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 subsequently 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

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. Since the oscillation period is the is a unit frequency ͓34͔. The radiated energy is normalized to laser period and no net displacement occurs, the radiation ␧ ␧ ␧ ␧ 0 throughout ¯ = / 0. The dimensionless amplitude func- pattern consists of only laser harmonics. In fact, the radiation tion is, in turn, given by energy per unit solid angle emitted normal to xˆ is known exactly ϱ ជ ϱ F͑␻͒ = ␻͵ ␤͑t͒exp͕i␻͓t − nˆ · x͑t͔͖͒dt. ͑4͒ d¯␧ ͑nˆ · xˆ =0͒ = ͚ 4␲2a2m2c2 , ͑10͒ −ϱ ⍀ 0 m d m=1 When ␤ is truly periodic with period ␶ and net displacement m odd per period x0 this integral can be shown to give an infinite where m denotes the harmonic number. Of course, since a0 is series of harmonics small, only the first few harmonics are significant. The angu-

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TABLE I. The first six Fourier coefficients for describing the TABLE II. Summary of the lowest-order solution to the linear- electron motion at the magnetic-field node in the low-field limit to ized single-particle model near the electric-field node equilibrium. 10 ͑ ͒m 2m⌫͓͑ ͒ ͔ ͓ͱ␲⌫͑ ͔͒ ␤ ␤ ͑ ͒⑀͑ ͒ order a0 . Am = −1 a0 2m+1 /2 / m+1 . This model assumes that terms of the form i j for i, j x,y,z are much less than one and that oscillations around the fixed points ͑ ͒␲ ⑀ Expansion coefficients for the low-field radiation pattern located at zn = 2n+1 /2 for n Z are small. The terms x0, y0, and ␩ represent the electron initial conditions. ͓ ͔ 0 c1 = 1+3A1 +10A2 +35A3 +126A4 +462A5 ͓ ͔ c3 =− A1 +5A2 +21A3 +84A4 +330A5 Linearized single-particle model lowest-order solution c =͓A +7A +36A +165A ͔ 5 2 3 4 5 ␤ ͑ ͒ ͑ ͒n ͑ ͒ ͓␩ ͑␻ ͔͒ ͓ ͔ x t =2 −1 a0 sin t sin 0 cos z0t c7 =− A3 +9A4 +55A5 ͑ ͒ ͑ ͒n+1͑ ␩ ␻2 ͕͒͑ ␻ ͒ ͓͑ ␻ ͒ ͔ c =͓A +11A ͔ x t = −1 a0 0 ր 1− z0 1+ z0 cos 1− z0 t 9 4 5 ͑ ␻ ͒ ͓͑ ␻ ͒ ͔ ͖ ͓ ͔ + 1− z0 cos 1+ z0 t −2 +x0 c11=− A5 ␤y͑t͒=0 ͑ ͒ y t =y0 ␤ ͑ ͒ ͱ ␩ ͑␻ ͒ lar distribution of this light can also be well approximated z t =− 2 0a0 sin z0t ͑ ͒ ␩ ͑␻ ͒ for the first two laser harmonics as z t = 0 cos z0t +zn d¯␧ ͑␮ ␻ ͒ ␲2 2 2͑ ␮2͒ ͑ ͒ , 0 =4 a0c1 1− , 11 d⍀ d2 ͓z͑t͒ −¯z ͔ +4a2 sin2͑t͓͒z͑t͒ −¯z ͔ =0, ͑14͒ dt2 n 0 n d¯␧ ͑␮,2␻ ͒ =64␲2a4͑c + c ͒2␮2͑1−␮2͒, ͑12͒ ⍀ 0 0 1 3 ͉ ͑ ͒ ͉Ӷ␲ d where z t −¯zn /2. This localization is not as restrictive where ␮=nˆ ·xˆ. Notice the characteristic Thomson-scattering as it appears. PIC simulations show that electrons will, in- lobed structure in the angular distribution of these harmonics deed, bunch near the bottom of the ponderomotive well, and the azimuthal symmetry built in by the inherent rota- thereby depleting the electron population away from this lin- ˆ earized solution. tional invariance about x of the coordinate system. ͑ ͒ As a becomes large, the velocity approximates a square The orbit is found by positing a series solution in z t such 0 ͑ ͒ ͑ ͒ 2 ͑ ͒ 4 ͑ ͒ ͑ ͒ ␤ ͑ ͒ ͑ ͒ ⑀͑ ␲͒ ͑ ͒ ⑀͑␲ ␲͒ that z t −¯zn =zl t +a g2 t +a g4 t + , where zl t is the wave, x t =−cos zN for t 0, and cos zN for t ,2 . 0 0 ¯ This high field orbit also results in a spectrum of laser har- low-frequency, time-averaged axial displacement, and the re- monics with the higher orders becoming increasingly promi- maining terms are small, high-frequency corrections. The nent as a approaches infinity. This limiting spectrum ͑a low-frequency solution is a simple harmonic motion form 0 0 ␻ →ϱ͒ is given by with characteristic ponderomotive oscillation frequency z0 ϵ ͱ ͑ ͒ ␩ ͑␻ ͒ ␩ ϵ ͑ ͒ a0 2, and zl t = 0 cos z0t , where 0 z t=0 −¯zn. The ϱ d¯␧ 8 cos͑m␮ ␲͒ cos͑m␮ ␲͒ high-frequency corrections are then given by the recursion = ͚ − − + + d⍀ ␮ ␮ 1 1 m=−ϱ ΄ + − ΂ ␮ ␮ ΃΅ g¨ ͑t͒ =2␩ cos͑2t͒cos͑␻ t͒͑15͒ 4 − 4 + 2 0 z0 ͑13͒ ¨ ͑ ͒ 2͑ ͒ ͑ ͒ ͑ ͒ ␮ gm t + 4 sin t gm−2 t =0, 16 where ± =1±nˆ ·x. The characteristic structure present in the low field approximation is retained and expanded to include all harmonics. Again, no emission is observed along the laser where m=4,6,8,.... The axial velocity then follows by simple differentiation, and the transverse velocity and dis- polarization and only odd harmonics are emitted normal to ͑ ͒ this, as is expected from one-dimensional, periodic motion. placement follow by direct substitution into Eq. 1 and integration in time as shown in Table II. The electron motion is dominated by two time scales forming so-called Lissajous patterns. The axial motion is a IV. ELECTRIC-FIELD-NODE EQUILIBRIUM simple oscillation at the ponderomotive bounce frequency ␻ The magnetic-field nodes represent unstable equilibria be- z0. The transverse motion, however, is a more complicated ing at local maxima of the ponderomotive potential. The mixture of the laser and ponderomotive bounce frequencies. nodes of the electric field, however, occur at the local With the orbit known, this can—in principle—be Fourier- minima and are, therefore, stable. The equations of motion transformed according to Eqs. ͑3͒ and ͑4͒ to give the associ- can then be solved in a neighborhood about these points, ated Thomson-scattering radiation pattern as a function of ␲ ␲ ␻ ¯zn = /2+n for integer n. Linearization of this system is frequency and direction of observation nˆ . The electron achieved by asserting that ␥=1 and expanding all sinusoids orbit is truly periodic and Eq. ͑5͒ can be used exactly when ͱ ͑ ͒ in z to first order about zn. This is equivalent to restricting the the field strength is chosen such that a0 = 2/ 2N , where N Ӷ ␻ electron to nonrelativistic motion, i.e., a0 1, and approxi- =2,3,4,.... In that case, the base frequency 1 is then sim- ␻ mating the minimum of the ponderomotive well quadrati- ply the bounce frequency z0. In fact, this formalism can be cally. This linearization yields the Mathieu equation used for any small a0, since the deviation from true period-

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icity is small. Numerical investigations have shown that the laser field and a second periodic potential. In ponderomotive difference in transverse and axial periods only acts to slightly intermodulation, this potential is the ponderomotive potential ␸ broaden the harmonics. p, instead of the plasma-wave electrostatic potential respon- The emitted radiation pattern can be approximated ana- sible for Raman scattering ͓37͔. Thus, intermodulation will lytically by expanding the retarded time phase term in Eq. exist whenever quiver motion driven by a laser interacts with ͑4͒ as an additional periodic potential, and ponderomotive intermodulation occurs when that potential is due to the interfer- e−inˆ ·x Ϸ 1−inˆ · x. ͑17͒ ence of overlapping laser pulses. ͑ ͒ ͑ ͒ Inclusion of the high-frequency corrections of Eqs. 15 Performing the integration of Eq. 3 in this way yields scat- and ͑16͒ acts to add additional harmonics of these two fre- tered radiation at both the bounce and laser fundamental frequencies. For example, the first correction as defined by Eq. quencies as well as their second harmonics ͑15͒ yields emission up to the sixth harmonic of the laser d¯␧ frequency as well as the first two intermodulation products, ͑␻ ͒ = ␲2␩2␻2͑1−n2͒, ͑18͒ N␻ ±␻ and N␻ ±2␻ . These higher-order modes, how- d⍀ z 0 z z 0 z0 0 z0 ever, scale strongly with the small parameter a0 and are not observed in numerical studies. d¯␧ ␻2 These formulas for the Thomson spectra correctly predict ͑2␻ ͒ =2␲2a2␩4͑2␻ ͒2ͫn2͑1−n2͒ −2n2n2ͩ z ͪ d⍀ z 0 0 z z z x z 1−␻2 the scattered frequencies and the qualitative behavior of the z angular distributions of each emitted line. The approximation ␻2 2 of Eq. ͑17͒, however, fails to capture enough detail to accu- + n2͑1−n2͒ͩ z ͪ ͬ, ͑19͒ x x 1−␻2 rately determine the absolute magnitude of the scattered sig- z nal. By integrating Eq. ͑3͒, including the full retarded time phase term exp͑−inˆ ·x͒, these magnitudes are obtained to d¯␧ 1 ␻2 ͑␻ ͒ = ␲2␩4ͫn2͑1−n2͒ −2n2n2n2ͩ z ͪ + n2͑1−n2͒ good approximation, as discussed later. ⍀ 0 0 z x x y z ␻2 x z d 2 1− z V. VALIDITY OF THE SINGLE-PARTICLE MODEL ␻2 2 ϫͩ z ͪ ͬ, ͑20͒ The electron dynamic equations and the resulting spectra ␻2 1− z have relied on several seemingly stringent assumptions— zero plasma density and no gradients in the laser envelope. d¯␧ ␻ 2 The question of the range of validity of this model then ͑2␻ ͒ =4␲2␩4ͩ z ͪ n2͑1−n2͒. ͑21͒ ⍀ 0 0 ␻2 x x naturally arises. In making the near-equilibrium linearization d 1− z at the electric-field nodes, the ponderomotive potential was The second harmonic of the ponderomotive frequency scales modeled quadratically in z. Electrons initially residing far- 2␩4 ≪ as a0 0 1. Thus, this emission line is not observed in nu- ther from the node sample a greater portion of the pondero- merical simulations of this problem. motive well and, hence, their bounce frequencies acquire an The laser and ponderomotive bounce frequencies are ␩ initial phase, 0, dependence. This can be calculated in the transferred directly to the spectrum through the fast electron nonrelativistic limit by averaging Eq. ͑2͒ over a laser cycle quiver motion and the slower ponderomotive well oscilla- and solving the resulting pendulum equation to yield tion, thereby generating the aforementioned harmonics. ␶ ␲ ␻ ␶ ͑␩ ͒ ␲ ͑ ͒ These frequencies also couple through a process we have z =2 / z = z0F„cos 0 , /2…, 25 termed ponderomotive intermodulation, creating the side- ␶ ␻ where z is the bounce period, z is the bounce angular fre- ␻ ␻ ␻ ␻ ␻ ␻ ␻ −1 band products = 0 ± z0 and = 0 ±2 z0, as well as quency, ␶ =2␲␻ is the linearized bounce period, ␩ re- ␻ ␻ z0 z0 0 =2 0 ±2 z0, as given by tains the meaning cited earlier, and F͑k,␲/2͒ is the elliptic ␧ integral of the first kind ͓38͔. d¯ 1 2 2 2 2 ͑␻ ± ␻ ͒ = ␲ ␩ ͑1±␻ ͒ ͑1−n ͒, ͑22͒ PIC simulations using the program OOPICPRO show that d⍀ 0 z 0 z x 2 for initially cold plasmas with electron densities as high as 1019 cm−3, electrons will tend to bunch to densities up to an d¯␧ 1 ͑␻ ±2␻ ͒ = ␲2␩4͑1±2␻ ͒4ͫn2͑1−n2͒ order of magnitude greater than the background plasma den- d⍀ 0 z 8 0 z z x sity in a range of one-eighth the laser wavelength about the ͓ ͔ ␻ ␻ 2 electric-field nodes 39,40 . In the notation of the solutions ϯ 2n2n2ͩ z ͪ + n2͑1−n2͒ͩ z ͪ ͬ, given earlier, this means that the majority of electrons have x z 1±␻ x z 1±␻ ͉␩ ͉ ␲ z z initial phases such that 0 ranges from 0 to /8. Approxi- ͑23͒ mating Eq. ͑25͒ using this restriction yields ϱ 1 ϫ 3 ϫ ͑2n −1͒ 2 d¯␧ 1 ␶ ␶ ͭ ͚ ͫ ¯ ͬ 2n͑␩ ͒ͮ = 1+ cos . ͑2␻ ±2␻ ͒ = ␲2a2␩4͑2±2␻ ͒2n2͑1−n2͒. ͑24͒ z z0 2 ϫ 4 ϫ ͑2n͒ 0 d⍀ 0 z 2 0 0 z x x n=1 ¯ ͑26͒ This intermodulation mechanism is, in fact, analogous to many classical nonlinear optical phenomena, including Ra- This then gives a maximum increase of ponderomotive fre- man scattering. The test electron couples to both the incident quency of 18.03% due to initial phase variation. This also

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FIG. 1. Schematic diagram of the OOPICPRO simulation box. The simulation was performed in two-dimensional rectangular coordinates.

provides an estimate for the upper bound of plasma densities for which this single-particle theory is valid. Single-particle FIG. 2. ͑Color online͒ Forward ͑z͒ light as a function of the effects remain independent of collective plasma behavior laser strength, a . The dashed line corresponds to a =0.05, and the provided they operate on faster time scales than the response 0 0 solid line to a0 =0.10. The laser is 300 fs in duration and focused to time of the bulk plasma. Since the plasma time scale is given a waist of 14 ␮m. The background plasma density is 1018 cm−3.A ␶ ␲␻−1 by p =2 p and ponderomotive intermodulation operates total of 116 test-particle spectra have been summed. ␶ տ␶ over a time z 0, this single-particle approach applies ␶ Ӷ␶ ␶ ␶ when z p. Taking the maximum value of z =1.1803 z0 laser propagation direction and 16 ␮m transverse to this, as shown in Fig. 1. A single 300-fs laser pulse of wavelength a 2 ͑ −3͒ Ӷ ͑ ϫ 21͒ͩ 0 ͪ ͑ ͒ 1 ␮m and focused to w =14 ␮m is launched from the z=0 n0 cm 1.6 10 , 27 0 ␭͑␮m͒ wall ͑left boundary͒ and propagates to the far wall, which is where ␭ is the laser wavelength given in ␮m. For example, a perfect conductor. Here the pulse is reflected, setting up a for a Nd:glass laser operating at ␭=1.053 ␮m and a strength counterpropagating geometry. Cold electrons are loaded throughout the entire simulation space, and an immobile neu- of a0 =0.2, the density must be less than about 5 ϫ 18 −3 tralizing ion background is included. A set of 116 test elec- 10 cm . ␮ ␮ The plane-wave model of the laser also gives rise to trons are randomly selected in a 10- m-by-1- m area within physical limits of applicability. For the temporal profile of the foci of the pulses. the laser pulse to not significantly affect this phenomenon, The initial laser pulse is allowed to propagate across the ␮ ͑ 1 ͒ ␮ the radiating electron must be able to traverse several bounce 64 m 2133 fs and reflect back an additional 30 m ͑ ͒ periods without the field strength changing significantly. 100 fs . OOPICPRO was modified so that the positions and ␶ ␶ ӷ␶ velocities of a random set of test electrons are tabulated in a Thus, the pulse duration l must satisfy the relation l z. This leads to the more specific requirement single output file 24 times per laser cycle for 100 fs while the pulses are overlapped. These data are then numerically inte- 5͑1.1803͒ͱ2 ␭͑␮m͒ grated according to Eq. ͑3͒ to yield the light scattered by ␶ ͑fs͒ ӷ ͫ ͬͫ ͬ. ͑28͒ l 3 a each electron. Each signal is then incoherently summed with 0 the other particles, producing an integrated signal. Two such ␮ Again, for a 1.053- m laser operating at a0 =0.2, the pulse spectra are shown in Fig. 2. Both of these calculations show duration must be greater than approximately 100 fs. Simi- the light scattered in the +zˆ direction. Notice that the spacing larly, during a bounce period, an electron can travel no more between the laser fundamental and the intermodulation prod- ␶ than a distance c z. For the laser field to be nearly constant to ucts increases linearly with a0, as predicted. an electron traveling this distance transverse to the laser, the Figure 3 shows the light scattered from 116 test particles waist w0 must satisfy in the x-z plane for a0 =0.2. The left-hand figure shows the light radiated 45° above the z axis, and the right-hand plot is w 1.1803 0 ӷ , ͑29͒ the light scattered along the laser polarization. Notice the ␭ ͱ a0 2 strong low-frequency signal along the polarization vector. ␮ ␮ which implies a տ10-␮m waist for the sample laser param- For the case of a 1- m laser, this corresponds to 3.54- m eters already considered. light with an intensity comparable to that of the fundamental scattered in the laser propagation direction. The top scatter image also shows weak emission at roughly the frequencies 2␻ ±␻ , which is predicted only when the first high- VI. PARTICLE-IN-CELL SIMULATIONS 0 z frequency correction is included in the particle orbits ͓see Since only the electron dynamics need be known, this Eq. ͑15͔͒. The small laser fundamental signal along the po- problem lends itself well to particle-in-cell modeling. Simu- larization is due to the spread in laser-wave vectors ͑k͒, lations have been run using OOPICPRO to span the parameter which is due to the laser focusing. space as defined in Eqs. ͑27͒–͑29͒, looking specifically for The analytic approximations for the Thomson-scattered the effects of the bulk plasma, a finite initial temperature, light derived in Eqs. ͑18͒–͑24͒ correctly predict the fre- finite duration, and focused laser pulses ͓39,40͔. All of the quency and overall angular distribution illustrated in Figs. 2 runs consist of a rectangular simulation box 64 ␮m in the and 3—that is, a strong laser fundamental as well as the

026501-5 SEPKE et al. PHYSICAL REVIEW E 72, 026501 ͑2005͒

FIG. 3. Thomson scattered light in the x-z plane. The same plasma and laser conditions as in Fig. 2 have been used with a0 =0.2, and 116 test particles have been summed. The left-hand figure shows the spectrum along n=͑ͱ2/2͒͑x+z͒. The peaks correspond to the frequencies ␻ ␻ ␻ ␻ ␻ ␻ ␻ ␻ ͑ ͒ ␻ ͑ ͒ ␻ z, 0 − z, 0, 0 + z, 0 +2 z weak , and 2 0. The right-hand plot is taken along the laser polarization n=x . The frequencies are z, ␻ ␻ ␻ ␻ ␻ 0,2 0 − z, and 2 0 + z. ␻ ␻ ␻ ␻ intermodulation products 0 ± z and 0 ±2 z along the k a small initial velocity shows that the intermodulation lines ␻ vector, and a dominant z signal along the laser polarization tend to be slightly offset from their theoretically predicted direction. The relative magnitudes of these lines predicted by value due to this initial motion. Taking the ensemble as this model, however, are poorly matched to the full simula- shown in these PIC results, the emitted lines tend to be tion results. Figure 4 compares the forward-scattered light slightly broadened due to the finite plasma temperature. for 57 particles under the same laser conditions as Fig. 2 and ͑ ͒ a0 =0.05 using the PIC data solid line and the single- electron orbits, now including the entire retarded-time phase VII. EXPERIMENTAL SIGNATURES ͑ ͒ ␩ term exp −inˆ ·x , using 57 evenly spaced initial phases 0 AND CONSEQUENCES between ␲/4 and 3␲/4 ͑dashed line͒. The agreement is quite good except that the PIC simulation produces a much larger By employing single-particle as well as particle-in-cell laser fundamental signal. This can be attributed to the con- numerical techniques, the process of ponderomotive inter- tribution from electrons in regions of marginal pulse overlap. modulation and its potential as a light source have been in- These particles see only a single laser and, therefore, radiate vestigated theoretically. Ponderomotive intermodulation has only at this frequency ͓28͔. several signature characteristics amenable to laboratory mea- Since the laser pulse propagates through the plasma be- surement. The intermodulation frequencies that straddle the fore reflecting and creating the standing laser wave, a plasma laser harmonics are intensity-dependent. That is, since the ϳ 2 wave is created, and the test electrons now no longer have a laser intensity I scales like I a0, the bounce frequency then ␻ ϳ 1/2 zero initial velocity when entering the standing wave. Exam- goes as z I , and the frequency separation between har- ining the spectra radiated by PIC electrons and electrons monics and intermodulations then also scales as I1/2. Thus, whose equation of motion has been solved numerically given by simply varying the laser intensity, the intermodulation products produced by this process will shift in frequency. Conversely, this process could be used to deduce the laser intensity by measuring the absolute offset of the intermodulation products. This would be important, for example, in a dynamic Hohlraum. In addition to intermodulation products, the preceding analysis and simulations show that the one-dimensional optical lattice created by overlapping two counterpropagating laser beams generates a significant long-wavelength signal along the laser polarization, as illustrated by Eq. ͑18͒. This light, for example, occurs at 3.76 ␮m for a Nd:Glass laser ␮ operating at 1.053 m and a0 =0.2. As this top-scattered fre- ͑ϳ 1/2͒ quency scales with a0 I , the absolute frequency is tunable by simply adjusting the laser intensity. Emission at this FIG. 4. A comparison of the spectra from 57 particles using the frequency will only occur as long as the lattice exists, that is, PIC simulation ͑solid line͒ and the single-particle equation of mo- as long as the laser pulses are overlapped. The duration of tion ͑dashed line͒. The relative agreement for each of the inter- this light source, then, is determined by the laser pulse modulation products is good. The laser fundamental line is en- duration—which, using chirped pulse amplification, can now hanced in the PIC simulation due to particles in regions of poor easily be on the order of hundreds of femtoseconds. Thus, overlap. the optical lattice created by two overlapped counterpropa-

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cludes pulse lengthening from being observed using bandwidth techniques. Such broadened pulses may also lend themselves to chirped pulse amplification as well and need to be taken into consideration in the interpretation of data as- cribed to other broadening mechanisms, such as relativistic self-modulation. Additionally, such spectrally broadened pulses hold the potential to be temporally compressed. This process is analogous to many parametric amplification processes, but now the laser fields themselves are being used as the nonlinear medium. For such a process to work, however, the phase characteristics of the Thomson-scattered light also must be considered. FIG. 5. The sum of three offset Gaussian signals yields a Ponderomotive intermodulation also generates a strong broader bell-shaped signal, which—if ponderomotive intermodula- ␻ ␻ 0 + z signal in the near-forward/near-backward direction, tion is not accounted for—can lead to incorrect pulse-duration esti- which can mimic a broad ϳ3/2 harmonic of a 1-␮m laser. ͑ ͒ mates. The three Gaussians dashed curves all have FWHM of The broadness of such emission is, of course, caused both by 4.00 nm. The sum of these three ͑solid curve͒, however, has a the temporal and spatial variation of a Gaussian laser pulse FWHM of 5.37 nm. ␩ ϳ and by the finite range of 0 values in a plasma. This 3/2 ␻ harmonic light can be further enhanced by the weaker 2 0 gating laser pulses acts as an ultrafast, all-laser-driven, near- ␻ ϳ −2 z scattering when a0 0.2. Similar features have been infrared source. Using the PIC results shown in Fig. 3, the observed experimentally ͓41͔. instantaneous scattered power radiated into a cone with slant ␸ ␶ angle m by a laser with a duration of fs, an energy of E Joules incident on a plasma with an electron number density −3 VIII. SUMMARY AND CONCLUSIONS of ne in cm is given by 8␸ We have here derived the equilibrium electron orbits in a P͑W͒ = ͑10−20͓͒␶͑fs͔͓͒E͑J͔͓͒n ͑cm−3͔͒ͩ m ͪ e 3 one-dimensional optical-lattice setup by counterpropagating laser pulses of equal amplitude and frequency. This motion, ␻ +⌬␻ 2 z d ¯␧ in general, includes both fast laser quiver motion and slower ϫsin ␸ ͑3 − sin2 ␸ ͒ͯ͵ ͯ d␻, m m ⍀ ␻ ␻ −⌬␻ d d ponderomotive oscillations. These time scales mix to yield z PIC harmonics of both the laser and ponderomotive frequencies ⌬␻Ϸ ␻ where 0.15 is half of the bandwidth of the z line as well as several intermodulation products. Emission of the normalized to the laser frequency, and the spectral energy ponderomotive bounce frequency has been shown to have density inside the integral is that shown in Fig. 3. Using two significant potential as an ultrafast near-infrared source. 1-ps laser pulses containing 20 J of energy each, incident on For completeness, the spectra emitted at the magnetic- ϫ 18 −3 a cold plasma with ne =3 10 cm and assuming a detec- field nodes have also been computed. The slow, axial bounce tor acceptance angle of 5° yields a power of ϳ616 W at motion is seen to disappear, and the orbit and, hence, the 3.76 ␮m. This power is low, but the tunability as well as the radiation pattern then no longer show mixing but only emis- ultrashort duration and directionality of such an optical- sion of laser harmonics. As this is an unstable equilibrium lattice source is encouraging. By utilizing other angles of point, however, the ponderomotive bounce motion will incidence, higher intensities and, therefore, higher scattered dominate the radiated light. powers may be reached. Ponderomotive intermodulation occurs independent of the The range of initial phases as well as the finite initial plasma response and is, therefore, significant in realistic temperature of any real set of electrons tend to broaden all of laboratory conditions, as demonstrated in Eqs. ͑27͒–͑29͒. the Thomson-scattered lines, as illustrated in the PIC simu- The resulting scattered light produces—in addition to the lation results shown in Figs. 2 and 3. When all of these low-frequency emission peaked along the laser factors are properly tuned, the first intermodulation products polarization—intensity-dependent spectral satellites about ␻ ␻ ␻ 0 ± z can overlap the laser fundamental mode, 0, anoma- the laser harmonics in much the same way Raman scattering ␻ lously broadening the 0 signal, as illustrated in Fig. 5. This does through the plasma wave. This not only gives a signa- can lead to incorrect pulse-duration calculations. For ex- ture dependence for diagnosis of this effect and the laser ample, an experiment utilizing counterpropagating, 750-nm intensity in the plasma but can also lead to anomalous spec- ϳ laser pulses with a0 0.002 generates a bounce wavelength tral broadening of laser harmonics. span from 2.15 to 2.50 nm, with the range being due to the In many plasma problems, multiple time scales act to ␩ ␩ spread in initial phases 0. Including both the 0 spread and drive electron motion. Associated with each mode is a natu- a 4-nm full width at half maximum ͑FWHM͒ bandwidth of ral frequency, which interacts with the others to produce the the laser—which is also imparted to the sidebands—the laser resulting orbit. This mixing can be seen directly through the and first intermodulation product signals overlap, which pre- Thomson-scattered light since the Larmor formula as shown

026501-7 SEPKE et al. PHYSICAL REVIEW E 72, 026501 ͑2005͒ in Eqs. ͑3͒ and ͑4͒ is nothing more than a Fourier decompo- ACKNOWLEDGMENTS sition of the electron velocity. We have here demonstrated a particular case of such intermodulation driven by the pon- The authors would like thank Kirk Flippo, Sudeep Baner- deromotive potential of beating laser beams. The idea can jee, and Matthew Rever for many stimulating conversations. now be extended to speciﬁc geometries and even other driv- This work has been supported by Sandia National Laborato- ing mechanisms to extend this method to additional scattered ries, the U.S. Department of Energy, the National Science wavelengths. Foundation, and the Ofﬁce of Naval Research.

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