## Abstract

The harmonics of the scattering of a femtosecond intense laser pulse by an electron has been numerically investigated. The harmonic spectrum shows interesting red shifts and parasitic lines in the blue sides of harmonic lines. The red shift of the lines is found to be caused by the dilation of laser oscillation experienced by an electron due to its relativistic drift motion along the direction of a driving laser propagation and the parasitic lines come from the variation of the laser intensity. The angular distribution of each higher harmonic line shows double peak patterns in the forward direction. The backward scattering has its own distinct pattern: line-shaped nodes perpendicular to the laser electric field, the number of which is the harmonic order number minus one. As the harmonic order increases, the primary peaks of higher harmonics move from the backward to the forward direction of the laser propagation. In the time domain, each radiation pulse in the case of a linearly-polarized laser pulse has a double peak structure due to the disappearance of the acceleration during the half cycle of an electron’s oscillation.

© 2003 Optical Society of America

## 1. Introduction

Recent advances in ultrashort intense laser technology [1, 2] has made it possible to investigate relativistic phenomena in a laboratory. Among them, relativistic, nonlinear Thomson scattering (RNTS) has been paid much attention to because of its potential as an ultrashort x-ray source. To generate intense, ultrashort x-ray radiation, physical mechanisms have been proposed: some of them are the relativistic Doppler shift with a relativistic electron beam [3, 4, 5], the harmonic frequency upshift [6, 7], and X-ray laser using innershell atomic processes [8, 9].

Linear Thomson scattering [10], the scattering of a low intensity laser by electrons, radiates the same frequency as that of the incident laser pulse in the direction perpendicular to the direction of the driving electric field. This has been widely used to diagnose the velocity distribution of electrons in plasmas. As the laser intensity increases, the dynamics of electrons become relativistically nonlinear, thus generating harmonic radiation, referred to as RNTS. The dynamics of electrons becomes relativistic as the following normalized vector potential approaches unity:

where λ is the laser wavelength in µm and *I* the laser intensity in W/cm^{2}.

Recently, RNTS was experimentally verified by observing angular patterns of the second and third harmonic lines with a laser intensity of 4.4 × 10^{18} W/cm^{2} (*a*
_{o} = 1.88) [17, 18]. The RNTS has been analytically investigated by several authors [4, 11, 12, 13, 14, 15, 16]. In particular, Sarachik, *et al.*[13] analyzed the RNTS in terms of Hamilton-Jacobi formalism in an average electron rest frame and Lorentzt ransformation. Yu, *et al.*[16] studied combined effects of the higher harmonic scattering and the relativistic Doppler shift with relativistic electrons. Coherent generation of the harmonics were also investigated with plasmas [14, 15]. The analytic studies on the RNTS have been limited to the case of a continuous laser due to the complexity of the analysis. This motivated us to investigate RNTS of a femtosecond pulsed laser by a numerical study, which also gives experimentally observable results. We investigated the harmonic structure of the scattering of an intense laser pulse of 20 fs pulse width by an electron initially at rest. The harmonic spectral lines and their angular distributions have been investigated, which agree well with analytical studies. The intensity-dependent frequency shift has been predicted using a Lorentzt ransformation [13, 19] and investigating a resonance function of the harmonic line [4]. Here we present a direct derivation of the phenomena considering the motion of an electron in an intense laser field, which gives a clear understanding of the red shift of the harmonic lines due to the relativistic drift motion of the electron in the direction of the laser propagation. The rapid variation of the laser intensity results in the generation of parasitic lines to the blue side of the harmonic lines.

## 2. Results and Discussions

In this paper, we focus on the RNTS of a linearly polarized laser by an electron initially at rest for the laser intensities of 10^{17} W/cm^{2} and 10^{18} W/cm^{2}. The generation of an ultrashort x-ray pulse using higher intensities are presented in Ref. [20], in which detailed descriptions on the calculation method are presented. For a driving laser pulse, typical parameters of current high-power Ti:Sapphire Chirped-Pulse-Amplification lasers have been adopted: a Gaussian envelope with a pulse duration of 20 fs FWHM(Full Width Half Maximum), the central wavelength at 800 nm, and a linear polarization (x direction), which propagates in the +z direction.

The quivering amplitude of an electron at a laser intensity of 10^{18} W/cm^{2} amounts to 0.1 µm, which is much smaller compared with a typical beam size of 10 um in diameter in real experiments. Hence the relative variation of laser intensity during the quivering motion of an electron can be neglected. The estimation of the ponderomotive force reveals that the ponderomotive force is about three orders of magnitude smaller than the Lorentz force. Thus the plane wave approximation is adopted for this calculation.

Figure 1(a) shows energy spectra of the scattered radiation integrated over solid angles for peak laser intensities of 10^{17} W/cm^{2} and 10^{18} W/cm^{2}. Harmonic structures are clearly seen. The normalized vector potentials of these intensities correspond to 0.21 and 0.68, respectively. At first, one can see that for a case of 10^{17} W/cm^{2}, the line shapes of the harmonic lines are broadened to the red side compared with the unshifted lines (the vertical lines). As the laser intensity increases, the harmonic lines shift more toward the red side and get broader. Also, small peaks between the harmonic lines can be noticed for a laser intensity of 10^{18} W/cm^{2}. The radiations in the direction of laser propagation (θ = 0 - 90°, θ is the polar angle measured from the +z axis) and the backward direction (θ = 90 - 180°) for the case of 10^{18} W/cm^{2} are separately plotted in Fig. 1(b), which shows clearly different harmonic spectral lines. The energy of the fundamental harmonic radiation, *E _{s}* with respect to the laser intensity and the direction from an electron initially at rest changes as [4, 13]

where *E _{o}* is the fundamental photon energy of a laser, 1.55 eV for 800 nm laser. Below, we will show that such a red shift can be explained by the drift motion of an electron due to β⃗ ×

*B*⃗ force which makes the electron feel a dilated laser oscillation period. β is the electron’s velocity divided by the speed of light and

*B*⃗ the magnetic field of a driving laser pulse.

In the direction of the laser electric field (θ = 90°, ϕ = 0°; ϕ is the azimuthal angle measured from the +x axis in the x-y plane), the harmonic spectrum for the case of 10^{18} W/cm^{2} is plotted in Fig. 2(a). An interesting feature in the spectrum is the low-intensity parasitic lines to the blue side of harmonic lines. This is caused by the variation of the laser intensity during a laser pulse. As seen from Eq. (2), the shift of a harmonic line increases as the laser intensity increases, and the principal line is generated during the peak laser intensity with a larger shift, while the parasitic lines are generated during the lower laser intensities with a less shift. The rapid variation of a laser intensity manifests itself in different ways in high harmonics generation by neutral atoms under a 25 fs TW laser pulse that the harmonic generation is enhanced and the degree of chirp on the rising edge of a pulse determines the direction as well as the amount of the shifts of harmonic lines [21]. In an actual experiment with an ultrashort laser pulse, since a measurement covers a certain range of solid angles, these parasitic lines would appear as a line broadening accompanied with the line broadening caused by the finite number of laser oscillations [4]. The radiation power in the same direction is shown in Fig. 2(b). In the time domain, the radiation has a periodic double-pulse structure and the period slightly increases as the laser intensity increases (not clearly seen in the figure). This variation of the period is manifested as parasitic lines in the frequency domain. The double-pulse structure is found to be caused by the disappearance of the acceleration during the half cycle of an electron’s oscillation [20].

The theta distributions of the radiations on the plane of electron’s motion are plotted in Fig. 3(a) and (b) for peak laser intensities of 10^{17} W/cm^{2} and 10^{18} W/cm^{2}, respectively. For a better comparison of the shape of the angular distributions between different harmonics, the intensities are normalized to their maxima. The peaks of the linear scattering, that is, the first order harmonic line lie on the plane perpendicular to the laser field (θ = 0°, 180°), as in usual dipole radiations. As the laser intensity increases, it radiates more in the direction of the laser propagation due to the relativistic effect. For the nonlinear scattering(higher harmonics), the primary peak directions of all the harmonics noticeable in the calculated spectra are plotted, up to the 8th order for 10^{17} W/cm^{2} and the 20th order for 10^{18} W/cm^{2}. The primary peaks are off the y-z plane (θ = 0°, 180°). As the harmonic order increases, they move from the backward to the forward direction of the laser propagation, approaching the direction of the electron’s velocity denoted by the horizontal line in Fig. 3(c). In the forward direction, there are two small peaks whereas in the backward direction, there exists the different number of peaks for different harmonic orders. The number of peaks, excluding two small forward peaks, is equal to the harmonic order number. The secondary peaks get weaker relative to the primary peaks as the harmonic order increases. The contour plots of the angular distributions (θ, ϕ) for the laser intensity of 10^{18} W/cm^{2} are plotted in Fig. 4. For the contour plot of each harmonic (Fig. 4(b)–(g)), the logarithmic scale is used to display detailed structures clearly. In the backward radiation, it can be noticed that there are line-shaped nodes perpendicular to the direction of the laser electric field and the number of the nodes is the harmonic order number minus one.

We can now better understand the harmonic spectrum(Fig. 1) with the help of the angular distributions(Figs. 3 and 4) and Eq. (2). Each harmonic radiates strongly in its own direction for each forward and backward direction with differently shifted frequencies according to Eq. (2). For a low laser intensity, the frequency shift on the radiation direction is too small to notice. The maximum frequency shift at θ = 180° in the case of 10^{17} W/cm^{2} is just 2 % of the unshifted frequency. This is why the small peaks that appear in the case of 10^{18} W/cm^{2} do not appear in the case of 10^{17} W/cm^{2} (Fig. 1(a)).

The red shift of the harmonic frequency of RNTS according to Eq. (2) tells us that at θ = 0°, the fundamental harmonic frequency is the same as the laser frequency and as θ increases, the fundamental harmonic frequency shifts to the red side and such a shift gets larger as the laser intensity increases. Such a red shift of harmonic lines originated from the relativistic Doppler effect [4, 13, 19] is investigated directly considering the dynamics of an electron in an intense laser field. This clearly shows that such a red shift is caused by the dilation of a laser oscillation period due to the drift motion of an electron along the laser propagation direction caused by the β⃗ × B⃗ force. At a high laser intensity, the drift velocity becomes significant, comparable to the speed of light. The peak drift velocity divided by the speed of light for the laser intensities of 10^{17} W/cm^{2} and 10^{18} W/cm^{2} amounts to 0.022 and 0.19, respectively. This drift motion of an electron along the laser propagation makes the electron feel a longer oscillation period (*T _{e}*) than that of the laser field oscillation (

*T*).

_{L}The phase of the laser oscillation at the electron’s position should advance by 2π for a single oscillation of the electron’s motion as

where *r*⃗(*t*) is the position of the electron. ω_{L} and *k*⃗_{L} = *k*
_{L}
*z*̂ are the angular frequency and the wave vector of the driving laser, respectively. With *t*′ = *t* + *T*
_{e} and *z*′ = *z* + Δ*z* (see Fig. 5), the oscillation period seen by the electron is given by

where *c* is the speed of light. This is just a mathematical statement that the laser pulse propagates more than the electron by its wavelength during one cycle of the electron’s motion. Obtaining Δ*z* form the solution of the electron’s motion under a laser pulse [4], the following relation between *T*
_{e} and *T*
_{L} is obtained from Eq. (4).

The period of a radiation, *T*
_{D}, detected by a detector located at a far distance is the difference between the arrival times of the radiations departed from A and B in Fig. 5:

In Eq. (7), Δ*l* = Δ*z* cosθ is used. Using Eq. (4) for Δ*z*, Eq. (7) becomes

which shows that at θ = 0°, *T _{D}* is the same as

*T*and at θ = 90°, the detector sees the same oscillation period as that of the electron oscillation. Then the insertion of Eq. (5) into Eq. (8) using

_{L}*E*=

*h*/

*T*reproduces Eq. (2). In Fig. 6 (a), the simulation and analytic results for the

*T*and

_{D}*E*at a laser intensity of 10

_{s}^{18}W/cm

^{2}are compared. In this figure, the simulation results for

*T*and the

_{D}*E*(solid squares and circles, respectively) are obtained from radiation power and spectrum, respectively. The oscillation of the electron plotted in Fig. 6(b) shows that the period of the electron’s oscillation is the same as

_{s}*T*at θ = 90°.

_{D}## 3. Summary

The temporal and spatial characteristics of the harmonic structure of the scattering of an intense ultrashort laser pulse (with a pulse width of 20 fs FWHM) by an electron was numerically investigated. Each radiation pulse in the case of a linearly-polarized laser pulse has a double peak structure due to the disappearance of the acceleration during the half cycle of an electron’s motion. The parasitic lines on the blue side of the harmonic lines are observed. These lines are found to be generated due to the significant variation of the laser intensity during the pulse. In addition to the Lorentz transformation and the resonance function investigation, the red shifts of the harmonic lines are derived by considering the dilation of a laser oscillation period experienced by an electron due to its drift motion, which becomes significant at a high laser intensity. The first harmonic radiates most strongly on the plane perpendicular to the laser field as in dipole radiation, whereas primary peak directions of higher harmonics are off the plane and move from the backward to the forward direction of the laser propagation, approaching the direction of the electron’s velocity, as the harmonic order increases. In the forward direction, there are two small peaks whereas in the backward direction, there exists a different number of peaks for different harmonic orders. The number of peaks, excluding two small forward peaks, is equal to the harmonic order number.

It has been discussed in a previous section that the spatial variation of a focused laser intensity over the quivering distance of an electron in the transverse direction does not affect the radiation pattern. In an actual experiment, however, radiations from electrons under different laser intensities could be integrated, which would then give a somewhat modified radiation pattern. In such a case, the coherence of the radiations from electrons under different intensities needs to be accounted for, given such a spatial distribution of laser intensity.

## Acknowledgment

This has been supported by Korea Research Foundation (Grant No. KRF-2000-015-DP0175).

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