial velocity has already been imposed since propagating lasers recently calculated by Salamin et al. ͓33͔, ␤ ͑ ͒ this is a Thomson-scattering model, j is the velocity in the which can be integrated directly to determine x t jth direction normalized to c, and ␥ is the relativistic Lorentz 2a cos͑t͒ 2a factor. x͑t͒ = arcsinͫ 0 ͬ − arcsinͫ 0 ͬ. ͑8͒ ͱ 2 ͱ 2 The far-field Thomson-scattered radiation follows directly 1+4a0 1+4a0 from the electron orbits defined by these equations through For a Ͻ0.5, this can be expanded and rearranged to yield the Lienard-Wiechert potentials as 0 ϱ 2 ␤ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ␧ x t =−2a0 cos ¯zN ͚ cp sin pt , 9 d 2 = ␧ ͉nˆ ϫ ͓nˆ ϫ F͑␻͔͉͒ , ͑3͒ p=1 d⍀d␻ 0 p odd ␧ ␧ 2 ␲2 where is the radiated energy in ergs, 0 =e /4 c, nˆ is the where cp are known expansion coefficients, the first few of unit direction of observation, d⍀ is a unit solid angle, and d␻ which are given in Table I.
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