PHYSICS OF PLASMAS VOLUME 10, NUMBER 5 MAY 2003 Nonlinear Thomson scattering: A tutoriala… Y. Y. Lau,b) Fei He, Donald P. Umstadter, and Richard Kowalczyk Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, Michigan 48109-2104 ͑Received 13 November 2002; accepted 10 February 2003͒ Recent advances in table-top, ultrahigh intensity lasers have led to significant renewed interest in the classic problem of Thomson scattering. An important current application of these scattering processes is the generation of ultrashort-pulse-duration x rays. In this tutorial, the classical theory of nonlinear Thomson scattering of an electron in an intense laser field is presented. It is found that the orbit, and therefore its nonlinear scattering spectra, depends on the amplitude and on the phase at which the electron sees the laser electric field. Novel, simple asymptotic expansions are obtained for the spectrum of radiation that is backscattered from a laser by a counter-propagating ͑or co-propagating͒ electron. The solutions are presented in such a way that they explicitly show—at least in the single particle regime—the relative merit of using an intense laser and of an energetic electron beam in x-ray production. The close analogy with free electron laser/synchrotron source is indicated. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1565115͔ I. INTRODUCTION atomic-scale spatial resolution, but also as a medical diag- nostic. Several proposals have been made to build a The widespread availability and the continuous develop- ‘‘gamma–gamma’’ collider for high-energy physics experi- ment of ultra-high intensity lasers have created the exciting ments, in which 200 GeV gamma rays are generated by field of ‘‘High Field Science.’’ Many regimes that were Compton scattering of 1 eV photons from 250 GeV energy deemed out of reach just a decade ago suddenly became electron beams. Colliding such energetic photons to create closer to reality. Important applications include table-top particles through the inverse process has advantages over electron and proton accelerators, advanced x-ray sources, direct particle collisions because of reduced beamstrahlung medical isotope production, ultrafast imaging, laser surgery, and disruption. In all cases of scattering, the electromagnetic and materials treatment.1–4 wave acts as an undulator/wiggler, replacing the alternating In this tutorial, we focus only on one aspect, namely, the static magnetic field used in conventional synchrotrons or generation of x rays by Thomson scattering of an intense free electron lasers ͑FEL͒.34 The field strength in the former laser by electrons.5–32 The energetic electrons may be gener- case can be orders-of-magnitude higher, and the length, ated separately by a conventional high energy accelerator, or orders-of-magnitude shorter, than in the latter case. The re- by another intense laser pulse via one of the several ad- sults of numerous experiments,37–42 theories,43–46 and vanced acceleration mechanisms.33 If a laser-based accelera- reviews4,31 related to these topics have been published. Par- tor were used in conjunction with a synchronized scattering allel efforts in the FEL/synchrotron community are docu- laser pulse, then an all-optical x-ray source would be mented in Refs. 47–83. Nonlinear Thomson and Compton possible.5,19,32 Applications of Thomson/Compton x-ray scattering was also studied in Refs. 84–90. sources include dynamical studies and imaging of solid, mo- While an x-ray source based on laser scatter- lecular, and biological systems.34–37 ing5,6,10,28,32,45 and one based on the conventional syn- In Thomson scattering, an electron that is initially at rest chrotron/FEL mechanism34,66,92 differ in the characteristics may acquire relativistic velocities in the fields of high- of undulator/wiggler used, their physical and mathematical intensity light and through this relativistic motion the elec- descriptions are strikingly similar. In fact, the normalized tron may emit radiation at high harmonics of the light fre- laser electric field, a, and the normalized wiggler parameter, quency. If the electron already possesses a relativistic energy K, are almost interchangeable in the description of Thomson before it encounters the high-intensity laser, there is an addi- scattering for head-on collisions between a relativistic elec- tional Doppler-shift of the scattered light. By Thomson scat- tron against a laser. Here we present the solution of the back- tering, we mean that the photon energy (h␷) of the scattered scatter spectrum for arbitrary laser intensity and arbitrary radiation is much less than the electron energy (mc2), or electron energy ͑including zero͒, and the simple asymptotic h␷Ӷmc2, while for Compton scattering, this condition is not expressions recently obtained for the various regimes.90 The met. As an example of Thomson scattering, electrons with solutions are presented in such a way that they explicitly only 100 MeV energy can upshifta1eVphoton to an energy show the relative merit of using an intense laser and of an of 50 keV, which is of interest not only as a probe with energetic electron beam in x-ray production in the single particle regime. We also indicate the origin of a number of a͒Paper GI2 1, Bull. Am. Phys. Soc. 47, 136 ͑2002͒. misconceptions that appear in the literature in the ultrain- b͒Invited speaker. tense laser community ͓cf. paragraph following Eq. ͑11͔͒. 1070-664X/2003/10(5)/2155/8/$20.002155 © 2003 American Institute of Physics Downloaded 01 May 2003 to 141.213.19.32. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp 2156 Phys. Plasmas, Vol. 10, No. 5, May 2003 Lau et al. The laser electric field is measured by the dimensionless Upon eliminating BF from these two Maxwell equations, one ϭ ␻ ϭ parameter, a eE0 /m 0c, where E0 is the electric field am- may find that EF MJF where M is a matrix which depends ␻ ␻ 34,92 plitude, 0 is the laser frequency, and m is the electron rest only on k, . The total work done, W, in ergs, performed mass. For a linearly polarized laser field, by the current J(r,t) on the electric field E(r,t) is given by ␭ 1/2 I 3 aϭ0.85ͩ ͪͩ ͪ Wϭ ͵ dtd rJ͑r,t͒"E͑r,t͒ 1 ␮m 1018 W/cm2 1 ␭ ␭ 1/2 ϭ ͵ ␻ 3 ͑ ␻͒" ͑ ␻͒ ͑ ͒ E0 P d d kJ k, E* k, , 2 ϭ0.31ͩ ͪͩ ͪ ϭ0.22ͩ ͪ ͩ ͪ ͑2␲͒4 F F 1 ␮m 1GW 100 V/Å rs where we have used the Parseval’s relation in writing the last ␭ in terms of the laser wavelength , laser intensity I, laser expression in which the asterisk denotes the complex conju- ͑ power P, and the rms radius rs of the laser spot size assum- gate. Upon using the matrix M and the k-space differential ͒ ing a Gaussian profile . Physically, a measures the transverse volume d3kϭk2dk d⍀ which is expressed in terms of the momentum, in units of mc, imparted by the oscillating laser solid angle ͑⍀͒ in the direction of the unit vector nϭck/␻, field upon an electron. Freund92 in effect obtains the general formula for Eq. ͑2͒, For lasers with low power intensities (aӶ1), an electron that is initially at rest undergoes a small amplitude, trans- d2W ␻ Wϵ ͵ d␻ d⍀ verse oscillation at the laser frequency 0 . The Thomson d␻ d⍀ ␻ scattering spectrum consists of a single frequency 0 in all directions and the radiation pattern is the same as that from a ␻2 ϭ ͵ ␻ ⍀ ͓͉ ͑ ␻͉͒2Ϫ͉ " ͑ ␻͉͒2͔ ͑ ͒ dipole antenna. As a is increased to a value of a few tens of d d JF k, n JF k, . 3 4␲2c3 percent, the electron’s oscillation frequency begins to deviate from the laser frequency. As the laser amplitude increases to Note that the square bracket in Eq. ͑3͒ is simply ϭ ͉ Ã Ã ␻ ͉2 a O(1), the Lorentz force associated with the laser’s mag- n ͓n JF(k, )͔ . netic field becomes significant, and the electron acquires an An electron with displacement r(t) and velocity v(t) oscillation along k, the direction of laser propagation, in ad- carries a current density J(r,t)ϭev(t)␦͓rϪr(t)͔, whose ␻ dition to the transverse oscillation. The electron also acquires Fourier transform JF(k, ) may easily be obtained from Eq. an average drift velocity along k. For aӷ1, the axial excur- ͑1͒. We immediately obtain from Eq. ͑3͒ the Jackson formula sion of electron oscillation greatly exceeds the transverse ex- under the far-field approximation,91 cursion, and the electron orbital period is much greater than 2 2 2 the laser optical period. At present, the achievable values of d W e ␻ ϭ ͉nÃ͓nÃF͑␻͔͉͒2, ͑4͒ a are in the single digits. Values of a in the tens or even d⍀ d␻ 4␲2c hundreds are being actively pursued in the ‘‘high field sci- ence’’ community.4,28,45 ϱ F͑␻͒ϭ ͵ dt ␤͑t͒ei␻͓tϪn"r͑t͒/c͔. ͑5͒ In Sec. II, we derive the general formula for the radiation Ϫϱ spectrum, and point out some subtlety in its evaluation. In ͑ ͒ Sec. III, we describe the electron orbit subject to an intense Equation 4 gives the energy radiated by the electron in the ⍀ laser field, displaying the potential importance of the phase. direction of the unit vector n, per unit solid angle , per unit ␻ In Sec. IV, we consider the backscatter spectra in detail. The frequency . Radiation damping is ignored throughout, and asymptotic scaling suggests the optimal combination of the all calculations are in the lab frame.